Jean-Pierre Boeuf

Andrei Smolyakov

Physics and instabilities of low temperature E×B plasmas for spacecraft propulsion and other applications

Low temperature 𝐸 × 𝐵 plasmas are used in various applications such as Hall thrusters for satellite propulsion, ion sources and magnetron discharges for plasma processing, and negative ion sources for neutral beam injection in fusion. The plasmas in these devices are partially magnetized, i.e. electrons are strongly magnetized while ions are not, and are subject to various micro and macro-instabilities that are very distinctive from instabilities in fusion plasmas. These instabilities are often triggered by the large difference in electron and ion drift velocities in the 𝑬 × 𝑩 direction. The possibility of maintaining a large electric field in the quasineutral plasma of Hall thrusters despite anomalous electron transport, or the presence of strong double layers associated with the azimuthal rotation of plasma structures ("rotating spokes") in magnetron discharges and Hall thrusters are examples of the very challenging and exciting physics of 𝑬 × 𝑩 devices. The turbulence and instabilities present in 𝑬 × 𝑩 plasma devices constitute a major obstacle to the quantitative description of these devices and to the development of predictive codes, and are the subject of intense research efforts. In this tutorial, we discuss some key aspects of the physics of low temperature, partially magnetized 𝑬 × 𝑩 plasmas, as well as recent advances made through simulations, theory, and experiments in our understanding of the various types of instabilities (such as gradient-drift/Simon-Hoh and lower hybrid instabilities, rotating ionization waves, electron cyclotron drift instability, modified two-stream instability, etc.) that occur in these plasmas.

I. INTRODUCTION

In low temperature 𝑬 × 𝑩 plasmas devices an external magnetic field is placed perpendicular to the discharge current or to the electric field applied between two electrodes. The perpendicular magnetic field increases the electron residence time in the device and the electron collisionality, allowing plasma sustainment by electron impact ionization at low pressures and moderate applied voltage. The perpendicular magnetic field decreases the electron conductivity parallel to the electric field, so an objective of the 𝑬 × 𝑩 configuration can also be to generate a large electric field in the quasineutral plasma, which can accelerate ions (application to gridless ion sources such as the Hall thruster). Lowering the electron conductivity with a perpendicular magnetic field is actually not trivial because of the development of instabilities that tend to increase electron transport across the magnetic field. This question is at the center of this tutorial.

In most applications of 𝑬 × 𝑩 plasmas the external magnetic field and the device dimensions are such that electrons are strongly magnetized (Larmor radius much smaller than the device dimensions), while ions are not or are only weakly magnetized. Such plasmas are called "partially magnetized plasmas". Electron transport along the electric field is in principle strongly reduced by the external magnetic field applied perpendicular to the electric field. In most devices the magnetic field lines intersect a wall (cathode or dielectric surface) so that along the magnetic field line the electrons are electrostatically confined by the wall sheath potential. Electron confinement is therefore a combination of magnetic and electrostatic confinement.

Collisionless electrons are trapped in the direction perpendicular to the magnetic field. Only collisions with neutrals can un-trap the electrons from their cyclotron trajectory and allow transport across the magnetic field ("cross-field transport") along the electric field. Electron confinement by the magnetic field in the direction of the applied field is generally not as good as predicted by the classical collisional theory because of turbulence and instabilities that lead to cross-field transport ("anomalous transport"). Because of their partial magnetization, the physics and instabilities of these plasmas are very distinct from those of other plasmas, e.g. for fusion applications.

The study of instabilities and anomalous transport in 𝑬 × 𝑩 plasmas has recently attracted much attention, largely driven by the need to better understand the operation of devices such as Hall thrusters and magnetron discharges, and to develop predictive models of these devices. Joint efforts by the low-temperature plasma community and the plasma propulsion community "to develop a rigorous understanding of the rich phenomena observed in 𝑬 × 𝑩 devices" are described in a recent perspective article by Kaganovich et al. [START_REF] Kaganovich | Physics of E × B discharges relevant to plasma propulsion and similar technologies[END_REF] that "presents the current understanding of the physics of these phenomena and state-of the-art computational results, identifies critical questions, and suggests directions for future research". The present tutorial article can be considered as complementary to the perspective article by Kaganovich et al., but is aimed at non-specialists, with more emphasis on the physics of instabilities in various 𝑬 × 𝑩 plasma devices.

In plasmas with perpendicular electric field and magnetic field, there is a charged particle drift in the 𝑬 × 𝑩 direction ("𝑬 × 𝑩 drift") with a velocity close to |𝑬 × 𝑩| 𝐵 [START_REF] Smolyakov | Fluid theory and simulations of instabilities, turbulent transport and coherent structures in partially-magnetized plasmas of E × B discharges[END_REF] ⁄ = 𝐸 𝐵 ⁄ . This drift plays an important role in regions of large electric fields. Electrons are subject to this drift while ions are accelerated out of the large electric field region before they have time to reach the 𝑬 × 𝑩 drift velocity (this is another way to say that the plasma is partially magnetized). The large 𝑬 × 𝑩 electron drift and the negligible ion drift in this direction can lead to charge separation and is often responsible for triggering instabilities. The electron pressure gradient 𝛁𝑃 𝑒 perpendicular to the magnetic field also creates a drift in the 𝛁𝑃 𝑒 × 𝑩 direction. This is the diamagnetic drift velocity, |𝛁𝑃 𝑒 × 𝑩| 𝑒𝑛𝐵 [START_REF] Smolyakov | Fluid theory and simulations of instabilities, turbulent transport and coherent structures in partially-magnetized plasmas of E × B discharges[END_REF]

= |𝛁(𝑛𝑇 𝑒 )| 𝑛𝐵 ⁄ ⁄
where 𝑛 is the plasma density, 𝑒 the elementary charge, and 𝑇 𝑒 the electron temperature in eV. 𝑬 × 𝑩 and 𝛁𝑃 𝑒 × 𝐵 electron drift can both contribute to the development of instabilities. Instabilities are ubiquitous in low temperature partially magnetized 𝑬 × 𝑩 plasmas and can significantly enhance cross-field electron transport. They manifest themselves in a wide range of unstable fluctuations, with frequencies from kilohertz to tens of megahertz and wavelengths from a fraction of millimeter to several centimeters [START_REF] Smolyakov | Fluid theory and simulations of instabilities, turbulent transport and coherent structures in partially-magnetized plasmas of E × B discharges[END_REF] .

The electron drift in the 𝑬 × 𝑩 direction gives rise to the Hall current. Efficient electron confinement parallel to the electric field requires that the Hall current does not intercept a wall (otherwise the Hall effect would destroy the confinement of electrons). This is possible if the Hall current is closed on itself, as, for example, in a cylindrical geometry where the electric field is axial and the magnetic field radial, or the opposite. The Hall current in such geometries is in the azimuthal direction, i.e. is closed on itself. Plasma devices with closed electron drift are called closed-drift devices [START_REF] Zhurin | Physics of Closed Drift Thrusters[END_REF] .

The concept of closed-drift devices has led to important applications in ion sources for space propulsion (Hall Thrusters, HT, also called Stationary Plasma Thrusters, SPT 4- 6 , 7 , and Thrusters with Anode Layer, TAL [START_REF] Goebel | Fundamentals of Electric Propulsion: Ion and Hall Thrusters[END_REF] ) and plasma processing (End-Hall ion sources [START_REF] Zhurin | Physics of Closed Drift Thrusters[END_REF][START_REF] Kaufman | End-Hall ion sources[END_REF][START_REF] Oudini | Physics and modeling of an end-Hall (gridless) ion source[END_REF] , closed-drift Anode Layer Ion Sources, ALIS [START_REF] Zhurin | Physics of Closed Drift Thrusters[END_REF][START_REF] Zhurin | Industrial ion sources -Broadbeam gidless ion source technology[END_REF] , or Anode Layer Plasma Accelerators, ALPA [START_REF] Zhurin | Physics of Closed Drift Thrusters[END_REF][START_REF] Dudnikov | Ion source with closed drift anode layer plasma acceleration[END_REF] ). The originality of these "Hall ion sources" lies in the fact that ions are extracted from the plasma not by a system of biased grids (they are "gridless ion sources"), but by the electric field generated in the quasineutral plasma by the local lowering of electron conductivity due to a magnetic field barrier placed between an emissive cathode and the anode. Magnetron discharges can also be considered as closed-drift ion sources where ions are extracted from the plasma and accelerated in the cathode sheath to sputter atoms from the cathode, with applications to thin film deposition. They can be operated at high power densities in a pulsed regime (HiPIMS, high power impulse magnetron sputtering) [START_REF] Sarakinos | High power pulsed magnetron sputtering: A review on scientific and engineering state of the art[END_REF][START_REF] Gudmundsson | High power impulse magnetron sputtering discharge[END_REF][START_REF] Gudmundsson | Physics and technology of magnetron sputtering discharges[END_REF][START_REF] Anders | Tutorial: Reactive high power impulse magnetron sputtering (R-HiPIMS)[END_REF] . In magnetron discharges as in Hall thrusters, electrons are heated by the applied dc electric field. They drift in the azimuthal 𝑬 × 𝑩 direction, generate the plasma by ionization and slowly drift to the anode due to collisions or instabilities. Penning discharges 16, 17 , 18, 19 , consisting of two facing cathodes surrounded by a cylindrical anode, with an axial magnetic field, are another type of closed-drift device that has been used in many ion source applications. Hall ion sources, magnetron discharges and Penning ion sources operate under pressure typically between a fraction of pascals and a few pascals but it is interesting to note that Penning discharges where first implemented as ion gauges [START_REF] Penning | A new manometer for low gas pressures especially between 10(-3) and 10(-5) mm[END_REF][START_REF] Penning | The spark discharge in low pressure between coaxial cylinders in an axial magnetic field[END_REF] in the 1930's, operating at much lower pressures. In these conditions, the 𝑬 × 𝑩 plasma can become nonneutral (electron density larger than ion density) and specific types of instabilities, called diocotron instabilities [START_REF] Knauer | Diocotron Instability in Plasmas and Gas Discharges[END_REF][START_REF] Redhead | Instabilities in crossed-field discharges at low pressures[END_REF][START_REF] Davidson | Physics of Nonneutral Plasmas[END_REF][START_REF] Kervalishvili | Formation and Dynamics of Vortex Structures in Pure and Gas-Discharge Nonneutral Collisionless Electron Plasmas[END_REF][START_REF] Boeuf | Rotating structures in low temperature magnetized plasmas; insight from particle simulations[END_REF] are likely to develop.

Historical references on the development of 𝑬 × 𝑩 devices for electric propulsion can be found in Refs. [START_REF] Zhurin | Physics of Closed Drift Thrusters[END_REF][START_REF] Morozov | The Conceptual Development of Stationary Plasma Thrusters[END_REF][START_REF] Kim | History of the Hall Thrusters Development in USSR[END_REF] , for magnetron discharges applications to sputtering and deposition in Refs. [START_REF] Sarakinos | High power pulsed magnetron sputtering: A review on scientific and engineering state of the art[END_REF][START_REF] Gudmundsson | High power impulse magnetron sputtering discharge[END_REF][START_REF] Gudmundsson | Physics and technology of magnetron sputtering discharges[END_REF][START_REF] Anders | Tutorial: Reactive high power impulse magnetron sputtering (R-HiPIMS)[END_REF] , and for various cross-field devices operating over a wide pressure range in Refs. [START_REF] Redhead | Instabilities in crossed-field discharges at low pressures[END_REF][START_REF] Abolmasov | Physics and engineering of crossed-field discharge devices[END_REF] . Recent review or perspective papers on electric propulsion including gridless ion sources can be found in Refs. [START_REF] Mazouffre | Electric propulsion for satellites and spacecraft: established technologies and novel approaches[END_REF][START_REF] Levchenko | Perspectives, frontiers, and new horizons for plasma-based space electric propulsion[END_REF][31] .

The question of cross-field electron transport or anomalous transport also arises in situations where the plasma simply diffuses across the magnetic field, in the absence of applied voltage perpendicular to the magnetic field. The plasma can be generated by inductive coupling, microwave, of by injection of an electron beam. This is the case of magnetized plasma sources where the plasma is confined by magnetic cusps (applications to plasma processing or gridded ion sources [START_REF] Sengupta | Magnetic confinement in a ring-cusp ion thruster discharge plasma[END_REF][START_REF] Wirz | Effects of magnetic field topography on ion thruster discharge performance[END_REF][START_REF] Dankongkakul | Miniature ion thruster ring-cusp discharge performance and behavior[END_REF] ), or of magnetized cylindrical plasma columns with axial magnetic field. Magnetized plasma columns have been used extensively to study turbulence and anomalous electron transport [START_REF] Sakawa | Nonlinear evolution of the modified Simon-Hoh instability via a cascade of sideband instabilities in a weak beam plasma system[END_REF][START_REF] Sakawa | Growth and nonlinear evolution of the modified Simon-Hoh instability in an electron beam-produced plasma[END_REF][START_REF] Rebont | Ion Velocity Distribution Function Investigated Inside an Unstable Magnetized Plasma Exhibiting a Rotating Nonlinear Structure[END_REF][START_REF] Claire | Ion velocity analysis of rotating structures in a magnetic linear plasma device[END_REF][START_REF] Jaeger | Direct observation of a crossfield current-carrying plasma rotating around an unstable magnetized plasma column[END_REF][START_REF] Gravier | Transitions between drift waves in a magnetized cylindrical plasma: experiments and fluid model, solved with a spectral method[END_REF][START_REF] Antar | The origin of convective structures in the scrape-off layer of linear magnetic fusion devices investigated by fast imaging[END_REF][START_REF] Schröder | Spatial mode structures of electrostatic drift waves in a collisional cylindrical helicon plasma[END_REF][START_REF] Light | Quiescent and unstable regimes of a helicon plasma[END_REF] . 𝑬 × 𝑩 and 𝛁𝑃 𝑒 × 𝑩 flows combine in these plasmas to ensure a zero net current to the walls. Diffusion is often nonambipolar and "short-circuit" effects [START_REF] Simon | Ambipolar Diffusion in a Magnetic Field[END_REF] are possible in the case of metallic walls, i.e. the electron current does not balance locally but globally the ion current to the walls. 𝑬 × 𝑩 plasmas have potential applications in mass separation, which requires the imposition and customization of an electric field perpendicular to the axial magnetic field in a plasma column to create an 𝑬 × 𝑩 flow [START_REF] Zweben | Plasma mass separation[END_REF][START_REF] Gueroult | E × B configurations for high-throughput plasma mass separation: An outlook on possibilities and challenges[END_REF] . Research is ongoing into ways to generate this radial electric field, such as through electrode biasing [START_REF] Gueroult | E × B configurations for high-throughput plasma mass separation: An outlook on possibilities and challenges[END_REF][START_REF] Lehnert | A partially ionized plasma centrifuge[END_REF][START_REF] Lehnert | The partially ionized plasma centrifuge[END_REF] , the use of emissive end-electrodes [START_REF] Trotabas | Trade-off in perpendicular electric field control using negatively biased emissive end-electrodes[END_REF] , or wave injection [START_REF] Fetterman | Alpha channeling in a rotating plasma[END_REF] .

𝑬 × 𝑩 plasma devices such as Hall ion sources or magnetron discharges are now currently used successfully in space propulsion and plasmas processing applications. However, despite important research efforts and progress in the last two or three decades, the physics of these devices is still poorly understood and models of these plasmas with predictive capabilities are still lacking. In this article we describe, for non-specialists, the basic physics of lowtemperature partially magnetized 𝑬 × 𝑩 plasmas, with emphasis on the development of instabilities, and we illustrate with some examples the recent progress in our understanding of these plasmas and their instabilities through experiments, simulation, and theory.

The basic physics, principles, and instabilities of 𝑬 × 𝑩 plasma devices are presented in section II. Section III focuses on two important 𝑬 × 𝑩 plasma devices, the Hall thruster and the magnetron discharge, with a more detailed discussion of the physics and of the evidence of the presence of instabilities in these devices. Examples of experimental, numerical, and theoretical approaches of the studies of these instabilities are presented in section IV.

II. BASIC PHYSICS AND PRINCIPLES OF 𝑬 × 𝑩 PLASMAS DEVICES

The typical conditions of low temperature partially magnetized 𝑬 × 𝑩 plasmas devices and the orders of magnitude of the plasma parameters are presented in section II.A. The classical, collisional transport theory is briefly recalled in section II.B. The geometry and principles of the main closed-drift devices are discussed in section II.C. The theoretical description of the linear phase of instabilities, and the derivation of dispersion relations of partially magnetized 𝑬 × 𝑩 plasmas are presented in section II.D. Section III.E shows some examples of instabilities in 𝑬 × 𝑩 plasma devices and of their interpretation with fluid dispersion relations. The Hall effect, which occurs in devices with nonclosed-drift, is discussed with two examples in section III.F.

A. Conditions and orders of magnitude

Under usual conditions of 𝑬 × 𝑩 plasmas applications, the effect of the plasma current on the magnetic field is negligible so we will assume that the magnetic field is purely external, i.e. is generated by a system of coils or magnets and a magnetic circuit. This condition may not be satisfied [START_REF] Bohlmark | Measurement of the magnetic field change in a pulsed high current magnetron discharge[END_REF] in the extreme conditions of HiPIMS where very high power densities are injected in the plasma through short pulses of several microseconds. Table 1 : Typical values of magnetic field intensities in lowtemperature partially magnetized 𝑬 × 𝑩 plasmas and corresponding values of electron and ion angular cyclotron frequencies, 𝜔 𝑐𝑒 and 𝜔 𝑐𝑖 , and mean Larmor radii, 𝜌 𝑒 and 𝜌 𝑖 , assuming Maxwellian distributions at 5 eV.

𝑩 (mT)

1 -100 𝜔 𝑐𝑒 (rad/s) 1.8 × 10 8 -1.8 × 10 10 𝜌 𝑒 (m) for 5 eV electrons 5300 -53 𝜔 𝑐𝑖 (rad/s) Ar + ions 2.4 × 10 3 -2.4 × 10 5 𝜌 𝑖 (cm) for 5 eV Ar + ions 140 -1.4

As said in the introduction, we consider partially magnetized plasmas, which implies that the mean electron Larmor radius 𝜌 𝑒 is much smaller than the device dimensions 𝜌 𝑒 ≪ 𝐿 while the ion Larmor radius is not. For a Maxwellian electron velocity distribution at temperature 𝑇 𝑒 , the mean Larmor radius is given by 𝜌 𝑒 = 𝑣 𝑒⊥ 𝜔 𝑐𝑒 ⁄ where 𝑣 𝑒⊥ = (𝑒𝑇 𝑒 𝑚 ⁄ ) 1/2 is the electron mean velocity perpendicular to the magnetic field, and 𝜔 𝑐𝑒 = 𝑒𝐵 𝑚 ⁄ is the electron cyclotron angular frequency. Orders of magnitude of magnetic field intensity, charged particles cyclotron frequencies and Larmor radii are given in Table 1. Electron-neutral collisions play an important role in applications of 𝑬 × 𝑩 plasmas because they control the electron magnetization and contribute to crossfield transport (and also because the plasma is sustained by electron impact ionization). Hall ion sources and magnetron discharges operate at pressures typically between 0.1 Pa and 5 Pa and with magnetic field usually smaller than 100 mT with plasma dimensions generally less than 5-10 centimeters. Table 2 gives some orders of magnitude of collision frequencies, and free paths of electron and argon ions. ). Note that the electron-neutral momentum cross-section in argon varies by about 2 orders of magnitudes for electron energies between 1 and 10 eV (Ramsauer minimum). Some 𝑬 × 𝑩 devices such as ion gauges can operate at gas pressures much below 0.1 Pa (in these conditions the plasma can be non-neutral).

𝒑 (Pa) -300 K 0.1 -5 𝑝 (mtorr) 0.75 -37.5 𝑁 (m -3 )
2.4 × 10 19 -1.2 × 10 21 𝜈 𝑒𝑛 (s -1 ) for 5 eV electron 2 × 10 6 -10 8 𝜈 𝑖𝑛 (s -1 ) for 5 eV Ar + ions 2 × 10 5 -10 7 𝜆 𝑒𝑛 (cm) for 5 eV electron 62 -1.2 𝜆 𝑖𝑛 (cm) for 5 eV Ar + ions 7. 7 -0.15 Usual values of plasma densities, plasma frequencies and electron Debye lengths are given in Table 3.

Table 3 : Plasma density range and corresponding values of the electron and ion plasma frequencies, 𝜔 𝑝𝑒 , 𝜔 𝑝𝑖 , and electron Debye length, 𝜆 𝐷𝑒 for an electron temperature of 5 eV. The plasma density can be larger than 10 18 m -3 in HiPIMS.

