# Large-Scale Atmosphere-Ocean Dynamics I by John Norbury, Ian Roulstone

By John Norbury, Ian Roulstone

The advanced flows within the surroundings and oceans are believed to be correctly modelled by way of the Navier-Stokes equations of fluid mechanics including classical thermodynamics. although, as a result huge, immense complexity of those equations, meteorologists and oceanographers have built approximate types of the dominant, large-scale flows that regulate the evolution of climate structures. The simplifications usually bring about versions which are amenable to resolution either analytically and numerically. This quantity and its significant other clarify why such simplifications to Newton's moment legislation produce actual, worthwhile types and, simply because the meteorologist seeks styles within the climate, mathematicians search constitution within the governing equations. They express how geometry and research facilitate answer innovations.

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Neither shall we treat the variations in gas constant R and principal speciﬁc heats cp and cv which accompany variations in the amount of water substance present; Gill (1982) gives a concise account. 13)–(3-16) may be written in many alternative forms by using either other equations of the set, or various thermodynamic relations. One of the most important is an alternative form of the thermodynamic equation involving the potential temperature θ deﬁned by θ=T R/cp pref p . 17) Here pref is a reference pressure (conventionally 1000hPa), cp is the speciﬁc heat at constant pressure and θ is the temperature that an element of air would have if it were to be brought adiabatically and reversibly to pressure pref .

1988) Semi-geostrophic theory as a Dirac bracket projection. J. , 196, 345–358. C. (1947) A contribution to the problem of development. R. Meteorol. , 73, 370–383. Z. J. (2000) The dynamical consequences for tropopause folding of PV anomalies induced by surface frontal collapse. R. Meteorol. , 126, 2747–2764. A. White 1 Introduction One of the attractions of meteorology is its many-faceted character. It invites study by mathematicians and statisticians as well as by physicists of either practical or theoretical disposition.

With this background, consider how the HPE momentum and continuity equations may be applied to a rotating incompressible, homogeneous ﬂuid of density ρˆ bounded by a rigid horizontal surface at z = 0 and having a free surface at z = h(λ, φ, t). 18) become Du Dt Dv Dt with uv tan φ g ∂h − + Fλ a a cos φ ∂λ u2 tan φ g ∂h = −2Ωu sin φ − − + Fφ a a ∂φ = 2Ωv sin φ + D ∂ u ∂ v ∂ ≡ + + . 39) If the horizontal velocity components u, v are initially independent of depth, they will remain so, since the pressure gradient is independent of depth.