ON LIMITING THE THICKNESS OF THE SOLAR TACHOCLINE

The numerical method we employ here is identical to that described in Rogers (2011), here we give a brief review and refer the reader there for more details. We solve the full set of axisymmetric MHD equations in the anelastic approximation:

$$\begin{equation} \nabla \cdot (\overline{\rho }{\bf u})=0. \end{equation} \tag{ 1 }$$

$$\begin{equation} \nabla \cdot {\bf B}=0 \end{equation} \tag{ 2 }$$

$$\begin{eqnarray} \frac{\partial {{\bf u}}}{\partial {t}}&+&({\bf u}\cdot \nabla){\bf u}=-\nabla P - C\overline{g}\hat{r}+2({\bf u}\times \Omega)\nonumber \\ &+&\frac{1}{\overline{\rho} }({\bf J}\times {\bf B})+\overline{\nu }\left(\nabla ^{2}{\bf u}+\frac{1}{3}\nabla (\nabla \cdot {\bf u})\right) \end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray} \frac{\partial {T}}{\partial {t}}&+&({\bf u}\cdot \nabla){T}=-u_{r}\left(\frac{d\overline{T}}{dr}-(\gamma -1)\overline{T}h_{\rho }\right)\nonumber \\ &+&(\gamma -1)Th_{\rho }u_{r}+\gamma \bar{\kappa }\left[\nabla ^{2}T+(h_{\rho }+h_{\kappa }){\frac{\partial {T}}{\partial {r}}}\right] \end{eqnarray} \tag{ 4 }$$

$$\begin{equation} {\frac{\partial {{\bf B}}}{\partial {t}}}=\nabla \times ({\bf u}\times {\bf B})+\eta \nabla ^{2}{\bf B.} \end{equation} \tag{ 5 }$$

Equations (1) and (2) ensure the mass and magnetic flux are conserved. In Equation (3), the momentum equation, C denotes the co-density (Braginsky & Roberts 1995; Rogers & Glatzmaier 2005), g the gravity, Ω the rotation rate, and $\bar{\nu}$ and $\bar{\kappa}$ the viscous and thermal diffusivity, respectively. The inverse density and thermal diffusivity scale heights are denoted by h_ρ and h_κ, γ is the adiabatic index, and all other variables take their usual meanings. In all of the above equations, reference state variables are denoted by an overbar. Equation (4) represents the energy equation written as a temperature equation, where u_r represents the radial velocity. The first term on the right-hand side of Equation (4) represents the difference between the reference state temperature gradient and the adiabatic temperature gradient and allows us to represent strongly subadiabatic regions in addition to convection zones. Equation (5) is the magnetic induction equation, in which η represents the magnetic diffusivity. The diffusivities have the following forms:

$$\begin{eqnarray} \bar{\kappa} &=&\kappa _{m}\frac{16\sigma \overline{T}^{3}}{3\overline{\rho }^{2}kc_{p}}\nonumber\\ \bar{\nu} &=&\frac{\nu _{m}}{\rho }\\ \eta &=&\rm const,\nonumber \end{eqnarray} \tag{ 6 }$$

where κ_m is 10⁶, ν_m is 5 × 10¹¹, and η is 5 × 10¹², giving units of cm² s^-1. Values at the base of the convection zone are shown in Table 1. It is worth mentioning here that because our thermal diffusivity, κ, is 10⁶ larger than solar the flux through the simulation is 10⁶× solar. Using mixing length theory this would imply velocities which are (10⁶)^1/3 ≈ 100× solar.