𝒏 (m -3 ) 10 15 -10 18 𝜔 𝑝𝑒 (s -1 )

1.8 × 10 9 -5.6 × 10 11 𝜔 𝑝𝑖 (s -1 ) for Ar + ions 6.6 × 10 6 -2.1 × 10 8

𝜆 𝐷𝑒 (m) for 𝑇 𝑒 =5 eV 530 -17

Since the electron trajectory around the magnetic field changes after each electron-neutral collision, leading to transport or diffusion across the magnetic field, an important parameter is the ratio of the electron Larmor frequency over the electron-neutral collision frequency. This ratio, ℎ = 𝜔 𝑐𝑒 𝜈 𝑒𝑛 ⁄ , is called the Hall parameter and characterizes the electron confinement by the magnetic field in the direction of the applied electric field. As said in the introduction, plasma fluctuations and instabilities also contribute to un-trapping the electrons from their trajectory around the magnetic field. Their effect can be considered as similar to the effect of collisions, and one can define an effective Hall parameter and an effective collision frequency to describe their effect on cross-field transport. It can be deduced from Table 1 and Table 2 that the Hall parameter associated with electronneutral collisions, for a magnetic field intensity of 10 mT, and a gas pressure of 0.1 Pa is ℎ = 𝜔 𝑐𝑒 𝜈 𝑒𝑛 ⁄ ≈ 1000 . Note finally that low temperature partially magnetized plasmas are far from thermal equilibrium, and that the electron temperature (defined as 2 3

⁄ of the thermal energy) is much larger than the ion temperature and the charged particle velocity distribution functions can be very far from Maxwellian.

B. Classical, collisional cross-field transport

In this section, the classical, collisional expression of the electron current across a magnetic field is derived from a simplified form of the steady-state, electron momentum equation. We neglect the inertia term and assume that the electron pressure tensor is scalar. We consider a plasma with an electric field is in the 𝑥 direction, the magnetic field in the perpendicular 𝑧 direction. We assume that the plasma density gradient is parallel to the electric field, in the 𝑥 direction. The notations are indicated in Figure 1. .

Neglecting inertia and assuming a scalar electron pressure, the electron momentum equation writes:

𝑒𝑛[𝑬 + 𝒗 𝒆 × 𝑩] + 𝑒𝛻(𝑛𝑇 𝑒 ) + 𝑚𝑛𝜈 𝑒𝑛 𝒗 𝒆 = 𝟎 (1)
𝒗 𝒆 is the electron mean velocity, 𝑒𝑛𝑇 𝑒 is the electron pressure, and the electron temperature 𝑇 𝑒 is expressed in eV. The projection of this equation on the 𝑥 and 𝑦 directions gives:

{ 𝑣 𝑒𝑥 = - 𝜇 𝑒0 1 + ℎ 2 (𝐸 + 1 𝑛 𝜕𝑛𝑇 𝑒 𝜕𝑥 ) ≈ - 1 ℎ𝐵 (𝐸 + 1 𝑛 𝜕𝑛𝑇 𝑒 𝜕𝑥 ) 𝑣 𝑒𝑦 = - 𝜇 𝑒0 ℎ 1 + ℎ 2 (𝐸 + 1 𝑛 𝜕𝑛𝑇 𝑒 𝜕𝑥 ) ≈ - 1 𝐵 (𝐸 + 1 𝑛 𝜕𝑛𝑇 𝑒 𝜕𝑥 ) } (2)
where 𝜇 𝑒0 = 𝑒 (𝑚𝜈 𝑒𝑛 ) ⁄ is the electron mobility without magnetic field, and we have assumed ℎ 2 ≫ 1 . Note that the ratio of these velocities is equal to the Hall parameter:

𝑣 𝑒𝑦 𝑣 𝑒𝑥 = ℎ (3)
so the mean velocity in the 𝑬 × 𝑩 direction can be significantly larger than the velocity parallel to the electric field. The two components of the mean velocity have two terms, one related to the electric field, and the other to the pressure gradient. The expression of the mean velocity parallel to the electric field is similar to the expression without magnetic field, with a drift term and a diffusion term, but with a mobility reduced by a factor (1 + ℎ 2 ) ≈ ℎ 2 with respect to the mobility without magnetic field. The mobility parallel to the electric field can be written as:

𝜇 𝑒∥𝐸 ≈ 𝜇 𝑒0 ℎ 2 = 1 ℎ𝐵 (4)
The mean velocity in the 𝑦 direction has also two components, one associated with the electric field, the 𝑬 × 𝑩 drift velocity: 5) and one related to the pressure gradient, the 𝛁𝑃 𝑒 × 𝑩 drift velocity or diamagnetic drift velocity:

𝑣 𝐸 = - 𝐸 𝐵 (
𝑣 𝑃 𝑒 = - 1 𝐵 1 𝑛 𝜕𝑛𝑇 𝑒 𝜕𝑥 (6)
where we have assumed ℎ 2 ≫ 1.

Along the magnetic field lines (in the 𝑧 direction) electrons move freely between collisions. The electron conductivity parallel to the magnetic field is therefore much larger than the electron conductivity parallel to the electric field. An important consequence is that the magnetic field lines tend to be equipotential. More precisely, from the momentum equation along the magnetic field lines, the electric field along these lines field must balance the electron pressure gradient, so the potential drop along these lines is small for low electron temperature or small electron pressure gradient (note that a magnetic field line should never connect cathode and anode in a magnetized plasma device since this would be equivalent to a short circuit at low pressure).

If the plasma is bounded by walls in the 𝑬 × 𝑩 direction, e.g. dielectric walls, there is an electron flow to one of the walls due to the 𝑬 × 𝑩 and 𝛁𝑃 𝑒 × 𝑩 drifts. The electron flow to the wall must be balanced by an ion flow. An electric field is therefore generated in the 𝑧 direction to draw the ions to the wall. This is the Hall effect. This electric field, combined with the magnetic field, will in turn create an 𝑬 × 𝑩 electron drift in the 𝑥 direction that will destroy the electron confinement. The question of Hall effect in low temperature partially magnetized plasmas has been illustrated analytically in the case of a plasma column in a transverse magnetic field by Ecker and Kanne [START_REF] Ecker | Cylindrical Plasma Column in a Transverse Magnetic Field[END_REF] and by Kunkel 53 , and with particle simulations [START_REF] Claustre | Particle-In-Cell Monte Carlo Collision Model on GPU-Application to a Low-Temperature Magnetized Plasma[END_REF] in the case of the negative ion source for the neutral beam injector of ITER. The solution to avoid the Hall effect is to choose a geometry where the 𝑬 × 𝑩 electron current is closed on itself. This is possible in a cylindrical geometry where the electric field is axial and the magnetic field is radial or the opposite. Devices with this configuration are called closed-drift devices [START_REF] Zhurin | Physics of Closed Drift Thrusters[END_REF] .

C. Closed drift devices

Figure 2 : Schematic of the principles of several 𝑬 × 𝑩 discharge devices with axial magnetic field and radial electric field (a, b,) or axial electric field and radial magnetic field (c, d). In all these devices, the Hall current is in the azimuthal direction.

Some classical examples of closed-drift devices are displayed in Figure 2. In Penning cells and cylindrical magnetrons discharges (Figure 2a, b) the external magnetic field is axial, and the applied electric field is radial. The concept of Penning discharge (Figure 2a) was introduced by Phillips at the end of the 19 th century [START_REF] Phillips | The Action of Magnetised Electrodes upon Electrical Discharge Phenomena in Rarefied Gases[END_REF] and was used for the first time in 1937 by Penning, with the construction of an ionization gauge [START_REF] Penning | A new manometer for low gas pressures especially between 10(-3) and 10(-5) mm[END_REF] . The different modes of a Penning discharge with a long anode cylinder at low pressure were studied by Knauer [START_REF] Knauer | Mechanism of the Penning Discharge at Low Pressures[END_REF] and Schuurmann [START_REF] Schuurman | Investigation of a low pressure Penning discharge[END_REF][START_REF] Schuurman | On the theory of electrical characteristics of a Penning discharge at low magnetic field[END_REF] and the diocotron instability that develops in Penning discharge at low pressures was described by Knauer [START_REF] Knauer | Diocotron Instability in Plasmas and Gas Discharges[END_REF] . More recent experimental work on Penning discharges and their different regimes can be found in the article by Mamedov et al. [START_REF] Mamedov | Discharge Characteristics of the Penning Plasma Source[END_REF] .

Penning discharges are currently used as ion sources 19 (ions can be extracted through an aperture in one of the two cathodes or through the anode) in various applications, e.g. to sputtering and deposition, ion implantation [START_REF] Cun | Characterization of a cold cathode Penning ion source for the implantation of noble gases beneath 2D monolayers on metals: Ions and neutrals[END_REF] , isotope separation, production of hydrogen negative ions [START_REF] Faircloth | An overview of negative hydrogen ion sources for accelerators[END_REF] , and neutron generators [START_REF] Das | Development of compact size penning ion source for compact neutron generator[END_REF] . The electron vortices associated with the diocotron instability, which develop in Penning discharges and in cylindrical magnetrons at very low pressures were extensively studied by Kervalishvili and his group [START_REF] Kervalishvili | Formation and Dynamics of Vortex Structures in Pure and Gas-Discharge Nonneutral Collisionless Electron Plasmas[END_REF][START_REF] Kervalishvili | Rotational Instability of a charged plasma in crossed fields and generation of electrons of anomalously high energy[END_REF][START_REF] Kervalishvili | Evolution of non-linear structures in charged plasmas in crossed fields[END_REF][START_REF] Kervalishvili | Electron vortices in a nonneutral plasma in crossed EXH fields[END_REF][START_REF] Kervalishvili | The mechanism of stability of long-lived, self-organized solitary vortex in nonenutral electron plasma[END_REF] with the help of probe measurements and described with particle simulations in Ref. [START_REF] Boeuf | Rotating structures in low temperature magnetized plasmas; insight from particle simulations[END_REF] . Cylindrical magnetrons (Figure 2b,) consist of two concentric cylindrical electrodes with an axial magnetic field. Early sputteringdeposition applications were demonstrated in cylindrical magnetrons [START_REF] Gudmundsson | Physics and technology of magnetron sputtering discharges[END_REF][START_REF] Thornton | Magnetron sputtering: basic physics and application to cylindrical magnetrons[END_REF] , but the planar magnetron configuration (Figure 2c) is now much more commonly used for deposition of metallic or compound thin films in many industrial applications. The planar magnetron can operate in a dc regime in the pressure range 0.2-4 Pa with applied voltages of 300-700 V and a magnetic field of 20-50 mT close to the cathode [START_REF] Gudmundsson | Physics and technology of magnetron sputtering discharges[END_REF] . Very high power densities can be achieved in a pulsed regime (HiPIMS) [START_REF] Gudmundsson | High power impulse magnetron sputtering discharge[END_REF][START_REF] Anders | Tutorial: Reactive high power impulse magnetron sputtering (R-HiPIMS)[END_REF] with repetition frequency in the range 50-5000 Hz and 1%-3% duty cycle [START_REF] Sarakinos | High power pulsed magnetron sputtering: A review on scientific and engineering state of the art[END_REF][START_REF] Gudmundsson | High power impulse magnetron sputtering discharge[END_REF][START_REF] Gudmundsson | Physics and technology of magnetron sputtering discharges[END_REF] , applied voltage in the range 500-2000 V and with peak power densities up to several kW/cm 2 . The plasma density is 10 15 to 10 17 m -3 in the dc case and can be larger than 10 18 -10 19 m -3 in the pulsed case. The physics of HiPIMS is extremely complex because of the high plasma density, ionization of the sputtered atoms, instabilities, neutral depletion etc…

The magnetic field in planar magnetrons is generated by magnets placed behind the cathode as schematically shown in Figure 2c. In a Hall ion source (Figure 2d), the objective is to generate an electric field in the quasineutral plasma and to use this field to extract ions from the plasma. The basic idea is therefore to place a magnetic barrier perpendicular to the applied electric field between an emissive cathode and the anode, as shown schematically in Figure 2d. Figure 3 shows some practical examples of Hall ion sources. In a Hall thruster [START_REF] Goebel | Fundamentals of Electric Propulsion: Ion and Hall Thrusters[END_REF][START_REF] Boeuf | Tutorial: Physics and modeling of Hall thrusters[END_REF][START_REF] Morozov | Reviews of Plasma Physics[END_REF] (Figure 3a) the plasma is formed in a channel between two concentric cylinders. The anode is at one end of the channel, and an electron emissive cathode is placed outside the channel. A radial magnetic field with a maximum at the other end of the channel ("exhaust plane") is generated by coils or magnets and a magnetic circuit. This magnetic field forms a magnetic barrier between cathode and anode. The applied electric field tends to be larger in the region of maximum magnetic field because of the lowering of the electron conductivity due to the magnetic barrier. Ions generated in the channel are extracted by this electric field. The external emissive cathode provides electrons to sustain the discharge as well as to neutralize the extracted ion beam. Several variations around this concept have been proposed and developed. Some of these designs are described in recent review papers [START_REF] Mazouffre | Electric propulsion for satellites and spacecraft: established technologies and novel approaches[END_REF][START_REF] Levchenko | Perspectives, frontiers, and new horizons for plasma-based space electric propulsion[END_REF][START_REF] Levchenko | Space micropropulsion systems for Cubesats and small satellites: From proximate targets to furthermost frontiers[END_REF] .

Here we only give a brief description of some of these designs to illustrate the different possible 𝑬 × 𝑩 configurations. The configuration of a magnetically-shielded Hall thruster [START_REF] Mikellides | Magnetic shielding of a Hall thruster. I. Theory and validation[END_REF][START_REF] Hofer | Magnetic shielding of a laboratory Hall thruster. II. Experiments[END_REF][START_REF] Su | Performance comparison of a 9-kW magnetically shielded Hall thruster operating on xenon and krypton[END_REF] is shown in Figure 3b. The magnetic barrier in this thruster is such that the magnetic field in the channel, close to the exhaust plane is parallel to the dielectric walls. Since the magnetic field lines tend to be equipotential (because of the large electron conductivity along these lines), this configuration limits the electron and ion fluxes to the walls and reduces ion erosion in this region. The cylindrical Hall thruster [START_REF] Raitses | Parametric investigations of a nonconventional Hall thruster[END_REF][START_REF] Raitses | Cylindrical Hall thrusters with permanent magnets[END_REF][START_REF] Doh | Structure of the ion acceleration region in cylindrical Hall thruster plasmas[END_REF] (Figure 3c), where the inner cylinder is shorter, is more adapted to small or low power thrusters since, according to the empirical scaling laws of Hall thrusters, the channel width of the channel between the two dielectric cylinder should decrease with decreasing power. The magnetic field barrier in this thruster is not perpendicular to the thruster axis so the electric field is not perfectly axial, and the divergence of the ion beam is larger than in standard Hall thrusters. The anode layer thruster (or thruster with anode layer, TAL) [START_REF] Choueiri | Fundamental difference between the two Hall thruster variants[END_REF][START_REF] Keidar | Modeling of a high-power thruster with anode layer[END_REF] is shown in (Figure 3d). In this thruster the anode is very close to the magnetic barrier and the dielectric walls are no longer needed and are replaced by metallic walls. Ions are accelerated in a thin region close to the anode. Ion sources based on the same concept as TAL are also used in plasma processing applications under the name of ALIS [START_REF] Zhurin | Physics of Closed Drift Thrusters[END_REF][START_REF] Zhurin | Industrial ion sources -Broadbeam gidless ion source technology[END_REF] (Anode Layer Ion Sources) or ALPA [START_REF] Dudnikov | Ion source with closed drift anode layer plasma acceleration[END_REF][START_REF] Kim | On the longitudinal distribution of electric field in the acceleration zones of plasma accelerators and thrusters with closed electron drift[END_REF] (Anode Layer Plasma Accelerators). Another thruster configuration where the anode is in the magnetic barrier is the wall-less thruster 77 of (Figure 3e) proposed by Kapulkin et al. 78 in the 1990's, and by Mazouffre et al. [START_REF] Vaudolon | Optimization of a wall-less Hall thruster[END_REF][START_REF] Mazouffre | Optimization of magnetic field topology and anode geometry for a wall-less Hall thruster[END_REF][START_REF] Mazouffre | Development and experimental characterization of a wall-less Hall thruster[END_REF] later. A variation of a wall-less thruster is based on a standard Hall thruster with a short channel but with a ring anode placed at the channel exit. The ion acceleration in this design takes place outside the channel. The advantage of the configuration of Figure 3e is its simplicity and reduced erosion problems, but the divergence of the ion beam can be an issue. Finally, gridless ion sources are not only used for spacecraft propulsion but also for plasma processing applications. This is the case of the ALIS and ALPA source mentioned above, and of the End-Hall ion sources [START_REF] Zhurin | Industrial ion sources -Broadbeam gidless ion source technology[END_REF] of Figure 3f, with a truncated conical anode and a magnetic field barrier along the anode surface. This ion source is used for ion assisted thin film deposition [START_REF] Anders | Plasma and ion sources in large area coating: A review[END_REF] . Particle simulations of End-Hall ion sources can be found in Refs. [START_REF] Oudini | Physics and modeling of an end-Hall (gridless) ion source[END_REF][START_REF] Oudini | Numerical study of the characteristics of the ion and fast atom beams in an end-Hall ion source[END_REF] .

A final remark on the magnetic field configuration of closed-drift devices is that in all cases, the magnetic field lines intersect the cathode or some dielectric surfaces. Electrons move freely along the magnetic field lines, so the electron confinement is not purely magnetic but is a combination of magnetic confinement in the direction perpendicular to the applied electric field and electrostatic confinement along the magnetic field lines. In configurations where the magnetic field lines end on the cathode (Penning or magnetron discharges), plasma electrons are efficiently confined electrostatically because of the large potential drop in the cathode sheaths. In configurations where the magnetic field lines intersect a dielectric surface (most of the Hall ion sources), low energy electrons are electrostatically confined by the wall sheath potential, but higher energy electrons can overcome this wall potential and reach the wall. Electron-wall interactions and the resulting secondary electron emission can play an important role in these devices [START_REF] Kaganovich | Physics of E × B discharges relevant to plasma propulsion and similar technologies[END_REF] .

From this brief review it appears that in closed-drift devices with axial electric field and radial magnetic field (magnetron discharges and Hall ion sources) there can be three generic types of axial profiles of the radial magnetic field, as shown in Figure 4. The three profiles correspond to 1), magnetron discharges, with a magnetic field decreasing from cathode to anode, 2), standard Hall thrusters, with a magnetic barrier between cathode and anode, and 3), ion sources with anode layers, where the magnetic field increases from cathode to anode. The choice between these three configurations depends on the application and on the objective to be achieved (performance, lifetime, simplicity, …). We will see in the rest of this paper that the magnetic field profile has also important consequences on the nature of the instabilities that develop in these devices and on their consequences on anomalous electron transport.

D. Theoretical description of instabilities -Fluid dispersion relations

In this section we describe some aspects of the derivation of the fluid dispersion relations of a low temperature partially magnetized 𝑬 × 𝑩 plasmas based on Refs. [START_REF] Smolyakov | Fluid theory and simulations of instabilities, turbulent transport and coherent structures in partially-magnetized plasmas of E × B discharges[END_REF][START_REF] Frias | Long wavelength gradient drift instability in Hall plasma devices. I. Fluid theory[END_REF][START_REF] Lakhin | Effects of finite electron temperature on gradient drift instabilities in partially magnetized plasmas[END_REF][START_REF] Lakhin | Marginal stability, characteristic frequencies, and growth rates of gradient drift modes in partially magnetized plasmas with finite electron temperature[END_REF] . We use the notations of Figure 1, where 𝑬 is in the 𝑥 direction 𝑩 in the 𝑧 direction, and 𝑬 × 𝑩 in the -𝑦 direction. The plasma density gradient is parallel to the electric field, in the 𝑥 direction.

