H J C Oliver
email: hjcoliver@utexas.edu

S E Sharapov

B N Breizman

A K Fontanilla

D A Spong

D Terranova

Jet Contributors

Modification of the Alfvén wave spectrum by pellet injection

Alfvén eigenmodes driven by energetic particles are routinely observed in tokamak plasmas. These modes consist of poloidal harmonics of shear Alfvén waves coupled by inhomogeneity in the magnetic field. Further coupling is introduced by 3D inhomogeneities in the ion density during the assimilation of injected pellets. This additional coupling modifies the Alfvén continuum and discrete eigenmode spectrum.

The frequencies of Alfvén eigenmodes drop dramatically when a pellet is injected in JET. From these observations, information about the changes in the ion density caused by a pellet can be inferred. To use Alfvén eigenmodes for MHD spectroscopy of pellet injected plasmas, the 3D MHD codes Stellgap and AE3D were generalised to incorporate 3D density profiles. A model for the expansion of the ionised pellet plasmoid along a magnetic field line was derived from the fluid equations. Thereby, the time evolution of the Alfvén eigenfrequency is reproduced. By comparing the numerical frequency drop of a toroidal Alfvén eigenmode (TAE) to experimental observations, the initial ion density of a cigar-shaped ablation region of length 4cm is estimated to be n * = 6.8×10 22 m -3 at the TAE location (r/a ≈ 0.75). The frequency sweeping of an Alfvén eigenmode ends when the ion density homogenises poloidally. Modelling suggests that the time for poloidal homogenisation of the ion density at the TAE position is τ h = 18 ± 4 ms for inboard pellet injection, and τ h = 26 ± 2 ms for outboard pellet injection. By reproducing the frequency evolution of the elliptical Alfvén eigenmode (EAE), the initial ion density at the EAE location (r/a ≈ 0.9) can be estimated to be n * = 4.8 × 10 22 m -3 . Poloidal homogenisation of the ion density takes 2.7 times longer at the EAE location than at the TAE location for both inboard and outboard pellet injection.

MHD spectroscopy, Alfvén eigenmodes, pellet injection

Introduction

Alfvén eigenmodes (AEs) driven unstable by energetic particles are ubiquitous in tokamaks with axisymmetric magnetic fields, and stellerators with 3D magnetic fields [START_REF] Wong | A review of Alfvén eigenmode observations in toroidal plasmas[END_REF]. The most frequently observed Alfvén eigenmodes are weakly damped modes with eigenfrequencies that lie inside gaps in the Alfvén continuum. These continuum gaps, and the gap modes within them, are created when poloidal harmonics of the shear Alfvén wave are coupled by poloidal modulation in the magnetic field. For example, the magnetic field is poloidally asymmetric in toroidal magnetic confinement devices. Therefore toroidicity couples neighbouring poloidal harmonics (denoted by the poloidal mode numbers m and m -1), creating the toroidal Alfvén eigenmode [START_REF] Cheng | High-n ideal and resistive shear Alfvén waves in tokamaks[END_REF] (TAE). In plasmas with elliptical crosssections, ellipticity of the magnetic surfaces introduces further poloidal asymmetry. Ellipticity couples the m and m -2 poloidal harmonics, producing the elliptical Alfvén eigenmode (EAE) [START_REF] Dewar | Long-wavelength kink instabilities in lowpressure, uniform axial current, cylindrical plasmas with elliptic cross sections[END_REF][START_REF] Betti | Ellipticity induced Alfvén eigenmodes[END_REF]. Poloidal and toroidal symmetries are broken in 3D magnetic configurations by poloidal and toroidal modulation of the magnetic field. As a result, toroidal harmonics are also coupled, producing helicity-induced gaps and helical Alfvén eigenmodes [START_REF] Nakajima | High-n helicity-induced shear Alfvén eigenmodes[END_REF] (HAEs).

The possible coupling of Alfvén eigenmode harmonics introduced by 3D inhomogeneities in the plasma mass density is examined in this paper. Such 3D density inhomogeneities are transiently caused by the injection of frozen pellets, which are used to refuel the core of the plasma, control edge localised modes (ELMs), and mitigate disruptions [START_REF] Lang | High-Efficiency Plasma Refuelling by Pellet Injection from the Magnetic High-Field Side into ASDEX Upgrade[END_REF][START_REF] Garzotti | Simulations of JET pellet fuelled ITB plasmas[END_REF][START_REF] Lang | ELM triggering by local pellet perturbations in type-I ELMy H-mode plasma at JET[END_REF]. These varied applications require pellets with drastically different parameters, for example, pellet size, material, trajectory, velocity, and injection frequency are all tailored to fit the application. As a result, the timescales of pellet assimilation can vary dramatically depending on the experiment and machine.

Diagnosing the effect of the pellet on the plasma represents an important but challenging task, because of the short life-time of the pellet and the complexity of the pellet assimilation into the plasma. We investigate whether key pellet parameters can be assessed using magnetohydrodynamic (MHD) spectroscopy [START_REF] Goedbloed | MHD spectroscopy: free boundary modes (ELMs) and external excitation of TAE modes[END_REF] based on Alfvén eigenmodes observed during pellet injection [START_REF] Sharapov | MHD spectroscopy of JET plasmas with pellets via Alfvén eigenmodes[END_REF]. MHD spectroscopy -the diagnosis of plasma properties through observations of MHD spectrum of waves and instabilities -is currently used to diagnose the time evolution of the safety factor profile in tokamaks [START_REF] Sharapov | Alfvén wave cascades in a tokamak[END_REF][START_REF] Fasoli | and contributors to the EFDA-JET Workprogramme[END_REF]. MHD spectroscopy based on Alfvén eigenmodes offers excellent time resolution, and provides information on the density of ions in the core, rather than the density of neutral atoms delivered by the pellet. For this reason, we apply MHD spectroscopy to pellet injected plasmas for the first time. MHD spectroscopy can complement other diagnostics that are commonly used to study the plasma density, such as interferometry, LIDAR and Thompson scattering, which measure the electron density, and D α spectroscopy measuring the density of deuterium atoms.

The ions deposited by the injected pellet introduce poloidal and toroidal dependence in the ion density, which causes the Alfvén velocity to be periodic in poloidal and toroidal angle.

Therefore, the refractive index is periodic, coupling the poloidal and toroidal harmonics of the Alfvén wave. This coupling significantly modifies the Alfvén continuum and discrete eigenmode spectrum. Information about the changes in the ion density caused by pellets can be inferred from observed changes in the frequency of Alfvén eigenmodes.

To better understand the relationship between the eigenfrequency and the 3D ion density profiles, we modified the 3D MHD codes Stellgap [START_REF] Spong | Shear Alfvén continua in stellarators[END_REF] and AE3D [START_REF] Spong | Clustered frequency analysis of shear Alfvén modes in stellarators[END_REF], which are routinely used to study Alfvén eigenmodes in stellerators. Stellgap and AE3D were generalised to incorporate 3D ion density profiles, in addition to the 3D magnetic field. A model that describes the 1D expansion of pellet material along magnetic field lines is derived from the fluid equations. The 3D ion density profiles resulting from pellet injection are obtained by mapping this 1D expansion onto the magnetic field lines. From these 3D density profiles, we use the modified AE3D to calculate the Alfvén eigenmodes for pellet injected plasmas.

First, the experimental motivation for this study is presented in Section 2. Next, analytical estimates for the Alfvén continuum and TAEs for a tokamak plasma with poloidal modulation of the ion mass density are derived in Section 3 and compared to numerical results in Sections 4. A model for the expansion of the pellet material is introduced in Section 5 to study the effect of pellets on Alfvén eigenmodes in JET plasmas in Section 6. Finally, these results are discussed in Section 7 and summarised in Section 8.

Experimental observations

In this article, we examine plasmas produced in the Joint European Torus (JET) [START_REF] Rebut | The Joint European Torus: installation, first results and prospects[END_REF] that are injected with frozen deuterium (D) pellets using the centrifugal pellet injector [START_REF] Jones | Results of JET Pellet Fuelling Experiments with Outboard and Inboard Launch[END_REF][START_REF] Lang | ELM triggering by local pellet perturbations in type-I ELMy H-mode plasma at JET[END_REF]. The plasmas in this experiment had on-axis safety factor, q 0 1; major radius, R 0 = 3.0 m; minor radius, a = 1.0 m; on-axis equilibrium magnetic field, B 0 = B(R 0 ) = 3.25 T; and plasma current, I p ≤ 2.5 MA. The pellet injector cuts and accelerates 4mm cubic pellets, before injecting them into the plasma from either the inboard or outboard track, as illustrated in Figure 1. The fuelling pellets under consideration were injected at speeds v pell ∼ 160 ms -1 and deposit up to 3.8 × 10 21 D atoms deep in the plasma. This injection of particles can approximately double the plasma density for JET plasmas with electron densities n e ≈ 1 -5 × 10 19 m -3 and plasma volume ∼ 100 m 3 . Neither of the injection tracks crossed the magnetic axis, as shown in Figure 2, and so hollow electron density profiles are frequently observed after pellet injection. The inboard trajectory is tangential to the flux surface at r/a ≈ 0.6. Most of the pellet material is deposited in the first 1.15m of the inboard trajectory, leading up to the tangent point [START_REF] Jones | Results of JET Pellet Fuelling Experiments with Outboard and Inboard Launch[END_REF].

