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:
[1] D. J. Acheson, Elementary Fluid Dynamics, Oxford University Press, Oxford, 1990. 
[2] G. K. Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press, Cambridge, 1967. 
[3]

J. P. Boyd, Dynamics of the Equatorial Ocean, Springer, Berlin, 2018.

[4] J. Y. Chemin, Perfect Incompressible Fluids, Clarendon Press, Oxford, 1998. 
[5]

A. J. Chorin and J. E. Marsden, A Mathematical Introduction to Fluid Mechanics, Springer, New York, 1990. doi: 10.1007/978-1-4684-0364-0.

[6]

A. Constantin and R. S. Johnson, The dynamics of waves interacting with the Equatorial Undercurrent, Geophys. Astrophys. Fluid Dyn., 109 (2015), 311-358.  doi: 10.1080/03091929.2015.1066785.

[7]

A. Constantin and R. S. Johnson, An exact, steady, purely azimuthal equatorial flow with a free surface, J. Phys. Oceanogr., 46 (2016), 1935-1945.  doi: 10.1175/JPO-D-15-0205.1.

[8]

A. Constantin and R. S. Johnson, An exact, steady, purely azimuthal flow as a model for the Antarctic Circumpolar Current, J. Phys. Oceanogr., 46 (2016), 3585-3594.  doi: 10.1175/JPO-D-16-0121.1.

[9]

A. Constantin and R. S. Johnson, Large gyres as a shallow-water asymptotic solution of Euler's equation in spherical coordinates, Proc. R. Soc. A, 473 (2017), 18 pp. doi: 10.1098/rspa. 2017.0063.

[10]

A. Constantin and R. S. Johnson, Steady large-scale ocean flows in spherical coordinates, Oceanography, 31 (2018), 42-50.  doi: 10.5670/oceanog.2018.308.

[11]

A. Constantin and R. S. Johnson, Atmospheric Ekman flows with variable eddy viscosity, Bound.-Layer Meterol., 170 (2019), 395-414.  doi: 10.1007/s10546-018-0404-0.

[12]

A. Constantin and R. S. Johnson, On the nonlinear, three-dimensional structure of equatorial oceanic flows, J. Phys. Oceanogr., 49 (2019), 2029-2042.  doi: 10.1175/JPO-D-19-0079.1.

[13]

A. Constantin and R. S. Johnson, Large-scale oceanic currents as shallow-water asymptotic solutions of the Navier-Stokes equation in rotating spherical coordinates, Deep Sea Res. Pt. Ⅱ, 160 (2019), 32-40.  doi: 10.1016/j.dsr2.2018.12.007.

[14]

A. Constantin and R. S. Johnson, On the modelling of large-scale atmospheric flows, J. Differ. Equ., 285 (2021), 751-798.  doi: 10.1016/j.jde.2021.03.019.

[15]

A. Constantin and R. S. Johnson, On the propagation of waves in the atmosphere, Proc. R. Soc. A, 477 (2021), 25 pp. doi: 10.1098/rspa. 2020.0424.

[16] J. A. Curry and P. J. Webster, Thermodynamics of Atmospheres and Oceans, Academic Press, London, 1999. 
[17]

V. W. Ekman, On the influence of the Earth's rotation on ocean currents, Ark. Mat. Astr. Fys., 2 (1905), 1-52. 

[18]

S. Fueglistaler, A. E. Dessler, T. J. Dunkerton, I. Folkins, Q. Fu and P. W. Mote, Tropical tropopause layer, Rev. Geophys., 47 (2009), 31 pp. doi: 10.1029/2008RG000267.

[19]

B. P. GallegoP. Cessi and J. C. McWilliams, The Antarctic Circumpolar Current in equilibrium, J. Phys. Oceanogr., 34 (2004), 1571-1587.  doi: 10.1175/1520-0485(2004)034<1571:TACCIE>2.0.CO;2.

[20]

W. L. Gates, Derivation of the equations of atmospheric motion in oblate spheroidal coordinates, J. Atmos. Sci., 61 (2004), 2478-2487.  doi: 10.1175/1520-0469(2004)061<2478:DOTEOA>2.0.CO;2.

[21] A. Gill, Atmosphere-Ocean Dynamics, International Geophysics Series, Vol. 30, Academic Press, New York, 1982. 
[22] J. R. Holton and G. J. Hakim, An Introduction to Dynamic Meteorology, Academic Press, New York, 2013. 
[23]

V. O. IvchenkoK. J. Richards and D. P. Stevens, The dynamics of the Antarctic Circumpolar Current, J. Phys. Oceanogr., 26 (1996), 753-774.  doi: 10.1175/1520-0485(1996)026<0753:TDOTAC>2.0.CO;2.