We first write the electron and ion continuity and momentum equations assuming quasineutrality, constant electron temperature, collisionless electrons and ions, nonmagnetized ions, and we neglect the ion pressure term and the electron inertia terms. We will also neglect the electron and ion mean velocities parallel to the electric field and consider purely azimuthal perturbations. This gives, for ions: 𝜕𝑛 𝜕𝑡 + 𝜵. (𝑛𝒗 𝒊 ) = 0 (7)

𝜕𝒗 𝒊 𝜕𝑡 + (𝒗 𝒊 . 𝜵)𝒗 𝒊 = - 𝑒 𝑀 𝜵𝜙 (8)
and for electrons:

𝜕𝑛 𝜕𝑡 + 𝜵. (𝑛𝒗 𝒆 ) = 0 (9)
𝑛𝒗 𝒆 = 𝑛𝒗 𝑬 + 𝑛𝒗 𝑃 𝑒 (10) with If we assume a uniform magnetic field, and since 𝜵 × 𝑬=0 and 𝜵 × 𝑩=0, eq. ( 9) can be written as:

𝒗
𝜕𝑛 𝜕𝑡 + 𝒗 𝑬 . 𝜵𝑛 = 0 (11)
We now consider small wave perturbations in the 𝑬 × 𝑩 direction and linearize the equations above for 𝑛 and 𝜙 around the equilibrium solutions 𝑛 0 , 𝜙 0 . As said above and shown in Figure 1 we consider a situation where the equilibrium electric field 𝑬 0 and density gradient 𝜵𝑛 0 are in the 𝑥 direction and we write, with the same notations as in Smolyakov et al. [START_REF] Smolyakov | Fluid theory and simulations of instabilities, turbulent transport and coherent structures in partially-magnetized plasmas of E × B discharges[END_REF] :

𝑬 0 = 𝐸 0 𝒙 ̂= - 𝜕𝜙 0 𝜕𝑥 𝒙 ̂; 𝜵𝑛 0 = 𝜕𝑛 0 𝜕𝑥 𝒙 ̂= 𝑛 0 𝐿 𝑛 𝒙 ̂ (12)
The equilibrium 𝑬 × 𝑩 drift and diamagnetic drift are noted 𝑣 0 and 𝑣 * , and are written (eqs. ( 5) and ( 6)):

𝑣 0 = - 𝐸 0 𝐵 𝑣 * = - 1 𝐵 1 𝑛 0 𝜕𝑛 0 𝑇 𝑒 𝜕𝑥 = - 𝑇 𝑒 𝐵 𝜕ln (𝑛 0 ) 𝜕𝑥 = - 𝑇 𝑒 𝐵 1 𝐿 𝑛 (13)
We look for solutions of eqs. ( 7)- (10) above, of the form 𝑛 = 𝑛 0 + 𝑛 ̃exp(-𝑖𝜔𝑡 + 𝑘𝑦), 𝜙 = 𝜙 0 + 𝜙 ̃exp(-𝑖𝜔𝑡 + 𝑘𝑦) where 𝜔 = 𝜔 𝑅 + 𝑖𝛾 is the complex wave frequency and 𝑘 the wavenumber in the 𝑦 direction (i.e. 𝑘 ≡ 𝑘 𝑦 ). 𝜔 𝑅 , the real part of 𝜔 , is the wave angular frequency and 𝛾, the imaginary part, is the growth rate. The instability grows when 𝛾 > 0.

The linearization of eqs. ( 7)- (10) for 𝑛 ̃≪ 𝑛 0 , 𝜙 ̃≪ 𝜙 0 is straightforward if one notes that the linearization of the term 𝒗 𝑬 . 𝜵𝑛 of eq. (11) gives the first order term 𝒗 0 . 𝜵𝑛 ̃+𝒗 ̃𝐸. 𝜵𝑛 0 where 𝒗 ̃𝐸 has a non-zero component in the 𝑥 direction resulting from the 𝑬 × 𝑩 drift related to the field perturbation along 𝑦, i.e. :

𝒗 𝐸 0 . 𝜵𝑛 ̃+𝒗 ̃𝐸. 𝜵𝑛 0 =𝑣 0 𝑖𝑘𝑛 ̃+ 𝑖𝑘 𝜙 B 𝜕𝑛 0 𝜕𝑥 = -𝑖𝑘𝑛 ̃𝑣0 -𝑖𝑘𝑛 0 𝑣 * 𝑒𝜙 T 𝑒
The linearization of the equations therefore gives, for ions:

𝑛 ñ0 = 𝑘 2 𝜔 2 𝑒𝜙 M (14)
and for electrons:

𝑛 ñ0 = - 𝑘𝑣 * 𝜔 + 𝑘𝑣 0 𝑒𝜙 T𝑒 (15)
Combining eqs. ( 14) and ( 15) with the notations:

𝜔 0 = -𝑘𝑣 0 , 𝜔 * = -𝑘𝑣 * , 𝑒𝑇 𝑒 𝑀 = 𝑐 𝑠 2 (16)
(𝑐 𝑠 is the ion sound speed), we get the dispersion relation:

𝑘 2 𝑐 𝑠 2 𝜔 2 =
𝜔 * 𝜔 -𝜔 0 (17) For 𝜔 0 = 0, the solution of eq.( 17) is purely real. This corresponds to the stable, anti-drift mode [START_REF] Smolyakov | Fluid theory and simulations of instabilities, turbulent transport and coherent structures in partially-magnetized plasmas of E × B discharges[END_REF][START_REF] Sakawa | Growth and nonlinear evolution of the modified Simon-Hoh instability in an electron beam-produced plasma[END_REF][START_REF] Fridman | Critical magnetic field and anomalous diffusion in slightly ionized plasmas[END_REF] . The (-𝜔 0 ) term in the denominator, corresponding to the𝑬 × 𝑩 flow, is a possible destabilizing term, since, for example, when | 𝜔 0 | ≫ 𝜔 and 𝜔 * 𝜔 0 > 0 ⁄ , 𝜔 2 becomes negative and eq.( 17) has a positive imaginary root (𝛾 > 0) and the instability grows. This instability, which occurs due to the destabilization of the anti-drift mode by the electron flow is called the gradient-drift instability or the collisionless Simon-Hoh instability or the modified Simon-Hoh instability [START_REF] Sakawa | Nonlinear evolution of the modified Simon-Hoh instability via a cascade of sideband instabilities in a weak beam plasma system[END_REF][START_REF] Sakawa | Growth and nonlinear evolution of the modified Simon-Hoh instability in an electron beam-produced plasma[END_REF] . In the collisionless case, the instability is reactive, i.e., the dispersion relation is real, and the complex roots come in complex conjugate pairs. This is an example of negative energy mode in flowing plasmas and fluids when the instability is caused by the coupling of positive and negative energy modes. Negative energy modes may become unstable solely due to dissipation. As discussed by Sakawa and Joshi [START_REF] Sakawa | Growth and nonlinear evolution of the modified Simon-Hoh instability in an electron beam-produced plasma[END_REF] , the dissipative Simon-Hoh instability was studied by Simon [START_REF] Simon | Instability of a Partially Ionized Plasma in Crossed Electric and Magnetic Fields[END_REF] , Hoh [START_REF] Hoh | Instability of Penning-Type Discharges[END_REF] , and Thomassen 90 in a weakly ionized, inhomogeneous, collisional, magnetized plasma column and was found to be unstable when the density gradient and electric field were in the same direction (i.e. 𝜔 * 𝜔 0 > 0 ⁄

). An important point in the Simon-Hoh instability is that the 𝑬 × 𝑩electron drift is much larger than the ion 𝑬 × 𝑩 drift. This difference in electron and ion drift can be due to the weak magnetization of ions or to collisions. The general solution of the dispersion relation ( 17) is:

2 𝜔 * (1 ± √1 - 4 𝜔 * 𝜔 0 𝑘 2 𝑐 𝑠 2 ) (18)
The condition for instability is that 𝜔 has a positive imaginary part, i.e. 𝜔 * 𝜔 0 > 𝑘 2 𝑐 𝑠 2 4 ⁄ , which is equivalent to 𝑣 0 𝑣 * > 𝑐 𝑠 2 4 ⁄ . This implies that the electric field 𝑬 0 and the density gradient 𝜵𝑛 0 are in the same direction (𝑬 0 . 𝜵𝑛 0 > 0), and since 𝑣 0 and 𝑣 * are generally much larger than 𝑐 𝑠 , the condition 𝜔 * 𝜔 0 > 0 ⁄ , or, 𝑣 0 𝑣 * > 0, or 𝑬 0 . 𝜵𝑛 0 is generally sufficient. Note that in Ref. [START_REF] Sakawa | Growth and nonlinear evolution of the modified Simon-Hoh instability in an electron beam-produced plasma[END_REF] , the effect of the ion velocity 𝑣 0i in the 𝑦 direction was included, so eq. ( 14) for the ion response is modified as follows:

𝑛 ñ0 = 𝑘 2 (𝜔 -𝑘 𝑣 0i ) 2 𝑒𝜙 M ( 19
)
and the corresponding dispersion relation takes the form:

𝜔 = 𝑘 𝑣 0i + 𝑘 2 𝑐 𝑠 2 2 𝜔 * (1 ± √1 + 4𝜔 * ( 𝑘 𝑣 0i -𝜔 0 ) 𝑘 2 𝑐 𝑠 2 ) (20)
Therefore, azimuthal ion velocity adds some real part to the frequency and may add some stabilizing or destabilizing effect to the growth rate, depending on the sign of 𝜔 * 𝑘 𝑣 0i .

This mode was named Modified Simon-Hoh Instability (MSHI) by Sakawa et al. [START_REF] Sakawa | Growth and nonlinear evolution of the modified Simon-Hoh instability in an electron beam-produced plasma[END_REF][START_REF] Sakawa | Excitation of the modified Simon-Hoh instability in an electron beam produced plasma[END_REF] .

Typically, for unmagnetized ions the ion azimuthal drift is neglected. In the case of weak magnetization 𝑘𝜌 𝑖 ≫ 1 , the ion drift can be estimated as 91

𝑣 0𝑖 = 𝑣 0 𝐼 0 [(𝑘𝜌 𝑖 ) 2 ]exp [-(𝑘𝜌 𝑖 ) 2 ]
where 𝐼 0 is the modified Bessel function (see Sakawa et al. 91 for the derivation of this expression, which is valid only in a slab geometry and does not include centrifugal effects). The dispersion relation (17) has been obtained assuming a uniform magnetic field. One can show [START_REF] Smolyakov | Fluid theory and simulations of instabilities, turbulent transport and coherent structures in partially-magnetized plasmas of E × B discharges[END_REF][START_REF] Frias | Long wavelength gradient drift instability in Hall plasma devices. I. Fluid theory[END_REF][START_REF] Kapulkin | Low-Frequency Instability in Near-Anode Region of Hall Thruster[END_REF] that if the magnetic field is non-uniform, the dispersion relation is modified as:

𝑘 2 𝑐 𝑠 2 𝜔 2 = 𝜔 * -𝜔 𝐷 𝜔 -𝜔 0 -𝜔 𝐷 (21)
where 𝜔 𝐷 is related to the magnetic field gradient:

𝜔 𝐷 = -𝑘𝑣 D and 𝑣 D = -2 𝑇 𝑒 𝐵 𝜕ln (B) 𝜕𝑥 = -2 𝑇 𝑒 𝐿 𝐷 𝐵 (22)
The condition for a Simon-Hoh instability in the presence of a magnetic field gradient is then ( 𝜔 * -𝜔 𝐷 ) ( 𝜔 0 + 𝜔 𝐷 ) > 0 ⁄ , which can also be written, if The magnetic field gradients effects enter the dispersion relation via the compressibility of the transverse electron drifts:

| 𝜔 𝐷 | ≪ | 𝜔 0 | : 𝑬 0 . 𝜵 𝑛 0 𝐵 2 > 0 (23)
𝛁 • 𝒗 𝑬 = -2𝒗 𝑬 • 𝛁 ln(𝐵 0 ) (24) 𝛁 • (𝑛𝒗 𝒑𝒆 ) = -2𝑛𝒗 𝒑𝒆 • 𝛁 ln(𝐵 0 ) (25)
These expressions are valid when the plasma current is neglected, i.e. it does not modify the magnetic field and 𝛁 × 𝑩 → 𝟎. These expressions also show that for a uniform magnetic field, electron temperature gradients and generally electron temperature dynamics are not coupled to the electron density directly (unless the electron temperature changes the density via ionization). For a nonuniform magnetic field, the compressibility of the electron diamagnetic flow in eq.( 25) may involve temperature evolution directly. In that case, the dispersion relation ( 21) becomes more complex and involve the equilibrium temperature gradients, as described in Frias et al. [START_REF] Frias | Long wavelength gradient drift instability in Hall plasma devices. I. Fluid theory[END_REF] . Among other effects, the temperature gradients affect the condition for the instability ( 23) and may result in the appearance of unstable modes in the direction opposite to the 𝑬 × 𝑩 drift [START_REF] Frias | Long wavelength gradient drift instability in Hall plasma devices. I. Fluid theory[END_REF] .

Because of the condition ( 25) , 𝛁 • (𝑛𝒗 𝒑𝒆 ) = 𝟎 in the case of a uniform magnetic field, the diamagnetic drift does not result in the frequency Doppler shift in the electron dynamics, as it occurs due to the 𝑬 × 𝑩 drift, 𝜔 → 𝜔 -𝜔 0 , and does not result in additional terms in the Simon-Hoh dispersion relation (20), contrary to the statement in Ref. [START_REF] Hara | Theory of gradient drift instabilities in low-temperature, partially magnetised plasmas[END_REF] . We note also that the condition 𝑣 𝑝𝑒 > 𝑣 𝑇𝑒 , obtained in Ref. [START_REF] Hara | Theory of gradient drift instabilities in low-temperature, partially magnetised plasmas[END_REF] is equivalent to the condition 𝜌 𝑒 > 𝐿 𝑛 , which is not possible in the fluid theory but requires a full kinetic treatment.

The derivation of the dispersion relations ( 17), ( 21) was made neglecting the inertia terms in the electron momentum equation. Also, the approach based on the standard momentum equation assuming scalar pressure cannot describe situations where the wavelength of the instability becomes close to the electron Larmor radius (i.e. when 𝑘 2 𝜌 𝑒 2 ≫ 1) and a kinetic approach is necessary. An alternative approach is to describe, in a more general electron momentum balance equation including the viscosity tensor, the finite Larmor radius (FLR) effects via the electron gyroviscosity [START_REF] Smolyakov | Fluid theory and simulations of instabilities, turbulent transport and coherent structures in partially-magnetized plasmas of E × B discharges[END_REF] . This was shown to provide qualitatively correct behaviour for 𝑘 2 𝜌 𝑒 2 ≫ 1 [START_REF] Pogutse | Magnetohydrodynamic equations for plasmas with finite Larmor radius effects[END_REF] . Here we will simply write the dispersion relation obtained with this approach and refer to Ref. [START_REF] Smolyakov | Fluid theory and simulations of instabilities, turbulent transport and coherent structures in partially-magnetized plasmas of E × B discharges[END_REF] for a complete derivation of the inertia and gyroviscosity contributions to this relation. In the absence of magnetic field gradient this equation writes 2 :

𝑘 2 𝜆 𝐷𝑒 2 + 𝜔 * + 𝑘 2 𝜌 𝑒 2 (𝜔 -𝜔 0 + 𝑖𝜈 𝑒𝑛 ) 𝜔 -𝜔 0 + 𝑘 2 𝜌 𝑒 2 (𝜔 -𝜔 0 + 𝑖𝜈 𝑒𝑛 ) = 𝑘 2 𝑐 𝑠 2 𝜔 2 (26)
The term 𝑘 2 𝜆 𝐷𝑒 2 in the left-hand side of eq. ( 26) accounts for finite Debye length effects and can be neglected if 𝜆 𝐷𝑒 2 ≪ 𝜌 𝑒 2 . The right-hand side of ( 26) can also be written:

𝑘 2 𝑐 𝑠 2 𝜔 2 = 𝑘 2 𝜌 𝑒 2 𝜔 𝐿𝐻 2 𝜔 2 , with 𝜔 𝐿𝐻 = 𝑐 𝑠 𝜌 𝑒 = √𝜔 𝑐𝑒 𝜔 𝑐𝑖
𝜔 𝐿𝐻 is the lower-hybrid-frequency.

In the presence of a magnetic field gradient the dispersion relation ( 26) must be modified as described by Lakhin et al. [START_REF] Lakhin | Marginal stability, characteristic frequencies, and growth rates of gradient drift modes in partially magnetized plasmas with finite electron temperature[END_REF] in eq. ( 5) of this reference, where the Debye length term and the electron collision frequency term are neglected. Including these terms, eq. ( 26) becomes:

𝑘 2 𝜆 𝐷𝑒 2 + 𝜔 * -𝜔 𝐷 + 𝑘 2 𝜌 𝑒 2 (𝜔 -𝜔 0 -2𝜔 𝐷 + 𝑖𝜈 𝑒𝑛 ) 𝜔 -𝜔 0 -𝜔 𝐷 + 𝑘 2 𝜌 𝑒 2 (𝜔 -𝜔 0 -2𝜔 𝐷 + 𝑖𝜈 𝑒𝑛 ) = 𝑘 2 𝑐 𝑠 2 𝜔 2 (27)
Note that the dispersion relation with magnetic field of Ref. [START_REF] Smolyakov | Fluid theory and simulations of instabilities, turbulent transport and coherent structures in partially-magnetized plasmas of E × B discharges[END_REF] (eq. ( 34) of this reference) is missing the higher order terms in 𝑘 2 𝜌 𝑒 2 𝜔 𝐷 and is less accurate than eq. ( 27).

The dispersion relations ( 26), ( 27) can be considered as a high frequency, small wavelength, and collisional extension of the collisionless Simon-Hoh or gradient-drift dispersion relation (17). The 𝑘 2 𝜌 𝑒 2 term on the numerator of ( 26) comes from the inertia term in the electron momentum equation, while the same term on the denominator comes from the gyroviscosity.

If the gyroviscosity term is omitted and assuming uniform magnetic field and quasineutrality, eq. ( 26) is modified into:

-𝜔 * + 𝑘 2 𝜌 𝑒 2 (𝜔 -𝜔 0 + 𝑖𝜈 𝑒𝑛 ) 𝜔 -𝜔 0 + 𝑖𝜈 𝑒𝑛 = 𝑘 2 𝑐 𝑠 2 𝜔 2 (28)
In the collisionless limit, equation ( 28) is equivalent to:

𝜔 2 𝜔 𝐿𝐻 2 = 1 - 𝜔 * (𝜔 -𝜔 0 )𝑘 2 𝜌 𝑒 2 (29)
This equation illustrates the transition from the collisionless Simon-Hoh mode to the lower-hybrid mode. For small values of 𝑘𝜌 𝑒 , i.e. for long wavelengths, equation (29) gives the collisionless Simon-Hoh dispersion relation (17), while it tends to the basic lower-hybrid mode, 𝜔 = ±𝜔 𝐿𝐻 when 𝑘 2 𝜌 𝑒 2 |𝜔 0 𝜔 * ⁄ | is sufficiently larger than 1 (assuming ω ≪ |ω 0 | ). The dispersion relation eq. ( 26) can therefore be also considered as a lower-hybrid mode destabilized by 𝑬 × 𝑩 flow, density gradient and collisions. If the diamagnetic drift is zero, 𝜔 * = 0 , and assuming |𝜔 0 | ≫ 𝜔 and 𝜈 𝑒𝑛 ≪ 𝜔 , the dispersion relation (28) becomes:

𝜔 = ±𝜔 𝐿𝐻 (1 + 𝑖𝜈 𝑒𝑛 2𝜔 0 ) (30)
This is the "resistive mode" of Ref. [START_REF] Litvak | Resistive instabilities in Hall current plasma discharge[END_REF] , which can be considered as a lower-hybrid mode destabilized by 𝑬 × 𝑩 drift and collisions only, and is another example of a negative energy mode destabilized by collisional dissipation.

For very large values of 𝑘𝜌 𝑒 , the general form of ( 26) gives the ion sound mode 𝜔 = 𝑘𝑐 𝑠 or 𝜔 = 𝑘𝑐 𝑠 √1 + 𝑘 2 𝜆 𝐷𝑒 2 ⁄ for finite Debye lengths.

Note that eq. ( 26) can be written in dimensionless units where, the frequencies are normalized by 𝜔 𝐿𝐻 , the lengths are normalized by 𝜌 𝑒 , and the velocities are normalized by 𝑐 𝑠 . If we write Ω = 𝜔 𝜔 𝐿𝐻 ⁄ , 𝐾 = 𝑘𝜌 𝑒 , the roots Ω(𝑘) of eq. ( 26) depend on three parameters: 𝑉 0 = 𝑣 0 𝑐 𝑠 ⁄ , 𝑉 * = 𝑣 * 𝑐 𝑠 ⁄ , and 𝑁 𝑒𝑛 = 𝜈 𝑒𝑛 𝜔 𝐿𝐻 ⁄ . If finite Debye length is taken into account, another parameter is 𝜆 𝐷𝑒 𝜌 𝑒 ⁄ (or, equivalently, 𝜔 𝑐𝑒 𝜔 𝑝𝑒 ⁄ ). We show in Figure 5 and Figure 6 some examples of solutions Ω(𝐾) of the dispersion relation (26) for different values of 𝑉 0 and 𝑉 * , with 𝜈 𝑒𝑛 𝜔 𝐿𝐻 ⁄ = 6.23 and 𝜆 𝐷𝑒 𝜌 𝑒 ⁄ = 𝜔 𝑐𝑒 𝜔 𝑝𝑒 ⁄ = 0.16 and a uniform magnetic field. In Figure 5, the dimensionless diamagnetic drift velocity 𝑉 * is fixed and the 𝑬 × 𝑩 drift velocity 𝑉 0 is varied. In Figure 6, 𝑉 0 is fixed and 𝑉 * is varied. The corresponding numerical values of 𝐸 0 , 𝐿 𝑛 , and other parameters are given in the figure captions, for a particular case corresponding to argon, a magnetic field intensity of 50 mT, and an electron temperature 𝑇 𝑒 =5 eV.

