On Intermediate Models for Stratified Flow

Intermediate models contain physics between that in the primitive equations and that in the quasigeostrophic equations. The specific objective here is to investigate the absolute and relative accuracy of several intermediate models for stratified flow by a comparison of numerical finite-difference solutions with those of the primitive equations (PE) and with those of the quasigeostrophic (QG) equations. The numerical experiments involve initial-value problems for the time-dependent development of an unstable baroclinic jet on an f plane in a doubly periodic domain with flat bottom. Although the geometry is idealized, the problem is set up so that the dynamics should be similar to that of the baroclinic jet observed off the northern California coast in the Coastal Transition Zone (CTZ) field program. Three numerical experiments are conducted where the flow fields are characterized by local Rossby numbers that range from moderately small to O(1). The unstable jet develops finite amplitude meanders that grow in amplitude until they pinch off to form detached eddies on either side of the jet. The instability process is characterized by the transfer of potential to kinetic energy accompanied by a large increase in the barotropic component of the flow. Although the initial jet velocity profiles are symmetric about the jet centerline, as the Rossby number of the jet increases the meander growth and eddy detachment process becomes more asymmetrical about the jet axis. A meander on the positive vorticity side of the jet pinches off first to form a relatively large anticyclonic eddy followed in time by the detachment of a smaller cyclonic eddy on the negative vorticity side. The intermediate models that we consider are the balance equations (BE), the balance equations based on momentum equations (BEM), the iterated geostrophic models (IG2 and IG3), the linear balance equations (LBE), the linear BEM (LBEM), and the geostrophic momentum approximation (GM). We also include a second-order quasigeostrophic approximation (QG2) and a primitive equation model with semi-implicit time differencing (PESI). The results of the numerical experiments for moderate Rossby number flow show that the QG, QG2, LBE, and GM models give large errors and produce flow fields that have substantial qualitative differences from the PE. The LBEM model is somewhat better, while IG2 gives considerably smaller errors. The BE, BEM, IG3, and PESI models give highly accurate approximate solutions to PE, and that result holds also for those models applied to the O(1) Rossby number flow