These equations are solved in axisymmetric spherical coordinates using a spherical harmonic decomposition in latitude and a finite difference scheme in the radial direction. The maximum spherical harmonic degree, l_max, is 340, so that the grid resolution is 512 zones in latitude. The radial resolution is 1500 zones, with 600 zones dedicated to the radiative interior and 900 zones covering the convection zone. We solve the above equations from 0.15 R_☉ to 0.95 R_☉. The radiative region extends from 0.15 R_☉ to 0.71 R_☉ and the convection zone extends from 0.71 R_☉ to 0.90 R_☉, for numerical reasons we impose an additional stably stratified region from 0.90 R_☉ to 0.95 R_☉. The reference state thermodynamic variables are taken from a polynomial fit to the standard solar model from Christensen-Dalsgaard et al. (1996). We should note that because of the convection and the gravity waves in the deep radiative interior our time step is severely limited and therefore, we are unable to run these simulations very long. Our average time step is approximately 10 s and all of these simulations have run ≈2 × 10⁸ s.

We recognize that without fully 3D simulations we would be unable to recover the observed solar differential rotation profile in the convection zone. However, we are mainly interested in studying the response of the underlying radiative interior to this differential rotation in the presence of physical phenomena such as convective overshoot and magnetic pumping. Therefore, as in Rogers (2011) we artificially force the differential rotation of the convection zone to be that inferred from helioseismology (Thompson et al. 1996) and described by

$$\begin{equation} \Omega _{cz}=A+B\cos ^{2}\theta +C\cos ^{4}\theta \end{equation} \tag{ 7 }$$

where A is 456 nHz, B is −72 nHz, and C is −42 nHz. Unlike Rogers (2011), we force the differential rotation only in the convection zone, with a step function to the uniform rotation of the radiative interior.¹ The differential rotation is imposed through a forcing term on the azimuthal component of the momentum equation, which is present only in the convection zone:

$$\begin{eqnarray} {\frac{\partial {}}{\partial {t}}}\left(\frac{u_{\phi }}{r \sin \theta }\right) &=&-\frac{(\nabla \cdot {\bf u}{u_{\phi }})_{\phi }}{r\sin \theta } +\frac{[(\nabla \times {\bf B})\times {\bf B}]_{\phi }}{\bar{\rho }\mu r\sin \theta }+\frac{(2{\bf u}\times \Omega)_{\phi }}{r\sin \theta }\nonumber \\ & &\mbox{}+\frac{\nu }{r\sin \theta }\nabla ^{2}u_{\phi }+\frac{\nu u_{\phi }}{r^{3}\sin ^{3}\theta }+F, \end{eqnarray} \tag{ 8 }$$

where the last term, F, represents the forcing function, defined as

$$\begin{equation} \begin{array}{rcll}F&=&\frac{1}{\tau }\left(\Omega ^{\prime }(r,\theta)-\frac{u_{\phi }(r,\theta)}{r\sin \theta }\right) & \mbox{for $\ r > 0.71\,R_{\odot }$}\\ F&=&0. & \mbox{for $\ r < 0.71\,R_{\odot }$},\\ \end{array} \end{equation} \tag{ 9 }$$

where Ω'(r, θ) represents the observed differential rotation relative to the constant Ω and τ is a forcing timescale. In all simulations, Ω = 441 nHz and τ is 10⁴ s.²

We ran three models: one purely hydrodynamic model and two magnetic models. In the magnetic models, we start with a hydrodynamic model until convection is sufficiently established at which point we imposed a dipolar field with the form

$$\begin{equation} B_{1}=B_{o}r^{2}\left(1-\frac{r}{0.71\,R_{\odot }}\right)^{2} \end{equation} \tag{ 10 }$$

so that initially all field lines close just beneath the convection zone. Figure 1 shows a schematic of the initial setup and model, reproduced from Rogers (2011). Although we ran models with two magnetic field strengths, B_o = 40G and B_o = 4000G, all of the results presented will be only for the strong field case because the weak field case is virtually identical to the non-magnetic case.

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In all models, the boundary conditions are stress free and impermeable at the top and bottom boundaries. The temperature boundary conditions are constant temperature perturbations on top and bottom and the magnetic field is matched to an internal potential field at the bottom of the domain and an external potential field at the top of the domain. The current density, J_l, is zero on the top and bottom boundary.

