axisymmetric mhd instabilities in solar/stellar tachoclines - IOPscience

The Astrophysical Journal, 692:1421–1431, 2009 February 20 c 2009. The American Astronomical Society. All rights reserved. Printed in the U.S.A. 

doi:10.1088/0004-637X/692/2/1421

AXISYMMETRIC MHD INSTABILITIES IN SOLAR/STELLAR TACHOCLINES Mausumi Dikpati1 , Peter A. Gilman1 , Paul S. Cally1,2 , and Mark S. Miesch1 1

High Altitude Observatory, National Center for Atmospheric Research, 3080 Center Green, Boulder, CO 80307-3000, USA; [email protected], [email protected], [email protected] Centre for Stellar and Planetary Astrophysics, School of Mathematical Sciences, Monash University, Clayton, VIC 3800, Australia; [email protected] Received 2008 September 18; accepted 2008 November 17; published 2009 February 24

2

ABSTRACT Extensive studies over the past decade showed that HD and MHD nonaxisymmetric instabilities exist in the solar tachocline for a wide range of toroidal field profiles, amplitudes, and latitude locations. Axisymmetric instabilities (m = 0) do not exist in two dimensions, and are excited in quasi-three-dimensional shallow-water systems only for very high field strengths (2 mG). We investigate here MHD axisymmetric instabilities in a three-dimensional thin-shell model of the solar/stellar tachocline, employing a hydrostatic, non-Boussinesq system of equations. We deduce a number of general properties of the instability by use of an integral theorem, as well as finding detailed numerical solutions for unstable modes. Toroidal bands become unstable to axisymmetric perturbations for solar-like field strengths (100 kG). The e-folding time can be months down to a few hours if the field strength is 1 mG or higher, which might occur in the solar core, white dwarfs, or neutron stars. These instabilities exist without rotation, with rotation, and with differential rotation, although both rotation and differential rotation have stabilizing effects. Broad toroidal fields are stable. The instability for modes with m = 0 is driven from the poleward shoulder of banded profiles by a perturbation magnetic curvature stress that overcomes the stabilizing Coriolis force. The nonaxisymmetric instability tips or deforms a band; with axisymmetric instability, the fluid can roll in latitude and radius, and can convert bands into tubes stacked in radius. The velocity produced by this instability in the case of low-latitude bands crosses the equator, and hence can provide a mechanism for interhemispheric coupling. Key words: stars: interiors – stars: magnetic fields – stars: rotation – Sun: interior – Sun: magnetic fields – Sun: rotation 1. INTRODUCTION

can also occur in rotating stars’ radiative zones away from tachoclines, both with or without differential rotation (Braithwaite 2006). All 2D, quasi-3D, and 3D MHD studies show that the tachocline latitudinal differential rotation coexisting with toroidal magnetic fields is unstable to disturbances with longitudinal wavenumbers m  1, for a wide range of toroidal field profiles, amplitudes, and latitude locations. In a pure hydrodynamic case, such instabilities exist in the overshoot part of the tachocline in quasi-3D and 3D models (Dikpati & Gilman 2001; Gilman & Dikpati 2002; Gilman et al. 2007), but not in 2D (Watson 1981; Dziembowski & Kosovichev 1987; Charbonneau et al. 1999; Garaud 2000). However, axisymmetric instabilities in the solar tachocline have not received much attention, although the history of the study of axisymmetric shear instabilities in the astrophysical context is much longer. Stability of a toroidal ring in stars had been investigated (Gough & Tayler 1966; Fricke 1969; Moss & Tayler 1969; Tayler 1973; Pitts & Tayler 1985). In particular, Goossens & Tayler (1980) found that the axisymmetric toroidal magnetic fields of strength 108 G in the core of solar-like stars are unstable to axisymmetric perturbations with e-folding growth times of a few hours. The poleward slip of toroidal bands was speculated to be an outcome of axisymmetric (m = 0) instability (Spruit & van Ballegooijen 1982). Spruit (1999) demonstrated the occurrence of the stratification-modified pinch-type Tayler instability of a strong azimuthal ring produced from some weak fields sheared by the differential rotation. Our present study of axisymmetric instabilities of differential rotation and magnetic fields in the interior of stars will differ from previous studies in several ways: (1) we will focus primarily on thin shells, such as for an interface between radiatively and convectively dominated domains, instead of deep radiatively dominated stellar interiors. There are readily available mecha-

Helioseismic inversions indicate that the solar tachocline, which contains a strong radial shear, straddles the base of the convection zone (0.713R). It is a slender layer of thickness ∼ 0.04R (Kosovichev 1996; Basu 1997) located at ∼ 0.7R (Charbonneau et al 1997). The transition from the latitudinal differential rotation of the convection zone to the solid rotation of the core takes place through the tachocline (Brown et al 1989; Goode et al 1991; Tomczyk et al. 1995; Corbard et al 1998). The properties of the tachocline and its importance in solar dynamics have been described in great detail in many papers, such as: Gilman (2000a); Forg´acs-Dajka & Petrovay (2002); Cally (2003); Dikpati (2006); we omit reviewing them here. A particular question about the tachocline has repeatedly been addressed—why is it so thin? Among various plausible theories, one is the anisotropic turbulent mixing of angular momentum in the latitudinal direction at a much higher rate than in the radial direction (Spiegel & Zahn 1992). Such anisotropic angular momentum mixing could happen due to the latitudinal shear instability. Gilman & Fox (1997) first showed that the tachocline latitudinal shear is unstable to 2D nonaxisymmetric disturbances when a toroidal magnetic field is present. Over the past decade, nonaxisymmetric hydrodynamic and magnetohydrodynamic instabilities have been extensively studied in 2D, quasi-3D shallow-water, and 3D thin-shell models of the solar tachocline (Gilman & Fox 1997, 1999a, 1999b; Dikpati & Gilman 1999; Gilman 2000b; Gilman & Dikpati 2000; Garaud 2000; Dikpati & Gilman 2001; Cally 2001, 2003; Garaud 2002; Gilman & Dikpati 2002; Cally et al. 2003; Rempel & Dikpati 2003; Dikpati et al. 2003, 2004; Tobias & Hughes 2004; Arlt et al. 2005; Miesch et al. 2007; Gilman et al. 2007; Arlt et al. 2007a, 2007b; Kitchatinov & R¨udiger 2008). Nonaxisymmetric 3D MHD instabilities 1421

