Stochastic Closures for Wave–Current Interaction Dynamics

Holm, Darryl D.

doi:10.1007/s00332-019-09565-0

Cited by 18 publications

(34 citation statements)

References 58 publications

(160 reference statements)

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“…In this case, the relation with Craik-Leibovich system can only be obtained by switching to the Itô setting or by considering additional random processes c.f. (Holm 2019). Note also that a Stratonovich decomposition requires stronger smoothness assumptions both in space and time of the transported quantities (see theorem C.2).…”

Section: General Remarks On the Lu Frameworkmentioning

confidence: 99%

Deciphering the Role of Small-Scale Inhomogeneity on Geophysical Flow Structuration: A Stochastic Approach

Bauer

Chandramouli

Chapron

et al. 2020

Journal of Physical Oceanography

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An important open question in fluid dynamics concerns the effect of small scales in structuring a fluid flow. In oceanic or atmospheric flows, this is aptly captured in wave–current interactions through the study of the well-known Langmuir secondary circulation. Such wave–current interactions are described by the Craik–Leibovich system, in which the action of a wave-induced velocity, the Stokes drift, produces a so-called “vortex force” that causes streaking in the flow. In this work, we show that these results can be generalized as a generic effect of the spatial inhomogeneity of the statistical properties of the small-scale flow components. As demonstrated, this is well captured through a stochastic representation of the flow.

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Section: General Remarks On the Lu Frameworkmentioning

confidence: 99%

Deciphering the Role of Small-Scale Inhomogeneity on Geophysical Flow Structuration: A Stochastic Approach

Bauer

Chandramouli

Chapron

et al. 2020

Journal of Physical Oceanography

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“…The SALT equations in this form have been studied extensively, for example, in wave-current interactions [Hol19a], uncertainty prediction [GBH19], solution properties of stochastic fluid dynamics [CFH19,AOBdLT18], and turbulent cascades [Hol19b], even when the spatial correlations are nonstationary [GBH18].…”

Section: Salt Equationsmentioning

confidence: 99%

Modelling the Climate and Weather of a 2D Lagrangian-Averaged Euler–Boussinesq Equation with Transport Noise

et al. 2020

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The prediction of climate change and its impact on extreme weather events is one of the great societal and intellectual challenges of our time. The first part of the problem is to make the distinction between weather and climate. The second part is to understand the dynamics of the fluctuations of the physical variables. The third part is to predict how the variances of the fluctuations are affected by statistical correlations in their fluctuating dynamics. This paper investigates a framework called LA SALT which can meet all three parts of the challenge for the problem of climate change. As a tractable example of this framework, we consider the Euler-Boussinesq (EB) equations for an incompressible stratified fluid flowing under gravity in a vertical plane with no other external forcing. All three parts of the problem are solved for this case. In fact, for this problem, the framework also delivers global well-posedness of the dynamics of the physical variables and closed dynamical equations for the moments of their fluctuations. Thus, in a well-posed mathematical setting, the framework developed in this paper shows that the mean field dynamics combines with an intricate array of correlations in the fluctuation dynamics to drive the evolution of the mean statistics. The results of the framework for 2D EB model analysis define its climate, as well as climate change, weather dynamics, and change of weather statistics, all in the context of a model system of SPDEs with unique global strong solutions.September 4, 2019Aims of the present paper. In this paper, we derive a stochastic version of the two-dimensional Euler-Boussinesq fluid system which is non-local in probability space, rather than in physical space, in the sense that the expected velocity is assumed to replace the drift velocity in the transport operator for the stochastic fluid flow. This stochastic fluid model is derived by exploiting a novel idea introduced in [DH19], of applying Lagrangian-averaging (LA) in probability space to the fluid equations governed by stochastic advection by Lie transport (SALT) which were introduced in [Hol15]. We follow the LA SALT approach to achieve three results of interest in climate modelling based on the Kelvin circulation theorem for stochastic transport of the Kelvin loop. The three results address the three components of the climate change problem discussed at the outset. First, it answers Lorenz's question about determinism in the affirmative. Namely, by replacing the drift velocity of the stochastic vector field by its expected value, one finds that the expected fluid motion becomes deterministic. This first step leads to the second result of interest in climate change modelling. Namely, it reduces the dynamical equations for the fluctuations to a linear stochastic transport problem with a deterministic drift velocity. Such problems are well-posed. We prove here that the LA SALT version of the 2D EB problem in a vertical plane possesses global strong solutions. The third result addresses the dynamics of the varianc...