The fundamental assumption of the Gough & McIntyre (1998) model is that meridional circulation driven by the differential rotation in the convection zone is able to penetrate (at least minimally) the radiative interior and confine the magnetic field. One of the persistent issues in testing this model has been regarding how deep this meridional circulation penetrates the radiative interior, if at all. Gilman & Miesch (2004) argued that the depth of penetration of the meridional circulation driven within the convection zone was limited to an Ekman depth and therefore, would penetrate no further than the overshoot layer. Garaud & Brummell (2008) showed that the circulation driven beneath the convection zone depends strongly on the boundary conditions imposed at the top of the radiative layer and that radial forcing at the interface could produce small but sustained flows deep within the interior.

The simulations presented here couple the convective and radiative regions. Moreover, the radiative interior is sufficiently "stiff," using the Brunt–Vaisala frequency from the standard solar model. Therefore, these simulations are able to address the penetration of meridional circulation in the most self-consistent manner to date. Furthermore, because we artificially impose the differential rotation in the convection zone we expect a faithful representation of the meridional circulation driven in the convection zone from the differential rotation.

In Figure 2 we show the latitudinal velocity, both in time average and time snapshots. In this figure, red represents flow toward the south pole, while blue represents flow toward the north pole. One can immediately see the poleward flow at the surface, with equatorward flow near the base of the convection zone (the overlaid white line in panels (a) and (b) shows the base of the convection zone). In both the magnetic and non-magnetic cases, the time-averaged meridional circulation shows more structure near the equator and is not well described as a single cell, but this is likely due to the need for longer time averages. The amplitudes of the meridional circulation within the convection zone show velocities of approximately 9 × 10⁴ cm s⁻¹ at the top of the convection zone and 5 × 10⁴ cm s⁻¹ near the base of the convection zone.³ One can also see hints of internal gravity waves, particularly in the non-magnetic case and in the time snapshots (these can be seen more clearly in Figure 3).

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While Figure 2 clearly shows the general circulation setup in the convection zone, it is hard to discern the amplitudes of the meridional flows within the radiative interior. In Figure 3, we show these flows more clearly. In that figure, we show the absolute magnitude of both the radial and latitudinal velocities as a function of radius below the convection zone for three different latitudes. Very little latitudinal dependence is seen. However, one can clearly see a rapid decrease in the depth of the flow velocities, which are down nearly four to five orders of magnitude by 0.60 R_☉. Figures 3(a) and (c) show this amplitude for non-magnetic cases while Figures 3(b) and (d) show it for the magnetic cases. One can see a slightly faster decay in the magnetic cases as well as shorter vertical wavelengths of the mixed gravity-Alfvén waves, which is expected given the dispersion relation for these mixed waves (MacGregor & Rogers 2011).

According to the analytic expression derived by Garaud & Acevedo-Arreguin (2009), the amplitude of the radial velocity should fall off exponentially with a length-scale determined by the parameter $\sigma =\sqrt{Pr}N/\Omega$, where N is the Brunt–Vaisala frequency. In these simulations the Prandtl number and Brunt–Vaisala frequency, N, vary substantially over the domain, so a direct comparison is difficult, but using N ≈ 100 μHz and a Prandtl number of 0.05 (the values just beneath the convection zone), the analytically predicted length-scale is ≈9 × 10⁸ cm. Investigation of the data plotted in Figure 3 gives a length-scale of ≈4 × 10⁸ cm, in fair agreement with the analytic prediction (given the simplified comparison).

