Shallow Water Magnetohydrodynamic Equations Research Articles

Lie Symmetries for the Shallow Water Magnetohydrodynamics Equations in a Rotating Reference Frame

Abstract We perform a detailed Lie symmetry analysis for the hyperbolic system of partial differential equations that describe the one-dimensional Shallow Water magnetohydrodynamics equations within a rotating reference frame. We consider a relaxing condition ∇(hB)!= 0 for the one-dimensional problem, which has been used to overcome unphysical behaviors. The hyperbolic system of partial differential equations depends on two parameters: the constant gravitational potential g and the Coriolis term f0, related to the constant rotation of the reference frame. For four different cases, namely g = 0, f0 = 0; g ?= 0 , f0 = 0; g = 0, f0 ?= 0; and g ?= 0, f0 ?= 0 the admitted Lie symmetries for the hyperbolic system form different Lie algebras. Specifically the admitted Lie algebras are the G10 = {A3,3 ⊗s A2,1} ⊗s A5a,34; G8 = A2,1 ⊗s A6,22; G7 = A3,5 ⊗s {A2,1 ⊗s A2,1}; and G6 = A3,5 ⊗s A3,3 respectively, where we use the Morozov-Mubarakzyanov-Patera

A well-balanced finite volume solver for the 2D shallow water magnetohydrodynamic equations with topography

In this paper, a second-order finite volume Non-Homogeneous Riemann Solver is used to obtain an approximate solution for the two-dimensional shallow water magnetohydrodynamic (SWMHD) equations considering non-flat bottom topography. We investigate the stability of a perturbed steady state, as well as the stability of energy in these equations after a perturbation of a steady state using a dispersive analysis. To address the elliptic constraint ∇⋅hB=0, the GLM (Generalized Lagrange Multiplier) method designed specifically for finite volume schemes, is used. The proposed solver is implemented on unstructured meshes and verifies the exact conservation property. Several numerical results are presented to validate the high accuracy of our schemes, the well-balanced, and the ability to resolve smooth and discontinuous solutions. The developed finite volume Non-Homogeneous Riemann Solver and the GLM method offer a reliable approach for solving the SWMHD equations, preserving numerical and physical equilibrium, and ensuring stability in the presence of perturbations.

Locally divergence-free well-balanced path-conservative central-upwind schemes for rotating shallow water MHD

We develop a new second-order flux globalization based path-conservative central-upwind (PCCU) scheme for rotating shallow water magnetohydrodynamic equations. The new scheme is designed not only to maintain the divergence-free constraint of the magnetic field at the discrete level but also to satisfy the well-balanced (WB) property by exactly preserving some physically relevant steady states of the underlying system. The locally divergence-free constraint of the magnetic field is enforced by following the method recently introduced in Chertock et al. (2024) [19]: we consider a Godunov-Powell modified version of the studied system, introduce additional equations by spatially differentiating the magnetic field equations, and modify the reconstruction procedures for magnetic field variables. The WB property is ensured by implementing a flux globalization approach within the PCCU scheme, leading to a method capable of preserving both still- and moving-water equilibria exactly. In addition to provably achieving both the WB and divergence-free properties, the new method is implemented on an unstaggered grid and does not require any (approximate) Riemann problem solvers. The performance of the proposed method is demonstrated in several numerical experiments that confirms robustness, a high resolution of obtained results, and a lack of spurious oscillations.

Invariant conservative finite-difference schemes for the one-dimensional shallow water magnetohydrodynamics equations in Lagrangian coordinates

Invariant finite-difference schemes for the one-dimensional shallow water equations in the presence of a magnetic field for various bottom topographies are constructed. Based on the results of the group classification recently carried out by the authors, finite-difference analogues of the conservation laws of the original differential model are obtained. Some typical problems are considered numerically, for which a comparison is made between the cases of a magnetic field presence and when it is absent (the standard shallow water model). The invariance of difference schemes in Lagrangian coordinates and the energy preservation on the obtained numerical solutions are also discussed.

Small Alfvén number limit for shallow water magnetohydrodynamics

Small Alfvén number limit of the shallow water magnetohydrodynamic equations with ill-prepared initial data is proved by applying fast averaging method. It is rigorously justified that the solutions of the original system tend to the solution of a one-dimensional system coupled with a linear system as the Alfvén number goes to zero. The approach is also used to study the small Alfvén number limit of the three-dimensional ideal incompressible magnetohydrodynamic equations.