The electron density radial profile, n e (r), was

Z (m) R (m)
Figure 2: Geometry of the inboard and outboard pellet injection trajectories in a typical JET equilibrium [START_REF] Jones | Results of JET Pellet Fuelling Experiments with Outboard and Inboard Launch[END_REF].

The typical ablation profile for a pellet injected from the inboard side is shown. measured using LIDAR [START_REF] Salzmann | The LIDAR Thomson scattering diagnostic on JET (invited)[END_REF], along the line of sight shown in Figure 1, at time intervals of 250 ms. This low time resolution was insufficient to consistently measure the electron density profile on timescales comparable to the characteristic timescales of pellet ablation and homogenisation. Highly poloidally asymmetric electron density profiles were observed despite this limitation. Figure 3 shows an electron density profile that was still highly asymmetric 22 ms after pellet injection from the inboard side. Similar poloidally asymmetric electron density profiles were observed up to 43 ms after pellet injection. Interferometry provides better time resolution than the LIDAR diagnostic, as demonstrated in Figure 4. However, the radial resolution of interferometry is limited, with only four lines of sight. Two of these lie at the plasma edge, and two lie close to the magnetic axis, as shown in Figure 1. None of the lines of sight intersect where most of the pellet material is deposited.

During this experiment, TAEs and EAEs were excited by fast ions produced by ion cyclotron resonance heating (ICRH) heating with high powers of P ICRH = 4 -8MW, as shown for discharge #49044 in Figure 4. The Alfvén eigenmodes studied here were excited by radial gradients of trapped energetic hydrogen minority ions with concentrations n H /n e ∼ 2 -5%.

Alfvén eigenmodes were detected using toroidally separated Mirnov coils positioned outside the plasma surface [START_REF] Heeter | Chaotic Regime of Alfvén Eigenmode Wave-Particle Interaction[END_REF]. These magnetic coils measure the timevarying perturbed poloidal magnetic field produced by the eigenmodes, d (δB θ ) /dt ∼ ωδB θ . Toroidal mode numbers were calculated using the relative phase shifts of the magnetic perturbations at the different coil toroidal positions. With sampling rates of 1 MHz -in contrast to a sampling rate of 4 Hz for the LIDAR diagnostic -magnetic coils facilitate MHD spectroscopy with much higher time resolution than many conventional diagnostics. When pellets were injected into plasmas in which TAEs were already excited, the frequency of the TAE dropped, as shown in Figure 5. There was a period ∼ 3 -8 ms after pellet injection where the toroidal mode numbers of the TAE could not be determined, as shown in Figure 5b. There was a longer period after pellet injection where toroidal mode numbers could be determined and dramatic changes in the TAE eigenfrequency were observed. There was a rapid drop in the TAE eigenfrequency in all discharges. This sweeping of the frequency occurs over ∼ 25 -45 ms.

The drop in frequency is larger for inboard injection versus outboard injection, as illustrated in Figures 5 and6. Additionally, the spectrum of excited toroidal mode numbers often broadens, as shown in Figure 5b. New toroidal mode numbers are excited after pellet injection. These new toroidal harmonics persist as long as existing toroidal harmonics. The frequencies of elliptical Alfvén eigenmodes (EAEs) also drop dramatically after the injection and assimilation of the pellet material [START_REF] Sharapov | MHD spectroscopy of JET plasmas with pellets via Alfvén eigenmodes[END_REF]. Occasionally, a delay in the drop in frequency is observed, as shown in Figure 7. The EAE frequencies continue to increase during this period. After a delay of ∼ 8 -18 ms, the frequency drops. The frequency sweeping of EAEs takes longer to stabilise than the frequency sweeping of TAEs. The period of frequency sweeping for EAEs continues for ∼ 55 -115 ms.

The differences in the timescale and magnitude of frequency sweeping between TAEs and EAEs, and between inboard (HFS) and outboard (LFS) pellet injection are summarised in Table 1. Table 1: A comparison of the timescales and fractional changes in frequency for frequency sweeping of TAEs and EAEs in plasmas where a pellet is injected from the inboard (HFS) or outboard (LFS). t 2 and t 1 are the time at the beginning and bottom of frequency sweeping, f 1 = f (t 1 ) is the initial frequency, and ∆f = f (t 1 ) -f (t 2 ) is the frequency decrease. The characteristic sweeping period τ is defined as f (τ ) = 1.01f (t 2 ). The mean value and standard deviation are calculated for each variable.

Analytical estimates of the Alfvén spectrum in toroidal plasmas with poloidal modulation of the ion density

We begin the analysis of the observations in Section 2 by deriving analytical estimates for the TAE radial structure and eigenfrequency. These provide a benchmark for the numerical work, and aid our understanding of the effect of ion density asymmetries on TAEs. We solve the high mode number shear Alfvén wave equation [START_REF] Berk | Continuum damping of low-n toroidicity-induced shear Alfvén eigenmodes[END_REF][START_REF] Ya | Alfvén continuum and high-frequency eigenmodes in optimized stellarators[END_REF]:

B 0 •∇ 1 B 0 ∇ 2 ⊥ B 0 B 0 • ∇φ +∇• ω 2 v 2 A ∇ ⊥ φ = 0. (1)
The wave electrostatic potential φ for TAE with toroidal mode number n is represented as a Fourier series:

φ(r) = e i(nζ-ωt) m φ m (r)e -imθ + c.c.. (2)
where m and n are the poloidal and toroidal mode numbers, and θ and ζ are the poloidal and toroidal angles. We can write the electrostatic potential as a Fourier series due to the periodicity of the eigenmode in poloidal and toroidal angle. The radial structure of a given poloidal harmonic, φ m (r), is localised in radius.

Working in flux coordinates (r, θ, ζ), the shear Alfvén wave equation can be rewritten using the contravariant metric tensor for a low β, large aspect ratio (a/R 0 1) tokamak with circular cross-section:

J g rr ≈ rR 0 1 + 2 r R 0 + ∆ cos θ , (3)
J g rθ ≈ -R 0 r R 0 + (r∆ ) sin θ (4)
J g θθ ≈ R 0 r (1 -2∆ cos θ) , (5)
where primes denote derivatives with respect to the flux coordinate r and the Jacobian is:

J ≈ rR 0 1 -2 r R 0 cos θ . (6)
The ion mass density i is no longer simply a function of the flux coordinate r. For ease, the poloidal inhomogeneity in ion density is assumed to be a modulation:

i (r, θ) = i0 (r) [1 + δ cos θ] , (7)
where δ is the size of poloidal modulation and i0 is the ion mass density at (r, θ, ζ) = (r, 0, 0). For an inhomogeneity peaking on the high field side, δ < 0, for an inhomogeneity peaking on the low field side, δ > 0.

The two poloidal harmonics φ m and φ m-1 are decoupled in a cylindrical plasma with circular cross-section and poloidally homogeneous ion density. Therefore, the two second-order eigenmode equations are singular at

ω 2 1 = k 2 m v 2 A and ω 2 2 = k 2 m-1 v 2 A , respectively.
The poloidal inhomogeneity due to toroidicity and ion density couples the m and m -1 poloidal harmonics, and the two eigenmode equations.

The two coupled eigenmode equations can be simplified by only considering frequencies close to ω

2 1 = k 2 m v 2 A and ω 2 2 = k 2 m-1 v 2
A , respectively. If magnetic shear S = r/q • dq/dr is small, we obtain two coupled equations for the m and m -1 poloidal harmonics:

L m φ m + (2ε + δ) 8q 2 R 2 0 d 2 φ m-1 dr 2 = 0, (8)
L m-1 φ m-1 + (2ε + δ) 8q 2 R 2 0 d 2 φ m dr 2 = 0, (9)
where ε = 5r m /2R 0 [START_REF] Candy | Nonideal theory of toroidal Alfvén eigenmodes[END_REF], and the operator L m is given by:

L m = d dr ω 2 v 2 A -k 2 m d dr - m 2 r 2 ω 2 v 2 A -k 2 m ( 10
)
Equations 8 and 9 describe two coupled "cylindrical" modes in the vicinity of the rational flux surface that satisfies the condition:

q(r m ) = m -1/2 n . ( 11
)
The Alfvén continuum is obtained when the determinant of the coefficients of the second order terms in the coupled eigenmode equations equals zero [START_REF] Fu | Excitation of the toroidicityinduced shear Alfvén eigenmode by fusion alpha particles in an ignited tokamak[END_REF]. This yields the normalised size of gaps in the continuum induced by toroidicity and the poloidal modulation of ion density is given by:

ω 2 + -ω 2 - ω 2 + + ω 2 - = ε + δ 2 ( 12
)
where ω + and ω -are the frequencies at the top and bottom of the continuum gap. The width of the gap is normalised to the frequency at the centre of the gap, ω 0 = v A /2qR 0 .

