On Geostrophic Adjustment and Numerical Procedures in a Rotating Fluid

Abstract

The geostrophic adjustment problem is considered for the case of a homogeneous fluid in a rotating cylindrical container. The formal solution for an arbitrary initial disturbance in the axisymmetrical case is obtained in terms of a series expansion in Bessel functions of zero order. The solution shows that the motion will consist of a time-independent part in geostrophic balance and a time-dependent part which is oscillatory. The general properties of the solution including the energeties are investigated. The degree to which the process is simulated by various finite-difference methods for the time derivatives is investigated using the leap-frog, a semi-implicit scheme, and a scheme which combines forward and backward time differences (mixed scheme). It is found that all schemes are acceptable provided the time step is sufficiently small, but in general the simulation of the process by the leap-frog and the mixed scheme is more realistic. The present conclusions are of importance in any scheme employed for the purposes of assimilation of meteorological data. The analysis can be expanded to a more general case than the axisymmetrical one.