## Torques and interior flow

If the wind stress acting on the ocean varies with latitude—as we see that it does in figure 4.3—then the wind provides a torque that tends to spin the ocean. In a steady state, not only do the forces on the ocean have to balance but so do the torques; otherwise the ocean would spin faster and faster. The torques on the ocean are provided by the wind, by friction, and by the Coriolis force (the pressure gradient does not provide a torque).2 Integrated over the entire ocean basin, the wind torque is balanced by the frictional torque, and since the frictional torque normally acts in a sense opposite to that of the spin of the fluid itself, the basin-scale circulation spins in the same sense as the wind. That is, for there to be a balance between wind and friction, the large-scale flow must have the same overall sense of rotation as the wind, producing the gyre shown in the left panel of figure 4.3.

However, in the interior of the basin, frictional effects are in fact very weak and the spin provided by the wind stress is locally balanced by the effects of the Coriolis force. Now, the Coriolis parameter increases northward, and it turns out that to locally balance the wind torque, a meridional flow must be produced in the ocean interior. The direction of the meridional flow depends on the sense of the spin provided by the wind, but in the subtropical gyre the meridional flow turns out to be equa-torward. Let's see why.

Consider a parcel of fluid in the middle of the ocean, as illustrated in figure 4.4. The wind blows zonally with a stronger eastward wind to the north and so provides a clockwise torque. We can balance this torque by a Coriolis force if there is a southward flow of water in the interior. In that case, the Coriolis force provides a westward force on all the parcels of fluid moving south. However, the force is stronger on the fluid that is in the northern part of the domain (because the Coriolis parameter increases northward), so the spin provided to the fluid is counterclockwise, opposing that spin provided by the wind. The southward flow adjusts itself so that the spin provided by the varying Coriolis force just balances the spin provided by the wind. (The balance is called Sverdrup balance, and the southward flow is called the Sverdrup interior.)

Now, the southward-flowing water must return northward somewhere, and this return must be at either the eastern or western boundaries because here the frictional effects of the water rubbing against the continental shelf and coast are potentially able to allow the flow to achieve a torque balance and move northward. However, only if the boundary layer is in the west (as illustrated in the right panel of figure 4.3) can such a balance be achieved: the

Wind

Wind

Figure 4.4. The production of a western boundary current. Schematic of the torques (namely, the spin-inducing forces: the wind, W; Coriolis, C; and friction, F) acting on parcels of water in the ocean interior (center) and western and eastern boundary layers (left and right), in a Northern Hemisphere subtropical gyre. In the interior, friction is small and the torques balance if the flow (denoted V) is southward. If the northward return flow is in the west, then a balance can be achieved between friction and Coriolis forces, as shown. If the northward return flow is in the east, no balance can be achieved.

Figure 4.4. The production of a western boundary current. Schematic of the torques (namely, the spin-inducing forces: the wind, W; Coriolis, C; and friction, F) acting on parcels of water in the ocean interior (center) and western and eastern boundary layers (left and right), in a Northern Hemisphere subtropical gyre. In the interior, friction is small and the torques balance if the flow (denoted V) is southward. If the northward return flow is in the west, then a balance can be achieved between friction and Coriolis forces, as shown. If the northward return flow is in the east, no balance can be achieved.

gyres then circulate in the same sense as the wind forcing, and the frictional forces at the western boundary act to oppose the wind forcing and achieve an overall balance. If the return flow were to be in the east, then the flow would, perversely, be circulating in the opposite sense to the torque provided by the wind, and no balance could be achieved. A local view of how the torque balances work in the boundary layer is provided in figure 4.4.

Suppose that the wind blew the opposite way. The balance of the wind torque and the Coriolis effect can now be achieved if interior flow is northward, and this is the case in the subpolar gyres. For the overall flow to have the same sense as the wind torque, the return flow still has to be in the west. Thus, we see that western boundary currents are a consequence of the differential rotation of Earth, not the way the wind blows. If Earth rotated in the opposite direction, the boundary currents would be in the east.

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