Leaky Rigid Lid: New Dissipative Modes in the Troposphere

An effective boundary condition is derived for the top of the troposphere, based on a wave radiation condition at the tropopause. This boundary condition, which can be formulated as a pseudo-differential equation, leads to new vertical dissipative modes. These modes can be computed explicitly in the classical setup of a hydrostatic, non-rotating atmosphere with a piecewise constant Brunt-Väisälä frequency.
In the limit of an infinitely strongly stratified stratosphere, these modes loose their dissipative nature and become the regular baroclinic tropospheric modes under the rigid-lid approximation. For realistic values of the stratification, the decay time-scales for the first few modes range from an hour to a week, suggesting that the time-scale for some atmospheric phenomena may be set up by the rate of energy loss through upwards propagating waves.