Barotropic modes, baroclinic modes and equivalent depths in the atmosphere

The equivalent depth of a rotating fluid layer determines the phase speed of all free (i.e. unforced) linear waves that propagate in it. The equivalent depth can be estimated from the eigenvalues of a second-order differential equation for the vertical dynamics where the mean temperature profile is a coefficient. The eigenfunctions dictate the vertical structure of the modes. This work combines analytic and numerical solutions of the vertical structure equation for various mean temperature profiles and for various combinations of top and bottom boundary conditions relevant to two atmospheric configurations: troposphere and combined troposphere-stratosphere. Our formulation provides a clear definition of the barotropic mode and the countable baroclinic modes (including the special n = 0 mode). The barotropic mode exists only when the lower boundary condition is the vanishing of the vertical velocity (i.e. when a solid boundary bounds the layer from below) and the equivalent depth of this mode is about 10 km. The n = 0 baroclinic mode does not exist in a layer whose thickness exceeds a threshold value of about 20 km and therefore this mode does not exist in the combined troposphere-stratosphere layer. The upper boundary condition affects the eigenvalues much more so than the details of the temperature profile, as the details of the temperature profile only affect the equivalent depth in the troposphere. Increasing the height of the upper boundary has little effect on the barotropic mode, but strongly influences the phase speed and vertical structure of the baroclinic modes; a general circulation model with a lid at or below the stratopause where, for example, planetary Rossby waves are present will therefore be incapable of correctly simulating the interaction of these waves with the mean flow. The values of the equivalent depth of baroclinic modes are approximately 10% of the actual layer's depth in all realistic cases.