New dissipative modes and decay time-scales in the troposphere with a leak

Much of our understanding of the tropical tropospheric dynamics is based on the concept of discrete internal modes. However, discrete modes are a signature of systems of finite extent, while the atmosphere should be modeled as infinite and "is characterized by a single isolated eigenmode and a continuous spectrum" [1]. Is it then unphysical to use the discrete internal rigid lid modes to model atmospheric phenomena? We propose the following approach to resolve this debate. We impose a radiation boundary condition at the top of the incompressible troposphere in hydrostatic balance, reformulate this boundary condition as a pseudo-differential equation, and use it to compute the new modes. 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 primarily set up by the rate of energy loss through upwards propagating waves. On the mathematical side, we show that though these modes are not orthogonal, they are complete, and there is an easy way to project initial conditions onto them.

[1] Lindzen, R., 2003: The interaction of waves and convection in the tropics. J. Atmos. Sci., 60, 3009-3020.

The work of the authors was partially supported by grants from the NSF, as follows: NSF 0903008 (LC), DMS 1007967 and DMS 0907955 (RRR), DMS 0908077 (ET).