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Variational Principles for Stochastic Fluid Dynamics
Darryl D. Holm
TL;DR
The paper addresses how to incorporate stochastic processes into variationally derived fluid equations while retaining geometric fluid-dynamical structure. It develops a stochastic variational approach and finds that Stratonovich formulations preserve key ideal-fluid properties, whereas equivalent Itô formulations obscure them through quadratic drift terms.
Problem
The paper seeks to include stochastic processes in variationally derived Euler–Poincaré systems, especially ideal fluid and geophysical fluid dynamics, while handling the difficulty that stochastic tangent vectors are not meaningful in usual Lagrangian flow relations.
Method
The paper formulates stochastic Euler–Poincaré fluid equations through a stochastically constrained variational principle, using Lie derivatives and spatially correlated noise modes selected from numerical or observational data.
Results
Stratonovich stochastic incompressible flows preserve Kelvin circulation, vorticity flux, and vorticity-line linkage, while the equivalent Itô representation masks these properties through an additional quadratic drift.
Takeaways & Limitations
Geometric conservation laws are transparent along Stratonovich stochastic paths, while interpreting Itô dynamics requires caution because representation-dependent quadratic drift can obscure them.
Takeaways & Limitations
The Itô formulation can appear to predict vorticity-line reconnection even when linkages are preserved in the Stratonovich representation, so conclusions about stochastic fluid behavior depend on the representation.
Abstract
from arXiv · showhide
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 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 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.
1 Introduction
The paper introduces a stochastic variational framework for deriving Eulerian fluid SPDEs with cylindrical noise and Lie-derivative advection. Its Stratonovich formulation preserves ideal-fluid circulation properties, while the equivalent Itô form exposes quadratic-covariation effects that mask those properties.
- Motivation: The paper extends variational derivations of Euler–Poincaré fluid equations to include cylindrical stochastic processes invariant under Lie-group actions.The framework targets compressible and incompressible fluids, including Euler and geophysical fluid dynamics.
- Variational formulation: The stochastic action uses a drift velocity plus spatially parameterized Stratonovich noise, with advected quantities evolving through a Lie derivative.The stochastic vector field is treated as a function of space rather than as a conventional stochastic flow satisfying an SDE.
- Noise construction: Proper Orthogonal Decompositions can select finitely many spatial modes representing unresolved, spatially correlated degrees of freedom coupled to resolved fluid variables.The selected modes may be drawn from numerical or observational velocity-field correlation data.
- Stochastic equations: The resulting SPDEs are first expressed in Stratonovich form and then converted to equivalent Itô equations using quadratic covariations of the temporal noise.For independent Brownian processes, only diagonal quadratic covariations remain, with [dW_j(t),dW_j(t)] = dt.
- Kelvin circulation: For incompressible flow, circulation is conserved along loops following the Stratonovich stochastic path.The equivalent Itô formulation uses different loop velocities and cannot represent the Itô terms as one Lie derivative, so the conservation law is masked; the induced circulation depends on the spatial correlation fields.
- Scope and assumptions: The approach formally assumes semimartingale objects and applies finite-dimensional stochastic reasoning pointwise in the spatial parameter.This formal treatment avoids addressing technical issues of stochastic analysis.
2 Stratonovich stochastic variational principle
The stochastic variational principle derives Stratonovich Euler–Poincaré fluid equations by imposing stochastic advection constraints and taking variations of the action.
- 2 Stratonovich stochastic variational principle: The variational derivation applies to both compressible and incompressible fluids in R3.The paper formulates the equations from a stochastically constrained action principle.
- 2 Stratonovich stochastic variational principle: The stochastic variational principle yields Stratonovich Euler–Poincaré equations for momentum, velocity, and advected quantities.Variations with respect to the action variables produce the governing relations, including the momentum equation recovered through a lemma.
- 2 Stratonovich stochastic variational principle: The momentum is identified as the variational derivative δℓ/δu, represented as a 1-form density.The diamond operation couples advected quantities to the momentum equation.
- 2 Stratonovich stochastic variational principle: The action constrains advected quantities to evolve by Lie transport along the Stratonovich stochastic vector field dxt.The vector field combines the drift velocity and cylindrical Stratonovich noise.
3 Stratonovich →Itˆo equations
The Stratonovich equations are converted to Itô form, where quadratic covariation produces double Lie-derivative operators that obscure the original advection interpretation.
- 3 Stratonovich →Itˆo equations: The Stratonovich Euler–Poincaré equations can be rewritten with stochastic terms separated on the right-hand sides, recovering deterministic ideal-fluid dynamics when noise vanishes.The stochastic vector fields are built from the prescribed ξi(x) modes.
