Geostrophic Turbulence

Strongly nonlinear, rapidly rotating, stably stratified flow is called geostrophic turbulence. This subject, which blends ideas from chapters 2,4, and 5, is relevant to the large-scale flow in the Earth’s oceans and atmosphere. The quasigeostrophic equations form the basis of the study of geostrophic turbulence. We view the quasigeostrophic equations as a generalization of the vorticity equation for two-dimensional turbulence to include the important effects of stratification, bottom topography, and varying Coriolis parameter. Thus the theory of geostrophic turbulence represents an extension of the theory of two dimensional turbulence. However, its richer physics and greater applicability to real geophysical flows make geostrophic turbulence a much more interesting and important subject. This chapter offers a very brief introduction to the theory of geostrophic turbulence. We illustrate the principal ideas by separately considering the effects of bottom topography, varying Coriolis parameter, and density stratification on highly nonlinear, quasigeostrophic flow. We make no attempt at a comprehensive review. In every case, the theory of geostrophic turbulence relies almost solely on two now-familiar components: a conservation principle that energy and potential vorticity are (nearly) conserved and an irreversibility principle in the form of an appealing assumption that breaks the time-reversal symmetry of the exact (inviscid) dynamics. This irreversibility assumption takes a great many superficially dissimilar forms, fostering the misleading impression of a great many competing explanations for the same phenomena. However, broadminded analysis inevitably reveals that these competing explanations are virtually equivalent. We begin by considering the quasigeostrophic flow of a single layer of homogeneous fluid over a bumpy bottom. No case better illustrates how diverse forms of the irreversibility principle lead to the same conclusions.

Introduction to Geophysical Fluid Dynamics

This second chapter offers a brief introduction to geophysical fluid dynamics—the dynamics of rotating, stratified flows. We start with the shallow water equations, which govern columnar motion in a thin layer of homogeneous fluid. Roughly speaking, the solutions of the shallow-water equations comprise two types of motion: ageostrophic motions, including inertia-gravity waves, on the one hand, and nearly geostrophic motions on the other. In rapidly rotating flow, these two types of motion may, in some sense, decouple. We seek simpler equations that describe only the nearly geostrophic motion. The simplest such equations are the quasigeostrophic equations. In the quasigcostrophic equations, potential vorticity plays the key role: The potential vorticity completely determines the velocity field that transports it, thereby controlling the whole dynamics. We begin by generalizing our previously derived fluid equations to a rotating coordinate frame.

Hamiltonian Fluid Dynamics

In this final chapter, we return to the subject of the first: the fundamental principles of fluid mechanics. In chapter 1, we derived the equations of fluid motion from Hamilton’s principle of stationary action, emphasizing its logical simplicity and the resulting close correspondence between mechanics and thermodynamics. Now we explore the Hamiltonian approach more fully, discovering its other advantages. The most important of these advantages arise from the correspondence between the symmetry properties of the Lagrangian and the conservation laws of the resulting dynamical equations. Therefore, we begin with a very brief introduction to symmetry and conservation laws. Noether’s theorem applies to the equations that arise from variational principles like Hamilton’s principle. According to Noether’s theorem : If a variational principle is invariant to a continuous transformation of its dependent and independent variables, then the equations arising from the variational principle possess a divergence-form conservation law. The invariance property is also called a symmetry property. Thus Noether’s theorem connects symmetry properties and conservation laws. We shall neither state nor prove the general form of Noether’s theorem; to do so would require a lengthy digression on continuous groups. Instead we illustrate the connection between symmetry and conservation laws with a series of increasingly complex and important examples. These examples convey the flavor of the general theory. Our first example is very simple. Consider a body of mass m moving in one dimension. The body is attached to the end of a spring with spring-constant K. Let x(t) be the displacement of the body from its location when the spring is unstretched.

Vorticity and Turbulence

Turbulence is an immense and controversial subject. Chapters 4, 5, and 6 present some ideas from turbulence theory that are relevant to flow in the oceans and atmosphere. In this chapter, we examine the connections between vorticity and turbulence. From ocean models that omit inertia, we turn to flows in which the inertia is a dominating factor. Vorticity is of central importance, and, in the case of threedimensional motion, we must take its vector character fully into account.

