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Linearly forced fluid flow on a rotating sphere

Published online by Cambridge University Press:  06 April 2020

Rohit Supekar
Department of Mechanical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA02139, USA Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA02139, USA
Vili Heinonen
Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA02139, USA
Keaton J. Burns
Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA02139, USA
Jörn Dunkel*
Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA02139, USA
Email address for correspondence:


We investigate generalized Navier–Stokes (GNS) equations that couple nonlinear advection with a generic linear instability. This analytically tractable minimal model for fluid flows driven by internal active stresses has recently been shown to permit exact solutions on a stationary two-dimensional sphere. Here, we extend the analysis to linearly driven flows on rotating spheres. We derive exact solutions of the GNS equations corresponding to time-independent zonal jets and superposed westward-propagating Rossby waves, qualitatively similar to those seen in planetary atmospheres. Direct numerical simulations with large rotation rates obtain statistically stationary states close to these exact solutions. The measured phase speeds of waves in the GNS simulations agree with analytical predictions for Rossby waves.

JFM Papers
© The Author(s), 2020. Published by Cambridge University Press

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Supekar et al. supplementary movie 1

Simulations showing linearly driven flows on a rotating sphere at different rotation rates, as seen in the corotating frame.

Download Supekar et al. supplementary movie 1(Video)
Video 9.9 MB

Supekar et al. supplementary movie 2

Wave propagation in linearly driven zonal flows and corresponding mode excitations of spherical harmonics.

Download Supekar et al. supplementary movie 2(Video)
Video 6.2 MB