# 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
##### References:

show all references

##### References:
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
 [1] Kersten Schmidt, Ralf Hiptmair. Asymptotic boundary element methods for thin conducting sheets. Discrete and Continuous Dynamical Systems - S, 2015, 8 (3) : 619-647. doi: 10.3934/dcdss.2015.8.619 [2] Bo You. Optimal distributed control of the three dimensional primitive equations of large-scale ocean and atmosphere dynamics. Evolution Equations and Control Theory, 2021, 10 (4) : 937-963. doi: 10.3934/eect.2020097 [3] David Julitz. Numerical approximation of atmospheric-ocean models with subdivision algorithm. Discrete and Continuous Dynamical Systems, 2007, 18 (2&3) : 429-447. doi: 10.3934/dcds.2007.18.429 [4] Zhiwen Zhao. Asymptotic analysis for the electric field concentration with geometry of the core-shell structure. Communications on Pure and Applied Analysis, 2022, 21 (4) : 1109-1137. doi: 10.3934/cpaa.2022012 [5] S. E. Pastukhova. Asymptotic analysis in elasticity problems on thin periodic structures. Networks and Heterogeneous Media, 2009, 4 (3) : 577-604. doi: 10.3934/nhm.2009.4.577 [6] Bo You, Chunxiang Zhao. Approximation of stationary statistical properties of the three dimensional autonomous planetary geostrophic equations of large-scale ocean circulation. Discrete and Continuous Dynamical Systems - B, 2020, 25 (8) : 3183-3198. doi: 10.3934/dcdsb.2020057 [7] Grigory Panasenko, Ruxandra Stavre. Asymptotic analysis of the Stokes flow with variable viscosity in a thin elastic channel. Networks and Heterogeneous Media, 2010, 5 (4) : 783-812. doi: 10.3934/nhm.2010.5.783 [8] Rejeb Hadiji, Ken Shirakawa. Asymptotic analysis for micromagnetics of thin films governed by indefinite material coefficients. Communications on Pure and Applied Analysis, 2010, 9 (5) : 1345-1361. doi: 10.3934/cpaa.2010.9.1345 [9] Mustapha El Jarroudi, Youness Filali, Aadil Lahrouz, Mustapha Er-Riani, Adel Settati. Asymptotic analysis of an elastic material reinforced with thin fractal strips. Networks and Heterogeneous Media, 2022, 17 (1) : 47-72. doi: 10.3934/nhm.2021023 [10] Martin Hanke, William Rundell. On rational approximation methods for inverse source problems. Inverse Problems and Imaging, 2011, 5 (1) : 185-202. doi: 10.3934/ipi.2011.5.185 [11] Per Christian Moan, Jitse Niesen. On an asymptotic method for computing the modified energy for symplectic methods. Discrete and Continuous Dynamical Systems, 2014, 34 (3) : 1105-1120. doi: 10.3934/dcds.2014.34.1105 [12] Martha Garlick, James Powell, David Eyre, Thomas Robbins. Mathematically modeling PCR: An asymptotic approximation with potential for optimization. Mathematical Biosciences & Engineering, 2010, 7 (2) : 363-384. doi: 10.3934/mbe.2010.7.363 [13] Thierry Paul, Mario Pulvirenti. Asymptotic expansion of the mean-field approximation. Discrete and Continuous Dynamical Systems, 2019, 39 (4) : 1891-1921. doi: 10.3934/dcds.2019080 [14] Caojin Zhang, George Yin, Qing Zhang, Le Yi Wang. Pollution control for switching diffusion models: Approximation methods and numerical results. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 3667-3687. doi: 10.3934/dcdsb.2018310 [15] Baoli Yin, Yang Liu, Hong Li, Zhimin Zhang. Approximation methods for the distributed order calculus using the convolution quadrature. Discrete and Continuous Dynamical Systems - B, 2021, 26 (3) : 1447-1468. doi: 10.3934/dcdsb.2020168 [16] Jake Bouvrie, Boumediene Hamzi. Kernel methods for the approximation of some key quantities of nonlinear systems. Journal of Computational Dynamics, 2017, 4 (1&2) : 1-19. doi: 10.3934/jcd.2017001 [17] Dingshi Li, Xiaohu Wang. Asymptotic behavior of stochastic complex Ginzburg-Landau equations with deterministic non-autonomous forcing on thin domains. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 449-465. doi: 10.3934/dcdsb.2018181 [18] Patrick Ballard, Bernadette Miara. Formal asymptotic analysis of elastic beams and thin-walled beams: A derivation of the Vlassov equations and their generalization to the anisotropic heterogeneous case. Discrete and Continuous Dynamical Systems - S, 2019, 12 (6) : 1547-1588. doi: 10.3934/dcdss.2019107 [19] Grigory Panasenko, Ruxandra Stavre. Asymptotic analysis of a non-periodic flow in a thin channel with visco-elastic wall. Networks and Heterogeneous Media, 2008, 3 (3) : 651-673. doi: 10.3934/nhm.2008.3.651 [20] Brian D. Ewald, Roger Témam. Maximum principles for the primitive equations of the atmosphere. Discrete and Continuous Dynamical Systems, 2001, 7 (2) : 343-362. doi: 10.3934/dcds.2001.7.343

2021 Impact Factor: 1.273