Because of the importance of the possible shoal- 

 ing of the mixed layer depth to a two dimensional 

 model, I sought a formulation including this 

 possibility. 



After reviewing the literature, I decided to use 

 the Denman (1973) formulation as applied by 

 Denman and Miyake (1973), and some elements 

 of the Pollard, Rhines and Thompson (1973) 

 model. 



I will outline the components of the Denman 

 (1973) derivation for application. His assump- 

 tions are similar to mine, with the added assump- 

 tion that all inputs to the mixed layer are redis- 

 tributed uniformly by turbulent diffusion, and 

 the time required for this distribution is small, 

 assumed to be instantaneous. Denman also as- 

 sumes that density is controlled only by tempera- 

 ture, as salinity is assumed to be constant 

 throughout the mixed layer. In applying his 

 derivation, I assume w = 0, because any mean 

 upwelling velocities are small compared to the 

 rate at which the mixed layer deepens. 



Using the equations for conservation of ther- 

 mal energy and conservation of mechanical 

 energy, he derives four equations for the thermal 

 behavior of the layer. Assuming H^ and H_., the 

 turbulent exchanges of heat at the sea surface 

 equal to zero, gives: 



(2.1) 



— =1. [-(G-D') + DB + R (D- 5 -1 + * ■' e- «D)] 

 9t D2 



aD 2[(G-D') + R5-Ml-e-«°)]-D[B + R(l+e «D)] 



at 



T.n = irRe 



at 



D(T3-To) 



^ aD aTl 



at dz l-D 

 SRe '^ 



(2.2) 



(2.3) 



(2.4) 



Where 5 is the extinction coefficient, .002 cm"\ 

 D is the mixed layer depth, R is the incident solar 

 radiation in cal cm"%ec"\ (G-D') is the wind energy 

 available for mixing, as defined by (G-D') = 

 nU^gro(po°=g)''. B is the back radiation from the 

 sea surface, T_, is the sea surface temperature, 

 and T p is the temperature below the thermo- 

 cline. H is a step function defined by: 



aD 



= if — < 0; no entrainment 



at 



= 1 if > 0; entramment at z = -D 



at 



The value of H controls the mode of the equa- 

 tions. When H is zero, the equations are in a heat 

 dominated mode, and a new thermocline will 

 form at a depth shallower than the original 

 thermocline, as in figure 2.1. H will equal zero 

 when the combination of mixing energy and inso- 

 lation is such that the old depth of the mixed 

 layer cannot be maintained under current condi- 

 tions. When H equals one, normal deepening of 

 the mixed layer occurs, as forced by wind energy 

 and controlled by the temperature gradient below 

 the mixed layer. 



3. Computational Scheme 



The problem of putting these two sets of equa- 

 tions into an integrated model was substantial, 

 because any inaccuracy in interfacing the two 

 sections would cause large differences in the final 

 solution. 



The equations of motion (1.9 and 1.10) were 

 used in a finite difference analog format, as was 

 equation 2.2 for use in the computer. 



To make the model as general as possible in a 

 predictive mode, and to satisfy a Coast Guard 

 operational constraint, a simple algorithm for 

 the radiative boundary condition was used. I used 

 a sine curve from to n, with the zero point being 

 0600 local time, the maximum at 7r/2 at 1200 

 local, and yrat 1800 local. From 1800 to 0600 the 

 insolation R = 0. Back radiation was a constant. 



To determine any deepening in the mixed 

 layer, the value of aD/at was solved for in 2.2 by 

 setting H = 1. If aD/at was greater than zero, 

 showing the assumption of H = 1 to be correct, the 

 system was shown to be in the wind driven mode, 

 and the model progresses to the next time step. If 

 aD/at was less than or equal to zero, the system 

 entered the heat driven mode. By setting H = in 

 2.2, and solving for (G-D'); 



(G-D') = % D [R(l+e-«D) + B] - RS ^ (1-e-*^) 



(3.1) 



This is similar to Denman's (21). D was solved for 

 by Newton's technique. The difference in mixed 

 layer temperature was found by integration of 

 2.1, substituting in 3.1, I obtain: 



AT =[R(l-e-*D) + B]At/D 



(3.2) 



