Joseph Pedlosky Geophysical Fluid Dynamics - CHAPTER 4 Friction and Viscous Flow
The observed persistence over several days of large-scale waves in the atmosphere and the oceans reinforces the impression that frictional forces are weak, almost everywhere, when compared with the Coriolis acceleration and the pressure gradient. Friction rarely upsets the geostrophic balance to lowest order. Indeed, for many flows it is probably also true that the dissipative time scale is long compared to the advective time scale, i.e., that the frictional forces are weak in comparison with the nonlinear relative acceleration. Nevertheless friction, and the dissipation of mechanical energy it implies, cannot be ignored. The reasons are simple yet fundamental. For the time-averaged flow, i.e., for the general circulation of both the atmosphere and the oceans, the fluid motions respond to a variety of essentially steady external forcing. The atmosphere, for example, is set in motion by the persistent but spatially nonuniform solar heating. This input of energy produces a mechanical response, i.e., kinetic energy of the large-scale motion, and eventually this must be dissipated if a steady state—or at least a statistically stable average state of motion—is to be maintained. This requires frictional dissipation. In addition, the driving force itself may be frictional, as in the case of the wind-driven oceanic circulation. There the wind stress on the ocean surface produces a major component of the oceanic circulation. Finally, even though friction may be weak compared with other forces, its dissipative nature, qualitatively distinct from the conservative nature of the inertial forces, require its consideration if questions of the decay of free motions are to be studied.
The presence of frictional dissipation arises ultimately from the random motion of fluid molecules. The random motion of the fluid molecules is not described by the continuum equations. Rather, only the resulting viscous forces they produce are considered, and empirically, for many fluids, this viscous force can be written in terms of a viscosity coefficient and the macroscopic velocity alone. This frictional force is given by (1.4.5). Although this force must ultimately be responsible for the dissipation of kinetic energy and its transformation into the kinetic energy of the disordered molecular motion (i.e., heat), its direct effect on the large-scale motion has been shown in Section 2.8 to be utterly negligible. The length scales of large-scale motions are too great for molecular viscosity to be directly significant in the force balance. Yet we have argued above that friction must be important.
The problem, of course, is that the large-scale motion does not exist in isolation. For the very reason that molecular viscosity is so small, both the atmosphere and the oceans contain an enormously broad spectrum of turbulent motions fed to a considerable extent by the energy of the largest-scale flows. These turbulent fluctuations, embedded in the larger flow, tend to drain the large-scale flow of energy by a variety of mechanical processes and in turn pass the energy to finer scales of motion where viscosity can act directly. This notion of the cascade of energy from the largest to the smallest scales of motion is far from clear and rigorous. Indeed, we already have seen in the discussion of Section 3.26 that in some cases the cascade must at least partially go from small to large scale. Nevertheless, the gross aspect of the notion of the energy cascade is probably correct and has the following implication. If we wish to formulate a mathematical framework that describes directly the dynamics of only the large-scale motions, the drain of large-scale energy by smaller-scale motions must be represented entirely in terms of the kinematic features of the large scale. The only alternative is the intractable task of describing in detail all scales of motion, from the very largest to the very smallest. Although the example of the representation of viscous forces of molecular origin in terms of the macroscopic velocity serves as an encouraging model, the situation with regard to the representation of the turbulent interactions of small- and large-scale motions is considerably less satisfactory. Indeed this problem is one of the most vexing in geophysical fluid dynamics. At this time, there seems to be no tractable theory of turbulence that provides a practical and accurate description of the effective frictional force due to the cascade of energy by turbulent fluctuations.
In the following sections of this chapter we instead will describe one very crude representation of the effect of the turbulent cascade on the large-scale flow that quite frankly follows the notions that are appropriate to the representation of the effect of molecular motions on the mean. This is quite clearly a necessary compromise, since considering only the molecular viscosity acting on the large scale severely underestimates the role of friction, while an accurate consideration of the details of turbulent motions is simply impractical to the point of impossibility.