Poloidal–toroidal decomposition in a finite cylinder. I: Influence matrices for the magnetohydrodynamic equations

Abstract

The Navier-Stokes equations and magnetohydrodynamics equations are written in terms of poloidal and toroidal potentials in a finite cylinder. This formulation insures that the velocity and magnetic fields are divergence-free by construction, but leads to systems of partial differential equations of higher order, whose boundary conditions are coupled. The influence matrix technique is used to transform these systems into decoupled parabolic and elliptic problems. The magnetic field in the induction equation is matched to that in an exterior vacuum by means of the Dirichlet-to-Neumann mapping, thus eliminating the need to discretize the exterior. The influence matrix is scaled in order to attain an acceptable condition number. Motivation and Governing EquationsThe requirement that velocity and magnetic fields be solenoidal, i.e. divergence-free, represents one of the most challenging difficulties in hydrodynamics and in magnetohydrodynamics [1,2,3,4,5,6,7]. For the velocity field, this condition is the fundamental approximation used in incompressible fluid dynamics. For the magnetic field, this condition is the statement of the non-existence of magnetic monopoles.Two main approaches exist for imposing this requirement. The first is to use three field components and to project three-dimensional fields onto a divergence-free field. In an incompressible fluid, the pressure serves to counterbalance the nonlinear term which is the source of the divergence in the Navier-Stokes equations; the pressure also plays this role numerically. The divergence of the Navier-Stokes equations is taken, leading to a Poisson problem for the pressure. However, the boundary conditions on the

Toroidal and poloidal energy in rotating Rayleigh–Bénard convection

“…Thus, there is a gauge freedom for the boundary conditions. The most simple and commonly used gauge condition for ξ is ξ| r=R = 0 (3.8) (Marques et al 1993;Boronski & Tuckerman 2007). This gauge in combination with (3.2) and the no-slip condition on the velocity, u r | r=R = 0, yields…”

Comparing seven spectral methods for interpolation and for solving the Poisson equation in a disk: Zernike polynomials, Logan–Shepp ridge polynomials, Chebyshev–Fourier Series, cylindrical Robert functions, Bessel–Fourier expansions, square-to-disk conformal mapping and radial basis functions

“…Several authors have applied Zernike polynomials to partial differential equations [101,102,72,86,15,16]. The Poisson equation is special because substituting the expansions…”

Poloidal–toroidal decomposition in a finite cylinder

“…We now describe the ways in which we have validated the hydrodynamic code described here and in our companion paper [20]. We have obtained exact polynomial solutions to the nested Helmholtz-Poisson solver, which is by far the most complicated portion of the code; we present its form in the hopes it may prove useful to other researchers.…”

“…Our main purpose here is to develop a mathematical and algorithmic tool which can be applied to von Kármán flow and to rotating turbulence, and which can be extended to the magnetohydrodynamic configuration of the VKS experiment [17]. In a companion article [20], we showed that this system could be reduced to the solution of a set of nested Helmholtz and Poisson problems with uncoupled Dirichlet boundary conditions, whose solutions could be superposed via the influence matrix technique. When applied to the Navier-Stokes equation in a finite cylinder, the resulting system has boundary conditions which are coupled and of high order.…”

“…Our article [20] was devoted to showing how the problem (1.4) and (1.5) can in turn be reduced to a sequence of five nested parabolic and elliptic problems, each with Dirichlet boundary conditions:…”