Fundamentals

This first chapter reviews the fundamental principles of fluid mechanics, emphasizing the relationship between the underlying microscopic description of the fluid as a swarm of molecules, and the much more useful (but less genuine) macroscopic description of the fluid as a set of continuous fields. Although it is certainly possible to study fluids without recognizing their true particulate nature, such an approach avoids important ideas about averaging that are needed later, especially in the study of turbulence. It is best to encounter these ideas at the earliest opportunity. Moreover, even if one adopts a strictly macroscopic viewpoint (as we eventually shall), one still has the choice between Eulerian and Lagrangian field theories. The Eulerian theory is the more useful and succinct, and most textbooks employ it exclusively. However, the Lagrangian theory, which regards the fluid as a continuous field of particles, is the more complete and illuminating, and it represents a natural extension of the ideas associated with the underlying molecular dynamics to the macroscopic level of description. All fluids are composed of molecules. We shall regard these molecules as point masses that exactly obey Newton’s laws of motion. This assumption is not precisely correct; the molecular motions are really governed by quantum mechanics. However, quantum effects are frequently unimportant, and the main ideas we want to develop are independent of the precise nature of the underlying molecular dynamics. It only matters that there be an exact underlying molecular dynamics, so that one could in principle predict the behavior of the whole fluid by solving the equations governing all of its molecules. It is of course utterly impractical to follow the motion of every molecule, because even the smallest volume of fluid contains an immense number of molecules. We are immediately forced to consider dynamical quantities that represent averages over many molecules.

Statistical Fluid Dynamics

In the previous chapter, we considered fluid turbulence from a heuristic viewpoint, combining ideas based on fundamental conservation laws with more qualitative ideas about irreversible behavior. However, there have been many attempts to study turbulence in a more quantitative and deductive fashion, beginning with the governing dynamical equations for a fluid. None of these attempts is, or could be, mathematically rigorous; all involve approximations that are difficult to justify and hard to test. In fact, these more deductive theories are perhaps best viewed as approximations in which the exact dynamical equations are replaced by stochastic-model equations having some (but not all) of the same physical properties as the exact equations, in much the same way as the quasigeostrophic equations contain only a part of the physics in the more exact primitive equations. In this chapter, we examine fluid turbulence from the standpoint of equilibrium and nonequilibrium statistical mechanics. This is a complicated and controversial subject, and our discussion will be introductory and elementary. In fact, we shall be much more concerned with the philosophy behind the statistical methods than with the detailed structure of the theory or with its successes and failures at describing real turbulence. This philosophy appears to be applicable to a much wider range of problems than so far considered. Readers who want a more thorough and complete description of statistical turbulence theory should consult more specialized sources. This chapter continues a theme, begun in chapter 4, to be continued in chapter 6, that much of our understanding of turbulence is based on two general principles: the conservation principle, which states that quantities like energy and potential vorticity are conserved (apart from the effects of dissipation), and the irreversibility principle, which holds that a turbulent system tends toward ever greater complexity. In chapter 4, we met the irreversibility principle in the assumptions that a narrow spectral peak spreads out and that nearby fluid particles tend to move apart. In this chapter, we encounter the irreversibility principle again, as a kind of macroscopic form of the Second Law of Thermodynamics.

Noninertial Theory of Ocean Circulation

This third chapter represents both a change in topic and a change in viewpoint. In chapter 2, we mainly considered the free (that is, unforced) motions of an ideal, horizontally unbounded fluid in which the inertia was always important. In this chapter, we examine the response of a horizontally bounded fluid to an external forcing. The presence of coastal boundaries is a complicating but indispensable feature of ocean circulation models. To compensate somewhat for the extra complications of the boundaries, we now simplify our dynamical equations by entirely neglecting the advection of momentum, or, more precisely, by replacing the inertia terms in the momentum equations with a large eddy viscosity of the kind discussed in chapter 1. Although this drastic step might be easier to justify for the ocean than tor the atmosphere—the ocean is in some sense more sluggish—our real motivation is a desire for tractable equations. Throughout most of this chapter, we also neglect the advection of buoyancy, and thus consider the linear theory of ocean circulation, which is relatively easy and reasonably complete. However, in the final two sections, we return to the much more challenging problem of properly incorporating nonlinear buoyancy advection.