scholarly journals Semi-martingale driven variational principles

2021 ◽  
Vol 477 (2247) ◽  
pp. 20200957
Author(s):  
O. D. Street ◽  
D. Crisan
Keyword(s):  
Variational Principles ◽  
Geophysical Fluid Dynamics ◽  
Deterministic Theory ◽  
Stochastic Fluid Models ◽  
Pressure Term ◽  
General Constraints ◽  
Stochastic Variational Principles ◽  
Dynamics Models ◽  
Stochastic Fluid ◽  
Special Case

Spearheaded by the recent efforts to derive stochastic geophysical fluid dynamics models, we present a general framework for introducing stochasticity into variational principles through the concept of a semi-martingale driven variational principle and constraining the component variables to be compatible with the driving semi-martingale. Within this framework and the corresponding choice of constraints, the Euler–Poincaré equation can be easily deduced. We show that the deterministic theory is a special case of this class of stochastic variational principles. Moreover, this is a natural framework that enables us to correctly characterize the pressure term in incompressible stochastic fluid models. Other general constraints can also be incorporated as long as they are compatible with the driving semi-martingale.

scholarly journals Variational principles for stochastic fluid dynamics

2015 ◽  
Vol 471 (2176) ◽  
pp. 20140963 ◽  
Author(s):  
Darryl D. Holm
Keyword(s):  
Fluid Dynamics ◽  
Variational Principles ◽  
Fluid Models ◽  
Motion Equations ◽  
Stochastic Fluid Models ◽  
Drift Term ◽  
Fluid Equations ◽  
Quadratic Covariation ◽  
Field Lines ◽  
Stochastic Fluid

This paper derives stochastic partial differential equations (SPDEs) for fluid dynamics from a stochastic variational principle (SVP). The paper proceeds by taking variations in the SVP to derive stochastic Stratonovich fluid equations; writing their Itô 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 Itô representation, because they are disguised by the quadratic covariation drift term arising in the Stratonovich to Itô transformation. This term is a geometric generalization 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; namely, the Euler–Boussinesq and quasi-geostropic approximations.

scholarly journals Circulation and Energy Theorem Preserving Stochastic Fluids

2019 ◽  
Vol 150 (6) ◽  
pp. 2776-2814 ◽  
Author(s):  
Theodore D. Drivas ◽  
Darryl D. Holm
Keyword(s):  
Variational Principle ◽  
Energy Conservation ◽  
Euler Equations ◽  
Navier Stokes ◽  
Fluid Models ◽  
Smooth Solutions ◽  
Stochastic Fluid Models ◽  
Natural Extensions ◽  
Stochastic Fluid ◽  
Incompressible Euler

AbstractSmooth solutions of the incompressible Euler equations are characterized by the property that circulation around material loops is conserved. This is the Kelvin theorem. Likewise, smooth solutions of Navier–Stokes are characterized by a generalized Kelvin's theorem, introduced by Constantin–Iyer (2008). In this note, we introduce a class of stochastic fluid equations, whose smooth solutions are characterized by natural extensions of the Kelvin theorems of their deterministic counterparts, which hold along certain noisy flows. These equations are called the stochastic Euler–Poincaré and stochastic Navier–Stokes–Poincaré equations respectively. The stochastic Euler–Poincaré equations were previously derived from a stochastic variational principle by Holm (2015), which we briefly review. Solutions of these equations do not obey pathwise energy conservation/dissipation in general. In contrast, we also discuss a class of stochastic fluid models, solutions of which possess energy theorems but do not, in general, preserve circulation theorems.

PERTURBATION ANALYSIS OF MULTICLASS STOCHASTIC FLUID MODELS

2002 ◽  
Vol 35 (1) ◽  
pp. 121-126 ◽  
Author(s):  
Christos G. Cassandras ◽  
Gang Sun ◽  
Christos G. Panayiotou ◽  
Yorai Wardi
Keyword(s):  
Perturbation Analysis ◽  
Fluid Models ◽  
Stochastic Fluid Models ◽  
Stochastic Fluid

Two Methods for Analyzing Waves in Composites with Random Microstructure

1991 ◽  
Vol 253 ◽  
Author(s):  
John R. Willis
Keyword(s):  
Variational Principle ◽  
Variational Principles ◽  
Statistical Information ◽  
Random Microstructure ◽  
Self Consistent ◽  
The Mean ◽  
Stochastic Variational Principles

ABSTRACTThe problem of calculating the mean wave in a composite with random microstructure is addressed. Exact characterizations of the problem can be given, in the form of stochastic variational principles. Substitution of simple configuration-dependent trial fields into these generates approximations which are, in a sense, ‘optimal’. It is necessary in practice to employ only trial fields which will generate, in the variational principle, no more statistical information than is actually available. Trial fields that require knowledge of two-point statistics generate equations that can also be obtained directly, through use of the QCA. The same fields can be substituted into an alternative variational principle to yield an approximation that makes use of three-point statistics – this approximation is less easy to obtain by direct reasoning. When not even two-point information is available, some more elementary approximation is needed. One such approximation, which is simple and direct in its application, is an extension to dynamics of a “self-consistent embedding” scheme which is widely used in static problems. This is also discussed, together with some illustrative results for a matrix containing inclusions and for a polycrystal.

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