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ABSTRACT
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 pseudodifferential equation, leads to new vertical dissipative modes. These modes can be computed explicitly in the classical setup of a hydrostatic, nonrotating atmosphere with a piecewise constant Brunt-Veuroaiseuroaleuroa frequency.
In the limit of an infinitely strongly stratified stratosphere, these modes lose 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 of the first few modes for mesoscale disturbances 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 upward-propagating waves.
1. Introduction
Much of our understanding of tropospheric dynamics is based on the concept of discrete internal modes. Internal gravity waves, such as those associated with convective systems, propagate at definite speeds, typically associated with the first to third baroclinic vertical modes, depending on the nature of the disturbance. Even though other effects such as nonlinearity, moist convection, and mean wind shear alter significantly the nature and speed of these waves, they remain nonetheless the dynamical backbone of the troposphere.
Yet discrete modes are the signature of systems of finite extent: a semi-infinite stratified atmosphere yields a continuum spectrum of modes, much as the Fourier transform in the infinite line, as opposed to the discrete Fourier series associated with finite intervals. This has led to arguments by R. Lindzen that these discrete tropospheric modes are just a fallacy of overly simplified theoretical models, and that the atmosphere ''is characterized by a single isolated eigenmode and a continuous spectrum'' (Lindzen 2003, p. 3009). On the other hand, the troposphere does seem to operate on distinct discrete modes [see, e.g., Hayashi (1976) for an early reference], and many phenomena, some of which we mention below, have been modeled successfully on such basis.
Replacing the tropopause by a rigid lid where the vertical velocity must vanish is the simplest and most conventional way to obtain a discrete set of tropospheric modes with realistic values for their speed and vertical structure. Two justifications are typically...