4.3 Maintenance of differential rotation

The most stringent observational constraints on solar interior dynamics come from helioseismic determinations of the solar differential rotation (reviewed in Section 3). In this subsection we address how this differential rotation is established and maintained.

4.3.1 Angular momentum redistribution

The angular momentum per unit mass is defined as

where $_O_0$ is the angular velocity of the rotating coordinate system and $c$ is the moment arm, $c = rsinh$ . An evolution equation for $L$ may be derived from the zonal component of the momentum equation, averaged over longitude, and the result may be written as

$-@L ( MC RS MS MT VD) r-@t = - \~/ . F + F + F + F + F . (5)$

The right-hand-side includes contributions from the meridional circulation, Reynolds stress, Maxwell stress, mean magnetic fields, and viscous diffusion. Complete expressions for each of these flux terms are given in Appendix A.4.

The first term represents the advection of angular momentum by the mean meridional circulation, having the form $FMC = r-<vM >L$ . The uniform rotation component of this, $r<vM >c2_O_0$ , represents the Coriolis force which redirects meridional flows into zonal flows. Within the anelastic approximation, the divergence of $MC F$ may also be expressed as

$- \~/ .F MC = - r<vM >. \~/ L. (6)$

Thus, meridional circulations perpendicular to $L$ contours redistribute angular momentum, tending to make $L$ constant along streamlines. If there were a global-scale circulation cell in the solar envelope extending from low to high latitudes, it would tend to “spin up” the poles relative to the equator. This is clearly not the case in the Sun (see Figure 1

), so there must be more to the story.

The net angular momentum transport through any closed surface of constant $L$ must vanish due to the divergenceless nature of the mass flux. For similar reasons, the component of $FMC$ due to the uniform rotation, $_O_0$ , cannot transport angular momentum across cylindrical surfaces aligned with the rotation axis. This result also applies to the more general case of a cylindrical rotation profile $_O_(r,h,t) = _O_(c, t)$ . Any net transport of angular momentum toward or away from the rotation axis by meridional circulation must come from the advection of the non-cylindrical component of the differential rotation (see also Section 4.3.2).

It may also be noted that angular momentum transport by meridional circulation alone cannot produce localized minima or maxima in $L$ . This follows from Equation (6 ), since $\~/ L$ vanishes at local extrema. Isolated features in the differential rotation profile such as jets must be produced by other means.

The main driver in maintaining the solar rotation profile is thought to be the Reynolds stress, $F RS$ . This term represents the redistribution of angular momentum by non-axisymmetric motions, particularly convection. Rotation, stratification, magnetic fields, and the spherical shell geometry all introduce anisotropies into the flow which give rise to systematic correlations between the fluctuating velocity components. Horizontal velocity correlations $< > v'hv'f$ produce latitudinal angular momentum transport whereas $<v' v'> r f$ correlations produce radial transport. Elucidating the nature of these correlations ranks among the greatest challenges in solar interior dynamics.

In the solar envelope, the Reynolds stress is dominated by turbulent convection, but other motions may also contribute in the tachocline and radiative interior. Convective overshoot excites a spectrum of internal wave modes, most notably gravity waves, which propagate throughout the radiative interior (see Section 8.4). In the absence of dissipation, linear waves cannot redistribute angular momentum. However, dissipation by thermal diffusion or wave breaking can induce a net angular momentum transport via the Reynolds stress which is generally long-range (non-local) and therefore difficult to model. A reliable model of wave transport requires a realistic depiction of wave generation, propagation, and dissipation, which is a formidable task due to the wide range of spatial scales involved. Other potential sources of Reynolds and Maxwell stresses include shear instabilities (see Section 8.2).

Magnetism can alter the rotation profile either by altering the Reynolds stress or by redistributing angular momentum directly via the Lorentz force. The angular momentum flux by the Lorentz force is here decomposed into contributions from fluctuating (non-axisymmetric) fields, $F MS$ , and mean (axisymmetric) fields, $F MT$ . The fluctuating component is known as the Maxwell stress and involves the nonlinear correlations $<B' B' > h f$ and $<B' B'> r f$ . Like the Reynolds stress, these may arise from turbulent convection, waves, or instabilities, and understanding their nature is every bit as challenging. The mean-field contribution is more straightforward and can be expressed as

$- \~/ .F MT = -1- . \~/ (c ). (7) 4p M f$

In this manner, a mean poloidal field $ M$ will resist deformation in the zonal ( $f$ ) direction because of the magnetic tension force. This “rubber band effect” will tend to reduce angular velocity gradients. The Maxwell stress may also have a similar “stiffening” effect due to magnetic tension (see Section 6.5).

