Piacsek 



^+ 



— (us^) + — (ws^) 



d r o z 



3T 



— + 



— + 0-U — +w — 



3t \ Br Bz 



.„(f).^,eli.v)-^ 



V2v - 



Br B z 



- + 1 u 



(12) 



(13) 

 (14) 



where 



aeAT • d 



4n2(b-a)2 2n(b-a)2 



and the operator a includes the geometric aspect ratio 



(15) 



Br "" Br 



+ \2 



Bz2 



(16) 



The boundary conditions on the system are taken as follows: 



1. At the rigid walls of the container all velocities vanish; hence both the 

 normal and tangential derivatives of ^ vanish, as does v. 



2. At the top free surface the normal velocity w and all stresses vanish. 

 This is equivalent to a "frictionless lid" approximation, and is designed to 

 eliminate external gravity waves and centrifugal effects on the surface. 



3. On the conducting cylindrical surfaces, the temperature is assumed to 

 be uniform; on the horizontal surfaces no heat flow is assumed, 



4. Since there is no in- or outflow into the annular cavity, the stream func- 

 tion may be set equal to zero on all surfaces. 



Before we write down the final set of equations that was programmed for the 

 computer, we must note that the advective term involving f is written as V • u^, 

 whereas those involving v and T are written as (u • V)v and (u.V)t, respec- 

 tively. The former is referred to in numerical weather forecasting as a "con- 

 servative" or "divergent" form, because its integral over r and z will reduce 

 to integrations on the boundaries only; furthermore, its finite difference ana- 

 logue preserves this property with respect to summation over the lattice of 

 gridpoints. In order to throw the remaining advective terms into a "conserva- 

 tive" form, we multiply through Eq. (13) by r^, and Eq. (14) by r, respectively, 

 and obtain 



— + V • urn = eP(m) - ur^ 

 Bt ^ ' 



BT '^ 



— + V • uT = - S(T) 

 Bt CT ^ ^ 



(17a) 

 (17b) 



758 



