108 Nonlinear Theories — Inertial 



THE POTENTIAL VORTICITY OF A LAYER 

 OF UNIFORM DENSITY 



In order to include nonlinear dynamical terms and density stratification, it 

 is helpful to introduce a quantity called potential vorticity, obtained in the 

 following manner. Using the coordinate systems employed in the preceding 

 chapter, let us consider a homogeneous layer of fluid A\dth density'- p^ and 

 thickness D, in which case the d3aiamical equations are 



at Pidx 



*+/„=_!??+ r, (2) 



dt p-^dy 



where dldt = {dldt) + u{dldx) + v{dldy), and X and Y include both frictional 

 driving and retarding forces, which we are not now interested in considering 

 in detail. The Coriohs parameter/ is regarded as a function of y. First, we 

 eHminate the pressure gradients by cross-differentiation: 



where ^, the relative vorticity, is defined as ^= (dv/dx) — {dujdy). The quantity 

 /+ ^ is called the absolute vorticity. The equation of continuity of the layer is 



dD 



dt \dx dyj 



Elimination of the horizontal divergence between the two equations results 

 in the equation of potential vorticity, {f+Q/D, 



dt\ D I D\dx dyJ' ^ ' 



If there are no frictional forces, this equation simply states that the 

 potential vorticity of a water column within the homogeneous layer cannot 

 change as it moves from one place to another. Thus, as a column moves 

 from one latitude to another, an adjustment must occur in the depth of the 

 layer and in the relative vorticity in such a way as to keep the potential 

 vorticity constant. 



THE POTENTIAL VORTICITY IN AN 

 ISOPYCNAL LAYER 



Any actual Gulf Stream cross section exhibits a marked decrease in the 

 thickness of layers (bounded by isotherms) in the high- velocity parts of the 

 current. This vertical shrinking is particularly noticeable in the isothermal 



