Observed dynamics near bounding upper surfaces in the atmosphere and ocean are interpreted in terms of quasi-geostrophic theory. The quasi-geostrophic equations consist of advection of linearized potential vorticity coupled with advection of temperature at the upper and lower bounding surfaces. We show that the standard vertical finite difference formulation of 3D quasi-geostrophic flow accurately represents the flow only down to a critical horizontal scale that decreases with vertical grid spacing. To overcome this constraint, we derive a surface-modal formulation which accurately and efficiently captures both the surface dynamics due to temperature anomalies on the upper and lower boundaries, and the interior dynamics due to potential vorticity anomalies, without the need for high vertical resolution.In the atmosphere, the horizontal wavenumber spectra of wind and temperature near the tropopause have a steep -3 slope at synoptic scales and a shallow -5/3 slope at mesoscales, with a smooth transition between the two regimes from 800km to 200km. We demonstrate that when the surface temperature anomalies are resolved, quasi-geostrophic flow driven by baroclinic instability exhibits such a transition near the tropopause. The horizontal scale of transition between -3 and -5/3 slopes depends on the relative magnitudes of the mean surface temperature gradient and the mean potential vorticity gradient.In the ocean, sea surface height anomalies measured by satellite altimetry exhibit shallower spectral slopes than quasi-geostrophic theory predicts, and faster than expected westward phase propagation of sea surface height in the midlatitudes. We argue that, in some regions, the shallow spectral slopes are due to surface quasi-geostrophic dynamics, and that the westward phase propagation in the midlatitudes is indicative of a transition from a linear Rossby wave regime in the tropics to a nonlinear turbulent regime in the midlatitudes.
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