The Frontogenetical Forcing of Secondary Circulations. Part II: Properties of Q Vectors in Exact Linear Solutions

Abstract

An exact solution of the primitive equations (PEs) and the corresponding exact solutions of the alternative balance (AB), geostrophic momentum (GM), and quasigeostrophic (QG) equations are presented. The PE solution illustrates how the temperature and horizontal vorticity fields evolve in a linear horizontal flow with constant deformation and vertical vorticity when the initial temperature field is also linear, as well as how ageostrophic circulations are produced. The other exact solutions show the errors produced by the various approximations and confirm that the AB solution is more accurate than the GM one and that the QG solution is almost always the most inexact. The utility of the Q vector and similar vectors is examined for each solution. The PE solution verifies that in a hyperbolic wind field (i) the isotherms eventually parallel the outflow axis, (ii) the ageostrophic circulation ultimately becomes normal to the outflow axis, (iii) thermal-wind balance becomes established in the direction normal to the isotherms, and (iv) the rotational component of the vector frontogenetical function decays.