The two poloidal harmonics are strongly coupled at r = r m . The region in the vicinity of the gap surface is referred to as the inner region. Equations 8 and 9 can be solved in the inner region by following a similar method to that of Breizman and Sharapov [START_REF] Breizman | Energetic particle drive for toroidicity-induced Alfvén eigenmodes and kinetic toroidicity-induced Alfvén eigenmodes in a lowshear tokamak[END_REF][START_REF] Mikhailovskii | Instabilities in a Confined Plasma[END_REF]. First, the equations are normalised by introducing the dimensionless frequency,

g = 2 2ε + δ 4q 2 R 2 0 ω 2 v 2 A -1 , (13)
and the dimensionless radial variable,

x = nq(r) -m + 1 2 ≈ mS r -r m r m . (14)
In the inner layer, derivatives of the potential dominate over derivatives of equilibrium variables (denoted by F), and non-derivative terms, dφ/dr mφ/r, dF/dr. By Taylor expanding

ω 2 /v 2 A -k 2 m and ω 2 /v 2 A -k 2
m-1 about x = 0, the eigenmode equations can be written:

d dz (g + z) dφ m dz + d 2 φ m-1 dz 2 0, (15)
d dz (g -z) dφ m-1 dz + d 2 φ m dz 2 0, (16)
where z = 8x/ (2ε + δ). Integrating once, and defining U (z) = dφ m /dz and V (z) = dφ m-1 /dz yields:

(g + z) U + V = C m , (g -z) V + U = -C m-1 , (17)
where C m and C m-1 are constants of integration. The solution for the wave potential in the inner region is found by re-arranging for U and V , and integrating once more:

φ in m = - C m-1 + gC m 1 -g 2 tan -1 z 1 -g 2 + C m 2 ln 1 -g 2 + z 2 + const., (18)
φ in m-1 = + C m + gC m-1 1 -g 2 tan -1 z 1 -g 2 + C m-1 2 ln 1 -g 2 + z 2 + const.. (19)
The inner width of the mode is described by

z ∼ 1 -g 2 : ∆ in ε + δ 2 r m m π 4 . (20)
Coupling effects are not important in the outer region [START_REF] Mikhailovskii | Instabilities in a Confined Plasma[END_REF] so terms containing coupling parameters δ and ε can be neglected from Equations 8 and 9. Therefore, we look for solutions of L m φ m = 0. For low shear S,

φ out m = C m exp - |x| S U 1 2 - S 2 x |x| , 1, 2|x| S , (21)
φ out m-1 = C m-1 exp - |x| S U 1 2 + S 2 x |x| , 1, 2|x| S , (22)
where U (a, b, y) is the confluent hypergeometric function [START_REF] Abramowitz | Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables[END_REF]. The outer width of the mode ∆ out = r m /m is also unaffected by the poloidal inhomogeneity in the plasma density.

The inner layer solutions (in the limit z → ±∞) are asymptotically matched to the outer layer solutions (in the limit x → 0). The TAE eigenfrequency is found by equating the jump of the inner solution to the jump of the outer solution for each of the two poloidal harmonics:

φ in | x=0 + -φ in | x=0 -= φ out | x=0 + -φ out | x=0 -. (23)
The two jump conditions yield a dispersion relation, which has the solutions:

g = ±1, ± -1 + 8 4 + π 2 S 2 . ( 24
)
These solutions correspond to the upper and lower continuum, and the odd and even TAE, respectively. Returning to dimensionalised variables, the solutions of the dispersion relation give the TAE eigenfrequencies:

ω = v A 2qR 0 1 ± 1 2 ε + δ 2 1 - π 2 S 2 2 , ( 25
)
where ± corresponds to the odd and even modes respectively. There are two distinct effects related to changes in density that change the TAE eigenfrequency. First, increasing the ion density causes the Alfvén velocity to decrease, and therefore the eigenfrequency to decrease. Second, a change in the inhomogeneity of the ion density changes the coupling of the poloidal harmonics in the TAE, causing the eigenfrequency to change. As the size of inhomogeneity peaked on the low field side (δ > 0) increases, the frequency of an even TAE will decrease. Conversely, for an inhomogeneity peaking on the high field side (δ < 0), the frequency of an even TAE will increase with size of inhomogeneity.

Numerical study of the Alfvén in toroidal plasmas with poloidal modulation of the ion density

To elucidate the effect of 3D inhomogeneities in the ion mass density on the Alfvén spectrum, we first consider the simpler problem of modulation in the ion mass density. We begin by using a test equilibrium with circular poloidal cross-section, major radius R 0 = 1 m, minor radius a = 0.3 m, on-axis equilibrium magnetic field B 0 = 1 T. The aspect ratio is therefore similar to JET, but the equilibrium lacks the ellipticity and triangularity required to produce significant higher frequency gaps and modes.

Effect of poloidal modulation in ion density on the Alfvén continuum

3D ion density profiles are introduced to the ideal MHD code Stellgap [START_REF] Spong | Shear Alfvén continua in stellarators[END_REF], which is usually used to study the shear Alfvén continuum in the 3D magnetic configuration of stellarators. Introducing poloidal and toroidal modulation couples poloidal and toroidal harmonics, modifying the continuum. This is demonstrated in Figure 8a and 8b for modulation of the ion mass density:

i = i0 (1 + δ cos θ) , (26)
i = i0 (1 + δ cos ζ) . (27)
The toroidal modulation in the ion density produces new gaps where the continua of the two toroidal harmonics were degenerate in the toroidally symmetric case. The toroidal coupling does not significantly affect existing gaps, for example the toroidicity-induced gap of interest.

The poloidal modulation in ion mass density widens existing continuum gaps induced by poloidal inhomogeneity in the magnetic field caused by toroidicity. Higher frequency gaps also widen due to higher order coupling effects, O δ i for the ith gap. The size of the TAE continuum gap increases with the size of poloidal modulation of the ion density, as shown in Figure 9. The numerical result for the size of the TAE gap agrees well with the analytical estimate given by Equation 12.

Effect of poloidal modulation in ion density on Alfvén eigenfrequencies

To understand how poloidally asymmetric ion mass density profiles affect Alfvén eigenmodes, 3D density profiles are introduced to the MHD code AE3D [START_REF] Spong | Clustered frequency analysis of shear Alfvén modes in stellarators[END_REF]. AE3D is usually used to calculate the Alfvén eigenmodes present in 3D magnetic configurations of stellerators. With 3D ion density profiles incorporated, AE3D is now suitable for the study of Alfvén eigenmodes during pellet injection. Four TAEs with toroidal mode number n = 2 are found within the continuum gap at frequencies 277 kHz, 312 kHz, 358 kHz, and 403 kHz for the circular equilibrium. All four TAEs are even TAEs -the two dominant poloidal harmonics (m = 2 and 3) have the same polarity. The modes differ due to the polarity of the m = 4 and 5 poloidal harmonics, with different permutations of the ± sign for the two harmonics.

Poloidal modulation of the ion mass density is introduced, = 0 (1 + δ cos θ). The eigenvalue of each TAE decreases as the size of poloidal inhomogeneity (δ) increases, as demonstrated in Figure 10. The decrease in eigenvalue with size of poloidal inhomogeneity found by AE3D closely matches that of the analytical estimate. The frequency of the gap centre does not decrease with δ, because the δ cos θ term averages to zero over the flux surface. The decrease in frequency shown in Figure 10 is entirely due to the increased coupling introduced by the poloidal inhomogeneity in mass density.

Model for the expansion of the ionised pellet wake with heating by thermal electrons

With good agreement between the analytical estimates and numerical results for a simplified equilibrium and density profile, we move to a more realistic model. The axisymmetric equilibrium is reconstructed using the 3D equilibrium code VMEC [START_REF] Hirshman | Steepest-descent moment method for three-dimensional magnetohydrodynamic equilibria[END_REF] based on an EFIT [START_REF] Lao | Reconstruction of current profile parameters and plasma shapes in tokamaks[END_REF] reconstruction for JET discharge #49044 at time t = 17.62 s. No constraints were available for the EFIT reconstruction, because the 1999 experiment predated the installation of the motional stark emission and Thomson scattering diagnostics. Constraint of the q profile reconstruction using MHD markers was not possible because neoclassical tearing modes and Alfvén cascades were not observed in the experiment. The equilibrium density profile is considered to be static. The ion density profile due to the pellet material is calculated using a model for the expansion of a pellet wake heated by background electrons described below. This model is applied to produce a 3D density profile that is used as input for the code AE3D.

As a pellet moves through the plasma, it deposits a wake of neutral material that rapidly ionises. The cold, overdense plasmoid expands along the flux tube at approximately the wake ion sound speed c s [START_REF] Pégourié | Review: Pellet injection experiments and modelling[END_REF]. As the pellet expands, it is heated by thermal electrons.

