235 research outputs found
K\'arm\'an--Howarth Theorem for the Lagrangian averaged Navier-Stokes alpha model
The K\'arm\'an--Howarth theorem is derived for the Lagrangian averaged
Navier-Stokes alpha (LANS) model of turbulence. Thus, the
LANS model's preservation of the fundamental transport structure of
the Navier-Stokes equations also includes preservation of the transport
relations for the velocity autocorrelation functions. This result implies that
the alpha-filtering in the LANS model of turbulence does not suppress
the intermittency of its solutions at separation distances large compared to
alpha.Comment: 11 pages, no figures. Includes an important remark by G. L. Eyink in
the conclusion
Variational Principles for Stochastic Fluid Dynamics
This paper derives stochastic partial differential equations (SPDEs) for
fluid dynamics from a stochastic variational principle (SVP). The Legendre
transform of the Lagrangian formulation of these SPDEs yields their Lie-Poisson
Hamiltonian form. The paper proceeds by: taking variations in the SVP to derive
stochastic Stratonovich fluid equations; writing their It\^o representation;
and then investigating the properties of these stochastic fluid models in
comparison with each other, and with the corresponding deterministic fluid
models. The circulation properties of the stochastic Stratonovich fluid
equations are found to closely mimic those of the deterministic ideal fluid
models. As with deterministic ideal flows, motion along the stochastic
Stratonovich paths also preserves the helicity of the vortex field lines in
incompressible stochastic flows. However, these Stratonovich properties are not
apparent in the equivalent It\^o representation, because they are disguised by
the quadratic covariation drift term arising in the Stratonovich to It\^o
transformation. This term is a geometric generalisation of the quadratic
covariation drift term already found for scalar densities in Stratonovich's
famous 1966 paper. The paper also derives motion equations for two examples of
stochastic geophysical fluid dynamics (SGFD); namely, the Euler-Boussinesq and
quasigeostropic approximations.Comment: 19 pages, no figures, 2nd version. To appear in Proc Roy Soc A.
Comments to author are still welcome
Bounds on solutions of the rotating, stratified, incompressible, non-hydrostatic, three-dimensional Boussinesq equations
We study the three-dimensional, incompressible, non-hydrostatic Boussinesq
fluid equations, which are applicable to the dynamics of the oceans and
atmosphere. These equations describe the interplay between velocity and
buoyancy in a rotating frame. A hierarchy of dynamical variables is introduced
whose members () are made up from the
respective sum of the -norms of vorticity and the density gradient.
Each has a lower bound in terms of the inverse Rossby number,
, that turns out to be crucial to the argument. For convenience, the
are also scaled into a new set of variables . By
assuming the existence and uniqueness of solutions, conditional upper bounds
are found on the in terms of and the Reynolds number .
These upper bounds vary across bands in the phase plane.
The boundaries of these bands depend subtly upon , , and the
inverse Froude number . For example, solutions in the lower band
conditionally live in an absorbing ball in which the maximum value of
deviates from as a function of and
.Comment: 24 pages, 3 figures and 1 tabl
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