636 



The Tropospheric Circulation 



Assuming that the specific volume a^ of the disturbed particles is conserved, then the 

 equations of motion for the displaced particles will take the form : 



dv 



where 



} 



(XIX.53) 



(XIX.54) 



X and z are the displacements of the small particles in the x- and 2-directions and 

 may be positive or negative. The coefficients a^x, ^xz and a^^ are given by 



\ dxf dx 8x 



du 



--/(/♦-^l-^. 



= +/* / 



(^•-9 



dP8a 



dx dz 



dP8a 



Tzd^ 



(XIX.55) 



J 



with axz= ^zx and/* = 2a; cos ^. 



It can then be shown that at a point in a geostrophic current at which there acts a 

 transverse disturbance, conditions will be stable, neutral or unstable according to 

 whether the quadratic form (Kleinschmidt) : 



x2 + 2a, 



+ a. 



^0. 



(XIX. 56) 



The sign of Q is determined firstly by that of the discriminant 



a ^ a^, — arr. a 



'XX "xz 



(XIX.57) 



and secondly by the sign of one of the coefficients of the quadratic terms in Q (for 

 instance a^^). The condition (XIX. 56) thus becomes 



a-0 or 



idu _ daldx ^\ , ^>Q 

 \dx daldz' 8z) ■' < ' 



(XIX.58) 



The last equation can be re-written with the help of (XIX.55) and by neglecting terms 

 of lower order one obtains 



4:^1 /(/+ir"- 



(Jadz 



(XIX.59) 



The expression - ^ 



p dz 



is the static stability (z-positive upwards; p. 196) and 



f{f-\- dujdx) is the expression for the inertial stability. The equation (XIX.59) gives 

 a hydrodynamic measure in as far as the geostrophic equilibrium in the current under 

 consideration is hydrodynamically stable or unstable when subject to external impulses 

 acting normal to the direction of the flow (in transverse or vertical direction). 



