Variational principles for fluid dynamics on rough paths

Crisan, Dan; Holm, Darryl D.; Leahy, James-Michael; Nilssen, Torstein

doi:10.48550/arxiv.2004.07829

Cited by 3 publications

(15 citation statements)

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“…Another collection of recent developments can be found in Castrillón López et al [2019]. An ongoing development is in the direction of rough geometric mechanics, initiated with a rough version of the Euler-Poincaré theorem in Crisan et al [2020]. Remarkably, variational principles which are driven by geometric rough paths again only require the product rule and the chain rule.…”

Section: The Circulation Mapmentioning

confidence: 99%

Stochastic geometric mechanics with diffeomorphisms

Holm¹,

Luesink²

2021

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Section: The Circulation Mapmentioning

confidence: 99%

Stochastic geometric mechanics with diffeomorphisms

Holm¹,

Luesink²

2021

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“…This approach was introduced in the stochastic setting in [Hol15] and subsequently many further developments have been made, [GBH18, dLHLT20, SC20, CCH + 19]. In [CHLN20], we extended [Hol15] and proposed a class of variational principles for fluid dynamics on geometric rough paths [LCL07,FV10,FH20] by constraining advection to incorporate a modeled temporally rough vector field. Critical points of the corresponding action functionals satisfy a system of rough partial differential equations (RPDEs) whose dynamics incorporate both the resolved-scale fluid velocity and the modeled effects of the unresolved fluctuations.…”

Section: Introductionmentioning

confidence: 99%

“…Critical points of the corresponding action functionals satisfy a system of rough partial differential equations (RPDEs) whose dynamics incorporate both the resolved-scale fluid velocity and the modeled effects of the unresolved fluctuations. The paper [CHLN20] provides a bridge between geophysical fluid dynamics and rough path theory. It draws upon knowledge from both areas, and we hope that it will impact both areas.…”

Section: Introductionmentioning

confidence: 99%

“…In our current endeavor, we establish the local well-posedness of Euler's equations for perfect incompressible fluid flow on geometric rough paths (see (3.1)) in 2 -Sobolev spaces with integer regularity ≥ ⌊ /2⌋ + 2. We have shown previously that this equation arises as a critical path of either Clebsch or Hamilton-Pontryagin action functionals and satisfies a Kelvin circulation balance law in [CHLN20] [Section 4.1 and 4.2] (see, also, Section 3.2.2).…”

Section: Introductionmentioning

confidence: 99%

“…[Hol15] or, more generally, driven by semi-martingales, see [SC20]. As already explained in [CHLN20], rough path perturbations include non-Markovian perturbations of the Lagrangian fluid trajectories. In particular, memory effects can be introduced through this formulation through a judicious choice of the rough path (e.g., by choosing a realization of a fractional Brownian motion).…”

Section: Introductionmentioning

confidence: 99%

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Solution properties of the incompressible Euler system with rough path advection

Crisan¹,

Holm²,

Leahy³

et al. 2021

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We consider the Euler equations for the incompressible flow of an ideal fluid with an additional rough-in-time, divergence-free, Lie-advecting vector field. In recent work, we have demonstrated that this system arises from Clebsch and Hamilton-Pontryagin variational principles with a perturbative geometric rough path Lie-advection constraint. In this paper, we prove local well-posedness of the system in 2 -Sobolev spaces with integer regularity ≥ ⌊ /2⌋ + 2 and establish a Beale-Kato-Majda (BKM) blow-up criterion in terms of the 1 ∞ -norm of the vorticity. In dimension two, we show that the -norms of the vorticity are conserved, which yields global well-posedness and a Wong-Zakai approximation theorem for the stochastic version of the equation.

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Stochastic mesoscale circulation dynamics in the thermal ocean

Holm,

Luesink,

Pan

2020

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The ocean surface boundary layer (OSBL) tends to be vertically well-mixed, but it can be horizontally inhomogeneous. For example, buoyancy fronts can develop as steep horizontal gradients of temperature, salinity and density in the OSBL. Misalignment of these steep horizontal buoyancy gradients with either the horizontal gradients of surface elevation or bathymetry is known to generate circulation which can deflect ocean currents which transport fluid properties, as well as pollution and debris. In turn, the generation of circulation itself can break up these fronts, thereby cascading horizontally circulating structures to smaller scales.Taking advantage of the vertically well-mixed property of the OSBL and working in the stochastic Euler-Poincaré variational framework introduced in [HL19], we derive the thermal rotating shallow water (TRSW) equations with stochastic advection by Lie transport (SALT), as a theoretical foundation for uncertainty quantification and data assimilation in computational models of the effects of submesoscale oceanic circulation using data-driven stochastic parametrisations, as in [CCH + 18, CCH + 19]. The key feature of SALT for geophysical fluid dynamics (GFD) is that SALT respects the Kelvin circulation theorem, which is the essence of cyclogenesis. Asymptotic expansion in the three small parameters present in the TRSW model in the neighbourhood of thermal quasi-geostrophic balance among the flow velocity and the gradients of free surface elevation and buoyancy leads first to the deterministic thermal versions of the Lagrangian 1 (TL1) equations and then to the thermal quasi-geostrophic (TQG) theory. We illustrate the instabilities of TQG which cascade circulation to smaller, typically unresolvable scales. We derive the stochastic version of this hierarchy of models TRSW/TL1/TQG in the framework of the stochastic Euler-Poincaré variational principle. Finally, we indicate the next steps in applying these results for uncertainty quantification and data assimilation of the cascading cyclogenetic effects of buoyancy fronts using the data-driven stochastic parametrisation algorithm based on SALT at these three levels of description.

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Variational principles for fluid dynamics on rough paths

Cited by 3 publications

References 0 publications

Stochastic geometric mechanics with diffeomorphisms

Stochastic geometric mechanics with diffeomorphisms

Solution properties of the incompressible Euler system with rough path advection

Stochastic mesoscale circulation dynamics in the thermal ocean

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