The effect of vertical temperature gradient on the equivalent depth in thin atmospheric layers

The equivalent depth of an atmospheric layer is of importance in determining the phase speed of gravity waves and characterizing wave phenomena. The value of the equivalent depth can be obtained from the eigenvalues of the vertical structure equation (the vertical part of the primitive equations) where the mean temperature profile is a coefficient. Both numerical solutions of the vertical structure equation and analytical considerations are employed to calculate the equivalent depth, $h_n$, as a function of the atmospheric layer's thickness, $\Delta z$. Our solutions for layers of thickness 100 $\le \Delta z\le$ 2000 m show that for baroclinic modes, $h_n$ can be over two orders of magnitudes smaller than $\Delta z$. Analytic expressions are derived for $h_n$ in layers of uniform temperature and numerical solutions are derived for layers in which the temperature changes linearly with height. A comparison between the two cases shows that a slight temperature gradient (of say 0.65 K across a 100 m layer) decreases $h_n$ by a factor of 3 (but can reach a factor of 10 for larger gradients) compared with its value in a layer of uniform temperature, while a change of 10 K in the layer's uniform temperature hardly changes $h_n$. The $n=0$ baroclinic mode exists in all combinations of boundary conditions top and bottom while the barotropic mode only exists when the vertical velocity vanishes at both boundaries of the layer.