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nisms for producing differential rotation and toroidal fields at such interfaces. (2) Our perturbation analysis will be global in latitude. (3) We will focus on the effects of latitudinal gradients in rotation and toroidal field, which facilitates the global analysis, but not yet radial and latitudinal gradients together. On the other hand, previous studies that used local analyses were able to consider both latitudinal and radial gradients (Fricke 1969; Goossens & Tayler 1980; Urpin 1996). (4) Any unstable modes we find should grow on dynamical timescales, since we include no diffusion in the problem. Most previous studies included one or more diffusivities (for diffusion of momentum, magnetic fields, and heat), and so generally found instabilities that occurred on timescales of diffusion across the local space scale (Fricke 1969; Tayler 1973; Acheson 1978; Acheson & Gibbons 1978). (5) We confine ourselves to reference state toroidal fields, while some previous studies could include poloidal fields, since the analysis was local (Fricke 1969). (6) In this paper, we study hydrostatic perturbations, whereas most previous studies were nonhydrostatic. Our equations include some effects of compressibility, namely the reference state density variation with height within the shell, so the equations are non-Boussinesq. The hydrostatic assumption can apply well to thin shells, and has the advantage of filtering out convective instability and most gravity waves, and thus we can focus on pure MHD and shearing instabilities in stably stratified parts of solar/stellar interiors. We will compare our results to those of Cally et al. (2008), who used a nonhydrostatic but Boussinesq system for the same problem. During previous attempts by us and others to explore the existence/nonexistence of m = 0 instability of a toroidal band in the solar tachocline, it was pointed out (Dikpati & Gilman 1999) that the instability cannot exist for modes with m = 0 in a purely 2D (longitude–latitude) system, because the only solution for that case will be the reference state magnetic field and flow—all other magnetic and flow fields would either violate mass conservation in the flow or create magnetic monopoles. Another way of understanding why m = 0 instability does not occur in the pure 2D system is that the fluid mass, pushed poleward of a toroidal band due to curvature stress, cannot move from the poleward side to the equatorward side of a band. In order for m = 0 instability to set in, the fluid mass needs to be pushed either upward over or downward below the poleward edge of the toroidal band toward its equatorward edge. This is not possible for a band with no radial extent. Quasi-3D shallow-water models allow the third dimension in a very simplified way—horizontal magnetic field and flow components are independent of depth, but radial components of the flow and field are zero at the rigid bottom boundary and vary linearly with height. Since the top boundary is deformable in such models, the fluid pushed poleward by the magnetic curvature stress of a toroidal band bulges upward and allows magnetic energy to be extracted from the reference state. Thus, axisymmetric instability can occur. Clearly this mechanism must do work against the stable stratification of the radiative tachocline. In the overshoot tachocline with a relatively weaker stratification, m = 0 modes can set in only for an extremely high field strength (∼ 2 mG) (Gilman & Dikpati 2002). Perhaps strong toroidal fields (> million Gauss) might be present in the solar/stellar radiative interior below the tachocline—they could be of primordial origin, due to an extinct dynamo or the fields from an active dynamo diffusing to the radiative interior, but m = 0 modes might be difficult to excite there in a shallow-water model, as argued in the previous paragraph. The questions that immediately arise are: (1) what

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happens in the fully 3D case? (2) Does m = 0 instability set in for solar-like field strength of ∼ 100 kG? (3) If it occurs for solar-like field strength, what pattern will it imprint at the surface? (4) Can the m = 0 mode compete with m = 0 modes? Furthermore, the detailed examination of nonaxisymmetric instability showed that, as a consequence of m = 0 instability in the solar tachocline, a broad toroidal magnetic field opens up into a clam-shell pattern (Cally 2001), whereas a banded toroidal field tips or deforms, but the toroidal field profile does not break up into narrower bands. If axisymmetric instability occurs for a solar-like tachocline toroidal band, does the fluid roll in latitude and radius as a consequence of this instability, creating narrower bands, stacked in depth, from broader toroidal fields? In order to seek answers to these questions, we investigate the axisymmetric instability of toroidal bands in the solar/stellar tachocline. In Section 2, we give the detailed mathematical formulation of the equations, as well as an integral theorem that can be derived from them. We present our results in Section 3, and close with comments and conclusions in Section 4. At various places in the text we compare our formulation and results with those of Cally et al. (2008). 2. GOVERNING EQUATIONS AND INTEGRAL THEOREMS To study the m = 0 instability problem, we start from the hydrostatic but non-Boussinesq equations shown in detail in Gilman et al. (2007, hereafter GDM07). Our starting point there is the set of Equations (16)–(25), for 3D perturbations about reference states defined by Equations (11)–(14) and (26). As in GDM07, we study in detail only reference states that are independent of z. The reduction of these equations to the m = 0 case is carried out by letting m → 0 while keeping the product mτ = σ finite. τ is the (complex) eigenvalue of the perturbation equations in GDM07. In this limit, it is readily seen that, when perturbing the reference state about purely toroidal fields, no perturbation poloidal fields (b, c) are produced. Then in GDM07, in Equations (7) and (24) all terms vanish, and Equations (18), (19), and (23) simplify. The m = 0 system then becomes 1 ∂ ∂w (v cos φ) + = 0, (1) cos φ ∂φ ∂z −iσ u − ω0 v sin φ + v

∂ (ω0 cos φ) = 0, ∂φ

−iσ v + 2(ω0 u − α0 a) sin φ = −

∂Π , ∂φ

∂θ0 + G1/2 w = 0, ∂φ   p = γ G1/2 δE −1 ρ + δS −1 θ , iσ θ + v

∂Π δS = − p + G1/2 θ, ∂z γ

(2) (3) (4) (5) (6)

∂α0 , ∂φ

(7)

Π = p + aα0 cos φ.

