The Astrophysical Journal Supplement Series, 170:203 Y 227, 2007 May # 2007. The American Astronomical Society. All rights reserved. Printed in U.S.A.
GLOBAL MHD INSTABILITIES IN A THREE-DIMENSIONAL THIN-SHELL MODEL OF SOLAR TACHOCLINE Peter A. Gilman, Mausumi Dikpati, and Mark S. Miesch High Altitude Observatory, National Center for Atmospheric Research,1 3080 Center Green, Boulder, CO 80301;
[email protected],
[email protected],
[email protected] Received 2006 October 9; accepted 2006 November 28
ABSTRACT We generalize the linear analysis of the global instability of coexisting differential rotation and toroidal magnetic fields in the solar tachocline to include continuous radial stratification, thermodynamics, and finite tachocline thickness, as perturbed by three-dimensional disturbances of longitudinal wavenumbers m ¼ 1, 2. For radiative tachocline stratification, the instability for both banded and broad toroidal field profiles is similar to the two-dimensional and shallow water cases studied previously, even though the unstable modes have substantial vertical structure. For overshoot tachocline stratification, instability for banded toroidal fields with peaks P20 kG is similar to the corresponding shallow water case, but for substantially higher ( perhaps unrealistic) toroidal fields there is no low-subadiabaticity cutoff, and modes appear with much higher growth rate and increasingly negative phase velocities in longitude, analogous to those found earlier by Cally. All of these results are only modestly sensitive to the tachocline thickness chosen. For broad toroidal field profiles, the instability results are similar to the two-dimensional and shallow water cases for toroidal field peaks up to at least 80 kG, unless the shell has a thickness that is a substantial fraction of a pressure scale height. For thinner shells, only above about 94 kG do the high growth rate, low phase velocity modes appear, with structure similar to the ‘‘polar kink’’ instability Cally found. But even in this case, we find that the slower growing ‘‘clam-shell’’ instability eventually replaces the faster growing polar kink modes. We conclude that for conditions most likely to occur in the solar tachocline, such as peak toroidal fields limited by the dynamo to 20 kG or less, the two-dimensional and shallow water type unstable modes are likely to predominate even when the tachocline has finite thickness and the modes can have radial structure. Subject headingg s: instabilities — MHD — Sun: interior — Sun: magnetic fields — Sun: rotation
1. INTRODUCTION
away from alignment with a latitude circle, but with no deformation of the ring. Within the 2D realm, Cally et al. (2003) have shown by extending the calculation into the nonlinear regime that the amount of this tipping increases with latitude, from just a few degrees in low latitudes, to greater than 10 poleward of sunspot latitudes. Of course, the solar tachocline is three-dimensional (3D), not 2D, although the subadiabatic temperature stratification there should tend to favor global flows in which the radial motion tends to be suppressed in favor of horizontal motions. On much smaller horizontal scales, rising flux tubes and associated fluid flow due to magnetic buoyancy should also occur (Fan et al. 1993; Schu¨ssler et al. 1994; Caligari et al. 1995; Magara 2004), but we deliberately filter out that set of modes here in order to concentrate on the purely global MHD. In reality, we expect that both scales and types of flows should occur, perhaps with the magnetic buoyancy driven flux tubes rising out of the global unstable patterns of the tachocline, into the convection zone above. Even within the global MHD domain, 3D effects must be present, since both the toroidal field and differential rotation vary in radius as well as latitude. But solving this more general problem is much more difficult. Cally (2003) is the first to explore the 3D global MHD tachocline instability under the Boussinesq approximation. Here we have taken a different approach by proceeding in steps that first include some 3D effects, without the full complexity. To this end, Gilman (2000) developed an MHD analog to the well known (in geophysical fluid mechanics) so-called ‘‘shallow water’’ equations ( Pedlosky 1987; Stoker 1957; Hough 1898). In this system, the spherical shell is allowed to have a deformable top (corresponding roughly to the interface between the stably
Gilman & Fox (1997) first demonstrated that combinations of differential rotation and toroidal fields that could be present in the solar tachocline are unstable to the growth of global nonaxisymmetric disturbances of longitudinal wavenumber m ¼ 1, even though the same profiles of differential rotation and toroidal field are stable when only one is present. This discovery has led to extensive further analyses of this instability for many different toroidal field profiles, including broad ones with or without nodes and sign changes within a single hemisphere (Gilman & Fox 1999a, 1999b), as well as Gaussian-profiled bands of toroidal field of a variety of widths and placed at many different latitudes (Dikpati & Gilman 1999; Gilman & Dikpati 2000). Cally (2001) first extended the linear results into the nonlinear regime and showed that the broad toroidal field profiles open up into a spectacular ‘‘clam-shell’’ pattern, as a result of the nonlinear evolution of this instability. These early studies used a model that is strictly two-dimensional (2D), which allows variations only in longitude and latitude, as well as time. All perturbations are confined to a spherical shell of uniform thickness. The principal results are that the system is virtually always unstable to m ¼ 1 modes, but higher longitudinal wavenumbers are also unstable for narrower banded toroidal fields, provided the peak fields are not larger than 10 Y20 kG. For higher fields, only m ¼ 1 is unstable. For strong banded toroidal fields, the instability therefore takes the form of a ‘‘tipping’’ of the toroidal ring 1
The National Center for Atmospheric Research is sponsored by the National Science Foundation.
