arXiv subjects

Hubert Baty

Publications and source records attributed to Hubert Baty.

15 recordsLinked to original sources

arXiv2026

Triggering physical plasmoids in forming current sheets: conditions and diagnostics

We investigate the conditions for triggering the plasmoid instability in a dynamically forming current sheet in the resistive magnetohydrodynamic framework, using a pseudo-spectral code applied to the Orszag-Tang vortex at Lundquist number $S \sim 10^5$. Following Garc\'ia Morillo \& Alexakis (2025), we use the power spectrum of the current density $E_J(k)$, complemented by the vorticity spectrum $E_\omega(k)$, to assess the convergence of our simulations, and show that this diagnostic remains valid even in the presence of physical plasmoids, allowing us to unambiguously distinguish them from spurious ones. We then show that physical plasmoids can be triggered in a well-resolved spectral simulation when three conditions are simultaneously met: a perturbation applied near the time of maximum current density, with amplitude above a critical threshold $\varepsilon_c \sim 10^{-5}$ for our numerical scheme, and with spectral content containing the unstable wavenumbers. These conditions are confirmed using continuous noise injection, which yields similar results at amplitudes one to two orders of magnitude lower. The resulting growth rates and plasmoid numbers are in good agreement with the theory of \citet{Comisso2017}. These results resolve the apparent paradox raised by Garc\'ia Morillo \& Alexakis (2025) and also clarify the role of numerical noise in the triggering of the plasmoid instability.

physics.plasm-ph↗

arXiv2025

Physics-Informed Neural Networks for Solving Forward and Inverse PDEs with Limited and Noisy Data: Application to Solar Corona Modeling

I will demonstrate the effectiveness of Physics-Informed Neural Networks (PINNs) in solving partial differential equations (PDEs) when training data are scarce or noisy. The training data can be located either at the boundaries or within the domain. Additionally, PINNs can be used as an inverse method to determine unknown coefficients in the equations. This study will highlight the application of PINNs in modeling magnetohydrodynamic processes relevant to strongly magnetized plasmas, such as those found in the solar corona.

astro-ph.SR↗

arXiv2024

A hands-on introduction to Physics-Informed Neural Networks for solving partial differential equations with benchmark tests taken from astrophysics and plasma physics

I provide an introduction to the application of deep learning and neural networks for solving partial differential equations (PDEs). The approach, known as physics-informed neural networks (PINNs), involves minimizing the residual of the equation evaluated at various points within the domain. Boundary conditions are incorporated either by introducing soft constraints with corresponding boundary data values in the minimization process or by strictly enforcing the solution with hard constraints. PINNs are tested on diverse PDEs extracted from two-dimensional physical/astrophysical problems. Specifically, we explore Grad-Shafranov-like equations that capture magnetohydrodynamic equilibria in magnetically dominated plasmas. Lane-Emden equations that model internal structure of stars in sef-gravitating hydrostatic equilibrium are also considered. The flexibility of the method to handle various boundary conditions is illustrated through various examples, as well as its ease in solving parametric and inverse problems. The corresponding Python codes based on PyTorch/TensorFlow libraries are made available.

physics.comp-ph↗

arXiv2023

Modelling solar coronal magnetic fields with physics-informed neural networks

We present a novel numerical approach aiming at computing equilibria and dynamics structures of magnetized plasmas in coronal environments. A technique based on the use of neural networks that integrates the partial differential equations of the model, and called Physics-Informed Neural Networks (PINNs), is introduced. The functionality of PINNs is explored via calculation of different magnetohydrodynamic (MHD) equilibrium configurations, and also obtention of exact two-dimensional steady-state magnetic reconnection solutions (Craig & Henton 1995). Advantages and drawbacks of PINNs compared to traditional numerical codes are discussed in order to propose future improvements. Interestingly, PINNs is a meshfree method in which the obtained solution and associated different order derivatives are quasi-instantaneously generated at any point of the spatial domain. We believe that our results can help to pave the way for future developments of time dependent MHD codes based on PINNs

astro-ph.SR↗

arXiv2023

Solving higher-order Lane-Emden-Fowler type equations using physics-informed neural networks: benchmark tests comparing soft and hard constraints

