The relations between the linear and bilinear atmospheric tidal fields in geometric height and in log-pressure coordinates

Abstract

The relationships between the linearized meteorological variables as expressed in geometric height and in log-pressure coordinates are derived from the assumptions of classical atmospheric tidal theory. While the horizontal velocity components are the same to first-order in the two coordinate systems, a linearized vertical velocity differencew′-H ₀ω′ occurs because of the periodic vertical displacement of the constant pressure surfaces due to time-dependent, hydrostatic density perturbations; a linearized temperature differenceT′-τ′ also results when these displacements occur in the presence of a zero-order vertical gradient of temperature. Both of these differences can be expressed in terms of the tidal geopotential field. For a given tidal mode, both differences are generally proportional to the square root of the ratio of the tidal mode's equivalent depth and the atmospheric scale height; the temperature difference is also proportional to the background temperature lapse rate. It is further shown that the two classical tidal vertical structure equations commonly derived in their respective geometric height and log-pressure coordinate systems are in fact identical to first-orderas long as the same thermotidal forcing function is used. Expressions for the zonal-mean components of the tidal bilinear fluxes, formed by zonally averaging the product of two longitudinally varying, linearized tidal fields, are also derived for the two coordinate systems. For the bilinear fields the largest relative differences (a few tens of percent) are for the tidal zonal-mean forcing per unit mass of the zonal wind. For Earth and Mars, differences between the tidal vertical velocity fields are generally less than 25% but may be significantly larger in the Martian atmosphere during one of its episodic planetary-scale dust storms. Tidal temperature differences are generally smaller.