Darryl D. Holm

Nonlinear stability of fluid and plasma equilibria

The Liapunov method for establishing stability has been used in a variety of fluid and plasma problems. For nondissipative systems, this stability method is related to well-known energy principles. A development of the Liapunov method for Hamiltonian systems due to Arnold uses the energy plus other conserved quantities, together with second variations and convexity estimates, to establish stability. For Hamiltonian systems, a useful class of these conserved quantities consists of the Casimir functionals, which Poisson-commute with all functionals of the given dynamical variables. Such conserved quantities, when added to the energy, help to provide convexity estimates bounding the growth of perturbations. These estimates enable one to prove nonlinear stability, whereas the commonly used second variation or spectral arguments only prove linearized stability. When combined with recent advances in the Hamiltonian structure of fluid and plasma systems, this convexity method proves to be widely and easily applicable. This paper obtains new nonlinear stability criteria for equilibria for MHD, multifluid plasmas and the Maxwell-Vlasov equations in two and three dimensions. Related systems, such as multilayer quasigeostrophic flow, adiabatic flow and the Poisson-Vlasov equation are also treated. Other related systems, such as stratified flow and reduced magnetohydrodynamic equilibria are mentioned where appropriate, but are treated in detail in other publications.

A New Integrable Shallow Water Equation

Publisher Summary This chapter discusses about a new integrable shallow water equation. Completely integrable nonlinear partial differential equations arise at various levels of approximation in shallow water theory. Such equations possess soliton solutions-coherent (spatially localized) structures that interact nonlinearly among themselves and then re-emerge, retaining their identity and showing particle-like scattering behavior. This chapter discusses a newly discovered, completely integrable dispersive shallow-water equation found by Camassa and Holm in 1993. This equation is obtained by using a small-wave-amplitude asymptotic expansion directly in the Hamiltonian for the vertically averaged incompressible Eulers equations, after substituting a solution ansatz of columnar fluid motion and restricting to an invariant manifold for unidirectional motion of waves at the free surface under the influence of gravity. Section II of the chapter derives the one-dimensional Green–Naghdi equations. Section III uses Hamiltonian methods to newly discovered equation for unidirectional waves. Section IV analyzes the behavior of the solutions of the equation and shows that certain initial conditions develop a vertical slope in finite time. It is also shown that there exist stable multisoliton solutions. Section V demonstrates the existence of an infinite number of conservation laws for the equation that follow from its bi-Hamiltonian property.

A new integrable equation with peakon solutions

We consider a new partial differential equation recently obtained by Degasperis and Procesi using the method of asymptotic integrability; this equation has a form similar to the Camassa–Holm shallow water wave equation. We prove the exact integrability of the new equation by constructing its Lax pair and explain its relation to a negative flow in the Kaup–Kupershmidt hierarchy via a reciprocal transformation. The infinite sequence of conserved quantities is derived together with a proposed bi-Hamiltonian structure. The equation admits exact solutions as a superposition of multipeakons, and we describe the integrable finite-dimensional peakon dynamics and compare it with the analogous results for Camassa–Holm peakons.

The Three Dimensional Viscous Camassa–Holm Equations, and Their Relation to the Navier–Stokes Equations and Turbulence Theory

We show here the global, in time, regularity of the three dimensional viscous Camassa–Holm (Navier–Stokes-alpha) (NS-α) equations. We also provide estimates, in terms of the physical parameters of the equations, for the Hausdorff and fractal dimensions of their global attractor. In analogy with the Kolmogorov theory of turbulence, we define a small spatial scale, ℓ∈, as the scale at which the balance occurs in the mean rates of nonlinear transport of energy and viscous dissipation of energy. Furthermore, we show that the number of degrees of freedom in the long-time behavior of the solutions to these equations is bounded from above by (L/ℓ∈)3, where L is a typical large spatial scale (e.g., the size of the domain). This estimate suggests that the Landau–Lifshitz classical theory of turbulence is suitable for interpreting the solutions of the NS-α equations. Hence, one may consider these equations as a closure model for the Reynolds averaged Navier–Stokes equations (NSE). We study this approach, further, in other related papers. Finally, we discuss the relation of the NS-α model to the NSE by proving a convergence theorem, that as the length scale α1 tends to zero a subsequence of solutions of the NS-α equations converges to a weak solution of the three dimensional NSE.

An Integrable Shallow Water Equation with Linear and Nonlinear Dispersion

We use asymptotic analysis and a near-identity normal form transformation from water wave theory to derive a 1+1 unidirectional nonlinear wave equation that combines the linear dispersion of the Korteweg-deVries (KdV) equation with the nonlinear/nonlocal dispersion of the Camassa-Holm (CH) equation. This equation is one order more accurate in asymptotic approximation beyond KdV, yet it still preserves complete integrability via the inverse scattering transform method. Its traveling wave solutions contain both the KdV solitons and the CH peakons as limiting cases.

Wave Structure and Nonlinear Balances in a Family of Evolutionary PDEs

We investigate the following family of evolutionary 1+1 PDEs that describes the balance between convection and stretching for small viscosity in the dynamics of one-dimensional nonlinear waves in fluids: \[ m_t\ +\ \underbrace{\ \ um_x\ \ }_{\text{convection}}\ +\ \underbrace{\ \ b\,u_xm\ \ }_{\text{stretching}}\ =\ \underbrace{\ \ \nu\,m_{xx}\ }_{\text{viscosity}} \quad\text{with}\quad u=g*m. \] Here u=g*m denotes

On a Leray–α model of turbulence

u(x)=\int_{-\infty}^\infty g(x-y)m(y)\,dy

Camassa-Holm, Korteweg-de Vries-5 and other asymptotically equivalent equations for shallow water waves

. This convolution (or filtering) relates velocity u to momentum density m by integration against the kernel g(x). We shall choose g(x) to be an even function so that u and m have the same parity under spatial reflection. When

A connection between the Camassa–Holm equations and turbulent flows in channels and pipes

\nu=0

The Camassa-Holm equations and turbulence

, this equation is both reversible in time and parity invariant. We shall study the effects of the balance parameter b and the kernel g(x) on the solitary wave structures and investigate their interactions analytically for