On the influence of gradients in the angular velocity on the solar meridional motions

If fluctuations in the density Ne neglected, the large-scale, axisymmetric azimuthal momentum equation for the solar convection zone (SCZ) contains only the velocity correlations [rho u(r), u(phi)] and [rho u(theta)u phi] where u are the turbulent convective velocities and the brackets denote a large-scale average. The angular velocity, Omega, and meridional motions are expanded in Legendre polynomials and in these expansions only the two leading terms are retained (for example, Omega = Omega(0)[omega(0)(r) + omega(2)(r)P-2(cos theta)] where theta is the polar angle). Per hemisphere, the meridional circulation is, in consequence, the superposition of two flows, characterized by one, and two cells in latitude respectively. Two equations can be derived fi om the azimuthal momentum equation. The first one expresses the conservation of angular momentum and essentially determines the stream function of the one-cell Flow in terms of integral(0)(pi)[rho u(r)u(phi)] sin(2) theta d theta : the convective motions feed angular momentum to the inner regions of the SCZ and in the steady state a meridional flow must be present to remove this angular momentum. The second equation contains also the integral integral(0)(pi)[rho u(theta)u(phi)] cot theta d theta indicative of a transport of angular momentum towards the equator. With the help of a formalism developed earlier we evaluate, for solid body rotation, the velocity correlations C-r phi = integral(0)(pi)[u(r)u(phi)] sin(2) theta d theta and C-theta phi = integral(0)(pi)[u(theta)u(phi)] cot theta d theta for several values of an arbitrary parameter, D, left unspecified by the theory. The most striking result of these calculations is the increase of C-theta phi with D. Next we calculate the turbulent viscosity coefficients defined by C-r phi = C-r phi(0)-nu(r phi)(1)r Omega(0) omega(0)'-nu(r phi)(2)r Omega(0) omega(2)'-nu(r phi)(3) Omega(0) omega(2) and C-theta phi =C-theta phi(0) -nu(theta phi)(1)r Omega(0) omega(0)'-nu(theta phi)(2)r Omega(0) omega(2)'-nu(theta phi)(3) Omega(0) omega(2) where C-r phi(0) and C-theta phi(0) are the velocity correlations for solid body rotation. In these calculations it was assumed that omega(2) was a linear function of r. The arbitrary parameter D was chosen so that the meridional flow vanishes at the surface for the rotation laws specified below. The coefficients nu(r phi)(i) and nu(theta phi)(i) that allow for the calculation of C-r phi and C-theta phi for any specified rotation law (with the proviso that omega(2) be linear) are the turbulent viscosity coefficients. These coefficients comply well with intuitive expectations: nu(r phi)(1) and nu(theta phi)(3) are the largest in each group, and nu(theta phi)(3) is negative. The equations for the meridional flow were first solved with omega(0) and omega(2) two linear functions of r (omega(0)' = -2 x 10(-12) cm(-1) and omega(2)' = -6 x 10(-12) cm(-1)). The corresponding angular velocity increases slightly inwards at the poles and decreases at the equator in broad agreement with heliosismic observations. The computed meridional motions are far too large (approximate to 150m s(-1)). Reasonable values for the meridional motions can only be obtained if omega(0) (and in consequence Omega), increase sharply with depth below the surface. The calculated meridional motion at the surface consists of a weak equatorward flow for theta < 29 degrees and of a stronger poleward flow for theta > 29 degrees. In the Sun, the Taylor-Proudman balance (the Coriolis force is balanced by the pressure gradient), must be altered to include the buoyancy force. The consequences of this modification an far reaching: Omega is not required, now, to be constant along cylinders. Instead, the latitudinal dependence of the superadiabatic gradient is determined by the rotation law. For the above rotation laws, the corresponding latitudinal variations of the convective flux are of the order of 7% in the lower SCZ.