arXiv subjects

Omar Maj

Publications and source records attributed to Omar Maj.

14 recordsLinked to original sources

arXiv2025

Time-splitting methods for the cold-plasma model using Finite Element Exterior Calculus

In this work we propose a high-order structure-preserving discretization of the cold plasma model which describes the propagation of electromagnetic waves in magnetized plasmas. By utilizing B-Splines Finite Elements Exterior Calculus, we derive a space discretization that preserves the underlying Hamiltonian structure of the model, and we study two stable time-splitting geometrical integrators. We approximate an incoming wave boundary condition in such a way that the resulting schemes are compatible with a time-harmonic / transient decomposition of the solution, which allows us to establish their long-time stability. This approach readily applies to curvilinear and complex domains. We perform a numerical study of these schemes which compares their cost and accuracy against a standard Crank-Nicolson time integrator, and we run realistic simulations where the long-term behaviour is assessed using frequency-domain solutions. Our solvers are implemented in the Python library Psydac which makes them memory-efficient, parallel and essentially three-dimensional.

math.NA↗

arXiv2024

Construction of an invertible mapping to boundary conforming coordinates for arbitrarily shaped toroidal domains

Boundary conforming coordinates are commonly used in plasma physics to describe the geometry of toroidal domains, for example, in three-dimensional magnetohydrodynamic equilibrium solvers. The magnetohydrodynamic equilibrium configuration can be approximated with an inverse map, defining nested surfaces of constant magnetic flux. For equilibrium solvers that solve for this inverse map iteratively, the initial guess for the inverse map must be well defined and invertible. Even if magnetic islands are to be included in the representation, boundary conforming coordinates can still be useful, for example to parametrize the interface surfaces in multi-region, relaxed magnetohydrodynamics or as a general-purpose, field-agnostic coordinate system in strongly shaped domains. Given a fixed boundary shape, finding a valid boundary conforming mapping can be challenging, especially for the non-convex boundaries from recent developments in stellarator optimization. In this work, we propose a new algorithm to construct such a mapping, by solving two Dirichlet-Laplace problems via a boundary integral method. We can prove that the generated harmonic map is always smooth and has a smooth inverse. Furthermore, we can find a discrete approximation of the mapping that preserves this property.

physics.plasm-ph↗

arXiv2024

Computing MHD equilibria of stellarators with a flexible coordinate frame

For the representation of axi-symmetric plasma configurations, it is natural to use cyl. coordinates (R,Z,$\phi$), where $\phi$ is an independent coordinate. The same cyl. coordinates have also been widely used for representing 3D MHD equilibria of non-axisymmetric configurations (stellarators), with cross-sections, defined in RZ-planes, that vary over $\phi$. Stellarator equilibria have been found, however, for which cyl. coordinates are not at all a natural choice, for instance certain stellarators obtained using the near-axis expansion (NAE), defined by a magn. axis curve and its Frenet frame. In this contribution, we propose an alternative approach for representing the boundary in a fixed-boundary 3D MHD equil. solver, moving away from cyl. coordinates. Instead, we use planar cross-sections whose orientation is determined by a general coordinate frame (G-Frame). This frame is similar to the conventional Frenet frame, but more flexible. As an additional part of the boundary representation, it becomes an input to the equil. solve, along with the geometry of the cross-sections. We see two advantages: 1) the capability to easily represent configurations where the magn. axis is highly non-planar or even knotted 2) a reduction in the degrees of freedom needed for the boundary surface, and thus the equil. solver, enabling progress in optimization of these configurations. We discuss the properties of the G-Frame, starting from the conventional Frenet frame. Then we show two exemplary ways of constructing it, first from a NAE solution and also from a given boundary surface. We present the details of the implementation of the new frame in the 3D MHD equil. solver GVEC. Furthermore, we demonstrate for a highly shaped QI-optimized stellarator that far fewer degrees of freedom are necessary to find a high quality equil. solution, compared to the solution computed in cyl. coordinates.

