Nidia Maldonado-Carmona

Murielle Baltazar

Joanne L Fothergill

Nelly Henry

M Palomares

Design of a millifluidic device for the evaluation of blue-light disinfection on Pseudomonas aeruginosa PA01 biofilm

Keywords: biofilm, blue-light irradiation, millifluidic device

Significance:

For the first time to our knowledge, a millifluidic device has been designed to observe the effects of blue-light irradiation in Pseudomonas aeruginosa PA01 bacterial biofilm.

Approach: A bacterial biofilm is formed under controlled flow of nutrients in microfabricated channels settled on a microscope stage. The setup is devised to apply defined irradiation and evaluate the photo-killing efficiency in situ using video-microscopy and fluorescent probes.

Results:

We investigated bacterial survival after delivering a precise, controlled and spatially defined blue light dose upon growing biofilm. We demonstrate in this report that the combined use of constitutive GFP and propidium iodide in a millifluidic setup enables to evidence light-induced mortality at the single cell and community level.

Conclusions:

Real-time monitoring of a bacterial biofilm growing in a millifluidic device after irradiation opens an avenue for a better understanding of the biofilm community response to blue-light, permitting researchers to approach photokilling efficiency of biofilm as a dynamic process.

Introduction and Background

Pathogenic biofilms proliferation represents a burden for human health and a challenge for the development of antibiotic alternatives [START_REF] Verderosa | Bacterial Biofilm Eradication Agents: A Current Review[END_REF] . Blue-light irradiation intends to use bacterial tetrapyrrolic metabolites (i.e., protoporphyrin IX) as photosensitizers, together with specific blue-light irradiation, for reactiven oxygen species generation [START_REF] Martegani | Effect of blue light at 410 and 455 nm on Pseudomonas aeruginosa biofilm[END_REF] . Millifluidic devices have already been used to describe the growth and complexity of biofilm communities [START_REF] Benyoussef | Flagellar Motility During E. coli Biofilm Formation Provides a Competitive Disadvantage Which Recedes in the Presence of Co-Colonizers[END_REF] , and, together with live/dead fluorescent labelling, it could be used to analyze the cellular and community response to blue-light irradiation. As part of the Light4Lungs initiative, the present work establishes for the first time the design of a millifluidic setup for biofilm irradiation.

Develop a millifluidic device for the growth and real-time observation of Pseudomonas aeruginosa PA01 biofilm development and response to in situ blue-light irradiation, using video microscopy and live/dead specific fluorescent probes.

Methods

Bacteria strains, culture media and reagents: Green-fluorescent Pseudomonas aeruginosa PA01-GFP [START_REF] Davies | Temperate phages both mediate and drive adaptive evolution in pathogen biofilms[END_REF] was provided by the University of Liverpool. Upon utilization a smear of the frozen stock was inoculated into M9 media [START_REF]M9 minimal medium (standard)[END_REF] supplemented with casaminoacids 2 g/L and propidium iodide (PI) 3 µM (M9PI). Device design and construction: The device was constructed as elsewhere described [START_REF] Benyoussef | Flagellar Motility During E. coli Biofilm Formation Provides a Competitive Disadvantage Which Recedes in the Presence of Co-Colonizers[END_REF] . Polydimethylsiloxane mixture (RTV615A+B 9:1 mixture, Momentive Performance Materials) was used to build a template with channels, which were then drilled and bound to glass coverslips using an oxygen plasma cleaner (Harrick) (Figure 1A). Imaging and light irradiation setup (Figure 1B): Samples were observed using an inverted NIKON TE300, equipped with motorized x, y and z displacements and shutters, equipped with a Hamamatsu ORCA-R2 EMCCD camera. Acquisitions: brightfield images were collected in direct illumination; PI channel, excitation: 550 nm, m-Cherry filter (Ex. 562/40nm, DM. 593, Em. 641/75); GFP channel, excitation: 490 nm, green filter for GFP filter (Ex. 482/35, DM 506 nm, FF01-536/40). Irradiation: lightsource: 405 nm laser diode (LDI-7, 89North); power: 0.0161 W (measured in situ with a powermeter (Thorlabs)); irradiation area: 0.00333 cm 2 ; irradiation time: 111 seconds; light dose: 536.7 J/cm 2 . Biofilm formation: Overnight bacteria cultures (500 rpm, 37 ºC) were diluted to obtain an optical density of 0.1, and further incubated during two hours. The culture was diluted in 5 ml of M9PI (OD 0.1, ~10 7 CFU/mL). Bacteria were injected into the millifluidic device and incubated at 37 ºC for 30 minutes. After incubation, the channel was connected to a syringe pumping M9PI at a 0.5 ml/hr flow rate and acquisitions started (Figure 1C).

Results

Previous efforts made in our group have unveiled the utility of using millifluidic devices for monitoring biofilm communities in a controlled environment [START_REF] Benyoussef | Flagellar Motility During E. coli Biofilm Formation Provides a Competitive Disadvantage Which Recedes in the Presence of Co-Colonizers[END_REF] . This device provides a constant supply of nutrients, and a water-biofilm interface, which needs to be considered when evaluating the biofilm's photokilling efficiency [START_REF] Treghini | Aerosol-based antimicrobial photoinactivation in the lungs: an action spectrum study[END_REF] . PA01-GFP biofilm development was observed close to the PDMS surface, and followed up during the different development stages, cells were seen as individual cells, growing to clusters and finally forming a continuous biofilm layer. Biofilm was irradiated at two hours (Figure 1D) and four hours of development in the channel (Figure 1E). Upon irradiation GFP fluorescence was dimed, due to photobleaching. Meanwhile, PI-labelled bacteria appeared in the red channel, indicating that PI is able to permeate through sensitized membranes and bind to nucleic acids inside the cell, a process closely correlated with bacterial death [START_REF] Nikparvar | A Diffusion Model to Quantify Membrane Repair Process in Listeria monocytogenes Exposed to High Pressure Processing Based on Fluorescence Microscopy Data[END_REF] . PI labelling reaches a maximum value after about one hour, suggesting membrane permeabilization due to blue-light induced damage is not an instantaneous process. (Figure 1D). At a community scale, the in situ irradiation is able to sensitize a precisely defined spot in the biofilm (Figure 1E), which appears defined by the PI-labelling in the irradiation site. At the geometry and power used, most of the cells in the field of view are PI-labelled after blue-light irradiation (Quantitative analysis to be published). Neighboring non-irradiated biofilm is able to proliferate and our results show that upstream bacteria are also able to colonize the irradiated site, leading to the recovery/establishment of a biofilm structure post-irradiation (data not shown).

Conclusion

This work presents for the first time the combination of a millifluidic device and blue-light irradiation, aiming for the disinfection of pathogenic biofilm. The devised setup enables to deliver a spatially defined light dose, enabling users to deliver light at precise timepoints. The real-time microscope monitoring allows to follow up the biofilm development at cellular and community levels, enabling us to observe the dynamics that follow up after light irradiation, a phenomena yet to explore. Now, it is possible to tune and analyze the importance of different parameters (i.e., thickness of the biofilm, length of the irradiation, power used for the irradiation, geometry of the irradiation spot), parameters which most of the time are neglected and/or impossible to control in traditional approaches. The number of applications of this device are likely to grow with time, enabling to assess different light sources and exogenous photosensitizers, as an example.

Figure 1 .

 1 Figure 1. A, scheme of the plasma-mediated binding of the PDMS patch and the glass slide. B, scheme of the acquisition and irradiation conditions. C, scheme of PA01-GFP biofilm formation and constant washing of unbound cells. D, follow-up of bacteria behavior and development before (gray square) and after (white square) light irradiation at 2 hours of biofilm development (40X magnification: 40XSFluor (NA: 0.9, WD: 0.3 mm)). E, 4X magnification (4XPlan (NA: 0.1, WD: 30 mm)) of the millifluidic channel, in non-irradiated position (gray square) and an irradiated position (white square) at 4 hours of development; the fluorescence intensity was measured along a line (dotted line).

Disclosures:

The authors declare no conflict of interest.

Acknowledgments

This research is part of the Light4Lungs project, which has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement nº 863102.


Nasr-Eddine Ouldei Tebina

Nacer- Eddine Zergainoh

Guillaume Hubert

Paolo Maistri

Simulation Methodology for Assessing X-Ray Effects on Digital Circuits

Keywords: leakage, TID, Spectre, Secure designs

Recently, X-Rays have proven to be an interesting and feasible mean to perform fault injection attacks against secure implementations. Their wavelength allows targeting single elements even with most recent fabrication technologies, and their high penetration power does not demand to invasively prepare the device for the attack. Unlike operations in harsh environment, a malicious attacker can choose the targeted region at will, from a single cell up to large regions. This aspect forces designers to consider hardening against X-Rays under a different perspective. In this work, we will present a design-time simulation flow, that addresses how the electrical characteristics of the logic are changed when irradiated, from the perspective of injected faults and of biased power consumption.

I. INTRODUCTION

It is well-known that a huge amount of digital information requires some protection from unauthorized use, either in terms of confidentiality or authentication features. Several cryptographic algorithms have been proposed from the very beginning in order to fulfill these requirements.Nonetheless, algorithms have to be implemented either in software or hardware, which can be vulnerable to physical attacks. Among them, Fault Attacks constitute a powerful and effective way to extract information from a system. These attacks are based on the fact that the device, subject to external perturbations, may exhibit an altered behavior (for example, a corrupted output) that might be leveraged to extract confidential data [START_REF] Boneh | On the importance of checking cryptographic protocols for faults (extended abstract)[END_REF], [START_REF] Piret | A differential fault attack technique against SPN structures, with application to the AES and KHAZAD[END_REF]. Several techniques exist, from perturbation on global inputs (clock or power supply [START_REF] Korak | On the effects of clock and power supply tampering on two microcontroller platforms[END_REF]), to more localized ones such as laser [START_REF] Dutertre | Laser fault injection at the cmos 28 nm technology node: an analysis of the fault model[END_REF] or electromagnetic [START_REF] Dehbaoui | Electromagnetic transient faults injection on a hardware and a software implementations of AES[END_REF] pulses.

