On the Inflection Point Instability of a Stratified Ekman Boundary Layer

Abstract

The neutral Ekman boundary layer is known to be dynamically unstable to infinitesimal perturbations under typical geophysical conditions. This paper discusses this instability to two-dimensional, simple-harmonic perturbations, for the stratified Ekman {layer.While} viscosity and Coriolis forces are generally important in setting up the basic mean profile, the inflection point instability can be investigated in the inviscid, non-rotating system limit. However, the singular nature of the resulting second-order characteristic equation makes it necessary to solve the non-singular sixth-order, viscous stratified equation. Since typically occurring Reynolds numbers are much larger than critical, emphasis has been placed on investigating the behavior of maximum growth rates versus stratification for large Re. The appropriate dimensionless parameters are found to be: =({2/Ro} Re){\textless}{SUP{\textgreater}} 1/2 {\textless}{/SUP{\textgreater}}, and Ra= {gS{\textless}SUP{\textgreater}4{\textless}/SUP{\textgreater}/K{\textless}SUB{\textgreater}mm{\textless}/SUB{\textgreater}K{\textless}SUB{\textgreater}h{\textless}/SUB{\textgreater}} [where =({2K/} f){\textless}{SUP{\textgreater}} 1/2 {\textless}{/SUP{\textgreater}}, Re= V{\textless}{SUB{\textgreater}g{\textless}/SUB{\textgreater}/K{\textless}SUB{\textgreater}m{\textless}/SUB{\textgreater}}, {Ro=V{\textless}SUB{\textgreater}g{\textless}/SUB{\textgreater}/f} and S=({\textless}{SUB{\textgreater}z{\textless}/SUB{\textgreater}+g/} cp)/] for the general case, or and {RI=gS/V{\textless}SUB{\textgreater}z{\textless}/SUB{\textgreater}} for the inviscid {case.Unstable} stratification shifts maximum growth rates toward a longitudinal orientation and shorter wave-lengths from the neutral stratification values of leftward orientation angle, =17°, and wavenumber, =0.5. The local Richardson number at the inflection point is found to he the pertinent parameter for the effects of stratification. This instability is damped completely for values of Ri{\textless}{SUB{\textgreater}i{\textless}/SUB{\textgreater}{\textgreater}0.25.} Unstable stratification tends to support the dynamic instability such that the growth rate for this mode is dominant significantly into the convective instability {regime.The} instability takes the form of counter-rotating circular motions which remain qualitatively similar for a wide range of the basic variables.