The internal variability of the stratosphere, in particular the vacillation of the polar vortex, in a shallow-water model is investigated. The combined effects of mechanical forcing and thermal forcing are the major factors involved in the vacillation of the polar vortex in the shallow water (SW) model examined here. Mechanical forcing is provided by a time-independent topography mimicking tropospheric excitation of the stratosphere. Thermal forcing is provided by a linear relaxation of the mass field to a time-independent equilibrium state mimicking the radiative relaxation taking place in the stratosphere. In this way, the setup of the problem is such that the barotropic effects arising from the horizontal structure of PV on the polar vortex in the real atmosphere can be examined.
The SW equations in potential vorticity (PV), velocity divergence and acceleration divergence representation are solved for a range of resolutions using the "diabatic contour-advective semi-Lagrangian" (DCASL) algorithm and a standard pure semi-Lagrangian (SL) algorithm. Using different numerical algorithms enables us to address the issues related to numerical sensitivity of the zonal vacillations in the SW model of the stratosphere. The equations for velocity and acceleration divergence are solved using spectral transform in longitude and compact fourth-order finite differencing in latitude. The spatial resolution is indicated by M?N, M and N being the number of grid points in the longitudinal and latitudinal directions, respectively.
As a first step in understanding the nature and robustness of the zonal wind vacillations in the SW for the stratosphere, the Eulerian diagnostics based on the terms of the zonal mean zonal momentum equation are calculated and analyzed. To this end, the results for the pure SL algorithm with spatial resolution of 256?256, 512?512, and 1024?1024 are presented and compared with the corresponding results for the DCASL algorithm with spatial resolution of 256?256. The results for all of the resolutions and algorithms indicate that the topographic forcing, the divergence of horizontal momentum flux and the Coriolis torque are the dominant factors determining the zonal mean zonal momentum time evolution. From the Eulerian perspective, overall, the zonal wind vacillations can be attributed to the out-of-phase variations of the topographic forcing, the divergence of horizontal momentum flux and the Coriolis torque. However, the irregularity of the cycles of the vacillations in the results for the PV-based algorithms in all of the resolutions examined is in clear contrast with the regular vacillations reported at T42 resolution by Rong and Waugh in 2004, where spectral transform algorithm used to solve the SW equations in vorticity, divergence, mass and dissipation is provided by explicitly damping vorticity using hyperdiffusion. Regarding the irregularity of the vacillations and the statistical difference between the results for the SL and DCASL, on the one hand, and between the various resolutions of the SL, on the other hand, further diagnostics to study the geometry of the vortex and its time evolution as well as additional numerical experiments are needed to assess the polar vortex oscillations.