Heating of the pellet wake by thermal electrons

Thermal electrons deposit energy in the wake due to collisional drag. The heating rate of the wake, Q, will vary along the wake depending on the local density. The energy deposited per unit volume per unit time by the background electrons due to drag is:

Q = - ∂ ∂t 1 2 m e n b v 2 f d 3 v = 4πm e n b v 4 νf dv, ( 28
)
where v is the thermal electron velocity, n b is the density of the background electrons, m e is the electron mass, f is the distribution function of the thermal electrons, ν = 8π( /m i )e 4 ln Λ/m 2 e v 3 is the collision frequency of thermal electrons with the wake [START_REF] Trubnikov | Particle Interactions in a Fully Ionized Plasma[END_REF], and is the ion mass density of the wake.

As the wake is heated, the background electrons cool down.

Hence their distribution function is described by a slowing down distribution. Assuming the background is initially Maxwellian, the distribution function of the incident electrons is:

f (v, t) = m e 2πT b 3 2 exp -3ν( , T b )t + v 3 v 3 b 2 3 , (29)
where v b = γT b /m e is the thermal velocity for background electrons at temperature T b . The distribution function f is normalised such that:

n b f d 3 v = 1.

Modelling the expansion of the pellet wake with the fluid equations

For the JET discharge under consideration, the background plasma temperature T b = 1 keV, and the background plasma density n b = 10 19 m -3 , and hence the equilibrium beta, β eq ≈ 4 × 10 -4 . Meanwhile, the wake temperature T ≈ 10 eV and wake density n w ≈ 10 23 m -3 , and so the wake beta, β w = 4 × 10 -2 1. Additionally, the wake is a good electrical conductor because the skin depth is much smaller than the size of the wake, so the frozen-in condition applies to the wake. Therefore, the wake expansion perpendicular to the magnetic field lines is inhibited by the j × B force.

The 1D wake expansion along a magnetic field line is modelled using the fluid equations for an ideal gas. The continuity, momentum and energy equations are, respectively:

∂ ∂t + ∇ • ( V) = 0 (30) ∂ ∂t ( V) + ∇ • ( VV) + ∇p = 0 (31)
∂ ∂t

1 2 V 2 + + ∇ • V 1 2 V 2 + h = Q (32)
where the following wake quantities are defined: mass of a wake ion, m i ; temperature, T ; velocity, V ; incident power density, Q; internal energy, = T /m i (γ -1); pressure, p = T /m i ; and enthalpy, h = + p/ .

By substituting the definitions for pressure, internal energy and enthalpy into Equations 31 and 32, we obtain:

∂V ∂t + (V • ∇) V + 1 m i ∇T + T m i ∇ log = 0 (33) 1 γ -1 ∂T ∂t + T (∇ • V) + 1 γ -1 (V • ∇) T = A ( 34
)
where γ = 5/3 is the adiabatic index, and A = m i Q/ is the heating rate per wake particle. A is independent of the wake mass density, so the heating rate is constant along the wake. The wake variables are normalised using characteristic quantities for a cigarshaped wake of length 4 cm:

t → tν * , L → L V * /ν * , → * , V → V V * , T → T T * , A → A ν * T b (35)
where L is the distance along the magnetic field line, * is the characteristic mass density, T * is the characteristic temperature, V * = γT * /m i is the characteristic sound speed, and ν * = 8π( * /m i )e 4 ln Λ/m 2 e v 3 b is the characteristic collision frequency.

We consider a solution of the fluid equations that assumes mass density is a Gaussian along the field line that varies in time, the wake expands linearly, and temperature is constant along the field line (due to constant heating):

(L, t) = k(t) exp -k(t)L 2 , (36)
V (L, t) = Ω(t)L, (37)

T (L, t) = T (t). ( 38
)
Equations 30, 33 and 34 reduce to the three ordinary differential equations for the functions of time k(t), Ω(t), and T (t):

dk dt + 2kΩ = 0, ( 39
)
dΩ dt + Ω 2 -2kT = 0, ( 40
) 1 γ -1 dT dt + ΩT = T b T * A. ( 41
)
Unknown functions k(t), Ω(t), and T (t) are determined numerically, and the background density and temperature are treated as constants.

Modelling the wake expansion in JET plasmas

Equations 28, 29, and 35 to 41 are applied to model the wake density, n w (r, θ, ζ) = /m i , due to a pellet injected on the inboard track of JET.

The radial density profile (displayed in Figure 11) approximates the D α emission profile [START_REF] Jones | Results of JET Pellet Fuelling Experiments with Outboard and Inboard Launch[END_REF], and the electron density profile measured by LIDAR 20 ms after pellet injection. The poloidal profile of the wake is initialised with a Gaussian that peaks at θ 0 = 5π/4 with width ∆θ = π/8. The toroidal profile of the wake is initialised as a highly peaked Gaussian with length 4 cm. The shape in the poloidal and toroidal directions at later times is determined by the model. The 1D expansion of the wake is then mapped onto the magnetic field line that corresponds to the local q value.

The wake expands from the high field side of the tokamak, and the diffuse tail expands around the tokamak to the low field side in both directions, as demonstrated in Figure 12. The corresponding ion density of a prolate spheroid (cigar-shaped) wake of length 4 cm [START_REF] Pégourié | Review: Pellet injection experiments and modelling[END_REF] is n * = 6.8 × 10 22 m -3 , although this is not resolved in our simulation due to the limited 3D equilibrium grid size.

The diffuse tail of the wake extends around the circumference of the tokamak, 2πR 0 , within ∼ 1 ms. However, a large proportion of the wake remains Equation 20 suggests the width of the inner layer -where coupling dominates -is ∆ in m 0.05 for a TAE with toroidal mode number n = 6. This is much smaller than the radial width of the pellet ∼ 0.2. Therefore Alfvén eigenmodes can be used to diagnose the local density at the mode location.

Modelling Alfvén eigenmodes in pellet injected plasmas in JET

The pellet material expands slowly compared to the Alfvénic time, therefore the non-uniform ion density profile can be treated as static. AE3D is run for a set of 3D density profiles to see how the Alfvén eigenfrequencies evolve with the expansion of the pellet wake. We use a Jacobi-Davidson QZ (JDQZ) algorithm [START_REF] Spong | Clustered frequency analysis of shear Alfvén modes in stellarators[END_REF] to calculate the Alfvén eigenmodes and the associated eigenfrequencies. This method significantly reduces the computational workload by only calculating a set of 40 eigenmodes that are clustered around a target frequency.

TAE frequency sweeping

The frequency evolution of a TAE as the pellet wake expands can be split into three stages: (1) the eigenfrequency increases slightly by ∼ kHz over a time scale of ∼ ms, (2) the eigenfrequency drops dramatically by ∼ 60 kHz over ∼ 10 ms, and (3) the frequency sweeping slows and the eigenfrequency stabilises. Henceforth, we define the time (t) from the beginning of the simulation, after the ablation and ionisation of the pellet material.

Initially the frequency of the n = 6 TAE under consideration, f 1 = 224 kHz, as shown in Figure 13. For the first 2 ms, the TAE eigenfrequency slightly increases (stage 1) as the wake expands from the initial poloidal position θ 0 = 5π/4. This increase in frequency is caused by the expansion of the wake to θ ≈ π, where the poloidal inhomogeneity in ion density produces the greatest increase in frequency. As the expansion proceeds and the wake expands to the low field side at time t = 2.0 ms, the TAE eigenfrequency begins to rapidly decrease (stage 2). The rate of frequency sweeping peaks at t = 6.6 ms. The TAE eigenfrequency stabilises as the wake homogenises in poloidal angle (stage 3). By t = 19.2 ms, the TAE frequency is stable at f 2 = 156 kHz. The frequency sweeping ends when the ion density is homogeneous in poloidal angle. Figure 14 shows the wake density averaged over toroidal angle ζ, n w ζ . Although still highly inhomogeneous in toroidal angle (as seen in Figure 12), the ion density is poloidally homogeneous to within 10% by t = 14.6 ms. The rate of change of the TAE frequency at t = 14.6 ms is ∼ 16 times slower than the maximum rate of frequency sweeping at t = 6.6 ms.

The magnitude of the frequency drop depends on the initial ion density of the pellet wake. Therefore, the experimental initial wake density can be estimated by matching the experimental TAE frequency drop after pellet injection with the numerical estimate. Using this new form of MHD spectroscopy, we estimate the initial pellet wake density, for cigar-shaped wake of length 4cm, to be n * = 6.8 × 10 22 m -3 for shot #49044 for the pellet injected at time t = 17.889 s.

Effect of q on TAE frequency sweeping

The reconstruction of the safety factor (q) profile is not accurate due to the lack of constraints on the solution generated by EFIT. This is expected to be the largest source of error in the estimation of the initial wake density. Therefore, q is varied numerically to study the effect on the TAE frequency sweeping. As q decreases, the fractional change in the TAE frequency increases. The fractional frequency change also increases with the initial wake density, n * . The effect of the inaccurate q profile on the inferred value of n * can be estimated by varying both q and n * , and matching the numerical frequency sweeping to the experiment. Figure 15 shows that a lower value of q at the TAE location will result in a higher inferred value for the initial wake density.