Recently, X-Ray illumination has been proposed as a mean to inject calibrated faults into a device as a security threat. Although already known and used in the spatial domain to reproduce harsh environments (eg, natural radiations), their application for physical attacks has been first proposed in [START_REF] Anceau | Nanofocused x-ray beam to reprogram secure circuits[END_REF], where the authors exploited a nanofocused beam available at a synchrotron facility to target a single transistor. Later, the same device was attacked using a laboratory source [START_REF] Maingault | Laboratory x-rays operando single bit attacks on flash memory cells[END_REF], demonstrating that attacks are possible, at a larger scale, even without very high-end facilities.

The challenges that X-Ray Fault Injection (XRFI) presents to secure designers are different from what they had to deal *Institute of Engineering Univ. Grenoble Alpes with so far. Unlike common fault injection techniques, the threat model of X-Rays is completely different on both spatial and temporal scale. Glitches and laser or EM pulses are precisely synchronized in time with the target device, and usually limited to one or very few clock cycles. XRFI, on the other hand, is a slow and continuous process, progressively affecting the characteristics of the circuit. Most importantly, the spatial granularity of XRFI may be completely tunable by the attacker: the first experiments were mostly focused on very few gates, but in principle the attacker can target specific regions or IPs in the design, under the assumption that the layout can be approximately known. As a term of comparison, the attacker has a complete control on the fault location when employing laser or EM pulses, but the targeted region is limited (due to resolution and focusing issues, and to the number of possible spots); no spatial control at all is possible when using glitches, where fault propagation is largely driven by layout and fabrication process. Protecting against XRFI might be achieved through well-known hardening approaches [START_REF] Tebina | X-ray fault injection: Reviewing defensive approaches from a security perspective[END_REF], but at a large cost and with little knowledge of new potential vulnerabilities.

For these reasons, it is important for the designer to have the right tools in order to analyse XRFI as soon as possible in the design flow. In this paper, we present a methodology based on Ray-Spect tool [START_REF] Tebina | Ray-spect: Local parametric degradation for secure designs[END_REF], targeted at simulating the effects of XRFI with configurable scalability of the targeted region. We will present how the tool can be exploited to simulate, early in the design flow, both the errors induced in the digital logic, as well as the deviations of the electrical characteristics induced by the TID.

The paper is structured as follows. We first introduce in Section II the necessary background on Total Ionizing Radiation (TID) on electronic systems and the state of the art on fault simulation methodologies. In Section III, we describe the simulation model used by the tool and its electrical equivalent. In Section IV, we provide details on the simulation flow used by Ray-Spect. Three use cases of the tool are presented in Section V-A. Finally, in Section VI, we summarize the main ideas discussed in this paper.

II. BACKGROUND A. TID Effects

It is known that TID radiations such as X-Rays have an impact on the drain current I d (V G ) characteristic of MOS 979-8-3503-1500-4/23/$31.00 ©2023 IEEE Fig. 1: Hole and interface traps formation in a MOS structure under ionizing radiation transistors. This impact can be resumed in 3 parametric shifts of the transistor: a shift in the threshold voltage, an increased static leakage current, and a trans-conductance degradation [START_REF]Ionizing Radiation Effects in Electronics: From Memories to Imagers[END_REF]. Hole trapping inside the oxide is the basic mechanism behind these phenomena. Figure 1 shows the basic mechanisms of hole and interface trapping.

The threshold voltage shift effect is a result of positive charge traps induced by the ionizing radiation in the gate oxides. These charges attract negative charge carriers at the canal level leading to lowering the threshold voltage of the N-type MOS, and (negatively) increasing it in case of a P-type MOS. Thus, an NMOS can be made permanently in a conductive state and the PMOS can be made into a permanently blocked state. This is know as the Semi-Permanent Fault Model; the Semi-Permanence property is due to the reversibility of the effect through heat-annealing.

The increased static leakage current effect is due to the TID on the Shallow Trench Isolation (STI) component. It only concerns N-Type transistors since the substrate is P-Type, and N-type parasitic canals can be formed on the STI walls. This leakage can be intra-component, between the source and the drain of the same transistor [START_REF] Esqueda | Modeling inter-device leakage in 90 nm bulk cmos devices[END_REF]; it can also be intercomponent, as a parasitic path of charge carriers can be formed on the STI connecting two MOS transistors, leading to an increase in the leakage current [START_REF] Zebrev | Physics-based modeling of tid induced global static leakage in different cmos circuits[END_REF].

The trans-conductance degradation is a result of the buildup of interface traps, which can actively exchange charge carrier with the transistor, and obstruct the movement of charge carriers. Interface traps are much slower to buildup than oxide traps: therefore, this effect is not discussed in this paper since we consider here only freshly irradiated components.

B. Simulation Tools for Radiation Assessment

Radiation hardening strategies have been developed to overcome single-event transients (SET) and upsets (SEU) triggered in cells, and for mitigating total ionizing dose (TID). Radiation Hardening By Design (RHBD) techniques for TID are only really designed to mitigate effects on parasitic field oxide transistors in the circuit. For SET mitigation, RHBD design techniques can help reduce the probability of SETs being generated and causing errors. Concerning the simulation contribution, development of SEE prediction tools has been a major and growing issue for a few years. The SEMM-2 [START_REF] Tang | Semm-2: a modeling system for single event upset analysis[END_REF], MRED/RADSAFE [START_REF] Weller | General framework for single event effects rate prediction in microelectronics[END_REF] and MUSCA SEP3 [START_REF] Hubert | Operational ser calculations on the sac-c orbit using the multi-scales single event phenomena predictive platform (musca sep 3 )[END_REF] are some examples of such initiatives. There is no similar approach concerning the TID.

C. Simulation Approaches for Fault Attacks

With respect to secure implementations, the goal of the designer is to develop countermeasures against known, and possibly unknown, attacks. Several methods exist in the literature exploiting either simulation or emulation approaches for fault injection, with application to evaluating the robustness of designs against defects or natural errors. The major difficulty in this task stems from the fact that fault injection techniques can be varied, as discussed in Section I. Depending on the chosen attack mean, the effects can be therefore different: sampling errors when clock glitching, corrupted signals created by laser pulses, etc. For this reason, one possibility is to abstract and model the effects of the faults at higher level: quite often, for example, fault injection attacks on microcontrollers and CPUs are modeled as instruction skips, replays, or corruptions [START_REF] Rivière | High precision fault injections on the instruction cache of armv7-m architectures[END_REF].

This holds for microcontrollers, however, and is not feasible when considering general circuits. In this case, when designing countermeasures it is important to delve deeper into lower abstraction levels, such as RTL. At this level, usual fault models are based on stuck-ats or bit-flips; then, qualification campaigns can be done on the basis of a statistical distribution of fault occurrences [START_REF] Leveugle | Statistical fault injection: Quantified error and confidence[END_REF], or on models based on the specific fault injection technique (such as selection of critical paths for delay faults, or spatial estimation of the targets for LFI [START_REF] Papadimitriou | Analysis of laser-induced errors: RTL fault models versus layout locality characteristics[END_REF]). Of course, the lower the abstraction, the more consistent the description will be, but this will also mean that the modeling will be quite tailored to the specific context.

Thus, specific approaches are required for different techniques when working at lower levels of abstraction. In the following, we will present our methodology addressing XRFI through two different approaches, and highlight what are the expected biases when irradiating a secure design.

III. SIMULATION MODELS

A. Parametric Degradation Model

In this approach, the parameters are injected directly into the SPICE-like netlist and are defined by the compact model of the transistor such as the BSIM3v3 or the BSIM4 models. New parameters can be inserted, and already existing parameters can also be used to emulate the desired effect. In the case of X-Ray effects, an inversion increment V i-pmos or a decrement V i-nmos in the threshold voltage parameter V th can be introduced in the equations of the compact model. This value can then be propagated through the SPICE-like netlist for the transistor level simulation.

B. Electrical Equivalent Model

Several approaches have been proposed over the years to emulate the effects of TID at the electrical level. Esqueda et al. [START_REF] Esqueda | Compact modeling of total ionizing dose and aging effects in mos technologies[END_REF] propose to use a Voltage Controlled Voltage Source (VCVS) connected to the grid of the transistor in order to emulate the threshold voltage shift. On the other hand, static leakage can be modeled as a transistor connected in parallel to the irradiated N-type MOS [START_REF] Esqueda | Two-dimensional methodology for modeling radiationinduced off-state leakage in cmos technologies[END_REF].

Our approach is to emulate both the dynamic leakage (Threshold voltage shift) and the static current leakage I OF F at the same time. The dynamic leakage can be emulated using a VCVS where the positive terminal of the voltage source is connected to the grid of either the N-Type or P-Type MOS. The static leakage can be emulated using an N-Type parasitic transistor connected to an independent VCVS and in parallel to the target NMOS. Figure 3 represents the equivalent schematic of an irradiated CMOS inverter.

IV. SIMULATION FLOW

A. Ray-Spect

Ray-Spect [START_REF] Tebina | Ray-spect: Local parametric degradation for secure designs[END_REF] is a Python framework designed specifically for assessing the sensitivity of secure designs against fault attacks and side channel attacks. It enables the simulation of localized parametric degradation in secure circuits. The simulation framework provided by Ray-Spect aims to capture Fig. 4: Overview of the simulation flow using Ray-Spect the impact of these attacks on circuit performance, with a particular emphasis on localized degradation. Ray-Spect is best suited for modifying a few to several instances at a time, such as in X-ray attacks using laboratory sources.

In this paper, the tool is used to assess the sensitivity of a secure design against X-Ray attacks. Firstly, from a fault injection perspective. Secondly, from a side channel attacks perspective, where X-Rays can be used not to fault an output, but to amplify the information leakage observed through the power consumption.

The simulation flow of Ray-Spect consists of several key steps:

1) Parsing: Ray-Spect starts by parsing a Cadence Spectre netlist using a netlist parser. The parsed netlist is organized into a class called NetlistElement, which represents different types of network elements in the netlist, such as sub-circuits, top instances, and comments.

2) Graph Generation: The tool generates a directed graph representing the netlist structure, starting from a given list of instances. The graph is constructed based on the connections between different components in the netlist, such as subcircuits and top instances. The graph generation algorithm defines the edges of the graph based on the grammar of the Spectre netlist components.

3) Model Propagation: Whether it is parameter-based or equivalent circuit, Ray-Spect propagates the desired model through the netlist using a traversal algorithm, such as Breadth First Search (BFS). The paths from the fetched instances to the base FET components are identified, and the modified values are injected into the netlist buffer based on the parameter inheritance grammar in the Spectre netlist. The propagation algorithm loops over all the fetched instances and modifies the netlist buffer accordingly.