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“…Furthermore, . For more discussion of the geometric properties of the GLM theory, see Gilbert and Vanneste ( 2018 ), Holm ( 2019b ).…”

Section: Deterministic Glm Background For Waves In the Oceanmentioning

confidence: 99%

“…Pseudomomentum is a 1-form density (dual to vector fields under pairing) which can be written in a variety of ways. For example, pseudomomentum can be written as Holm ( 2019b ) In the next section, the last variant in ( 2.8 ) will allow us to consider and as canonically conjugate wave field variables. It will follow that is a momentum map (Holm et al.…”

Section: Deterministic Glm Background For Waves In the Oceanmentioning

confidence: 99%

“…For more in-depth discussions of recent developments of GLM, see Bühler and McIntyre ( 1998 ), Bühler ( 2010 , 2014 ), Gilbert and Vanneste ( 2018 ), Holm ( 2019b ), Thomas ( 2017 ). In particular, recent refinements of GLM include its formulation for flows on manifolds (Gilbert and Vanneste 2018 ), and its extension from deterministic to stochastic dynamics, within the Euler–Poincar’e variational framework of geometric mechanics (Holm 2019b ).…”

Section: Deterministic Glm Background For Waves In the Oceanmentioning

confidence: 99%

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Stochastic Variational Formulations of Fluid Wave–Current Interaction

Holm

2020

J Nonlinear Sci

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We are modelling multiscale, multi-physics uncertainty in wave–current interaction (WCI). To model uncertainty in WCI, we introduce stochasticity into the wave dynamics of two classic models of WCI, namely the generalised Lagrangian mean (GLM) model and the 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 reduced Lagrangian for the current flow and a phase-space Lagrangian for the wave field. WCI in the GLM model involves the nonlinear Doppler shift in frequency of the Hamiltonian wave subsystem, which arises because the waves propagate in the frame of motion of the Lagrangian-mean velocity of the current. In contrast, WCI in the CL model arises because the fluid velocity is defined relative to the frame of motion of the Stokes mean drift velocity, which is usually taken to be prescribed, time independent and driven externally. We compare the GLM and CL theories by placing them both into the general framework of a stochastic Hamilton’s principle for a 3D Euler–Boussinesq (EB) fluid in a rotating frame. In other examples, we also apply the GLM and CL methods to add wave physics and stochasticity to the familiar 1D and 2D shallow water flow models. The differences in the types of stochasticity which arise for GLM and CL models can be seen by comparing the Kelvin circulation theorems for the two models. The GLM model acquires stochasticity in its Lagrangian transport velocity for the currents and also in its group velocity for the waves. However, the CL model is based on defining the Eulerian velocity in the integrand of the Kelvin circulation relative to the Stokes drift velocity induced by waves driven externally. Thus, 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. In an “Appendix”, we also discuss dynamical systems analogues of WCI.

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Stochastic Closures for Wave–Current Interaction Dynamics

Cited by 18 publications

References 58 publications

Deciphering the Role of Small-Scale Inhomogeneity on Geophysical Flow Structuration: A Stochastic Approach

Deciphering the Role of Small-Scale Inhomogeneity on Geophysical Flow Structuration: A Stochastic Approach

Modelling the Climate and Weather of a 2D Lagrangian-Averaged Euler–Boussinesq Equation with Transport Noise

Stochastic Variational Formulations of Fluid Wave–Current Interaction

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