## I

higher vetoclty (momentum)

lower velocity (momentum lower velocity (momentum

Figure 3.5 Schematic diagram to illustrate the difference between (a) molecular viscosity and (b) eddy viscosity. In (a), the momentum transferred between layers is that associated with individual molecules, whereas in (b) it is that associated with parcels of fluid. (For simplicity, we have only shown two layers of differing velocity; in reality, of course, there are an infinite number of layers.)

Turbulent eddies in the upper layer of the ocean act as a 'gearing' mechanism that transmits motion at the surface to deeper levels. The extent to which there is turbulent mixing, and hence the magnitude of the eddy viscosity, depend on how well stratified the water column is. If the water column is well mixed and hence fairly homogeneous, density will vary little with depth and the water column will be easily overturned by turbulent mixing; if the water column is well stratified so that density increases relatively sharply with depth, the situation is stable and turbulent mixing is suppressed.

QUESTION 3.3 Between the warm well-mixed surface layer and the cold waters of the main body of the ocean is the ihermoclme. the /one within which temperature decreases markedly with depth. Expluin whether you would expect eddy viscosity to be greater in the ihermocline or m the mixed surface layer.

An extreme manifestation of the answer to Question 3.3 is the phenomenon of the slippery sea. If the surface layers of the ocean are exceptionally warm or fresh, so that density increases abruptly not far below the sea-surface, there is very little frictional coupling between the thin surface layer and the underlying water. The energy and momentum of the wind are then transmitted only to this thin surface layer, which effectively slides over the water below.

As you might expect from the foregoing discussion, values of eddy viscosity in the ocean vary widely, depending on the degree of turbulence. Eddy viscosity (strictly, the coefficient of eddy viscosity) is usually given the symbol A, and we may distinguish between A:. which is the eddy viscosity resulting from vertical mixing (as discussed above), and /\h, which is the eddy viscosity resulting from horizontal mixing - for example, that caused by the turbulence between two adjacent currents, or between a current and a coastal boundary. Values of Az typically range from 10"5 m2 s~' in the deep ocean to 10~' m2s 1 in the surface layers of a stormy sea; those for Ah are generally much greater, ranging from 10 to 10sm2s_l. (By contrast, the molecular viscosity of seawater is ~10~6 m2 s~'.)

Wh\ ,uv the \ ulues ot It, m> much higher than values <>l \ '

The high values of Ah and the relatively low values of Az reflect the differing extents to which mixing can occur in the vertical and horizontal directions. The ocean is stably stratified nearly everywhere, most of the time, and stable stratification acts to suppress vertical mixing; motion in the ocean is nearly always in a horizontal or near-horizontal direction. In addition, the oceans are many thousands of times wider than they are deep and so the spatial extent of horizontal eddying motions is much less constrained than is vertical mixing.

The fact that frictional coupling in the oceans occurs through turbulence rather than through molecular viscosity has great significance for those seeking to understand wind-driven currents. Currents developing in response to wind increase to their maximum strength many times faster than would be possible through molecular processes alone. For example, in the absence of turbulence, the effect of a 10 m s~' wind would hardly be discernible 2 m below the surface, even after the wind had been blowing steadily for two days; in reality, wind-driven currents develop very quickly. Similarly, when a wind driving a current ceases to blow, the current is slowed down by friction from turbulence many times faster than would otherwise be possible. Turbulence redistributes and dissipates the kinetic energy of the current; ultimately, it is converted into heat through molecular viscosity.

The types of current motion that are easiest to study, and that will be considered here, are those that result when the surface ocean has had time to adjust to the wind, and ocean and atmosphere have locally reached a state of equilibrium. When a wind starts to blow over a motionless sea-surface, the surface current which is generated takes some time to attain the maximum speed that can result from that particular wind speed; in other words, it first accelerates. The situations that are generally studied are those in which acceleration has ceased and the forces acting on the water are in balance.

The first satisfactory theory for wind-driven currents was published by V.W. Ekman in 1905. It is to his ideas, and their surprising implications, that we now turn.