Experiments on Convective Flows in Geophysical Fluid Systems 



Fourier inversion method was used for all three experiments. In experiment A, 

 the expansion was made in the z -direction, and the resulting ordinary differ- 

 ence equations had variable coefficients in the radius r . In experiments B and 

 C expansion was made in the horizontal x-direction. 



We must turn now to solutions of the "forecast" equations, where one 

 marches forward in time and finds the new values of <f, T, and m at a later time 

 step from their values at earlier time steps. Each transport equation may be 

 put in the general form 



— . -V-ucp, + c.V^cp, . F.^cp^.cp,, ...;--,...;-_. u,r, ...j . (34) 



In cases B and C, Fj = -Ra* 3t/bx and F2 = 0; in case A, Fj = (l/r^). 

 (20m/r3 + l)«Bm/3z - dj/dr, F^ = -r^'u, F^ = 0, and v^ is replaced by 

 the more complicated diffusion operators a, P and s (see Eqs. (16- 18)). Sup- 

 pose the time coordinate has been discretized as t = n • At and the iteration 

 has progressed through step n; we would like to compute the next values at 

 t = (n + l)At. In general, the time iteration methods may be divided into two 

 classes: explicit methods, in which all terms on the right-hand side of Eq. (34) 

 are evaluated at previous time steps n, n - 1, etc.; and implicit methods, in 

 which some terms may be evaluated at the step n + 1 to be computed. In the 

 latter case an iterative procedure is required to find the values at all grid- 

 points; for, in general, these become coupled in Eq. (34). In all the numerical 

 experiments the author has seen, the terms Fj are evaluated at the step n , 

 and the time derivatives as B(p/3t = (qp^^^ - <Pi"M/^t or {(p"*^ - cp-")/At. 



Lilly (1965) has given a summary of the more widely used time iteration 

 methods in geophysical fluid dynamics. Both he and Henrici (1962) have shown 

 that weak instabilities are associated with some of the multistep methods (i.e., 

 involving more than two time levels), due to the fact that the difference equation 

 admits spurious solutions that are not present in the original equation. In fact, 

 the solution may become decoupled on odd and even time levels. In addition, 

 strong instabilities may arise if care is not taken in differencing the advective 

 formas, or if too large a time step is used in the explicit schemes. 



The existence of "aliasing errors," due to misrepresentation of the shorter 

 waves because of the inability of the finite grid to properly resolve them, may 

 lead to computational instability. It may occur due to the nonlinear advective 

 terms; but it may also occur in linear equations with nonconstant coefficients. 

 Arakawa (1966) has shown that by a proper form of space-differencing the ad- 

 vective terms the nonlinear instability may be overcome. He presented several 

 schemes which simulate several important properties of continuous fluids, such 

 as conservation of vorticity, kinetic energy, mean-square vorticity, and con- 

 straint on the spectral distribution of energy. 



In two-dimensional incompressible flow, the expression V • u(p = (u 'V) qp 

 may be written as 



ciKp B(p Bi// 3qj d(\p,cp) 



"^ ■ ^S~ - ^ ■ ^~ = ^T : = 3(0-(p) • (35) 



OZ dx dx dz 3(X,Z) vr.-r/ V" / 



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