[24]

R. S. Johnson, Some problems in physical oceanography (including the use of rotating spherical coordinates) treated as exercises in classical fluid mechanics: Methods and examples, Deep Sea Res. Pt. Ⅱ, 160 (2019), 48-57.  doi: 10.1016/j.dsr2.2018.08.010.

[25]

W. S. Kessler and M. J. McPhaden, Oceanic equatorial waves and the 1991–93 El Niño, J. Climate, 8 (1995), 1757-1774.  doi: 10.1175/1520-0442(1995)008<1757:OEWATE>2.0.CO;2.

[26] L. D. Landau and E. M. Lifshitz, Fluid Mechanics, Pergamon Press, Oxford, 1959. 
[27]

K. M. Lau and S. Yang, Walker circulation, in Encyclopaedia of Atmospheric Sciences (eds. G. R. North, J. Pyle and F. Zhang), Elsevier, Amsterdam, (2015), 177–181.

[28] M. J. Lighthill, An Informal Introduction to Theoretical Fluid Mechanics, Clarendon Press, Oxford, 1986. 
[29] P. L. Lions, Mathematical Topics in Fluid Mechanics, Clarendon Press, Oxford, 1996. 
[30] M. Mak, Atmospheric Dynamics, Cambridge University Press, Cambridge, 2011. 
[31] J. Marshall and R. A. Plumb, Atmosphere, Ocean and Climate Dynamics: An Introductory Text, Academic Press, New York, 2016. 
[32]

J. P. McCreary, Modelling equatorial ocean circulation, Ann. Rev. Fluid Mech., 17 (1985), 359-409.  doi: 10.1146/annurev.fl.17.010185.002043.

[33]

M. J. McPhadenJ. A. Proehl and L. M. Rothstein, The interaction of equatorial Kelvin waves with realistically sheared zonal currents, J. Phys. Oceanogr., 16 (1986), 1499-1515.  doi: 10.1175/1520-0485(1986)016<1499:TIOEKW>2.0.CO;2.

[34]

D. OlbersD. BorowskiC. Völker and J. O. Wölff, The dynamical balance, transport and circulation of the Antarctic Circumpolar Current, Antarct. Sci., 16 (2004), 439-470.  doi: 10.1017/S0954102004002251.

[35]

N. Paldor, Shallow Water Waves on the Rotating Earth, Springer, Cham, Switzerland, 2015.

[36]

J. Pedlosky, Geophysical Fluid Dynamics, Springer, Berlin, 1979.

[37]

J. P. Peixoto and A. H. Oort, Physics of Climate, Springer, Berlin, 1992.

[38]

S. R. Rintoul, C. Hughes and D. Olbers, The Antarctic Circumpolar Current system, in Ocean Circulation and Climate: Observing and Modelling the Global Ocean (eds. G. Siedler, J. Church and J. Gould), Academic Press, San Francisco, (2001), 271–302.

[39] R. M. Samelson, The Theory of Large-scale Ocean Circulation, Cambridge University Press, Cambridge, 2011.  doi: 10.1017/CBO9780511736605.
[40]

B. Sandor and K. G. Szabo, Simple vortex models and integrals of two dimensional gyres, in Proc. Second Conf. Junior Res. Civil Eng., (2013), 284–291, Budapest, Hungary.

[41] C. A. Swanson, Comparison and Oscillation Theory of Linear Differential Equations, Academic Press, New York, 1968. 
[42] L. D. TalleyG. L. PickardW. J. Emery and J. H. Swift, Descriptive Physical Oceanography: An Introduction, Elsevier Press, London, 2011. 
[43] G.K. Vallis, Atmospheric and Oceanic Fluid Dynamics, Cambridge University Press, Cambridge, 2017. 
[44]

Á. Viúdez and D. G. Dritschel, Vertical velocity in mesoscale geophysical flows, J. Fluid Mech., 483 (2015), 199-223.  doi: 10.1017/S0022112003004191.