An initial-boundary value problem for the one-dimensional rotating shallow water magnetohydrodynamic equations

We investigate an initial-boundary value problem for the one-dimensional rotating shallow water magnetohydrodynamic equations. The Dirichlet boundary conditions are imposed only on the velocity, while no boundary condition is imposed on the height of the fluid or the magnetic field. We derive a series of a priori estimates for the approximate solution sequences to show that they are Cauchy in a suitable Sobolev space. The local well-posedness in time of strong solutions for the initial-boundary value problem is established by the strong convergence of the approximate solution sequences.

Symmetries and conservation laws of the one-dimensional shallow water magnetohydrodynamics equations in Lagrangian coordinates

Symmetries of the one-dimensional shallow water magnetohydrodynamics equations (SMHD) in Gilman’s approximation are studied. The SMHD equations are considered in case of a plane and uneven bottom topography in Lagrangian and Eulerian coordinates. Symmetry classification separates out all bottom topographies which yields substantially different admitted symmetries. The SMHD equations in Lagrangian coordinates were reduced to a single second order PDE. The Lagrangian formalism and Noether’s theorem are used to construct conservation laws of the SMHD equations. Some new conservation laws for various bottom topographies are obtained. The results are also represented in Eulerian coordinates. Invariant and partially invariant solutions are constructed.

High-order accurate entropy stable finite difference schemes for the shallow water magnetohydrodynamics

This paper develops the high-order accurate entropy stable (ES) finite difference schemes for the shallow water magnetohydrodynamic (SWMHD) equations. They are built on the numerical approximation of the modified SWMHD equations with the Janhunen source term. First, the second-order accurate well-balanced semi-discrete entropy conservative (EC) schemes are constructed, satisfying the entropy identity for the given convex entropy function and preserving the steady states of the lake at rest (with zero magnetic field). The key is to match both discretizations for the fluxes and the non-flat river bed bottom and Janhunen source terms, and to find the affordable EC fluxes of the second-order EC schemes. Next, by using the second-order EC schemes as building block, high-order accurate well-balanced semi-discrete EC schemes are proposed. Then, the high-order accurate well-balanced semi-discrete ES schemes are derived by adding a suitable dissipation term to the EC scheme with the WENO reconstruction of the scaled entropy variables in order to suppress the numerical oscillations of the EC schemes. After that, the semi-discrete schemes are integrated in time by using the high-order strong stability preserving explicit Runge-Kutta schemes to obtain the fully-discrete high-order well-balanced schemes. The ES property of the Lax-Friedrichs flux is also proved and then the positivity-preserving ES schemes are studied by using the positivity-preserving flux limiters. Finally, extensive numerical tests are conducted to validate the accuracy, the well-balanced, ES and positivity-preserving properties, and the ability to capture discontinuities of our schemes.

Structural stability of shock waves and current-vortex sheets in shallow water magnetohydrodynamics

We study the structural stability of shock waves and current-vortex sheets in shallow water magnetohydrodynamics (SMHD) in the sense of the local-in-time existence and uniqueness of discontinuous solutions satisfying corresponding jump conditions. The equations of SMHD form a symmetric hyperbolic system which is formally analogous to the system of 2D compressible elastodynamics for particular nonphysical deformations. Using this analogy and the recent results in [Morando A., Trakhinin Y., Trebeschi P. Math. Ann. (2019), https://doi.org/10.1007/s00208-019-01920-6] for shock waves in 2D compressible elastodynamics, we prove that shock waves in SMHD are structurally stable if and only if the fluid height increases across the shock front. For current-vortex sheets the fluid height is continuous whereas the tangential components of the velocity and the magnetic field may have a jump. Applying a so-called secondary symmetrization of the symmetric system of SMHD equations, we find a condition sufficient for the structural stability of current-vortex sheets.

