Wind shear and stratification effects on baroclinic adjustment

The baroclinic adjustment is defined as the neutralization of the mean flow by eddies. With the amplitude growth of the baroclinic waves generated by an unstable mean flow, the eddy fluxes increase and act to reduce the meridional temperature gradient or vertical wind shear of the mean flow to subcritical value. This is believed to constitute the basic mechanism of baroclinic adjustment. The concomitant reduction of the potential vorticity (PV) along isentropic surfaces during the baroclinic adjustment process can be accommodated within the quasi-geostrophic theory by removing meridional gradient of quasigeostrophic PV in the interior of the domain and satisfying the requirements of the Charny–Stern–Pedlosky theorem. The result is a group of models that extend the work of Eady and thus called Eady-like models. In addition to uniform interior PV, these models share a rigid lid as the upper boundary mimicking the tropopause.  In fact, the upper rigid helps to resolve a major issue regarding an unphysical aspect of the state obtained through 
baroclinic adjustment. For the problem with a constant buoyancy frequencyfor the mean flow, as results of previous studies suggest, the major issue is that baroclinic neutrality requires that the surface value of buoyancy frequency become infinite or exceedingly large, even if frictional damping is taking into account.
The mathematical tractability, physical simplicity, and yet the ability to explain some of the important features of baroclinic eddies in the real atmosphere and oceans, have made the Eady-like models of baroclinic instability the ideal setting to study baroclinic adjustment. In particular, the presence of a short-wave cut-off for instability, an immediate effect of using uniform interior PV, can provide a route to reach baroclinic neutrality. This can be achieved through limiting the interaction between the lower and upper edge waves by, for example, moving the lid to upper levels and thus displacing the short-wave cut-off to longer wavelengths. Further, confining the baroclinic eddies in meridional direction leads to a lower limit for the horizontal wavenumber.  Making the latter lower limit equal to the short-wave cut-off leads to a state of neutrality for all of the zonal modes.
Within the extent of Eady-like models, this paper is devoted to study the effect of vertical wind shear and stratification on the baroclinic adjustment. Disregarding the density variation with height, analytical solutions are obtained in the form of the Bessel and Neumann functions for the instability problem with a quadratic profile for the zonal mean flow and a general form for the height variation of the squared buoyancy frequency.  The instability problem is investigated by varying  and the level of maximum zonal mean flow, that is, the location of the jet. Varying the latter parameter help us determine the sensitivity to the changes of vertical shear in the upper boundary relative to that in the lower boundary.  The relation between the eddy flux of PV and the stratification parameters including is also explored at a representative mid-tropospheric level  it is shown that the eddy flux of PV is minimum (maximum) for positive (negative) values of   which correspond to decreasing (increasing) values of with height.  Therefore, it can be concluded that the process of baroclinic adjustment is associated with negative values of
Using a properly determined vertical profile for , the meridional gradient of PV is removed in the interior of the domain. It is shown that the adjusted buoyancy frequency decreases with increasing altitude. With stronger winds, the reduction with height in buoyancy frequency becomes more significant.
With the meridional gradient of PV eliminated, the necessary condition for instability based on the Charny–Stern–Pedlosky criterion requires that the wind shear on the lower and upper boundaries be of the same sign. However, even in the case of equal-sign wind shear on the two boundaries, neutrality may be achieved depending on the structure of the wind shear throughout the domain and not just on the boundaries. In the Eady problem, wind shear has no effect in the condition for instability. Considering the effects of wind shear on the adjusted buoyancy frequency, it is shown that instability can occur if wind shear increases even when the surface value of is. Without considering the variation with height of, as the case with no rigid lid, neutrality can only happen for exceedingly large values of surface buoyancy frequency. It is shown that for baroclinic adjustment to occur, the two following conditions are necessary:
1- a decreasing with height of the buoyancy frequency;
2- a weak wind shear.
Nevertheless, baroclinic adjustment can also occur in the presence of strong wind shear, if there is sufficiently large surface value of . Further, it is shown that the effect of changing the wind shear difference between the lower and upper boundaries is compensated by a corresponding change in the reduction of buoyancy frequency with height.