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stratified tachocline and the unstably stratified convection zone above), with corresponding small but nonzero radial motions and magnetic fields. The effective gravity G of this system, proportional to the departure of the temperature gradient from the adiabatic gradient (which would have an effective gravity of zero), becomes a parameter of the problem; it takes on a relatively small value for the ‘‘overshoot’’ part of the tachocline, which is significantly influenced by penetrating convection from above, and a relatively large value in the radiative part of the tachocline below. Dikpati & Gilman (2001a) showed that even without a toroidal field, solar-like differential rotation in the shallow water system can be unstable to low m modes when the effective gravity G is characteristic of the overshoot part of the tachocline. Furthermore, they showed that the unstable disturbances all contain kinetic helicity, arising from the correlation between radial motion and the radial component of disturbance vorticity, and therefore could contribute to the driving of the solar dynamo, by generating new poloidal field in and near the tachocline, where toroidal field can be generated by shearing by the differential rotation there. Dikpati & Gilman (2001c) have shown that in flux-transport dynamo models this source of kinetic helicity could be extremely important for determining the correct symmetry of magnetic field about the solar equator. With the addition of toroidal fields (Gilman & Dikpati 2002, hereafter GD02; Dikpati et al. 2003), this hydrodynamic instability is suppressed if the toroidal field is broad and contains peak values above10Y20 kG, but is replaced by the same global MHD instability found in the 2D system. The shallow water results show that the 2D instability results summarized above extend far into the domain of finite vertical stratification. Bands of toroidal field placed at almost all latitudes are unstable; the particular symmetry of unstable mode about the equator depends on the latitude of the band placement. In addition, prograde fluid jets placed inside the toroidal bands, of a magnitude large enough that the associated Coriolis force toward the equator balances the curvature stress pointed toward the poles, are very stabilizing, cutting off the instability for peak toroidal fields larger than about 40 kG, depending on latitude of band placement. The same result applies in the strictly 2D case. In the MHD case, the kinetic helicity of unstable hydrodynamic disturbances remains present if the toroidal field is not strong at latitudes where the disturbance amplitude peaks, generally in mid to high latitudes. But in addition, there is kinetic helicity associated with the MHD disturbances themselves, in the neighborhood of the toroidal bands. Thus, in the MHD case, there are two sources for kinetic helicity, one which is fixed in latitude by the hydrodynamics of the problem, and the other which migrates with change in latitude of the peak toroidal field. This result opens the possibility of a nonlinear -effect in the solar dynamo, in which the -effect amplitude and location in latitude depends on the induced toroidal fields. This would be an effect in addition to the nonlinearity introduced by -quenching due to the toroidal field (Blackman & Brandenburg 2002; Field & Blackman 2002). There has also been extensive study of the influence of poloidal fields on global tachocline MHD (Charbonneau & MacGregor 1992; Ru¨diger & Kitchatinov 1997; MacGregor 2000; Forga´csDajka & Petrovay 2002; Garaud 2002). Here we take the next step in generality of model, moving us still closer to the full 3D MHD problem in a very similar way as Cally (2003) did. However, there are substantial differences in emphasis. In particular, while Cally (2003) focuses almost entirely on instability for broad toroidal profiles, here we focus primarily on banded profiles. Cally (2003) elaborates in considerable detail on the so-called ‘‘polar-kink’’ instability, which occurs in his formulation for peak toroidal fields in
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excess of 84 kG. While we also examine instability for toroidal fields this high, we put more emphasis on results for lower field strengths, particularly in the neighborhood of 20 kG. Although the rising thin flux tube simulations indicate that the field strength of 100 kG is necessary to match the emergence latitude and the tilt patterns of bipolar spots, our emphasis on 20 kG field for this study comes from the following arguments. Recent calculation by Rempel (2006) has shown that a fluxtransport dynamo with j < B-type feedbacks can produce no more than 30 kG toroidal fields at the tachocline. Otherwise the feedback will cause substantial changes in the differential rotation that are not seen in helioseismic measurements. Therefore, certain other mechanisms (beyond the scope of the present calculation) are needed to ensure that the spot properties at the surface are matched with observations. One such mechanism can be the flux concentration from a 20 Y30 kG field to 100 kG flux tube by etaquenching (Gilman & Rempel 2005). Other possible mechanisms, such as whether a thick flux tube of 20Y30 kG field strength rising to the surface would match the spot properties, needs to be investigated in future. In addition, we study the effects of boundary conditions at the top of the shell, while Cally (2003) omits that case. This is necessary for comparing with nonlinear simulations, which cannot handle a semi-infinite shell thickness, such as used by Cally (2003). We also make explicit mathematical connections to the shallow water system, which does not allow for a ‘‘polar kink’’ instability, and take advantage of these connections to solve the eigenvalue problem without having to develop a completely new eigenvalue code. Our presentation of the eigenfunctions of unstable modes is also much different