# American Institute of Mathematical Sciences

July  2022, 21(7): 2357-2381. doi: 10.3934/cpaa.2022040

## The ocean and the atmosphere: An applied mathematician's view

 School of Mathematics, Statistics and Physics, Newcastle University, Newcastle, NE1 7RU, UK

Received  December 2021 Revised  January 2021 Published  July 2022 Early access  February 2022

In this survey article, we provide a mathematical description of oceanic and atmospheric flows, based on the incompressible Navier–Stokes equation (for the ocean), and the compressible version with an equation of state and the first law of thermodynamics for the atmosphere. We show that, in both cases, the only fundamental assumption that we need to make is that of a thin shell on a (nearly) spherical Earth, so that the main elements of spherical geometry are included, with all other attributes of the fluid motion retained at leading order. (The small geometrical correction that is needed to represent the Earth's geoid as an oblate spheroid is briefly described.) We argue that this is the only reliable theoretical approach to these types of fluid problem. A generic formulation is presented for the ocean, and for the steady and unsteady atmosphere, these latter two differing slightly in the details. Based on these governing equations, a number of examples are presented (in outline only), some of which provide new insights into familiar flows. The examples include the Ekman flow and large gyres in the ocean; and in the atmosphere: Ekman flow, geostrophic balance, Brunt–Väisälä frequency, Hadley–Ferrel–polar cells, harmonic waves, equatorially trapped waves.

Citation: R. S. Johnson. The ocean and the atmosphere: An applied mathematician's view. Communications on Pure and Applied Analysis, 2022, 21 (7) : 2357-2381. doi: 10.3934/cpaa.2022040
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The spherical coordinate system, where $\theta$ is the polar angle measured from the Equator, $\phi$ the azimuthal angle and $r'$ the distance from the origin
Streamlines for constant vorticity in the Northern hemisphere for $a=12$ and $b=-3.5$; the blue line is the Equator
Two views of the same solution of a jet-like velocity profile represented as a surface. Nearest edges are zeros on the bottom (that follows the Earth's curvature) and along the outer edges of the jet; the surface flow is parabolic with maximum at the centre
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