(8)

iσ a = v cos φ

Note that Equation (7) is much reduced, with two terms eliminated, compared to Equation (23) of Gilman et al. (2007).

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To arrive at a single equation for m = 0 unstable modes, we first reduce the system of Equations (1)–(8) to two coupled equations for the perturbation latitudinal velocity v and the perturbation total pressure Π. These equations are   ∂ ∂Π σ 2 − 4ω0 2 sin2 φ + sin φ cos φ (ω0 2 − α0 2 ) v = −iσ , ∂φ ∂φ (9) 1 ∂ δS ∂ (v cos φ) − cos2 φ cos φ ∂φ G ∂φ   iσ ∂ 2 Π ∂Π . =− + δS G ∂z2 ∂z Figure 1. Growth rate σi as function of m for 1 > m → 0, showing rate of convergence to m = 0 solutions.

It is easy to see that Equation (7) here can be obtained from Equation (23) of Gilman et al. (2007) by invoking mass conservation as implied by Equation (1) above. Except for the eigenvalue σ , all quantities in Equations (1)–(8) are the same as defined in GDM07. Latitude is φ and the vertical coordinate is z. ω0 is the differential rotation, α0 is an angular measure of the reference state toroidal field, and δS is a measure of the reference state radial density variation. We thus have eight equations that determine the eight variables u, v, w, a, θ , p, Π, and ρ, respectively the perturbation longitudinal, latitudinal and vertical velocities, the toroidal field, the potential temperature, the gas and gas plus magnetic pressures, and the density. Since the density ρ is present only in Equation (5), we can use this equation only for determining density, and leave it out of the system for determining the eigenvalues and eigenfunctions. In numerically computing the eigenvalues and eigenfunctions of the m = 0 problem, we do not actually solve Equations (1)– (8); instead we use the code already developed for the system with general m used in GDM07, and numerically compute the asymptotic values of the eigenvalues as m → 0 numerically. Figure 1 shows the rate of convergence of the growth rate σi as the longitudinal wavenumber is reduced from 1 to close to zero. For G = 0.1 and n = 1, we reach convergence for m ∼ 10−1 − 10−2 , while for G = 10 and n = 5, it is converged by m ∼ 10−3 − 10−4 . This approach eliminates the possibility of new errors being introduced. We can also use as initial guesses solutions already found for m = 1 cases with the same reference state. Finding the “basin of attraction” that contains the eigenvalue corresponding to the fastest growing unstable mode can be quite tricky; the approach we have adopted appears to minimize such difficulties. Another way to solve the m = 0 instability equations is to reduce them to a single higher order equation. It is useful to carry out the reduction in order to compute the eigenvalue by solving directly the m = 0 eigensystem and compare with the solution obtained by the method described above. Furthermore, we can also investigate from this reduced higher order equation what general statements can be made about the instability. In particular, certain integrals of the single equation can usually be derived that provide a necessary condition for instability. Such was the case for the 2D instability problem of coexisting differential rotation and toroidal fields studied in Gilman & Fox (1997) and in the shallow-water system (Gilman & Dikpati 2002).



α0 2 2



∂v ∂z (10)

Next, we apply the normal mode analysis for the separation of variables in z; with v and π ∝ einπz , n real, we eliminate the variable Π from the resulting forms of Equations (9) and (10), transform the dependent variable by defining W = v cos φ, transform the independent variable φ to μ by the relation μ = sin φ, and obtain a single second-order governing equation 2

d 2W iδS nπ (1 − μ2 ) dα0 2 dW (1 − μ2 ) − 2 dμ 2G dμ dμ    2 nπ (iδS − nπ ) (1 − μ )iδS nπ d d (1 − μ2 ) − α0 2 + 2G dμ dμ G    d ω0 2 − α0 2 × σ 2 − 4ω0 2 μ2 + μ(1 − μ2 ) W = 0. dμ (11) We solve Equation (11) using a shooting method and compute a few selected unstable, axisymmetric modes for a 10◦ band with peak field strength of 400 kG (a = 4), placed at 30◦ latitude. The effective gravity value and the differential rotation amplitude are respectively G = 0.1 and s = 0.18 (i.e., ω = r − sμ2 as used in Gilman et al. 2007). We will study the cases with other values of G, but the pole-to-equator differential rotation amplitude will always be kept at 18%. Table 1 shows the comparison of eigenvalues obtained by solving respectively the full set of Equations (16)–(25) of GDM07 with m → 0 (as demonstrated in Figure 1) and the reduced higher order Equation (11) for m = 0. Table 1 reveals that the two different approaches for solving the hydrostatic non-Boussinesq system produce the same results within ∼ 0.3% accuracy. To find a useful integral relation from Equation (11), it is necessary to ‘suppress’ the dW/dμ term by the method described on page 20 of Carrier & Pearson (1991), by introducing the further transformation, W = U e−

iδS nπ 4G



μ (1−μ

2

d ) dμ α0 2 dμ

,

(12)

with which Equation (11) reduces to the form, d 2U + rU = 0, dμ2

(13)

in which    2 n2 π 2 2 2 2 2 2 d σ − 4ω0 μ + μ(1 − μ ) ω0 − α0 r= (1 − μ2 )G dμ   2 d  2 inπ δS 3 2 d 2 2 (1 − μ ) 2 α0 + μ ω0 − 4α0 + G 4 dμ dμ 2

n2 π 2 δS 2 (1 − μ2 ) d α0 2 . + 16G2 dμ

(14)

