By Norbury J., Roulstone I. (eds.)

The advanced flows within the surroundings and oceans are believed to be thoroughly modelled via the Navier-Stokes equations of fluid mechanics including classical thermodynamics. notwithstanding, end result of the huge, immense complexity of those equations, meteorologists and oceanographers have developed 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 legislations produce actual, necessary types and, simply because the meteorologist seeks styles within the climate, mathematicians search constitution within the governing equations. They convey how geometry and research facilitate resolution options.

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U = ∂u ∂ ∂w 1 + (v cos φ) + . 19), g is properly considered constant. 18) is usually referred to as the Coriolis parameter and accredited the symbol f . The other Coriolis parameter, 2Ω cos φ, which is absent from the HPEs, has no universally accepted title. 17), is ρ D ∂p {(u + Ωa cos φ) a cos φ} = ρFλ a cos φ − . 23) The energy conservation law (Lagrangian form) is ρ D 1 2 v + gz + cv T = −∇. (pu) + ρ(Q + v. Fh ). 24) The equations of meteorological dynamics and various approximations 27 Here v is the horizontal ﬂow (the ‘wind’).

Phys. A. , 30, L63–L68. , Roulstone, I. (2001) Holomorphic structures in hydrodynamical models of nearly geostrophic ﬂow. Proc. R. Soc. Lond. A 457, 1519–1531. Roulstone, I. and Norbury, J. (1994) A Hamiltonian structure with contact geometry for the semi-geostrophic equations. J. , 272, 211–233. ¨ Runge, C. (1895) Uber die numerische Auﬂ¨ osung vor Diﬀerentialgleichungen. Math. Ann. 46. Salmon, R. (1983) Practical use of Hamilton’s principle. J. , 132, 431–444. Salmon, R. (1985) New equations for nearly geostrophic ﬂow.

35) from z = zs upwards, assuming a reasonable lower boundary condition (such as w = 0 at a ﬂat lower boundary). The explicit expression for w(z) so obtained is known as Richardson’s equation from its use in the ﬁrst numerical weather prediction experiment (Richardson 1922). A diﬀerent treatment is necessary if an upper boundary condition is applied at a ﬁnite height (Kasahara and Washington 1967). 35) leads to a form that does not contain a vertical integral: γ ∂ ∂z p ∂w Q + ∇z . v − ∂z T cp = ∂v ∂p ∇z .