By forcing the differential rotation in the convection zone, we are artificially breaking the Taylor–Proudman constraint there. As predicted by Rempel (2005) and tested by Miesch et al. (2006) in 3D simulations, breaking the Taylor–Proudman constraint could be achieved by a latitudinal temperature gradient within the tachocline (or as a bottom boundary condition in the case of the simulations). Here we can ask the question in reverse; given the imposed differential rotation, is a latitudinal temperature gradient recovered? In Figure 4 we show the temperature fluctuation as a function of latitude at three radii within the tachocline (the tachocline will be defined in the next section; see figure caption for actual depths beneath the convection zone). Here we see poles which are much warmer than the equator at the top of the tachocline (solid line) and slightly lower in the tachocline (dotted line). However, we also see a fair amount of variation at lower latitudes. In addition, near the middle of the tachocline (dashed line) this temperature variation in latitude switches sign (particularly at high latitudes), so that the poles are slightly cooler than the equator.

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Investigation of Figure 2 indicates that the middle tachocline (where the latitudinal temperature gradient changes sign) is also associated with a meridional circulation cell which rotates counter to the main meridional circulation driven in the convection zone. This counterrotating cell (CRC) is robust in the polar regions. Near the equator the cell is more time dependent, but longer time averages show the CRC becoming more prevalent. We should note that such a CRC has been seen in 3D simulations by Miesch et al. (2010); however, in that work they attribute this cell to their artificial impermeable boundary at the base of their convection zone. We confirm here the presence of such a cell even under realistic boundary conditions. This CRC has important implications for the spread of the tachocline as we will discuss in Section 3.4.

In the simulations discussed here we do not impose a tachocline, but instead want to investigate the spread of differential rotation from the convection zone into the radiative interior. Thus, we look at the formation of the tachocline and its evolution in time. Figure 5 shows both time-averaged and time snapshots of the azimuthal velocity. One can clearly see that the differential rotation quickly penetrates into the radiative interior. However, between Figures 5(e) and (f), there is very little change in that penetration, although Figure 5(f) represents a substantial increase in time. Given that the observationally determined tachocline coincides with the region of strongest radial shear in the azimuthal velocity, one way to measure the tachocline depth is by looking at this shear. Looking at |∂(v_ϕ/r)/∂r| as a function of radius and time we can then measure the tachocline penetration depth as the depth beneath the convection zone at which this is half its peak value. This is shown in Figure 6, which shows the radial gradient of angular velocity as a function of radius at both high (left-hand panels) and low (right-hand panels) latitudes at various times. The dashed lines represent the secondary peak and the solid lines represent half of that peak. If the tachocline penetration depth is measured as the distance between the base of the convection zone and the position of half of the secondary peak, the average tachocline depth is 0.021 R_☉ (1.46 ×  10⁹ cm) at high latitudes and 0.022 R_☉ (1.53 × 10⁹ cm) at low latitudes.⁴ Similarly, the magnetic models have tachocline depths of 0.022 R_☉ at low latitudes and 0.023 R_☉ at high latitudes. Measured this way, the tachocline penetration depths we measure are consistent with helioseismic results.

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Another way to define tachocline thickness at a given latitude is by the depth beneath the convection zone at which the angular velocity perturbation (initially zero) is X% of its forced value at that latitude within the convection zone. We plot this value for X = 50% in Figure 7, and note that the general behavior in time at different values of X is the same even if the amplitude is not. Figure 7 shows the tachocline thickness, measured this way, as a function of time, averaged over polar regions (0°−30°) and equatorial regions (60°−90°) for the non-magnetic case (solid line) and the strong field case (dashed line). Also shown in that figure is the time derivative of the tachocline penetration depth as a function of time (panels (b) and (d), and zoomed in over the last half of the simulation in the insets). The first obvious feature of these results is that the tachocline thickness increases rapidly, and subsequent evolution is slow. In fact, in the polar regions the tachocline penetration depth has stopped increasing, as can be seen in Figure 7(a), as well as Figure 7(b), where the time derivative is sometimes negative. In the equatorial regions the tachocline penetration depth is still increasing, though at a very slow rate, approximately 1–5 cm s⁻¹ (as seen in the insets of panels (b) and (d) of Figure 7), which would take longer than the solar age to spread through the entire solar radiative interior. Measuring the tachocline depth this way, we recover tachocline depths of ≈8 × 10⁹–10¹⁰ cm or 0.12–0.16 R_☉.

