**Ross Tulloch**

New York Institute

in collaboration with K. Shafer Smith

### A Baroclinic Model for the Atmospheric Energy Spectrum

The horizontal wavenumber spectra of wind and temperature near the tropopause
have a steep -3 slope at synoptic scales and a shallower -5/3 slope at mesoscales,
with a transition between the two regimes at a wavelength of about 600 km. Here
it is demonstrated that a quasigeostrophic model driven by baroclinic instability
exhibits such a transition near its upper boundary (analogous to the tropopause) when surface temperature advection at that boundary is properly resolved and forced.

In order to accurately represent surface advection at the upper and lower boundaries, the vertical structure of the model streamfunction is decomposed the into four parts, representing the interior flow with the first two neutral modes, and each surface with its Green's function solution, resulting in a system with four prognostic equations.

Mean temperature gradients are applied at each surface, and a mean potential vorticity gradient consisting both of β and vertical shear is applied in the interior.
The system exhibits three fundamental types of baroclinic instability: interactions
between the upper and lower surfaces (Eady-type), interactions between one surface
and the interior (Charney-type) and interactions between the barotropic and baroclinic
interior modes (Phillips-type). The turbulent steady-states that result from each of
these instabilities are distinct, and those of the former two types yield shallow
kinetic energy spectra at small scales along those boundaries where mean temperature
gradients are present. When both mean interior and surface gradients are present,
the surface spectrum reflects a superposition of the interior-dominated -3 slope
cascade at large scales, and the surface-dominated -5/3 slope cascade at small scales.

The transition wavenumber depends linearly on the ratio of the interior potential
vorticity gradient to the surface temperature gradient, and scales with the inverse
of the deformation scale when β.