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Table 1 Comparison of Eigenvalues Obtained by Solving HyPE Equations (16)–(25) of GDM07 with m → 0 and Reduced Higher Order Equation (11) for m = 0 Radial Wavenumber n

Eigenvalues from Lower Order HyPE Equations with m → 0

Eigenvalues from Higher Order Reduced Equation (11)

4.4756 5.2316 5.3839

4.4613 5.2196 5.3729

1 3 5

An integral theorem can be derived by multiplying the Equation (13) by U , the complex conjugate, and integrating by parts and then applying boundary conditions at the poles as follows:   dU 2 dμ, r|U |2 dμ = (15) dμ in which r = rr + iri ,  n2 π 2 rr = σr 2 − σi 2 − 4ω0 2 μ2 + μ(1 − μ2 ) (1 − μ2 )G  2 d  2 δS 2 n2 π 2 (1 − μ2 ) d 2 ω0 − α0 α0 2 , × + dμ 16G2 dμ (16) and ri =

 n2 π 2 nπ δS 3 d2 σ (1 − μ2 ) 2 α0 2 σ + r i 2 (1 − μ )G G 4 dμ  d  2 ω0 − 4α0 2 . +μ dμ

(17)

It follows from Equation (15) that nontrivial solutions exist for Equation (13) only when    dU 2 2 dμ; ri |U |2 dμ = 0. (18) rr |U | dμ = dμ The nonhydrostatic but Boussinesq analog of Equation (18) is Equation (15) of Cally et al. (2008). One limiting special case of Equation (13) is of particular interest. σ = 0 corresponds to marginal instability in this system. Does our system have such stability boundaries? Following the approach of Cally et al. (2008), we can show that when toroidal field gradients are large compared to rotational effects and compressibility effects are neglected (δS = 0), Equation (13) reduces to, d 2U μn2 π 2 d α0 2 U = 0. − dμ2 G dμ

(19)

If we represent α0 by aA0 , where a is the peak field and A0 is a “shape profile,” then Equation (19) reduces to the eigenvalue equation, d 2U  − νμA0 2 U = 0, (20) dμ2 in which ν = n2 π 2 a 2 /G is the eigenvalue. So nontrivial solutions to Equation (20) should lead to a sequence of eigenvalues ν0 , ν1 , and so on, which contain parameter combinations for which σ = 0. 3. RESULTS In the following three subsections, we present the inferences from the integral theorems, followed by instability results obtained from the hydrostatic system, together with a few comparisons with corresponding nonhydrostatic but Boussinesq results shown in Cally et al. (2008).

3.1. Common Corollaries from the Integral Theorems of Hydrostatic and Nonhydrostatic Systems We derived above the governing equations and integral theorems for the hydrostatic compressible or non-Boussinesq system; the corresponding results for the nonhydrostatic system are given in Cally et al. (2008, Section 3). Since the approximations made to derive the these two systems are different, the respective integral relations may give different results. However, in the Boussinesq limit for the non-Boussinesq equation, found by letting δS → 0, certain corollaries are the same as follows. Corollary 1. σ 2 is real, so there are only neutral oscillations and exponentially growing modes. This result follows from the description of ri in Equation (17) used in Equation (18), when δS → 0. Corollary 2. A necessary condition for instability in both d systems is that the quantity μ(1−μ2 ) dμ (ω0 2 −α0 2 )−4ω0 2 μ2 > 0 somewhere in the domain. This comes from the requirement that rr > 0 somewhere. This necessary condition for instability applies for all vertical wavenumbers n or k in the two systems. Corollary 3. It follows from Corollary 2 that the angular momentum per unit mass, (1−μ2 )ω0 , decreasing monotonically toward the poles, such as in the case of the solar differential rotation, is stabilizing. Corollary 4. Simple toroidal field profiles such as α0 = aμ (antisymmetric about the equator, peaks at 45◦ N and S latitude) are also stabilizing. For example, the combination α0 = aμ and ω0 = (1 − sμ2 ) (0  μ  1) is stable in both systems, no matter what the peak field strength. But banded toroidal fields of sufficiently large field strength can be unstable, since d 2 μ dμ α < 0 on the poleward side of such bands. This feature is the source of unstable modes we show in later sections. This effect was also noted in Pitts & Tayler (1985). Corollary 5. For given toroidal field and differential rotation profiles, the growth rate of an unstable mode should be linearly proportional to the peak field strength for large fields. This can be inferred from the necessary condition for instability in both systems, which also implies an upper bound to the growth rate for a given peak toroidal field. Corollary 6. From Equation (20) above, analogous to Equation (35) of Cally et al. (2008), we can deduce that the eigenvalues of these equations define lines in parameter space on which there are marginally unstable solutions. In the hydrostatic case, these are lines of constant π 2 n2 a 2 /G, and in the nonhydrostatic case, lines of constant k 2 a 2 /N 2 . If we denote ν0 as the minimum positive eigenvalue of these equations, then for a given G or N and given a, there are minimum ver1/2 1/2 tical wavenumbers for instability given by nmin = ν0 πaG and 1/2 kmin = ν0 a N .

