Experiments on Convective Flows in Geophysical Fluid Systems 



1 _32^ d^ 



^ , ^2v20 , v2 = 4r^ + ^ ' • ^''"^'- /"^ ' (27) 



Ra = -^^ , cr = ^ , 7 = -, ■ -I-. 28) 



where d is the depth of the fluid, a the coefficient of thermal expansion, and 

 AT the applied temperature contrast. The boundary conditions are u = w = 

 at z = and 1, and T = 1 at z = 0, and T = -1 at z = 1. The lateral ex- 

 tension of the system, though infinite in principle, must be restricted for com- 

 putational purposes. Since this dimension influences the horizontal wavelengths 

 admitted by the system, we will discuss below the special conditions assumed 

 in connection with the instability problem. 



The final dominant mode is expected to depend on the Rayleigh number Ra, 

 the Prandtl number a, and the geometry of the container. Furthermore, because 

 in finite amplitude flows the various horizontal wave numbers interact in a non- 

 linear fashion, the final mode will also depend on the amplitude of the total initial 

 perturbation, the relative amplitudes of the various constituent harmonics, and 

 the initial state of the system. The complete examination of this problem is a 

 long and tedious task; in this paper only token results are presented that, in the 

 author's opinion, serve to illustrate some of the interesting and difficult aspects 

 of this problem. 



An excellent review of most analytical studies on Benard convection appears 

 in a recent article by Brindley (1967). Rayleigh (1916) has shown that in the 

 case of rigid- rigid boundaries convection will only occur if the value of Ra ex- 

 ceeds 1708; below this, friction is able to overrule the weak destabilizing tem- 

 perature gradient. The preferred horizontal wavelength at the onset of convec- 

 tion was predicted by Pellew and Southwell (1940), based on linearized theory. 

 A summary of all linearized work appears in Chandrasekhar (1961). For two- 

 dimensional rolls, the nondimensional wavelength is \' = \/d = 2.016. Later 

 workers included nonlinear effects in several ways. Malkus and Veronis (1958), 

 Palm and C^iann (1960), Segel and Stuart (1962), Segel (1965a,b), Schluter, Lortz, 

 and Busse (1965), and Busse (1967), have included the nonlinear interaction of 

 many harmonics by working with finite amplitude flows; many workers have in- 

 cluded the variation of viscosity with temperature. Most of the studies have 

 been based on expansion in a small parameter e, being --Ra - Ra^, so that the 

 results are valid only for Ra not too much larger than the critical value. Near 

 Ra^, hexagonal shapes were predicted, and for higher Ra two-dimensional rolls. 



Roberts (1965) has used a different approach: he assumed a simple sinus^. 

 oidal variation in time, e''^*, and in the horizontal direction, sin nrrx, and 

 searched for the eigenvalues of the resulting system of nonlinear ordinary dif- 

 ferential equations in z. The criterion in determining the critical wave number 

 used was to find the value of \' for which da/d\' = 0. In this manner, he found 

 the optimum wavelengths for Ra up to 5000; at Ra = 4000 a value of \' = 2.004 

 was found, slightly smaller than the critical value. The behavior of \' with 

 increasing Ra showed a monotonic decrease. 



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