Introduction

The atmosphere and the ocean are the two most important fluid systems of our planet. The bulk of the atmosphere is a thin layer of air 10 km thick that engulfs the earth, and the oceans cover about 70% of the surface of our planet. Both the atmosphere and the ocean are in states of constant motion where the main source of energy is supplied by the radiation of the sun. The large-scale motions of the atmosphere and the ocean constitute geophysical flows and the science that studies them is geophysical fluid dynamics. The motions of the atmosphere and the ocean become powerful mechanisms for the transport and redistribution of energy and matter. For example, the motion of cold and warm atmospheric fronts determine the local weather conditions; the warm waters of the Gulf Stream are responsible for the temperate climate in northern Europe; the winds and the currents transport the pollutants produced by industries. It is clear that the motions of the atmosphere and the ocean play a fundamental role in the dynamics of our planet and greatly affect the activities of mankind.

It is apparent that the dynamical processes involved in the description of geophysical flows in the atmosphere and the ocean are extremely complex. This is due to the large number of physical variables needed to describe the state of the system and the wide range of space and time scales involved in these processes.

This book is an introduction to the fascinating and important interplay between non-linear dynamics and statistical theories for geophysical flows. The book is designed for a multi-disciplinary audience ranging from beginning graduate students to senior researchers in applied mathematics as well as theoretically inclined graduate students and researchers in atmosphere/ocean science. The approach in this book emphasizes the serendipity between physical phenomena and modern applied mathematics, including rigorous mathematical analysis, qualitative models, and numerical simulations. The book includes more conventional topics for non-linear dynamics applied to geophysical flows, such as long time selective decay, the effect of large-scale forcing, non-linear stability and fluid flow on the sphere, as well as emerging contemporary research topics involving applications of chaotic dynamics, equilibrium statistical mechanics, and information theory. The various competing approaches for equilibrium statistical theories for geophysical flows are compared and contrasted systematically from the viewpoint of modern applied mathematics, including an application for predicting the Great Red Spot of Jupiter in a fashion consistent with the observational record. Novel applications of information theory are utilized to simplify, unify, and compare the equilibrium statistical theories and also to quantify aspects of predictability in non-linear dynamical systems with many degrees of freedom. No previous background in geophysical flows, probability theory, information theory, or equilibrium statistical mechanics is needed to read the text. These topics and related background concepts are all introduced and developed through elementary examples and discussion throughout the text as they arise.

Introduction

In the previous Chapters 6, 8, and 9, various statistical theories have been introduced to predict large-scale structures for geophysical flows. These statistical theories range from the simple energy–enstrophy theory (EEST) developed in Chapters 6 and 8, point vortex theory (PVST) from Section 9.3, empirical statistical theory with a prior distribution and a few external constraints (ESTP) introduced in Chapter 9 and recalled in Chapter 10, to the elaborate equilibrium statistical theory involving many constraints (ESTMC) developed in Section 9.4. In the presence of such a wide variety of theories, a central question is the applicability of these statistical theories to geophysical flows. As indicated in Chapter 10, the purpose of this chapter is to address several practical as well as theoretical issues pertinent to the potential applicability of various equilibrium statistical theories. In particular we will address three applied issues, (A-1)–(A-3), and two theoretical issues, (T-1)–(T-2) of Chapter 10. We will provide strong evidence, both numerical and analytical, in supporting the central role of equilibrium statistical theories with judicious prior distribution and few external constraints.

Introduction

In this chapter we continue to study the dynamic behavior of the barotropic quasi-geostrophic equations in the absence of dissipation and external forcing, paying special attention to the non-linear interaction of the large-scale mean flow and the small-scale flow through topographic stress. Situations of obvious importance in atmosphere and ocean science occur when smaller-scale motions have a significant feedback and interaction with a larger-scale mean flow. One prototype situation of this sort occurs in the interaction of large-scale and small-scale components of barotropic flow over topography via topographic stress. In two influential papers, Charney and DeVore (1979) and Hart (1979) studied the multiple equilibrium states of this system with dissipation and single mode topography, and suggested their possible importance as model states for atmospheric blocking (see also Carnevale and Frederiksen, 1987; Vallis, 1985 for further developments).

In oceanography, in the special case of single mode topography as well as damping and driving, these equations have been used as a model for large-scale mean flow modification through topographic stress for flow along a continental shelf with smaller-scale topographic ridges (Allen etal., 1991; Samelson and Allen, 1987); also recently Holloway (Holloway, 1987; Edy and Holloway, 1994) has emphasized the possible dynamical significance of topographic stress in modifying coastal currents in many oceanographic contexts.

Introduction

A major theme of Chapters 6–13 of this book is to illustrate that systematic application of ideas from equilibrium statistical mechanics leads to novel promising strategies for assessing the unresolved scales of motion in geophysical flows. The various theories range from the simplest energy–enstrophy statistical theory (EEST) discussed in Chapters 6 and 8 to the empirical statistical theories discussed in Section 9.4 attempting to encode all the conserved quantities (ESTMC), to point vortex statistical theories discussed in Section 9.3, and, finally, to the empirical statistical theories with a few large-scale constraints and a judicious small-scale prior distribution (ESTP) formulated in Section 9.2. It was established in Chapters 11, 12, and 13 that the ESTP theories have a wide range of applicability in predicting large-scale behavior in damped and driven flows, as well as for observations such as the Great Red Spot of Jupiter. The ESTP formulation also includes the predictions of the energy–enstrophy statistical theory for the mean flow from Chapters 6 and 8 by utilizing a simple Gaussian prior probability distribution for potential vorticity fluctuations.