4) Netlist Generation: Once the desired parameters have been propagated throughout the netlist, Ray-Spect can generate a new netlist using templates. The tool utilizes a Spectre component template to generate the Spectre netlist, providing flexibility and high rendering speed. Figure 4 provides a global overview of the Ray-Spect simulation flow.

B. Fault injection use

Fault injection occurrences can be simulated by propagating the fault model into the netlist and sweeping its values until observing faulted outputs. In the case of TID effects, the propagated model can be purely parametric, meaning that we propagate an increment V i-pmos or a decrement V i-nmos to the threshold voltage of the transistor model until obtaining the inversion threshold voltage V thi-nmos (where the N-MOS becomes permanently conductive), and the inversion threshold voltage V thi-pmos (where the P-MOS becomes permanently blocked). Equations 1 and 2 describe the relationship between the inversion threshold voltage and the inversion voltage, where V th is the original threshold voltage of the transistor model:

V thi-nmos = V th -V i-nmos (1)
V thi-pmos = V th + V i-pmos (2)
The propagated model can be the electrical equivalent of the TID effect. The object-oriented design of the tool makes it easy to insert additional components: we can for example insert instances of DC voltage sources in a certain "sub-circuit" and modify the nets so that they are connected to the grid of the target transistors. A sweep can be performed on the values of the DC voltages until observing faulty outputs.

C. Side channel use

Optically induced static leakages are a recent technique that was introduced in [START_REF] Bělohoubek | Optically induced static power in combinational logic: Vulnerabilities and countermeasures[END_REF], where a laser transient pulse is injected in a combinational gate in the moment of switching in order to amplify its data dependant leakage. However, this technique is very expensive in terms of spatial and temporal accuracy. In this paper, we describe a novel technique to amplify the data dependency of the leakages. A certain area of the circuit can be targeted by X-Rays in order to increase its dynamic and static power consumption due to effects of TID radiations (detailed in Section II-A). The collection of power traces from the Spectre simulator is facilitated using Monaco [START_REF] Gutiérrez | Open automation framework for complex parametric electrical simulations[END_REF], a tool designed to automate the simulation of multiple netlists.

Side channel leakage assessment, most notably power analysis, can be performed in Ray-Spect using a similar approach to fault injection, by sweeping the parameters in a range that does not induce faulty behaviour. The higher abstraction level of Ray-Spect provides a mean to assess the contribution of each component. For example, we can propagate the parametric degradation over a specific type of standard-cells to evaluate if certain logic gates are more leaky than others. In another scenario, one can also assess which transistor type is more prominent to target, by propagating the model over only a certain transistor type. Additionally, the assessment can be area dependent, by sweeping over chosen areas of a circuit to identify the most sensitive spots to attack. In this section, we will discuss three different use cases for our flow: the first concerning TID fault emulation, whereas the second and the third address side channel leakage variations.

A. Fault Injection on AES S-Box

For this case, a simple 8-bit Rijndael Substitution Box (S-Box) circuit for the Advanced Encryption Standard (AES) has been implemented (see figure 5). The goal in this case is to reach the inversion threshold voltage and observe flipped bit values at the output, when propagating the changes throughout selected regions of the circuit. The inversion voltage depends on the technology used; in our case, the circuit has been implemented with the AMS 350nm technology, but the methodology applies to any process. A simple threshold voltage sweep simulation can be performed in order to estimate the suitable values. Table I details different values extracted from the simulation.

The S-Box layout measures 400.4 × 400.4µm. We divided the area into an 8 × 8 grid, with each square measuring 50 × 50µm and matching a simulated X-Ray beam. The chosen size is configurable and is based on our threat model, which considers the possibility of an attacker targeting an area of interest rather than just a single transistor. It is worth noting that this choice does not take into account the recent advancements in laboratory X-Ray targeted beams, which can reach sizes of approximately 1µm. Figure 6a shows the definition of each region to be targeted by the parameter injection flow in Ray-Spect. Each square represents a netlist generated after the parameter propagation within the square, that is then simulated using Spectre. Figure 6b represents the number of flipped bits in the output cipher text after targeting the corresponding region.

B. Side Channel Analysis on an AES S-Box

For the following case study, the threat model considers that an attacker may be able to bias the power consumption in a certain area of the circuit and thus amplify the information leakage. Power analysis methods such as the Correlation Power Analysis (CPA) and Differential Power Analysis (DPA) are applied to our design in order to assess potential information leakages. A shunt resistor is connected in series with the power distribution pin (vdd) in order to observe the simulated power traces.

1) Correlation Power Analysis: A CPA attack has been conducted on the 64 generated 8-bit S-Box netlists. A set of 70 random plaintexts among the possible 256 have been fed to the inputs of the circuit. If the key guess equals the correct encryption key, the Pearson Correlation Coefficient is stored.

For each netlist, CPA is performed 1000 times with 70 plaintexts and the correlation coefficient is averaged for statistical accuracy. Figure 7a shows the result of CPA after the irradiation of each region, whereas Figure 7b depicts the number of flip-flops contained in the region. The pre-rad coefficient was uniformly equal to 0.50, but after irradiation the behavior is region-dependent: these results suggest that Flip-Flops containing the current state and the key are the most sensitive components when targeted with radiations.

2) Differential Power Analysis: A DPA analysis was performed on two netlists: under the normal behaviour (figure 8a), and after irradiating the most sensitive spot discovered in the previous scenario (Figure 8b). The DPA was performed on several selection bits. For specific bits, the difference between the power consumption of the logical state '0' and the logical state '1' increases, thus resulting in a higher differential leakage. Figure 8 shows the difference between the two behaviours: the '+' waveform is the correct key guess. In the selection bit 5, the correct guess could not be retrieved in normal operation, but became identifiable after irradiation due to the leakage amplification caused by X-Rays. In the other two cases, the correct key is still hidden, but its differential feature is largely improved.

C. Side Channel Analysis on 32-bit AES SubBytes

In this scenario, a design with four AES S-Boxes is implemented (S-Box(0-3)). An arbitrary region in S-Box(2) is targeted by X-Ray irradiation. A set of fixed 500 plaintexts are used to attack the design in both cases, before and after the attack. Figure 9 shows the location of the 4 S-Box circuits and the targeted area in the S-Box [START_REF] Piret | A differential fault attack technique against SPN structures, with application to the AES and KHAZAD[END_REF].

The results of the CPA attack before and after radiation are shown in Table II. The same 500 plaintexts were applied in both cases. Byte 3 was not retrievable initially, but was successfully found after the irradiation. While the TID effects introduced in Byte 2 actually reduced the corresponding

VI. CONCLUSION

Ray-Spect is a Python framework designed specifically for assessing the sensitivity of secure designs against fault attacks and side channel attacks. It provides a simulation flow that captures the impact of these attacks on circuit performance, with a focus on localized parametric degradation. The framework offers capabilities specifically targeted at X-Ray attacks.

Three case studies were presented to demonstrate the capabilities of Ray-Spect. In the first case study, the sensitivity of an AES S-Box circuit against fault injection was assessed. The inversion threshold voltage was propagated throughout selected regions of the circuit to observe flipped bit values at the output. The second and third cases studies are focused on side channel attacks, where power analysis methods were applied to assess information leakages. CPA and DPA attacks were conducted, and the effects of X-Ray irradiation on power consumption and leakage amplification were analyzed.

  Fig 2 shows an example of the parameter propagation across different sub-circuits in a Cadence® Spectre netlist.

Fig. 2 :

 2 Fig. 2: Spectre sub-circuit parameter inheritance grammar description

Fig. 3 :

 3 Fig. 3: Schematic representation of the TID leakage model

Fig. 5 :Fig. 6 :

 56 Fig. 5: Schematic view of the implemented S-Box design

Fig. 7 :

 7 Fig. 7: (a) CPA correlations coefficients after each region irradiation (b) Number of Flip-Flop instances existing in each region

Fig. 8 :

 8 Fig. 8: DPA analysis on the selection bits 0, 5 and 6 of: (a) the original circuit; and (b) when region 5 is irradiated.

Fig. 9 :

 9 Fig. 9: Quadruple S-Box and targeted region area distribution

TABLE I :

 I Simulation results of the threshold voltage sweep

	Technology node	vdd (V)	W/L (um)	V th (mV)	V i-nmos (V)
	AMS 350 nm	3.3	0.35/10	567	1.54

TABLE II :

 II CPA attack results on the quadruple S-Box

	Actual key	0x0A	0x0D	0x82 0x80
	Key guess pre-rad	0x0A	0x2D	0x82 0x24
	Coefficients pre-rad	0.23	0.21	0.24	0.22
	Key guess post-rad	0x0A 0xAB 0x82 0x80
	Coefficients post-rad	0.23	0.22	0.23	0.21

ACKNOWLEDGMENT

This work has been partially funded by French National Research Agency in the frame of the ANR project MITIX (ANR-20-CE39-0012). TIMA Laboratory is part of the Cybersecurity Institute of Grenoble Alpes (ANR-15-IDEX-02).


Anaïs Servais
email: anais.servais@cnrs.fr

Raphaël Poveda

Hanna Gerony

Emmanuel Barbeau

Christophe Hurter

Can we use EOG to identify when your attention switches away from the world to focus on your memories?

INTRODUCTION

Observe what's going on in your mind while answering the following question: "How did you celebrate your last birthday?". Answering this question required the creation of a temporary internal mental space where your attention was focused [START_REF] Tulving | Episodic memory: From mind to brain[END_REF]. Thus, there was an attentional switch from the external world to the internal worlda change in mental state commonly known as mind-wandering-a general term popularly used to designate daydreaming and zoning-out [START_REF] Smallwood | The Restless Mind[END_REF]. Indeed, since attention is a limited resource, when it is focused on the internal world, the external world vanishes [START_REF] Fernandes | Divided attention and memory: Evidence of substantial interference effects at retrieval and encoding[END_REF], a phenomenon known as perceptual decoupling. Though useful for managing attentional resources, this decoupling has a concerning aspectof impairing performance and safety in operational tasks (such as driving a car or piloting a plane) where attention must be directed to the external world [START_REF] Gouraud | Out of the Loop, in Your Bubble: Mind Wandering Is Independent From Automation Reliability, but Influences Task Engagement[END_REF][START_REF] Smallwood | Pupillometric evidence for the decoupling of attention from perceptual input during offline thought[END_REF]. Reports show that aviation accidents related to attentional lapses occur (NTSB, 2014), but the origin of these lapses remains unknown. Given perceptual decoupling, mind-wandering could be a potential cause but objective behavioral markers to monitor attention remain elusive.

