A Statistical-Dynamical Model for Stationary Perturbations in the Atmosphere

Abstract

The governing equations for stationary perturbations are derived by subtracting the equations for the mean zonally averaged flow from the equations for the time-averaged flow. The averaging period for the ensemble is chosen such that the terms involving the time derivatives can be neglected compared to the other terms in the equations. These equations contain the first moments (means and gradients of means) of stationary perturbations and the zonally averaged flow, second moments such as eddy transports of momentum and heat, heating due to diabatic processes, and frictional forces. The closure for the equations is sought by relating the second moments and frictional forces to the first moments, parameterizing the diabatic heating in terms of first moments and radiation-convection parameters, and evaluating the stationary perturbation from a quasi-geostrophic approximation. The equations are linearized by neglecting the products of the perturbation quantities and the products of perturbations and the mean meridional circulations. The vertical variation of the perturbation in the atmosphere is represented by a two-layer model. The zonally averaged variables are assumed to be uniform with latitude and prescribed from observations. In addition, we prescribe subsurface temperature, cloudiness and convection radiation parameters as well as the surface cover, such as ice, snow and vegetation. The simplified equations are solved by expanding the variables as a sum of spherical harmonics. The model equations yield solutions for stationary perturbations of the geopotential field at 250 and 750 mb; temperature, vertical velocity and diabatic heating at 500 mb; and the temperature of the earth's surface in the Northern Hemisphere for January conditions. The phase and the amplitude of the 250 mb geopotential field agree well with observations. The phase of the 750 mb geopotential field agrees favorably with observations but the amplitude is slightly smaller than observed. The amplitude of the temperature field at 500 mb agrees well with observations and is approximately in phase with the geopotential field. The average amplitude of the vertical velocity is about 2 mm s?1. The rising motion is on the eastward side of the trough and the sinking motion on the westward side. The average amplitude of the heating field is equivalent to about 1K day?1 and qualitatively agrees with observations.