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The tachocline widths measured this way are larger and inconsistent with helioseismic results. However, given the diffusivities used in these simulations a thicker tachocline is not surprising. Furthermore, the most important feature is that the tachocline thickness has virtually stopped spreading (particularly at high latitudes). This is seen in both methods for determining tachocline thickness (Figures 6 and 7). At high latitudes, the tachocline thickness at the end of the simulation is 97% that at half the simulation time for the non-magnetic case and 98% for the magnetic case. At low latitudes the tachocline thickness is 2% higher at the end of the simulation compared to that at half the simulation time for both magnetic and non-magnetic cases.

The viscous spread of the tachocline should give a tachocline thickness Δ ≈ R(t/t_ν)^1/2. After 4 ×  10⁷ s (the time of initial rapid tachocline spread in these simulations) and using the viscous diffusivity at the top of the tachocline, the viscous depth is approximately 0.16 R_☉ (10¹⁰ cm), using the viscous diffusivity at the bottom of the tachocline this depth is approximately 0.08 R_☉ (5.5 × 10⁹ cm), in excellent agreement with the tachocline thicknesses seen here.⁵ However, if we were to assume viscous spread over the entire simulation time, we would get tachocline penetration approximately equal to the depth of the radiative interior.

If, on the other hand, the subsequent evolution of the tachocline (after the rapid initial spread) obeyed the circulation-enhanced, "hyper-diffusive" spreading described in Spiegel & Zahn (1992) in which Δ ∝ t^1/4, then the tachocline penetration depth would be approximately 1.7 times larger at the end of the simulation than it is at one-third of the simulation time. This is also not seen, clearly indicating that something is preventing both the viscous and the thermal hyper-diffusion spread of the tachocline, even in the non-magnetic case.⁶

In the following sections, we refer to the "tachocline" in these models as defined by the radial gradient of the azimuthal velocity or 0.02 R_☉. Although we recognize that there are other ways to define the tachocline depth (as studied above) we find that the relevant force balance which stops the tachocline spread occurs in this narrow depth. We now turn to the question of what is stopping the spread of the tachocline.

The defining work on the tachocline by Spiegel & Zahn (1992) proposed that the spread of the tachocline could be stopped by an anisotropic turbulence, akin to that seen in Earth's atmosphere. This is argued based on the fact that the bulk of the tachocline lies within the radiative interior, where the Brunt–Vaisala frequency is large and likely inhibits substantial radial motion. Since our non-magnetic model also shows a halt of the tachocline penetration depth this seems a likely candidate. We can test this proposal by investigating the stresses within the tachocline. The Reynolds stress can be broken into mean and fluctuating components, such that

$$\begin{equation} \nabla \cdot ({\bf u}u_{\phi })= \nabla \cdot ({\bf u} u_{\phi }^{\prime })+\nabla \cdot ({\bf u} \bar{u}_{\phi }), \end{equation} \tag{ 11 }$$

where $\bar{u_{\phi }}$ represents a time average and u'_ϕ represents the fluctuation, so that $u_{\phi }=\bar{u}_{\phi }+u_{\phi }^{\prime }$.

The first term on the right-hand side of Equation (11), the divergence of the Reynolds stress, can be split into horizontal and radial terms and we can ask if there is any anisotropy:

$$\begin{eqnarray} \nabla \cdot ({\bf u} u_{\phi }^{\prime })&=&\frac{1}{r^{2}}{\frac{\partial {(r^{2}u_{\phi }^{\prime }u_{r})}}{\partial {r}}}+\frac{u_{\phi }^{\prime }u_{r}}{r}\nonumber\\ && + \frac{1}{r\sin \theta }{\frac{\partial {(u_{\phi }^{\prime }u_{\theta }\sin \theta)}}{\partial {\theta }}} +\frac{u_{\phi }^{\prime }u_{\theta }\cot \theta }{r}. \end{eqnarray} \tag{ 12 }$$