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3.2. Corollaries Specific to One System of Equations Corollary 1. In the hydrostatic system with some compressibility (δS = 0), σ 2 is in general no longer real, since from Equations (17) and (18) both σr and σi must not equal zero for instability to occur. We can estimate the size of σr from Equation (18) if we approximate σi for the δS > 0 case with its value when δS = 0. By inspection σr σi in most cases, and σr → 0 as σi increases with peak field strength. Corollary 2. Also in the hydrostatic system, the larger the factor n2 /G is, e.g., the weaker the subadiabatic stratification is, the more unstable the system should be. Modes for the same n2 /G and other parameters the same should have the same growth rate. Corollary 3. Unstable modes can be found from the nonhydrostatic system (Cally et al. 2008) even when N → 0. But such modes cannot be found from the hydrostatic system when the corresponding parameter G → 0, because the hydrostatic approximation breaks down in that case. In the nonhydrostatic system, there are unstable modes even when N < 0, whether or not there is rotation or magnetic fields. These are convective modes, relevant only to stellar convection zones, not interiors in radiative equilibrium. Corollary 4. When N > 0, the nonhydrostatic system contains an infinite number of neutral gravity waves, which the hydrostatic system mostly filters out. These waves may be modified by the presence of rotation and/or magnetic field, but if these effects are small, then the upper limit on gravity wave frequencies is N. 3.3. Properties of Unstable Modes The governing equations for this instability problem contain a wide variety of physical effects, as represented by the various functions and parameters we must specify to get numerical results. These include the toroidal field α0 , the differential rotation ω0 , effective gravity G (or Br¨unt–V¨ais¨al¨a parameter N in the nonhydrostatic system), and the compressibility parameter δS , as well as the width and latitude of a toroidal band. Here, we focus on choices that are plausible for the solar tachocline, and so we specify a differential rotation of 0.18 relative to the reference frame, choose G values appropriate for the tachocline, ranging between 0.1 for the overshoot part and 10 for the radiative part, toroidal band widths of 10◦ (FWHM) placed at latitudes between 15◦ and 60◦ . Toroidal field strengths in the tachocline are unknown from observations, so we scan a wide range of peak values, from well below 100 kG, for which m = 0 modes are always stable, up to 1MG, probably well above what is likely to occur in the Sun. 3.3.1. Growth Rates

As stated in Section 2, we actually solve the full m = 0 equations by taking the limit numerically as m approaches zero, in effect to solve Equations (1)–(8) above. Cally et al. (2008) solve the nonhydrostatic version of Equation (11) as a generalized eigensystem by implementing a spectral method. Eigenvalues obtained by solving a hydrostatic non-Boussinesq system and a nonhydrostatic Boussinesq system, for a 10◦ band at 30◦ latitude (G = 0.1, a = 4), are presented in Table 2 for a few selected cases. It reveals that the systems derived from two different approximations yield eigenvalues within about 1% of each other.

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Table 2 Sample Comparison of Eigenvalues Obtained by Solving a Hydrostatic non-Boussinesq System and a Nonhydrostatic Boussinesq System Radial Wavenumber n

1 3 5

Eigenvalues from Hydrostatic Non-Boussinesq system

Eigenvalues from Nonhydrostatic Boussinesq system

4.4756 5.2316 5.3839

4.3729 5.1836 5.3532

We demonstrate in more detail in Figures 2(a) and (b) that the two different approximations of the equations give similar disturbance growth rates as functions of toroidal field and stratification parameters G or N, indicating the robustness of both the methods and the instability. In both cases, instability occurs for toroidal fields above threshold values (the straight diagonal lines) predicted in Corollary 6 of the corollaries common to both systems. In both cases, the presence of rotation and differential rotation leads to a diagonal band of stability (shown in black) immediately to the right of the white line. Further to the right, for higher toroidal fields, the growth rates linearly increase with a, as also deduced from the integral theorems. Here, a growth rate of 1 corresponds to an e-folding time for growth of just 3.6 days (one year/100), so the instability is very powerful. The largest differences in growth rate for the two systems occur for high (megaGauss) toroidal field and low subadiabatic stratification (very small G and N; the lower right of each figure). For high toroidal fields and low effective gravity G, the hydrostatic approximation starts to break down, so in this part of the parameter space the nonhydrostatic equations are probably more accurate. But it takes rather extreme conditions to see this difference. Detailed growth rate plots for the instability are shown in Figures 3(a) and (b), respectively for the overshoot (G = 0.1) and radiative (G = 10) tachoclines. We chose δS = 0.5, thereby including half a density scale height of the density variation within the tachocline. Here n is the vertical wavenumber of the unstable mode. In Figure 3(a), we see that the instability sets in near a = 1, higher for low latitude bands, and lower for high latitude bands, higher for lower n, lower for higher n. These differences are preserved and even expanded for still higher a. All growth rates linearly increase with a in accordance with Corollary 5 of the common corollaries. Figure 3(b) shows growth rates for the same band placements for the radiative tachocline. Here, the instability sets in for higher a than for the overshoot layer, and requires higher vertical wavenumbers to be unstable at all. By comparing curves in Figures 3(a) and (b), such as for bands at 30◦ for n = 1 (overshoot) and n = 10 (radiative), we can verify that Corollary 2 for the hydrostatic system is verified; these curves, if superimposed, coincide. Instability is stronger for higher n than lower n because the fluid displacements in the vertical are smaller, resulting in less work required to be done against the negative stabilizing buoyancy force. This effect is sufficiently strong for the radiative tachocline that the low-n modes there are not unstable. Instability is stronger for bands at higher latitudes because there the perturbation magnetic curvature force is stronger for the same field strength, since the angle between latitude and the perpendicular to the rotation axis is smaller, i.e., a larger fraction of the curvature force is directed in latitude.

