3 results
Comparison of variational balance models for the rotating shallow water equations
- David G. Dritschel, Georg A. Gottwald, Marcel Oliver
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- Journal:
- Journal of Fluid Mechanics / Volume 822 / 10 July 2017
- Published online by Cambridge University Press:
- 07 June 2017, pp. 689-716
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We present an extensive numerical comparison of a family of balance models appropriate to the semi-geostrophic limit of the rotating shallow water equations, and derived by variational asymptotics in Oliver (J. Fluid Mech., vol. 551, 2006, pp. 197–234) for small Rossby numbers $Ro$. This family of generalized large-scale semi-geostrophic (GLSG) models contains the $L_{1}$-model introduced by Salmon (J. Fluid Mech., vol. 132, 1983, pp. 431–444) as a special case. We use these models to produce balanced initial states for the full shallow water equations. We then numerically investigate how well these models capture the dynamics of an initially balanced shallow water flow. It is shown that, whereas the $L_{1}$-member of the GLSG family is able to reproduce the balanced dynamics of the full shallow water equations on time scales of $O(1/Ro)$ very well, all other members develop significant unphysical high wavenumber contributions in the ageostrophic vorticity which spoil the dynamics.
8 - Stochastic Climate Theory
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- By Georg A. Gottwald, The University of Sydney, Daan T. Crommelin, University of Amsterdam, Christian L. E. Franzke, University of Hamburg
- Edited by Christian L. E. Franzke, Universität Hamburg, Terence J. O'Kane
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- Book:
- Nonlinear and Stochastic Climate Dynamics
- Published online:
- 26 January 2017
- Print publication:
- 19 January 2017, pp 209-240
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Summary
Abstract
In this chapter we review stochastic modeling methods in climate science. First we provide a conceptual framework for stochastic modeling of deterministic dynamical systems based on the Mori-Zwanzig formalism. The Mori-Zwanzig equations contain a Markov term, a memory term and a term suggestive of stochastic noise. Within this framework we express standard model reduction methods such as averaging and homogenization which eliminate the memory term. We further discuss ways to deal with the memory term and how the type of noise depends on the underlying deterministic chaotic system. Second, we review current approaches in stochastic data-driven models.We discuss how the drift and diffusion coefficients of models in the form of stochastic differential equations can be estimated from observational data. We pay attention to situations where the data stems from multi-scale systems, a relevant topic in the context of data from the climate system. Furthermore, we discuss the use of discrete stochastic processes (Markov chains) for example, stochastic subgrid-scale modeling and other topics in climate science.
Introduction
The climate system is characterized by the mutual interaction of complex systems each involving entangled processes running on spatial scales from millimeters to thousands of kilometers, and temporal scales from seconds to millennia. Given current computer power it is impossible to capture the whole range of spatial and temporal scales and this will also not be possible in the foreseeable future. Depending on the question we pose to the climate system – be it forecasting regimes in the atmosphere or simulating past ice ages – we have to make a decision as to what components to include in the analysis and as to what scales to resolve. A corollary of this decision is that each numerical scheme inevitably fails to resolve so-called unresolved scales or subgrid-scales. However, typically one is only interested at the slow processes active on large spatial scales. For example, for weather forecasts it is sufficient to resolve large-scale high and low pressure fields rather than smallscale fast oscillations of the stratification surfaces, whereas for climate predictions with a coupled ocean-atmosphere model we may want to distill the slow dynamics of the ocean ignoring weather systems interacting with the ocean on fast time-scales of days.
Late time evolution of unforced inviscid two-dimensional turbulence
- DAVID G. DRITSCHEL, RICHARD K. SCOTT, CHARLIE MACASKILL, GEORG A. GOTTWALD, CHUONG V. TRAN
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- Journal:
- Journal of Fluid Mechanics / Volume 640 / 10 December 2009
- Published online by Cambridge University Press:
- 19 October 2009, pp. 215-233
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We propose a new unified model for the small, intermediate and large-scale evolution of freely decaying two-dimensional turbulence in the inviscid limit. The new model's centerpiece is a recent theory of vortex self-similarity (Dritschel et al., Phys. Rev. Lett., vol. 101, 2008, no. 094501), applicable to the intermediate range of scales spanned by an expanding population of vortices. This range is predicted to have a steep k−5 energy spectrum. At small scales, this gives way to Batchelor's (Batchelor, Phys. Fluids, vol. 12, 1969, p. 233) k−3 energy spectrum, corresponding to the (forward) enstrophy (mean square vorticity) cascade or, physically, to thinning filamentary debris produced by vortex collisions. This small-scale range carries with it nearly all of the enstrophy but negligible energy. At large scales, the slow growth of the maximum vortex size (~t1/6 in radius) implies a correspondingly slow inverse energy cascade. We argue that this exceedingly slow growth allows the large scales to approach equipartition (Kraichnan, Phys. Fluids, vol. 10, 1967, p. 1417; Fox & Orszag, Phys. Fluids, vol. 12, 1973, p. 169), ultimately leading to a k1 energy spectrum there. Put together, our proposed model has an energy spectrum ℰ(k, t) ∝ t1/3k1 at large scales, together with ℰ(k, t) ∝ t−2/3k−5 over the vortex population, and finally ℰ(k, t) ∝ t−1k−3 over an exponentially widening small-scale range dominated by incoherent filamentary debris.
Support for our model is provided in two parts. First, we address the evolution of large and ultra-large scales (much greater than any vortex) using a novel high-resolution vortex-in-cell simulation. This verifies equipartition, but more importantly allows us to better understand the approach to equipartition. Second, we address the intermediate and small scales by an ensemble of especially high-resolution direct numerical simulations.