Here, we propose eye movements as one such potential candidate. Mind-wandering is associated with oculomotor features reducing visual processing: more and longer blinks, fewer and longer fixations, longer saccades durations, and so on [START_REF] Benedek | Eye behavior associated with internally versus externally directed cognition[END_REF]. However, these eye behaviors cannot be used as markers to detect mind-wandering in real-time because they require posteriori averages. Instead, here we suggest investigating another eye behaviorgaze aversiona very common behavior while answering memory questions. People routinely look away as if they were searching for the answer on the ceiling or "in the sky" [START_REF] Doherty-Sneddon | Gaze aversion: A response to cognitive or social difficulty?[END_REF][START_REF] Glenberg | Averting the gaze disengages the environment and facilitates remembering[END_REF].

So far, mind-wandering has been seldomly studied in aeronautics, perhaps because it is a spontaneous mental state and therefore difficult to assess. The current methods are based on self-reporting which has several biases including social desirability for pilots [START_REF] Casner | Thoughts in flight: Automation use and pilots 'taskrelated and task-unrelated thought[END_REF]. To overcome this problem, since people tend to retrieve events that occurred in their past during 60% of their zoning out time [START_REF] Mildner | Spontaneous Thought as an Unconstrained Memory Process[END_REF], we propose to trigger attentional switches in the lab using questions like the one you answeredautobiographical memory questions [START_REF] Conway | Sensory-perceptual episodic memory and its context: autobiographical memory[END_REF] while recording eye movements with electro-oculography (EOG). EOG records the electrical potential between electrodes placed on the muscles around the eyes and allows to quantify vertical and horizontal eye movements. In airplane cockpits, the EOG provides a less invasive alternative to infrared cameras, which generally obstruct part of the visual field. Recent studies have also shown that EOG can be recorded around the ears [START_REF] Favre-Félix | Real-time estimation of eye gaze by in-ear electrodes[END_REF] opening prospects for integration of the electrodes in the pilot's helmet. In this, currently preliminary, work we aim to estimate the relevance of EOG to study gaze aversion during internal attention.

MATERIALS AND METHODS

Data were collected on 4 participants (2 females, aged between 23 and 28, 1 left-handed) without oculomotor, visual, neurological, or psychiatric disorders.

We designed a task where a participant answers autobiographical memory questions. Each trial starts with a fixation cross (5 sec) followed by a question delivered through loudspeakers (5 sec). After the question, the participant completes a short visual task to keep attention towards the external world: for 8 to 12 letters, the participant presses a key if the letter contains a curve line and another if it does not. When there is no more letter, the participant starts to search for autobiographical memory. This "reflection phase" is divided into an earlier phase of access (max 12 sec) and a later phase of elaboration (5 sec) where the participant respectively selects and explores a personal memory corresponding to a unique and short event (< 24 hours), with a defined spatiotemporal context. An audio "debriefing" indicates the start of the verbalization. Then, the patient presses a key to start the next trial (see Figure 1 for details).

To isolate the impact of autobiographical memories, we include two control conditions: a semantic memory task (also involving internal attention) concerning famous events or people, and a visual task (external attention) that requires finding the digit "5" hidden in a visual scene. The whole experiment is divided into 5 blocks of 15 questions (5/condition) presented in random order. Each participant completed 25 trials per cognitive condition.

EOG recording

Stimuli presentation was controlled by OpenSesame 3.2.8. BioSemi Active Two amplifier was used to record the EOG with 5 electrodes: a reference on the forehead, one electrode at each extern canthus for horizontal movement, one electrode above the right eyebrow, and one electrode on the top of the right cheek for vertical movements. The frequency rate is 2048 Hz. To guarantee the quality of the EOG data throughout the session, a calibration was performed at the beginning of each block. As ground truth, we recorded a video of the participant's face (with SONY FDR-AX33 Handycam, 1920x1080px resolution).

Data analyses

The signal was preprocessed-resampled to 256Hz, and filtered with a low-pass at 40Hz. All the saccades were labeled manually by two students doing their internship in the lab. While labeling the saccades, they were blind concerning the part of the signal they were processing.

Only the saccades labeled by both judges were kept for analysis.

The periods of interest are the moments of visual and memory retrieval (for the 2 memory conditions) trials. The visual trials allow us to get a sample of visual saccades, usually done while exploring the environment. During memory retrieval, we focus on the trials where gaze aversion occurs. With the video as ground truth, the saccades initiating gaze aversion were noted as aversion saccades. The synchronization between the EOG and the video was performed on BrainStorm (https://neuroimage.usc.edu/brainstorm/).

ERPs were calculated (using BrainStorm) on epochs aligned on the aversion time (settled to zero), a baseline of -300ms before (because 300ms is the min. time between the start of the memory retrieval and a gaze aversion), and duration of 600ms (the min. duration of a gaze aversion). ERPs allow observing if a similar EOG pattern is repeated between aversions.

The sensitivity of Eogert (Toivanen, 2015), a probabilistic online method to detect EOG saccades, was compared for visual and aversion saccades. Compared to the saccades labeled manually, false negatives and true positives were counted: sensitivity=TP/(TP+FN). For visual saccades, the sensitivity is 36% against only 9% for the aversions saccades. The algorithm is more accurate on visual saccades than aversion saccades, pleading for different EOG features between these types of eye movements. See Figure 2 for an illustration of the signal during a memory trial with gaze aversion and a visual trial with visual saccades. To investigate the differences between the EOG features of visual compared to aversion saccades, we conduct exploratory analyzes comparing the gaze angle and velocity of the two kinds of saccades with the Mann-Whitney U-test (the size effect is the rank-biserial correlation (r), α is corrected with Bonferroni to .016). For each gaze direction, a constant of reference is calculated based on the calibration (in µV/degree). Gaze angle is calculated following a linear relation: .

𝑮𝒂𝒛𝒆

RESULTS

Out of 100, Gaze aversion was observed in 35 trials for autobiographical memory and 26 trials for semantic memory questions. There is inter-individual variability in the occurrence of gaze aversion since the proportion of questions where gaze aversion is observed varies between participants from 4% to 68% for autobiographical questions and from 0% to 56% for semantic questions. During all the visual trials, a total of 1062 visual saccades were counted.

The visualization of the ERPs is possible only for 3 subjects because the fourth one does not do enough gaze aversions. It shows, per individual, a similarity in the vertical EOG pattern of the different aversions (see Figure 3).

The aversion saccades show different gaze angle and velocity than visual saccades (see Figure 4). For the gaze angle, Mann-Whitney U-tests show significant differences between visual and aversion saccades. Gaze angles are much larger for aversions, both in autobiographical (EOGh: U= 7429, p<.001, r=.48; EOGv: U=3669, p<.001, r=.74) and semantic memory trials (EOGh:

DISCUSSION

It seems that the EOG signal associated with eye saccades initiating a gaze aversion during attentional switch differs from the EOG signal associated with visual eye saccades that people do when they explore their environment. Saccades initiating gaze aversion are faster and have larger amplitude than visual saccades. As expected, aversions seem to be more distinguishable from the visual saccades on the vertical rather than on the horizontal EOG.

However, the present study is far from real-time detection of attentional switches. We are facing many limits. First, we show that even on a small sample of participants, there is huge individual variability in the gaze aversion behavior. The ERP analysis shows that the pattern of the aversion differs between participants. To be transferred to an application, an individual personal calibration of the system would be necessary. Here, we use a small sample size because we aim to determine the potential interest of the method, but a validation of the method would require a higher sample. Second, to refine the results, future studies should include recordings of head movements and a better calibration to infer gaze angle more accurately. Here, we show that gaze aversions induce a large gaze angle, and the linear relation with EOG is not true for gaze at high eccentricities. Third, it is too early to extrapolate our results to ecological situations because we currently don't have any proof that gaze aversion happens during spontaneous switches. Fourth, the results for gaze aversion during autobiographical memory do not differ significantly from the one about gaze aversion during semantic memory. It seems therefore linked to internal attention in general, and maybe not specific to autobiographical memory.

We want to emphasize the importance of studying such behavior in aeronautics. Although here we have focused on gaze aversion occurring during memory retrieval for the sake of developing an experimental paradigm, in our view such behavior is similar to what occurs during mindwandering and in fact any behavior requiring access to internal thoughts including when one is speaking to an interlocutor. Monotonous tasks generate higher rates of mind-wandering, which is a problem given the increasing automation in the cockpits [START_REF] Gouraud | Out of the Loop, in Your Bubble: Mind Wandering Is Independent From Automation Reliability, but Influences Task Engagement[END_REF]. Therefore, we urgently need an objective marker allowing real-time detection of internal thoughts switching to monitor attention in critical situations. In this context, our work aims to open discussions and perspectives.

CONCLUSION

Although our results are preliminary, they are encouraging. First, we propose a protocol to trigger attentional switches in the lab. Second, we show that these switches are associated with gaze aversions. Third, since we observe that gaze aversion have different EOG features compared to visual saccades, we think that EOG could be a potential method to study and detect attentional switch. In an early laboratory phase, EOG could be coupled with augmented reality helmets to characterize gaze aversion better before reaching a reliable detection in EOG only.

Figure 1 :

 1 Figure 1: This figure illustrates the method used to collect the data. On the left, is the position of the electrodes. On the right, is the description of the cognitive task.

Figure 2 :

 2 Figure 2: Illustration of the EOG signal during a memory trial (left) and a visual trial (right).

  .007, r=.33; EOGv: U = 713, p<.001, r=.93). The angle does not differ significantly between the aversions from the two memory conditions memory (EOGh: U = 263, p=.22, r=.20; EOGv: U =424, p=.08, r=-.28). The velocity of the saccades is also significantly different between visual saccades and aversion saccades, which appear either in autobiographical memory (EOGh: U= 10957, p=.03, r=.23; EOGv: U=5901, p<.001, r=.59) or in semantic memory trials (EOGh: U = 11754, p=.32, r=-.12; EOGv: U= 2486, p=<.001, r=.76). Vertical, but not horizontal, saccades are much faster during gaze aversion than visual saccades. The velocity does not differ significantly between the aversions from the two memory conditions memory (EOGh: U = 218, p=.04, r=.34; EOGv: U =405, p=.17, r=-.23).