The viscous contribution, $F VD$ , is negligible in the Sun but can be significant in numerical and theoretical models (see Section 6.3). This term opposes angular velocity gradients, $FVD oc - \~/ _O_$ , driving the system toward uniform rotation.

The primary angular momentum balance in the Sun is thought to be between the Reynolds stress and meridional circulation, with a lesser role played by the Lorentz force. Thus, if the differential rotation is in a statistically steady state, we expect the following to hold, at least in an approximate and time-averaged sense:

$\~/ .F RS = - \~/ .F MC. (8)$

It has been realized for decades that this balance is likely to hold in the solar envelope (e.g., Tassoul, 1978

; Zahn, 1992

, and references therein) but there had been little further progress until recently, thanks to new insights from helioseismology and high-resolution numerical simulations. Now the specific angular momentum profile, $L$ , is well-established from global helioseismic inversions (see Figure 1

). The meridional circulation is still only known reliably in the solar surface layers (see Section 3.4) but plausible profiles which are consistent with these surface results can be used to compute possible forms for $FMC$ . Equation (8

) may then be used to determine the corresponding Reynolds stress divergence. In other words, if we take the inferred differential rotation profile from helioseismology, we can determine what the Reynolds stress must be doing in order to maintain that profile against redistribution by some assumed meridional circulation. An illustrative example is shown in Figure 6

Although the angular velocity in the solar envelope, $_O_$ , varies by $~ 30%$ from equator to pole and exhibits nearly radial contours at mid-latitudes (Figure 6 , panel a), the corresponding specific angular momentum, $2 L = c _O_$ , is approximately cylindrical (Figure 6 , panel b). The hypothetical meridional circulation pattern shown in panel c of Figure 6 would redistribute this angular momentum as shown in panel d of Figure 6 . Thus, if the balance expressed in Equation (8 ) holds, the Reynolds stress must act to accelerate the lower convection zone and equatorial regions and to decelerate the upper convection zone in order to offset the advection of angular momentum by the meridional circulation. Any self-consistent mean-field model which exhibits a solar-like differential rotation profile as shown in panel a of Figure 6 and a single-celled meridional circulation pattern as shown in panel c of Figure 6 must include a Reynolds stress parameterization which redistributes angular momentum as shown in panel d of Figure 6 (unless the Lorentz force plays a significant role).

The results shown in Figure 6 are easily generalized to more complicated circulation patterns. If the angular momentum transport by Reynolds stress is to maintain a balance, it must converge wherever the circulation is away from the rotation axis and diverge wherever it is toward the rotation axis. This is best demonstrated by expressing the meridional circulation flux divergence as in Equation (6 ) and by noting that $\~/ L$ is directed away from the rotation axis. Another perspective can be gained by turning the problem around. For a given model of the Reynolds stress, helioseismic rotation profiles can be used to deduce the meridional circulation needed to maintain an equilibrium. This has been done by Durney (2000a ).

Figure 6:

(a) Angular velocity profile based on helioseismic inversions. This is a 2D SOLA inversion based on MDI data similar to that shown in panel a of Figure 1

. Solid and dotted lines denote prograde and retrograde rotation relative to $-6 _O_0 = 2.6 × 10$ . (b) The specific angular momentum profile given by the rotation profile in (a). (c) A hypothetical meridional circulation pattern, illustrated in terms of the mass-flux streamfunction defined in Equation (13

). The circulation in the northern hemisphere is counter-clockwise. (d) Divergence of the angular momentum flux $F MC$ carried by the hypothetical meridional circulation. Solid and dotted lines denote positive and negative values respectively. If Equation (8

) were satisfied, this would be equal to the convergence of the angular momentum transport by the Reynolds stress, $RS F$ .

If the anelastic equations are solved in a spherical shell with impenetrable, stress-free boundaries, and if the magnetic field is assumed to be radial at the boundaries, then there is no net torque and the total angular momentum of the shell, $integral -- rLdV$ , is conserved. This is of course just an approximation. In actuality, coupling between the convective envelope and the radiative interior may play a role in the global angular momentum balance (Section 7.3). Angular momentum exchange between the convection zone and the solar atmosphere is likely less important on dynamical timescales, although it is believed that the Sun has lost a large fraction of its initial angular momentum over the course of its lifetime via the solar wind.

