## Momentum transfer and drag

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The main driving force of the ocean surface circulation is the wind. Blowing over the surface, the wind exerts a stress on the ocean, transferring momentum

Fig. 2.23. The balance of forces in the atmospheric boundary layer. The surface wind is deflected in the direction of the pressure gradient force (PGF) due to the retarding effect of drag, which also reduces the size of the Coriolis force, as this is proportional to speed. Note that the Coriolis force must always be at right angles to the velocity.

Fig. 2.23. The balance of forces in the atmospheric boundary layer. The surface wind is deflected in the direction of the pressure gradient force (PGF) due to the retarding effect of drag, which also reduces the size of the Coriolis force, as this is proportional to speed. Note that the Coriolis force must always be at right angles to the velocity.

1 Coriolis Force from the air to the water. This is then, in ways to be discussed in the next three sections, mixed into the fluid, producing motion of various scales.

At the same time as the wind pushes the sea, the water, in extracting energy from the wind, is acting as a drag force on the atmosphere. This slows the wind near the surface, which acts as a drag higher in the air. It also turns the near-surface wind away from its geostrophic direction to give cross-isobaric flow towards the lower pressure (Fig. 2.23). The direct influence of surface drag in the atmosphere extends to about a kilometre in height, although it can be much less in light winds. The vertical profile of horizontal wind speed in this boundary layer is logarithmic in many circumstances. As the wind speed above the boundary layer is independent of the ground immediately below it, the vertical gradient of the natural logarithm depends on the roughness of the underlying surface and the turbulence of the atmosphere at the surface:

In equation (2.12) z0 is the roughness length of the surface, k ~ 0.4, and u+ is the friction velocity. The roughness length gives the height at which the surface drag brings the atmosphere to rest, and can be very crudely thought of as the typical height of a perturbation to a flat surface (this is not quite correct, as z0 is usually smaller than this, but it gives a first approximation). The friction velocity is a measure of atmospheric turbulence near the ground. In a strongly turbulent atmosphere the horizontal and vertical velocities will be similar in magnitude: in this extreme, u+ is large and the horizontal velocity will change rapidly with height. In a nearly laminar atmosphere the horizontal velocity is much greater than the vertical velocity: u+ is small and the horizontal velocity will change slowly with height.

Equation (2.12) gives the vertical profile of horizontal velocity over any surface. The roughness length and friction velocity are often determined by taking two measurements of velocity at different heights and solving the two simultaneous equations that (2.12) then provides. Over the sea the friction velocity is about a twentieth of the 10 m wind speed; z0 varies from 10-6 m at wind speeds of 3 ms-1 to 10-4 m at 10 ms-1. Over the land z0 is a few centimetres above grass, and metres in a city. Land surfaces, having greater roughness which induces turbulence, exert considerably greater drag on the atmosphere than does the sea.

The logarithmic profile gives the wind near the surface a vertical gradient, or shear. This provides the mechanism for momentum to be transferred down towards the surface. A shear flow is not stable because small disturbances tend to grow, making the fluid turbulent. This turbulence, consisting of small eddies which provide the gustiness of the wind, acts to alter the shear. Faster moving parcels of air from above tend to move downwards, while slower moving eddies from below move upwards. This produces a net transfer of momentum downwards. This momentum is then captured by the surface as its drag force operates. The stress is clearly related to the mean wind speed as well as shear. Conventionally, it is empirically evaluated relative to the 10 m wind speed, u.

The stress, t (measured in kgm-1s-2), acting on the sea surface is given by

where CD, the drag coefficient, is a non-dimensional coefficient. The drag coefficient is an important parameter, as it determines the proportion of the atmospheric boundary layer momentum that is converted into ocean currents. As a first approximation CD can be taken to be a constant: CD ~ 1.5 x 10-3. This is reasonable for wind speeds of moderate strength, between 5 and 15 ms-1. However, CD will, in practice, depend on the type of logarithmic profile, and, hence, the roughness length and wind speed. Many experiments have been performed to find a relationship between the wind speed and CD; most produce a linear fit, such as

There is much scatter in the experimental results, and such equations as (2.14) must be regarded as approximations only, especially at low and high wind speeds. In both light and strong winds the degree of gustiness, for example, will be very important for downward transmission of momentum; current study of the provision of stress to the sea surface is concentrating on relating this stress to such atmospheric variability. Note that the magnitude of CD shows that the transfer of momentum from atmosphere to ocean is very inefficient; the surface ocean currents are thus typically only 1-2% of the wind speed.