Stochastic Parametrization of the Richardson Triple

Holm, Darryl D.

doi:10.1007/s00332-018-9478-6

Cited by 13 publications

(23 citation statements)

References 53 publications

(143 reference statements)

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“…In the stochastic representation of fluctuating wave effects in the GLM picture, the stochastic pressure fluctuations in (3.10) might arguably be dropped because they cause no circulation. In that case, the stochasticity of the GLM group velocity in (3.9) would coincide with the existing theory of Stochastic Advection by Lie Transport (SALT) [36,37,11,16,9,10] which introduces the same type of Hamiltonian stochastic transport into the material fluid evolution.…”

Section: Stochastic Glm Closure #1asupporting

confidence: 58%

“…(3.15) Remark 3.2 In the class of closures #1a and #1b, with prescribed noise, it still remains to determine the set of vectors {ζ a (x t )} in the stochastic part of the Lagrangian trajectory given by dx t in equation (3.14). For this, it may be advisable to model the effects of wave fluctuations in the GLM equations (3.13) and (3.14) the same way as for any other high frequency transport effect in the SALT modelling approach of [11,36,37]. This approach would also simplify the calibration procedure for the correlation eigenvectors in ζ(x) • dW t , which is required in the application of SALT, because it would consolidate the stochastic effects of the wave transport with those of the material transport.…”

Section: Stochastic Glm Closure #1bmentioning

confidence: 99%

“…After discovering the elusiveness of the data required in formulating the stochastic WCI closure for GLM, in the second part of the paper, we propose an alternative stochastic closure for WCI. The alternative stochastic closure proposed here is a variant of the existing theory in [20] of Stochastic Advection by Lie Transport (SALT) [36,37,11,16] which introduces Hamiltonian stochastic transport into the material fluid evolution as a constraint in Hamilton's variational principle for fluid dynamics. The SALT approach separates the slow and fast time scales of the fluid transport velocity into drift and stochastic parts.…”

Section: Introductionmentioning

confidence: 99%

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Stochastic Closures for Wave–Current Interaction Dynamics

Holm

2019

J Nonlinear Sci

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Wave-current interaction (WCI) dynamics energizes and mixes the ocean thermocline by producing a combination of Langmuir circulation, internal waves and turbulent shear flows, which interact over a wide range of time scales. Two complementary approaches exist for approximating different aspects of WCI dynamics. These are the Generalized Lagrangian Mean (GLM) approach and the Gent-McWilliams (GM) approach. Their complementarity is evident in their Kelvin circulation theorems. GLM introduces a wave pseudomomentum per unit mass into its Kelvin circulation integrand, while GM introduces a an additional 'bolus velocity' to transport its Kelvin circulation loop. The GLM approach models Eulerian momentum, while the GM approach models Lagrangian transport. In principle, both GLM and GM are based on the Euler-Boussinesq (EB) equations for an incompressible, stratified, rotating flow. The differences in their Kelvin theorems arise from differences in how they model the flow map in the Lagrangian for the Hamilton variational principle underlying the EB equations.A recently developed approach for uncertainty quantification in fluid dynamics constrains fluid variational principles to require that Lagrangian trajectories undergo Stochastic Advection by Lie Transport (SALT). Here we introduce stochastic closure strategies for quantifying uncertainty in WCI by adapting the SALT approach to both the GLM and GM approximations of the EB variational principle. In the GLM framework, we introduce a stochastic group velocity for transport of wave properties, relative to the frame of motion of the Lagrangian mean flow velocity and a stochastic pressure contribution from the fluctuating kinetic energy. In the GM framework we introduce a stochastic bolus velocity in addition to the mean drift velocity by imposing the SALT constraint in the GM variational principle.

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Section: Stochastic Glm Closure #1asupporting

confidence: 58%

Section: Stochastic Glm Closure #1bmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Stochastic Closures for Wave–Current Interaction Dynamics

Holm

2019

J Nonlinear Sci

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“…At the same time, we regard the stochastic transport velocity of the circulation loop as the result of rapid, small-scale effects which perturb the Lagrangian trajectories of the CL model. This dual viewpoint is consistent with the stochastic modelling approach to Richardson’s metaphor of three-way interactions among Big, Little and Lesser Whorls whose stochastic theory was developed in Holm ( 2019a ).…”

Section: Introductionsupporting

confidence: 74%

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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“…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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Stochastic Parametrization of the Richardson Triple

Cited by 13 publications

References 53 publications

Stochastic Closures for Wave–Current Interaction Dynamics

Stochastic Closures for Wave–Current Interaction Dynamics

Stochastic Variational Formulations of Fluid Wave–Current Interaction

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

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