than those in Cally (2003), much more closely connected to prior 2D and shallow water cases. Finally, we also explore the role played in unstable disturbance structure by the singular points of the problem, which carry over from the shallow water case. This is not done in Cally (2003). We start from the so-called hydrostatic primitive equations (HPEs) of Miesch & Gilman (2004, hereafter MG04), which are derived for a tachocline with continuous vertical stratification and differential rotation and toroidal fields that can be functions of both latitude and radius, and which allow for disturbances that are continuous functions of longitude, latitude and radius. Here we focus exclusively on the linear instability problem as an eigenvalue problem. Avoiding the nonlinear problem here allows us to do an extensive parameter survey to explore the full range of instability behavior for all unstable modes. This approach also provides guidance for nonlinear simulations, which must necessarily be far more restricted in parameter choices. In parallel, Miesch et al. (2007, hereafter MGD07) have carried out numerical simulations of the nonlinear evolution of this instability. At various points below we compare our linear results with the nonlinear results reported there. 2. MODEL EQUATIONS FOR THIN STRATIFIED SPHERICAL SHELL 2.1. Transformation of HPE System To establish the link between the HPEs of MG04 and ‘‘shallow water’’ global MHD equations of Gilman (2000) and GD02, we begin from equations (1)Y(6) of MG04, and make almost all the same approximations. In particular, we assume that the vertical scale dTR, the radius of the shell at tachocline depth, so that ¼ d/RT1, but use somewhat different scaling, as well as an inertial rather than a rotating coordinate frame, longitude-latitude (k, ) coordinates rather than spherical polar coordinates; we also use potential temperature rather than temperature and retain the
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full equation of state. The potential temperature, , is defined as cp ln ¼ specific entropy, in which cp is the specific heat at constant pressure. This last change allows us to retain accuracy when the magnetic field is strong. We scale longitudinal and latitudinal velocities by R and vertical velocity by R , where is the solid rotation of the radiative interior. Then time is scaled by
1 , horizontal and vertical magnetic fields by (4m )1/2 R and (4m )1/2 R , respectively, where m is the average density of the shell; pressure by m (R ) 2, density by m (R )N / R? and temperature by R /N, where ¼ @Tm /@r þ g/cp , the subadiabatic temperature gradient (g is the gravity), N is the buoyancy or Bru¨nt-Vaisala frequency; N ¼ ( g/Tm )1/2 and R? is the universal gas constant. We also need two other ratios: S d/H, where H is the pressure scale height; and E ¼ m /m , where m ¼ d , the potential temperature difference across the shell, and Tm is its spherically symmetric temperature. Then if u, v, and w are the dimensionless velocities in the longitude, latitude, and z-directions, a, b, and c are the corresponding magnetic field components, is the dimensionless potential temperature, and p is the dimensionless pressure, then the dimensionless HPEs (12)Y (20) of MG04 become 1 @u 1 @ @w þ (v cos ) þ ¼ 0; cos @k cos @ @z
ð1Þ
1 @a 1 @ @c þ (b cos ) þ ¼ 0; cos @k cos @ @z
ð2Þ
Du 1 @ þ (ab uv) tan ¼ (B = : )a ; Dt cos @k Dv 2 @ þ u a 2 tan ¼ (B = : )b ; Dt @ @ S S ¼ G1=2 ¼ p þ G1=2 ; @z E D þ G1=2 w ¼ 0; Dt p ¼ G1=2 E1 þ S1 ;
ð11Þ
@0 S S ¼ G1=2 0 ¼ p0 þ G1=2 0 ; @z E 1=2 1 p0 ¼ G E 0 þ S1 0 ; 0 ¼ p0 þ
02 cos2 : 2
ð12Þ ð13Þ ð14Þ
In equations (11)Y(14) there are six variables: 0 , !0 , 0 , 0 , p0 , and 0 . Therefore two variables can be specified at will. We usually take them to be the differential rotation !0 and the toroidal field 0 . Then if we assume all perturbation variables are of the form u; v; w; a; b; c; ; p; ; e im(kt) ;
ð15Þ
where m is the integer longitudinal wavenumber, ¼ r þ ii contains the longitudinal phase velocity r and the disturbance growth rate mi , and substitute equation (15) into equations (1)Y (10), subtracting out the reference state equations (11)Y(14) yields ð16Þ
ð4Þ
ima 1 @ @c þ (b cos ) þ ¼ 0; cos cos @ @z
ð17Þ
ð5Þ
@ @!0 (!0 cos ) þ w @ @z @ im @0 (0 cos ) þc ; ð18Þ ¼ im0 a þ b @ cos @z
im(!0 )u þ (0 b !0 v) sin þ v
ð6Þ ð7Þ
@b 1 @ @ ¼ (va ub) (wb vc); @t cos @k @z
ð9Þ
im(!0 )v þ 2(!0 u 0 a) sin ¼ im0 b @0 þ G1=2 w ¼ 0; @ p ¼ G1=2 E1 þ S1 ;
@ S S ¼ G1=2 ¼ p þ G1=2 ; @z E ima ¼
D @ u @ @ @ ¼ þ þv þw ; Dt @t cos @k @ @z a @ @ @ B=: ¼ þb þc ; cos @k @ @z Nd : R
2.2. Linearization about Unperturbed States Equations (1) Y(10) can be perturbed about a reference state containing differential rotation, toroidal field, pressure, and temperature that are in latitudinal and vertical force balance and are all functions of latitude and height z. To that end, if we assume
@ ; @
im(!0 ) þ v
ð10Þ
in which
G1=2 ¼
@0 ; 02 !02 cos sin ¼ @
imu 1 @ @w þ (v cos ) þ ¼ 0; cos cos @ @z
ð8Þ
a2 þ b2 ; 2
a reference state with differential rotation u0 ¼ !0 cos and toroidal field a0 ¼ 0 cos , then the reference state variables must satisfy the unperturbed force balance equations,
ð3Þ
@a @ @ ¼ (va ub) (wa uc); @t @ @z
¼pþ
205
@ @ ½(0 v !0 b) cos þ cos ð0 w !0 cÞ; @ @z
ð19Þ ð20Þ ð21Þ ð22Þ ð23Þ
im(!0 )b ¼ im0 v;
ð24Þ
¼ p þ a0 cos :
ð25Þ
Equations (16)Y(25) can also be solved for axisymmetric modes (m ¼ 0), provided the product m ¼ is taken to be finite. In that case, b, c ¼ 0 everywhere, because there is no longitudinal dependence in the velocity to turn magnetic field into the meridional plane. They will still have three components of motion, plus perturbations to the toroidal field. These m ¼ 0 modes have no counterpart in the 2D case, and are extremely difficult to excite in the shallow water case. Preliminary investigation shows they do exist in equations (16)Y(25) for solar-like values of G, !0 , 0 and other parameters; we save them for a later paper.
