In analogy with similar effects in adiabatic compressible fluid dynamics, the effects of buoyancy gradients on incompressible stratified flows are said to be “thermal.” The thermal rotating shallow water (TRSW) model equations contain three small nondimensional parameters. These are the Rossby number, the Froude number, and the buoyancy parameter. Asymptotic expansion of the TRSW model equations in these three small parameters leads to the deterministic thermal versions of the Salmon's L1 (TL1) model and the thermal quasi-geostrophic (TQG) model, upon expanding in the neighborhood of thermal quasi-geostrophic balance among the flow velocity and the gradients of free surface elevation and buoyancy. The linear instability of TQG at high wavenumber tends to create circulation at small scales. Such a high-wavenumber instability could be unresolvable in many computational simulations, but its presence at small scales may contribute significantly to fluid transport at resolvable scales. Sometimes, such effects are modeled via “stochastic backscatter of kinetic energy.” Here, we try another approach. Namely, we model “stochastic transport” in the hierarchy of models TRSW/TL1/TQG. The models are derived via the approach of stochastic advection by Lie transport (SALT) as obtained from a recently introduced stochastic version of the Euler–Poincaré variational principle. We also indicate the potential next steps for applying these models in uncertainty quantification and data assimilation of the rapid, high-wavenumber effects of buoyancy fronts at these three levels of description by using the data-driven stochastic parametrization algorithms derived previously using the SALT approach.

Holm (Proc R Soc A Math Phys Eng Sci 471(2176):20140963, 2015) introduced a variational framework for stochastically parametrising unresolved scales of hydrodynamic motion. This variational framework preserves fundamental features of fluid dynamics, such as Kelvin’s circulation theorem, while also allowing for dispersive nonlinear wave propagation, both within a stratified fluid and at its free surface. The present paper combines asymptotic expansions and vertical averaging with the stochastic variational framework to formulate a new approach for developing stochastic parametrisation schemes for nonlinear waves in fluid dynamics. The approach is applied to two sequences of shallow water models which descend from Euler’s three-dimensional fluid equations with rotation and stratification under approximation by asymptotic expansions and vertical averaging. In the entire family of nonlinear stochastic wave–current interaction equations derived here using this approach, Kelvin’s circulation theorem reveals a barotropic mechanism for wave generation of horizontal circulation or convection (cyclogenesis) which is activated whenever the gradients of wave elevation and/or topography are not aligned with the gradient of the vertically averaged buoyancy.

Two different types of perturbations of the Lorenz 63 dynamical system for Rayleigh-Bénard convection by multiplicative noise -called stochastic advection by Lie transport (SALT) noise and fluctuationdissipation (FD) noise -are found to produce qualitatively different effects. For example, SALT noise preserves the sum of the deterministic Lyapunov exponents at any time, while FD noise does not. In the process of making this comparison between effects of SALT and FD noise on the Lorenz 63 system, a stochastic version of a robust deterministic numerical algorithm for obtaining the individual numerical Lyapunov exponents was developed. With this stochastic version of the algorithm, the value of the sum of the Lyapunov exponents differs from the deterministic value for the FD noise, whereas SALT noise retains this value with high accuracy.arXiv:1706.05882v6 [math.DS]

We extend the Itô-Wentzell formula for the evolution of a time-dependent stochastic field along a semimartingale to k-form-valued stochastic processes. The result is the Kunita-Itô-Wentzell (KIW) formula for k-forms. We also establish a correspondence between the KIW formula for k-forms derived here and a certain class of stochastic fluid dynamics models which preserve the geometric structure of deterministic ideal fluid dynamics. This geometric structure includes Eulerian and Lagrangian variational principles, Lie-Poisson Hamiltonian formulations and natural analogues of the Kelvin circulation theorem, all derived in the stochastic setting.Keywords Stochastic geometric mechanics · Lie derivatives with respect to stochastic vector fields · Pull-back by smooth maps with stochastic time parameterization Communicated byPurpose of this paper. This paper aims to derive stochastic partial differential equations (SPDEs) for continuum dynamics with stochastic advective Lie transport (SALT). These derivations require stochastic counterparts of the deterministic approaches for deriving fluid equations (PDEs). The approach we follow is the stochastic counterpart of the Euler-Poincaré variational principle as in Holm et al. (1998) which reveals the geometric structure of deterministic ideal fluid dynamics. Our goal is to create stochastic counterparts which preserve this geometric structure. Such variational formulations of stochastic fluid PDEs will also possess auxiliary SPDEs for stochastic advection of material properties which will correspond to various differential k-forms. The auxiliary SPDE for advective transport of a given material property by a stochastic fluid flow corresponds to a type of stochastic "chain rule" in which a k-form-valued semimartingale K (t, x) is evaluated (via pullback) along a stochastic flow φ t . The stochastic aspect of the flow map φ t represents uncertainty in the Lagrange-to-Euler map for the fluid. Examples of the k-form-valued process K (t, x) advected by the pullback of the stochastic flow map φ t include mass density, regarded as a volume form, and magnetic field, interpreted as a two-form for ideal magnetohydrodynamics (MHD) (Holm et al. 1998). We denote pullback of a k-form K by φ * t K , which for scalar functions f takes the simple form φ * t f := f • φ t . We will refer to this "chain rule" that gives us the auxiliary SPDE for SALT satisfied by the k-form-valued process φ * t K as the Kunita-Itô-Wentzell (KIW) formula for k-forms.In Kunita (1981Kunita ( , 1997, Kunita derived an extension of Itô's formula showing that if K is a sufficiently regular (l, m)-tensor and φ t is the flow of the SDE(1.1) with sufficiently regular coefficients, then an analogue of Itô's formula holds for tensor fields, namelyRemark 1.1 In applications, we will sometimes express (1.6) using the differential notation

This work involves theoretical and numerical analysis of the Thermal Quasi-Geostrophic (TQG) model of submesoscale geophysical fluid dynamics (GFD). Physically, the TQG model involves thermal geostrophic balance, in which the Rossby number, the Froude number and the stratification parameter are all of the same asymptotic order. The main analytical contribution of this paper is to construct local-in-time unique strong solutions for the TQG model. For this, we show that solutions of its regularized version α-TQG converge to solutions of TQG as its smoothing parameter α → 0 and we obtain blowup criteria for the α-TQG model. The main contribution of the computational analysis is to verify the rate of convergence of α-TQG solutions to TQG solutions as α → 0 for example simulations in appropriate GFD regimes.