1,262 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
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