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2.3. Reduction to Reference States That Are Independent of z Equations (16)Y(25) are quite general, allowing calculation of perturbations of reference states that are functions of both latitude and height. This perturbation problem is much more general and difficult to solve than we attempt here, because it involves perturbation equations with coefficients that vary in both dimensions, requiring a large matrix inversion calculation for the eigenvalues and eigenfunctions. We save this full calculation for a later paper, and instead pursue a more limited goal: establishing a connection between perturbation solutions for the shallow water problem studied in GD02 and Dikpati et al. (2003) and the continuously stratified problem for the case of reference state differential rotation and toroidal fields that are functions of latitude only. From equation (11), since the left-hand side is independent of z, the right-hand side must be also, so @0 /@ is independent of z. If we differentiate equation (12) with respect to z, we can infer that @0 /@ is also zero. Then from equation (13), latitudinal pressure gradients in the reference state imply latitudinal temperature gradients. The condition @0 /@ ¼ 0 implies that there is no net magnetic buoyancy in the reference state. Under these conditions, equations (11)Y(14) are analogous to the shallow water system, solutions of which have been discussed extensively in Dikpati & Gilman (2001b) and Rempel & Dikpati (2003). With a reference state that varies only in latitude, equations (16)Y(25) can be quickly reduced further, to five equations for u, v, a, b, and . In doing so, we find it convenient to switch the latitudinal variable to ¼ sin , as we have done in all the previous papers on global MHD instabilities of the tachocline. Then for specified differential rotation !0 and toroidal field 0 (independent of z), the potential temperature gradient in becomes @0 S ¼ @
G1=2
@ 02 ð1 2 Þ 02 !02 : @
2
ð26Þ
Equation (26) implies that a latitudinal potential temperature gradient appears in the reference state only if the ratio of the shell thickness to the scale height is finite—a strictly compressible or ‘‘non-Boussinesq’’ effect. Equations (16) and (17) are then used to eliminate w and c, and equations (20), (21), and (25) are used to eliminate p, , and in favor of . The reduced perturbation equations, rearranged in order, are im(!0 )u im0 a 2 ð!0 v 0 bÞ @ !0 @0 im v b þ ¼ 0; þ 1 2 @
@
ð1 2 Þ1=2
ð27Þ
1=2 @ ¼ 0; im(!0 )v im0 b 2 ð0 a !0 uÞ þ 1 2 @
ð28Þ @0 @!0 im(!0 )a im0 u þ v 1 2 b 1 2 ¼ 0; @
@
ð29Þ im(!0 )b im0 v ¼ 0; ð30Þ 2 S @ S @a 1=2 @ 2 1=2 im(!0 ) G þ 1
0 @z 2 G1=2 @z G1=2 @z 1=2 @0 @v G1=2 þ 1 2 @ @z i imu @ h 2 1=2 1
; þ v ¼ 0: ð31Þ (1 2 )1=2 @
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Equations (27)Y(30) are extremely similar in form to the corresponding equations in the shallow water system (GD02), except that now all the variables are functions of z. By contrast, equation (31) is rather different from the equation for prediction of shell thickness variation in the shallow water system. This is because the formulation here contains thermodynamics explicitly. To carry out the final separation of variables, z from , we find it convenient, for applying boundary conditions at the bottom and top of the shell, to represent each variable u in the form u ¼ uc ( ) cos (nz) þ us ( ) sin (nz);
0 z 1;
ð32Þ
where uc and us are both complex. The parameter n is then a vertical ‘‘wavenumber.’’ There are uses for both integer and halfinteger values of n, for satisfying different boundary conditions. If we apply no boundary conditions in z, we can treat n as varying continuously, leading to solutions that are periodic in z with an infinite range of vertical wavelengths. From substitution of equation (32) for all variables u, v, a, b, and into equations (27)Y(31), collecting terms in sin (nz) and cos (nz), and setting them separately equal to zero in each equation, we end up with a total of 10 equations for 10 complex variables that are functions of latitude. The five equations from coefficients of cos (nz) are im(!0 )uc im0 ac 2 ð!0 vc 0 bc Þ @ !0 @0 im 2 vc bc þ c ¼ 0; þ 1
@
@
ð1 2 Þ1=2 ð33Þ 1=2 @c ¼ 0; im(!0 )vc im0 bc 2 (0 ac !0 uc ) þ 1 2 @
ð34Þ @0 @!0 im(!0 )ac im0 uc þ vc 1 2 bc 1 2 ¼ 0; @
@
ð35Þ im(!0 )bc im0 vc ¼ 0;
ð36Þ
S (n)s G1=2 1=2 S 1=2 1 2 0 (n)as G 1=2 @0 þ 1 2 (n)vs G1=2 @
( ) i im @ h 1=2 1 2 ; uc þ vc ¼ 0; @
ð1 2 Þ1=2
im(!0 ) G1=2 (n) 2 c þ
ð37Þ and the five from coefficients of sin (nz) are im(!0 )us im0 as 2 (!0 vs 0 bs ) @ !0 @0 im vs bs þ s ¼ 0; þ 1 2 @
@
ð1 2 Þ1=2 ð38Þ 1=2 @s ¼ 0; im(!0 )vs im0 bs 2 (0 as !0 us )þ 1 2 @
ð39Þ