Shallow Water Magnetohydrodynamics in Plasma Astrophysics. Waves, Turbulence, and Zonal Flows

The purpose of plasma astrophysics is the study and description of the flow of rotating plasma in order to understand the evolution of various objects in the universe, from stars and planetary systems to galaxies and galaxy clusters. A number of new applications and observations have appeared in recent years and actualized the problem of studying large-scale magnetohydrodynamic flows, such as a thin layer under the convective zone of the sun (solar tachocline), propagation of accreting matter in neutron stars, accretion disks in astrophysics, dynamics of neutron star atmospheres, and magnetoactive atmospheres of exoplanets tidally locked with their host star. The article aims to discuss a fundamental problem in the description and study of multiscale astrophysical plasma flows by studying its general properties characterizing different objects in the universe. We are dealing with the development of geophysical hydrodynamic ideas concerning substantial differences in plasma flow behavior due to the presence of magnetic fields and stratification. We discuss shallow water magnetohydrodynamic equations (one-layer and two-layer models) and two-dimensional magnetohydrodynamic equations as a basis for studying large-scale flows in plasma astrophysics. We discuss the novel set of equations in the external magnetic field. The following topics will be addressed: Linear theory of magneto-Rossby waves, three-wave interactions and related parametric instabilities, zonal flows, and turbulence.

The Higher-Order CESE Method for Two-dimensional Shallow Water Magnetohydrodynamics Equations

The numerical solution of two-dimensional shallow water magnetohydrodynamics model is obtained using the $4^{th}$-order conservation element solution element method (CESE). The method is based on unified treatment of spatial and temporal dimensions contrary to the finite difference and finite volume methods. The higher-order CESE scheme is constructed using same definitions of conservation and solution elements that are used for $2^{nd}$-order CESE scheme formulation. Hence it is more convenient to increase accuracy of CESE methods as compared to the finite difference and finite volume methods. Moreover the scheme is developed using the conservative formulation and do not require change in the source term for treating the degenerate hyperbolic nature of shallow water magnetohydrodynamics system due to divergence constraint. The spatial and temporal derivatives have been obtained by incorporating $4^{th}$-order Taylor expansion and the projection method is used to handle the divergence constraint. The accuracy and robustness of the extended method is tested by performing a benchmark numerical test taken from the literature. Numerical experiment revealed the accuracy and computational efficiency of the scheme.

Dynamics of nonlinear Alfvén waves in the shallow water magnetohydrodynamic equations

Vortices in a model of the solar tachocline decay into pairs of stable donutlike Alfv\'en waves. These propagate zonally in opposite directions, colliding periodically. Nonlinear effects distort the waves significantly after each collision, but between collisions their shapes and speed remain fixed.

Rossby waves in the magnetic fluid dynamics of a rotating plasma in the shallow-water approximation

We have studied rotating magnetohydrodynamic flows of a thin layer of astrophysical plasma with a free boundary in the β-plane. Nonlinear interactions of the Rossby waves have been analyzed in the shallow-water approximation based on the averaging of the initial equations of the magnetic fluid dynamics of the plasma over the depth. The shallow-water magnetohydrodynamic equations have been generalized to the case of a plasma layer in an external vertical magnetic field. We have considered two types of the flow, viz., the flow in an external vertical magnetic field and the flow in the presence of a horizontal magnetic field. Qualitative analysis of the dispersion curves shows the presence of three-wave nonlinear interactions of the magnetic Rossby waves in both cases. In the particular case of zero external magnetic field, the wave dynamics in the layer of a plasma is analogous to the wave dynamics in a neutral fluid. The asymptotic method of multiscale expansions has been used for deriving the nonlinear equations of interaction in an external vertical magnetic field for slowly varying amplitudes, which describe three-wave interactions in a vertical external magnetic field as well as three-wave interactions of waves in a horizontal magnetic field. It is shown that decay instabilities and parametric wave amplification mechanisms exist in each case under investigation. The instability increments and the parametric gain coefficients have been determined for the relevant processes.

A space–time CESE scheme for shallow water magnetohydrodynamics equations with variable bottom topography

The dynamics of thin layer incompressible and electrically conducting fluids is described by shallow water magnetohydrodynamics equations (SWMHD), for which the evolution is nearly two dimensional and the magnetic equilibrium lies in the third direction. In this article, a space–time conservation element and solution element (CESE) scheme is extended for numerical investigation of one- and two-dimension shallow water magnetohydrodynamics (SWMHD) with variable bottom topography. The presence of non-conservative terms and the inclusion of variable bottom topography on the right hand side of SWMHD equations poses some major challenges for any numerical scheme. The proposed CESE scheme differs from the previous techniques because of global and local flux conservation in a space–time domain without restoring to interpolation and extrapolation. To check the validity of the scheme, a number of test problems are considered. The results are also compared with the already existing results in the literature by central-upwind scheme.