We see on Figure 5 that the maximum growth rate is obtained for decreasing values of the wave number, i.e., increasing wavelength, when 𝑉 0 increases at constant 𝑉 * . It is equivalent to say that the wavenumber at maximum growth rate increases with the electric field 𝐸 0 for a constant plasma density gradient defined by 𝐿 𝑛 . On Figure 6, the wavenumber at maximum growth rate decreases with decreasing diamagnetic drift at constant 𝑬 × 𝑩 drift. Long wavelengths are therefore obtained for low values of the diamagnetic drift or large values of the 𝑬 × 𝑩 drift. At the limit of zero density gradient or zero diamagnetic drift (|𝐿 𝑛 | → ∞ or 𝑉 * = 0), the lower-hybrid mode is destabilized by 𝑬 × 𝑩 drift and collisions only in coherence with the simplified relation (30).

From the general dispersion relation ( 26), one can calculate, for each value of 𝑉 0 and 𝑉 * , the maximum growth rate 𝛾 𝑚𝑎𝑥 , the wave angular frequency at the maximum growth rate, 𝜔 𝑅,𝛾𝑚𝑎𝑥 , and the wavenumber 𝑘 𝛾 𝑚𝑎𝑥 at the maximum growth rate, or the wavelength 𝜆 𝛾 𝑚𝑎𝑥 at maximum growth rate.

The corresponding dimensionless values, 𝛾 𝑚𝑎𝑥 𝜔 𝐿𝐻 ⁄ , 𝜔 𝑅,𝛾𝑚𝑎𝑥 𝜔 𝐿𝐻 ⁄ , and 𝜆 𝛾 𝑚𝑎𝑥 𝜌 𝑒 ⁄ are displayed in the form of contour plots as a function of 𝑉 0 and 𝑉 * , in Figure 7 (similar plots were presented in Ref. [START_REF] Boeuf | Micro instabilities and rotating spokes in the nearanode region of partially magnetized plasmas[END_REF] ).

On each of these plots, the Simon-Hoh or gradient drift instability is obtained in the lower right part of the figure (long wavelengths, lower angular frequencies) and the lower hybrid mode in the upper left part (shorter wavelengths, higher frequencies). The resistive mode is obtained for 𝑉 * → 0 (not seen on the plots because of the log scale), while the lower-hybrid mode destabilized by density gradient and collisions is obtained for 𝑉 0 → 0.

The instabilities discussed in this section are closely related to Farley-Buneman and gradient-drift instabilities in ionospheric plasmas [START_REF] Kelley | The Earth's Ionosphere: Plasma Physics and Electrodynamics[END_REF] .

The dispersion relations above have been obtained assuming perturbations in the 𝑦 (𝑬 × 𝑩 ) direction only. More comprehensive treatment of the fluid dispersion relations in low temperature partially magnetized 𝑬 × 𝑩 plasmas can be found in the literature, which consider effects such as electron or ion flow in the 𝑥 direction [START_REF] Frias | Long wavelength gradient drift instability in Hall plasma devices. I. Fluid theory[END_REF] , electron temperature perturbations [START_REF] Frias | Long wavelength gradient drift instability in Hall plasma devices. I. Fluid theory[END_REF] , sheath effects [START_REF] Morin | Modification of the Simon-Hoh Instability by the sheath effects in partially magnetized E × B plasmas[END_REF] , etc… Finally, dispersion relations give information on the development of the instabilities in the linear stage only. Analyzing the growth of the instability and the saturation mechanisms is more challenging and requires the use of full fluid or kinetic models. Examples of fluid description of the evolution of the instabilities based on the BOUT++ code 99 can be found in Refs. [START_REF] Smolyakov | Fluid theory and simulations of instabilities, turbulent transport and coherent structures in partially-magnetized plasmas of E × B discharges[END_REF]100 .

E. Example of evidence for rotating azimuthal instabilities in 𝑬 × 𝑩 plasma devices and interpretation with dispersion relations

In this section, we briefly show a few examples of evidence for the presence of rotating azimuthal instabilities in low temperature partially magnetized plasmas, drawn from experiments or simulations. We also discuss some of the efforts to understand the physics behind these rotating structures. A more detailed discussion of these instabilities will be given in section III, with the example of the Hall thruster and of the magnetron discharge, and with simulation results in section IV.

We start with the Penning discharge of Raitses et al. 101,102 , which is schematically represented in Figure 8. The principles and conditions of operation of this discharge are indicated in the caption to Figure 8. In these conditions, rotating plasma non-uniformities ("spokes") in a plane perpendicular to the magnetic field are observed in the experiments with both fast imaging techniques (CCD camera) and Langmuir probes measurements. Rotating spokes have been observed in many other 𝑬 × 𝑩 plasma devices including standard Hall thrusters [103][104][105][106][107] , cylindrical Hall thrusters 108 , wall-less Hall thrusters 109 , anode layer Hall thrusters 110,111 , low power dc or rf as well as high power pulsed magnetron discharges [112][113][114][115][116][117][118][119] , and, of course in magnetized plasma columns dedicated to the study of turbulent phenomena 37-39, 120, 121 . 8 in a plane (radial-azimuthal) perpendicular to the magnetic field, showing the space distribution of the electron density at two different times. The anode radius R0 is 2.5 cm. Electrons are injected in a circular area of radius Ri=0.1 cm around the discharge axis, with a directed (beam) energy of 15 eV. The applied magnetic field is 10 mT, and the neutral gas pressure is 1 mtorr (293 K). The simulations are performed in a gas with helium mass and approximate crosssections of xenon (electron momentum cross-section, no excitation). Ions are collisionless. A pseudo-permittivity 𝜀 𝑟 = 400 is used in the simulations. This means that the "real density" simulated is actually 400 times smaller than indicated in the scale of this figure (i.e. the plotted density is actually 𝜀 𝑟 𝑛 𝑒 so the maximum "real" density in the simulation is about 10 16 m -3 ), but Coulomb collisions are taken into account for the density indicated in the scale. Adapted from Powis et al., Phys. Plasmas 25, 072110 (2018), with permission of AIP Publishing (Fig. 6 of Ref. 123 ).

A number of articles has also been devoted to modeling and simulation of rotating spokes in the recent years, with kinetic approaches (Particle-In-Cell Monte Carlo Collisions, PIC-MCC) [START_REF] Boeuf | Rotating structures in low temperature magnetized plasmas; insight from particle simulations[END_REF][START_REF] Boeuf | Micro instabilities and rotating spokes in the nearanode region of partially magnetized plasmas[END_REF][122][123][124][125][126][127][128][129][130] or fluid 131 and hybrid fluid-particle simulations [132][133][134] . An example of results of particle simulations of the Penning discharge of Figure 8 is shown in Figure 9 while Figure 11 displays some high speed camera images of a rotating spoke in a wall-less Hall thruster 109 . Figure 9 shows an example of PIC-MCC simulation results of the Penning discharge of Figure 8 taken from the article of Powis et al. 123 . The conditions of the simulations are indicated in the caption to this figure. The figure illustrates the formation of a spoke in argon at 1 mtorr, and its rotation in the 𝑬 × 𝑩 direction at a velocity at mid-radius of 4.9 km/s. Simulations have also been performed in a purely collisionless situation and compared with this collisional case with self-consistent ionization (in the collisionless case, positive ions are injected in the central region of the discharge). The main results of the paper by Powis et al. can be summarized as follows. The PIC-MCC simulation predicts the formation and rotation of a spoke under both the collisionless and collisional conditions, with similar rotation velocities. Anomalous electron transport to the anode can be analyzed by looking at the time averaged profiles of the electric field and plasma density. Figure 10 shows the radial distribution of the electric potential, radial electric field, and electron density in the collisionless and collisional case. We see that the injection of electrons and the trapping of electrons by the magnetic field tend to lower the potential in the plasma center.

The instability and resulting deformation of the electric potential allows magnetically trapped electrons to reach the anode even in the absence of collisions as well as electrostatically trapped ions to flow to the anode (no axial losses are taken into account in the simulation). The fact that the plasma density gradient and electric field are in the same direction suggests that the conditions necessary for a Simon-Hoh or gradient-drift instability, as described in the previous subsection, are met. The scaling of the rotational frequency deduced from the PIC-MCC simulation was compared with the scaling given by the dispersion relation of the collisionless Simon-Hoh mode (eqs. ( 17), ( 18) above) and good agreement was found. It was however shown in a more recent paper by Tyushev et al. 129 that in this cylindrical geometry, the Simon-Hoh dispersion relation must be modified to account for the stationary centrifugal ion rotation with frequency 𝜔 𝑖 ≈ √-𝑒𝐸 𝑟 𝑀𝑟 ⁄ . The anomalous electron mobility or conductivity can be quantified by writing the expression of the radial mean electron velocity, i.e. parallel to the time averaged electric field. Using the expression derived in eq. ( 2), we obtain:

𝑣 𝑒𝑟 = - 𝜇 𝑒,𝑒𝑓𝑓 ℎ 𝑒𝑓𝑓 2 (𝐸 𝑟 + 1 𝑛 𝜕𝑛𝑇 𝑒 𝜕𝑟 ) (31)
Equation ( 2) was derived for situations where electron transport across the magnetic field was due to collisions with neutrals. The index "e𝑓𝑓 » in eq. ( 31), indicates that we consider an effective electron collision frequency 𝜈 𝑒𝑓𝑓 to represent the effect of instabilities on cross-field transport in the same way as in the purely collisional case.

The effective mobility and Hall parameter 𝜇 𝑒,𝑒𝑓𝑓 = 𝑒/(𝑚𝜈 𝑒𝑓𝑓 ) and ℎ 𝑒𝑓𝑓 = 𝜔 𝑐𝑒 𝜈 𝑒𝑓𝑓 ⁄ are obtained simply by equating the expression (31) of the mean cross-field velocity (with 𝐸 𝑟 (𝑟) and 1 𝑛 𝜕𝑛 𝜕𝑟 from Figure 10) to the mean velocity deduced from the PIC-MCC simulation (a similar approach was used in Ref. [START_REF] Boeuf | Micro instabilities and rotating spokes in the nearanode region of partially magnetized plasmas[END_REF] ).

The radially (and time) averaged effective collision frequency calculated by Powis et al. in the conditions of Figure 9 and Figure 10 was on the order of 10 8 s -1 both in the collisionless and collisional cases. For comparison the real electron-neutral collision frequency in the collisional case (1 mtorr pressure) was about ten times smaller, indicating that electron cross-field transport is also dominated by fluctuations at 1 mtorr. The calculated effective Hall parameter was between 13 and 16 in both cases.

One of the conclusions of the article by Powis et al. is that, since similar rotating spokes are obtained in the Penning discharge model for both collisional and collisionless cases, ionization is not a necessary condition for the existence of spokes. We will see however in section IV that under magnetron discharge conditions, ionization may play a major role in the physics of a rotating spoke and that an instability triggered by the Simon-Hoh criterion can evolve into an ionization wave. 3D PIC-MCC simulations 125 performed in the conditions of the wall-less Hall thruster of Figure 11 In the case of a deep potential well they found a 𝑚 = 1 spoke regime as in the results of Powis et al. 123 , Figure 9. When the potential profile was flatter, they observed a transition into a rotating small scale spiral arm with 𝑚 ≫ 1 , as illustrated in Figure 12a. This is a clear illustration of a transition from a long wavelength Simon-Hoh instability (spoke), where the instability is triggered by the density gradient and radial electric field being in the same direction, to a small-scale lower hybrid instability driven by density gradient and electron-neutral collisions. Similar conclusions were drawn in Ref. [START_REF] Boeuf | Micro instabilities and rotating spokes in the nearanode region of partially magnetized plasmas[END_REF] , where the 2D PIC-MCC simulation was performed in an axial-azimuthal geometry rather than in a radial-azimuthal geometry. In these simulations the profile of the electric potential was controlled by the magnetic field intensity. An example from this reference is shown in Figure 12b in the conditions of small-scale instabilities. Lucken et al. 135 also found the formation of rotating spiral structures with 𝑚 ≫ 1 in 2D PIC-MCC simulations under conditions where the plasma was generated by radiofrequency electron heating in the direction parallel to the magnetic field, i.e. perpendicular to the simulation domain. This is illustrated in Figure 12c.

In all three models. [START_REF] Boeuf | Micro instabilities and rotating spokes in the nearanode region of partially magnetized plasmas[END_REF]129,135 , the effective Hall parameter was found to decrease, i.e. the anomalous transport was increasing, from the spiral regime (low magnetic field or quasi-flat electric potential) to the spoke regime (larger magnetic field or deeper potential well), with a saturation at large magnetic field in Refs. [START_REF] Boeuf | Micro instabilities and rotating spokes in the nearanode region of partially magnetized plasmas[END_REF]135 .

F. Hall effect in a low temperature partially magnetized plasma

The Hall effect occurs when the 𝑬 × 𝑩 current is not closed on itself but flows to a wall. There is not Hall effect in closed-drift devices like the Hall thruster even though Hall thrusters are sometimes called "Hall effect thrusters". To illustrate the Hall effect, we consider, as in Kunkel [START_REF] Kunkel | Hall effect in a plasma[END_REF] (see also Ecker 52 for the case of a cylindrical column) a plasma between two dielectric or floating walls parallel to the 𝑥 direction and perpendicular to the 𝑦 direction, an electric field 𝑬 = 𝐸 𝑥 𝒙 ̂ in the 𝑥 direction and a perpendicular magnetic field 𝑩 = 𝐵𝒛 ̂ in the 𝑦 direction. The electric field 𝐸 𝑥 is given and must adjust so that charged particle losses to the walls are balanced by ionization, as in a positive column plasma. The 𝑬 × 𝑩 electron current, in the -𝑦 direction, flows to the wall. We assume that the plasma column is uniform in the 𝑥 direction (𝜕𝑛 𝜕𝑥 ⁄ = 0), with a constant electron temperature, and that the plasma density is a function of 𝑦 only (𝜕𝑛 𝜕𝑦 ⁄ is non-zero due to the presence of the walls). The electron current to the wall must be balanced by an ion current so an electric field must form to draw the ions to the walls and ensure a zero net current. This electric field is the Hall electric field. The Hall electric field, combined with the perpendicular magnetic field can in turn generate an 𝑬 × 𝑩 drift in the 𝑥 direction. One can easily write a simplified momentum equation as eq.( 1), except that there is now a component of the electric field, 𝐸 𝑦 , and a density gradient 𝜕𝑛 𝜕𝑦 ⁄ in the 𝑦 direction, and we assume a zero-density gradient in the 𝑥 direction. We obtain:

{ 𝑣 𝑒𝑥 = - 𝜇 𝑒0 1 + ℎ 2 𝐸 𝑥 + 𝜇 𝑒0 ℎ 1 + ℎ 2 𝐸 𝑦 * 𝑣 𝑒𝑦 = - 𝜇 𝑒0 1 + ℎ 2 𝐸 𝑦 * - 𝜇 𝑒0 ℎ 1 + ℎ 2 𝐸 𝑥 } ( 32
)
where 𝐸 𝑦 * = 𝐸 𝑦 + 𝑇 𝑒 𝑛 𝜕𝑛 𝜕𝑦 The plasma is quasineutral so the condition for a zero current to the wall is that the electron mean velocity 𝑣 𝑒𝑧 , and ion mean velocity 𝑣 𝑖𝑧 to the walls are equal. We assume collisional, non-magnetized ions with a mobility 𝜇 𝑖 and we neglect ion diffusion. Writing 𝑣 𝑖𝑦 = 𝜇 𝑖 𝐸 𝑦 in eq. ( 32), we obtain the following expression for 𝐸 𝑧 :

𝐸 𝑦 = - ℎ 𝛼 𝐸 𝑥 - 1 𝛼 𝑇 𝑒 𝑛 𝜕𝑛 𝜕𝑦 , with 𝛼 = 1 + 𝜇 𝑖 𝜇 𝑒0 (1 + ℎ 2 ) (33)
The first term on the right-hand-side of eq. ( 33) is the Hall electric field, 𝐸 𝐻 = -ℎ 𝛼 𝐸 𝑥 , the second term is the ambipolar field. The average field in the 𝑦 direction is therefore 𝐸 𝐻 .

Using the expression ( 33) of 𝐸 𝑦 in eq.( 32), we obtain the following expression for the mean electron velocity 𝑣 𝑒𝑥 parallel to the axial electric field :

𝑣 𝑒𝑥 = - 𝜇 𝑒0 (1 + ℎ 2 ) (1 + ℎ 2 𝛼 ) 𝐸 𝑥 - 𝜇 𝑒0 ℎ (1 + ℎ 2 ) (1 - 1 𝛼 ) 𝑇 𝑒 𝑛 𝜕𝑛 𝜕𝑦 (34)
The mean velocity parallel to the electric field 𝐸 𝑥 , averaged in the z direction is

𝑣 𝑒𝑥 ̅̅̅̅ = - 𝜇 𝑒0 (1+ℎ 2 ) (1 + ℎ 2 𝛼 ) 𝐸 𝑥 .
This is to be compared to the mean velocity in the absence of Hall effect, given by eq. ( 2), 𝑣 𝑒𝑥 ̅̅̅̅ = - A consequence of the presence of the Hall electric field in the 𝑦 direction is that the plasma density is no longer symmetric in this direction. This can be seen by writing a continuity equation assuming a constant ionization frequency. The charged particle flux in this equation contains an ambipolar term and a term related to the Hall current. One can easily show (see Kunkel 53 ) that the density profile along 𝑧 is of the following form (with an eigenvalue condition that connects ionization frequency, ambipolar diffusion and parameters related to the Hall effect [START_REF] Kunkel | Hall effect in a plasma[END_REF] ) :

𝑛(𝑦) = 𝑛 0 exp (ℎ 𝐸 𝑥 2𝑇 𝑒 𝑦) sin(𝜋 𝑦 𝐿 ⁄ ) (35)
𝐿 Is the distance between the walls. The usual sinusoidal solution is obtained in the absence of Hall effect (ℎ = 0). The simple example above corresponds to an axially uniform plasma column between parallel walls in the presence of a perpendicular magnetic field.

We consider now the more complex example of a negative ion source used for the neutral beam injection system of ITER 136 . The design of this source is schematically represented in Figure 13. The source consists of a driver where electrons are heated by inductive coupling, a diffusion region with a magnetic filter, and an extraction region. The magnetic field is perpendicular to the plane of Figure 13. The plasma is produced in hydrogen or deuterium at 0.3 Pa. The negative ions of hydrogen or deuterium are generated in the plasma volume by attachment on vibrationally excited states and on the plasma grid covered with a caesium layer.

Hydrogen or deuterium atoms hitting the caesiated grid are converted into negative ions. Negative ions are extracted through a system of biased grids. Negative ions can be destroyed by fast electrons coming from the driver. The objective of the magnetic filter in the diffusion region is to increase the residence time of electrons and increase their energy losses through collisions with neutrals. A second objective is also to lower the density of electrons close to the extraction region and to limit their extraction through the grids (in addition to the magnetic filter, a magnetic field is also generated by magnets near and parallel to each grid hole to limit this flux).

Figure 13 : Two-dimensional rectangular geometry considered in the 2D PIC MCC simulations 137 of the ITER negative ion source for neutral beam injection 136 . The chamber walls are metallic and grounded, and the right electrode ("plasma grid") is biased at a voltage 𝑉 𝑏 . The magnetic field is perpendicular to the simulation domain (upward) and is constant along the 𝑦 direction.

The plasma density gradient from the driver to the magnetic filter due to the large electron pressure in the driver creates a charged particle flux toward the extraction grids. The grid voltage is positive with respect to the grounded walls to enhance the flux of negative ions to the extraction system. In this configuration, there is an axial flux of charged particles perpendicular to the magnetic field. We therefore expect to have a combination of diamagnetic drift and 𝑬 × 𝑩 drift in the 𝑦 direction of Figure 13. These drifts are not closed and are perpendicular to the rectangular chamber walls, so we expect a Hall effect to occur. This Hall effect is discussed in Refs. [137][138][139][140] on the basis of PIC-MCC simulations. We present here a few results from Ref. 137 . Figure 14 shows the 2D distributions of the electron density and plasma potential in the ion source. The plasma potential in the driver is much larger than in the diffusion chamber because electrons are heated in the driver and lose energy in the filter.

We observe on the potential and plasma density distributions an asymmetry which can be attributed to the Hall effect.

