Semi-martingale driven variational principles

Street, Oliver D.; Crisan, Dan

doi:10.48550/arxiv.2001.10105

Cited by 2 publications

(5 citation statements)

References 19 publications

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“…The stochastic dynamics of the advected quantities imposes a constrant on the variations known as a driving martingale relation in which the operation d in (3.1) may be regarded as a stochastic differential. For more discussion of this notation and the concept of driving martingales, see [73].…”

Section: Refinements Of Glmmentioning

confidence: 99%

“…which is required in order to impose preservation of volume when the transport velocity dt is stochastic, as discussed in [73].…”

Section: Including Stochastic Nonlinear Wave Propagation (Snwp) For Wcimentioning

confidence: 99%

“…However, we shall forgo this technical feature in favour of keeping the notation transparent. For a full explanation of semimartingale-driven variational principles, see [73].…”

Section: Deriving the Ou Craik-leibovich (Ou Cl) Equationsmentioning

confidence: 99%

“…) admits stochastic advection by Lie transport (SALT), as well as the OU process. Namely, by following[47] we find a modification of the Kelvin circulation theorem in (C.18) given byd c( u) v • dx = − c( u) gbdz dt , (C.21)with stochastic transport velocity in the SALT form[47],u := u(x, t)dt + ξ(x) • dW t .The corresponding SALT CL motion equation is given bydv − u × curl v + ∇ dp− 1 2 |u| 2 dt + u • u = − g∇b × ∇z dt , (C.22)in which now the pressure dp is a semimartingale, see[73]. The corresponding vorticity equation keeps its form, as in (C.19), dω + u • ∇ω − ω • ∇ u = 0 = − g∇b × ∇z dt , (C.23) as well as the conservation of potential vorticity (PV) on Lagrangian particles.…”

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confidence: 99%

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Stochastic Variational Formulations of Fluid Wave-Current Interaction

Holm

2020

Preprint

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We are dealing with multi-scale, multi-physics uncertainty in modelling wave-current interaction (WCI) in a stochastic fluid flow. The objective is to introduce stochasticity into WCI for the purpose of quantifying the uncertainty associated with the wave physics in either the generalised Lagrangian mean (GLM) model, or the alternative Craik-Leibovich (CL) model.The key idea for the GLM approach is the separation of the Lagrangian (fluid) and Eulerian (wave) degrees of freedom in Hamilton's principle. This is done by coupling an Euler-Poincaré vector field Lagrangian for the current flow and a phase-space Lagrangian for the wave field. The wave-current coupling is accomplished for GLM by pairing the Lagrangian-mean velocity of the current flow with the momentum map of the Hamiltonian wave system. To demonstrate the applicability of this hybrid approach, we use our wave-current Hamilton's principle approach to close both the deterministic and stochastic GLM equations for a 3D Euler-Boussinesq (EB) fluid. We also apply the method to add wave physics and stochasticity to the familiar 1D shallow water flow model.The appendices discuss finite dimensional analogues of WCI as well as comparing the differences in approaches for the stochastic GLM and CL models. The differences in the types of stochasticity can be seen in the Kelvin circulation theorems for the two theories. The GLM model acquires stochasticity in its Lagrangian transport velocity for the currents and in its group velocity for the waves. However, the CL model is based on modifying the Eulerian velocity in the integrand of the Kelvin circulation. Appendix C shows that the Kelvin theorem for the stochastic CL model can accept stochasticity in its both its integrand and in the Lagrangian transport velocity of its circulation loop.

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Section: Refinements Of Glmmentioning

confidence: 99%

“…which is required in order to impose preservation of volume when the transport velocity dt is stochastic, as discussed in [73].…”

Section: Including Stochastic Nonlinear Wave Propagation (Snwp) For Wcimentioning

confidence: 99%

“…However, we shall forgo this technical feature in favour of keeping the notation transparent. For a full explanation of semimartingale-driven variational principles, see [73].…”

Section: Deriving the Ou Craik-leibovich (Ou Cl) Equationsmentioning

confidence: 99%

mentioning

confidence: 99%

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Stochastic Variational Formulations of Fluid Wave-Current Interaction

Holm

2020

Preprint

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“…The integrals in (1) are of Stratonovitch type. The system (1) belongs to a class of stochastic models derived using the Stochastic Advection by Lie Transport Approach (SALT) approach, as described in [17], [34], [18]. A detailed derivation of this specific system can be found in [22], following [18], [17].…”

Section: Introductionmentioning

confidence: 99%

Well-posedness Properties for a Stochastic Rotating Shallow Water Model

Crisan¹,

Lang²

2021

Preprint

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In this paper, we study the well-posedness properties of a stochastic rotating shallow water system. An inviscid version of this model has first been derived in [17] and the noise is chosen according to the Stochastic Advection by Lie Transport theory presented in [17]. The system is perturbed by noise modulated by a function that is not Lipschitz in the norm where the wellposedness is sought. We show that the system admits a unique maximal solution which depends continuously on the initial condition. We also show that the interval of existence is strictly positive and the solution is global with positive probability.

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Semi-martingale driven variational principles

Cited by 2 publications

References 19 publications

Stochastic Variational Formulations of Fluid Wave-Current Interaction

Stochastic Variational Formulations of Fluid Wave-Current Interaction

Well-posedness Properties for a Stochastic Rotating Shallow Water Model

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