The anisotropy envisioned by Spiegel & Zahn (1992) would then require that the ratio of the radial to latitudinal terms on the right-hand side of Equation (12) be small. We show this ratio in Figure 8 as a function of latitude at various depths within the tachocline. In both models and at all radii evaluated in the tachocline there is no detectable anisotropy of the kind envisioned in Spiegel & Zahn (1992). This could be due to our lack of resolution, so we cannot rule this model out completely, but we can say it is not responsible for the halt of the tachocline spread seen here. In fact, in much of the tachocline these Reynolds stresses are small compared to other terms in Equation (8) (see the next section).

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The above discussion then implies that the Spiegel & Zahn (1992) model is not at work in these simulations. Furthermore, given that the tachocline spread is halted in the non-magnetic cases, we also see that a magnetic field is not necessary in stopping the tachocline spread.⁷ This leads to the important question of what is responsible for limiting the tachocline penetration depth. For an answer, we can look directly at the force balance within the tachocline to determine which forces are responsible for stopping this spread. The relevant equation is the azimuthal component of the momentum equation given by

$$\begin{eqnarray} {\frac{\partial {}}{\partial {t}}}\left(\frac{u_{\phi }}{r \sin \theta }\right) &=&-\frac{1}{r\sin \theta }(\nabla \cdot \mathbf {u}\bar{u_{\phi }}+\nabla \cdot \mathbf {u}u_{\phi }^{\prime })\nonumber \\ && +\frac{[(\nabla \times {\bf B})\times {\bf B}]_{\phi }}{\bar{\rho }\mu r\sin \theta } +\frac{(2{\bf u}\times \Omega)_{\phi }}{r\sin \theta }\nonumber \\ &&\mbox{}+\frac{\nu }{r\sin \theta }\nabla ^{2}u_{\phi }+\frac{\nu u_{\phi }}{r^{3}\sin ^{3}\theta }, \end{eqnarray} \tag{ 13 }$$

where F has been omitted because there is no forcing in this region and, as in Equation (11), the total azimuthal velocity, u_ϕ, is made up of the mean (time-averaged, $\bar{u_{\phi }}$) plus the fluctuating component, u'_ϕ.

In Figure 9 we show the various force terms as a function of latitude, at two different radii within the tachocline, with the top four panels representing the non-magnetic case and the bottom four panels representing the strong magnetic case.⁸ The left-hand panels represent high latitudes, while the right-hand panels represent low latitudes. In the following discussion we will concentrate on the non-magnetic case, but note here that the magnetic case shows the same behavior. At high latitudes, at the top of the tachocline (top left panel) the total force (denoted by the solid line) is very small. The force there is dominated by the Coriolis force (dashed line, defined as the sum of the first and fourth terms on the right-hand side of Equation (13)) and the divergence of the Reynolds stress (dotted line, defined as the second term on the right-hand side of Equation (13)). The divergence of the Reynolds stress in this region is largely due to convective overshoot and is therefore, unsurprisingly large. The sign of the Coriolis force at the top of the tachocline is negative indicating its tendency to slow the poles. However, this Coriolis term is offset entirely by the Reynolds stress and diffusive terms, leading to very little net angular momentum transport there.

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At the bottom of the tachocline, again at high latitudes, the Coriolis term which is still dominant has changed sign and leads to an acceleration of the azimuthal flow at high latitudes. The change of sign of the Coriolis force is because, at the bottom of the tachocline, the Coriolis force is acting on the counterrotating meridional circulation cell discussed in Section 3.1, which is poleward. Therefore, this CRC acts to change the sign of the dominant force in the tachocline, effectively turning off the spread of the tachocline in radius.