[45] C. Wunsch, Modern Observational Physical Oceanography, Princeton University Press, New Jersey, 2015. 
[46]

W. ZhongJ. ZhaoJ. Shi and Y. Cao, The Beaufort Gyre variation and its impact on the Canada Basin in 2003–2012, Acta Oceanol. Sin., 34 (2015), 19-31.  doi: 10.1007/s13131-015-0657-0.

show all references

References:
[1] D. J. Acheson, Elementary Fluid Dynamics, Oxford University Press, Oxford, 1990. 
[2] G. K. Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press, Cambridge, 1967. 
[3]

J. P. Boyd, Dynamics of the Equatorial Ocean, Springer, Berlin, 2018.

[4] J. Y. Chemin, Perfect Incompressible Fluids, Clarendon Press, Oxford, 1998. 
[5]

A. J. Chorin and J. E. Marsden, A Mathematical Introduction to Fluid Mechanics, Springer, New York, 1990. doi: 10.1007/978-1-4684-0364-0.

[6]

A. Constantin and R. S. Johnson, The dynamics of waves interacting with the Equatorial Undercurrent, Geophys. Astrophys. Fluid Dyn., 109 (2015), 311-358.  doi: 10.1080/03091929.2015.1066785.

[7]

A. Constantin and R. S. Johnson, An exact, steady, purely azimuthal equatorial flow with a free surface, J. Phys. Oceanogr., 46 (2016), 1935-1945.  doi: 10.1175/JPO-D-15-0205.1.

[8]

A. Constantin and R. S. Johnson, An exact, steady, purely azimuthal flow as a model for the Antarctic Circumpolar Current, J. Phys. Oceanogr., 46 (2016), 3585-3594.  doi: 10.1175/JPO-D-16-0121.1.

[9]

A. Constantin and R. S. Johnson, Large gyres as a shallow-water asymptotic solution of Euler's equation in spherical coordinates, Proc. R. Soc. A, 473 (2017), 18 pp. doi: 10.1098/rspa. 2017.0063.

[10]

A. Constantin and R. S. Johnson, Steady large-scale ocean flows in spherical coordinates, Oceanography, 31 (2018), 42-50.  doi: 10.5670/oceanog.2018.308.

[11]

A. Constantin and R. S. Johnson, Atmospheric Ekman flows with variable eddy viscosity, Bound.-Layer Meterol., 170 (2019), 395-414.  doi: 10.1007/s10546-018-0404-0.

[12]

A. Constantin and R. S. Johnson, On the nonlinear, three-dimensional structure of equatorial oceanic flows, J. Phys. Oceanogr., 49 (2019), 2029-2042.  doi: 10.1175/JPO-D-19-0079.1.

[13]

A. Constantin and R. S. Johnson, Large-scale oceanic currents as shallow-water asymptotic solutions of the Navier-Stokes equation in rotating spherical coordinates, Deep Sea Res. Pt. Ⅱ, 160 (2019), 32-40.  doi: 10.1016/j.dsr2.2018.12.007.

[14]

A. Constantin and R. S. Johnson, On the modelling of large-scale atmospheric flows, J. Differ. Equ., 285 (2021), 751-798.  doi: 10.1016/j.jde.2021.03.019.

[15]

A. Constantin and R. S. Johnson, On the propagation of waves in the atmosphere, Proc. R. Soc. A, 477 (2021), 25 pp. doi: 10.1098/rspa. 2020.0424.

[16] J. A. Curry and P. J. Webster, Thermodynamics of Atmospheres and Oceans, Academic Press, London, 1999. 
[17]

V. W. Ekman, On the influence of the Earth's rotation on ocean currents, Ark. Mat. Astr. Fys., 2 (1905), 1-52. 

[18]

S. Fueglistaler, A. E. Dessler, T. J. Dunkerton, I. Folkins, Q. Fu and P. W. Mote, Tropical tropopause layer, Rev. Geophys., 47 (2009), 31 pp. doi: 10.1029/2008RG000267.

[19]

B. P. GallegoP. Cessi and J. C. McWilliams, The Antarctic Circumpolar Current in equilibrium, J. Phys. Oceanogr., 34 (2004), 1571-1587.  doi: 10.1175/1520-0485(2004)034<1571:TACCIE>2.0.CO;2.

[20]

W. L. Gates, Derivation of the equations of atmospheric motion in oblate spheroidal coordinates, J. Atmos. Sci., 61 (2004), 2478-2487.  doi: 10.1175/1520-0469(2004)061<2478:DOTEOA>2.0.CO;2.

[21] A. Gill, Atmosphere-Ocean Dynamics, International Geophysics Series, Vol. 30, Academic Press, New York, 1982. 
[22] J. R. Holton and G. J. Hakim, An Introduction to Dynamic Meteorology, Academic Press, New York, 2013. 
[23]

V. O. IvchenkoK. J. Richards and D. P. Stevens, The dynamics of the Antarctic Circumpolar Current, J. Phys. Oceanogr., 26 (1996), 753-774.  doi: 10.1175/1520-0485(1996)026<0753:TDOTAC>2.0.CO;2.