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(a)

(b)

Figure 2. Growth rate contours in G–a plane for hydrostatic non-Boussinesq system (top) and nonhydrostatic Boussinesq system (bottom) for a 10◦ toroidal band at 30◦ latitude.

In Figure 4, we plot growth rates as functions of the vertical wavenumber n for selected latitudes of band placement for peak toroidal fields of 100 and 200 kG for the overshoot tachocline parameter values. An analogous plot could be made for the radiative tachocline. We see in Figure 4 that in all cases the growth rate is asymptotic to a constant value as the vertical wavenumber increases. Once the vertical displacements are small enough that not much work needs to be done against the negative buoyancy force, the vertical structure does not matter much in determining the growth rate. Energy for the instability is coming from the latitudinal gradients of the toroidal field, for which the vertical mode structure does not matter. 3.3.2. Disturbance Planforms

In order to show the structure of unstable disturbances in a linear calculation, we construct the total (reference state + perturbation) flow fields and magnetic fields by using the following steps: (1) we first normalize the arbitrary amplitude of the eigensystem by setting the amplitude of the perturbation

Figure 3. Growth rates as a function of peak field strength (a) for four selected latitude placements (60◦ , 45◦ , 30◦ , and 15◦ ) of the band and for various radial wavenumbers n: the top frame for G = 0.1 and the bottom frame for G = 10. Four different colors denote four selected latitudes, and solid and dashed lines denote two different radial wavenumbers in each frame. For these modes, the top and bottom are taken to be rigid, in the sense that the boundaries do not deform, and there is no flow through them. These are equivalent to stress-free boundaries in a system with viscosity present.

longitudinal component of the magnetic field to unity, (2) set the amplitude of the longitudinal component of the perturbation magnetic field to 17% of the amplitude of the reference state toroidal field, and that fixes the amplitude of all other magnetic field components and flow components, (3) add the perturbation components to the reference states, and (4) plot in meridional cuts (latitude–depth planes). Note that there is no longitude dependence in this calculation. Figure 5 displays velocity and magnetic patterns of unstable disturbances for a band placed at 30◦ shown in a pole-to-equator meridional cut, and a suitable enlargement for a limited latitude range centered on the reference state toroidal band. The upper frames show the overshoot tachocline (G = 0.1) and the lower ones show the radiative tachocline (G = 10). Several features of these unstable disturbances are evident. In both cases, the disturbance deforms the toroidal field in the radial direction, leading to the possibility of formation of separate toroidal rings stacked in radius (a nonlinear calculation would be required to verify this possibility, but see discussion of Figure 7 below).

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Figure 4. Growth rates of unstable modes as a function of radial wavenumber n for three selected latitude placements (60◦ , 45◦ , and 30◦ ) and for peak fields a = 2 and 1.

We also see that most of the deformation occurs on the poleward side of the peak in the undisturbed band; this is consistent with the integral theorem result that showed that instability can occur where the toroidal field amplitude decreases rapidly toward the poles. Another prominent feature of the unstable mode is that its velocity patterns extend much farther in latitude away from the toroidal field peak than do the magnetic patterns. They also extend much farther in latitude in the radiative tachocline than in the overshoot tachocline. This latter effect is caused by the more subadiabatic stratification in the radiative case, which tends to suppress radial motions, forcing the circulation patterns to be much flatter. The figure suggests that the flows are carrying field away from or toward the maximum in the undisturbed Gaussian toroidal band. But in fact the opposite is occurring: the magnetic curvature forces are pushing the fluid along with the moving field. We describe in Section 5 conceptually how this force causes the instability. Figure 6 compares unstable disturbances on toroidal bands at 60◦ and 15◦ latitude for the radiative tachocline. We see that the latitudinal extent of the disturbance on both sides of the peak of the band is much smaller for the polar band than for the band near the equator. This occurs because the restoring Coriolis force is weaker in low latitudes, as is the restoring magnetic curvature force on the equatorward side of each band (because the total magnetic curvature force always points toward the axis, not the pole). An interesting feature of the unstable mode on the band at 15◦ is that the velocities actually cross the equator, linking up to a similar disturbance in the southern hemisphere acting on the band there. 3.3.3. Possible Consequences of Nonlinear Evolution

In Figures 5 and 6, we considered a 17% amplitude of the perturbation (measured with respect to the reference state amplitude) and saw how it can modify the reference state. If we allow the perturbation to grow with time, we can get a hint as to the nonlinear evolution of this instability. For example, 17% amplitude of the perturbation will grow to ∼ 80% in 4.2 days for a growth rate, σi = 1.34, for a 10◦ band at 30◦ latitude (G = 10 and n = 10). The deformations in latitude and radius of the toroidal field will amplify. Figure 7 demonstrates two such

