332 



FOFONOFF 



[sect. 3 



where R'e = poUoLlfXH can be considered analogous to the Reynolds number 

 for molecular friction. In order that frictional forces be comparable to the 

 acceleration terms, the coefficient R'e must be of unit magnitude, i.e. 



fiH ~ pqUqL ~ po{UolL)L'^, 



where L is the horizontal scale of the flow and UqIL is a measure of the strain 

 or deformation rate of the mean flow. Similarly, for friction due to vertical 

 shear to become important, we must have 



fiv - tiH{HILY- ~ po{UolL)H\ 



The eddy viscosity terms will dominate over the acceleration terms if R'e is 

 less than the Rossby number. This occurs if [xh > pofoL'^- For L of the order of 

 20 to 30 km, we obtain hh> 10^ g cm~i sec~i. Values of /x// reported in the 

 literature range from 10^ g cm^i sec~i for small current systems to lO^o g cm~i 

 sec~i for the Antarctic Circumpolar Current (Hidaka and Tsuchiya, 1953). 



The time-dependent equations (13) and (14) can be converted to non- 

 dimensional form using the characteristic magnitudes already introduced for 

 the steady-state equations. We need, in addition, a time-scale To to charac- 

 terize the rate of variation and a density difference Apo to indicate density 

 variations. Using these characteristic magnitudes, we obtain the time-dependent 

 equations in the form : 



1 du"i 



F'r 



Up 



+ Rc 



du"i „ dU'i 

 ^ i -r-r + u j — — 





dx'i Re 



+ eijicQ'jU"ic 



L\2 du"i 



dx'j 



dx z' 



{i = 1,2) (22) 



L(C/o/L)T dt 



du"z , jj, c>u"3 „ 



dp" 



dx's 



dx'i 



P" + — 

 Rp 



Up 



Uo 



dU's 



dx'j 



Up 



Uo 



dr" 



3; 



dx'j 

 , pouofo „, „ 



+ — ; €ijlci'-i jU k 



^pog 



VV3+(^ 



L\2 d^u"z 



dx'?!^ 



1 



Apo 



pp {UpIL)Tp dt' 



dp" dp'u'j 

 dx'j 



= 0, 



(23) 



f24l 



where the variables depending on time are denoted by double primes and F'r 

 is introduced for Up^l{Apolpp)gH, the internal Froude number. 



The derivative with respect to time in (22) is comparable to unity if the time- 

 scale or period of the variation is of the order of a half-pendulum day (27r//o) or 

 less. If fpTp is small compared with unity, the acceleration term will dominate 

 over the Coriolis term and can be balanced only by the pressure gradient. For 

 this reason, short-period fluctuations {To<27rjfo) are sometimes referred to as 

 inertio-gravitational motion. For longer periods (T'o> 277-//o), the balance will 