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@0 @ !0 bs 1 2 ¼ 0; im(!0 )as im0 us þ vs 1 2 @
@
ð40Þ im(!0 )bs im0 vs ¼ 0;
ð41Þ
S (n)c G1=2 1=2 S 1=2 1 2 0 (n)ac G 1=2 @0 (n)vc þ G1=2 þ 1 2 @
( ) i im @ h 1=2 1 2 ; us þ vs ¼ 0: @
ð1 2 Þ1=2
im(!0 ) G1=2 (n) 2 s þ
ð42Þ The c- and s-subscripted variables separate completely in equations (33)Y(42) if we take the ‘‘Boussinesq’’ limit of letting S ! 0 in equations (37) and (42) (discussed in MGD07), leading to (n) 2 im c; s þ uc; s G ð1 2 Þ1=2 1=2 i @ h 1 2 þ vc; s ¼ 0: @
im(!0 )
ð43Þ
Equation (43) is mathematically identical to equation (9) of GD02, with h0 ¼ 0 there and with the variable transformation s;c ¼ Ghs; c /(n) 2 . This property implies that our instability results for different n as functions of G should be ‘‘self-similar,’’ simply displaced toward higher G with increasing n. Therefore in this limit equations (33)Y(37) and (38)Y(42) are separately identical to the shallow water system, equations (7)Y(11) of GD02, if we set the reference state departure from constant shell thickness equal to zero there. The 10 equations (33)Y(42) can be further separated into twenty equations for the real and imaginary parts of each equation, which is what we actually solve. We omit those equations here. A restricted version of the equations above was presented in Gilman et al. (2004).
and growth rates. For many reference states, modes of only one symmetry are unstable at all. 2.4.2. Conditions at the Poles
The poles do not represent physical boundaries, but because of the spherical coordinate system we use, care must be taken to avoid introducing singularities or multiple values of variables there. What conditions we must apply depend on the longitudinal wavenumber m of the mode: For m ¼ 0: u, v, a ¼ 0 at poles; w, , p, , all can be nonzero (recall that b, c ¼ 0 everywhere when m ¼ 0). For m ¼ 1: u, a, w, c, , p, , ¼ 0 at poles; v, b can be nonzero, to allow flow and field to cross the poles. For m > 1, all variables must vanish at the poles to avoid multiple values there. 2.4.3. Boundary Conditions in z
It is possible to solve equations (33)Y(42) for unstable modes with longitudinal wavenumber m, longitudinal phase velocity r and growth rate mi as a function of vertical wavenumber n, without actually applying any boundary conditions in z. In that case the solutions are periodic in z in the range 1 < z < þ1, with the period determined by n. This is physically unrealistic in a global sense, since the sphere has finite dimensions, but it does have meaning in a local sense in z. Alternatively, we can specify conditions at fixed z, denoting the bottom and top of a spherical shell of finite thickness. Alternative choices for such boundary conditions are as follows: 1. Rigid and perfectly conducting top (placed at z ¼ 1) and bottom (z ¼ 0), requiring w, c ¼ 0 there. This condition can be fulfilled by setting wc, cc ¼ 0 and taking n to be an integer. 2. Deformable top and bottom, with the total perturbation pressure vanishing there. This is achieved by setting c ¼ 0 with n an integer. Application of the above boundary conditions can have profound effects on the equations we actually solve. Boundary conditions 1 are on variables already eliminated from equations (33)Y (42). We can relate them to variables we are solving for by substituting the forms defined in equation (32) into equations (16) and (17) and separating terms with sines from those with cosines. This results in the following constraints on the variables u, v, a, and b: 1=2 i @ h 1 2 vs ¼ 0; @
ð1 1=2 i im @ h 1 2 as þ bs ¼ 0: 1=2 @
ð1 2 Þ im
2.4. Symmetry and Boundary Conditions 2.4.1. Separable Symmetries of Solutions
As in the 2D and shallow water linear instability calculations referred to above, all solutions to equations (33)Y(42) separate into two sets of variables with opposite symmetries about the equator, for reference state differential rotation and potential temperatures that are symmetric about the equator and toroidal field that is antisymmetric. These reference state symmetries are the dominant ones on the Sun, so we focus on these. In terms of the original variables listed in expression (15), these symmetry pairings are u; w; b; ; p; ; u; w; b; ; p; ;
symmetric; v; a; c antisymmetric; antisymmetric; v; a; c symmetric:
In agreement with our convention in prior papers, we call the first set the antisymmetric modes and the second set the symmetric modes. For a given longitudinal wavenumber m, modes of these two symmetries generally have distinct phase velocities
207
2 Þ1=2
us þ
ð44aÞ ð44bÞ
