The Earth surface and the tropopause wind shear effects on baroclinic instability

The behavior of the oceans and the atmosphere in mid-latitudes may be considered as a small departure from the background rotation of the Earth as a solid body. This provides a ground for the quasigeostrophic (QG) approximation, which is obtained by a formal expansion of the primitive equations in Rossby number that measures the intensity of such departures from the background rotation. The resulting equations, though much simpler than the full set, are still complex enough that it is not always clear what they imply about the nature of their solutions. Therefore, further simplifications have been sought in particular contexts, looking for more tractable models. A model of this kind constructed based on the assumption of a uniform interior QG potential vorticity is discussed in this paper. A further simplification may be obtained by assuming uniform stratification in the atmosphere/ocean. This model has been proposed for explaining some aspects of instability in the atmosphere by Charney and Eady and is used in this paper for studying some effects of wind shear on baroclinic instability.
    In the Eady model, the wind shear on the lower (ground surface) and upper (tropopause surface) boundaries plays a determining role in the occurrence of instability. However, in the classic form of the baroclinic instability theory of Charney and Eady, wind shear is considered constant with height, and therefore the effect of variations in wind shear on the ground and tropopause surfaces are not covered. According to the Charney–Stern–Pedlosky theorem, with uniform interior potential vorticity, for instability to occur, the wind shear at the upper and lower boundaries must be of the same sign. This theorem provides the necessary condition for instability but gives no further information on the effect of the wind shear at the two boundaries.
    Then here, the objective is to assess the effects of the wind shear on Eady-like models, that is, models with uniform interior QG potential vorticity.  After examining a quadratic vertical zonal wind profile for the basic state as a special case, the arbitrary variation of the wind shear at the two boundaries is studied in an Eady-like model. It is shown that for each wavenumber, there are upper and lower bounds, respectively denoted by  and, for the ratio of the tropopause wind shear  to the earth's surface wind shear , beyond which instability cannot occur. That is, for instability the ratio must be in the interval
 
which serves as an additional necessary condition for instability.  Considering all the wavenumbers, the lowest value for is found to be  With nondimensional wavenumbers in whichandare respectively the dimensional wavelength and Rossby deformation radius, for , instability occurs provided that the wind shear at the lower boundary is greater than that at the upper boundary. For between 1 and 1.4, tends to infinity which means that for instability there is no restriction on the magnitude of the wind shear at the upper boundary.