There is a potential difference (Hall voltage) between the sheath edges on the top and bottom walls. The plasma density is higher on the lower side of the chamber (while the electron temperature is lower on the lower side, see Ref. 141 ).

Figure 15 displays the intensity and streamlines of the electron and ion current densities in the chamber. We can clearly see the combination of the different electron drifts in the 𝑧 direction at the driver exit (diamagnetic and 𝑬 × 𝑩 drift; a drift due to the magnetic field gradient, proportional to 𝛁𝑩 × 𝑩 is also present). The more important plasma asymmetry predicted by the model at the driver exit has been confirmed by experiments 142 . Compared with a closed-drift system, the Hall effect enhances the electron flow through the magnetic filter and no instability is observed in these conditions. If the walls are removed in the 𝑧 direction and periodic conditions are used instead (see Ref. 143 ), the drift is closed on itself and electron confinement in the axial direction is much more efficient. In these conditions instabilities are present, which is not the case in the bounded conditions of Figure 14 and Figure 15. Finally, we note, by comparing Figure 15a and Figure 15b, that the plasma diffusion in this source is clearly non-ambipolar except in the magnetic field free driver. Nonmagnetized ions simply go down the potential while the streamlines of the electron current density are much more complex. This feature is typical of partially magnetized plasmas.

III. HALL THRUSTERS AND MAGNETRON DISCHARGES

In this section we describe in more details the physics and instabilities of two important 𝑬 × 𝑩 plasma devices, the Hall thruster and the magnetron discharge.

A. Hall thrusters

The plasma of a Hall thruster (Figure 16) is generated in a channel between two concentric dielectric cylinders. The anode is at one end of the channel. Xenon (or krypton) is injected through the anode. An electron emissive cathode is placed outside the channel exhaust. A radial magnetic field, with a maximum (about 20 mT) close to the exhaust plane, is generated by coils or magnets and a magnetic circuit. A voltage of 300 V is applied between cathode and anode under nominal operations. The xenon mass flow rate, and radii of the inner and outer dielectric cylinders are about 5 mg/s and 3.5 and 5 cm for a 1 kW thruster. Because of the lowering of the axial electron mobility in the region of maximum magnetic field, the axial electric field in that region is expected to increase. The purpose of a Hall thruster is to extract ions from a plasma and generate thrust without using biased grids like in usual ion sources, but by creating a large accelerating field within the plasma. The Hall thruster is a very interesting and unique plasma device where a potential drop of several hundred volts can indeed be generated in a quasineutral plasma. The electric field in this potential drop can be as large as 400 V/cm over a distance of about 1 cm, much larger than the Debye length. This is quite remarkable since one could expect the cross-field electron mobility to be greatly increased by instabilities and turbulence, leading to a collapse of the electric field.

One thing that helps maintaining a low electron conductivity in the acceleration region and outside the channel is the fact that the flow of neutral atoms coming from the anode side is practically fully ionized by the energetic electrons in the ionization region upstream of the channel exhaust (see the location of the ionization region in Figure 16b). This has two consequences: (1) the contribution of electron-neutral collisions to cross-field transport is reduced (anomalous transport due to instabilities is present but we will see that the effective Hall parameter does not collapse in this region); (2) Since ions are practically collisionless due to almost complete ionization of the gas flow, they can be freely accelerated by the electric field resulting from the lowering of the electron conductivity, hence a decrease of the ion density or plasma density and electron conductivity. A larger neutral atom density in the region outside the channel would significantly decrease the mean ion velocity due to charge exchange collisions and thus increase the plasma density leading to an increase of electron conductivity and a collapse of the electric field. Nevertheless, there are some instabilities that increase the electron cross-field transport, and this is very difficult to quantify. An example of experimental evidence of the presence of instabilities responsible for anomalous electron transport is shown in Figure 17 where the inverse effective Hall parameter, ℎ 𝑒𝑓𝑓 -1 , deduced from different measurements of plasma parameters, is compared with the inverse Hall parameter calculated for electron-neutral collisions only. ℎ 𝑒𝑓𝑓 -1 is deduced from indirect estimations of the effective electron mobility 𝜇 𝑒,𝑒𝑓𝑓 based on measurements of the axial electric field 𝐸 𝑥 , electron current density 𝐽 𝑒𝑥 and electron density 𝑛 𝑒 :

ℎ 𝑒𝑓𝑓 -1 = 𝜇 𝑒,𝑒𝑓𝑓 𝐵 𝑟 = 𝐽 𝑒𝑥 𝑒𝑛 𝑒 𝐸 𝑥 𝐵 𝑟 (36)
The inverse Hall parameter due to electron-neutral collisions, ℎ -1 = 𝜈 𝑒𝑛 𝜔 𝑐𝑒 ⁄ is deduced from measurements of the neutral atom density and electron temperature. Although this comparison involves many complex and delicate measurements and the accuracy of the electron mobility obtained in this way is difficult to estimate, there is clear evidence in Figure 17 that the effective collision frequency is much larger than the electron-collision frequency. This is especially true close to the exhaust plane where the inverse effective Hall parameter (or effective collision frequency) is larger than the inverse collisional Hall parameter (or electroncollision frequency) by almost two orders of magnitude. In this region the neutral atom density is very low due to ionization upstream, and cross-field electron is clearly not due to electron-neutral collisions alone. Although instabilities certainly play an important role in the enhancement of cross-field electron transport in a Hall thruster, they are not the only cause of the larger than expected electron mobility. Electrons interactions with the channel walls also contribute to the increase of electron effective mobility inside the channel. The scattering of fast electrons by the channel walls and the secondary electron emission resulting from this interaction are also important causes of cross-field electron transport. The interactions of electrons with the channel walls also have important impact on the electron energy balance in the channel. There is a considerable literature on the effect of electron-wall interactions on the conductivity in Hall thrusters and a discussion of these effects is beyond the scope of this paper. Discussions and references on this question can be found in Kaganovich et al. [START_REF] Kaganovich | Physics of E × B discharges relevant to plasma propulsion and similar technologies[END_REF] .

Figure 18 : Schematic axial profiles of the radial external magnetic field, 𝐵 𝑟 , axial electric field, 𝐸 𝑥 , and plasma density 𝑛 along a line through the mid-channel of a Hall thruster.

Controlling the electron conductivity in a Hall thruster is an important issue because the efficiency of the thruster decreases if the electron current entering the channel increases [START_REF] Goebel | Fundamentals of Electric Propulsion: Ion and Hall Thrusters[END_REF][START_REF] Boeuf | Tutorial: Physics and modeling of Hall thrusters[END_REF] (i.e. if the electron conductivity increases in the exhaust region). The operations and performance of a Hall thruster are extremely sensitive to the distribution and axial profile of the radial magnetic field which is still empirically optimized (models with predictive capabilities are not available). This sensitivity is largely due to the fact that the magnetic field intensity profile controls the instabilities and cross-field electron transport. For example, the nature of azimuthal instabilities may be different in different parts of the thruster, as illustrated in Figure 18, which shows schematic axial profiles of the radial magnetic field, axial electric field, and plasma density. The plasma density decreases outside the channel and toward the anode. On the anode side of the thruster the plasma density and axial electric field are in the same direction. This suggests that long wavelengths instabilities of the Simon-Hoh or gradient-drift type may be present in this region. Rotating spokes have indeed been observed in this region 104-108, 145, 146 and their presence depends on the magnetic field intensity. On the cathode side of the maximum magnetic field the electric field and plasma density gradient are in opposite direction so a gradient-drift instability cannot develop in that region. We will see in section IV.A that a short wavelength kinetic instability, the Electron Cyclotron Drift Instability, and a fluid instability, the Modified Two-Stream Instability can be present in the acceleration region of a Hall thruster. Figure 19 shows the different types of instabilities that can develop in a Hall thruster. Rotating spokes, in the frequency range 5-20 kHz, Electron Cyclotron Drift instabilities around 2-5 MHz and Modified Two-Stream instabilities at 5-10 MHz form in the azimuthal direction and are directly related to the 𝑬 × 𝑩 configuration, but axial instabilities such as the breathing mode and the ion transit time can also be present and sometimes coupled with the azimuthal fluctuations. An early review of Hall thruster oscillations can be found in Ref. 147 . Although progress in understanding these oscillations has been made over the past twenty years and is described in Ref. [START_REF] Kaganovich | Physics of E × B discharges relevant to plasma propulsion and similar technologies[END_REF] , a fully self-consistent and predictive description of Hall thruster instabilities has yet to be achieved. We briefly summarize these instabilities below, and describe some of them in more details in section IV.A.

The breathing mode is an ionization instability related to a periodic depletion of the neutral density due to strong ionization of the neutral flow. This is one of the most common and most studied instabilities in Hall thrusters [START_REF] Morozov | Review of Plasma Physics[END_REF] . This ionization instability can lead to large amplitude oscillations of the discharge current. The breathing mode is strongly related to the cross-field electron transport and to the electron energy balance. Strong ionization by electrons trapped in the large magnetic field region and drifting in the 𝑬 × 𝑩 direction induces a depletion of the neutral density in the ionization region. The neutral density depletion changes the momentum and energy balance of the electrons, leading to a decrease in electron conductivity and in the current of electron entering the channel. The ionization decreases until the flow of neutral atoms from the anode replenishes the neutral depletion. The oscillation frequency is therefore related to the time necessary for the neutrals to refill the ionization region. This is a low-frequency mode typically in the 5-20 kHz range. This qualitative description actually hides a very complex and non-linear physics that is still under discussion. Excitation of the breathing mode is affected by several interrelated mechanisms that depend in a complex way on the magnetic field configuration and are not easily quantifiable. These mechanisms include electron momentum and energy losses to the walls, anomalous electron transport, ion backflow and recombination on the walls and anode, etc… Attempts to untangle these complex physical mechanisms have been made in a number of papers based on 1D 148-157 numerical models. A simple 0D predator-prey model has been proposed 149,153,155,158 to describe the ionization-depletion-replenishment process but fails to identify the conditions for the instability. We will not discuss these research efforts further here and we refer to some early or more recent review articles [START_REF] Kaganovich | Physics of E × B discharges relevant to plasma propulsion and similar technologies[END_REF]159 and to recently published papers on this subject 156,157,160 .

Under certain conditions, the low-frequency breathing mode coexists with high-frequency oscillations in the range of 100 to 500 kHz. These high frequency oscillations correspond to the characteristic time-of-flight of ions in the channel and have been evidenced in hybrid Hall thruster models and in experiments 157,[161][162][163][164][165][166][167][168] . Using 1D fluid and hybrid models, Chapurin et al. 157 identified two distinct regimes of breathing oscillations depending on the assumption on the electron energy losses to the walls. The regime with higher energy losses exhibits a low-frequency mode of about 14 kHz that coexists with high-frequency ion transit time oscillations around 150 kHz. In the second regime with low electron energy losses, pure breathing oscillations are observed.

The higher frequency oscillations of Figure 19, the Electron Cyclotron Drift Instability and the Modified Two-Stream Instability are of kinetic nature and will be described in section IV.A, on the basis of PIC-MCC simulations.

B. Magnetron discharges

We have seen in section II.C that in a magnetron discharge, the magnetic field is generated by magnets placed behind the cathode (Figure 20). This creates a point-cusped magnetic field. Electrons are mirroring at the cusp regions and are also electrostatically confined by the cathode sheath. The plasma forms in the region above the cathode, where the magnetic field is parallel to the cathode. This is an 𝑬 × 𝑩 configuration with axial electric field and radial magnetic field. The magnetic field intensity decreases from cathode to anode (as in the channel of a Hall thruster). Magnetron discharges can be operated in dc or rf regimes, but also in a pulsed regime with very high-power densities (HiPIMS) at gas pressures below a few Pa. They are used in sputter deposition applications. The ions are accelerated at high energies toward the target cathode in the quasi-collisionless sheath and sputter atoms from the cathode surface. These atoms then deposit on a substrate, forming a thin layer.

A number of papers report the presence of instabilities in magnetron discharges, which take the form of large-scale plasma non-uniformities rotating in the azimuthal direction and associated with rotating regions of enhanced light emission and ionization. These rotating non-uniformities were first detected in Hall thrusters by Janes and Lowder 169 using Langmuir probes and were called "spokes". More recent observations with high speed camera imaging in Hall thrusters 104, 107-109, 170, 171 as well as in magnetron discharges 112,115,116,[172][173][174][175][176] have shown that these rotating non-uniformities are associated with a strong light emission.

Interestingly, it has been shown in several articles 116,174,[176][177][178] that the spoke can rotate in the -𝑬 × 𝑩 direction in low power dc magnetrons and in the +𝑬 × 𝑩 direction in high power pulsed magnetrons used in plasma processing applications (HiPIMS), at velocities more than ten times smaller than the 𝐸 × 𝐵 drift velocity. The images of rotating spokes in a dc magnetron discharge obtained by Panjan and Anders 178 and displayed in Figure 21 show that the number of spokes increases with gas pressure. Stable plasma patterns with four and up to seven spokes were observed by these authors under rf (radiofrequency) voltages with self-bias. At high enough pressure collisional electron transport becomes dominant and the spokes disappear 179 .

An important issue that has been widely discussed in the literature on magnetron discharges [START_REF] Anders | Tutorial: Reactive high power impulse magnetron sputtering (R-HiPIMS)[END_REF]113,176,180,181 relates to the electron heating mechanisms leading to the formation of these luminous regions, one conclusion being that electrons are not heated in the cathode sheath as in non-magnetized glow discharges but rather in the presheath, by ohmic heating. We will show in section IV.B that particle simulations provide a new insight in the electron heating mechanisms in the spokes of magnetron discharges.

Space and time resolved Langmuir probe measurements of the plasma and floating potentials in dc magnetron discharges by Panjan and Anders 178 have revealed the existence of strong electric fields parallel and perpendicular to the cathode surface, forming a double layer structure at the leading edge of the ionization zone. Figure 22 shows an example of measured plasma potential distribution in a plane parallel to the cathode and 7.5 mm above it. A sharp drop of the potential of several tens of volts can be seen at the spoke front. The magnetron discharges in Figure 21 and Figure 22 operate in a low power dc regime and the spokes rotate in the -𝑬 × 𝑩 direction.

These measurements suggest that the double layer at the leading edge of the spoke plays a crucial role in the energization of electrons in the spoke. Held al. 119 measured the plasma potential, electron density, electron temperature, and electron energy probability function (EEPF) with Langmuir probes, as a function of time at a given location above the cathode, on the spoke path. The sharp increase of the plasma potential when the spoke front reaches the probe, seen in the measurements of held et al. Figure 23) is consistent with the 2D potential distribution of Figure 22 and is associated with a very fast increase and decay of the electron temperature. The electron density present large amplitude oscillations as a function of time in the spoke region. Held al. 119 also measured the electron energy distribution at different times and observed a fast increase of the tail of this energy distribution at the spoke front, followed by a quick decay. We will see in section IV.B that the particle simulations of magnetron plasma are qualitatively consistent with these measurements.

IV. SIMULATIONS, THEORY, AND EXPERIMENTS ON INSTABILITIES IN 𝑬 × 𝑩 PLASMAS

In this section we present some examples of electrostatic Particle-In-Cell Monte Carlo Collisions simulations of instabilities in Hall thrusters (section IV.A) and magnetron discharges (section IV.B). In PIC-MCC simulations, the trajectories of a large representative number of particles (super-particles) are followed in phase space under the influence of the electric forces due to the self-consistent electric field and external magnetic field, and of collisions. The electric field is calculated at each time step on a grid from the charged particle densities on this grid, deduced from the positions of the super-particles The electric forces are then calculated on the grid and the equations of motion of the super-particles are integrated over time. A Monte Carlo module is used to decide which particles are undergoing collisions during the time step, the nature of the collisions, the change in velocity, etc… We will not discuss here in detail the principles of PIC-MCC and simply refer to standard text books 182,183 and review papers [184][185][186][187] . Three-dimensional particle models are still very computationally intensive, especially if one wants to resolve the large plasma frequencies, small Debye lengths, and long time to reach steady-state of Hall thrusters or magnetron discharges. Most of the particle simulations of the 𝑬 × 𝑩 plasmas of Hall thrusters and magnetron discharges have been performed with 1D3V or 2D3V models, i.e. one or two dimensions in space, three dimensions in velocity, and we will limit our discussion of this section to these models (we note also the recent attempts at reducing the computational cost of 2D PIC simulations of Hall thrusters by introducing the concept of pseudo-2D PIC scheme 188,189 ). As said above, the dominant instabilities develop in the azimuthal direction because of the large difference between electron and ion drift velocities in this direction. Therefore, the 1D and 2D models must include the azimuthal direction. The 1D models are purely azimuthal, and the 2D models can be axial-azimuthal or radialazimuthal.

𝑬 × 𝑩 plasma devices are complex because of the different non-linear and coupled physical mechanisms involved (instabilities, ionization, neutral depletion, interaction of charged particles with neutral gas and sputtered atoms in the case of magnetron discharges, …). Instead of trying to deal with the full complexity of the problem it is often useful to simplify or reduce the problem and to decouple some of these mechanisms. In this section we give some examples of reduced problems aimed at isolating and studying some specific aspects of 𝑬 × 𝑩 plasma devices.

The first examples (section IV.A) concern the instabilities that develop in the ionization and acceleration regions of a Hall thruster, i.e. the right side of Figure 18, where the magnetic field increases from cathode to anode. Then, in section IV.B, we consider a simplified model of a magnetron discharge (or the inside of the channel of a Hall thruster) where the magnetic field decreases from cathode to anode (left side of Figure 18).

A. « Micro-instabilities » : Electron cyclotron drift instability (ECDI) and Modified Two-Stream Instability (MTSI) in Hall thrusters

The existence of the Electron Cyclotron Drift Instability (ECDI) in the particle simulations of Hall thrusters has been demonstrated for the first time by Adam et al. 190 in 2004 with a 2D PIC-MCC axial-azimuthal model including a fluid description of the neutral atom transport, ionization, and neutral atom depletion. The model was able to describe the azimuthal ECDI and the breathing mode mentioned above. The ECDI was characterized, in these simulations, by the presence of large amplitude, small wavelengths azimuthal oscillations of the azimuthal electric field and plasma density in the acceleration region of a Hall thruster. The EDCI was then studied in more details by Ducrocq et al. 191,192 with dispersion relations and a simpler 1D PIC-MCC simulation. An analysis and a parameter study of the 3D dispersion relation of the ECDI in Hall thrusters can be found in the paper by Cavalier et al. 193 . The dispersion relation for the ECDI had been derived and studied earlier [194][195][196][197] , in the 1970's, in other contexts such as collisionless shocks in the magnetosphere.

The dispersion relation for the ECDI is described in subsection 1. In subsection 2, we discuss the results of 1D PIC and Vlasov simulations of the ECDI under conditions typical of Hall thrusters. Results from 2D PIC simulations of the ECDI in the axial-azimuthal plane are reported in subsection 3. 2D radial-azimuthal PIC simulations showing the combination of the Electron Cyclotron Drift Instability and Modified Two-Stream Instability, ECDI and MTSI, are described in subsection 4. We conclude this section by some remarks on experimental evidence of the presence of the ECDI in Hall thrusters in subsection 5.