With substitution and rearrangement of terms, it is possible to show that if equation (44a) is imposed, equation (44b) will also be true. With boundary condition 2, the effect of applying the above boundary conditions reduces the number of equations we actually solve. In particular, if c is zero, it follows from equations (38)Y (41) that uc, vc, ac, and bc are also all zero at all latitudes. This is because under these assumptions, all these equations are algebraic rather than differential, with coefficients that vary with latitude. Then the remaining equations to be solved are reduced to equations (37)Y(42). This system appears to be overdetermined, because we must solve six equations for five unknown variables. But we are actually solving five eigenvalue equations (two ODE’s and three algebraic equations), plus one additional equation (containing no
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Fig. 1.—Growth rates (mci ; left column) and longitudinal phase velocities (cr ; right column) for unstable modes of longitudinal wavenumber m ¼ 1 perturbing bands of toroidal field centered at 30 latitude, plotted as functions of the effective gravity parameter G, for selected peak toroidal fields (a ¼ 0:2, 0.5, 0.8). Modes symmetric about the equator are denoted by heavy solid curves, antisymmetric modes by various dashed curves. In panels a, b, and d, we use multiple dashed line styles to track curves for different n that are not continuous. Curves are shown for vertical wavenumbers n ¼ 1, 3, and 10. The corresponding results for the previously studied shallow water (SW ) mode are also shown. Shaded domains are for G values representative of the radiative and overshoot parts of the tachocline.
eigenvalue) that defines a constraint on the total pressure perturbation and the horizontal magnetic field components. Then for boundary condition 2, each variable is represented by only a sine. In compact form, these constraint equations can be obtained from equations (37) and (42), with substitution for the potential temperature gradient from equation (26), in the form h
1=2 0 as;c S im0 s;c 1 2
þ 1
2 1=2
02
!02
i
@ 02 ð1 2 Þ
bs;c ¼ 0: @
2 ð45Þ
In this form, this constraint can only be used when S is not zero.
For all S , including zero, boundary condition 2 leads to reduced forms of equations (37) and (42), in the compact form shown in equation (43). 3. SCANNING THE PARAMETER SPACE Equations (33)Y(43) can yield a very large number of solutions, since there are so many parameters to specify. Therefore we must be judicious in our choice of parameters in order to find the range of unstable modes, while not being overwhelmed with detail. The reference state we are perturbing is defined by the profiles of differential rotation and toroidal field, from which the thermodynamic structure follows from equations (11)Y(14), once we have specified also the effective gravity G of the stratification, and S , the ratio of the shell thickness to the pressure scale height. The parameter E is needed only if we wish to calculate the reference state density.
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Fig. 2.—Same as Fig. 1, but for toroidal bands centered at 45 latitude.
Once the reference state is defined, then we solve the perturbation equations (33)Y(43) to look for the growth rates and longitudinal phase velocities of unstable modes of specified symmetry about the equator, longitudinal wavenumber m, vertical ‘‘wavenumber’’ n, for a particular boundary condition choice. We also construct the disturbance planforms for typical cases. To limit the problem further, we confine the differential rotation to the form !0 ¼ r s 2 , and choose s ¼ 0:18, a typical tachocline value. Normalizing with respect to the core rotation rate yields r ¼ 0:052 in the rotating frame. As in previous papers cited above, we consider both banded and broad toroidal field profiles, the former of the form 0 ¼ ap(e ( 0 ) e ( þ 0 ) ), where determines the width of the band and 0 the location of the band; p is chosen to ensure max ½0 (1 2 )1/2 ¼ a/2 (peak field strength independent of latitude). a ¼ 1 corresponds to a
peak field strength of 105 G. We limit our study to bands with full widths at half maximum of 10 latitude. For the broad profile, we use 0 ¼ a , which yields a peak toroidal field at 45 . 4. RESULTS 4.1. Banded Toroidal Field Profiles 4.1.1. Growth Rates and Phase Velocities
Figures 1, 2, and 3 show growth rates (left-hand columns) and longitudinal phase velocities (right-hand columns) for unstable modes for toroidal band placements at 30 , 45 , and 60 , respectively, for peak toroidal field parameter a ¼ 0:2, 0.5, and 0.8 (corresponding to dimensional toroidal fields at tachocline depth of 20, 50, and 80 kG). Figures 1Y3 are for the stratification parameter S ¼ 0, and no boundary conditions are applied to the top or
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Fig. 3.—Same as Fig. 1, but for toroidal bands centered at 60 latitude.
