Asymptotic limit analysis for numerical models of atmospheric frontogenesis

Accurate prediction of the future state of the atmosphere is important throughout society, ranging from the weather forecast in a few days time to modelling the effects of a changing climate over decades and generations. The equations which govern how the atmosphere evolves have long been known; these are the Navier-Stokes equations, the laws of thermodynamics and the equation of state. Unfortunately the nonlinearity of the equations prohibits analytic solutions, so simplified models of particular flow phenomena have historically been, and continue to be, used alongside numerical models of the full equations. In this thesis, the two-dimensional Eady model of shear-driven frontogenesis (the creation of atmospheric fronts) was used to investigate how errors made in a localised region can affect the global solution. Atmospheric fronts are the boundary of two different air masses, typically characterised by a sharp change in air temperature and wind direction. This occurs across a small length of O(10 km), whereas the extent of the front itself can be O(1000 km). Fronts are a prominent feature of mid-latitude weather systems and, despite their narrow width, are part of the large-scale, global solution. Any errors made locally in the treatment of fronts will therefore affect the global solution. This thesis uses the convergence of the Euler equations to the semigeostrophic equations, a simplified model which is representative of the large-scale flow, including fronts. The Euler equations were solved numerically using current operational techniques. It was shown that highly predictable solutions could be obtained, and the theoretical convergence rate maintained, even with the presence of near-discontinuous solutions given by intense fronts. Numerical solutions with successively increased resolution showed that the potential vorticity, which is a fundamental quantity in determining the large-scale, balanced flow, approached the semigeostrophic limit solution. Regions of negative potential vorticity, indicative of local areas of instability, were reduced at high resolution. In all cases, the width of the front reduced to the grid-scale. While qualitative features of the limit solution were reproduced, a stark contrast in amplitude was found. The results of this thesis were approximately half in amplitude of the limit solution. Some attempts were made at increasing the intensity of the front through spatial- and temporal-averaging. A scheme was proposed that conserves the potential vorticity within the Eady model.Open Acces

Quasi-convergence of an implementation of optimal balance by backward-forward nudging

Optimal balance is a non-asymptotic numerical method to compute a point on the slow manifold for certain two-scale dynamical systems. It works by solving a modified version of the system as a boundary value problem in time, where the nonlinear terms are adiabatically ramped up from zero to the fully nonlinear dynamics. A dedicated boundary value solver, however, is often not directly available. The most natural alternative is a nudging solver, where the problem is repeatedly solved forward and backward in time and the respective boundary conditions are restored whenever one of the temporal end points is visited. In this paper, we show quasi-convergence of this scheme in the sense that the termination residual of the nudging iteration is as small as the asymptotic error of the method itself, i.e., under appropriate assumptions exponentially small. This confirms that optimal balance in its nudging formulation is an effective algorithm. Further, it shows that the boundary value problem formulation of optimal balance is well posed up at most a residual error as small as the asymptotic error of the method itself. The key step in our proof is a careful two-component Gronwall inequality

Rotating convection : 1995 Summer Study Program in Geophysical Fluid Dynamics

The 1995 program in Geophysical Fluid Dynamics addressed "Rotating Convection," with particular emphasis on high-Rayleigh-number convection and on convection in the ocean.Funding was provided by the National Science Foundation under Grant No. OCE-8901012

Exponential smallness of inertia-gravity wave generation at small Rossby number

This paper discusses some of the mechanisms whereby fast inertia-gravity waves can be generated spontaneously by slow, balanced atmospheric and oceanic flows. In the small-Rossby-number regime relevant to mid-latitude dynamics, high-accuracy balanced models, which filter out inertia-gravity waves completely, can in principle describe the evolution of suitably initialised flows up to terms that are exponentially small in the Rossby number , i.e, of the form exp(−α/) for some α> 0. This suggests that the mechanisms of inertia-gravity-wave generation, which are not captured by these balanced models, are also exponentially weak. This has been confirmed by explicit analytical results obtained for a few highly-simplified models. We review these results and present some of the exponential-asymptotic techniques that have been used in their derivation. We examine both spontaneous-generation mechanisms which generate exponentially small waves from perfectly balanced initial conditions, and unbalanced instability mechanisms which amplify unbalanced initial perturbations of steady flows. The relevance of the results to realistic flows is discussed. 2

Mathematical Theory and Modelling in Atmosphere-Ocean Science

Mathematical theory and modelling in atmosphere-ocean science combines a broad range of advanced mathematical and numerical techniques and research directions. This includes the asymptotic analysis of multiscale systems, the deterministic and stochastic modelling of sub-grid-scale processes, and the numerical analysis of nonlinear PDEs over a broad range of spatial and temporal scales. This workshop brought together applied mathematicians and experts in the disciplinary fields of meteorology and oceanography for a wide-ranging exchange of ideas and results in this area with the aim of fostering fundamental interdisciplinary work in this important science area