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

Abstract

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...

A Hamiltonian Interacting Particle System for Compressible Flow

“…In Section 3.1 it is explained how the Hamiltonian H N is a sum of terms involving: translational kinetic energy of the particle ensemble, equilibrium internal energy associated to the hydrostatic pressure, equilibrium internal energy associated to capillary forces, non-equilibrium internal energy due to expansion/compression along the flow, and stochastic energy associated to molecular bombardment. The HIPS equations of motion are derived in Section 3.2, see (25)- (27). Under the assumption that the mean field limit exists, as N → ∞, the stochastic mean field equations are obtained in Section 3.3.…”

“…Recent advances in the stochastic modeling of geophysical flows include the "location uncertainty" approach of Mémin, Resseguier and collaborators [23][24][25][26], as-well as the "stochastic advection by Lie transport (SALT)" theory of Holm and collaborators [6,[27][28][29][30]. The SALT approach is based on the observation that subgrid phenomena represent unknown physical processes, and should therefore be derived from a stochastic variational principle.…”

“…Assume the mean field limit of (25)- (27) exists, for N → ∞. Since all sub-volumes ∆V α and their enclosed fluid elements are identical, it suffices to consider α = 1,…”

“…of [27,29], where u L t is a stochastic velocity field. However, there are a few crucial differences-while (50) is the starting point for the LA SALT theory, the HIPS formulation is based on the ensemble Hamiltonian (22), and the Lie transport along (49) in the mean field Equations (32)-(33) is a consequence of the Hamiltonian structure of the IPS (and the ensuing passage to the mean field limit).…”

Introduction to the Special Issue on the Statistical Mechanics of Climate

“…Alonso-Oran et al [4] propose in the case of two-dimensional Euler-Boussinesq equations a closed theory of weather and climate-intended as statistics of fluctuations and expectation value of the quantities of interest, respectively. This is achieved by taking into account the corresponding Lagrangian averaged stochastic advection by Lie transport equations, which are nonlinear and non-local, in both physical and probability space [28].…”