bottom. Unstable modes symmetric about the equator in all cases are denoted by solid curves, and antisymmetric modes by various broken curves. Several features of these results immediately stand out. We see that in all cases the phase velocity and growth rate curves are selfsimilar in n. The higher the n, the more the curves move toward higher G, meaning generally that modes with high n are excited only when the stratification is more subadiabatic. As pointed out near the end of x 2.3, this self-similar behavior follows immediately from the fact that the vertical wavenumber n appears in the equations solved only in combination with the effective gravity G, in the form G/(n 2 ), thus ‘‘reducing’’ the effective gravity as n increases. Since any unstable mode with vertical motion has to do work against this stratification to be unstable, this result implies that the larger the n, the less vertical motion relative to horizontal motion there is in the disturbance. This follows immediately from the mass continuity equation. More nodes in the vertical, from
higher n, means less vertical motion for the same amplitude horizontal motion and the same horizontal wavenumber and latitudinal disturbance structure. This property is guaranteed by the fact that the n dependence separates out from the longitudinal and latitudinal dependence in the equations we have solved. In MGD07, we demonstrate that the vertical motion in the dominant modes is even more horizontal than required by mass continuity, as G is increased. Most of the flow is in the form of vorticies with radial axes, with very little horizontal divergence to which all the vertical motion is tied. There is an n ¼ 1 structure in the vertical, but the flow is almost entirely horizontal. The practical effect of this result is that in the overshoot part of the tachocline (G ¼ 0:1) only modes with low n are excited, while in the radiative part of the tachocline, modes up to much higher n are excited and have nearly the same growth rates. The low G modes will have more vertical velocity in them, allowed by the smaller subadiabaticity. The near independence of the growth
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Fig. 4.—Growth rates and phase velocities for unstable modes for toroidal field peak a ¼ 0:2, band at 30 , m ¼ 1 and 2 ( both symmetries), for the ratio of the shell thickness to pressure scale height, S ¼ 0 and 0.5. In all cases the lower growth rate occurs with S ¼ 0:5.
rate from n for many modes at high G is of course a result of having no diffusion in the problem. With diffusion added, the diffusion terms due to n-dependence go like n 2 , so an n ¼ 10 mode would have 100 times more diffusion acting on it than would an n ¼ 1 mode, most likely significantly reducing its growth rate compared to the nondiffusive case. For comparison purposes, we plot in Figures 1Y3 the growth rates and phase velocities for the ‘‘shallow water’’ modes of Dikpati et al. (2003). These modes by definition have no vertical structure in their horizontal velocities, but do have some vertical velocity, associated with the deformation of the top boundary. We see that, in and near the high-G limit, the growth rates and phase velocities for shallow water modes and for modes in our present calculation are the same. This is again evidence that in this limit the modes found here have very little vertical velocity in them, which would be true in the shallow water case, too, because at high G the deformation of the top boundary is very small due to the high subadiabaticity. By contrast, at low G the preferred modes with vertical structure found here acquire growth rates much higher than those of shallow water modes as the toroidal field strength is increased from 0.2 to 0.5 to 0.8, reaching e-folding growth times as short as a few days for a ¼ 0:8. Similar behavior was found by Cally (2003). These high growth rate modes are another manifestation
of a kink instability for polar magnetic loops first identified by Tayler (1973; see also Spruit 1999), and occur even without differential rotation or even rotation. Both rotation and subadiabatic stratification tend to stabilize modes. In these modes the vertical motion and vertical structure must be playing a very important role in the instability. But as stated in the introduction, we know from dynamo studies with hydromagnetic feedbacks (e.g., Rempel 2006) that achieving such high-amplitude toroidal fields in the tachocline may be very difficult. At these low G values Dikpati & Gilman (2001a), Gilman & Dikpati (2002), and Dikpati et al. (2003) showed that both hydrodynamic and MHD instabilities disappear because of the shrinkage of the shell thickness that is required by the force balance of the unperturbed state, an effect that has no real counterpart in the continuously stratified case we are considering here. Another prominent effect seen with increasing toroidal field is the increasingly negative phase velocities of unstable modes, as measured relative to the rotation rate of the solar interior. No such effect is seen in shallow water modes. The effect is more pronounced for toroidal bands at higher latitudes. By a ¼ 0:8 for a band at 60 , the phase velocity at low G of about 0.6 means that such modes would be propagating in longitude at only about half the rate of the high-latitude rotation in the tachocline (or at the solar surface). Such low pattern rotation rates have to our knowledge
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Fig. 5.—Growth rates and phase velocities of unstable modes for various boundary conditions at top and bottom, plotted as functions of S , for toroidal bands at 30 latitude with a ¼ 0:2 and 0.8.
never been seen in high-latitude magnetic patterns in the photosphere, but since we cannot observe patterns in the tachocline directly, we cannot be sure such low speeds are not seen in the high-latitude tachocline, although they do require very high amplitude toroidal fields there. What must be happening in these modes is that the perturbation j < B forces are large enough to swamp the effects of Coriolis forces and longitudinal advection by the differential rotation, which is consistent with the extremely high growth rates, much shorter than the timescales associated with rotational effects near the poles. But given the high toroidal fields required, it is not clear that such modes would actually be found in the Sun. This phase velocity behavior was also reported in Cally (2003). All of the above results are for the case of S ¼ 0, which from equation (26) means that in the unperturbed reference state, there is no latitudinal potential temperature gradient. Here S > 0 means we have such a gradient, and that in the thermodynamics we are including the fact that a tachocline of finite thickness spans some fraction of a pressure scale height. What happens to the instability when these effects are included? Figure 4 gives an example. Here we compare the growth rates and phase velocities for un-
stable modes with m ¼ 1 and 2 (both symmetries) as a function of G, for S ¼ 0 and 0.5 (half a pressure scale height within the tachocline, a large value). Again, no boundary conditions are applied at the top or bottom. These results are for easily realized solar values, namely a band at 30 latitude, and a peak toroidal field of 0.2 (20 kG). We see that for high G the differences between these two cases are too small to see in the plot, while for low G, they are discernible, but modest. In all cases, the growth rates for S ¼ 0:5 are a little smaller. Figure 5 amplifies on the low G result, showing growth rates and phase velocities as functions of S for G ¼ 0:1, for all the boundary conditions we have considered, and including both a ¼ 0:2 and 0.8. We see that the least realistic assumption, namely no boundary condition, gives the largest variation with S . With either rigid top and bottom, or deformable top and bottom, the variations are much smaller. Note particularly that the scale of the phase velocity has been expanded to show the variation, which in all cases is less than 1% of the interior rotation. We conclude from Figures 4 and 5 that the instability of this system is much less sensitive to changes in S that it is to changes in G, a, or the latitude of band placement. This result is confirmed by the nonlinear simulations reported in MGD07.
