Let us now consider what forces and balances occur in the horizontal direction, for these forces give rise to the ocean currents and the great circulation patterns in the ocean. Away from regions where the direct effects of wind and friction are important (usually at the top and bottom of the ocean), the two dominant forces in the horizontal direction are the pressure gradient force and the Coriolis force; if the flow is steady, these two forces almost balance each other. This balance is called geostrophic balance. The evolution of a flow into geostrophic balance may be envisioned as follows. Suppose that initially there is a pressure gradient in the fluid. The pressure gradient generates fluid flow from the high-pressure region to the low-pressure region, but as the fluid moves it is deflected by the Coriolis force. In the Northern Hemisphere, the fluid comes into equilibrium, with the Coriolis force trying to deflect the fluid to the right of its direction of motion and the pressure force trying to deflect it to the left. The direction of motion is perpendicular to both the pressure force and the Coriolis force.
Mathematically, we equate the Coriolis force given in equation 3.4 with the pressure gradient force of equation 3.5 and find
Zonal direction: fv 1(3.11) J p dx
Meridional direction: fu 1 . (3.12)
These are the equations of geostrophic balance, and these equations hold to a good approximation for most large-scale flow in the ocean, especially away from boundaries.
How do horizontal pressure gradients arise in the ocean? Recall that the pressure at a point is primarily determined by the weight of the water above it. Thus, there is a horizontal gradient of pressure in the ocean interior either if the surface of the ocean is sloping or if the density of the seawater is varying. The latter effect may happen if the water has lateral gradients of temperature or salinity, as indeed is the case in the real ocean. We talk more about these effects in chapter 4, but let us perform a simple calculation to see how much slope of the ocean surface we need to produce a decent ocean current. The pressure at a point under the ocean is given by the weight of the column above it, so that p = pgh where h is the height of the column. Using this replacement in equation 3.12 gives fu = —gdh/dy, where dh/dy is the slope of the sea surface. Simple arithmetic then reveals that if the sea surface slopes by just 1 meter over a horizontal distance of 1,000 km, a geostrophic current of 10 cm s—1 is generated.
Finally, let's consider the question of why geostrophic balance should be dominant. Its dominance stems from the fact that the wind and currents are relatively weak compared to the speed of rotation of Earth itself. If the current changes by an amount U over a distance L, then the acceleration that the current undergoes is U2/L.
In comparison, the Coriolis acceleration is fU, so that the ratio of the local acceleration to the Coriolis acceleration is U/fL. This ratio is called the Rossby number, after the Swedish meteorologist Carl-Gustav Rossby, and it is a very small number for currents in the ocean (the reader should calculate a representative value with U = 0.1 m s—', L = 1,000 km, and f = 10—4 s—'. Thus, the Coriolis effect is far larger than the local effect of the flow changing speed or direction, and so most flows in the ocean are, in fact, geostrophic flows satisfying equations 3.11 and 3.12.
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