Experiments on Convective Flows in Geophysical Fluid Systems 



equation that was applicable to three-dimensional problems. In this case, 

 however, no set of acceleration parameters was found. Later, Douglas (1961, 

 1962) has extended the method to nonlinear parabolic equations of the type 

 it = V2(p + f((p, X, z). The scheme is as follows: 



(q,n+l/4_q,n)/(^^/2) =i — + U — ^ + F" , 



^ \3x2 3x2 / Bz2 ../- 



,n+ 1 /2 



. — )(Ar/2) -^[ — 



B2(p 

 .3/2 



d-'cp d-'m \ d^c 



(41) 



■^ \ Bx2 3x2 



1 /32(p 



(^n+l_q,n.3/4) (Ar/2) =1 



^ \3z2 



1+ 1 



32(p 



3z2 



It consists of two pairs of row-column iterations with a correction for the non- 

 linear term sandwiched in between, and has a truncation error of 0(At2 4- Ax2). 

 Subject to the condition that | 3f/3(p| is bounded for all times, the scheme has 

 unconditional stability regarding the time steps. In view of the high order of 

 truncation error and the unlimited stability, it is most promising for applica- 

 tion to the Navier-Stokes equations, at least for flows in which the nonlinear 

 terms are not larger by orders of magnitude than the diffusive terms. It must 

 be noted that the column iteration involves only the second derivative in the 

 z-direction, so that the additional arithmetic due to row-column iteration is 

 minimal. Although the advective terms in the Navier-Stokes equations are not 

 exactly of the form f(x, z,cp ), they can be taken care of in several ways. One 

 way is to iterate more frequently in the predictor-corrector fashion of Eq. (41) 

 for the advective terms, and with each of such iterations solve the Poisson equa- 

 tion for the stream function. The other method, as introduced by Wilkes and 

 Churchill, is to include the u 'd/dx and w -d/dz terms into joint operators 

 d2/dx2 4- u 'd/dx and d2/dz2 4. w 'd/dz, respectively, and then proceed as in 

 Eq. (41), with F" now including only the buoyancy and Coriolis terms. 



A sure way of extending the Douglas method to the Navier-Stokes equations 

 is to regard the nonlinear terms as the only ones giving rise to limitations on 

 the time step. Thus, instead of unlimited stability we recover that of Scheme 

 II, given by Eq. (40). This is justified, because with respect to the nonlinear 

 term the scheme is a simple explicit one, and because unconditional stability 

 holds for the pure diffusion terms. It is to overcome even this limitation on 

 the time step that Wilkes and Churchill (1966) have evaluated the nonlinear 

 terms implicitly. Aziz and Heliums (1967) have successfully used this method 

 in a three-dimensional convection problem. 



The general scheme for the whole numerical procedure, therefore, is as 

 follows: 



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