The frequency initially increases by ∼ kHz as the wake expands further to the inboard side of the tokamak, before decreasing dramatically. This initial frequency increase becomes less prominent for lower q values. For q 2, the initial increase in frequency disappears, and the TAE eigenfrequency instead drops immediately. The initial increase in frequency shown in the numerical results of Figure 13 is not seen for TAEs in the experiment, which suggests q 2 at the location of the TAE.

Poloidal homogenisation of ion density

In Section 3, the size of poloidal inhomogeneity, δ, was used to derive analytical estimates for the TAE eigenfrequency. The size of poloidal inhomogeneity can be estimated by fitting the toroidal angle averaged wake density profile, n w ζ , to δ cos θ. The pellet is injected from the high field side, with the initial poloidal profile highly peaked around θ 0 = 5π/4. Therefore, δ is initially a large negative number, as shown in Figure 16. As the wake expands and the ion density homogenises, δ → 0. Experimentally, the rapid increase in the flux surface average of the ion density as the pellet ablates occurs within τ ∼ ms. The resultant decrease in the TAE eigenfrequency, f (t) ∝ v A (t) ∝ B 0 / n i (t), would occur on the same timescale. This rapid drop in frequency is balanced by the large increase in frequency due to the density inhomogeneity coupling term ∝ δ(t). As δ(t) → 0, the coupling term also decreases to zero. Therefore, the drop in frequency due to the flux surface average of the ion density is slowed from a timescale of τ ∼ ms to τ ≈ 15 ms.

The TAE eigenfrequency f (t) can be estimated from the analytical expression given by Equation 25and the estimate of the size of poloidal inhomogeneity shown in Figure 16. The analytical estimate matches the numerical result from AE3D well for t > 5 ms, as demonstrated by Figure 17. For t < 5 ms, |δ| is large and the linear expression for the eigenfrequency is insufficient.

We can define time in the TAE frequency sweeping at which the ion density can be said to be homogeneous, τ h . We define this as the time at which the ratio of the maximum and minimum in the toroidal angle averaged wake density ( n w ζ ) is equal to 1.1. At this time there is no more than 10% variation in density with poloidal angle. This degree of inhomogeneity corresponds to δ ≈ 0.04.

A relationship between the experimental TAE frequency and the poloidal homogenisation time can be found from the expression for the analytical eigenfrequency (Equation 25). For an even TAE at The TAE eigenfrequency, f , decreases with time, t. The analytical estimate of the frequency (solid red) agrees well with the numerical estimate from AE3D (dashed black) for t > 5 ms. For t < 5 ms, δ is large and the linear analytical estimate breaks down.

r m /a ≈ 0.75:

f T AE (t) -f T AE (t → ∞) f T AE (t → ∞) = - δ/4 1 -ε/2 ≈ - δ 4 . (42)
For δ(τ h ) ≈ 0.04:

f T AE (τ h ) ≈ 1.01f T AE (t → ∞). (43)
Equation 43 can be used to obtain an estimate for τ h from the experimental TAE eigenfrequency. The time for homogenisation calculated from the experimental TAE frequency, τ h ≈ 14.56 ± 1.03 ms, for shot #49044. This is close to the time for homogenisation calculated numerically (Figure 13 and 17), τ h ≈ 14.58 ± 0.66 ms.

Frequency sweeping of EAEs

The frequency of elliptical Alfvén eigenmodes (EAEs) also sweep in response to the expansion of the pellet wake, as demonstrated by Figure 18. The frequency drop of the EAE is reproduced for an initial ion density n * = 4.8 × 10 22 m -3 (for a 4 cm cigarshaped wake). The initial wake density at the EAE location (r/a ≈ 0.90) is 30% lower than the initial wake density at the TAE location (r/a ≈ 0.75). The background temperature is lower at the EAE location, so less material ablates from the pellet than at the TAE location. The time evolution of the EAE frequency follows a similar pattern to that of the TAE frequency (described in Section 6.1).

The frequency sweeping of the EAE also ends when the wake density homogenises poloidally. However, poloidal homogenisation is slower at the EAE location (r/a ≈ 0.90). q at the EAE location is higher than for TAEs. Homogenisation of the wake density in the poloidal direction is slower at higher q values. Additionally, the lower background temperature near the edge also results in a slower expansion velocity. As a result, the frequency sweeping of EAEs occurs over τ h ≈ 50 ms, 2.7 longer than for TAEs.

The initial increase in frequency (stage 1 of the frequency sweeping) occurs over a longer period for higher q values due to the slower expansion to the low field side. Therefore, there is a longer delay before the drop in EAE frequency. There is a delay of 9.3 ms before frequency sweeping begins for q = 5.35. This qualitatively matches the experimental observation of a delay in EAE frequency sweeping after pellet injection. However, the experimentally observed delay was longer, averaging 12.8 ms. The difference between the numerical and experimental result may be due to lower background plasma temperature at the EAE position compared to the TAE position. This decreases the rate of power transfer from the background electrons to the wake electrons, resulting in a lower wake temperature and slower expansion.

Discussion

Only the 1D expansion of the wake along the field line is included in this study.

The cross-field movement of the wake due to the E × B force is not considered. Additionally, cross-field diffusion is not included and may be important at late times. The frequency sweeping is still well reproduced despite these limitations.

The timescale for frequency sweeping and the fractional change in frequency for TAEs and EAEs were compared for inboard and outboard pellet injection in Table 1. As shown in the previous section, these timescales are related to the time required for poloidal homogenisation of the ion density profile. This suggests the average time required for poloidal homogenisation of the ion density at the TAE location is 18 ms for inboard pellet injection.

This relationship suggests that the ion density profile produced by outboard pellet injection takes 1.4 times longer to homogenise poloidally than for inboard pellet injection, at the location of both TAEs and EAEs. The pellet density at the EAE location (r/a ≈ 0.9) takes 2.7 times longer to homogenise poloidally than the material at the TAE location (r/a ≈ 0.75), for both inboard and outboard pellet injection. This is likely to be due to the higher q at higher radii. Consequently, shallow pellets -which will expand along higher q field lines -could take longer to homogenise poloidally.

The fractional change in eigenfrequency depends on the initial ion density of the wake at the mode location. The relative fuelling efficiency of inboard and outboard pellet injection can be calculated by matching the numerical fractional frequency change to the experimental values for inboard and outboard pellet injection. Inboard fuelling deposits more 4 times more material than outboard fuelling at the TAE location. The discrepancy in fuelling efficiency with the injection track [START_REF] Lang | High-Efficiency Plasma Refuelling by Pellet Injection from the Magnetic High-Field Side into ASDEX Upgrade[END_REF] is caused by the ∇B-induced drift of the deposited plasmoid, which causes an acceleration to the outboard side. As a result, material injected on the outboard track is more easily ejected from the plasma compared to material injected on the inboard track.

EAEs do not show the discrepancy in fuelling efficiency between inboard and outboard injection. This suggests the change in ion density due to a pellet at the plasma edge is similar for inboard and outboard injection. Additionally, the fuelling efficiency for outboard injection is equally poor at the location of the TAEs and EAEs. Conversely, only the fuelling efficiency in the core is improved for inboard injection.

The experimental fractional change in TAE frequency for shot #49044 was replicated by a simulation with an initial ion density for a 4cm cigarshaped ablatant n * = 6.8×10 22 m -3 . This value agrees well with experimental observations made on different machines, n * ∼ 10 23 m -3 [START_REF] Tfr Group | Pellet injection experiments on the TFR tokamak[END_REF][START_REF] Durst | Experimental observations of the dynamics of pellet ablation on the Texas Experimental Tokamak (TEXT)[END_REF][START_REF] Sakakita | Measurement of ablation cloud parameters and plasma potential with on/offaxis pellet injection in the JIPP T-IIU tokamak[END_REF][START_REF] Müller | High β plasmoid formation, drift and striations during pellet ablation in ASDEX Upgrade[END_REF]. The value found here is slightly lower because the TAEs under consideration were not located at the peak of the radial ablation profile (r/a ≈ 0.84). Additionally, a more accurate reconstruction of the q profile may increase the inferred value of the initial ion density.

Previous simulations have suggested the plasma returns to a steady state after 28 ms [START_REF] Garzotti | Simulations of JET pellet fuelled ITB plasmas[END_REF]. This is longer than the poloidal homogenisation time found in this study, τ h ≈ 15 ms, but includes radial transport of the pellet material.

During the experiment, there is a short period ∼ 3-8 ms where the toroidal mode numbers of the Alfvén eigenmodes can not be determined. The time scale of this period is consistent with the time taken for the pellet to travel through the plasma. For inboard (HFS) injection, most of the pellet material was deposited in the first 1.15m of the track, which suggests the time of flight of the pellet was τ T oF 1.15/v pell ≈ 7 ms. Magnetic reconnection will occur as the pellet ablates and ionises [START_REF] Strauss | Magnetohydrodynamic effects on pellet injection in tokamaks[END_REF]. The broadband magnetic perturbation seen on the spectrograph during this period may be a signal of this magnetic reconnection. The effect of the wake on the equilibrium is not considered here, and therefore the effect of magnetic reconnection is outside the scope of our study.