In this paper, numerical methods using Physics-Informed Neural Networks (PINNs) are presented with the aim to solve higher-order ordinary differential equations (ODEs). Indeed, this deep-learning technique is successfully applied for solving different classes of singular ODEs, namely the well known second-order Lane-Emden equations, third order-order Emden-Fowler equations, and fourth-order Lane-Emden-Fowler equations. Two variants of PINNs technique are considered and compared. First, a minimization procedure is used to constrain the total loss function of the neural network, in which the equation residual is considered with some weight to form a physics-based loss and added to the training data loss that contains the initial/boundary conditions. Second, a specific choice of trial solutions ensuring these conditions as hard constraints is done in order to satisfy the differential equation, contrary to the first variant based on training data where the constraints appear as soft ones. Advantages and drawbacks of PINNs variants are highlighted.

physics.comp-ph↗

arXiv2023

Solving stiff ordinary differential equations using physics informed neural networks (PINNs): simple recipes to improve training of vanilla-PINNs

Physics informed neural networks (PINNs) are nowadays used as efficient machine learning methods for solving differential equations. However, vanilla-PINNs fail to learn complex problems as ones involving stiff ordinary differential equations (ODEs). This is the case of some initial value problems (IVPs) when the amount of training data is too small and/or the integration interval (for the variable like the time) is too large. We propose very simple recipes to improve the training process in cases where only prior knowledge at initial time of training data is known for IVPs. For example, more physics can be easily embedded in the loss function in problems for which the total energy is conserved. A better definition of the training data loss taking into account all the initial conditions can be done. In a progressive learning approach, it is also possible to use a growing time interval with a moving grid (of collocation points) where the differential equation residual is minimized. These improvements are also shown to be efficient in PINNs modeling for solving boundary value problems (BVPs) as for the high Reynolds steady-state solution of advection-diffusion equation.

physics.comp-ph↗

arXiv2023

Solving differential equations using physics informed deep learning: a hand-on tutorial with benchmark tests

We revisit the original approach of using deep learning and neural networks to solve differential equations by incorporating the knowledge of the equation. This is done by adding a dedicated term to the loss function during the optimization procedure in the training process. The so-called physics-informed neural networks (PINNs) are tested on a variety of academic ordinary differential equations in order to highlight the benefits and drawbacks of this approach with respect to standard integration methods. We focus on the possibility to use the least possible amount of data into the training process. The principles of PINNs for solving differential equations by enforcing physical laws via penalizing terms are reviewed. A tutorial on a simple equation model illustrates how to put into practice the method for ordinary differential equations. Benchmark tests show that a very small amount of training data is sufficient to predict the solution when the non linearity of the problem is weak. However, this is not the case in strongly non linear problems where a priori knowledge of training data over some partial or the whole time integration interval is necessary.

cs.LG↗

arXiv2022

Hyperbolic method to explore multiplicity flow solutions in a four-sided lid-driven cavity

In this study, the hyperbolic method is adopted to explore the flow field states of incompressible flow in a four-sided lid-driven square cavity. In particular, we focus on the flow bifurcation obtained at the critical Reynolds number $R_e \simeq 130$. In the hyperbolic method, the diffusive term is transformed into an hyperbolic one by introducing a diffusion flux term, which is the solution of an additional equation. A classical Riemann-like solver with a finite-volume discretization is thus employed for the full flux (splitted into advective and diffusive parts), in order to solve the steady-state incompressible Navier-Stokes equation. The incompressibility of the flow is treated via the artificial pseudo-compressibility method. It is shown that our numerical code is able to detect the bifurcation, by the analysis of the residual term relaxation during the pseudo-time iteration procedure. Moreover, depending on the combination choice of slope limiters for the two spatial directions, our method is able to select the first or the second stable solution among the double flow field state obtained when the Reynolds number is higher than the critical value that is estimated to be $129.4$ in our study.

physics.flu-dyn↗

arXiv2022

On the Sweet-Parker model for incompressible visco-resistive magnetic reconnection in two dimensions associated to ideal magnetohydrodynamic instabilities

We revisit the well known Sweet-Parker (SP) model for magnetic reconnection in the framework of two dimensional incompressible magnetohydrodynamics. The steady-state solution is re-derived by considering a non zero viscosity via the magnetic Prandtl number $P_m$. Moreover, contrary to the original SP model, a particular attention is paid to the possibility that the inflowing magnetic field $B_e$ and the length of the current layer $L$ are not necessarily fixed and may depend on the dissipation parameters. Using two different ideally unstable setups to form the current sheet, namely the tilt and coalescence modes, we numerically explore the scaling relations with resistivity $\eta$ and Prandtl number $P_m$ during the magnetic reconnection phase, and compare to the generalized steady-state SP theoretical solution. The usual Sweet-Parker relations are recovered in the limit of small $P_m$ and $\eta$ values, with in particular the normalized reconnection rate being simply $S^{-1/2} (1 + P_m)^{-1/4}$, where $S$ represents the Lundquist number $S = LV_A/\eta$ ($V_A$ being the characteristic Alfv\'en speed). In the opposite limit of higher $P_m$ and/or $\eta$ values, a significant deviation from the SP model is obtained with a complex dependence $B_e (\eta, P_m)$ that is explored depending on the setup considered. We discuss the importance of these results in order to correctly interpret the numerous exponentially increasing numerical studies published in the literature, with the aim of explaining eruptive phenomena observed in the solar corona.

