Semi-martingale driven variational principles

D., Street, Oliver; Crisan, Dan

doi:10.1098/rspa.2020.0957

Cited by 30 publications

(22 citation statements)

References 37 publications

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“…After this introduction of this stochastic transport constraint, the action integral becomes a semimartingale driven variational principle. 53 Consequently, the pressure Lagrange multiplier must be compatible with the noise introduced in the advection. This is required because one cannot enforce a variable in a stochastic system to remain constant without also requiring the Lagrange multiplier to also be a semimartingale, to control both the deterministic part of the system as well as the random fluctuations.…”

Section: Stochastic Advection By Lie Transport (Salt)mentioning

confidence: 99%

“…For models where we are considering incompressible flow, the pressure must act as a Lagrange multiplier to enforce the advected quantity

D

, the volume element, to be constant. More specifically, the advection constraint enforces that the advected quantities obey a stochastic partial differential equation given by

false(normald + L_{normald x_{t}} false) q : = d q + {scriptL}_{\hat{boldu}} q d t + \sum_{i} {scriptL}_{{\overset{ξ}{true}}_{i}} q \circ d W_{t}^{i} = 0,

where the vector field

\hat{boldu}

has been perturbed in the following way:

d {bold-italicx}_{t} = \overset{u}{true} d t + \sum_{i} {\tilde{boldξ}}_{i} \circ d W_{t}^{i} .

After this introduction of this stochastic transport constraint, the action integral becomes a semimartingale driven variational principle 53 . Consequently, the pressure Lagrange multiplier must be compatible with the noise introduced in the advection.…”

Section: Stochastic Wave Modellingmentioning

confidence: 99%

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Wave–current interaction on a free surface

Crisan

Holm

2021

Stud Appl Math

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The classical water wave equations (CWWEs) comprise two boundary conditions for the two-dimensional flow on the free surface of a bulk three-dimensional (3D) incompressible potential flow in the volume bounded by the free surface, which itself moves under the restoring force of gravity. One of these two boundary conditionsThis is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

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Section: Stochastic Advection By Lie Transport (Salt)mentioning

confidence: 99%

“…For models where we are considering incompressible flow, the pressure must act as a Lagrange multiplier to enforce the advected quantity

D

, the volume element, to be constant. More specifically, the advection constraint enforces that the advected quantities obey a stochastic partial differential equation given by

false(normald + L_{normald x_{t}} false) q : = d q + {scriptL}_{\hat{boldu}} q d t + \sum_{i} {scriptL}_{{\overset{ξ}{true}}_{i}} q \circ d W_{t}^{i} = 0,

where the vector field

\hat{boldu}

has been perturbed in the following way:

d {bold-italicx}_{t} = \overset{u}{true} d t + \sum_{i} {\tilde{boldξ}}_{i} \circ d W_{t}^{i} .

Section: Stochastic Wave Modellingmentioning

confidence: 99%

Wave–current interaction on a free surface

Crisan

Holm

2021

Stud Appl Math

Self Cite

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“…The action integral is, after the introduction of the stochastic transport constraint, thus in the form of a semi-martingale driven variational principle [47] and thus the pressure constraint must be compatible with the noise introduced in the advection. This is since one cannot enforce a variable in a stochastic system to remain constant without the Lagrange multiplier controlling both the deterministic part of the system as well as the random fluctuations.…”

Section: Jpψ Qqmentioning

confidence: 99%

Wave-current interaction on a free surface

Crisan

Holm

2020

Preprint

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The classic evolution equations for potential flow on the free surface of a fluid flow are not closed and the wave dynamics do not cause circulation of the fluid velocity on the free surface. The equations for free-surface motion we derive here are closed and they are not restricted to potential flow. Hence, true wave-current interaction occurs. In particular, the Kelvin-Noether theorem demonstrates that wave activity can induce fluid circulation and vorticity dynamics on the free surface. The wave-current interaction equations introduced here open new vistas for both the deterministic and stochastic analysis of nonlinear waves on free surfaces. A Transformation theory for fluid dynamics -Kelvin theorem 38 B Hamilton's principle for 3D fluid dynamics with a free surface 40 C Legendre transformation to the Hamiltonian for ACWWE 43

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“…The well-posedness of the Euler fluid version of the Euler-Poincaré SALT equations in three dimensions was established in [16] for initial conditions in appropriate Sobolev spaces. The mathematical framework of semimartingale-driven stochastic variational principles was established in [65].…”

Section: Plan Of the Papermentioning

confidence: 99%

“…The symbol d in (2.6) abbreviates stochastic time integrations. The action integral S in (2.6) is defined in the framework of variational principles with semimartingale constraints which was established in [65]. As we shall see below, the semimartingale nature of a Lagrange multiplier which imposes one of these semimartingale constraints emerges in the context of the full system of equations, which is obtained after the variations have been taken.…”

Section: Stochastic Semidirect-product Coadjoint Fluid Motion With Sa...mentioning

confidence: 99%

Stochastic effects of waves on currents in the ocean mixed layer

Holm,

2020

Preprint

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How does one guarantee that a stochastic approach for modelling fluid dynamics will preserve its fundamental deterministic properties, such as (i) energy conservation, (ii) Kelvin circulation theorem and (iii) conserved quantities arising from the Lagrangian particle relabelling symmetry? In fact, a choice must be made. For example, the approach of stochastic advection by Lie transport (SALT) preserves the latter two properties, but SALT does not conserve the deterministic energy. This paper introduces an energypreserving stochastic model for studying wave effects on currents in the ocean mixing layer. The model is called stochastic forcing by Lie transport (SFLT). The SFLT model is derived here from a stochastic constrained variational principle, so it has a Kelvin circulation theorem. The examples of SFLT given here treat 3D Euler fluid flow, rotating shallow water dynamics and the Euler-Boussinesq equations. In each example, one sees the effect of stochastic Stokes drift and material entrainment in the generation of fluid circulation. Moreover, we present an Eulerian-averaged SFLT model (EA SFLT), based on the Eulerian decomposition of solutions of the energy-conserving SFLT model into the sums of their expectations and fluctuations.

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Semi-martingale driven variational principles

Cited by 30 publications

References 37 publications

Wave–current interaction on a free surface

Wave–current interaction on a free surface

Wave-current interaction on a free surface

Stochastic effects of waves on currents in the ocean mixed layer

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