Introduction to Hamiltonian Fluid Dynamics and Stability Theory

This book provides an introduction to Hamiltonian fluid dynamics and describes aspects of hydrodynamic stability theory within the context of the Hamiltonian formalism. It is assumed that the reader has had some (but not necessarily an exhaustive) exposure to introductory fluid dynamics, hydrodynamic stability theory, finite dimensional Hamiltonian mechanics and variational calculus. As such, this book is written at a level appropriate for senior undergraduate or beginning graduate students in applied mathematics, engineering, physics and specific physical sciences such as fluid and plasma dynamics or dynamical meteorology and oceanography. There is more than enough material for a single semester course. We take a tutorial approach to presenting the subject material.

Our study begins with a review of the Hamiltonian structure of the nonlinear pendulum. A thorough linear and nonlinear stability analysis of the equilibrium solutions of the pendulum is given in the context of the Hamiltonian formalism. This simple physical example serves to introduce the reader, within the context of a finite dimensional phase space, to many of the ideas associated with the mathematical argument required in infinite dimensional Hamiltonian theory needed for fluid mechanics.

A detailed derivation of the noncanonical Hamiltonian structure for the vorticity equation formulation of the two dimensional incompressible Euler equations is given. Included is a rigorous derivation of the noncanonical Poisson bracket for the vorticity equation (written in terms of Eulerian variables) as an appropriate reduction of the canonical Poisson bracket for the two dimensional Euler equations (written in terms of Lagrangian variables). The complete family of Casimir invariants is systematically derived. Noether's Theorem is introduced and is used to construct time invariant functionals associated with various invariances of the Hamiltonian structure.

The Hamiltonian structure is exploited to derive a variational principle for general steady flows to the vorticity equation associated with the two dimensional Euler equations for an incompressible fluid. For general steady flows, the variational principle is based on an appropriately constrained energy functional. However, for steady flows with special spatial symmetries additional variational principles can be introduced. For example, parallel shear flows and circular flows are shown to satisfy the first order necessary conditions for an extremal of, respectively, an appropriately constrained linear momentum and angular momentum functional.

The variational principles are then used to determine conditions that can establish, in principle, the linear stability in the sense of Liapunov of the steady flows. It is shown that Fjortoft's Stability Theorem corresponds to the parallel shear flow limit of the stability conditions obtained from the variational principle associated with the constrained energy. It is shown that Rayleigh's Stability Theorem corresponds to the stability conditions derived from the variational principle associated with the constrained linear momentum. The stability conditions based on the constrained energy correspond to Arnol'd's First and Second Stability Theorems, respectively.

The linear stability conditions are used to motivate introducing appropriate convexity hypotheses on the functionals associated with the variational principles that can, in principle, establish the nonlinear stability in the sense of Liapunov of the steady flows. The convexity hypotheses ensure that the steady solutions correspond to a strict extrema of the functional associated with the variational principles and are required due to the loss of compactness in infinite dimensional vector spaces.

It is shown, however, that there are symmetry restrictions on the class steady flows that can, in principle, satisfy the stability conditions. This result is known as Andrews's Theorem. We examine the implications of Andrews's Theorem for steady flows in a periodic channel, steady flows with finite area-integrated energy and enstrophy on the plane, and steady flows in a circular annulus.

One unsatisfactory feature of the two-dimensional vorticity equation is that it is, of course, incapable of modelling any three-dimensional effects such as vortex tube stretching/compression. Another physically relevant feature that the two-dimensional vorticity equation does not include, in the absence of a mean flow, is a term corresponding to a background vorticity gradient. The most important consequence of this property is that there are no nontrivial linear plane wave solutions to the two-dimensional vorticity equation. The simplest generalization of the two dimensional vorticity equation that includes these effects is the Charney-Hasegawa-Mima (CHM) equation. We derive the CHM equation in the context of describing the leading order low frequency/wave number dynamics of a rapidly rotating fluid of finite depth. The CHM equation possesses, in addition to steady flow solutions, linear dispersive wave solutions and solitary steadily travelling vortex solutions.

The Hamiltonian structure of the CHM equation is developed and exploited to give variational principles for arbitrary steady and steadily travelling solutions. For the steady parallel shear flow solutions, the normal mode problem is examined and the generalizations of Rayleigh's and Fjortoft's stability theorems are derived. These stability conditions are generalized, in the context of the Hamiltonian structure, to establish conditions for the linear and nonlinear stability in the sense of Liapunov for general steady solutions.

The analogue of Andrews's Theorem is established for the CHM equation. It is shown that Andrews's Theorem implies that there is no nontrivial steadily travelling solution with finite area-integrated energy and enstrophy in the plane or a periodic channel that can satisfy the conditions of the Arnol'd-like stability argument developed. In addition to establishing this general result, we specifically examine in some detail the breakdown of the stability theory for the periodic solutions of the CHM} equation known as Rossby waves and the steadily travelling dipole vortex solutions known as modons.

One of the major successes of infinite-dimensional Hamiltonian mechanics has been to provide the framework for a proof of the nonlinear stability of solitons. Here, we present an account of the Hamiltonian structure of the Korteweg-de Vries (KdV) equation and present the linear and nonlinear stability theory for the KdV soliton as a consequence of the Hamiltonian formalism. The KdV equation is a canonical equation of mathematical physics in that it invariably arises in the mathematical modelling of small-but-finite amplitude weakly dispersive wave phenomena. We include a brief derivation of the KdV equation and its periodic and soliton solutions in the context of internal gravity waves in a continuously stratified fluid of finite depth.

The KdV has two Hamiltonian formulations associated with it. We describe in full one of the Hamiltonian formulations. Based on the Hamiltonian structure, variational principles are established for the periodic and soliton solutions. The stability theory associated with the KdV soliton is described. Our discussion begins with the normal mode stability problem that, it is shown, has no unstable solutions associated with it. The main mathematical problem associated with developing the rigorous (linear and nonlinear) stability theory for the KdV soliton is that it is not obvious that the second variation of the functional associated the soliton variational principle (evaluated at the soliton) is definite for all perturbations. By exploiting the spectral properties associated the operator associated with the integrand of the second variation, it is possible to establish appropriate bounds on the second variation and thereby establish stability. We develop this theory with sufficient generality that the reader can see how it applies to other soliton equations.

Reviews in the Proceedings of the Edinburgh Mathematical Society, Applied Mechanics Reviews and MathSciNet.