astro-ph.SR↗

arXiv2021

On the explosive phase of the tearing mode in double current sheet plasmas: effect of the equilibrium magnetic configuration on the onset threshold and growth rate

Magnetic reconnection associated with the tearing instability occurring in double-current sheet systems is investigated within the framework of reduced resistive magnetohydrodynamics (MHD) in a two-dimensional Cartesian geometry. The explosive non linear phase is particularly explored using the adaptive finite-element FINMHD code. The critical aspect ratio, that is defined as the minimum $L/x_s$ ratio (with $L$ and $x_s$ being the periodic system length and half-distance between the two current layers respectively) necessary for non linear destabilization after the linear and early non linear saturation phases, is obtained. The latter threshold is independent of the details of the chosen initial equilibrium (double Harris-like magnetic profile) and of the resistivity. Its value is shown to be $4.7$, that is close and slightly smaller than the value of order $5$ deduced using a more particular equilibrium configuration in previous studies. The time dependence of the kinetic energy ($E_K$) is shown to follow a double exponential law, $E_K \propto \exp \ [e^{(\gamma^* t)} ]$, with a pseudo-growth rate $\gamma^* \simeq 0.1 \ t_A^ {-1}$ ($t_A$ being the characteristic Alfv\'en time) that is again independent of the configuration and resistivity. The mechanism offers a possible explanation for the sudden onset of explosive magnetic energy release occurring on the fast Alfv\'en time scale in disruptive events of astrophysical plasmas with pre-existing double current sheets like in the solar corona.

astro-ph.SR↗

arXiv2020

Magnetic reconnection and plasmoid formation driven by the two-dimensional resistive kink instability in a current hole configuration

We investigate the nonlinear evolution of the $m = 1$ resistive internal kink mode in a two-dimensional (2D) configuration containing a central region of negative current density, also known as the "current hole" setup. The finite-element code FINMHD is used to solve a reduced set of incompressible Magnetohydrodynamic (MHD) equations with a current-vorticity formulation. First, the kink instability linearly develops in agreement with the general theory of resistive internal kink mode, and it subsequently leads to the formation of a current sheet. At relatively low Lundquist number $S$, a magnetic reconnection process proceeds with a rate predicted by the Sweet-Parker regime. Conversely, when $S$ exceeds a critical value that is $S_c \simeq 10^4$, the current sheet is disrupted by the formation of plasmoids on a slightly sub-Alfv\'enic time scale. In the latter case, a stochastic reconnection regime exhibiting Petshek-type features enriched by plasmoids is reached. A relatively fast normalized reconnection rate value of order $0.02$ is also measured. Finally, we compare our results with those obtained in similar 2D previous studies using ideal MHD instabilities to initiate the process, and discuss their relevance for the general theory of plasmoid chains formation and associated fast reconnection regime.

physics.plasm-ph↗

arXiv2020

Petschek-type reconnection in the high-Lundquist-number regime during nonlinear evolution on the tilt instability

The process of fast magnetic reconnection supported by the formation of plasmoid chains in the high Lundquist number ($S$) regime is investigated using a recently developed adaptive finite-element magnetohydrodynamic (MHD) code. We employ a two-dimensional incompressible model with a set of reduced visco-resistive MHD equations. The tilt instability setup is chosen to provide a three-step mechanism, where two curved current sheets initially form on an Alfv\'enic time scale followed by a second phase of super-Alfv\'enic growth of plasmoid chains for $S \ge S_c$ (Baty 2020). A third phase is reached where an ensuing stochastic time-dependent reconnection regime with a fast time-averaged rate independent of $S$ is obtained. We reveal the multi-scale current structures during magnetic reconnection, where merging events of plasmoids give rise to monster plasmoids with shocks bounding the outflow regions. At high enough $S$ values (typically for $S \sim 100 S_c$), a dynamical Petschek-type reconnection is achieved with pairs of slow-mode shocks emanating from a small central region containing a few plasmoids. Finally, we briefly discuss the relevance of our results to explain the flaring activity in solar corona and internal disruptions in tokamaks.