Dispersion relation of the ECDI

The dispersion relation of the ECDI is a kinetic dispersion relation obtained from the linearization of the Vlasov and Poisson equations. Here we will only recall the assumptions leading to this dispersion relation, and its general form. We use the notations of Figure 1 for the configuration of electric and magnetic field. The wave is supposed to be electrostatic. Ions are assumed to be unmagnetized and cold, with an axial velocity (beam velocity) 𝑣 𝑏𝑖 . The electron dynamics is described using the Vlasov equation and assuming a Maxwellian velocity distribution at temperature 𝑇 e . The electrons drift azimuthally (-𝑦 direction) in a static axial electric field 𝑬 0 (𝑥 direction) and perpendicular magnetic field 𝑩 (𝑧 direction) at a velocity 𝑣 0 = -𝐸 0 𝐵 ⁄ . The drifts due to plasma density and magnetic field gradients are neglected (the 𝑬 × 𝑩 drift velocity 𝑣 0 is much larger than the azimuthal drifts due to magnetic field gradient and density gradient in the conditions of Hall thrusters). The electron thermal velocity is noted 𝑣 𝑇𝑒 . In these conditions, the 3D dispersion relation, obtained by linearization of the electron Vlasov equation, ion fluid equations, and Poisson's equation writes 193,195,[198][199][200] :

1 - 𝜔 𝑝𝑖 2 (𝜔 -𝑘 𝑥 𝑣 𝑏𝑖 ) 2 + 1 𝑘 2 𝜆 𝐷𝑒 2 × (37) [1 + Ω 𝑘 𝑧 𝑣 𝑇𝑒 √2 ∑ 𝑍 ( Ω + 𝑚𝜔 𝑐𝑒 𝑘 𝑧 𝑣 𝑇𝑒 √2 ) exp(-𝑏)𝐼 𝑚 (𝑏) +∞ 𝑚=-∞ ] = 0 with Ω = 𝜔 -𝑘 𝑦 𝑣 0 , 𝑏 = (𝑘 𝑥 2 + 𝑘 𝑦 2 )𝜌 𝑒 2 , 𝜔 𝑝𝑖 = ( 𝑒 2 𝑛 𝜀 0 𝑀 ) 1 2
𝑣 𝑇𝑒 = ( 𝑒𝑇 𝑒 𝑚 ) Some examples of solutions of this dispersion relation are shown in Figure 24 as a function of the wave number in the azimuthal direction 𝑘 𝑦 , for three values of the wave number in the magnetic field direction, 𝑘 𝑧 , and for a zero wave number 𝑘 𝑥 in the direction of the applied electric field. We see on Figure 24 that the angular frequency of the instability at maximum growth rate is slightly below the ion plasma frequency (about 10 8 rad/s for xenon and a plasma density of 10 18 m -3 , i.e. a wave frequency in the (5-10 MHz range). For a wave vector strictly orthogonal to the magnetic field (which was the case of the 2D PIC-MCC simulation of Adam et al. 190 ), i.e. for 𝑘 𝑧 = 0, instabilities are present only in small intervals in 𝑘 𝑦 around each multiple of the inverse Larmor radius of an electron with velocity 𝑣 0 , i.e. |𝑘 𝑦,𝑚 | = 𝑚 𝜔 𝑐𝑒 𝑣 0 ⁄ , with 𝑚 integer.

The dispersion relation for 𝑘 𝑧 = 𝑘 𝑥 = 0 can be written as 198 :

𝑘 𝑦 2 𝜆 𝐷𝑒 2 [1 - 𝜔 𝑝𝑖 2 𝜔 2 ] + 1 -exp(-𝑏)𝐼 0 (𝑏) (38)
-2(𝜔 -𝑘 𝑦 𝑣 0 ) 2 ∑ exp(-𝑏)𝐼 𝑚 (𝑏)

(𝜔 -𝑘 𝑦 𝑣 0 ) 2 -𝑚 2 𝜔 𝑐𝑒 2 +∞ 𝑚=1 = 0
For 𝑘 𝑧 = 0, eq. (38) shows that the instability occurs as a reactive mode due to the overlap of the Bernstein modes Doppler shifted by the 𝑬 × 𝑩 flow with the ion sound mode. In this case, the dispersion relation is fully real and the instability appears in complex conjugate pairs with the growth rates narrowly peaked around the cyclotron resonances at -𝑘 𝑦 = 𝑚 𝜔 𝑐𝑒 𝑣 0 ⁄ (with 𝑚 = 1, 2, 3, …).

For finite but small 𝑘 𝑧 the instability still appears as a sequence of narrowly spaced resonant peaks, the cyclotron resonances, as can be seen in Figure 24 (blue curves). With increasing 𝑘 𝑧 , the instability gradually changes to the dissipative mode driven by inverse Landau damping on the electrons. For larger values of 𝑘 𝑧 (red curves of Figure 24), the resonant peaks overlap and the instability eventually switches into the ion sound unmagnetized plasma regime (black curve of Figure 24) but driven by the electron 𝑬 × 𝑩 flow in the 𝑦 direction. The transition to the ion sound regime is also possible with 𝑘 𝑧 = 0 due to collisions and/or nonlinear resonance broadening. It can be shown that the angular frequency and growth rate of this "modified ion acoustic mode" can be written as 193,[203][204][205] :

{ 𝜔 𝑅 = 𝑘 𝑥 𝑣 𝑏𝑖 + 𝑘𝑐 𝑠 (1 + 𝑘 2 𝜆 𝐷𝑒 2 ) 1/2 𝛾 = √ 𝜋𝑚 8𝑀 𝑘𝑣 0 (1 + 𝑘 2 𝜆 𝐷𝑒 2 ) 3/2
(39)

1D PIC simulation of the ECDI

The ECDI can be studied with reduced models containing the azimuthal direction, i.e. one-dimensional azimuthal models (this subsection), and two-dimensional axialazimuthal or radial-azimuthal models (next subsections). In these PIC-MCC simulations, periodic boundary conditions are assumed in the azimuthal direction and the perpendicular magnetic field is given and is constant.

In 1D models, the electric field 𝐸 0 (perpendicular to the azimuthal direction of the simulation and to the magnetic field) is given. The simulation is collisionless and starts with given Maxwellian velocity distributions of electrons and ions. As the instability forms, anomalous transport across the magnetic field occurs and electrons can gain energy from the axial electric field. In order to limit the electron mean energy, a finite dimension can be assumed in the direction of the electric field, i.e. each electrons that has travelled a given length along the electric field (the acceleration region) is removed from the simulation and replaced by an electron with the initial velocity distribution. Figure 25 shows an example of result from a 1D Particle-In-Cell simulation of the ECDI, from Ref. 199 . We see on this figure that the instability is dominated by the 𝑚 = 1 resonance with wavelength corresponding to the condition -𝑘 𝑦 𝑣 0 = 𝜔 𝑐𝑒 and appears as a quasi-coherent mode. Although higher 𝑚 modes can have larger growth rates (see Figure 24) and may start to grow first, the increase of electron temperature and nonlinear inverse cascade make the 𝑚 = 1 mode dominant in the nonlinear stage. We also observe in Figure 25 the generation of large-scale structures, at the length scale of the simulation box. These long wavelength modes were also seen in Ref. 200 . A possible explanation of the presence of large-scale structures is that while the energy input via the resonant ECDI occurs at small scales, nonlinear inverse cascade mechanisms may result in energy condensation in large scale structures, as shown in the analytical results of Ref. 206 .

In some earlier 1D PIC studies it was shown that at some finite amplitude, the cyclotron resonances are washed out by nonlinear effects and the instability occurs in the ion sound regime. This is not the case in the simulation results of Figure 25 where the numerical noise has been reduced by using more than 1000 particles per cell (10000 particles per cell were used in the 1D simulation of Janhunen et al., with similar results). Similar conclusions were drawn in Ref. 207 . It is important to note that numerical collisions due to the inherent numerical noise in particle simulations may also lead to resonance overlap and the transition to the ions sound regime. Recent simulations with a noise-free Vlasov code 208 have shown that in the 1D case, the nonlinear broadening does not produce the transition to the ion sound and the cyclotron resonances remain most active so that the fluctuations energy is concentrated in the lowest m=1 resonance and in the lower wavelength (due to the inverse cascade condensation). This is illustrated in Figure 26, which shows the frequency spectrum of the azimuthal electric field in the nonlinear regime obtained with a 1D Vlasov simulation of the ECDI (comparisons between Vlasov and PIC simulations of the ECDI are discussed in Ref. 209 ).

The cyclotron harmonics are clearly pronounced in Figure 26 . Note the backward wave (with positive value of the wavevector) propagating in the direction opposite to the 𝑬 × 𝑩 drift. These waves are usually a result of the strong modification of the electron distribution function due to trapping in high-amplitude potential wells of the electric field (see, eg. Refs. [210][211][212][213] ). A similar feature of the backward waves has also been observed experimentally in the Hall thruster plasma (see Fig.

2D axial-azimuthal PIC simulation of the ECDI

More realistic, but still reduced models of the ECDI in the context of Hall thrusters, based on 2D axial-azimuthal PIC-MCC simulations have also been developed. We present in this subsection some PIC simulations results corresponding to a reduced model defined in Refs. 167,215 and which was used in the benchmark simulations of Ref. 216 . In these models the aim is to study the ECDI independently of the question of self-consistent ionization and complications related to the ionization instability mentioned above (the breathing mode). Only the ionization region, acceleration region, and a small part of the near plume are simulated in the model. The gas flow is supposed to be fully ionized so the model is purely collisionless. Compared with the 1D model, this 2D reduced model allows to describe self-consistently the formation of the electric field in the acceleration region as well as the effect of the azimuthal instability on cross-field electron transport. The principles of the 2D model are shown in Figure 27. The axial length is 2.5 cm and contains a small part of the channel (generally situated on the left side of the maximum magnetic field 𝐵), the acceleration region, which is supposed to develop around the maximum magnetic field, and part of the near plume. The profiles of the radial magnetic field 𝐵(𝑥) and of the given ionization rate 𝑆(𝑥) (cosine profile) are shown in Figure 27. The simulation domain is periodic in the azimuthal direction, with a length of 1.28 cm. In the results shown in Figure 28, the maximum magnetic field is 10 mT, the integral of the ionization rate over the axial length corresponds to a current density of 400 A/m 2 (the current density in standard Hall thrusters [START_REF] Boeuf | Tutorial: Physics and modeling of Hall thrusters[END_REF] is on the order of 1000 A/m 2 ). To represent the effect of the electron emissive cathode, a flux of electrons is injected on the right side of the simulation domain as described in the caption to Figure 27 and in Refs. 167,216 .

Figure 28 shows the axial-azimuthal distribution of the azimuthal electric field and of the ion density obtained with the PIC simulation in the steady-state regime (at a simulation time of 20 s). An azimuthal wave moving in the 𝑬 × 𝑩 direction is observed in the simulations. Very large amplitude oscillations of the azimuthal electric field, on the order of 400 V/cm, can be seen on Figure 28, with wavelengths below 1 mm and frequency in the MHz range. One can actually distinguish in Figure 28, two regions with different wavelengths on the left (smaller wavelength) and on the right side (larger wavelength) of the maximum magnetic field at 𝑥 =0.75 cm.

A Fast Fourier transform of the azimuthal electric field of Figure 28 at two different locations is shown in Figure 29. The dispersion relation of the modified ion acoustic mode is also shown on these plots. We see that at the location 𝑥 = 0.3 cm, i.e. on the left of the maximum magnetic field, the angular frequency from the PIC simulation follows the angular frequency of the modified ion acoustic instability, eq.( 39), with a dominant mode corresponding to the mode with maximum growth rate (the maximum of the growth rate of eq. ( 39) is obtained for 𝑘 𝑥 𝜆 𝐷𝑒 = 2 -1/2 ). Similar results were obtained in Ref. 215 . Although it is known that nonlinear effects can potentially lead to the transition to an ion acoustic instability, the exact reason for this transition in the 2D simulations, in contrast to the 1D case, and in spite of the fact that 𝑘 𝑧 is necessarily zero in this axial-azimuthal model, is unclear and need further study. The spectrum of the modified ion acoustic mode is less visible at the location 𝑥 = 1.5 cm of Figure 29, where the dispersion relation from the PIC simulation exhibits a more discrete structure. 39)). The green dashed lines indicate the wavelength and frequency of the dominant mode. Adapted with permission from Charoy et al., Plasma Sources Sci. Technol. 28, 105010 (2019).

Copyright 2023 IOP Publishing Ltd. (Fig. 8 of Ref. 216 ).

The space distribution of the ion density in the axialazimuthal plane of Figure 28 presents a maximum in the ionization region on the left of the maximum magnetic field and decreases, due to ion acceleration by the axial electric field, on the right of the maximum field, as expected in a Hall thruster. The axial profiles of the azimuthally and time averaged axial electric field and ion density are shown in Figure 30. Results from different PIC simulations performed by different groups 216 are represented on this figure. Although the principles of PIC simulations are relatively simple, the different numerical constraints (time step, grid spacing, number of particle per cell…) control the accuracy and numerical noise, and it is important to perform convergence tests and benchmarks (see Ref. 217 for a list of the past and current benchmarks simulations organized by the low temperature plasma community to study instabilities in 𝑬 × 𝑩 partially magnetized plasmas). The aim of the benchmark of the 2D axial-azimuthal ECDI reported in Ref. 216 was to increase the confidence in the simulation results and their physical interpretation. Good agreement between the different codes (Figure 30) was obtained when 250 particles per cell were used in the simulations.

The axial profile of the electric field in Figure 30a shows a maximum of about 500 V/cm with a shift of a few mm to the right of the maximum magnetic field. The width of the acceleration region is less than 1 cm. Ions generated in the ionization region are accelerated by the axial electric field, resulting in the decay of the ion density from 4 × 10 17 m -3 in the ionization region to 1.5 × 10 17 m -3 at the end of the acceleration region (Figure 30b).

The ECDI instability described in this section is responsible for the collisionless electron transport across the magnetic field. An effective Hall parameter or collision frequency due to the instability can be deduced from the simulations. The effective Hall parameter (see section II.A) deduced from the simulation at the location of the maximum magnetic field, was ℎ 𝑒𝑓𝑓 ≈192 for a total current density of 400 A/m 2 in the simulations of Ref. 167 , under conditions close to those reported in this subsection, and was found to decrease with decreasing total current density. This corresponds to an effective collision frequency 𝜈 𝑒𝑓𝑓 ≈ 8.3 × 10 6 s -1 . These values of ℎ 𝑒𝑓𝑓 and 𝜈 𝑒𝑓𝑓 seem consistent with the experimental measurements. For example, the inverse Hall parameter ℎ 𝑒𝑓𝑓 -1 deduced from measurements in Figure 17 is on the order of 10 -2 . Also, the ratio of electron to ion current in the exhaust region that can deduced from the PIC simulations is on the same order as in the experiments. However, the amplitude of the azimuthal electric field oscillations predicted by the PIC simulations are much larger than those measured experimentally, and the question arises as to whether the mechanisms leading to anomalous transport are well described by this reduced PIC model. This issue is briefly discussed in subsection IV.A.5.

2D radial-azimuthal PIC simulation: Electron Cyclotron Drift Instability (ECDI) and Modified Two-Stream Instability (MTSI)

In order to study the role of electron transport parallel to the magnetic field, and of the presence of dielectric walls and sheaths in this direction, 2D PIC simulations in the radialazimuthal plane have been performed. In these PIC models as well as in the 1D models of section IV.A.2, the axial electric field cannot be calculated self-consistently and is imposed. Different ways of describing the axial transport (e.g. finite length of the acceleration region or not, re-injection of particles, ionization) and the electron interactions with the dielectric walls perpendicular to the radial direction (i.e. to the magnetic field) have been used by different authors [218][219][220][221][222][223] , leading to various conclusions. Heron et al. 224 conclude that the balance due to the ECDI and the SEE (Secondary Electron Emission at the walls) is extremely complex and depends on the emissivity of the wall as well as on the thruster geometry. Beam-plasma instabilities can even be present when the interaction of the SEE with the instability is too violent. Croes et al. 220 found that the wall effects play only an auxiliary role and suggest that the fact that there is experimental evidence that the nature of the walls has a strong influence on the characteristics of a thruster, is due to the modification of the electron distribution function by the SEE. Tavant et al. 221 showed that anomalous electron cross-field transport decreases in the presence of SEE because of the electron cooling associated with secondary electron emission. This was also a result of the simulations of Taccogna et al. 222 . From their 2D radial-azimuthal simulations without SEE in the direction of the applied electric field), Janhunen et al. 219 show that the ECDI proceeds as a coherent mode driven by the main cyclotron resonance (𝑘 𝑦 𝑣 0 = 𝜔 𝑐𝑒 ) and does not evolve in the ion sound instability well into the nonlinear stage. They also found a strong energy cascade towards the longer wavelength and that the electron current was dominated by the long wavelength modes. They showed the development of a Modified Two-Stream Instability (MTSI) resulting in rapid heating parallel to the magnetic field due to the finite electric field along the magnetic field. The MTSI was shown to amplify the inverse energy cascade due to its long wavelength nature. Similar conclusions were drawn by Taccogna et al. 222 (where SEE was included). We illustrate below the radial-azimuthal models with the work of Villafana et al. 223 which reports benchmark PIC simulations without SEE, in a radial-azimuthal (𝑧, 𝑦) Cartesian domain. Before doing that, we recall the definition and properties of the MTSI.

The MTSI can be obtained by considering the asymptotic case of cold electrons (𝑇 𝑒 → 0) in the ECDI dispersion relation (37) with non-zero wavenumber parallel to the magnetic field, 𝑘 𝑧 ≠ 0, and assuming 𝑘 𝑥 = 0. This implies that 𝑏 → 0 and 𝜉 → ∞, where 𝜉 is the argument of the 𝑍 function in eq. (37). Lashmore-Davies 225 derived the cold electrons asymptotic form of the ECDI by keeping only the 𝑚 = 0 term in the sum of eq. ( 37), and obtained the following form of the MTSI:

1 - 𝜔 𝑝𝑖 2 𝜔 2 - 𝜔 𝑝𝑒 2 𝑘 𝑧 2 (𝜔 -𝑘 𝑦 𝑣 0 ) 2 𝑘 2 + 𝜔 𝑝𝑒 2 𝑘 𝑦 2 𝜔 𝑐𝑒 2 𝑘 2 = 0 (40)
A more accurate asymptotic limit of eq.( 37) can be obtained by keeping the 𝑚 = ±1 terms in the sum of eq. (37). The limits of the Bessel functions when 𝑏 → 0 are 𝐼 0 (𝑏) → The exponential term gives (1 -𝑏) for small 𝑏 and the asymptotic form of the plasma function 𝑍(𝜉) for large values of 𝜉 (since 𝑣 𝑇𝑒 → 0) is:

𝑍(𝜉) → - 1 𝜉 (1 + 1 2𝜉 2 )
Therefore, the expression of the dispersion relation (37) in the cold electrons approximation takes the following asymptotic form (to the first order in 𝑇 𝑒 ) : We illustrate below the development of ECDI and MTSI in with the radial-azimuthal simulations of Villafana et al. 223 . The principles and conditions of the simulations in the model of Villafana et al. 223 are as follows (see Figure 31). The radial magnetic field is given and uniform. The electric field in the radial-azimuthal plane is obtained by solving Poisson's equation at each time step in this plane, and the electric field in the axial direction is imposed. Electron and ion trajectories are followed not only in the (𝑦, 𝑧) plane but also in the axial direction. The simulation domain is bounded by two grounded walls in the radial direction, and periodic boundary conditions are used in the azimuthal directions. Electrons and ions reaching the walls are removed from the simulation. A finite length 2𝐿 𝑥 is considered in the axial, 𝑥 direction, and each charged particle exiting the [-𝐿 𝑥 , 𝐿 𝑥 ] interval is replaced by a new electron at 𝑥 = 0 and at the same (𝑦, 𝑧) location, according to a Maxwellian distribution at fixed temperatures, 𝑇 𝑒,0 and 𝑇 𝑖,0 for electrons and ions respectively. The value of 𝐿 𝑥 , together with the given axial electric field 𝐸 𝑥 , imposes the maximum energy gain (𝐿 𝑥 𝐸 𝑥 ) from this applied electric field. To balance the charged particle losses to the walls, a fixed ionization rate, with a cosine profile in an interval [𝑧 1 , 𝑧 2 ] in the radial direction is imposed. The integral of the given ionization rate over the radial direction gives the total discharge current density (100 A/m 2 ). These conditions are summarized in Figure 31 (see Ref. 223 for more details). The results of the radial-azimuthal PIC simulations in these conditions show the clear development of the ECDI with a dominant 𝑚 = 1 mode (𝑘 𝑦 = 𝜔 𝑐𝑒 𝑣 0 ⁄ ) and a less visible 𝑚 = 2 mode, as can be seen in the FFT plot of Figure 32. No evolution towards an ion sound instability was found in these simulations. The MTSI at low wavenumber is also clearly apparent on Figure 32, and its amplitude seems to decay in time, so the inverse cascade is not observed in these conditions. The relatively small length of the simulation domain in the azimuthal direction could limit the development of long wavelength modes in these simulations (this is confirmed by the simulations of Refs. 219,226 ).

1 - 𝜔 𝑝𝑖 2 𝜔 2 + 1 𝑘 2 𝜆 𝐷𝑒 2 [𝑏 -

Experimental evidence of the ECDI, and conclusion

We can conclude from the previous subsections that Electron Cyclotron Instabilities are ubiquitous in particle simulations of 𝑬 × 𝑩 plasmas under Hall thruster conditions and that they result in large amplitude oscillations of the azimuthal electric field with wavelengths around 1 mm and frequency in the 5-10 MHz range. Soon after the first evidence of the presence of ECDI in PIC simulations by Adam et al. 190 , experiments based on collective light scattering were designed to detect these instabilities in Hall thrusters. Electron density fluctuations were measured by coherent Thomson scattering by Tsikata et al. 214,[227][228][229] in Hall thrusters, and were found to be consistent with the presence of the ECDI microinstabilities predicted by the PIC simulations and with their transition into an ion acoustic instability. The measurements showed the existence of a finite wave component along the magnetic field, which could explain the transition to the ion sound behavior 214 . However, the amplitudes of the fluctuations measured by collective light scattering were much smaller than those observed in the simulations and displayed in Figure 28. Possible reasons for this discrepancy are 3D effects, coupling of wall electronemission to the ECDI, limits in the azimuthal dimension of the 2D models which prevent inverse energy cascades, etc… The results of laser experiments show the presence of ECDI, but it is uncertain if the PIC simulations correctly identify ECDI or its modified ion acoustic version as the main source of anomalous transport in the acceleration and near-plume region of a Hall thruster. A discussion of the experimental measurements of the microinstabilities in Hall thrusters and of their interpretation, and of the need for joint experimental and modelling work to clarify the relationship between different instabilities and anomalous transport can be found in the section by Jorns and Tsikata of Kaganovich et al. [START_REF] Kaganovich | Physics of E × B discharges relevant to plasma propulsion and similar technologies[END_REF] .

