A TWO DIMENSIONAL MODEL OF WIND 

 FORCED INERTIAL OSCILLATIONS 



BRUCE E. VIEKMAN 



INTRODUCTION 



Recently several papers have investigated 

 inertial oscillations in the mixed layer. Pollard 

 (1970) showed that, for times less than seasonal, 

 the wind stress at the surface is confined to the 

 layer bounded below by the thermocline, and that 

 the wind stress can be applied as a body force in 

 that layer. Pollard and Millard (1970) used Pol- 

 lard's development and a linear damping term to 

 successfully model inertial oscillations in the 

 North Atlantic 130 miles south of Nantucket. 



Other papers have investigated the deepening 

 of the mixed layer. Pollard, Rhines and Thomp- 

 son (1973) investigated the deepening of the 

 mixed layer for times up to inertial, using the 

 temperature gradient below the mixed layer to 

 quantify the stability of the fluid below the layer 

 and entrainment rate of that fluid to deepen the 

 wind forced layer. Niiler (1977) modeled the 

 behavior of the mixed layer on a seasonal time 

 scale. Denman (1973) used a mechanism similar 

 to that used by Pollard, Rhines and Thompson to 

 determine the deepening of the mixed layer 

 using solar radiation and wind stress as forcing 

 functions. Denman and Miyake (1973) used 

 Denman's formulation to successfully model the 

 behavior of the thermocline at Ocean Station 

 Papa (50°N, 145°W). 



My objective is to combine these two separate 

 trains of research and formulate an integrated, 

 two dimensional model of mixed layer behavior 

 for times of up to two weeks. This integration is 

 important because the changing nature of the 

 thermocline affects directly the currents in it, as 

 we shall later find. To my knowledge, this is the 

 first attempt to combine these two lines of 

 investigation. 



My motivation in pursuing this research was 

 two-fold. First was the pure science of the sub- 



ject, and second was to fill a gap in the Coast 

 Guard's Search and Rescue problem. The deter- 

 mination of the wind current as part of a predic- 

 tion of the location of an object in the water has 

 long been a problem to SAR planners, and I hope 

 to give a more scientific aspect to this prediction. 

 This goal of formulating an operational model 

 imposed several constraints on its development 

 which I will discuss throughout this paper. 



THE MODEL 



The formulation of the model is divided into 

 two sections. One concerns the currents in the 

 mixed layer, the second concerns the changes in 

 the thermal structure of the upper ocean. 



1. Inertial Oscillations in the Mixed Layer. 



The mixed layer is assumed to be incompressi- 

 ble, in hydrostatic equilibrium, with a Rossby 

 number less than one. It obeys the Boussinesq 

 approximation. Vertical mean velocities are 

 assumed to be zero. The mixed layer is bounded 

 above by the sea surface and below by a density 

 gradient. The ocean is assumed to be laterally 

 unbounded and horizontally homogeneous. The 

 wind stress is assumed to be zonal. The entire 

 mixed layer is assumed to be highly turbulent 

 and homogeneous. The distribution of all stresses 

 and the diffusion of all properties in the layer are 

 rapid, because of the highly turbulent nature of 

 the layer. 



The assumption that the mixed layer is bounded 

 by a density gradient is based on the work of 

 Pollard (1970). He showed that only a small part 

 of the wind energy penetrates below the buoyancy 

 frequency gradient, for times under my consid- 

 eration. Here the thermocline is considered to be 

 synonymous with the buoyancy frequency and 

 density gradients under discussion. 



