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R. Ibragimov and D. Pelinovsky

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Effects of rotation on stability of viscous stationary flows on a spherical surface

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Physics of Fluids 22, 126602 (2010)

**Abstract:**

We study the incompressible viscous fluid flows within a thin rotating atmospheric
shell. The model uses the two-dimensional Navier–Stokes equations on a spherical surface
and serves as a simple mathematical description of a general atmospheric circulation caused
by the difference in temperature between the equator and the poles. Linearized stability
of a particular stationary flow is considered. Under the assumption of no friction and a
distribution of temperature dependent only upon latitude, the stationary flow models a
zonal distribution of pressure corresponding to atmospheric currents parallel to the circles
of latitude. We prove analytically that the stationary flow is asymptotically stable in the
time evolution of the Navier–Stokes equations. When the spherical surface is truncated
between two symmetrical rings near the North and South poles, the asymptotic stability of
the stationary flow is verified numerically.

**Keywords**:

Navier-Stokes equations, spherical shell, stability of stationary flows,
associated Legendre equation, Sturm-Liouville theory