4.4 Maintenance of meridional circulation

The axisymmetric circulation in the meridional plane may be described in terms of the zonal component of the curl of the mass flux

$dr- p =_ ( \~/ × <rvM>) .f^ = r<wf > +---<vh> , (12) dr$

where $wf$ is the zonal component of the vorticity and $vM$ denotes the meridional component of the velocity: $v = v ^r + v ^h M r h$ . If we wish to take advantage of the vanishing divergence of the mass flux under the anelastic approximation, we may also introduce a streamfunction $Y$ , defined such that

$-- ( ) <rvM > =_ \~/ × Yf^ . (13)$

This implies

$2 ---Y---- p = - \~/ Y + r2sin2h . (14)$

The evolution equation for $p$ may be expressed as follows (see Appendix A.5):

$( ) ^ @p-- = - rsinh ~ \~/ . -G---- = - \~/ .G + -G.c--, (15) @t rsinh rsin h$

where

The Reynolds stress has three distinct components [see Equation (76 )]:

The first term is the most straightforward; it represents advection of zonal vorticity by the fluctuating meridional flow. The second term is easier to interpret if we consider its divergence: $(-< '> ') < ' (--')> \~/ . r vf w M = w M. \~/ rvf$ . Vortex structures which lie in the meridional plane $' w M$ may be tilted out of the plane by radial and latitudinal gradients in the longitudinal momentum, $rvf$ , thus generating longitudinal vorticity, $wf$ . The final term in Equation (17

) arises from the density stratification and its divergence is proportional to latitudinal kinetic energy gradients. It cannot generate longitudinal vorticity, $wf$ , but it can modify $p$ through the second term on the right-hand-side of Equation (12

), inducing a net mass flux circulation by altering $vh$ .

The mean-flow term likewise involves three components due to the advection of longitudinal vorticity by the meridional circulation, the tipping of the absolute vorticity (relative to an inertial frame) associated with the mean rotation, $wrot = \~/ × (_O_c) = <wM > + 2_O_0$ , and latitudinal kinetic energy gradients [see Equation (77 )]:

The contribution from $wrot$ may also be regarded as the generation of meridional circulation via the action of the Coriolis force on the differential rotation.

In a compressible fluid, buoyancy cannot generate vorticity directly. However, if the mass flux is divergenceless as in the anelastic approximation, buoyancy can induce overturning circulations as reflected by the term $GBF$ . In the present context, these may be regarded as axisymmetric convection cells. The Lorentz force may only induce mass flux circulations through magnetic tension, $(B.\ ~/ )B$ . This effect is contained in the term $GMT$ which includes contributions both from fluctuating fields (the Maxwell stress) and from mean fields.

In the Sun the rotational component of $AD G$ (that involving $wrot$ ) plays an important role, particularly at low latitudes where the prograde differential rotation is forced outward by the Coriolis force and subsequently turns poleward in the surface layers (see Section 6.4). The buoyancy and Reynolds stress terms ( $BF G$ , $RS G$ ) are also likely to be important (see Section 6.4).

In our anelastic formulation, we have neglected the centrifugal force. It is known that the centrifugal force can produce axisymmetric motions, often called Eddington-Sweet circulations, in the radiative zones of stellar interiors due to the distortion of the gravitational potential surfaces relative to surfaces of constant temperature (e.g., Tassoul, 1978 ). The mixing of chemicals and angular momentum by such circulations may have important consequences for stellar evolution models or for the relatively ”slow” dynamics which may contribute to tachocline confinement (Section 8.5). However, Eddington-Sweet circulations are insignificant in the convection zone and upper tachocline. Measured meridional flows in the solar surface layers imply turnover timescales of years to decades, much longer than the Eddington-Sweet timescale which is more than $106 yr$ .

Equation (15 ) quantifies the relative importance of processes which redistribute meridional momentum but, as with the differential rotation (cf. Section 4.3.2), other balance equations can often provide further insight into the meridional circulation amplitude and profile which may ultimately be achieved in equilibrium. In this respect, the mean thermal energy equation is particularly important:

$[ -- ( -)] -- ' ' -- 1 [ -- ( --)] \~/ . <rvM >ft <S>ft + S = - \~/ . <rv S >ft + T \~/ . krrCP \~/ <T >ft + T + Q. (19)$

Here $Q$ represents viscous and Ohmic heating. Equation (19

) has been derived by averaging Equation (41

) in Appendix A.2 over longitude and time (denoted by $<>ft$ ) and assuming a steady state. In the radiative zone below the solar tachocline, non-axisymmetric fluctuations and dissipation $Q$ are negligible so advective heat transport by the meridional circulation balances radiative diffusion (Spiegel and Zahn, 1992

). In the convection zone there is an additional contribution from the convective heat flux, represented by the first term on the left-hand-side of Equation (19

). Thus if the thermal structure is known and the convective heat flux is parameterized via mean-field theory or otherwise given, then Equation (19

) may be used to determine the equilibrium meridional circulation. In a more sophisticated mean-field model, Equation (19

) may be solved simultaneously with the zonal and meridional momentum equations to obtain a self-consistent equilibrium state.

In the solar convection zone, the advection of angular momentum by meridional circulation is thought to balance angular momentum transport by the Reynolds stress as expressed by Equation (8 ). Thus, if the Reynolds stress and rotation profile are given, this equation may similarly be used to determine the equilibrium meridional circulation. However, the thermal wind component of the differential rotation discussed in Section 4.3.2 is independent of the meridional circulation profile. The equation for thermal wind balance (11 ) may be derived from the meridional circulation maintenance Equation (15 ) if the uniform rotation component of $AD G$ ( $- 2r-<vf >_O_0$ ) balances the buoyancy term $GBF$ and if geostrophic balance [Equation (10 )] is assumed. Under these conditions, the maintenance Equation (15 ) becomes independent of $p$ .