[24]

R. S. Johnson, Some problems in physical oceanography (including the use of rotating spherical coordinates) treated as exercises in classical fluid mechanics: Methods and examples, Deep Sea Res. Pt. Ⅱ, 160 (2019), 48-57.  doi: 10.1016/j.dsr2.2018.08.010.

[25]

W. S. Kessler and M. J. McPhaden, Oceanic equatorial waves and the 1991–93 El Niño, J. Climate, 8 (1995), 1757-1774.  doi: 10.1175/1520-0442(1995)008<1757:OEWATE>2.0.CO;2.

[26] L. D. Landau and E. M. Lifshitz, Fluid Mechanics, Pergamon Press, Oxford, 1959. 
[27]

K. M. Lau and S. Yang, Walker circulation, in Encyclopaedia of Atmospheric Sciences (eds. G. R. North, J. Pyle and F. Zhang), Elsevier, Amsterdam, (2015), 177–181.

[28] M. J. Lighthill, An Informal Introduction to Theoretical Fluid Mechanics, Clarendon Press, Oxford, 1986. 
[29] P. L. Lions, Mathematical Topics in Fluid Mechanics, Clarendon Press, Oxford, 1996. 
[30] M. Mak, Atmospheric Dynamics, Cambridge University Press, Cambridge, 2011. 
[31] J. Marshall and R. A. Plumb, Atmosphere, Ocean and Climate Dynamics: An Introductory Text, Academic Press, New York, 2016. 
[32]

J. P. McCreary, Modelling equatorial ocean circulation, Ann. Rev. Fluid Mech., 17 (1985), 359-409.  doi: 10.1146/annurev.fl.17.010185.002043.

[33]

M. J. McPhadenJ. A. Proehl and L. M. Rothstein, The interaction of equatorial Kelvin waves with realistically sheared zonal currents, J. Phys. Oceanogr., 16 (1986), 1499-1515.  doi: 10.1175/1520-0485(1986)016<1499:TIOEKW>2.0.CO;2.

[34]

D. OlbersD. BorowskiC. Völker and J. O. Wölff, The dynamical balance, transport and circulation of the Antarctic Circumpolar Current, Antarct. Sci., 16 (2004), 439-470.  doi: 10.1017/S0954102004002251.

[35]

N. Paldor, Shallow Water Waves on the Rotating Earth, Springer, Cham, Switzerland, 2015.

[36]

J. Pedlosky, Geophysical Fluid Dynamics, Springer, Berlin, 1979.

[37]

J. P. Peixoto and A. H. Oort, Physics of Climate, Springer, Berlin, 1992.

[38]

S. R. Rintoul, C. Hughes and D. Olbers, The Antarctic Circumpolar Current system, in Ocean Circulation and Climate: Observing and Modelling the Global Ocean (eds. G. Siedler, J. Church and J. Gould), Academic Press, San Francisco, (2001), 271–302.

[39] R. M. Samelson, The Theory of Large-scale Ocean Circulation, Cambridge University Press, Cambridge, 2011.  doi: 10.1017/CBO9780511736605.
[40]

B. Sandor and K. G. Szabo, Simple vortex models and integrals of two dimensional gyres, in Proc. Second Conf. Junior Res. Civil Eng., (2013), 284–291, Budapest, Hungary.

[41] C. A. Swanson, Comparison and Oscillation Theory of Linear Differential Equations, Academic Press, New York, 1968. 
[42] L. D. TalleyG. L. PickardW. J. Emery and J. H. Swift, Descriptive Physical Oceanography: An Introduction, Elsevier Press, London, 2011. 
[43] G.K. Vallis, Atmospheric and Oceanic Fluid Dynamics, Cambridge University Press, Cambridge, 2017. 
[44]

Á. Viúdez and D. G. Dritschel, Vertical velocity in mesoscale geophysical flows, J. Fluid Mech., 483 (2015), 199-223.  doi: 10.1017/S0022112003004191.

[45] C. Wunsch, Modern Observational Physical Oceanography, Princeton University Press, New Jersey, 2015. 
[46]

W. ZhongJ. ZhaoJ. Shi and Y. Cao, The Beaufort Gyre variation and its impact on the Canada Basin in 2003–2012, Acta Oceanol. Sin., 34 (2015), 19-31.  doi: 10.1007/s13131-015-0657-0.

Figure 1.  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
Figure 2.  Streamlines for constant vorticity in the Northern hemisphere for $ a=12 $ and $ b=-3.5 $; the blue line is the Equator
Figure 3.  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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