- 3 Stratonovich →Itˆo equations: The double Lie derivatives define a second-order operator associated with the POD vector fields ξj(x).The paper interprets this operator as a Lie-Laplacian-type construction.
- 3 Stratonovich →Itˆo equations: The Lie Laplacian commutes with the exterior derivative.This follows from the nilpotence of the exterior derivative and commutation of Lie and exterior derivatives.
- 3 Stratonovich →Itˆo equations: Itô conversion introduces quadratic covariation drift terms that become double Lie derivatives rather than a single Lie derivative of q.For Brownian motion, the covariations reduce using [dWi(t), dWj] = δijdt.
4 Abstract Kelvin theorem
The abstract Kelvin–Noether theorem gives circulation laws for loops transported by stochastic flows, with conservation transparent in Stratonovich form but altered in Itô form.
- 4 Abstract Kelvin theorem: The Kelvin–Noether circulation map is defined by integrating the momentum one-form around a material loop transported by the stochastic flow.For a momentum density m and mass density D, the integrated one-form is m/D.
- 4 Abstract Kelvin theorem: Stratonovich circulation conservation holds for loops moving along the Stratonovich stochastic path.The theorem applies to loops satisfying the stochastic flow relation.
- 4 Abstract Kelvin theorem: In incompressible flow with volume preservation and no relevant advected-variable contribution, the circulation map is conserved for co-moving loops.The loops follow the Stratonovich stochastic vector field dxt.
- 4 Abstract Kelvin theorem: In the Itô representation, extra circulation sources arise because quadratic covariation terms cannot be represented by a single Lie derivative.Misalignment of the correlation eigenvectors creates or destroys circulation when viewed along the Itô path.
5 Examples
The examples formulate stochastic Euler, Euler–Boussinesq, and quasigeostrophic dynamics using geometric and variational structures. Stratonovich paths preserve circulation-related quantities, while the Itô representation can obscure these properties through quadratic drift terms.
- Stochastic Euler–Poincaré flows: The stochastic Euler–Poincaré equations use a divergence-free velocity and stochastic vector fields, with vorticity obtained by taking the curl of the motion equation.The momentum 1-form is dual to the velocity vector field, and the vorticity is represented as a 2-form.
- Stochastic Euler–Poincaré flows: The stochastic flow preserves volume elements when both the drift velocity and stochastic vector fields are divergence-free.This is the incompressible setting used for the geometric conservation results.
- Helicity preservation: Stratonovich stochastic Euler flows preserve helicity under homogeneous or periodic boundary conditions.The result follows from a conservation law whose integrated boundary contribution vanishes under those conditions.
- Helicity preservation: The Itô representation masks helicity preservation because its quadratic drift leaves a non-vanishing contribution even when boundary divergences vanish.This makes the Itô dynamics appear to allow vorticity-line reconnection, although linkages remain preserved in the Stratonovich representation.
- Euler–Boussinesq approximation: The Euler–Boussinesq model yields stochastic vorticity and buoyancy equations, with potential vorticity preserved along Stratonovich paths but not along the corresponding Itô paths.The Itô equation contains a double-Lie-derivative term; for independent unit stochastic vectors, this operator reduces to the metric Laplacian.
- Quasigeostrophic approximation: Stochastic quasigeostrophic dynamics is obtained by adding a Stratonovich stochastic term to the QG Hamiltonian, while potential-vorticity Casimirs remain conserved along Stratonovich paths.This conservation follows from the Lie–Poisson bracket independently of the stochastic Hamiltonian choice.
6 Conclusions
The paper’s stochastically constrained variational principle produces incompressible stochastic flows retaining key ideal-fluid conservation properties in the Stratonovich formulation. Their Itô form introduces quadratic drift and can mask circulation and potential-vorticity conservation, while the associated Lie Laplacian generalizes a scalar-density covariation drift.
- Conclusions: The variational principle yields stochastic incompressible flows preserving Kelvin circulation, vorticity-line linkage, and helicity properties associated with relabelling symmetry.The conclusions connect these results to the same Stratonovich-path interpretation of material loops and vortex field lines.
- Conclusions: Potential vorticity is preserved along each Stratonovich stochastic path in the Euler–Boussinesq equations.The conclusion extends the same pathwise conservation interpretation used for the Euler examples.
- Conclusions: The Itô representation masks these ideal-fluid conservation properties because its quadratic drift introduces terms that are not single Lie derivatives.The resulting interpretation is representation-dependent rather than a direct change in the Stratonovich conservation statement.
- Conclusions: The Lie Laplacian is a sum of double Lie derivatives over POD vector fields and generalizes Stratonovich’s quadratic covariation drift for scalar densities.It is not generally the standard Laplacian, although it reduces to the metric Laplacian for suitable independent stochastic vector fields.