Piacsek 



Since the contributions from the advection terms in vorticity equations were 

 generally small, and in the case of rotation completely negligible, the finite dif- 

 ference form of the Jacobian was used that conserves only kinetic energy and 

 vorticity. In all cases the total absolute value of the vorticity converged to four 

 or more significant figures, so there was no need to use a scheme which also 

 conserves mean- square vorticity; such a scheme is the superposition of two 

 differently differenced Jacobians and represents twice the computer time. The 

 scheme used is given by 





(36) 



Three schemes were used in evaluating the advective and diffusive terms 

 during the model experiments: 



I. 



2At 



= c(v2(p)" + 5(0, cp)" + F" 



(37) 



II. 



,n+l _ ^n-1 



2At 



i + i,i 



■1, J 



n+l 



(...) 



(Ax): 



(Az)- 



+ 3(V^.<P)" + F" • (^^) 



The combination of the time-derivative evaluated at levels n+l and n- 1 with 

 the advective terms evaluated at level n is called the "leap-frog" procedure. 

 The particular way of differencing the diffusive terms in Scheme II is the well- 

 known DuFort-Frankel method (for a detailed description of this and other 

 schemes for the diffusion terms, the reader is referred to Richtmeyer and 

 Morton (1967), p. 189). Scheme III is described below and is given essentially 

 by Eqs. (41). Based on linearized stability analysis (see Richtmeyer and 

 Morton (1967)), Scheme I has a limit on the time step for the case Ax = Az 

 given by 



At < 



lc+ (|u|+|w|)-A 



while the stability of Scheme II is not affected by the diffusion terms, but is 



(39) 



At < 



(40) 



The truncation errors of the two schemes are 0(At + Ax 2) and 0(At/AX)2, so 

 that the gain in computer time is slightly offset by the poorer accuracy of 

 Scheme II. Nevertheless, in the cases considered, the truncation error could 

 be kept the same and still gain a factor of 10 in the allowable time step. 



After noting that a cycle of iteration for elliptic partial differential equa- 

 tions is analogous to a time step in parabolic equations such as the diffusion 

 equation, Douglas and Rachford (1956) devised an ADI method for the diffusion 



778 



