Numerical simulation of axisymmetric unsteady incompressible flow by a vorticity-velocity method

A new numerical method for solving the axisymmetric unsteady incompressible Navier-Stokes equations using vorticity-velocity variables and a staggered grid is presented. The solution is advanced in time with an explicit two-stage Runge-Kutta method. At each stage a vector Poisson equation for velocity is solved. Some important aspects of staggering of the variable location, divergence-free correction to the velocity held by means of a suitably chosen scalar potential and numerical treatment of the vorticity boundary condition are examined. The axisymmetric spherical Couette flow between two concentric differentially rotating spheres is computed as an initial value problem. Comparison of the computational results using a staggered grid with those using a non-staggered grid shows that the staggered grid is superior to the non-staggered grid. The computed scenario of the transition from zero-vortex to two-vortex flow at moderate Reynolds number agrees with that simulated using a pseudospectral method, thus validating the temporal accuracy of our method.

Energy conversions and storage caused by an unsteady poloidal flow in active solar regions

In this paper we discuss coupling processes between a magnetic field and an unsteady plasma motion, and analyze the features of energy storage and conversions in active region. It is pointed out that the static force-free field is insufficient for a discussion of storage processes, and also the pure unsteady plasma rotation is not a perfect approach. In order to analyze the energy storage, we must consider the addition of poloidal plasma motion. The paper shows that because the unsteady poloidal flow is added and coupling occurs between the magnetic field and both the toroidal and the poloidal plasma flows, an unsteady process is maintained which changes the force-free factor with time. Hence, the energy in the lower levels can be transferred to the upper levels, and a considerable energy can be stored in the active region. Finally, another storage process is given which is due to the pure poloidal flow. The article shows that even if there is no twisted magnetic line of force, the energy in the lower levels may still be transferred to the upper levels and stored there.