On the Hamiltonian approach: Applications to geophysical flows

Abstract

Abstract. This paper presents developments of the Harniltonian Approach to problems of fluid dynamics, and also considers some specific applications of the general method to hydrodynamical models. Nonlinear gauge transformations are found to result in a reduction to a minimum number of degrees of freedom, i.e. the number of pairs of canonically conjugated variables used in a given hydrodynamical system. It is shown that any conservative hydrodynamic model with additional fields which are in involution may be always reduced to the canonical Hamiltonian system with three degrees of freedom only. These gauge transformations are associated with the law of helicity conservation. Constraints imposed on the corresponding Clebsch representation are determined for some particular cases, such as, for example. when fluid motions develop in the absence of helicity. For a long time the process of the introduction of canonical variables into hydrodynamics has remained more of an intuitive foresight than a logical finding. The special attention is allocated to the problem of the elaboration of the corresponding regular procedure. The Harniltonian Approach is applied to geophysical models including incompressible (3D and 2D) fluid motion models in curvilinear and lagrangian coordinates. The problems of the canonical description of the Rossby waves on a rotating sphere and of the evolution of a system consisting of N singular vortices are investigated.

Errors, correlations and fidelity for noisy Hamilton flows. Theory and numerical examples

“…We notice that for = 0 the fundamental solution of the Fokker-Planck reduces to δ(x − S t (x 0 )), namely the fundamental solution of the deterministic continuity equation (3). We remark also that the root mean square deviation for β > 0 has an exponential growth σ(t) ∼ e βt .…”

“…We denote with G (x, x 0 ; t) the fundamental solution of the Fokker-Planck equation, namely its solution with initial condition G (x, x 0 ; 0) = δ(x − x 0 ). In the limit → 0 we recover the fundamental solution of equation (3). We write the solution of the Langevin equation as x(t) = S , t (x 0 ) where…”

“…When the noise is present we can still write a stochastic continuity equation given by (3) where Φ is replaced by x t ( ) ( ) ξ Φ +ε . In this case after averaging over the noise the distribution function satisfies the Fokker-Planck equation (see [10]).…”

“…The Hamiltonian description of geophysical fluids has been extensively developed [1][2][3][4] and is suited to analyse the conservation laws and to investigate approximation schemes which preserve the geometric invariants. These methods are also applicable to the Poisson-Vlasov equations for a collisionless plasma [5,6].…”

Null modes effect in Rossby wave model

“…For this reason, functional derivatives with respect to field variables are used. To address the issue we use the version of the HA given in Goncharov and Pavlov (1984, 1993, 1998, 2002 and . Information on supplementary bibliography can be found in Zakharov et al (1985).…”