Astronomy and Astrophysics – Astronomy
Scientific paper
Jan 2000
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2000apj...528..486d&link_type=abstract
The Astrophysical Journal, Volume 528, Issue 1, pp. 486-492.
Astronomy and Astrophysics
Astronomy
15
Sun: Interior, Sun: Rotation
Scientific paper
The solar angular velocity, Ω, and meridional motions in the solar convection zone (SCZ) are expanded in Legendre polynomials. If the velocity correlations ruφ>,θuφ> and the angular velocity are known, then the azimuthal momentum equation determines the meridional flow; here u stands for the turbulent convective velocities and the bracket denotes an appropriate average; θ and φ are the polar angle and longitude. The velocity correlation ruφ> transports angular momentum to the inner regions of the SCZ. This angular momentum can either spin-up the inner regions, or be removed by a meridional motion that rises at the equator and sinks at the poles; the stream function for this motion will be designated by ψ2. For slowly rotating stars, the inner regions must spin-up. As the angular velocity increases, a transition must take place to the second option: in the Sun the angular velocity does not increase sharply with depth. This transition should occur at a value for Ω at which the Taylor-Proudman balance (a balance between the pressure, Coriolis, and buoyancy forces) becomes valid. In the SCZ, this balance determines the latitudinal variations of the superadiabatic gradient (∇ΔT) from the rotation law, and it provides, therefore, a link between the energy equation and the azimuthal momentum equation. The solar meridional motion also has a component, with stream function ψ4, that rises at the equator and poles and sinks at midlatitudes; its contribution to the removal of angular momentum from the inner regions of the SCZ is negligible. In the Sun, ψ2 depends mainly on ruφ> and ψ4~-4ψ2/3 (this expression for ψ4 is not as robust as that of ψ2, which is an excellent approximation). Therefore, the meridional motions are essentially determined by ruφ>. However, the ψ2-meridional circulation transports angular momentum toward the polar regions of the Sun which must be balanced by θuφ> and ψ4. Globally, the conservation of angular momentum in the latitudinal direction requires that the sum of the terms in ruφ> and in θuφ> of an integral over the entire SCZ cancels. For this to be the case, θuφ> must be positive since ruφ> is negative (which is a very robust result). For stars satisfying the Taylor-Proudman balance, a fast rotating equator appears to be an unavoidable necessity. An equation is derived that clarifies the reasons for the existence of the relation ψ4~-4ψ2/3 and for the weak dependence of ψ4 on θuφ>. A simple model for the velocity correlations is studied. In this simple model, if the latitudinal differential rotation increases, θuφ> must decrease for the integral relation defined above to remain valid. This dependence of θuφ> on ∂Ω/∂θ agrees with what can be inferred from physical considerations.
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