If we assume, as an aside, that the motion is 

 frictionless, and the wind stress is limited to the 

 +x direction, the equations can be easily solved 

 analytically by taking the time derivative of the 

 frictionless form of 1.4. Substituting in the friction- 

 less form of 1.3 I obtain a linear second order 

 partial differential equation, which is solved to 

 be: 



V - (I-cos ft) 



Df ,1.5) 



u= — (SIN ft) 



Df (1.6) 



These equations describe an oscillatory flow 90 

 degrees cum sole to the wind, in a "hopping" 

 motion. Each "hop" takes one inertial period, and 

 will persist into infinity as long as the wind stress 

 is constant (Kollmeyer, 1978). 



The form of the Reynolds stress terms imply a 

 dependence on velocity squared for the amount of 

 turbulent dissipation of energy within the fluid. 

 A proportionality factor, assumed to be linear, is 

 also necessary. To make the equations stable in a 

 computer time stepped format, the following 

 form was used: 



f — u'u' + — u'v'i=-K,vlu^ + v-'u 



\dx dy J (1.7) 



/ a — 9 — .\ 



I — u'v' + — v'v' =-KJu2 + v2'v 



\3x 9y / ' (1.8) 



The terms defining the drag on the bottom of 

 the layer, to, were quantified in a similar manner 

 to the Reynolds stress terms. The D in the 

 denominator was dropped because the drag at 

 the bottom of the layer should be dependent on 

 the stratification below the mixed layer, not on 

 the depth of the layer itself. Since the current 

 velocities in the mixed layer are a function of its 

 depth, any effects of the depth of the layer on the 

 dissipation function would be taken into account 

 by the coefficient and the velocity on which it 

 acts. 



— =K,u 



D ^ (1.9) 



Toy 



— = K„v 



D (1.10) 



Substituting 1.7 through 1.10 into 1.3 and 1.4, 

 we have final form of the motion equations. 



^ =fv+ — -K,J^FTT^u-K„u 

 3t D ' (1.11) 



^ = -fu+— -KJu2 + v2'v-K,v 

 3t D ' (1.12) 



Several facets of these equations deserve men- 

 tion. The wind stress is modeled as a body force 

 over the entire mixed layer, as was shown to be 

 valid by Pollard (1970). The equations obey the 

 concept of "slab flow", as have most models deve- 

 loped recently. 



The formulation of the dissipation functions is 

 designed to take into account first and second 

 order dissipations that could be acting on the 

 mixed layer. The terms are designed to span the 

 large gap in the current knowledge of regarding 

 downward momentum transport and turbulent 

 energy dissipation. In testing the model, the K^ 

 and Kg coefficients will be adjusted to fit the data 

 base. 



Another aspect of the equations is the shifting 

 of the dominant frequency of the model to some- 

 what greater than f'\ because of the subtraction 

 of the dissipation terms (Pollard and Millard, 

 1970). This effect would be maximum during a 

 period of steady, unidirectional winds. In testing 

 the model, it was found that this effect was not 

 significant enough to warrant an alteration to the 

 coriolis parameter, as was contemplated to bring 

 the model's dominant frequency into line with the 

 natural inertial frequency at the latitude modeled. 



The surface wind stress, To, was determined 

 using the Bodine (1971) formulation. His equa- 

 tions for the wind stress are: 



ro = u X 1.1x10 



for winds of less than 715 cm sec"' and 



(1.13) 



ro = uio((l-715/u,J x2.5x 10 )+ 1.1 x 10 



(1.14) 



for winds greater than 715 cm sec"'. Bodine's 

 fomulation has a transition point at 14 knots, 

 which takes into account the more turbulent 

 nature of higher wind velocities, and, to a more 

 limited degree, the higher pressure differentials 

 on the leeward and windward sides of surface 

 waves at the higher wind velocities. 