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examples: one for radial wavenumber n = 5 and G = 0.1 (see frame 7(a)) and the other for n = 10 and G = 10 (see frame 7(b)). In both cases, concentrated structures result in the radial direction. In Figure 7(a), for the low G case (with n = 5), we find that two and one half flux concentrations, stacked in depth, are formed on the poleward side of the band, whereas for the high G case, such localized flux concentrations occur on both the poleward and the equatorward sides of the toroidal band (see Figure 7(b)). If such flux concentrations, stacked in depth, are formed in the timescale of days to weeks, it might explain why we see the repeated emergence of magnetic flux at nearly the same latitude and longitude over a period of several months. The flux erupting from the top tube of concentrated flux could suck upward the next tube at the same latitude and longitude. From a linear calculation, we can illustrate only the tendency to form such separate tubes. In the case of a 2D MHD calculation of the nonaxisymmetric instabilities, the tendency of the tipping of a toroidal band was noted from the linear calculations, by studying how the perturbations can modify the reference state as a result of the first-order effect, and actual simulations of the nonlinear evolution of that instability confirmed this speculative result (Cally et al. 2003). The result that a band tips as the consequence of the nonlinear evolution of the m  1 instabilities was also retained in the 3D nonlinear simulations (Miesch et al. 2007). However, the nonlinear calculations showed that the band does not continue to tip indefinitely; the Coriolis force limits the tipping to a certain tip angle. A nonlinear calculation of the 3D MHD axisymmetric instability is necessary to investigate how large the amplitude of the perturbation can grow, and what limits the aforementioned flux concentration mechanism. The scenario described above takes into account only the processes included in our model. In reality, many competing processes are going on. It does not account, for example, for what the solar dynamo would be doing to produce new toroidal field before the instability we have studied acts, or for the role played by instability to other longitude wavenumbers. Only a full 3D MHD nonlinear simulation including the dynamo effects can tell us the relative importance of all the various processes that could be contributing to the physics of a full system. The present calculation can provide the motivation and some insight for the step toward building the full system. Nevertheless, the appearance of concentrated flux tubes in Figure 7 leads to another important question. Could these tubes be magnetically buoyant, and therefore tend to rise out of the tachocline into the convection zone and photosphere above? From the original argument of Parker (1955) that established the concept of magnetic buoyancy, we can say that if such a tube reaches both total pressure and temperature equilibrium with its surroundings, then it will be magnetically buoyant. Reaching total pressure equilibrium takes only a sound travel time across the tube, which is extremely short. Temperature equilibrium takes longer, determined by a radiative or turbulent diffusion time across the short radial dimension of the tube. This is still possible on a timescale of a month or two. Only a detailed nonlinear calculation would show if this works, but the patterns of toroidal field seen in Figure 7 are highly suggestive. If magnetic buoyancy is achieved in these tubes, it leads to a nonhydrostatic process for their rise, which is beyond the scope of the hydrostatic equations used here. But the close agreement between our results and that of Cally et al. (2008) from nonhydrostatic equations implies that the tubes should become magnetically buoyant and rise.

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We started here from a toroidal band already concentrated in latitude, but uniformly distributed in depth, and showed that the 3D MHD axisymmetric instability could provide a mechanism for forming concentrated structures stacked in depth. How to produce the banded structures in latitude is an issue in solar physics, but that is beyond the scope of the present calculation. This issue has been addressed by studying the undular arching of a flat magnetic sheet with neutral buoyancy by Fan (2001) who showed that the arching of the magnetic region produces draining of the fluid eventually leading to more flux concentration in the arched magnetic region. A similar study has recently been performed by Kersal´e et al (2007) who started from the thermal equilibrium of a magnetic sheet and showed that the flux concentration can occur at the regions of two adjacent counter flow cells. Brummell et al (2002) suggested another mechanism by applying the horizontal shear into the magnetic sheet and showed that the continuous shearing leads to the break up of the magnetic layer into concentrated rolls. Perhaps in the solar interior all these processes are taking place simultaneously to produce concentrated structures in latitude and depth, before they manifest at the surface as active regions. 4. CONCEPTUAL PHYSICS OF THE INSTABILITY To understand more intuitively the nature of the instability we have studied, we developed in Figure 8 a schematic representa-

tion of what the perturbation forces do to cause the instability, and where it is most pronounced. In the reference state for a banded toroidal field, at all latitudes there is a balance in latitude among three forces, namely a poleward magnetic curvature force, and equatorward Coriolis and total (gas plus magnetic) pressure gradient forces (see Equation (11) of Gilman et al. 2007). (The balance among all forces in the radial direction is not shown, because it is always hydrostatic and does not contribute to the dynamics of the instability.) Using Figure 8 (top), we can show how this force balance can be made unstable, in accordance with common Corollaries 2, 3, and 4 above, by conceptually interchanging rings of toroidal magnetic flux and fluid of equal but small cross-sectional area between points A and B on the poleward side of the band. By contrast, exchanging toroidal rings between points C and D on the equatorward side leads to continued stability of the band. This “ring” argument is analogous to the classical “parcel” arguments used to demonstrate convective instability in a fluid with superadiabatic temperature gradient, and the magnetic buoyancy first demonstrated by Parker (1955). The reader is reminded that in Figure 8, as well as in Figures 5 and 6, the radial scale is greatly expanded. In reality, points A, B, C, and D are virtually at the same radial distance from the center. Thus, all the exchanges of rings mean displacements in latitude, not radius. Consider first the exchange of rings between locations A and B. During this exchange the total toroidal flux and the angular

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momentum of the rings are conserved. With rings of small crosssection, the total pressure inside the rings equilibrates quickly to the total pressure of the surroundings. Therefore there should be no change in the ambient total pressure gradient acting on the rings. But the new ring at A (B) now has a higher (lower)

toroidal field than its surroundings, which implies a net magnetic curvature force (MCF) directed poleward at A and equatorward at B. Because each ring conserves its angular momentum, there is a net Coriolis force (CF) toward the equator at A, and toward the pole at B. Then at both A and B, if the peak toroidal field in