Figure 3 .

 3 Figure 3. ERPs per participant. Each color represents the epoch of a trial. Aversions from autobiographical and semantic trials have been pooled together. Vertical red lines indicate the start of the aversion separating the baseline from the gaze aversion period.

Figure 4 .

 4 Figure 4. Gaze angle and velocity for visual and aversion saccades. One color stands for all the saccades of a single participant. ** p-value < .001


Michel Nabaa

Cyrille Bertelle

Antoine Dutot

Damien Olivier

Exploitation of a displacement survey to detect road network use vulnerability

Keywords: vulnerability, risk, self organization, detection of organizations, GIS, household displacement survey

Many agglomeration are faced to multiple technological and natural hazards. The use of the road network by vehicles may cause big problems especially under evacuation situations. An accident on a main road may endanger many lives. A household survey was conducted in CODAH 1 about people displacement in the seine estuary. In this paper, we extracted and treated the survey result. We also compared results with a detection of organizations in large graphs algorithm.

I. INTRODUCTION

A S many agglomerations, CODAH is faced to natural and technological hazards. 33 establishments are classified SEVESO 2 with high threshold. To minimize risk effects, we must localize the population at any time. In this context, the major risk management team (DIRM) of CODAH has developed a model to estimate the diurnal / nocturnal population distribution at buildings scale (PRET RESSE) [START_REF] Bourcier | Allocation spatio-temporelle de la population exposée aux risques majeurs. contribution à l'expologie sur le bassin de risques majeurs de l'estuaire de la seine: modèle pret-resse[END_REF]. This model does not take into account the population displacement on the road network. In previous works, we proposed an approach to dynamically assess the vulnerability related to the road network use by vehicles [START_REF] Bertelle | Modeling of the vulnerability related to the dynamic road traffic[END_REF], [START_REF] Bertelle | Detection of organizations in large graphs : Application on risk management[END_REF]. To complete this work, we have exploited a household survey about the displacement of the population of CODAH in the Seine estuary. In this paper, we exploit the survey and we compute vehicles path declared displacements to dispatch the population on the network. We present also a detection of organizations on large graphs algorithm. The aim of this algorithm is to detect zones 1 COmmunauté De l'Agglomération Havraise: it groups 17 communes including Le Havre. The goal is to to develop many common projects in territory management, health and hygiene, public transport, risk management... 2 Directive SEVESO is an European directive, it lays down to the states to identify potential dangerous site. It intends to prevent major accidents involving dangerous substances and limit their consequences for man and the environment, with a view to ensuring high levels of protection throughout the Community.

susceptible to be congested, this by taking into account the topological aspect of the graph. Finally, the results of the survey will be compared to those obtained from the algorithm.

II. RISK MANAGEMENT

Real exercises in an evacuation planning are so expensive (people, resources, logistics...) and sometimes not realistic : they can not take into account each individual behavior as the panic effect and initial response to an evacuation in a dangerous area. Panic generally results from the lack of coordination and dialog between individuals. Hence the use of the simulation and in particular Multi-Agent Systems (MAS) is necessary to model each agent behavior and more understand the evolution of a critical situation due to interactions between evacuees and the propagation of a danger : an accident on the road network may cause dangerous traffic jams especially in a case of a danger that spreads quickly. Once we understand the evolution of a critical situation, we try to fight against organizations of vehicles and try to allocate vehicles on different roads to avoid bottlenecks. Recently, many researchers tried to couple a MAS with a Geographical Information System (GIS), especially in risk management and vulnerability assessment. For computer modelers, this integration provides the ability to have agents linked to real geographical locations. For GIS users, it provides the ability to model the emergence of phenomena by various interactions of agents in time and space by using a GIS [START_REF] Najlis | Repast for gis[END_REF]. Thanks to a GIS, one can combine different vulnerability layers (a layer per danger : geophysical one, social one, access to the network...) to construct a global layer which leads to a better assessment of vulnerability; we think particularly about the work of [START_REF] Cutter | Revealing the vulnerability of people and places: a case study of georgetown county, south carolina[END_REF]. In geography, the representation of a MAS coexists n levels of organizations and use several classes of agents (e.g. Level 1: individuals or companies, Level 2 and three: economic, urban communities).There will be rules at every level and the approach is not necessarily a bottom up one as in the models of self-organization [START_REF] Daudé | Systèmes multi-agents pour la simulation en géographie : vers une Géographie Artificielle[END_REF]. In [START_REF] Provitolo | A proposition for a classification of the catastrophe systems based on complexity criteria[END_REF], the author modeled the spread of the panic from a group of individuals in danger situation to non panicked ones. She used a dynamic system (differential equations) to simulate the behavior of individuals. Many execution scenarios showed that the emergence of panic has not occurred in all scenarios when changing some parameters. The emergence depends on the rate of transmission from a population susceptible to panic to a panicked population , the time taken to return to a normal behavior (the population is more panicked after the disaster) and the number of initially panicked people. The principle of resilience was also discussed; it represents the time that the system makes to return to its initial state after a period of instability due to a disaster.

At neighborhood scale, many researches where developed models based on microsimulation. In their paper [START_REF] Church | Mapping evacuation risk on transposion networks using a spatial optimization model[END_REF], the authors have presented a model to estimate the time required to evacuate a neighborhood according to the population, the number of vehicles and roads network capacity. The model is based on optimization in order to find the dangerous area around a critical point. This model was coupled with a GIS (ArcInfo) to visualize the results (identifying evacuation plans) and to establish an evacuation map for the town (Santa Barbara). The same authors [START_REF] Cova | Modelling community evacuation vulnerability using gis[END_REF] opened the way to the study based on geographic information systems to evacuate people. Their study identified communities that may face transportation difficulties during an evacuation. They modeled the city by a graph to estimate lanes (Lane) occupation by vehicles during an evacuation. A graph partitioning model was adopted in order to detect vulnerable neighborhoods around each node of the graph and to build a vulnerability map around the nodes of the graph. A constructive heuristic was used to calculate the best cluster around each node. The result was displayed on a map with ARCINFO GIS. An area is increasingly vulnerable according to the number of evacuees per lane (number of people to evacuate in a neighborhood / number of exit roads). The author believes that the combining of an evacuation based vulnerability layer and a hazard one allows to build a general risk map, so we can explore the various issues related to risk. However, in this approach, we predefine the maximum number of nodes in a neighborhood, which may not always be realistic and does not take into account the traffic evolution during evacuation. Secondly, the city of Santa Barbara is not a typical example of urban metropolitan areas in the United States with an important population facing great difficulties during a evacuation.

III. DISPLACEMENT SURVEY

Lifestyles and travel practices are changing. Traveled distances have increased. New infrastructures have been established and new services are proposed: these developments must meet the expectations and new patterns of migration of our territory inhabitants. That is why a household displacement survey was conducted, with a standard model developed by the Center for Studies on networks, transport, urban planning and public constructions (the CERTU). The survey will pro-vide shared data on current displacement practices and their evolution.

A. Survey utility

The last displacement survey occurred in 1992 at Le Havre agglomeration. Over time, taken paths become more complex and new infrastructures exist; hence the need to a displacement survey to respond to the population expectations, this by :

• considering the travel conditions between the city and its catchment area of life: people come from further far away to work, study or consume.

• Interview people about their expectations for transport.

• defining the transport policy of the CODAH for the coming years: how to increase the use of public transport, where and how to adapt the supply, how to relate the supply of transport lanes with the rest of the territory, what are the new flows, new trends, future developments areas of housing and economy ... • promote environmental issues, defined in the law of Solidarity and Urban Renewal. Communities undertake increasingly the development of a sustainable mobility. In this work, the main utility of the survey is to localize the population on the network at any time of the day. This will help us to detect the congested areas and to estimate the vulnerability related to the road network use by vehicles.

C. Environment modeling

The road network is integrated as a layer in the Geographic Information System (GIS). From this layer, we extract the data by using the open source java GIS toolkit Geotools. This toolkit provides several methods to manipulate Geo spatial data and implements Open Geospatial Consortium (OGC) specifications, so we can read and write to ESRI shapefile format. Once data road network are extracted, we use the GraphStream tool [START_REF] Dutot | Graphstream: A tool for bridging the gap between complex systems and dynamic graphs[END_REF] developed within LITIS laboratory of Le Havre to construct a graph corresponding to the GIS network layer. This tool is designed for modeling; processing and visualizing graphs. The data extracted from network layer contains the roads circulation direction, roads id, roads type, their lengths and geometry.

The extracted multigraph G = (V, E) represents the road network where V is the set of nodes and E the set of arcs. We deal with a multigraph because we have sometimes more than one oriented arc in the same direction between two adjacent nodes due to multiple routes between two points in the Seine Estuary road network. GraphStream facilitates this task because it is adapted to model and visualize multigraphs. In the constructed multigraph:

• The nodes represent roads intersections,

• The arcs represent the roads taken by vehicles,

D. Compute vehicles paths

In our work, we must repatriate people having their personal vehicle on the road network. The main goal is to detect routes with high vehicles density. It helps us in the detection of organizations and estimation of the vulnerability related to the road network using. Generally, people tend to choose the shortest path in time when using their own vehicles. This assumption, which may seem audacious and even unrealistic, however, is found in almost all mobility models implemented in most countries since 1960 [START_REF] Banos | À la conquête de la fourmilière urbaine : quand le géographe se fait explorateur de mondes artificiels[END_REF]. For each person displacement from a centroid c1 to a centroid c2, we compute the shortest path between the two nodes by using A-Star algorithm. It has the advantage to be fastest than Dijkstra algorithm when using an efficient heuristic. The last is an exact algorithm which compute the shortest path between a source node to all other nodes in a graph. In future work, we will use the dynamic version of A-Star (D-star) to avoid congested routes when computing paths. Another problem occurred when defining just one centroid per zone; obviously the must connected one. In figure 4, taken routes by vehicles are colored in red. At the left of the figure, we adopt one centroid per zone : we have 4772 taken routes and those routes are quickly congested. At the right, we used 15 centroids per zone to overcome this limitation : 17684 routes was taken. It is the maximum centroids number per zone if we use a machine having 2GO of ram. We can conclude that the more routes are taken and the less maximum roads capacity is violated. We conducted a simple experiment using random walks inside the city road network, seen as a graph. In this experiment we take a given number of entities placed randomly on nodes of the city graph. Then we run an iterative loop where, at each step, these entities either traverse a randomly chosen edge or wait.