At lower latitudes, the situation is more complicated. There is substantially more time and latitudinal variation. This is because a persistent CRC has not developed at lower latitudes. We believe that over longer time averages the low-latitude CRC will become more persistent and halt the spread there as well. This expectation is supported by Figure 7, which shows the spread of the tachocline nearing zero, even at low latitudes.

As stated above we ran three models: two with a magnetic field and one without. Although we ran two magnetic cases, as stated previously we only present results from our strong field case, for ease of presentation, but also because the results in the weak field case are virtually identical to the non-magnetic case. The results for the tachocline penetration depth shown in Figures 6 and 7 indicate that the Gough & McIntyre (1998) model cannot be solely responsible for halting the spread of the tachocline. However, it is also expected that an internal poloidal field once "connected" to the differentially rotating convection zone would cause the differential rotation to be spread into the radiative interior on approximately an Alfvén crossing time. In Figure 10 we show the evolution of the poloidal (and toroidal) field, where we can clearly see that the poloidal field connects to the convection zone. Yet, despite this connection we see no additional spread of the tachocline in the magnetic case.

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For our strong field case, with an initial internal field strength of 4 kG, assuming no diffusion, nor any turbulent diffusion due to small-scale motion, the Alfvén velocity varies from a few cm s⁻¹ at the base of the convection zone to a few hundred cm s⁻¹ in the deep radiative interior. After the initial rapid spread of the tachocline as seen in Figure 7, that is, at ≈0.6 R_☉, the Alfvén velocity (v_a) is approximately 100 cm s⁻¹. Therefore, in the subsequent 10⁸ s after the initial spread of differential rotation, one would (very simply) expect an additional spread of v_at ≈ 10¹⁰ cm if that spread were being mediated by a magnetic field. However, we see no spread of this magnitude. Indeed, we see very little effect of the magnetic field at all.

There are (at least) three reasons for this (that this author can think of). The first, and most obvious, is that these simulations have not run long enough to see the additional effects of the magnetic field. The estimate above is just that, a very crude estimate. Because we wanted our magnetic Prandtl number to be less than one, our magnetic diffusivity is rather large. Compounding this problem, we have no dynamo and therefore, the field strength and Alfvén velocity are decreasing (after an initial rise; see Figure 11). Estimating the decay of the magnetic field strength using the imposed diffusivity, the amplitude of the magnetic field (and hence, Alfvén velocity) would have dropped by approximately 30% and hence, the spread might have only amounted to 7 × 10⁹ cm, but this would still be measurable. The picture is more complicated though, because the evolution of the magnetic field is not governed solely by diffusion. As can be seen in Figure 11, initially the magnetic energy increases due to differential rotation and convection and only later does exponential decay set in. Depending on the field component, the field decay requires nearly half of the simulation time to decay back to its original amplitude. Therefore, estimating how far the differential rotation should have spread if mediated by the magnetic field is not straightforward, given the complications of a time and space dependent field. In light of the uncertainty in estimating the timescale of tachocline spread mediated by magnetic field, it is wise to at least investigate the trend of the magnetic stresses.

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3.5.1. The Lorentz Force in the Tachocline

3.5.2. Sense of the Lorentz Force

Finally, the spread of differential rotation by an internal magnetic field depends on the configuration of the magnetic field generated at the convective-radiative interface. Simply looking at the azimuthal component of the induction equation (neglecting diffusion),

$$\begin{eqnarray} {\frac{\partial {B_{\phi }}}{\partial {t}}}&=&rB_{r}{\frac{\partial {}}{\partial {r}}}\!\left(\frac{u_{\phi }}{r}\right)+\frac{\sin \theta B_{\theta }}{r}{\frac{\partial {}}{\partial {\theta }}}\!\left(\frac{u_{\phi }}{\sin \theta }\right)\nonumber\\ && -ru_{r}{\frac{\partial {}}{\partial {r}}}\!\left(\frac{B_{\phi }}{r}\right)-\frac{\sin \theta u_{\theta }}{r}{\frac{\partial {}}{\partial {\theta }}}\!\left(\frac{B_{\phi }}{\sin \theta }\right), \end{eqnarray} \tag{ 14 }$$