physics.plasm-ph↗

arXiv2024

Modulational instability of Geodesic-Acoustic-Mode packets

Isolated, undamped geodesic-acoustic-mode (GAM) packets have been demonstrated to obey a (focusing) nonlinear Schr\"odinger equation (NLSE) [E. Poli, Phys. Plasmas 2021]. This equation predicts susceptibility of GAM packets to the modulational instability (MI). The necessary conditions for this instability are analyzed analytically and numerically using the NLSE model. The predictions of the NLSE are compared to gyrokinetic simulations performed with the global particle-in-cell code ORB5, where GAM packets are created from initial perturbations of the axisymmetric radial electric field $E_r$. An instability of the GAM packets with respect to modulations is observed both in cases in which an initial perturbation is imposed and when the instability develops spontaneously. However, significant differences in the dynamics of the small scales are discerned between the NLSE and gyrokinetic simulations. These discrepancies are mainly due to the radial dependence of the strength of the nonlinear term, which we do not retain in the solution of the NLSE, and to the damping of higher spectral components. The damping of the high-$k_r$ components, which develop as a consequence of the nonlinearity, can be understood in terms of Landau damping. The influence of the ion Larmor radius $\rho_i$ as well as the perturbation wavevector $k_{\text{pert}}$ on this effect is studied. For the parameters considered here the aforementioned damping mechanism hinders the MI process significantly from developing to its full extent and is strong enough to stabilize some of the (according to the undamped NLSE model) unstable wavevectors.

physics.plasm-ph↗

arXiv2022

Uniqueness of the Cauchy datum for the tempered-in-time response and conductivity operator of a plasma

We study the linear Vlasov equation with a given electric field $E \in \mathcal{S}$, where $\mathcal{S}$ is the space of Schwartz functions. The associated damped partial differential equation has a unique tempered solution, which fixes the needed Cauchy datum. This tempered solution then converges to the causal solution of the linear Vlasov equation when the damping parameter goes to zero. This result allows us to define the plasma conductivity operator $σ$, which gives the current density $j = σ(E)$ induced by the electric field $E$. We prove that $σ$ is continuous from $\mathcal{S}$ to its dual $\mathcal{S}^\prime$. We can treat rigorously the case of uniform non-magnetized non-relativistic plasma (linear Landau damping) and the case of uniform magnetized relativistic plasma (cyclotron damping). In both cases, we demonstrate that the main part of the conductivity operator is a pseudo-differential operator and we give its expression rigorously. This matches the formal results widely used in the theoretical physics community.

math.AP↗

arXiv2018

The deteriorating effect of plasma density fluctuations on microwave beam quality

Turbulent plasma edge density fluctuations can broaden a traversing microwave beam degrading its quality. This can be a problem for scenarios relying on a high spatial localization of the deposition of injected microwave power, like controlling MHD instabilities. Here we present numerical estimations of the scattering of a microwave beam by density fluctuations over a large parameter range, including extrapolations to ITER. Two codes are used, the full-wave code IPF-FDMC and the wave kinetic equation solver WKBeam. A successful comparison between beam broadening obtained from DIII-D experiments and corresponding full-wave simulations is shown.

physics.plasm-ph↗

arXiv2018

Microwave beam broadening due to turbulent plasma density fluctuations within the limit of the Born approximation and beyond

Plasma turbulence, and edge density fluctuations in particular, can under certain conditions broaden the cross-section of injected microwave beams significantly. This can be a severe problem for applications relying on well-localized deposition of the microwave power, like the control of MHD instabilities. Here we investigate this broadening mechanism as a function of fluctuation level, background density and propagation length in a fusion-relevant scenario using two numerical codes, the full-wave code IPF-FDMC and the novel wave kinetic equation solver WKBeam. The latter treats the effects of fluctuations using a statistical approach, based on an iterative solution of the scattering problem (Born approximation). The full-wave simulations are used to benchmark this approach. The Born approximation is shown to be valid over a large parameter range, including ITER-relevant scenarios.

physics.plasm-ph↗

arXiv2017

Variational Integrators for Ideal Magnetohydrodynamics

A variational integrator for ideal magnetohydrodynamics is derived by applying a discrete action principle to a formal Lagrangian. Discrete exterior calculus is used for the discretisation of the field variables in order to preserve their geometrical character. The resulting numerical method is free of numerical resistivity, thus the magnetic field line topology is preserved and unphysical reconnection is absent. In 2D numerical examples we find that important conservation laws like total energy, magnetic helicity and cross helicity are satisfied within machine accuracy.

math.NA↗

arXiv2015

The field line map approach for simulations of magnetically confined plasmas

In the presented field line map approach the simulation domain of a tokamak is covered with a cylindrical grid, which is Cartesian within poloidal planes. Standard finite-difference methods can be used for the discretisation of perpendicular (w.r.t.~magnetic field lines) operators. The characteristic flute mode property $\left(k_{\parallel}\ll k_{\perp}\right)$ of structures is exploited computationally by a grid sparsification in the toroidal direction. A field line following discretisation of parallel operators is then required, which is achieved via a finite difference along magnetic field lines. This includes field line tracing and interpolation or integration. The main emphasis of this paper is on the discretisation of the parallel diffusion operator. Based on the support operator method a scheme is constructed which exhibits only very low numerical perpendicular diffusion. The schemes are implemented in the new code GRILLIX, and extensive benchmarks are presented which show the validity of the approach in general and GRILLIX in particular. The main advantage of the approach is that it does not rely on field/flux-aligned, which become singular on the separatrix/X-point. Most tokamaks are based on the divertor concept, and the numerical treatment of the separatrix is therefore of importance for simulations of the edge and scrape-off layer.

