L/ieber and Rintel 



For considering stability with respect to convective vortices, the standard 

 form of the perturbations in dimensioniess form is (Pellew and Southwell, 1940): 



\ = -e'^t i(^,v) a)*(0 , 



T^ = AT e^t f (^,^) T*(0 , 



(7) 



C^ = AC e-t i(S^V) C*(0 , ^ = — - V 



i = ^ 



where f is the solution of the membrane equation for the particular form of 

 horizontal periodicity of the perturbations 



^'f ^'f ,2. n 

 + + k'' f = , 



3^2 3^2 



AT is a characteristic temperature difference, AC is a characteristic differ- 

 ence of concentration, and h is the depth of the fluid layer. The constant k 

 arises from the separation of variables and depends on the particular geom- 

 etry of periodicity of the perturbations. This geometry is not specified in the 

 present investigation, so that the result is valid for hexagonal cellular vortices 

 as well as for longitudinal rolls arising when a small shear is applied to the 

 fluid. Equations (4) and (6) then reduce to 



a - — (D2-k2) 



h2 



HAT 



fi'c 



a -^ (D2-k2) 

 h2 



C* = B" 



hAC 



a - — (D2-k2) (D2-k2)a;* 



(8) 



k2g (a'AT * - a"CC*) 



d^ 



By construction, the stability or instability of the basic solution of Eqs. (3) is 

 determined by the sign of the real part of a. Thus the margin of instability 

 will be characterized by the vanishing of the real part of a. Concerning the 

 imaginary part, two possibilities are considered: (a) the marginal instability 

 is characterized by the principle of exchange of stabilities, i.e., the imaginary 

 part of a also vanishes for the state of transition; and (b) overstable oscilla- 

 tions characterized by the nonvanishing imaginary part of a are relevant to 

 the instability. By using a method developed by Chandrasekhar (1963), Mr. H. 

 Weinberger (1962) has shown that in the case considered here overstability 

 can be the first kind of instability to evolve. However, experiments performed 



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