An Entropy Stable Finite Volume Scheme for the Equations of Shallow Water Magnetohydrodynamics

In this work, we design an entropy stable, finite volume approximation for the shallow water magnetohydrodynamics (SWMHD) equations. The method is novel as we design an affordable analytical expression of the numerical interface flux function that exactly preserves the entropy, which is also the total energy for the SWMHD equations. To guarantee the discrete conservation of entropy requires a special treatment of a consistent source term for the SWMHD equations. With the goal of solving problems that may develop shocks, we determine a dissipation term to guarantee entropy stability for the numerical scheme. Numerical tests are performed to demonstrate the theoretical findings of entropy conservation and robustness.

Weakly nonlinear shallow water magnetohydrodynamic waves

We consider an electrically conducting rotating fluid governed by the shallow water magnetohydrodynamic equations with no diffusion. We use an a priori asymptotic technique (the method of geometric optics or ray method) to study weakly nonlinear hydromagnetic waves. These waves are intermediate in length in the following sense: they are much longer than the fluid depth but much shorter than the radius of the earth. The time scale for the waves is much longer than that of the free surface oscillations and the approximation varies on an even longer timescale. The waves we are considering are studied in the beta plane approximation for an ambient magnetic field parallel to the equator which varies in the direction perpendicular to the equator. The leading order approximation gives a dispersion relation for the waves, which are generally found to be confined to bands about the equator as well as in bands at higher and lower latitudes. At the next order of approximation, a conservation law is found for the wave amplitude. We also obtain an equation governing the behavior of the leading order mean azimuthal velocity which is forced to grow linearly with time.

Numerical solution of shallow water magnetohydrodynamic equations with non-flat bottom topography

The governing equations of shallow water magnetohydrodynamics describe the dynamics of a thin layer of nearly incompressible and electrically conducting fluids for which the evolution is nearly two-dimensional with magnetic equilibrium in the third direction. A high-resolution central-upwind scheme is applied to solve the model equations considering non-flat bottom topography. The suggested method is an upwind biased non-oscillatory finite volume scheme which doées not require a Riemann solver at each time step. To satisfy the divergence-free constraint, the projection method is used. Several case studies are carried out. For validation, a gas kinetic flux vector splitting scheme is also applied to the same model.

Global solutions to one-dimensional shallow water magnetohydrodynamic equations

In this paper, we study the Cauchy problem for the one-dimensional shallow water magnetohydrodynamic equations. The main difficulty is the case of zero depth (h=0) since the nonlinear flux function P(h) is singular and the definition of solution is not clear near h=0. First, assuming that h has a positive and lower bound, we establish the pointwise convergence of the viscosity solutions by using the div–curl lemma from the compensated compactness theory to special pairs of functions (c,fε), and obtain a global weak entropy solution. Second, under some technical conditions on the initial data such that the Riemann invariants (w,z) are monotonic and increasing, we introduce a “variant” of the vanishing artificial viscosity to select a weak solution. Finally, we extend the results to two special cases, where P(h) is for the polytropic gas or for the Chaplygin gas.

An initial boundary value problem for one-dimensional shallow water magnetohydrodynamics in the solar tachocline

In this article we investigate the shallow water magnetohydrodynamic equations in space dimension one with Dirichlet boundary conditions only for the velocity. This model has been proposed to study the phenomena in the solar tachocline. In this article, the local well-posedness in time of the model is proven by constructing the approximate solutions and showing the strong convergence of the approximate solutions.

Rossby waves and polar spots in rapidly rotating stars: implications for stellar wind evolution

Rapidly rotating stars show short-period oscillations in magnetic activity and polar appearance of starspots. The aim of this paper is to study large-scale shallow water waves in the tachoclines of rapidly rotating stars and their connection to the periodicity and the formation of starspots at high latitudes. Shallow-water magnetohydrodynamic equations were used to study the dynamics of large-scale waves at the rapidly rotating stellar tachoclines in the presence of toroidal magnetic field. Dispersion relations and latitudinal distribution of wave modes were derived. We found that low-frequency magnetic Rossby waves tend to be located at poles, but high-frequency magnetic Poincare waves are concentrated near the equator in rapidly rotating stars. These results have important implications for the evolution of the stellar wind in young Sun-like stars. Unstable magnetic Rossby waves may lead to the local enhancement of magnetic flux at high latitudes of tachoclines in rapidly rotating stars. The enhanced magnetic flux may rise upwards owing to the magnetic buoyancy in the form of tubes and appear as starspots at polar regions. Magnetic Rossby waves may also cause observed short-term periodicity in the stellar magnetic activity.