Each time an entity crosses an edge e ∈ E, a special variable e np (number of passes) bound to the edge is incremented. At the end of such a simulation, the number of passes on each edge by entities will therefore be stored in these variables.

The entities may either cross an edge or wait because the displacements must consider road traversing times. The time an entity waits is proportional to the average time used to travel on the corresponding road: for two edges of the same length, an entity will wait longer if the road is a city road than if it is an highway.

After an entity crossed an edge, it chooses the next edge to cross randomly, but do not consider any more the edge it came from. The edges the entity used are stored in a memory, and the entity tries to avoid reusing them. This memory is a FIFO stack and ultimately, roads in the memory are forgotten, so that the entity can reuse them later.

As the road graph we use is bounded, some roads on the border are cut. If an entity reaches such a dead end (or a real bag end), it "jumps" in another position in the graph, chosen randomly.

In the figure 5 we show one run of this experiment. We in fact did several runs with a varying random seed, and always obtained very similar results. Edges are colored using a color graduation going from blue (almost no pass) to red (a lot of passes) passing by green, yellow and orange. The scale is geometric. To pass from blue to green you have to double the number of passes, and to pass from green to yellow you anew have to double this number of passes.

There are as many entities as nodes in the network. After some setup, we set the entity memory to 40 nodes. We stop the algorithm when a given maximum number of passes is reached on one of the edges, here 4000. We chosen such a high number to ensure the exploration of entities is significant enough. Fig. 5: Random walks in Le Havre, blueish to greenish roads are less used and yellowish to reddish roads are the more used.

For a better understanding, the same results are shown on figure 6 with labels that help to locate roads. The road that are highly used are labelled and highlighted. They correspond to town centers and highways. The A29 and A13 are the two main highways passing by and going to Le Havre. The road labelled "main city entry" is one of the most used road when coming to Le Havre (due to the fact the city is a harbour and have its south and west sides bordered by water, and at the north a plateau ("ville haute") with smaller accesses. The "Breque" label indicates a very large interchange that is, for the same reasons, one of the most used section of road when coming to and leaving Le Havre. Fig. 6: Figure 5 with labels on important areas.

The number of passes found with this random walk is an indication of the roads in the network were vehicles are inclined to pass, due to the topology of the road graph. This can be an indicator on areas where vehicle will most probably pass often.

We also run this simulation anew, but with some roads removed. Namely, we removed the A29 and A13 highways, as well as the "main city entry". The figure 7 shows the result. These road "absorbed" a large part of the traffic, and therefore all the entities use a lot more other roads. The "Breque" interchange is completely saturated. This place is well known by Le Havre inhabitants for the traffic jams occurring in the morning and the evening when people go to or leave their work. "La Breque" is one of the mandatory road to use in order to leave or enter the city. If we run the same experiment on figure 8, but only disabling the "main city entry", the results are more similar to the one without any road removed. This may mean that this road could be removed and people redirected to adjacent roads without impacting largely the traffic.

V. CONCLUSION

We have extracted a household survey on the motorized population displacement in the Seine Estuary. For each displacement from a zone to another one, we computed a shortest path from a centroid of the first zone to a centroid to the other zone. We used an A-Star algorithm to do that. Results showed that the more we have centroids per zone, the more we explore more routes and the less we violate routes capacity. We also applied on the Seine estuary graph, a detection of organizations algorithm based on random walks. The last showed that we can predict the possibility to have traffic jams in some areas on the basis of the network structure, this without having any data on the traffic. In the near future, we will adopt a microsimulation of the traffic with the use of D-Star algorithm to dynamically compute paths and to respect the road capacities at any time.

Fig. 1 :

 1 Fig. 1: zoning the Seine estuary communes

Fig. 2 :

 2 Fig. 2: UML schema

Fig. 3 :

 3 Fig. 3: Extract the graph and compute vehicles path

Fig. 4 :

 4 Fig. 4: Taken routes with one centroid and 15 centroids per zone

Fig. 7 :

 7 Fig. 7: Random walks in Le Havre, with the A29 and A13 highways removed as well as the main city entry.

Fig. 8 :

 8 Fig. 8: Random walks in Le Havre, with only the main city entry removed.


Anton Svirsky

Corentin Herbert

Anna Frishman

Two-dimensional turbulence with local interactions: statistics of the condensate

Two-dimensional turbulence self-organizes through a process of energy accumulation at large scales, forming a coherent flow termed a condensate. We study the condensate in a model with local dynamics, the large-scale quasi-geostrophic equation, observed here for the first time. We obtain analytical results for the mean flow and the two-point, second-order correlation functions, and validate them numerically. The condensate state requires parity+time-reversal symmetry breaking. We demonstrate distinct universal mechanisms for the even and odd correlators under this symmetry. We find that the model locality is imprinted in the small scale dynamics, which the condensate spatially confines.

Introduction Understanding the interactions between turbulent fluctuations and a mean flow is a central problem in fluid mechanics. Such interactions usually prevent a prediction of the mean flow due to their nonlinear nature. Recently, some progress was made in this direction for two-dimensional turbulence [START_REF] Laurie | Universal Profile of the Vortex Condensate in Two-Dimensional Turbulence[END_REF][START_REF] Frishman | Turbulence Statistics in a Two-Dimensional Vortex Condensate[END_REF]. In this case, the mean flow spontaneously emerges from smallscale fluctuations in an out-of-equilibrium process of selforganization. The underlying mechanism is that of an inverse transfer of energy -from small to large scales, leading to the accumulation of energy at large scales and the establishment of a system-size mean flow termed a condensate [START_REF] Robert | Inertial Ranges in Two-Dimensional Turbulence[END_REF].

Two-dimensional flow minimalistically encapsulates the main dynamical constraints imposed by rotation and density stratification, encountered in nature or in laboratory experiments [START_REF] Xia | Spectrally condensed turbulence in thin layers[END_REF]. In fact, a cornerstone of the theory of large-scale dynamics of geophysical flows, including the formation of coherent structures such as jets and vortices, is the quasi-geostrophic (QG) approximation [START_REF] Vallis | Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-Scale Circulation[END_REF], which shares the same fundamental structure as two-dimensional flow. However, a general understanding of the feedback between such structures and turbulence within the QG framework is currently lacking.

In QG, there is a typical scale for the range of fluid elements interactions, called the Rossby radius of deformation L d . The special cases which have been studied in detail so far correspond to dynamics at scales much smaller than L d : the vortex condensate in two-dimensional incompressible Navier-Stokes (2DNS) [START_REF] Laurie | Universal Profile of the Vortex Condensate in Two-Dimensional Turbulence[END_REF][START_REF] Frishman | Turbulence Statistics in a Two-Dimensional Vortex Condensate[END_REF][START_REF] Smith | Bose condensation and small-scale structure generation in a random force driven 2d turbulence[END_REF][START_REF] Chertkov | Dynamics of energy condensation in twodimensional turbulence[END_REF][START_REF] Chertkov | Universal velocity profile for coherent vortices in two-dimensional turbulence[END_REF][START_REF] Kolokolov | Structure of coherent vortices generated by the inverse cascade of twodimensional turbulence in a finite box[END_REF][START_REF] Frishman | The culmination of an inverse cascade: Mean flow and fluctuations[END_REF], and jets in the presence of strong differential rotation (beta effect) [START_REF] Farrell | Structure and Spacing of Jets in Barotropic Turbulence[END_REF][START_REF] Srinivasan | Zonostrophic Instability[END_REF][START_REF] Tobias | Direct Statistical Simulation of Out-of-Equilibrium Jets[END_REF][START_REF] Woillez | Theoretical prediction of Reynolds stresses and velocity profiles for barotropic turbulent jets[END_REF][START_REF] Woillez | Barotropic theory for the velocity profile of Jupiter's turbulent jets: An example for an exact turbulent closure[END_REF]. Here we address the opposite limit, of scales much larger than L d (without differential rotation): the large-scale quasi-geostrophic (LQG) equation [START_REF] Vitaly | Weakly decaying turbulence in an equivalent-barotropic fluid[END_REF]. We observe the condensate regime in LQG for the first time in numerical simulations, and provide a detailed theoretical picture of the mean-flow/turbulence interactions. Although the flow is turbulent and the mean flow is sustained due to non-linear interactions, we show that the statistics can be captured within a perturbative approach, the quasi-linear approximation [START_REF] Marston | Recent developments in theories of inhomogeneous and anisotropic turbulence[END_REF]. This rein-forces the relevance of the approach for low-order turbulence statistics in the presence of a strong mean flow [START_REF] Laurie | Universal Profile of the Vortex Condensate in Two-Dimensional Turbulence[END_REF][START_REF] Frishman | Turbulence Statistics in a Two-Dimensional Vortex Condensate[END_REF]. While in 2DNS fluid-element interactions are long-range, in LQG they are local. Comparing the two cases, our work sheds light on the influence of the range of interactions on the condensate state. Indeed, we show that the statistics at large scales are determined by universal underlying mechanisms. On the other hand, at small scales we find that the locality of interactions has a dramatic influence, with small-scale fluxes and dissipation suppressed in regions where the mean flow is strong.