astro-ph.SR↗

arXiv2020

On the growth rate of plasmoid chains during nonlinear viscoresistive evolution of the tilt instability

We investigate by means of two-dimensional incompressible magnetohydrodynamic (MHD) numerical simulations, the onset phase of the fast collisional magnetic reconnection regime that is supported by the formation of plasmoid chains when the Lundquist number S exceeds a critical value. The present study extends previous results obtained at magnetic Prandtl number Pm = 1 (Baty 2020) to a range of different Pm values. We use FINMHD code where a set of reduced visco-resistive MHD equations is employed to form two quasi-singular current layers as a consequence of the tilt instability. The results reinforce the conclusion that, a phase of sudden super-Alfvenic growth (when Pm is not too high) of plasmoid chains is obtained, following a previous quiescent phase during current sheet formation on a slower Alfvenic time scale. We compare our results with predictions from the general theory of the plasmoid instability. We also discuss the importance of this onset phase to reach the ensuing stochastic time-dependent reconnection regime, where a fast time-averaged rate independent of S is obtained. Finally, we briefly discuss the relevance of our results to explain the flaring activity in solar corona and internal disruptions in tokamaks.

astro-ph.SR↗

arXiv2020

Formation of plasmoid chains and fast magnetic reconnection during nonlinear evolution of the tilt instability

We investigate, by means of two-dimensional incompressible magnetohydrodynamic (MHD) numerical simulations, the fast collisional magnetic reconnection regime that is supported by the formation of plasmoid chains when the Lundquist number $S$ exceeds a critical value (at magnetic Prandtl number, $P_m = 1$). A recently developed characteristic-Galerkin finite-element code, FINMHD, that is specifically designed for this aim in a reduced visco-resistive MHD framework, is employed. Contrary to previous studies, a different initial setup of two repelling current channels is chosen in order to form two quasi-singular current layers on an Alfv\'enic time scale as a consequence of the tilt instability. If $S < 5 \times 10^3$, a subsequent stationary reconnection process is obtained with a rate scaling as $S^{-1/2}$ as predicted by the classical Sweet-Parker model. Otherwise, a stochastic time-dependent reconnection regime occurs, with a fast time-averaged rate independent of $S$ and having a normalized value of $0.014$. The latter regime is triggered by the formation of two chains of plasmoids disrupting the current sheets with a sudden super-Alfv\'enic growth following a quiescent phase, in agreement with the general theory of the plasmoid instability proposed by Comisso et al. [Phys. Plasmas 23, 100702 (2016)]. Moreover, the non-monotonic dependence of the plasmoid growth rate with $S$ following an asymptotically decreasing logarithmic law in the infinite $S$-limit is confirmed. We also closely compare our results to those obtained during the development of the coalescence instability setup in order to assess the generality of the mechanism. Finally, we briefly discuss the relevance of our results to explain the flaring activity in solar corona and internal disruptions in tokamaks.

astro-ph.SR↗

arXiv2019

FINMHD: an adaptive finite element code for magnetic reconnection and plasmoid chains formation in Magnetohydrodynamics

Solving the problem of fast eruptive events in magnetically dominated astrophysical plasmas requires the use of particularly well adapted numerical tools. Indeed, the central mechanism based on magnetic reconnection is determined by a complex behavior with quasi-singular forming current layers enriched by their associated small scale magnetic islands called plasmoids. A new code for the solution of two dimensional dissipative magnetohydrodynamics (MHD) equations in cartesian geometry specifically developed to this end is thus presented. A current-vorticity formulation representative of an incompressible model is chosen in order to follow the formation of the current sheets and the ensuing magnetic reconnection process. A finite element discretization using triangles with quadratic basis functions on an unstructured grid is employed, and implemented via a highly adaptive characteristic- Galerkin scheme. The adaptivity of the code is illustrated on simplified test equations and finally for magnetic reconnection associated to the non linear development of the tilt instability between two repelling current channels. Varying the Lundquist number S, has allowed to study the transition between the steady-state Sweet-Parker reconnection regime (for S < 10^4) and plasmoids dominated reconnection one (for S > 10^5). The implications for the understanding of the mechanism explaining the fast conversion of free magnetic energy in astrophysical environments such as in solar corona are briefly discussed.

astro-ph.HE↗