Shallow Water Magnetohydrodynamic Equations Research Articles

A kinetic flux-vector splitting method for the shallow water magnetohydrodynamics

A kinetic flux-vector splitting (KFVS) scheme for the shallow water magnetohydrodynamic (SWMHD) equations in one- and two-space dimensions is formulated and applied. These equations model 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. The proposed numerical scheme is based on the direct splitting of macroscopic flux functions of the SWMHD equations. In two-space dimensions the scheme is derived in a usual dimensionally split manner; that is, the formulae for the fluxes can be used along each coordinate direction. The high-order resolution of the scheme is achieved by using a MUSCL-type initial reconstruction and Runge–Kutta time stepping method. Both one- and two-dimensional test computations are presented. For validation, the results of KFVS scheme are compared with those obtained from the space–time conservation element and solution element (CE/SE) method. The accuracy, efficiency and simplicity of the KFVS scheme demonstrate its potential in modeling SWMHD equations.

A wave propagation method for hyperbolic systems on the sphere

Presented in this work is an explicit finite volume method for solving general hyperbolic systems on the surface of a sphere. Applications where such systems arise include passive tracer advection in the atmosphere, shallow water models of the ocean and atmosphere, and shallow water magnetohydrodynamic models of the solar tachocline. The method is based on the curved manifold wave propagation algorithm of Rossmanith, Bale, and LeVeque [A wave propagation algorithm for hyperbolic systems on curved manifolds, J. Comput. Phys. 199 (2004) 631–662], which makes use of parallel transport to approximate geometric source terms and orthonormal Riemann solvers to carry out characteristic decompositions. This approach employs TVD wave limiters, which allows the method to be accurate for both smooth solutions and solutions in which large gradients or discontinuities can occur in the form of material interfaces or shock waves. The numerical grid used in this work is the cubed sphere grid of Ronchi, Iacono, and Paolucci [The ‘cubed sphere’: a new method for the solution of partial differential equations in spherical geometry, J. Comput. Phys. 124 (1996) 93–114], which covers the sphere with nearly uniform resolution using six identical grid patches with grid lines lying on great circles. Boundary conditions across grid patches are applied either through direct copying from neighboring grid cells in the case of scalar equations or 1D interpolation along great circles in the case of more complicated systems. The resulting numerical method is applied to several test problems for the advection equation, the shallow water equations, and the shallow water magnetohydrodynamic (SMHD) equations. For the SMHD equations, we make use of an unstaggered constrained transport method to maintain a discrete divergence-free magnetic field.

Application of space–time CE/SE method to shallow water magnetohydrodynamic equations

In this article we apply a space–time conservation element and solution element (CE/SE) method for the approximate solution of shallow water magnetohydrodynamic (SMHD) equations in one and two space dimensions. These equations model the dynamics of nearly incompressible conducting fluids for which the evolution is nearly two-dimensional with magnetic equilibrium in the third direction. In this article we are using a variant of the CE/SE method developed by Zhang et al. [A space–time conservation element and solution element method for solving the two-dimensional unsteady Euler equations using quadrilateral and hexahedral meshes, J. Comput. Phys. 175 (2002) 168–199]. This method uses structured and unstructured quadrilateral and hexahedral meshes in two and three space dimensions, respectively. In this method, a single conservation element at each grid point is employed for solving conservation laws no matter in one, two, and three space dimensions. The present scheme use the conservation element to calculate flow variables only, while the gradients of flow variables are calculated by central differencing reconstruction procedure. We give both one- and two-dimensional test computations. A qualitative comparison reveals an excellent agreement with previous published results of wave propagation method and evolution Galerkin schemes. The one- and two-dimensional computations reported in this paper demonstrate the remarkable versatility of the present CE/SE scheme.

An evolution Galerkin scheme for the shallow water magnetohydrodynamic equations in two space dimensions

In this paper we propose a new finite volume evolution Galerkin (FVEG) scheme for the shallow water magnetohydrodynamic (SMHD) equations. We apply the exact integral equations already used in our earlier publications to the SMHD system. Then, we approximate these integral equation in a general way which does not exploit any particular property of the SMHD equations and should thus be applicable to arbitrary systems of hyperbolic conservation laws in two space dimensions. In particular, we investigate more deeply the approximation of the spatial derivatives which appear in the integral equations. The divergence free condition is satisfied discretely, i.e. at each vertex. First numerical results confirm reliability of the numerical scheme.

Dispersive shallow water magnetohydrodynamics

Shallow water magnetohydrodynamics (SWMHD) is a recently proposed model for a thin layer of incompressible, electrically conducting fluid. The velocity and magnetic field are taken to be nearly two dimensional, with approximate magnetohydrostatic balance in the perpendicular direction, leading to a reduced two-dimensional model. The SWMHD equations have been found previously to admit unphysical cusp-like singularities in finite amplitude magnetogravity waves. This paper extends the Hamiltonian formulation of SWMHD to construct a dispersively regularized system, analogous to the Green–Naghdi equations of hydrodynamics, that supports smooth solitary waves and cnoidal wave trains, and shares the potential vorticity conservation properties of SWMHD.

Hyperbolic theory of the “shallow water” magnetohydrodynamics equations

Recently the shallow water magnetohydrodynamic (SMHD) equations have been proposed for describing the dynamics of nearly incompressible conducting fluids for which the evolution is nearly two-dimensional (2D) with magnetohydrostatic equilibrium in the third direction. In the present paper the properties of the SMHD equations as a nonlinear system of hyperbolic conservation laws are described. Characteristics and Riemann invariants are studied for 1D unsteady and 2D steady flow. Simple wave solutions are derived, and the nonlinear character of the wave modes is investigated. The ∇⋅(h B)=0 constraint and its role in obtaining a regularized Galilean invariant conservation law form of the SMHD equations is discussed. Solutions of the Rankine–Hugoniot relations are classified and their properties are investigated. The derived properties of the wave modes are illustrated by 1D numerical simulation results of SMHD Riemann problems. A Roe-type linearization of the SMHD equations is given which can serve as a building block for accurate shock-capturing numerical schemes. The SMHD equations are presently being used in the study of the dynamics of layers in the solar interior, but they may also be applicable to problems involving the free surface flow of conducting fluids in laboratory and industrial environments.