one can see that in the tachocline, where the radial gradient of differential rotation is initially dominant in inducing toroidal field (first term in Equation (14)), a toroidal field which switches sign at mid-latitudes is induced (Figure 10(b)). Considering the azimuthal component of the Lorentz force,

$$\begin{equation} [(\nabla \times \mathbf {B})\times {\mathbf {B}}]_{\phi }=\frac{B_{\theta }}{r\sin {\theta }}{\frac{\partial {(B_{\phi }\sin \theta)}}{\partial {\theta }}}-\frac{B_{r}}{r}{\frac{\partial {(rB_{\phi })}}{\partial {r}}}, \end{equation} \tag{ 15 }$$

a toroidal field that switches sign at mid-latitudes, coupled with a latitudinal field which is positive, generates a Lorentz force which promotes decelerating angular velocity at high latitudes and accelerating angular velocity at low latitudes, thereby promoting the spread of the differential rotation of the convection zone into the radiative interior.⁹

In these simulations, we indeed find that the radial gradient of differential rotation dominates the initial induction of the toroidal field, leading to a toroidal field that switches sign at mid-latitude (Figure 10(b)). Later in the simulation, however, the latitudinal component of differential rotation becomes dominant. This leads to a toroidal field configuration that has one sign toroidal field in the northern hemisphere (positive) and one sign in the southern hemisphere (negative); see Figure 10(c). The amplitude of the toroidal field peaks at around mid-latitudes (see Figure 10(d)), so that the latitudinal gradient of toroidal field changes sign at mid-latitudes. Furthermore, as the field diffuses out of the radiative interior and connects with the convection zone, the sign of the latitudinal magnetic field changes at high latitudes. Whereas initially the latitudinal field was positive everywhere in the tachocline, later in the simulation the latitudinal field is negative at high latitudes and positive at low latitudes. Combining the toroidal field configuration seen in Figure 10(d), with positive latitudinal field at low latitudes and negative latitudinal field at high latitudes, one can then simply see that this field configuration gives acceleration at high latitudes and deceleration at low latitudes, as seen in Figure 12. This field configuration would then act to stop tachocline spread (if Lorentz forces were dominant over long timescales).

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Of course, it is not clear that such a field configuration would develop in the Sun. But this highly simplified simulation indicates that the magnetic field configuration and associated Lorentz force in a tachocline that is bounded on top by convection and fed from below by a space and time-dependent magnetic field is quite complicated and possibly not well described without considering instabilities and poloidal field evolution.

Clearly, this model has several weaknesses. The first is that it is essentially 2D and there is some possibility that 2D convection conspires to limit tachocline spread. It is impossible to test this until 3D simulations, using similar parameters as those described here are conducted. Such simulations are likely not too far off.

The second major limitation of these simulations is that the diffusivities are too large. This prevents us from directly testing the case in which the thermal hyper-diffusion described by Spiegel & Zahn (1992) dominates the viscous diffusion. It also prevents us from getting the force balance in the tachocline exactly right. For instance, our meridional circulation appears to be only about 45 times too large, leading to Reynolds stresses and Coriolis forces in the tachocline which are larger than expected. However, this should be offset by the larger than solar viscous diffusivity. In the end, the viscous forces are not dominant in the tachocline in these simulations and that is the first order goal.

Finally, because our time step is severely limited by gravity waves and convection we are not able to run this simulation very long. We have only run these simulations approximately 20% of a viscous diffusion time or 40% of a magnetic diffusion time. Although only a small time, we still would have expected more tachocline spread. Furthermore, over this short time we can still investigate the trend of the stresses in the tachocline region. The following discussion then proceeds with these weaknesses in mind.