physics.plasm-ph↗

arXiv2014

Variational Integrators for Nonvariational Partial Differential Equations

Variational integrators for Lagrangian dynamical systems provide a systematic way to derive geometric numerical methods. These methods preserve a discrete multisymplectic form as well as momenta associated to symmetries of the Lagrangian via Noether's theorem. An inevitable prerequisite for the derivation of variational integrators is the existence of a variational formulation for the considered problem. Even though for a large class of systems this requirement is fulfilled, there are many interesting examples which do not belong to this class, e.g., equations of advection-diffusion type frequently encountered in fluid dynamics or plasma physics. On the other hand, it is always possible to embed an arbitrary dynamical system into a larger Lagrangian system using the method of formal (or adjoint) Lagrangians. We investigate the application of the variational integrator method to formal Lagrangians, and thereby extend the application domain of variational integrators to include potentially all dynamical systems. The theory is supported by physically relevant examples, such as the advection equation and the vorticity equation, and numerically verified. Remarkably, the integrator for the vorticity equation combines Arakawa's discretisation of the Poisson brackets with a symplectic time stepping scheme in a fully covariant way such that the discrete energy is exactly preserved. In the presentation of the results, we try to make the geometric framework of variational integrators accessible to non specialists.

math.NA↗

arXiv2008

Complex geometric optics for symmetric hyperbolic systems I: linear theory

We obtain an asymptotic solution for $\ep \to 0$ of the Cauchy problem for linear first-order symmetric hyperbolic systems with oscillatory initial values written in the eikonal form of geometric optics with frequency $1/\ep$, but with complex phases. For the most common linear wave propagation models, this kind on Cauchy problems are well-known in the applied literature and their asymptotic theory, referred to as complex geometric optics, is attracting interest for applications. In this work, which is the first of a series of papers dedicated to complex geometric optics for nonlinear symmetric hyperbolic systems, we develop a rigorous linear theory and set the basis for the subsequent nonlinear analysis.

math-ph↗

arXiv2008

Complex geometric optics for symmetric hyperbolic systems II: nonlinear theory in one space dimension

This is the second part of a work aimed to study complex-phase oscillatory solutions of nonlinear symmetric hyperbolic systems. We consider, in particular, the case of one space dimension. That is a remarkable case, since one can always satisfy the \emph{naive} coherence condition on the complex phases, which is required in the construction of the approximate solution. Formally the theory applies also in several space dimensions, but the \emph{naive} coherence condition appears to be too restrictive; the identification of the optimal coherence condition is still an open problem.

math-ph↗

arXiv2005

The relationship between the Wigner-Weyl kinetic formalism and the complex geometrical optics method

The relationship between two different asymptotic techniques developed in order to describe the propagation of waves beyond the standard geometrical optics approximation, namely, the Wigner-Weyl kinetic formalism and the complex geometrical optics method, is addressed. More specifically, a solution of the wave kinetic equation, relevant to the Wigner-Weyl formalism, is obtained which yields the same wavefield intensity as the complex geometrical optics method. Such a relationship is also discussed on the basis of the analytical solution of the wave kinetic equation specific to Gaussian beams of electromagnetic waves propagating in a ``lens-like'' medium for which the complex geometrical optics solution is already available.

math-ph↗

arXiv2004

Asymptotic solutions of pseudodifferential wave equations

The aim of this paper is to give an account of some applications of pseudodifferential calculus for solving linear wave equations in the limit of high frequency/short wavelength waves. More specifically, on using as a benchmark the case of electromagnetic waves propagating in a cold isotropic slowly space- and time-varying plasma, it is shown that, in general, linear plasma waves are governed by pseudodifferential operators. Thereafter, the asymptotic techniques for solving the corresponding pseudodifferential wave equations are presented with emphasis on the paraxial propagation of Gaussian wave trains in a cold isotropic plasma. Finally, it is addressed the unicity of the dispersion tensor in terms of which the considered asymptotic solutions are determined.

math-ph↗