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...

In this work, we examine the solution properties of the Burgers' equation with stochastic transport. First, we prove results on the formation of shocks in the stochastic equation and then obtain a stochastic Rankine-Hugoniot condition that the shocks satisfy. Next, we establish the local existence and uniqueness of smooth solutions in the inviscid case and construct a blow-up criterion. Finally, in the viscous case, we prove global existence and uniqueness of smooth solutions.

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

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In atmospheric science, slice models are frequently used to study the behaviour of weather, and specifically the formation of atmospheric fronts, whose prediction is fundamental in meteorology. In 2013, Cotter and Holm introduced a new slice model, which they formulated using Hamilton's variational principle, modified for this purpose. In this paper, we show the local existence and uniqueness of strong solutions of the related ISM (Incompressible Slice Model). The ISM is a modified version of the Cotter-Holm Slice Model (CHSM) that we have obtained by adapting the Lagrangian function in Hamilton's principle for CHSM to the Euler-Boussinesq Eady incompressible case. Besides proving local existence and uniqueness, in this paper we also construct a blow-up criterion for the ISM, and study Arnold's stability around a restricted class of equilibrium solutions. These results establish the potential applicability of the ISM equations in physically meaningful situations.