Numerical simulation of the transition from three- to two-dimensional turbulence under a uniform magnetic field

Abstract

The transition of homogeneous turbulence from an initially isotropic three-dimensional to a quasi-two-dimensional state is simulated numerically for a conducting, incompressible fluid under a uniform magnetic field B₀. The magnetic Reynolds number is assumed to be small, so that the induced fluctuations of the magnetic field are small compared with the imposed magnetic field B₀, and can be computed from a quasi-static approximation. If the imposed magnetic field is strong enough, all variations of the flow field in the direction of B₀ are damped out. This effect is important e.g. in the design of liquid-metal cooling systems for fusion reactors, and the properties of the final state are relevant to atmospheric turbulence. An extended version of the code of Orszag and Patterson (1972) is used to integrate the Navier-Stokes equations for an incompressible fluid. The initial hydrodynamic Reynolds number is 60. The magnetic interaction number N is varied between zero and 50. Periodic boundary conditions are used. The resolution corresponds to 323 points in real space. The full nonlinear simulations are compared with otherwise identical linear simulations; the linear results agree with the nonlinear ones within 3% for about one-fifth of the large-scale turnover time. This departure is a consequence of the return-to-equilibrium tendencies caused mainly by energy transfer towards high wavenumbers. The angular energy transfer and the energy exchange between different components are smaller, and become virtually zero for large values of N. For N [approximate] 50 we reach a quasi-two-dimensional state. Here, the energy transfer towards high wavenumbers is reduced for the velocity components perpendicular to B₀ but relatively increased for the component parallel to B₀. The overall behaviour is more similar to three-than to purely two-dimensional turbulence. This finding is of great importance for turbulence models of the atmosphere. The realization of a purely two-dimensional state does not seem to be possible for decaying turbulence. The magnetic field causes highly intensified pressure fluctuations, which contribute to the redistribution of the anisotropic Lorentz forcing.