Simulations of Barotropic and Baroclinic Waves of the Linearized Shallow Water Equations by Global Scale Models

A recently developed wave theory of the Linearized Shallow Water Equations (LSWE) on the rotating spherical Earth provides closed-form analytic expressions for the phase speeds and spatial structures of Planetary (Rossby) and Inertia-Gravity (Poincaré) waves. These expressions were derived by formulating the boundary value problem associated with zonally propagating waves of the LSWE as an approximate (time-independent) Schrödinger equation and finding explicit expressions for the eigenfunctions and energy levels. These expressions were previously used as initial conditions in simulations by specifically designed numerical solvers of both barotropic (i.e. gH ≈ 5·10⁴ m²s^-2) and baroclinic (i.e. gH ≈ 5 m²s^-2) wave-modes of the LSWE. In this work we examine the feasibility of simulating these waves by operational global-scale models. The model we use is the spectral Eulerian dynamical core of the atmospheric component of NCAR-Community Earth System Model. Simulations of barotropic modes using spectral resolution of T85 preserve the initial fields accurately for at least 100 wave periods while propagating with phase speeds that differ from the analytic phase speeds by 1(±1)%. In contrast, baroclinic wave modes simulated using the same spectral resolution become unstable after only O(1) wave period, while simulations with higher resolution of T341 preserve the fields for O(10) wave periods while propagating with phase speeds that differ from the analytic phase speeds by 40(±10)%. The instability of the baroclinic wave modes at low resolutions seems to originate at the poles as a low wavenumber disturbance.