4.3.2 The Taylor-Proudman theorem and thermal wind balance

In the previous section we discussed the mechanisms which can redistribute angular momentum in the solar interior, giving rise to differential rotation. There is more we can say about the angular momentum balance which may eventually be achieved if we consider the limit of rapid rotation³ such that $Ro « 1$ , where the Rossby number is defined as

where $U$ is a characteristic velocity scale relative to the rotating reference frame. We neglect viscous diffusion and the Lorentz force, and we assume that the mean flows are in a statistically steady state. With these approximations, the momentum Equation (40

) expresses what is called geostrophic (or heliostrophic) and hydrostatic balance:

$\~/ f,t <r>f,tg 2_O_0 × <v>f,t + ---------+ -------^r = 0. (10) r r$

If we compute the zonal component of the curl of Equation (10

), we obtain, with a little manipulation:

$( ) 1 - 1@ f,t @ <r>f,t g @ <S>f,t _O_0. \~/ _O_ = 2rrc- H r --@h----- g---@h-- = 2C--cr---@h---. (11) P$

The final equality in Equation (11

) holds if the reference state is approximately adiabatic and hydrostatic. A more general reference state can be incorporated by interpreting the latitudinal gradient on the right-hand-side as the mean gradient on isobaric (constant pressure) surfaces. Equation (11

) is the well-known Taylor-Proudman theorem (e.g., Pedlosky, 1987

), as it applies to the solar differential rotation. If the stratification is perfectly adiabatic ( $@S/@h = 0$ ), this equation implies that the rotation profile should be cylindrical, i.e., contours of angular velocity $_O_$ should be parallel to the rotation axis, $_O_0$ . Alternatively, if significant latitudinal entropy gradients are present, then the Taylor-Proudman balance expressed by Equation (11

) implies non-cylindrical rotation profiles, such that relatively warm poles ( $@S/@h < 0$ in the northern hemisphere) correspond to a decrease in angular velocity toward higher latitudes ( $_O_0. \~/ _O_ < 0$ ). In other words, latitudinal gradients of entropy (or density or temperature) on isobaric surfaces will tend to establish a non-cylindrical differential rotation.

If the rotation profile satisfies Equation (11 ) it is said to be in thermal wind balance, in analogy with the thermal wind of geophysical fluid dynamics (Pedlosky, 1987 ). More specifically, the thermal wind component of the differential rotation is the component which is non-cylindrical and which satisfies Equation (11 ).

In a thermal wind, departures from cylindrical symmetry are maintained by latitudinal entropy gradients. This is consistent with the angular momentum Equation (5 ) because if the Taylor-Proudman balance is satisfied perfectly, then both the Reynolds stress and the meridional circulation are negligible (as are the Lorentz and viscous forces), so Equation (5 ) becomes degenerate. However, meridional circulations are the means by which the thermal wind balance is established and maintained in a rapidly-rotating fluid shell. An imbalance in Equation (11 ) will drive circulations which will redistribute angular momentum until balance is achieved.

In the solar envelope, latitudinal entropy gradients may be established by the influence of rotation on the efficiency of the convection. For example, if convection is more efficient in the polar regions where the rotation vector is nearly vertical, then these regions will be relatively warm. In radiative equilibrium (the net energy flux into the convection zone equals the net flux out through the surface), such efficiency variations must be balanced by latitudinal energy transport as reflected by Equation (2 ). Thus, the role played by anisotropic energy transport in maintaining the solar differential rotation may potentially be as important as that played by the Reynolds stress, and may be just as enigmatic.

Figure 7:

Shown are the thermal variations implied by Equation (11

) based on an angular velocity profile, $_O_$ , obtained from helioseismic inversions (Figure 6

, panel a), and on other parameters ( $g$ , $CP$ , $-- T$ ) obtained from a solar structure model (model S of Christensen-Dalsgaard, 1996

). Frame (a) illustrates the normalized latitudinal entropy gradient, $-1 - CP @S/@h$ , consistent with thermal wind balance. Frame (b) illustrates the corresponding temperature perturbation, assuming $C -1@S/@h ~~ T--1@T /@h P$ (cf. Equation (43

) in Appendix A.2).

If the solar differential rotation were in thermal wind balance, we would expect thermal variations of a few parts in $6 10$ as shown in Figure 7

. If we neglect the pressure contribution to the latitudinal entropy gradient as a first approximation, the resulting temperature variations are about 5 K, increasing from equator to pole (Figure 7

, panel b). Thus, if helioseismic inversions were to detect relatively warm poles near the base of the convection zone, this could be interpreted as evidence for thermal wind balance. However, the implied variations are still below the sensitivity limits of current inversions (Section 3.7).