Large Oceanic Gyres: Lagrangian Description

  • Anatoly AbrashkinEmail author


A hydrodynamic model of an oceanic gyre is proposed. The fluid motion is considered in the leading-order shallow-water approximation in the spherical Lagrangian coordinates. Motion of liquid particles at the spherical surfaces is studied versus latitude and longitude as unknown variables. The boundary condition at the edge of the gyre is not formulated. An approximation of the “averaged latitude” is introduced when the coefficients of the momentum equation are replaced by constant values corresponding to the latitude of the gyre’s center. It is shown that the resulting set of equations is similar to the equations of plane hydrodynamics. Its analytical solutions containing two arbitrary functions and two arbitrary constants (time frequencies) are obtained. The trajectories of liquid particles represent a superposition of two rotational motions, and their general properties are discussed. A family of the gyres with invariable shape in time is selected. Their outer boundaries either remain motionless or rotate uniformly. An example of the unsteady gyre both rotating and deforming in its shape is studied numerically.


Oceanic gyres Lagrangian spherical coordinates Trajectories of liquid particles 

Mathematics Subject Classification




The publication was prepared within the framework of the Academic Fund Program at the National Research University Higher School of Economics (HSE) in 2018-2019 (Grant No 18-01- 0006) and by the Russian Academic Excellence Project “5-100”. The author has the pleasure to thank Prof. A. Constantin for stimulating discussions.

Compliance with ethical standards

Conflict of interest

The author declares that he has no conflict of interest.


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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.National Research University Higher School of EconomicsNizhny NovgorodRussia

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