Summary

To use Alfvén eigenmodes for MHD spectroscopy of plasmas fuelled with pellet injection, the 3D MHD codes Stellgap and AE3D were generalised to incorporate 3D density profiles. A model for the 1D expansion of the ionised pellet wake along a magnetic field line was derived from the fluid equations. This model of the wake expansion provided a 3D density profile as input for AE3D. Thereby, the time evolution of the Alfvén eigenfrequency was reproduced.

By comparing the numerical frequency drop to that of the experiment, the initial ion density of a cigar-shaped ablation region of length 4cm is estimated to be n * = 6.8 × 10 22 m -3 at the TAE location (r/a ≈ 0.75). This value is in good agreement with previously measured values. By reproducing the frequency evolution of the elliptical Alfvén eigenmode (EAE), the initial ion density at the EAE location (r/a ≈ 0.9) is estimated to be n * = 4.8 × 10 22 m -3 .

The frequency sweeping of the Alfvén eigenfrequency ends when the ion density homogenises in poloidal angle. Therefore, the time for poloidal homogenisation of the ion density after pellet ablation is τ h ≈ 15 ms for the analysed discharge. We can determine the experimental homogenisation time τ h from the TAE frequency evolution using our analytical expression. For poloidal homogenisation of the wake density to within 10%, f (τ h ) = 1.01f (t 2 ), where f (t 2 ) is the frequency at the end of the frequency sweeping. This modelling suggests that the time for poloidal homogenisation of the ion density at the TAE position is τ h = 18 ± 4 ms for inboard pellet injection, and τ h = 26 ± 2 ms for outboard pellet injection. Poloidal homogenisation of the ion density takes 2.7 times longer at the EAE location than at the TAE location for both inboard and outboard pellet injection.

Figure 1 :

 1 Figure 1: The poloidal cross-section of JET, showing the magnetic flux surfaces (blue), vertical lines of sight of the far infrared interferometer (black), LIDAR line of sight (green), inboard pellet injection track (red) and outboard pellet injection track (magenta) for shot #49044 at time t = 17.89 s.

Figure 3 :

 3 Figure 3: The electron density n e (black solid) as a function of radius (r) normalised to minor radius (a), as measured by LIDAR at time t = 19.63 s in discharge #49032. Positive and negative radii correspond to the outboard and inboard side, respectively. The difference between the high field side and low field side is shown in dashed red.

Figure 4 :

 4 Figure 4: Heating waveforms and plasma parameters for JET discharge #49044. Top: ICRH (blue solid) and NBI (red dashed) heating power, P . Middle: Electron temperature T e as measured by ECE radiometry at R = 3.06, 3.29, 3.40, 3.54, and 3.78 m (from top to bottom). Bottom: Line integrated electron density n e (R I )dZ as measured by far infrared interferometry along the line of sights at radii R I , shown in Figure 1.

Figure 5 :

 5 Figure 5: (a) Magnetic spectrograph showing the amplitude of the magnetic perturbation caused by a TAE before and after pellet injection in discharge #49044; (b) a phase magnetic spectrograph showing the toroidal mode numbers of the same TAE. The pellet was injected on the inboard track at time t = 17.89 s.

Figure 6 :

 6 Figure 6: Magnetic spectrograph showing the amplitude of the magnetic perturbation caused by a TAE before and after pellet injection in discharge #49221. The pellet was injected on the outboard track at time t = 18.342 s.

Figure 7 :

 7 Figure 7: Magnetic spectrograph showing the amplitude of the magnetic perturbation due to an EAE before and after inboard pellet injection in discharge #49043 at time t = 19.615 s.

Figure 8 :

 8 Figure 8: The Alfvén continua for the n = 2 (yellow) and n = 3 (purple) toroidal harmonics for a plasma with circular poloidal cross-section and (a) toroidal or (b) poloidal modulation of the ion mass density, with the size of inhomogeneity δ = 3/4. The continuum for a plasma with homogeneous mass density is displayed in black.

Figure 9 :

 9 Figure 9: The size of the n = 2 TAE continuum gap increases with the size of poloidal modulation, δ. The frequency at the top and bottom of the TAE gap is denoted by ω + and ω -, respectively. The numerical estimate (solid blue) agrees well with the analytical estimate (dashed orange).

Figure 10 :

 10 Figure 10: The eigenvalue (f ) decreases with increasing size of poloidal modulation (δ) for poloidal modulation peaking on the low field side. The trend predicted analytically in Equation 25 (red dashed) agrees well with the results from AE3D (black dots). Modes at the top and bottom of the figure (f > 410 kHz and f < 235 kHz) are continuum modes.

Figure 11 :

 11 Figure 11: The radial ion density profile of the pellet wake n w (r), where the density is normalised to the maximum value at r max ≈ 0.84.

Figure 12 :

 12 Figure 12: The 3D ion density profile of the pellet wake at times (a) t = 0.66 ms, (b) t = 5.30 ms, (c) t = 14.58 ms, and (d) t = 25.85 ms after ablation. Colour shows log 10 (n w /n b ), where n w is the wake density and n b = 10 19 m -3 is the equilibrium density, at the position of the maxima of the radial density profile. The initial density for a cigar-shaped wake of length 4 cm would be n * = 6.8 × 10 22 m -3 . The pellet material expands along the q = 3.35 field line in a plasma with background temperature T b = 1 keV.

Figure 13 :

 13 Figure 13: As the wake expands around the tokamak, the frequency of the n = 6 TAE drops. Numerical estimates of the TAE eigenfrequency (black) are overlaid on the experimental magnetic spectrograph for shot #49044. The eigenfrequency is calculated for four estimates of the initial ion density n * .

Figure 14 :

 14 Figure 14: The logarithm of the wake density averaged over toroidal angle ζ normalised to the background density n b , log 10 ( n w ζ /n b ), as a function of poloidal angle, θ.

Figure 15 :

 15 Figure15: The inferred value of the initial wake density n * as a function of the safety factor q at the TAE location for shot #49044.

Figure 16 :Figure 17 :

 1617 Figure16: The size of poloidal inhomogeneity, δ, is initially a large negative number due to the initial position of the wake on the high field side. δ → 0 as the wake expands with time, t.

Figure 18 :

 18 Figure 18: The EAE frequency drops as the wake expands along the q = 5.35 magnetic field line. The numerical reconstruction of the EAE frequency sweeping (black) is overlaid on the experimental magnetic spectrograph for shot #49043. The frequency drop matches for an initial ion density n * = 4.8 × 10 22 m -3 (for a 4 cm cigar-shaped wake).

Acknowledgments

This paper benefited from comments by D Keeling and J Mailloux. This work has been carried out within the framework of the EUROfusion Consortium and has received funding from the Euratom research and training programme 2014-2018 and 2019-2020 under grant agreement No 633053 and from the RCUK Energy Programme [grant number EP/P012450/1]. The views and opinions expressed herein do not necessarily reflect those of the European Commission. This research was also supported by the U.S. Department of Energy under Grant No. DE-FG02-04ER54742 (IFS).


Sepideh Stewart
email: sepidehstewart@ou.edu

Jonathan Epstein
email: jepstein@ou.edu

Jonathan Troup
email: jtroup@ou.edu

David Mcknight

A mathematician's deliberation in reaching the formal world and students' world views of the eigentheory

Keywords: three worlds of mathematical thinking, eigentheory, concept images, reflections

) three world model, to follow the mathematician's and his students' movements between the three worlds. The study revealed that despite the mathematician's efforts in demonstrating a more holistic view of the concepts, many students found linear algebra very abstract and gravitated more toward symbolic thinking.

Theoretical Background

Building on [START_REF] Tall | Concept image and concept definition in mathematics with particular reference to limits and continuity[END_REF] notions of concept images and concept definitions, Vinner (1991, p. 69) added that, "We assume that to acquire a concept means to form a concept image for it. To know by heart a concept definition does not guarantee understanding of the concept. To understand, so we believe, means to have a concept image". Developing these two notions further, [START_REF] Tall | Perceptions Operations and Proof in Undergraduate Mathematics, Community for Undergraduate Learning in the Mathematical Sciences[END_REF]2013) three-world model of mathematical thinking (embodied, symbolic, and formal) endeavored to lay out an individual's mathematical journey from childhood to research mathematician. According to [START_REF] Tall | Perceptions Operations and Proof in Undergraduate Mathematics, Community for Undergraduate Learning in the Mathematical Sciences[END_REF], the embodied world is based on "our operation as biological creatures, with gestures that convey meaning, perception of objects that recognize properties and patterns…and other forms of figures and diagrams" (p. 22). In Tall's (2010, p. 22) words, "The world of operational symbolism involves practicing sequences of actions until we can perform them accurately with little conscious effort. It develops beyond the learning of procedures to carry out a given process (such as counting) to the concept created by that process (such as number)". Tall defines thinking in the formal world as that which "builds from lists of axioms expressed formally through sequences of theorems proved deductively with the intention of building a coherent formal knowledge structure" (p. 22). The overall goal of the first author's research program is to: (a) Examine the three-world model of mathematical thinking as a possible lens for understanding how mathematical content is conveyed by a mathematician-teacher with a view toward serving students' learning; (b) Develop and extend the three-world model further. In this study we are focusing on goal (a).