B. « Macro-instability »: Rotating spokes in magnetron discharges

Two-dimensional PIC-MCC simulations were conducted to examine the physics of spokes in magnetron discharges. The following are the key findings from these simulations.

As in section IV.A, the simulations are done in a rectangular axial-azimuthal domain and periodic boundary conditions are used in the azimuthal direction to represent the closed-drift. The simulation domain is shown in Figure 33a. The cathode-anode gap is 𝑑 𝑥 =2 cm and the length of the domain in the azimuthal direction is 𝑑 𝑦 =8 cm. Simulations results will be presented below for two profiles of the axial magnetic field, 𝐵 1 (𝑥) and 𝐵 2 (𝑥) , shown in Figure 33b. The magnetic field profile 𝐵 1 (𝑥) is of the form:

With 𝜎 = 0.35𝑑 𝑥 and 𝑎 and 𝑏 chosen so that 𝐵 1 (0) = 0.1 T, 𝐵 1 (𝑑 𝑥 ) = 0, The magnetic field 𝐵 2 (𝑥) is:

𝐵 2 (𝑥) = 𝐵 2 (0)𝑒 -𝑥 𝛼 with 𝐵 2 (0) = 0.12 T, 𝛼 = 0.3𝑑 𝑥 (43)
We first present some simulation results for a pressure of 2 Pa, with the magnetic field profile 𝐵 1 of Figure 33b. The conditions (argon, 2 Pa, 260 V dc voltage, secondary emission 25 mA/m 2 ) are similar to those of Refs. 126,127 except that the azimuthal length of the simulation is 8 cm instead of 4 cm and the magnetic field profile is slightly different (𝐵 1 (𝑑) = 0.01 T in Refs. 126,127 , instead of 𝐵 1 (𝑑) = 0). These conditions are similar to those of the experimental work of Ito et al. 116 , with a scaling factor of 10 (pressure and magnetic field divided by 10, dimensions multiplied by 10).

The PIC-MCC simulations in these conditions predict the formation of two rotating spokes along the azimuthal direction. Figure 34 displays the 2D distribution of the electric potential, ion density, ionization rate and electron current density at a given time after the spoke regime is established. The spokes are moving in the -𝑬 × 𝑩 direction at a velocity of about 10 km/s. The spoke is characterized by a region of intense ionization, which is typically observed as a luminous region in experiments. It is located at the edge and on top of a region with higher plasma density, and is oriented in the -𝑬 × 𝑩 direction. By extension, we will refer to the combination of the region of intense ionization and the adjacent region of high plasma density as the "spoke". The region of high plasma density is characterized by a low electric field. In contrast, the region of intense ionization is located in an area where the electric potential drops abruptly. The deformation of the potential forms a long wavelength (~ 4 cm) azimuthal wave that moves upward. The region of high plasma density moves in the direction of the region of intense ionization, and therefore the spoke can be considered as an ionization wave under these conditions. The electric field is perpendicular to the equipotential lines so the 𝑬 × 𝑩 electron current follows the equipotential contours. This can be seen in the 𝑗 𝑒 plot of Figure 34. The maximum electron current density roughly follows the equipotential lines at the boundary between the low electric field region (red region on the potential plot of Figure 29) and the region (green-yellow regions on the potential plot) where a large electric field is present in the quasineutral plasma. This boundary is characterized by a significant potential drop and can be called a double layer. The deformation of the equipotential contours by the spoke acts as a short-circuit between the cathode and anode regions, enabling cross-field electron transport. The axial profiles of the charged particles densities and axial electric field at two different azimuthal locations noted 𝑦 1 and 𝑦 2 in Figure 34 (dashed lines on the potential plot) are shown in Figure 35. The large cathode sheath electric field is apparent on this figure but one can also see that the electric field penetrates the quasineutral plasma with a modulation of the penetration depth that depends on the azimuthal location and is associated with the electric potential wave. The axial electric field and plasma density gradient are in the same direction, which suggests that the plasma non-uniformity is triggered by gradient-drift or Simon-Hoh instability.

The spoke instability is highly coherent in both the simulations and in many experiments on magnetron discharges. The position of the ionization region is welldefined on the side of the potential wave where the axial component of the 𝑬 × 𝑩 drift is directed towards the cathode. Electrons are clearly heated in the spoke region and the question arises as to the mechanisms leading to this electron heating. Collisional heating is possible along the line of intense electron current density and large potential drop seen on the 𝑗 𝑒 plot of Figure 34, but it is unclear why electron heating would only occur on one side of the potential wave. A detailed analysis shows that another mechanism is responsible for a large part of this electron heating. This is illustrated in Figure 36, which shows an electron trajectory in the double layer region. We said above that electrons follow the equipotential lines in their 𝑬 × 𝑩 drift between collisions and therefore should not gain energy on the average, along their path. Figure 36b shows a magnified view of part of the electron trajectory of Figure 36a. We see that the electron slightly deviates from the equipotential contour, so its mean energy (red line of Figure 36c) does not stay constant along its trajectory. The electron gains energy on the upper part of the potential wave (leading edge of the spoke, or spoke front) and loses energy on the other side (trailing edge of the spoke). These successive energy gain and losses are related to the 𝛁𝑩 drift. This drift is due to the gradient of magnetic field and is in the -𝑩 × 𝛁𝑩 direction (it is included in the 𝑣 𝐷 and 𝜔 𝐷 terms of the fluid dispersion relation, see eq.( 22)). It can be written as 230 : 44) where 𝑣 ⊥ is the electron velocity and 𝜀 ⊥ the electron kinetic energy in eV, perpendicular to the magnetic field. In the conditions of the simulations considered here 𝒗 𝛁𝐵 is directed in the azimuthal direction, in the positive 𝑦 direction, i.e downward. The electron heating or cooling per electron per unit time due to the 𝛁𝐵 drift is given (in eV/s) by:

𝒗 𝛁𝐵 = -1 2 ⁄ 𝜌 𝑒 𝑣 ⊥ 𝑩 × 𝛁𝐵/𝐵 2 = -𝜀 ⊥ 𝑩 × 𝛁𝐵/𝐵 3 (
𝜃 𝒗 𝛁𝐵 = -𝒗 𝛁𝐵 . 𝑬 = 𝜀 ⊥ 𝐸 𝑦 (𝐵𝐿 𝐵 ) ⁄ (45)
As illustrated in Figure 36a, the scalar product 𝒗 𝛁𝐵 . 𝑬 is negative on the top side of the potential wave and positive on the bottom side so that 𝛁𝐵 drift heating takes place at the leading edge of the spoke while cooling occurs at the trailing edge. Electron heating and cooling would cancel each-other in the absence of collisions. The net heating due to 𝛁𝐵 drift in our conditions is due to collisions: electrons can lose energy through inelastic collisions before 𝛁𝐵 drift cooling becomes important. Simulations under conditions close to those of the experiments and Langmuir probe measurements of Held et al. 119 (see Figure 23) have been performed and are described in detail in Ref. 130 . The simulation domain is the same as above (Figure 33a), the argon pressure is 0.6 Pa, and the magnetic field distribution corresponds to the 𝐵 2 curve of Figure 33b. In these conditions only one spoke is observed in the simulations (as in the experiments, where the closed drift of the magnetron is also on the order of 8 cm). The spoke rotates, as in the previous example, in the -𝑬 × 𝑩 direction but at a smaller velocity, on the order of 1.3 km/s (2.5 km/s in the experiment of Held et al. 119 ). The 1D stable, steady state solution of this problem was used as initial condition for the 2D simulation. This means that the initial axial positions and velocities of the charged particles in the 2D simulations are identical to those of the 1D steady state solution. This allows the study of the linear development of the instability around the unperturbed, stable, steady state 1D solutions. The plasma density gradient and electric field of the 1D solution were in the same direction, indicating that the condition for a gradient-drift or Simon-Hoh instability was satisfied. An interesting aspect of the 2D solutions was that a gradient-drift instability forms in a few hundred nanoseconds, leading to the rotation of plasma non-uniformities in the +𝑬 × 𝑩 direction.

After about 1 s, ionization starts to play an important role and the instability evolves into an ionization wave, rotating in the -𝑬 × 𝑩 direction (details are shown in Ref. 130 ). Figure 37 displays color maps of the ion density, electron temperature, and ionization rate at a given time, when the instability has reached a steady state, saturated regime. Equipotential contours are also shown, superimposed to the ion density color map. A plot of the electron heating rates due to 𝛁𝐵 drift, 𝜃 𝒗 𝛁𝐵 , and to collisions, 𝜃 𝑬 , is also represented in Figure 37, as a function of azimuthal position and at the axial position x =1 cm. The collisional heating rate, 𝜃 𝑬 , can be written:

𝜃 𝑬 = -𝒗 ∥𝐄 . 𝑬 = 𝝂 𝛚 𝒄𝒆 𝑬 𝟐 𝑩 ( 46) where 𝒗 ∥𝐄 is the collisional electron drift velocity parallel to the electric field. We see on Figure 37 the sharp increase of the electron temperature at the leading edge of the spoke. The maximum ionization rate is shifted downward along the double layer, with respect to the electron temperature (the mean free path for ionization is a few cm). The plots of the 𝛁𝐵 drift and collisional heating rates clearly indicate that the 𝛁𝐵 drift heating is largely dominant in the double layer. The time variation of the plasma parameters, electron density, electron temperature, and plasma potential at a location 6 mm above the cathode surface, i.e. on the spoke path, are shown in Figure 38 and can be compared to the Langmuir probe measurements of Held et al. 119 (reproduced in Figure 23). These parameters exhibit large amplitude oscillations with very sharp variations of the electron temperature when the leading edge of the spoke reach the point where they are recorded in the simulation (𝑥 = 6 mm). There are some clear discrepancies between the experiments and the simulation results, but the qualitative agreement is reasonably good. The plasma density is lower in the simulation, by a factor of about 10. This is because the electron emission current at the cathode has been limited in order to reduce the simulation time (which increases with increasing plasma density). The large increase of the electron temperature at the leading edge of the spoke is a common feature of experiments and simulations but the rise and decay of the electron temperature have different time constants. The period of variations of the plasma parameters is longer in the simulations (60 s, instead of about 32 s in the experiment, see Figure 23). This is consistent with the higher spoke velocity in the experiment (2.5 km/s, vs 1.3 km/s in the simulations).

Held et al. have also measured the electron energy probability function at different times near the spoke front and found a strong modulation of the tail of the distribution. This is also observed in the simulations as shown in Figure 39. The population of high energy electrons increases abruptly in the double layer region and decays in the high plasma density and low field region of the spoke. This is consistent with the heating of electrons flowing along the double layer in the spoke front. The residence time of electrons in the high-density plasma behind the spoke front is long because these electrons are trapped in this low electric field region. This explains the fast decay of the electron temperature in the simulations and the large population of low energy electrons in that region (Coulomb collisions, not taken into account in the model, could reduce this effect). Finally, the simulations show that the number of spokes increases with pressure, in agreement with the experiments of Panjan et al. 117 who showed that this is related to a decrease of the ionization mean free path with increasing pressure.

To conclude this section, we can say that the simulations tend to show that the very coherent mode of spoke rotation observed in dc magnetron discharges in experiments and simulations is due to the strong and localized electron heating associated with the 𝛁𝐵 drift, and that the -𝑬 × 𝑩 rotation is a consequence of the evolution of a gradient-drift instability into an ionization wave resulting from this mechanism of electron heating. Simulations performed under uniform magnetic field in similar conditions exhibit much less coherent instabilities rotating in the +𝑬 × 𝑩 direction.

CONCLUSION

Low-temperature 𝑬 × 𝑩 plasmas with closed-drift configurations are widely used in space propulsion and plasma processing, however, their design and development still relies on a semi-empirical approach due to a lack of selfconsistent engineering models. The availability of predictive models would be an extremely valuable asset for the industry as it would greatly reduce development costs. Even though understanding the fascinating physics of 𝑬 × 𝑩 plasmas can be a goal in itself, the important research effort devoted, for example, to Hall thrusters in the last twenty years, is largely driven by the development of predictive models. The physics of these plasmas is complex and non-linear, and the presence of instabilities at a wide range of frequencies and wavelengths makes it challenging to develop predictive models. In this tutorial we have shown that, despite these difficulties, significant progress has been made over the years in characterizing and understanding the various types of instabilities that can occur in closed-drift devices. This progress has been made possible through a collaborative research effort involving theory, simulations and experiments. The theory, based on the linearization of fluid or kinetic transport equations, has resulted in the derivation of dispersion relations that define the various types of instabilities that may develop in the partially magnetized plasmas of closed-drift devices. Computer simulations, primarily using particle methods, have been used to verify or to initiate the theoretical developments, and to describe the non-linear evolution of instabilities. Experiments have revealed the presence of some of the instabilities predicted by theory and simulation. Despite these important achievements, the development of fully predictive models is still not within our reach for the following reasons. 1) although the simulations can predict both the linear and non-linear evolution of some instabilities, they are based on reduced or over-simplified models that focus on one particular aspect or one particular type of instability. Additionally, these models are primary two-dimensional (and still computationally intensive) with limitations in the dimensions of the simulation domain that may prevent the inverse energy cascade to longer wavelengths. 2) three-dimensional particle models can be run in massively parallel simulations but their use under realistic plasma densities and device dimensions remains challenging and does not allow parameter studies at a reasonable cost. 3) An alternative approach is using hybrid models, in which electrons are treated as a fluid and ions and neutral atoms are treated as particles. This approach is less computationally intensive and is better suited for engineering applications. It has been used in the last twenty years, but it is not selfconsistent, as it cannot accurately describe instabilities. These models use anomalous transport coefficients, such as electron mobility, as input to fit experimental data. It is yet to be determined whether anomalous transport coefficients derived from particle simulations can be used to improve the predictivity of hybrid models.

The strong electric fields in some regions of 𝑬 × 𝑩 devices result in 𝑬 × 𝑩 drift velocities that approach thermal particle velocities. In these conditions, kinetic theory and simulations are essential for studying anomalous transport. Results from existing 1D and 2D simulations, as well as emerging 3D simulations 125,231,232 , suggest not only quantitative changes in multidimensional effects, such as the reduction of fluctuation amplitudes and associated transport, but also qualitative changes due to the emergence of new instability regimes (e.g. MTSI) and complex plasma-wall interactions. These investigations require comprehensive 3D simulations, the importance of which cannot be underestimated.

The physics of low temperature partially magnetized plasmas presents a formidable intellectual challenge in the field of general nonlinear physics. The magnetic configurations, though simple in appearance, give rise to turbulence and transport in 𝑬 × 𝑩 plasmas that are just as complex as those observed in high temperature fusion plasmas. This makes it a topic that deserves significant attention and focused research efforts.

Despite the challenges in achieving a fully self-consistent description of 𝑬 × 𝑩 plasmas for closed-drift devices, we are optimistic that our capabilities in understanding and modeling these plasmas will continue to improve. This will be achieved through: 1) increasing model power and complexity with larger simulation domains and 3D simulations (investments in exascale computing at a small fraction of the cost of those spent on fusion plasmas would certainly bring us closer to fully predictive codes, see, e.g. the project WarpX, an exascale computing platform for beam-plasma simulations 233 ), 2) continuing international collaboration through 𝑬 × 𝑩 workshops and benchmarking of simulations and experiments, 3) strengthening the synergy between theory, simulations and experiments.

Figure 1 :

 1 Figure 1 : Notations of the 𝑬 × 𝑩 configuration: electric field 𝑬 = 𝐸𝒙 ̂, external magnetic field 𝑩 = 𝐵𝒛 ̂. In this configuration, the 𝑬 × 𝑩 drift is in the -𝑦 direction. The plasma density gradient is supposed to be parallel to the electric field, in the 𝑥 direction, with a characteristic length (positive or negative) 𝐿 𝑛 = ( 1 𝑛

Figure 3 :

 3 Figure 3 : Examples of Hall ion sources or gridless ion sources. (a), (b), (c), (d), (e) are different types of gridless ion thrusters, (f) is an ion source used for plasma processing (e.g. for ion assist thin film coating). These figures only show the basic principles of these devices (magnetic circuit is not represented).

Figure 4 :

 4 Figure 4 : Axial profiles of the normalized radial magnetic field in Hall thrusters, magnetron discharges, and anode layer ion sources (at the radial position where the axial component of the magnetic field is zero).

Figure 5 :

 5 Figure 5 : (a) growth rate, and, (b), angular frequency normalized to the lower-hybrid frequency, as a function of 𝑘𝜌 𝑒 for a fixed dimensionless diamagnetic drift velocity 𝑉 * = 10 , and for three values of the dimensionless 𝑬 × 𝑩 drift velocity, 𝑉 0 =2, 10, 50. The dimensionless collision frequency and Debye length are fixed to 𝜈 𝑒𝑛 𝜔 𝐿𝐻 ⁄ = 6.23 and 𝜆 𝐷𝑒 𝜌 𝑒 ⁄ = 0.16. The magnetic field is uniform, (𝜔 D = 0). Numerically, these curves can correspond to argon, with a magnetic field intensity 𝐵 = 5 mT, an electron temperature 𝑇 𝑒 = 5 eV, and a plasma density 𝑛 = 10 16 m -3 . In these conditions, 𝜔 𝐿𝐻 = 3.2 × 10 6 rd/s, 𝑐 𝑠 = 3.46 × 10 3 m/s , 𝜌 𝑒 = 1.1 mm, and 𝜈 𝑒𝑛 = 2 × 10 7 , 𝜆 𝐷𝑒 =0.17 mm. 𝑉 * = 10 corresponds to 𝑣 * = 3.5 × 10 4 m/s and 𝐿 𝑛 = 2.89 cm. 𝑉 0 =2, 10, 50 correspond respectively to 𝑣 0 = 6.9 × 10 3 , 3.46 × 10 4 and 1.73 × 10 5 m/s, and 𝐸 0 = 35, 173, and 865 V/m.

Figure 6 :

 6 Figure 6 : (a) growth rate, and , (b), angular frequency normalized to the lower-hybrid frequency, as a function of 𝑘𝜌 𝑒 for a fixed dimensionless 𝑬 × 𝑩 drift velocity 𝑉 0 = 10 , and for three values of the dimensionless diamagnetic drift velocity, 𝑉 * =2, 10, 50. The dimensionless collision frequency and Debye length are, as in Figure 5, fixed to 𝜈 𝑒𝑛 𝜔 𝐿𝐻 ⁄ = 6.23 and 𝜆 𝐷𝑒 𝜌 𝑒 ⁄ = 0.16 , and the magnetic field is uniform, (𝜔 D = 0). The numerical example is the same as in in Figure 5: argon, 𝐵 = 50 mT, 𝑇 𝑒 = 5 eV, 𝑛 = 10 16 m -3 , giving 𝜔 𝐿𝐻 = 3.2 × 10 6 rd/s, 𝑐 𝑠 = 3.46 × 10 3 m/s , 𝜌 𝑒 = 1.1 mm, and 𝜈 𝑒𝑛 = 2 × 10 7 , 𝜆 𝐷𝑒 =0.17 mm. 𝑉 0 = 10 corresponds to 𝑣 0 = 3.5 × 10 4 m/s and 𝐸 0 = 173 V/m. 𝑉 * =2, 10, 50 correspond respectively to 𝑣 * = 6.9 × 10 3 , 3.46 × 10 4 and 1.73 × 10 5 m/s, and 𝐿 𝑛 = 14.4 cm, 2.89 cm, and 0.58 cm.

Figure 7 :

 7 Figure 7 : Contour plots of dimensionless, (a), maximum growth rate, 𝛾 𝑚𝑎𝑥 𝜔 𝐿𝐻 ⁄ , (b), angular frequency at the maximum growth rate, 𝜔 𝑅,𝛾𝑚𝑎𝑥 𝜔 𝐿𝐻 ⁄ , and (c), wavelength 𝜆 𝛾 𝑚𝑎𝑥 𝜌 𝑒 ⁄ at the maximum growth rate as a function of normalized 𝑬 × 𝑩 drift 𝑉 0 , and diamagnetic drift 𝑉 * , calculated from the dispersion relation (26). The dimensionless collision frequency and Debye length are fixed to 𝜈 𝑒𝑛 𝜔 𝐿𝐻 ⁄ = 6.23 and 𝜆 𝐷𝑒 𝜌 𝑒 ⁄ = 0.16 (as in Figure 5 and Figure 6) , and the magnetic field is uniform, (𝜔 D = 0). The up triangle symbols correspond to the three cases of Figure 5 (𝑉 * = 10 with 𝑉 0 =2, 10, 50) and the down triangle symbols to the three cases of Figure 6 (𝑉 0 = 10 with 𝑉 * =2, 10, 50).

