The Reynolds stress has three distinct components [see Equation (76)]:
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, , and latitudinal kinetic energy gradients [see Equation (77)]:
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 . In the present context, these may be regarded as axisymmetric convection cells. The Lorentz force may only induce mass flux circulations through magnetic tension, . This effect is contained in the term which includes contributions both from fluctuating fields (the Maxwell stress) and from mean fields.
In the Sun the rotational component of (that involving ) 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 (,) 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 .
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:
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 () balances the buoyancy term and if geostrophic balance [Equation (10)] is assumed. Under these conditions, the maintenance Equation (15) becomes independent of .
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