Many researchers have maintained that reflection is an essential part of teaching. For example, according to [START_REF] Dewey | How we think: A restatement of the relation of reflective thinking to the educative process[END_REF], reflection is "active, persistent, and careful consideration of any belief or form of knowledge in the light of the grounds that support it and the further conclusions to which it tends" (p. 9). Using Tall's (2013) model, Stewart, Thompson, and[START_REF] Stewart | Navigating through the mathematical world: Uncovering a Geometer's thought processes through his handouts and teaching journals[END_REF] investigated a mathematician's (and co-author) movements between the three worlds while teaching algebraic topology. The instructor reported that students experienced the most difficulty in moving from the embodied world into the formal world. Believing the struggle would stimulate mathematical growth in his students, this instructor "refused to give students proofs that were pre-packaged. More specifically, he desired to provide students with intuitions and pictures that would help them understand the conceptual nature of the proof and ultimately lead them to it" (p. 2262). In a similar study, [START_REF] Stewart | Teaching Linear Algebra: Modeling One Instructor's Decisions to Move between the Worlds of Mathematical Thinking[END_REF] examined a mathematics educator's (and co-author) movements as well as decision making moments while teaching linear algebra. In a different study, [START_REF] Stewart | Moving between the embodied, symbolic and formal worlds of mathematical thinking in linear algebra[END_REF] created a set of linear algebra tasks designed to help students move between the three worlds. These studies indicate that movements between the worlds is worthy of ongoing investigation.

Since the present study was focused on eigentheory, naturally recent literature on students' reasoning with eigentheory is acknowledged. While several studies have highlighted students' difficulties learning the eigenvalue and eigenvector concept, they have also provided methods whereby these difficulties can be alleviated. For example, [START_REF] Thomas | Eigenvalues and eigenvectors: Embodied, symbolic and formal thinking[END_REF] investigated students' conceptual understanding of eigenvalues and eigenvectors. They discovered that students appeared confident with symbolic procedures (i.e., calculating the characteristic polynomial), but not embodied ones (linking diagrams and eigenvector properties). While the students referred to "being stretched," or "unchanging direction," when describing eigenvectors, it appeared very few were able to implement this definition in context. [START_REF] Thomas | Eigenvalues and eigenvectors: Embodied, symbolic and formal thinking[END_REF] also noted that despite their relative strength in symbolic manipulation, the students were not able to give justifications for the change of the equation into the equivalent equation . They claimed that thinking about eigenvectors as invariant appeared to be a useful embodied concept for the students, even while they operated within the symbolic world. Gol Tabaghi and Sinclair (2013), and [START_REF] Caglayan | Making sense of eigenvalue-eigenvector relationships: Math majors' linear algebra-Geometry connections in a dynamic environment[END_REF], reported that usage of dynamic geometric environments (DGEs) appeared to help students learn the eigenvectors and eigenvalues concepts, perhaps by encouraging related embodied thought as [START_REF] Thomas | Eigenvalues and eigenvectors: Embodied, symbolic and formal thinking[END_REF] suggested. [START_REF] Salgado | Teaching eigenvalues and eigenvectors using models and APOS Theory[END_REF] claimed that via a model designed to capture student interest, the students constructed an object conception of eigenvalues and eigenvectors. They presented modeling as a powerful tool that can elucidate the problems students are having with the associated concepts, make learning the concepts easier and more approachable for the students, and supply students with a broader range of strategies with which to approach problems. These studies suggested that learning eigenvalues and eigenvectors requires students to be able to consider multiple aspects of the concept: geometric, algebraic, and structural. The presentation of eigenvectors and eigenvalues geometrically as a stretching direction and a stretching factor seems to be considered particularly useful. Thus, the research questions guiding this study were:

(a) When and why did the teacher decide to move between the three worlds? (b) Were the students willing to move with him and what were their challenges? (c) What were some of the challenges for the mathematician-teacher in moving the class to the formal world?

Methods

This qualitative narrative study is intended to examine a linear algebra instructor's and his students' mathematical thought processes using [START_REF] Tall | How humans learn to think mathematically: Exploring the three worlds of mathematics[END_REF] model. The study took place over the course of a semester at a Southwestern research university in the US. The research team consisted of a mathematician specializing in differential geometry (the instructor, postdoctoral fellow, and coauthor), two mathematics educators, and an undergraduate research assistant. The analysis of the data focused on the instructor's observations, as recorded through journal entries, over a five-day period, while implementing tasks from the Inquiry-Oriented Linear Algebra (IOLA) curriculum [START_REF] Wawro | Inquiry oriented linear algebra: Course materials[END_REF]. These 50-minute classes were structured around a sequence of four IOLA tasks using the ideas of "stretch direction" and "stretch factor" of a linear transformation to develop the formal notions of eigenvector and eigenvalue. Several of the requisite concepts, such as bases, coordinates and matrix representations of linear transformations, were covered earlier, so that the IOLA sequence could be used. The instructor was introduced to the three-world model, but taught the class as he originally planned and the research team never made any comments on how or what to teach. Introducing the three worlds was done to help him articulate his thoughts and to provide a language for reflection in his teaching journals. His dual role as a teacher as well as a researcher was indispensable in every aspect of this research.

In addition, the instructor presented Tall's (2013) framework to his class. Throughout the semester, the instructor recorded his thought processes as well as observations on how his class reacted to his teaching. He also met with the research team regularly throughout the semester and the following summer to discuss these experiences and reflections. This allowed the research team to triangulate data via member checking with the instructor directly and additionally afforded him ample time to share a wide variety of teaching experiences, as well as his reasoning while making these decisions. To collect additional data on the student's perspective, the research team administered a survey (see Figure 1), and provided some excerpts defining the three worlds. The students (16 from a class of 30) were mainly engineering students and were already familiar with the notions of embodied, symbolic and formal. To analyze the instructor's journal, in line with a narrative study, the research team performed a retrospective analysis of the journal [START_REF] Creswell | Qualitative Inquiry and research design: choosing among five approaches[END_REF] by iteratively coding the data. The team started with a combination of themes developed from the previous study [START_REF] Stewart | Teaching Linear Algebra: Modeling One Instructor's Decisions to Move between the Worlds of Mathematical Thinking[END_REF] and an open coding [START_REF] Strauss | Basics of qualitative research: Grounded theory procedures and techniques[END_REF] scheme to allow for the possibility of discovering new themes unique to this study. The themes were: Teaching, Students, Class Activities, Math (instructor's math, students' math), Reflection, and the Three Worlds. By instructor's math, we mean the math he was doing and talking about, and by students' math, we mean his reflections on students' mathematical abilities, and mathematical conversations in class. Under each theme we considered further fine-grained ideas and assigned codes. For Teaching, we considered IOLA tasks (TtIOLA); other tasks (TtOther); pedagogical decisions (Tpd); responses (Tr); lecture style (Tsl); group work style (Tsgw); and class discussion style (Tscd). The themes that emerged from students' survey were, movements between the three worlds as well as static presence in each world. Other themes were students' elaboration on the worlds' appearance in their own line of thinking, or the course itself, as well as the connections between the concepts.

Results

In analyzing his 5-day teaching segments, we examined the instructor's (a) movements between [START_REF] Tall | How humans learn to think mathematically: Exploring the three worlds of mathematics[END_REF] worlds, (b) pedagogical decisions, and (c) reflections on self and students. In the students' survey we examined (a) their views and preferences on each world, and (b) their thoughts on eigenvalues and eigenvectors and their connections to other concepts. The IOLA unit on eigentheory consists of four one-page worksheets designed to introduce the notions of eigenvalue and eigenvector through the ideas of "stretching factor" and "stretching direction" for a linear transformation.

An analysis of the instructor's journals on teaching eigenvalues and eigenvectors

Day 1: Blending embodiment and symbolism. The first IOLA task started by describing a linear transformation geometrically (see Figure 2), in terms of "stretch directions" and "stretch factors". First, the students were asked to sketch the image of a figure "Z" centered at the origin. Next, they were asked to sketch the image of two vectors and then compute the images numerically. Lastly, they were asked to produce a matrix representation of the linear transformation. This task is primarily situated in the embodied and symbolic worlds. The students must grapple with the action of the linear transformation before considering a matrix representation.