Figure 8 :

 8 Figure 8 : Schematic of the Penning discharge in the PIC-MCC simulations of Carlsson et al. 122 and Powis et al. 123 (some results are shown in Figure 9 and Figure 10). The simulations are aimed at modeling the Penning discharge experiment of Raitses et al. 101, 102 , where electrons are injected by a radiofrequency plasma cathode around the discharge axis. An axial magnetic field is generated by Helmoltz coils. A dc voltage of 20-200 V is applied between the rf cathode and the cylindrical anode. The discharge is operated in xenon at a pressure in the 0.1-1 mtorr range. The anode diameter is 10 cm in the experiments and the axial length is 40 cm 102 .

Figure 9 :

 9 Figure 9 : 2D PIC-MCC simulation of the Penning discharge of Figure8in a plane (radial-azimuthal) perpendicular to the magnetic field, showing the space distribution of the electron density at two different times. The anode radius R0 is 2.5 cm. Electrons are injected in a circular area of radius Ri=0.1 cm around the discharge axis, with a directed (beam) energy of 15 eV. The applied magnetic field is 10 mT, and the neutral gas pressure is 1 mtorr (293 K). The simulations are performed in a gas with helium mass and approximate crosssections of xenon (electron momentum cross-section, no excitation). Ions are collisionless. A pseudo-permittivity 𝜀 𝑟 = 400 is used in the simulations. This means that the "real density" simulated is actually 400 times smaller than indicated in the scale of this figure (i.e. the plotted density is actually 𝜀 𝑟 𝑛 𝑒 so the maximum "real" density in the simulation is about 10 16 m -3 ), but Coulomb collisions are taken into account for the density indicated in the scale. Adapted fromPowis et al., Phys. Plasmas 25, 072110 (2018), with permission of AIP Publishing (Fig.6of Ref.123 ).

Figure 10 :

 10 Figure 10 : Time averaged radial profiles of the plasma potential and electron density (top) and electric field (bottom) in the collisionless and collisional cases in the conditions of Figure 9. Adapted from Powis et al., Phys. Plasmas 25, 072110 (2018), with permission of AIP Publishing (Fig.7 of Ref. 123 ).

Figure 11 :

 11 Figure 11 : Images of a rotating spoke at 4 different times in a wallless Hall thruster 109 . The thruster was operated at 110 V applied voltage, 1.3 mg/s xenon mass flow rate, and 1.16 A discharge current. The exposure time of the images was 7.66 s. The images were processed. 109 , and a mask was applied to delete the light outside the channel boundaries. The measured velocity of the rotating spoke was 1290 m/s. Adapted with permission from Mazouffre et al., Plasma Sources Sci. and Technol. 28, 054002 (2019). Copyright 2019 IOP Publishing Ltd. (Fig. 5 from Ref. 109 ).

  also suggest that the rotating spoke is associated with an ionization wave coupled with a periodic depletion of the neutral atom density (breathing mode, see section III.A).Tyushev et al.129 studied a Penning discharge under conditions similar to those of Powis et al., with collisional electrons and collisionless ions. By varying the power input by the electron beam into the discharge, they could control the radial profile of the time averaged electric potential.

Figure 12 :

 12 Figure 12: Examples of spiral structures of the plasma density characteristics of a lower-hybrid instability destabilized by density gradient and collisions, obtained in 2D PIC-MCC simulations. The magnetic field is perpendicular to the simulation domain. (a) Electron beam sustained Penning discharge, radial-azimuthal plane 129 , (b) closed-drift discharge between two biased electrodes at low uniform magnetic field 96 (axial-azimuthal plane; the azimuthal direction is vertical) , (c) discharge sustained by radiofrequency electric field perpendicular to the simulation domain, radialazimuthal plane 135 . The dimensions, units and other conditions can be found in the references. (a), adapted from Tyushev et al. to be published, Phys. Plasmas (2023), with permission of AIP publishing (Fig. 20a of Ref. 129 ); (b) adapted from Boeuf, Phys. Plasmas 26 072113 (2019), with permission of AIP Publishing (Fig. 16 of Ref. 96 ); (c) adapted from Lucken et al., Phys. Plasmas 26 070702 (2019), with permission of AIP Publishing (Fig. 1b of Ref. 135 ).

1 .

 1 𝑥 . Since the ratio of ion to electron mobility without magnetic field, 𝜇 𝑖 𝜇 𝑒0 ⁄ can be on the order of 10 -2 or less, we have generally The electron mobility parallel to the electric field 𝐸 𝑥 in the presence of Hall effect is therefore considerably larger than without Hall effect and, in the limit 𝜇 𝑖 𝜇 𝑒0 ⁄ → 0, 𝛼 → 1, and we have 𝑣 𝑒𝑥 ̅̅̅̅ → -𝜇 𝑒0 𝐸 𝑥 . We can conclude that the Hall effect destroys the electron confinement perpendicular to the magnetic field.

Figure 14 :

 14 Figure 14 : 2D distributions of (a), electron density (b), electric potential in the source in the conditions of Figure 13. The color scale is linear for 𝜙, and log on three decades for 𝑛 𝑒 . H2, pressure 0.3 Pa, maximum magnetic field 𝐵 𝑚𝑎𝑥 = 3 mT, bias voltage 𝑉 𝑏 = 20 V, power absorbed in the driver 100 kW/m. The scale factor of the PIC-MCC simulation is 𝜂 = 2 × 10 4 (i.e. the simulations are performed with a plasma density 𝜂 times lower than indicated in (a) in Poisson equation). Adapted from Boeuf et al., Phys. Plasmas 19, 113510 (2012), with permission of AIP Publishing (Fig. 3 of Ref. 137 ).

Figure 15 :

 15 Figure 15 : Intensity and streamlines of (a) electron current density (unit 8 kA/m 2 ), and (b), ion current density (unit 2.8 k A/m 2 ) in the source in the conditions of Figure 13 and Figure 14. H2, pressure 0.3 Pa, maximum magnetic field 𝐵 𝑚𝑎𝑥 = 3 mT, bias voltage 𝑉 𝑏 = 20 V, power absorbed in the driver 100 kW/m. Reproduced from Boeuf et al., Phys. Plasmas 19, 113510 (2012), with permission of AIP Publishing (Fig. 7 of Ref. 137 ).

Figure 16 :

 16 Figure 16 : Schematic of a Hall thruster; (a), 3D view, (b) radial-axial view showing the profiles, in the middle of the channel, of the radial magnetic field 𝐵 𝑟 , and of the ionization rate 𝑆 and axial electric field 𝐸 𝑥 defining the ionization and acceleration zones. Adapted from Boeuf, J. Appl. Phys. 121, 011101 (2017), with permission from AIP Publishing (Fig. 5 of Ref. 7 ).

Figure 17 :

 17 Figure 17 : Effective inverse Hall parameter ℎ 𝑒𝑓𝑓 -1 = 𝜇 𝑒,𝑒𝑓𝑓 𝐵 𝑟 = 𝜈 𝑒𝑓𝑓 𝜔 𝑐𝑒 ⁄ (triangles) deduced from experiments, compared with the inverse classical, collisional Hall parameter ℎ -1 = 𝜈 𝑒𝑛 𝜔 𝑐𝑒 ⁄ (circles) as a function of axial position along the channel. The channel length is 7 cm in this thruster. Adapted with permission from Meezan et al., Phys. Rev. E 63 (2001). Copyright 2001 by the American Physical Society (Fig. 12 of Ref. 144 ).

Figure 19 :

 19 Figure 19 : Different types of instabilities occurring in a Hall thruster and their frequencies.

Figure 20 :

 20 Figure 20 : Schematic of a magnetron discharge. Adapted from Gudmundson, Plasma Sources Sci. and Technol. 29, 113001 (2020) (Fig. 2 of Ref. 14 ); licensed under a Creative Commons Attribution (CC BY) license.

Figure 21 :

 21 Figure 21 : Images of rotating spokes in a dc magnetron discharge at two different gas pressures in argon. The spokes rotate in the -𝑬 × 𝑩 direction. Adapted from Panjan et al., J. Appl. Phys. 125, 203303 (2019), with permission of AIP (Fig. 4 of Ref. 118 ).

Figure 22 :

 22 Figure 22 : 2D radial-azimuthal distribution of the plasma potential measured by Langmuir probe in a dc magnetron discharge, 7.5 mm above the cathode. Adapted from Panjan and Anders, J. Appl. Phys. 121, 063302 (2017), with permission from AIP (Fig.5 of ref. 178 ).

Figure 23 :

 23 Figure 23 : Probe measurements by Held et al. 119 of the time evolution of electron density, 𝑛 𝑒 , electron temperature, 𝑇 𝑒 , and plasma potential 𝜙 in a dc magnetron discharge in argon , 0.56 Pa pressure and applied dc voltage 255 V. The probe is located 6 mm above the cathode surface on the spoke path. The gray areas indicate positions where the validity of the electron temperature and plasma potential are in question. Reproduced from Held et al., Plasma Sources Sci. Technol 31, 085013 (2022) (Fig.6 of Ref. 119 ); licensed under a Creative Commons Attribution (CC BY) license.

2 𝑍 21 𝑍

 221 is the plasma dispersion function 201 defined by: 𝐼 𝑚 the modified Bessel function of the first kind.

Figure 24 :

 24 Figure 24 : Solutions of the 3D dispersion relation for the Electron Cyclotron Drift Instability as a function of the wave number in the azimuthal direction, 𝑘 𝑦 , and for three values of the wave number in the magnetic field direction. The component parallel to the electric field, 𝑘 𝑥 is set to zero. 𝜔 𝑅 and 𝛾 are the angular frequency and the growth rate of the perturbation, respectively. 𝑣 0 is the 𝑬 × 𝑩 drift, 𝑣 𝑇𝑒 the electron thermal velocity, 𝜔 pi the ion plasma frequency. The maximum growth rate, around 𝑘 𝑦 𝜆 𝐷𝑒 ≈ 0.7, corresponds to an azimuthal wavelength 𝜆 = 2 𝜋 𝑘 𝑦 ⁄ on the order of 1 mm in the conditions of a Hall thruster. Adapted from Cavalier et al., Phys. Plasmas 20, 082107, (2013) with permission of AIP Publishing (Fig. 2 of Ref. 202 ).

Figure 25 :

 25 Figure 25 : Propagating ion density perturbation around 𝑡 = 25 s, from a 1D PIC-MCC simulation of the azimuthal direction of a Hall thruster, with perpendicular magnetic field 𝐵 = 20 mT and electric field 𝐸 0 = 20 kV/m, and a plasma density 𝑛 = 10 17 m -3 . Reproduced with permission from Smolyakov et al., Plasma Physics reports, 46 496 (2020). Copyright 2020 Pleiades Publishing, Ltd. (Fig. 3 of Ref. 199 ).

  3 of Ref. 214 and Fig. 4 of Ref. 214 ).

Figure 26 :

 26 Figure 26: frequency spectrum of the azimuthal electric field (in V/m) in the nonlinear regime, obtained with a noise free Vlasov simulation.The green solid line and the green dashed line show the ion-sound dispersion relation with the initial temperature and large average temperature later in simulations, respectively. Reproduced from Tavassoli et al., Phys. Plasmas 29 030701 (2022), with permission of AIP Publishing (Fig. 4 of Ref. 208 ).

Figure 27 :

 27 Figure 27 : Axial-azimuthal domain used in the benchmark 2D PIC-MCC simulations of Charoy et al. 216 and Boeuf and Garrigues 167 (top) and axial profiles of the perpendicular magnetic field and of the fixed ionization rate 𝑆 (bottom). Collisionless simulation, with axial length 𝑑 𝑥 = 2.5 cm, and azimuthal length 𝑑 𝑦 = 1.28 cm. The simulation domain is periodic in the azimuthal direction. Electrons from the emissive cathode are injected uniformly along the dashed line on the right of the simulation domain. The total emitted electron flux Γ 𝑒𝑐 (flux entering the channel, Γ 𝑒𝑐1 , plus flux neutralizing the ion beam, Γ 𝑒𝑐2 ) is equal to the net charged particle flux at the anode (Γ 𝑒𝑎 -Γ 𝑖𝑎 ).

Figure 28 :

 28 Figure 28 : 2D axial-azimuthal color maps of the azimuthal electric field (top) and ion density (bottom) at a given time, from PIC-MCC simulations, illustrating the Electron Cyclotron Drift Instability. Conditions of Figure 27. The instability moves in the 𝑬 × 𝑩 direction Adapted with permission from Charoy et al., Plasma Sources Sci. Technol. 28, 105010 (2019). Copyright 2023 IOP Publishing Ltd. (Fig.4 of Ref. 216 ).

Figure 29 :

 29 Figure 29: Fast Fourier Transform (FFT) of the azimuthal electric field (color map) of Figure 28, at 𝑥 = 0.3 cm, (a), and at 𝑥 = 1.5 cm, (b). The solid line is the angular frequency of the ion acoustic instability (eq.(39)). The green dashed lines indicate the wavelength and frequency of the dominant mode. Adapted with permission from Charoy et al., Plasma Sources Sci. Technol. 28, 105010 (2019). Copyright 2023 IOP Publishing Ltd. (Fig.8 of Ref. 216 ).

Figure 30 :

 30 Figure 30 : Azimuthally and time averaged axial profiles of the axial electric field (a), and ion density (b) deduced from the PIC-MCC simulations of Figure 28. The different curves correspond to benchmark simulations by different groups. Adapted with permission from Charoy et al., Plasma Sources Sci. Technol. 28, 105010 (2019). Copyright 2023 IOP Publishing Ltd. (Fig.5 of Ref. 216 ).

1 2,

 1 and 𝐼 ±1 (𝑏) → 𝑏/2 .

Figure 31 :

 31 Figure 31 : Simulation domain for the radial-azimuthal model. The axial electric field and radial magnetic field are given, 𝐸 = 10 4 V/m and 𝐵 =20 mT in the simulations reported here 223 . The dimensions in the radial and azimuthal directions are 𝑑 𝑧 = 𝑑 𝑦 = 1.28 cm. The virtual length in the axial direction is 𝐿 𝑥 = 1 cm. The ionization rate 𝑆 has a cosine profile between 𝑧 1 =0.09 cm, and 𝑧 2 =1.19 cm. The integral of 𝑆(𝑦, 𝑧) in the radial direction correspond to a current density of 100 A/m 2 . The walls in the radial direction are grounded and secondary electron emission is not taken into account.

Figure 32 :

 32 Figure 32 : 1D azimuthal Fast Fourier Transform of the azimuthal electric field , averaged over radial positions and over two time intervals, (a) 5-10 s, (b), 15-20 s. 𝑘 0 = 𝜔 𝑐𝑒 𝑣 0 ⁄ is the wavenumber corresponding to the first cyclotron resonance. The different curves correspond to simulations by different groups carried out within a benchmark of different codes. Adapted from Villafana et al., Phys. Plasmas 30 075002 (2021), with permission of AIP Publishing (Fig. 9 of Ref. 223 ).

Figure 33 :

 33 Figure 33 : (a) Axial-azimuthal simulation domain for a magnetron discharge. The magnetic field is perpendicular to the simulation domain. A 260 V voltage is applied between cathode and anode, and a fixed electron current density je0 is emitted at the cathode to represent the secondary electron emission by ion impact. (b) Two different axial profiles, 𝐵 1 (𝑥) and 𝐵 2 (𝑥) , of the perpendicular magnetic field used in the simulations.

Figure 34 :

 34 Figure 34 : 2D color maps of the electric potential, 𝜙 , ion density, 𝑛 𝑖 , ionization rate, 𝑛 𝑒 𝜈 𝑖 , and current density 𝑗 𝑒 in the simulation domain of Figure 33a, with the magnetic profile 𝐵 1 of Figure 33b. Argon, 2 Pa, 260 V dc voltage, net current density of emitted electron 25 mA/m 2 . The units are 260 V for 𝜙, 0.5 × 10 16 m -3 for 𝑛 𝑖 , 1.2 × 10 22 m -3 s -1 for 𝑛 𝑒 𝜈 𝑖 , and 200 A/m 2 for 𝑗 𝑒 .The equipotential contours are plotted from 20 to 240 V and are separated by 20 V. Some streamlines are plotted for the electron current density. The initial plasma density in the simulation was uniform and equal to 5 × 10 14 m -3 and the plasma parameters of this figure are plotted at time t=18 s. The axial plots of Figure 35 correspond to the azimuthal positions 𝑦 1 and 𝑦 2 indicated by dashed lines on the potential plot.

Figure 35 :

 35 Figure 35 : Axial profiles of the axial electric field, and the electron and ion densities at two different azimuthal positions 𝑦 1 , (a), and 𝑦 2 , (b) indicated by dashed lines on the potential plot of Figure 34.

Figure 36 :

 36 Figure 36 : (a) Example of electron trajectory along an equipotential line in the double layer region in the axial-azimuthal plane; (b) magnified view of part of Figure 36a illustrating the fact that the electron trajectory slightly deviates from the equipotential line due to 𝛁𝐵 drift; (c) variations of the electron kinetic energy as a function of azimuthal position along the trajectory of Figure 36a. The red line is the averaged energy over the cyclotron oscillations. Same conditions as Figure 34.

Figure 37 :

 37 Figure 37 : 2D color maps of the ion density, 𝑛 𝑖 , electron temperature, 𝑇 𝑒 , and ionization rate, 𝑛 𝑒 𝜈 𝑖 at a given time, under conditions close to the experiments of Held et al. 119 . Equipotential contours are superimposed on the ion density color map. The right plot shows the azimuthal profile of the collisional and 𝛁𝐵 drift heating rates along the dashed line of the electron temperature plot, at x =1 cm. The conditions are: argon, 0.6 Pa, 260 V dc voltage, net current density of emitted electron 30 mA/m 2 . The units are 2 × 10 16 m -3 for 𝑛 𝑖 , 20 eV for 𝑇 𝑒 and 1.4 × 10 22 m -3 s -1 for 𝑛 𝑒 𝜈 𝑖 .The equipotential contours are plotted from 20 to 260 V and are separated by 20 V. Adapted from Fig. 5 and Fig. 11 of Ref. 130 .

Figure 38 :

 38 Figure 38 : Time variations of electron density, 𝑛 𝑒 , electron temperature, 𝑇 𝑒 , and plasma potential 𝜙 from the 2D PIC-MCC simulation, 6 mm above the cathode surface (𝑥 = 6 mm), in the same conditions as Figure 37 (argon, 0.6 Pa, dc voltage 260 V). To be compared with the experimental measurements of Held et al. 119 reproduced in Figure 23 of the present paper). Adapted from Boeuf, Phys. Plasmas, to be published (2023), with permission of AIP (Fig. 9 of Ref. 130 ).

Figure 39 :

 39 Figure 39 : (a) Normalized electron energy probability function (EEPF) at four different azimuthal locations around the spoke front, 8.8 mm above the cathode surface (𝑥 = 8.8 mm) obtained from the PIC-MCC simulation at a given time in the conditions of Figure 37. The EEPF are integrated over squares of 80 m side (indicated on (c)) in space, and over 10 ns in time; (b) azimuthal profile of the electron temperature at 𝑥 = 8.8 mm around the spoke front in the conditions of (a); (c) 2D distribution of ion density. The azimuthal locations corresponding to the EEPF of (a) are indicated by colored squares and numbers on (c), and by lines on (b), with the same colors and numbers as the EEPF plots on (a). Adapted from Boeuf, Phys. Plasmas, to be published (2023) with permission of AIP (Fig. 10 of Ref. 130 ).

Table 2 :

 2 Gas pressure range and corresponding values of gas density, collision frequencies with neutrals, 𝜈 𝑒𝑛 , 𝜈 𝑖𝑛 , and free paths, 𝜆 𝑒𝑛 , 𝜆 𝑖𝑛 of electrons and ions with 5 eV kinetic energy. The ion parameters are calculated for argon mass (𝑀 𝑚 = 7.35 × 10 4 ⁄

Scientific Research, and The Digital Research Alliance of Canada.

ACKNOWLEDGEMENTS

JPB thanks the French Spatial Agency CNES for continuous support, and the RTRA STAE foundation under the project IMPULSE. Part of the work of AS in this tutorial was supported by NSERC Canada, US Air Force Office of

DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.