As discussed earlier, in Chapters 8, 9, and 10, the different equilibrium statistical theories all attempt to predict the coarse-grained behavior at large scales through the use of some of the formally infinite list of conserved quantities for idealized geophysical flows derived in Chapter 1.

Introduction

In this chapter we will lay down the foundations for statistical theories of geophysical flows and predictability that will be developed in the following chapters. The mathematical underpinning for these applications is a systematic use of information theory following Shannon (1948) Shannon and Weaver (1949) and Jaynes (1957). We develop such ideas beginning with elementary examples in this chapter. The motivation for the statistical studies of barotropic quasi-geostrophic flows lies in the generic occurrence of coherent large-scale flow patterns in physical observations and numerical simulations of flows that are approximately two dimensional. Examples of these flows are large isolated eddies in the atmosphere (a well-known example is the great red spot in Jupiter's atmosphere, see Chapter 13), and the discovery of mesoscale eddies in the ocean. Large-scale organized flow patterns emerge under a wide variety of initial conditions of the flow and topography. The robustness of these patterns seems to indicate that these large-scale coherent flows do not depend on the fine details in the dynamics of the flow or the topography. In addition, these flow patterns persist for a long time, and are essentially steady in nature. Therefore, it is plausible that an explanation for the observed two-dimensional coherent patterns can be found with considerations from equilibrium statistical mechanics, where we are interested in the large-scale features of the flow rather than all the fine details, and where only a few bulk properties of the flow, such as averaged conserved quantities like the energy and enstrophy enter the analysis.

Introduction

The observation of the geophysical flows in the atmosphere and the ocean reveal the existence of large-scale coherent flow structures. Examples of these structures are the atmospheric cyclonic and anti-cyclonic flow patterns, mesoscale ocean eddies, currents, and jets. These structures develop under fairly broad conditions and are characterized by their essentially steady nature, as well as their robustness and persistence in time. Possibly the most dramatic example of such coherent flow structures is exemplified by the Great Red Spot of Jupiter, discovered by Robert Hooke in 1664, which has persisted for at least 300 years.

From a dynamical point of view, such robust and persistent steady states must be non-linearly stable; small but finite initial perturbations of the steady states must remain small in time for the coherent flow structures to be observable. It is therefore clear that a fundamental problem is the study of the non-linear dynamical stability of the steady geophysical flows under small initial perturbations of the flow. This chapter and the next are devoted to the study of non-linear stability or instability of several classes of steady flows introduced earlier in Chapter 1. This study considers geophysical flows with topography and beta-plane effects, but without external forcing and dissipation mechanisms. In particular, we are interested in gaining a better understanding of what is the role played in the stability of the steady states by the beta-plane effect, and by the non-linear interaction of the large-scale mean flow and the small-scale flow through topographic stress.

Introduction

Prominent examples of long-lived large-scale vortices in geophysical flows are those such as the Great Red Spot (GRS) on Jupiter (Dowling, 1995; Marcus, 1993; Rogers, 1995). The emergence and persistence of such coherent structures at specific latitudes such as 22.4° S for the GRS in a background zonal shear flow that seems to violate all of the standard stability criteria (Dowling, 1995) are a genuine puzzle. Here we show how to utilize the equilibrium statistical theory with a suitable prior distribution, ESTP, introduced in Section 9.2 and discussed in Chapters 10, 11, and 12 to predict the actual coherent structures on Jupiter in a fashion which is consistent with the known observational record. As discussed in Section 9.2, the statistical theory is based on a few judiciously chosen conserved quantities for the inviscid dynamics such as energy and circulation and does not involve any detailed resolution of the fluid dynamics. The key ingredient of the ESTP is the prior distribution, Π(λ), for the one-point statistics of the potential vorticity which parameterizes the unresolved small-scale turbulent eddies that produce the large-scale coherent structures. Below, we show how the observational record of Jupiter from the recent Galileo mission suggests a special structure for such a prior distribution resulting from intense small-scale forcing. Recall from the studies discussed in Chapter 12 (DiBattista, Majda, and Grote, 2001) that ESTP can potentially describe the meta-stable large-scale coherent structures occurring from strong small-scale forcing, provided that the flux of energy to large scales is sufficiently weak.

A principal objective of any theory of fluid motion is the prediction of the spread of matter or “tracer” within the fluid. The problem is trivial for the fluid particles themselves in steady flow: they follow streamlines. It is nontrivial if the motion is time dependent, or if the tracer is dissolved in the fluid but diffusing through it. The time dependence of general interest is turbulence. The next four chapters develop a coherent framework for considering inhomogeneous and nonstationary turbulence, with elaboration in detail for the homogeneous, stationary and incompressible case, excluding and including tracer diffusion. Applications to the spread of phytoplankton are of special interest to oceanographers; these marine organisms are modeled as reacting tracers having nonlinear reaction rates. Absolute dispersion is considered first. This is the problem of predicting the path of a single fluid particle, or the path of the centroid of a cluster of particles, in turbulent flow. Turbulence being conceived as a random process, the problem is the prediction of the probability distribution function or pdf for the particle path. The mathematical difficulty is the closure of the infinite heirarchy of moments of the nonlinear kinematics, that is, the relating of certain high-order moments of particle displacement to low-order moments. There are any number of workings of this task in the literature, most of which close at second order, that is, second moments are related to first.