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In the special case of no rotation in the system, the same conceptual argument applies, but there is no Coriolis force to oppose the displacement of the rings, and therefore the instability occurs for weaker toroidal field peaks. In Figures 2(a) and (b), we can see that the onset of this instability without rotation occurs for field strengths that are in between the “zero” growth rate contour and the white stability boundary line. This is also in accordance with Corollary 6. This conceptual experiment shows that instability, if it occurs, should originate from the poleward side of the toroidal band, as stated in the corollaries. To see more quantitative detail about the forces responsible for the instability requires solving the eigenvalue system explicitly, results from which we present below. There is a small change in the peak toroidal field of the rings when they are exchanged between points A and B, and between C and D. The toroidal field of the ring at A is slightly smaller than it was at B, because the circumference of the ring, being closer to the pole, has to shrink. Similarly, the ring at B after the interchange has a higher toroidal field than it had when it was at A. The same arguments apply to new rings at C and D. For toroidal bands that are narrow in latitude, this change in the toroidal field is small compared to the difference between the toroidal field of the ring at its new location and the field of its new surroundings. Therefore, we ignored this effect in the “ring” arguments above. But in the limit of large band width, such as for the toroidal field profile α0 = aμ, toroidal rings at different latitudes on the poleward side of the peak (which occurs at 45◦ latitude) when exchanged experience a large enough change in strength of the correct sign to eliminate the instability by nearly eliminating the magnetic curvature force, while the stabilizing Coriolis force remains essentially the same. This is the reason why the toroidal field profile α0 = aμ is stable. 5. SUMMARY AND CONCLUSIONS

Figure 8. Top: a schematic diagram depicting the latitudinal forces acting on toroidal rings exchanged between points A and B, and between points C and D. MCF denotes the magnetic curvature force, CF denotes the Coriolis force. Background radial lines are schematic contours of the Gaussian toroidal field amplitude for a band placed at 45◦ latitude. Bottom: deformed contours of the band of toroidal field for initial displacements (red and blue solid curves) and subsequent movement of the same contours (dashed curves) due to instability on the poleward side (the red curve), and stability on the equatorward side (the blue curve).

the original toroidal band is large enough, and the band narrow enough, the magnetic curvature force is not balanced by the Coriolis (or any other) force so the new rings at both A and B would continue to move away from their locations before the exchange. Therefore, the original toroidal band would be unstable to such interchanges. On the other hand, exchanging rings between points C and D results in magnetic curvature forces toward the equator on the new ring at C, and toward the pole on the new ring at D. In each case, the Coriolis force points in the opposite direction to the magnetic curvature force, but is smaller. So at points C and D, the net forces push the new rings back toward their original locations, so there is no instability.

The axisymmetric instability of toroidal field bands in differentially rotating shells or “tachoclines” found between convective and radiative domains of stellar interiors could be important for a wide variety of stars. Two different systems of governing equations, one hydrostatic and the other nonhydrostatic, give similar results for this instability. Differences that do occur can be attributed to the different approximations made. In this paper, we have emphasized the hydrostatic system. Integrals of the perturbation equations together with the equations themselves yield a number of general properties of the instability that are subsequently verified and quantified by actual numerical solutions. In the absence of a radial density variation in the unperturbed state, all unstable modes grow exponentially without oscillation or propagation. But with such a gradient, unstable modes do propagate slowly in latitude, at a rate that declines with increasing toroidal field strength. Toroidal field bands that are sufficiently narrow in latitude are unstable while broad toroidal fields are not. Bands at high latitudes are more unstable than low-latitude bands. The instability is always driven by the sharp decline in toroidal field amplitude on the poleward side of such bands. The destabilizing force is the perturbation magnetic curvature stress that creates a poleward force on poleward moving rings and an equatorward force on equatorward moving rings. This arrangement of forces leads to perturbations in velocities and toroidal fields that are much stronger on the poleward than the equatorward side of the band.

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For solar parameters, instability first sets in for toroidal field peaks in the range 80–120 kG. Above these levels, mode growth rates rapidly climb to e-folding growth times of down to a few days or even hours with increased field. For sufficiently large peak fields, the growth rate increases linearly with field strength. In contrast to nonaxisymmetric modes, differential rotation stabilizes axisymmetric modes. Nonaxisymmetric modes become unstable at much lower toroidal field strengths, but their growth rates are much lower than for the m = 0 modes. The minimum vertical wavenumber of unstable modes increases with the subadiabaticity of the stratification, but decreases with the toroidal field peak. Unstable modes with higher vertical wavenumbers contain smaller vertical fluid displacements, and therefore do less work against the stable stratification, enhancing the instability. But for a given stratification and peak toroidal field, the growth rate becomes independent of vertical wavenumber for large enough wavenumbers. This is because the energy of the instability is coming from the latitude gradient of the field. The stronger the stratification, the greater the latitudinal extent of the velocity perturbations of an unstable mode. This is because vertical motion is inhibited by the more negative buoyancy force arising from the stronger stratification. For a given stratification and peak toroidal field, the higher the latitude of the band, the smaller the latitudinal extent of perturbation velocities and toroidal fields on the poleward side of the band, and the smaller the velocities and perturbation fields on the equatorward side. This occurs because for a given peak toroidal field, the closer the band is to the poles, the larger is the destabilizing magnetic curvature stress on the poleward, leading to more vertical motion and less latitudinal motion, and the larger the stabilizing curvature stress is on the equatorward side leading to less extent and lower amplitude velocities. If the instability is allowed to grow to finite amplitude at which nonlinear effects become important, the toroidal band should fragment in the radial coordinate into a sequence of smaller, radially stacked bands or tubes. By contrast, instability to nonaxisymmetric modes can only tip or deform the band, not fragment it. This fragmentation process may be important for creating tubes that rise to the surface to form active regions. Because the tubes are stacked in radius, there could be a succession of flux emergence events at the same longitude, a possible cause of repeated emergence of magnetic flux observed at the site of long-lived active regions in the photosphere. We thank Keith MacGregor for reviewing the entire manuscript and for his helpful comments. We extend our thanks to the referee for a very thorough review and for several constructive comments on mathematical as well as conceptual points on an earlier version of the manuscript, taking care of which has improved the paper. This work is partially supported by NASA grants NNH05AB521 and NNX08AQ34G. The

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