A Mathematical Theory of Large-scale Atmosphere/ocean Flow by Michael J P Cullen

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By Michael J P Cullen

This booklet counteracts the present model for theories of "chaos" and unpredictability through describing a idea that underpins the wonderful accuracy of present deterministic climate forecasts, and it means that additional advancements are attainable. The e-book does this by way of creating a special hyperlink among a thrilling new department of arithmetic known as "optimal transportation" and current classical theories of the large-scale surroundings and ocean move. it really is then attainable to resolve a suite of easy equations proposed decades in the past via Hoskins that are asymptotically legitimate on huge scales, and use them to derive quantitative predictions approximately many large-scale atmospheric and oceanic phenomena. a specific function is that the easy equations used have hugely predictable options, therefore suggesting that the bounds of deterministic predictability of the elements would possibly not but were reached. it's also attainable to make rigorous statements in regards to the large-scale behaviour of the ambience and ocean by means of proving effects utilizing those basic equations and using them to the true approach taking into account the error within the approximation. there are many different titles during this box yet they don't deal with this large-scale regime.

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2, assume that U/L = T. Then we use the ratio of the flow speed to the inertia-gravity wave speed: The governing equations 27 as the small parameter. Note that 1 e = 2 VFr~ =. 45) Thus e «; 1 if either Fr C 1 or Ro < 1. Using the condition e

1 Top: back trajectories from Mace Head for 12UTC on 11 January 2005. The total time covered is 96 hours, with points marked at 12 hour intervals. The trajectories were computed from 3-hourly data. Bottom: the vertical position of the trajectories (m). Source: Atmospheric Dispersion Group, Met Office. ©Crown copyright 2005 Published by the Met Office. 1 Various approximations t o the shallow water equations The shallow water equations In this section we place the semi-geostrophic approximation in context by comparing it to other commonly used approximate systems of equations.

75) by first rearranging them into a single equation for the evolution of h, in the manner of [Schubert (1985)]. )+9mVh I _dh Q = fgl $ dx 9 x 9 Ju J fUUg g f2 _ f , V I- Q=l 'J dx J (2-77) J Use of the third equation then gives dh ¥- V ! d dt Vf •t hQ-'g-Vh Q 9 - = -V • hQ-tfg I $ I. 78) The governing equations 39 This equation is elliptic if Q is positive definite. It has flow-dependent coefficients, as always occurs for approximations valid for L > LR. 77) multiplied by Q _ 1 yields the boundary condition Q_lv dh\ UJ- n=Q 1 dh ' ^J-n (2 79) - to ^ j .

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