Dynamics of Circulation and Vorticity

Abstract

In the preceding two chapters, we looked at the dynamics of the atmosphere in terms of the motion of fluid parcels, discrete bits of fluid that obey Newton’s laws of motion. Large-scale motions in the atmosphere and ocean—and, in fact, persistent motions in any fluid—are typically characterized by flow around closed paths. The strength of the flow around a closed path, denoted the circulation, is loosely analogous to the angular momentum of a solid body. More precisely, the circulation is the integral of the along-path component of the fluid velocity around a closed path. Just as the angular momentum is altered by torques, the circulation is altered by quantities akin to torques. The models in this chapter range from convection—the vertical overturning of a fluid heated from below—to the Gulf Stream—the western boundary current in the North Atlantic Ocean. They are unified by the fact that the dynamics in all cases are treated in terms of sources and sinks of the circulation, or its local manifestation, the vorticity. In Chapter 3, we considered the stability of a parcel of air, and we found that, for a dry parcel, the motion was unstable if the decrease in temperature with height exceeded some critical value. Including the circulation permits a more fluid-dynamical view of the process of convection. Here, we model a fluid circuit that starts off with cooler, denser, fluid above and warmer, less dense fluid, below. This situation is shown schematically in Figure 5.1. It is closely analogous to a bicycle wheel with a weight attached to one spoke near the rim, and starting with the weight directly above the hub. When the wheel is given a small push, the weight is moved to one side, and gravity then provides a net torque that increases the rotation of the wheel, moving the weight further off center, increasing the torque, and thus the acceleration, and so on. This is a positive feedback on the motion, and it yields instability. A fluid that starts with warmer fluid below and cooler above is similarly unstable to overturning motions.