Leaky Rigid Lid: New Dissipative Modes in the Troposphere

Lyubov G. Chumakova Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts

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Rodolfo R. Rosales Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts

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Esteban G. Tabak Courant Institute of Mathematical Sciences, New York University, New York, New York

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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–Väisälä 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.

Corresponding author address: Lyubov Chumakova, MIT 2-376, 77 Massachusetts Ave., Cambridge, MA 02139. E-mail: lyuba@math.mit.edu

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–Väisälä 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.

Corresponding author address: Lyubov Chumakova, MIT 2-376, 77 Massachusetts Ave., Cambridge, MA 02139. E-mail: lyuba@math.mit.edu
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