Framework Both the LQG equation and 2DNS can be derived as limiting cases of the shallow water quasigeostrophic (SWQG) equation. This is an idealized model widely used for flows in the atmosphere and oceans [START_REF] Vallis | Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-Scale Circulation[END_REF], and for magnetically confined plasmas [START_REF] Diamond | Zonal flows in plasma-a review[END_REF]. In the geophysical context, it captures the dynamics of the free surface of a rapidly rotating shallow fluid layer under the influence of gravity. The inviscid SWQG equation reads ∂ t q +v •∇q = ∂ t q +J(ψ, q) = 0, where q = ∇ 2 -L -2 d ψ, is the potential vorticity, J(ψ, q) = ∂ x ψ∂ y q-∂ y ψ∂ x q, and ψ is the stream-function related to the velocity field by v = ẑ × ∇ψ. The flow is assumed to be in geostrophic balance. Thus, ψ is proportional to the deviation of the fluid layer from its mean. The system has L d , as a characteristic length scale, determining the range of influence of a height perturbation. If the domain size L ≪ L d then every fluid element will influence every other, and the flow will become incompressible, giving the 2DNS. The opposite limit L d /L → 0 giving LQG (which can be made consistent with the SWQG approximation [START_REF] Svirsky | Two-dimensional turbulence with local interactions: statistics of the condensate[END_REF]), corresponds to a very rapidly rotating fluid where surface perturbations remain completely localized. The dynamics become slow in this limit, requiring the re-scaling of time as τ = t (L d /L) 2 (and ψ accordingly), resulting in the LQG equation:

∂ τ ψ + v ω • ∇ψ = ∂ τ ψ + J(ω, ψ) = f + α∇ 2 ψ -ν(-∇ 2 ) p ψ, (1)
where we defined an effective velocity v ω = ẑ × ∇ω, f is a forcing term, and the last two terms are dissipativea viscous term referred to as drag in the following, and a hyper-viscosity. The former provides the dominant dissipation mechanism at large scales, while the latter will dominate at small scales. The inviscid invariants of (1) are the kinetic energy Z = 1 2 (∇ψ)

2 d 2 x = 1 2 v 2 d 2
x and all moments of ψ, in particular the potential energy E = 1 2 ψ 2 d 2 x. The existence of two positive quadratic invariants results in the inverse cascade of E and a direct cascade (from large to small scales) of Z [START_REF] Smith | Turbulent diffusion in the geostrophic inverse cascade[END_REF].

The LQG advection equation is similar to 2DNS but with the roles of the vorticity ω = ∇ 2 ψ and the streamfunction reversed. Indeed, LQG and 2DNS are part of a class of active scalar equations where a scalar q is advected by a velocity with stream-function ϕ, and q k = |k| m ϕ k [START_REF] Pierrehumbert | Spectra of local and nonlocal two-dimensional turbulence[END_REF], m controlling the range of the dynamics [START_REF] Venaille | Violent relaxation in two-dimensional flows with varying interaction range[END_REF]. For 2DNS, m = 2 and the dynamics is long range. For LQG m = -2, so that the advecting velocity is determined by active scalar gradients, making the dynamics local.

Simulations Direct Numerical Simulations (DNS) are performed by integrating (1) using the Dedalus framework [START_REF] Keaton | Dedalus: A flexible framework for numerical simulations with spectral methods[END_REF]. Simulating in a doubly periodic box of dimensions L ≡ L y = 2L x = 2π, we enforce the formation of a jet-type condensate by breaking the symmetry between the x y directions [START_REF] Bouchet | Random Changes of Flow Topology in Two-Dimensional and Geophysical Turbulence[END_REF][START_REF] Frishman | Jets or vortices-What flows are generated by an inverse turbulent cascade?[END_REF]. We use a spatial resolution of 64 × 128, a white-in-time forcing with characteristic length scale l f = 2π/13 , hyper-viscosity with p = 7 and ν = 7.3×10 -19 and take α = (0.5, 1, 2)×10 -3 .

Condensation of E at large scales requires a slow dissipation rate compared to the eddy-turnover time at the box scale, amounting to the condition δ ≡ α(L 2 /ϵ) 1/3 ≪ 1, where ϵ = ⟨f ψ⟩ is the energy injection rate. Additional simulation details, including for DNS with other parameters, are in [START_REF] Svirsky | Two-dimensional turbulence with local interactions: statistics of the condensate[END_REF].

The steady state condensate takes the form of two alternating jets along the short side (x direction) of the domain, as demonstrated in Fig. 1. Here the jet structure is selected by the domain geometry, unlike in betaplane turbulence. Between the jets, there are two small vortices, similarly to what was found in 2DNS [START_REF] Frishman | Jets or vortices-What flows are generated by an inverse turbulent cascade?[END_REF].

Mean flow We now set out to obtain a statistical description of the steady state LQG jet condensate. We start by decomposing the flow into the mean Ψ = ⟨ψ⟩ and fluctuations ψ ′ = ψ -Ψ. We will focus on the jet region, where the flow is statistically homogeneous in x, and the mean flow depends on y only. In particular, the mean flow is in the x direction: U = -∂ y Ψ. In the steady state, averaging (1) and neglecting the influence of hyper-viscosity at large scales we obtain

∂ y v ω′ y ψ ′ -α∂ y Ψ = 0, (2)
where we have used that V ω y = ∂ x ⟨ω⟩ = 0. As mentioned before, ψ is proportional to the layer height, thus (2) can be interpreted as a mass balance and ⟨v ω′ y ψ ′ ⟩ as the turbulent mass flux, analogously to the momentum flux or Reynolds stress in 2DNS. We note that a constant non-zero total flux is inconsistent with the symmetries of our system, as there is no preferred direction aside from the one imposed by the emerging mean flow, giving

v ω′ y ψ ′ = α∂ y Ψ. (3)
Next, we obtain the potential energy balance of the mean flow by multiplying (1) by Ψ and averaging:

∂ y Ψ v ω′ y ψ ′ -αΨ∂ y Ψ = v ω′ y ψ ′ ∂ y Ψ-α (∂ y Ψ) 2 . (4)
Similarly, we obtain the potential energy balance of fluctuations by multiplying (1) by ψ ′ and averaging, giving:

∂ y ⟨v ω′ y ψ ′2 2 ⟩ -α⟨ψ ′ ∂ y ψ ′ ⟩ = ϵ -⟨v ω′ y ψ ′ ⟩∂ y Ψ. (5)
In writing [START_REF] Vallis | Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-Scale Circulation[END_REF] we have neglected the dissipation of potential energy by fluctuations, since the development of the condensate implies dissipation by fluctuations is inefficient and thus small. Equations ( 4) and ( 5) reflect the flow of potential energy in the system, where the interaction term ⟨v ω′ y ψ ′ ⟩∂ y Ψ = -U ⟨v ω′ y ψ ′ ⟩ controls the out-ofequilibrium transfer of energy between the fluctuations and the mean flow. The inverse nature of potential energy transfer in this system implies that it goes from the fluctuations to the mean-flow. This requires mass-flux up the gradient of the mean height:

⟨ψ ′ v ω′ ⟩ • ∇Ψ > 0.
Due to relation (3) the left hand side of equation ( 4), corresponding to the spatial flux of mean potential energy Ψ 2 , vanishes. Thus, here unlike in 2DNS [START_REF] Laurie | Universal Profile of the Vortex Condensate in Two-Dimensional Turbulence[END_REF], all the energy input from the fluctuations is dissipated locally by the mean flow. Note that the transfer of potential energy to the mean flow cannot be the dominant process in regions where ∂ y Ψ = -U ≈ 0 since ⟨v ω′ y ψ ′ ⟩∂ y Ψ is small there. For jets in a periodic domain there always must be such a region as ∇Ψd 2 x = 0 identically. Finally, to close the system of equations we assume that interactions with the mean flow are much faster than nonlinear interactions between fluctuations, ⟨v ω′ y ψ ′ ⟩∂ y Ψ ≫ |⟨ψ ′ v ω′ • ∇ψ ′ ⟩|, justified by the development of a strong condensate for δ ≪ 1. This is called the quasi-linear approximation, which has a long history [START_REF] Marston | Recent developments in theories of inhomogeneous and anisotropic turbulence[END_REF], and was recently successfully used in a similar context for 2DNS [START_REF] Laurie | Universal Profile of the Vortex Condensate in Two-Dimensional Turbulence[END_REF]. Unlike 2DNS, in LQG the dynamics are local (e.g. no pressure term), and thus so is the analysis, obviating the need for further approximations.

Using (3) together with [START_REF] Vallis | Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-Scale Circulation[END_REF] in the quasi-linear approximation gives the leading order solution

∂ y Ψ = -U = ± ϵ α , (6)
⟨ψ ′ v ω′ y ⟩ = ± √ ϵα, (7)
implying a flat mean velocity profile U (and mass flux) for each of the jets. We compare these predictions to results from DNS in Fig. 2. For the averaging procedure, after reaching steady state, we perform a temporal (over ∼ 1000 large scale turnover-time) as well as a spatial average along the x-axis, utilizing the homogeneity of the system. For the mean velocity profile, good agreement with ( 6) can be seen in Fig. 2(a), demonstrating convergence to the leading order prediction with decreasing δ. The main difference between DNS and the theoretical prediction is the oscillations of the jet velocity seen in DNS (see also Fig. 1), which persist after averaging over long times. They appear to be a sub-leading correction to the mean flow profile U (controlled by a small parameter other than δ), not captured at leading order. The mass flux ⟨ψ ′ v ω′ y ⟩ is also computed from DNS, shown in Fig. 2(b). It fluctuates about the theoretical prediction, due to rapid fluctuations of v ω′ y requiring additional statistics for convergence. Still, the sign of the mass flux always matches the sign of ∂ y Ψ (in the jet region), as is necessary for the inverse transfer.

Two-point correlation function Having obtained the mean flow profile, we now examine the twopoint (single-time) correlation function ⟨ψ ′ 1 ψ ′ 2 ⟩ ≡ ⟨ψ ′ (r 1 )ψ ′ (r 2 )⟩ where r i = (x i , y i ). We obtain an equation for ⟨ψ ′ 1 ψ ′ 2 ⟩ at leading order in perturbation theory from (1), neglecting dissipation of the fluctuations, using the statistical homogeneity in x, and the quasi-linear approximation to neglect higher order moments [START_REF] Svirsky | Two-dimensional turbulence with local interactions: statistics of the condensate[END_REF]. In addition, we use that from Eq. ( 6), V ω x = 0 at leading order. We thus get

[L 1 + L 2 ] ⟨ψ ′ 1 ψ ′ 2 ⟩ = 2ϵΦ 12 , (8)
where the operator

L i = ∂ yi Ψ(y i )∇ 2 ∂ yi (no summation is implied here) and ⟨f (r 1 , t)f (r 2 , t ′ )⟩ = 2ϵΦ 12 δ(t -t ′ )
is the force two-point correlation function with Φ 12 ≡ Φ(∆x/l f , ∆y/l f ). Using the mean profile Eq. 6, this equation reads where we have defined the variables y + = y1+y2 2 and y -= ∆y/2 = y1-y2 2 . Integrating ( 9) we get:

∂ y+ ∂ y-∂ x1 ⟨ψ ′ 1 ψ ′ 2 ⟩ = 2 √ ϵαΦ 12 , (9)
⟨ψ ′ 1 ψ ′ 2 ⟩ = C(∆x, ∆y)+2y + √ αϵl 2 f ∆x l f 0 dz ∆y 2l f 0 dz ′ Φ (z, z ′ ) , (10
) with Φ(x, y) = Φ(x, y) -Φ(k x = 0, y) -Φ(x, k y = 0), subtracting the contribution to the forcing from k x = 0 and k y = 0 modes [START_REF] Svirsky | Two-dimensional turbulence with local interactions: statistics of the condensate[END_REF]. The second term in [START_REF] Frishman | The culmination of an inverse cascade: Mean flow and fluctuations[END_REF] is the inhomogeneous solution to equation ( 9), due to a balance between the forcing and the mean flow. The first term, C(∆x, ∆y), is a homogeneous solution of (10), i.e. a zero mode of the advection operator

L 1 + L 2 = ∂ y+ ∂ y-∂ x1 .
This family of zero modes is chosen assuming decaying correlations with ∆y → L/2, and using the result from DNS that the variance ⟨(ψ ′ ) 2 ⟩ is constant in the jet region [START_REF] Svirsky | Two-dimensional turbulence with local interactions: statistics of the condensate[END_REF]. Deriving the functional form and amplitude of the zero modes is beyond the scope of this work. Empirically, our DNS results [START_REF] Svirsky | Two-dimensional turbulence with local interactions: statistics of the condensate[END_REF] point to the scaling ⟨u ′2 ⟩ = -∂ 2 ∆y C| (0,0) ∼ (ϵL) 2/3 δ -1/2 and ⟨v ′2 ⟩ = -∂ 2 ∆x C| (0,0) ∼ (ϵL) 2/3 δ 1/4 , while from Eq. ( 7) the off-diagonal term ⟨u ′ v ′ ⟩ = (ϵL) 2/3 (y/L)δ 1/2 . Thus, the perturbation theory is indeed consistent, at most

⟨u ′2 ⟩/U 2 ∼ δ 1/2 ≪ 1.
PT symmetry breaking The condensate state owes its existence to out-of-equilibrium fluxes which break time-reversal symmetry. On the level of single time correlators such symmetry breaking will manifest itself through a combined parity+time reversal (PT), x → -x and t → -t, breaking. Indeed, the inviscid LQG dynam- ics with a homogeneous in x mean flow is statistically invariant under PT. However, the mass flux ⟨ψ ′ v ω′ y ⟩ (responsible for the inverse transfer of energy) is odd under this symmetry and can be non-zero only if it is broken.

We propose that even and odd under PT correlators are determined through different mechanisms, reflected in the decomposition of the two-point correlator [START_REF] Frishman | The culmination of an inverse cascade: Mean flow and fluctuations[END_REF] into, respectively, the zero modes and inhomogeneous solution. The inhomogeneous solution is determined by a balance between the forcing, which breaks time reversal symmetry, and the mean flow, which breaks parity (but preserves PT). The homogeneous solution, on the other hand, is insensitive to the forcing and instead is a zero mode of the advection operator L 1 + L 2 . A similar structure was inferred in 2DNS [START_REF] Frishman | Turbulence Statistics in a Two-Dimensional Vortex Condensate[END_REF][START_REF] Kolokolov | Structure of coherent vortices generated by the inverse cascade of twodimensional turbulence in a finite box[END_REF][START_REF] Frishman | The culmination of an inverse cascade: Mean flow and fluctuations[END_REF].

The inhomogeneous term in [START_REF] Frishman | The culmination of an inverse cascade: Mean flow and fluctuations[END_REF] is indeed odd under PT: it changes sign under reflection x → -x (so that ∆x → -∆x) and, being a single-time quantity, does not explicitly depend on time. We propose that it fully determines odd correlators. The mass flux ⟨ψ ′ v ω′ y ⟩ can indeed be directly computed from it, in agreement with [START_REF] Chertkov | Dynamics of energy condensation in twodimensional turbulence[END_REF] [START_REF] Svirsky | Two-dimensional turbulence with local interactions: statistics of the condensate[END_REF]. The limit of large scale-separation L/l f ≫ 1 is interesting to consider at this point, as in this limit we expect universal behaviour, independent of forcing details. This is manifested in Eq. ( 10): the influence of the inhomogeneous term becomes limited to separations ∆x, ∆y ≲ l f , meaning that its main role becomes to determine singlepoint (odd) correlation functions [START_REF] Svirsky | Two-dimensional turbulence with local interactions: statistics of the condensate[END_REF].

Turning to even-under-PT correlators, if they are determined by zero modes then those should be even under PT, implying C(-∆x, ∆y) = C(∆x, ∆y). We verify this property, as well as that the even part of ⟨ψ ′ 1 ψ ′ 2 ⟩ is independent of y + in the jet region, as expected for the zero modes, in [START_REF] Svirsky | Two-dimensional turbulence with local interactions: statistics of the condensate[END_REF].

Arrest of the direct cascade The condensate necessarily coexists with a direct cascade to small scales of a second inviscidly conserved quantity-the kinetic energy in LQG. As the direct cascade involves scales from the forcing scale and smaller, we generally expect the mean flow to have a negligible effect on it. This is indeed the case in 2DNS [START_REF] Frishman | Jets or vortices-What flows are generated by an inverse turbulent cascade?[END_REF] where the direct cascade of enstrophy (squared vorticity) remains homogeneous and isotropic, as reflected in an essentially spatially uniform enstrophy dissipation rate. Surprisingly, we find this is not the case in LQG, as seen in Fig. 3(a): the kinetic energy dissipation is concentrated in regions in between the jets, being close to zero elsewhere. To explain this inhomogeneity we turn to the kinetic energy balance, obtained by multiplying the derivative ∂ i of (1) by ∂ i ψ ′ and averaging. In the steady state, the balance of kinetic energy for the fluctuations reads [START_REF] Svirsky | Two-dimensional turbulence with local interactions: statistics of the condensate[END_REF]:

∂ y J ′ y + I ′ D = η -D ′ + T, (11)
where η is the kinetic energy injection rate, I ′ D and D ′ originate from the viscous terms and are the respective flux and dissipation rate, T = ∂ y U ∂ y ⟨v ω′ y ψ ′ ⟩ is a transfer term between the mean flow and the fluctuations, and J ′ y is a spatial kinetic energy flux given by

J ′ y = U ⟨v ω′ y u ′ ⟩ + ⟨∂ y ψ ′ v ω′ • ∇ψ ′ ⟩ -⟨ω ′ v ω′ y ψ ′ ⟩. (12)
The transfer term T both here and in 2DNS can be estimated to generically be much smaller than the injection rate η [START_REF] Svirsky | Two-dimensional turbulence with local interactions: statistics of the condensate[END_REF]. It identically vanishes for the leading order solution ( 6), [START_REF] Chertkov | Dynamics of energy condensation in twodimensional turbulence[END_REF]. Furthermore, we expect the kinetic energy fluctuations to have homogeneous statistics at small scales (where they reside), so that spatial fluxes involving solely fluctuations should be small. We thus expect the balance

∂ y U ⟨v ω′ y u⟩ = η -D ′ . (13)
The key difference between Eq. ( 13) and the analogous balance in 2DNS is in the spatial flux due to the mean flow, preventing a local balance between injection and dissipation. It is this spatial flux which arrests the direct cascade in regions where the mean flow U is strong, carrying the injected kinetic energy away from those regions. By a direct computation using [START_REF] Frishman | The culmination of an inverse cascade: Mean flow and fluctuations[END_REF], see [START_REF] Svirsky | Two-dimensional turbulence with local interactions: statistics of the condensate[END_REF], we find that

U ⟨v ω′ y u ′ ⟩ = ∂ y Ψ⟨∂ y ψ ′ v ω′ y ⟩ = ηy. ( 14
)
This is confirmed by a direct comparison to DNS in the jet region, Fig. 3(b). Combining [START_REF] Farrell | Structure and Spacing of Jets in Barotropic Turbulence[END_REF] and ( 14) we find that the spatial flux carries all the injected kinetic energy before it has time to cascade to small scales and dissipate there, in agreement with Fig. 3(a).

Conclusion

We presented results from a perturbative analysis and direct simulations of LQG in the condensate regime, observed here for the first time. Obtaining the two-point correlation functions, we have revealed the different mechanisms determining the odd and even correlators under parity+time reversal. While the former are determined at small scales by a balance between the forcing (breaking time reversal) and the mean flow (breaking parity), the latter are constrained to be zero modes of an advection operator. In fact, this is also the underlying structure in 2DNS [START_REF] Frishman | Turbulence Statistics in a Two-Dimensional Vortex Condensate[END_REF][START_REF] Kolokolov | Structure of coherent vortices generated by the inverse cascade of twodimensional turbulence in a finite box[END_REF][START_REF] Frishman | The culmination of an inverse cascade: Mean flow and fluctuations[END_REF], suggesting a universal picture across different 2D flows. We also find an unexpected influence of the mean flow on the direct cascade, which is locally arrested in regions with a strong inverse energy transfer. This phenomenon, absent in 2DNS, is explained by the formation of an outof-equilibrium spatial flux out of regions where the mean flow is strong. This flux term is a consequence of the locality of the dynamics, and is absent for active scalar models with m > 0 and for SWQG with L d > l f [START_REF] Svirsky | Two-dimensional turbulence with local interactions: statistics of the condensate[END_REF]. It is however not ruled out for SWQG with L d < l f , of which LQG is a limiting case, and it will be interesting to see if a similar expulsion of the direct cascade could be observed there.

FIG. 1 .

 1 FIG. 1. LQG jet condensate in a periodic, rectangular box with δ = 0.052. (a) Vorticity snapshot ω = ∇ 2 ψ and (b) Velocity snapshot v = ẑ × ∇ψ.

FIG. 2 .

 2 FIG. 2. (a) Horizontal velocity profile ∂yΨ, dashed line is the prediction (6); (b) Mass flux ⟨v ω′ y ψ ′ ⟩, dashed line is the prediction (7).

5 FIG. 3 .

 53 FIG. 3. (a) Mean kinetic energy dissipation normalized by η; (b) The term v ω′ y v ′x as measured from DNS, the theoretical prediction (14) is marked by the dashed line.

Acknowledgments: A.F. would like to thank Guido Boffettta and Stefano Musacchio, and acknowledge the 2017 SCGP workshop on fluid flows, for nucleating the idea for this project.