The instructor noted that "They had a lot of trouble with this. So, after a few minutes, we went through the exercise collectively." Among the concepts that he recalled were "for every linear transformation the zero vectors gets sent to the zero vector," and "points are identified with vectors." In order to show how the image of any vector can be computed, "we converted the two vectors into linear combinations of vectors in the stretching direction. Then used the linearity of the transformation to find their images. I'm not sure if this made sense to them." Day 2: Embodied, symbolic and formal thinking. The second IOLA task continued with the same linear transformation introduced in task 1. It presents R 2 with the standard coordinate grid (referred to as the "black" coordinates) overlaid on the one determined by the stretch directions (referred to as "blue" coordinates) together with a discrete collection of points (see Figure 3). The students were asked to 1) label each point with its "black" and "blue" coordinates, 2) determine two matrices that will systematically rename points from the blue coordinate system as points in the black coordinate system and vice versa, and 3) compute the images of new points in the black and blue coordinate systems. The instructor quickly noted that "most students don't have a facility with coordinate vectors. Even if they remembered how to compute the coordinates of a vector, they have not connected that to the picture of a grid determined by a basis." After expressing some frustration that "Even after the class, another student asked…how was I finding the 'blue' coordinates," he related that "his follow-up question was very good: what does the symbol 'a' represent when we write [a]_{blue}. This indicated that he was starting to abstract the notion of a vector and separate it from its various coordinate representations." Prior to the class, the instructor made the pedagogical decision to present the formal definitions of eigenvalue and eigenvector after this task, and the last sentence of the journal entry for this day is, "Finally, I was able to define eigenvalue and eigenvector." Day 3: Reinforcing Day 2. The instructor made the pedagogical decision to use day 3 to recap and consolidate the various embodied, symbolic and formal aspects of eigentheory that the students have thus far encountered. His journal entries for this day contain almost no mention of the students, but instead focused on the mathematical connections that he aimed to convey to the students. First, he showed how the two matrix representations of the linear transformations are related by conjugation by the change of coordinates matrix. Then, to demonstrate how "The 'stretch factors' and 'stretch directions' correspond to eigenvalues and eigenvectors, respectively," he used GeoGebra to show "how you can spot the stretch directions by moving around a vector, or by looking at what happens to the unit circle. Through several examples, he "presented the definition of eigenvalue/vector as a way to find the stretch factors and directions." He even pushed these ideas into the realm of infinite dimensional vector spaces by considering differentiation operators on function spaces. Day 4: Symbolic and formal thinking. He returned to the IOLA sequence with task 3 (a standard textbook exercise). For three distinct two-by-two matrices, the students were asked to 1) find the stretch factors given the stretch directions 2) find the stretch directions given the stretch factors, and 3) find both the stretch factors and directions. The instructor "expected most students launch themselves into finding the eigenvalues and eigenvectors." But, "Instead, I was surprised to see how many were unsure where to start." He made the pedagogical decision to guide the class and used the opportunity to present several connections. Day 5: Symbolic and formal thinking. The fourth IOLA task considers a single linear transformation of R 3 , presented as a matrix. It is found that a certain stretch factor has two stretch directions, i.e. the corresponding eigenspace is two-dimensional. The third and final part posed a rather provocative question: given that 2 and 3 are stretch factors and the former has two distinct stretch directions, could there be additional stretch factors? He observed that "every student's work that I saw was the same. To decide if there was another eigenvalue or stretch direction they all computed the characteristic polynomial to see if there was another root." Although this was a valid approach he was eager to present a more sophisticated approach that connected to earlier concepts. "I then presented a solution that crucially uses the fact that all three eigenvectors form a basis for R 3 . I did not get very much feedback from the class on whether they were internalizing this." In summary, the instructor used four IOLA tasks, supplementing with lecture where necessary. His goal was to build a concept image of eigentheory by presenting the fundamental notions via all three worlds of mathematical thinking. Moreover, he emphasized the formal definition of eigenvalues. For activities situated in the embodied and symbolic worlds the students were encouraged to explore independently, while the active guidance provided by the instructor was primarily to connect those worlds back to the formal. Although, many of his pedagogical decisions were dictated by time pressure, he made sure each day to bring the class closer and closer to the formal world.

An analysis of the student survey

In response to describing the meaning of eigenvalues and eigenvectors most students gave a symbolic view of the concepts (9/16). Most students also included a symbolic representation, namely the equation , with the exception of S14 and S16, who only used words. We noticed that their reasoning within this world (symbolic) were mostly reasonable.

S11:

det(λI-A) = 0. An eigenvalue is the variable λ in the equations det (λI -A) = 0 or in the equation Lx = λx. Eigenvector is the variable x in Lx = λx.

Eigenvector of L associated to the eigenvalue λ.

S16:

An eigenvector is a vector associated with a matrix. An eigenvector multiplied by this matrix equals a scalar multiple of the eigenvector. The values that eigenvector is scaled by are called eigenvalues.

The symbolic world seemed the most comfortable world for students in which to express their thoughts, and some even felt that moving from the formal world to the symbolic world "solves most problems" (S10). In general students used more positive language in describing this world. For example one students wrote, I feel as though I thrive in the symbolic world. For linear algebra, the matrices and vectors make the most sense to me in symbolic form (S14). The students' views on the usefulness of the embodied world were mixed. For example, one student wrote, "I am able to understand easier if I can visualize something" (S15). "The embodied world really does not help me understand what is going on in problems" (S14). At one end of the spectrum, a student wrote, "my actions in the symbolic world are determined by understanding in embodied world. I refer to the embodied world to initially understand the formal world, but usually don't go back." (S10). Another student claimed that "the embodied world is useful for getting the big idea" (S12) and understanding the question before moving to the symbolic. In agreement another student wrote: "we took complex problems and simplified them to embody and represent a bigger picture" (S9). At the other end of the spectrum, one student went so far as to isolate the embodied world completely, writing that the "embodied and symbolic worlds" (S16), seldom interacted in her/his viewpoint. Students did not elaborate as much on the formal world. One student believed that the formal world is useful, or at least prevalent in class, but difficult to work with (S12). Others believed that "most of the class was abstract proofs" (S8), "learning concepts required a lot of formal proofs" (S13), and "linear algebra relies on a formal understanding" (S16). As for moving between the worlds, one student wrote: "I start in symbolic and move to formal for proofs" (S14). Although, most comments were directed toward the class or the course in general, one student conveyed her/his thoughts as, "Formal is the worst that I have a problem with" (S15). In connecting between the concepts, students referred to determinants, linear operators, linear transformations, basis, special solutions, system response, system stability, and even music.

Discussion and concluding remarks

The mathematician-teacher in this study has negotiated the mathematical journey himself, and knows the path well. He viewed the formal world as the destination and wished to bring his students there. However, this was not straightforward and required time and perseverance. Since the instructor's objective was a mathematical treatment of eigentheory, he used IOLA tasks to present a web of connections surrounding the formal definitions. Although, he valued the embodied and symbolic worlds as part of the concept image, the instructor's goal of reaching the formal world became apparent in many of his journal writings. For example, his decision to present the definitions of eigenvalue and eigenvector at precisely the midpoint of the unit reflected their significance, representing a single idea uniting the various notions from all three worlds. A mathematical understanding of eigentheory (to him) involved primarily the definitions and how those definitions manifested themselves in the embodied, symbolic, and formal worlds. For example, he was able to think of a definition of eigenvector in symbols as described by the equation , in the embodied world as a picture of an image vector collinear with its preimage, and additionally various properties related to eigenvalues and eigenvectors in the formal world. The instructor believed the more connections between eigentheory and other linear algebraic concepts that he can convey to the students, the more robust their concept image. In [START_REF] Tall | How humans learn to think mathematically: Exploring the three worlds of mathematics[END_REF] view "formal mathematics is more powerful than the mathematics of embodiment and symbolism, which are constrained by the context in which the mathematics is used" (p. 18). It was interesting to notice that even a mathematician that values all three worlds of mathematical thinking, still gravitates toward the formal world as the most important part of a mathematical concept. He seemed to consider them complementary to the formal world. Also, what appears "rote" and part of his "everyday" mode of thinking is completely foreign to the typical undergraduate linear algebra student. Hence the connections between the formal definitions and surrounding concepts that appeared so strong to the instructor were quite tenuous with the students. Although the IOLA curriculum was utilized to highlight the embodied world and draw out its connections to the other two worlds of mathematical thinking, nonetheless, the students gravitated toward symbolic thinking. We speculate on two reasons for this. First, living comfortably in all three worlds and moving between them is a big hurdle for most novice students of mathematics. Second, a significant part of students' motivation is derived from "answering the question". As such, students may value the world that seems most helpful in doing so. From the mathematician-instructor's point of view the world most amenable to evaluation is the symbolic world. Hence the class setting establishes an incentive for students to remain safely in the symbolic world, and any desire to branch out must be internally motivated.

Our study suggests exploring ways of motivating students to achieve a more holistic understanding of linear algebra concepts across the three worlds. The three-world model gave the mathematician-instructor a language that he felt accurately reflected his own thought processes. It also empowered the students by providing them a language to express their mathematical thought processes.

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  Figure 1: The Survey

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  Figure 2: IOLA Task 1