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Fig. 6.—Growth rates and phase velocities of unstable modes plotted as functions of the latitude of band placement, for overshoot (G ¼ 0:1) and radiative (G ¼ 104 ) tachocline subadiabatic stratifications for toroidal band peak amplitude a ¼ 0:2 (20 kG). Rigid boundary applied at the top and bottom.
We have sampled the instability for band placement at 30 , 45 , and 60 latitude. Figures 6 and 7 show growth rates and phase velocities as functions of the latitude of band placement for all n ¼ 1 unstable modes for the overshoot (G ¼ 0:1) and the radiative (G ¼ 104 ) parts of the tachocline, for peak toroidal fields of 0.2 (Fig. 6) and 0.8 (Fig. 7). Both are for rigid top and bottom. Results with no boundary condition applied at the top and bottom for a ¼ 0:2 are very similar to Figure 6, and so are not shown. From Figure 6 we see that, for a ¼ 0:2, the m ¼ 2 mode is unstable in a range of midlatitudes, from about 22 to about 55 in the G ¼ 0:1 case, and from about 27 Y 43 in the G ¼ 104 case. Thus, the m ¼ 2 mode is confined between latitudes where a strong toroidal field might be expected to occur in the Sun somewhat before the start of a new sunspot cycle and latitudes of the rising phase of such a cycle. By contrast, the m ¼ 1 mode can occur at all latitudes above about 15 in both the G ¼ 0:1 and 104 cases. Thus, the ‘‘tipping’’ instability of the toroidal field should be seen first, and probably be more prominent, given its larger growth rate. We note that the m ¼ 1 mode is always dominant in the nonlinear simulations shown in MGD07. For G ¼ 0:1, the growth rate and phase velocity curves coincide for toroidal bands poleward of about 30 . This means that above this latitude, the disturbances do not ‘‘feel’’ the opposite hemisphere. By contrast, with G ¼ 104 , which symmetry is more unstable, or unstable at all, depends on the latitude of band placement. This is consistent with the hydrodynamic disturbances being global and in contact with the opposite hemisphere. We show evidence of this global property in the disturbance planform plots given below in x 4.1.2, particularly Figure 12 below. We see also in Figure 6 the small beginnings of higher growth rate modes near the pole when there is low effective gravity, when the toroidal band is located poleward of 70 latitude.
In all cases, the longitudinal phase velocities of the unstable modes are quite close to the local tachocline rotation rate of the band, denoted by the solid black line in Figures 6b and 6d, which is what we should expect. The presence of high-latitude high growth rate modes at low G is much more prominent in Figure 7, where we show results similar to those in Figure 6, but for a peak toroidal field amplitude a ¼ 0:8. These high growth rate modes occur for all toroidal bands poleward of about 50 . As in other cases already studied, these modes also have very low longitudinal phase speeds relative to the rotating frame. By contrast, the growth rates for modes with the same a remain small for high-latitude toroidal bands when G ¼ 104 , characteristic of the radiative tachocline. Their phase velocities also remain close to the local rotation rate. As also seen in Figure 6 for a ¼ 0:2, at low G, in Figure 7, modes of both symmetries about the equator show the same growth rate and phase velocities poleward of about 30 , which is evidence that they do not feel the effects of the other hemisphere. Similarly, at high G, which symmetry of mode is more unstable, or unstable at all, varies with the latitude of band placement. Finally, we see that for a ¼ 0:8 only the m ¼ 1 mode is unstable, consistent with earlier studies that show that for these peak fields, the band is simply too rigid to deform. The m ¼ 1 mode has no deformation, just tipping. 4.1.2. Horizontal Structure of Unstable Modes
We focus first and primarily on mode structures for both the radiative (G ¼ 104 ) and the overshoot (G ¼ 0:1) parts of the tachocline, for the case of toroidal bands placed at 30 latitude, perhaps most interesting in terms of the sunspot cycle. We choose a ¼ 0:2 ( plausible amplitude from dynamo theory) for which both m ¼ 1 and 2 modes are unstable, and rigid top and bottom
Fig. 7.—Same as Fig. 6, but for a ¼ 0:8.
Fig. 8.—Horizontal perturbation velocities (arrows in left column) magnetic fields (arrows in right column) and total pressure perturbations (color contours in all figures — red, high pressure; blue, low pressure) for the m ¼ 1 symmetric mode for toroidal bands at 30 latitude, for the radiative (G ¼ 104 ) and overshoot (G ¼ 0:1) tachocline, with rigid top and bottom boundary conditions and peak toroidal field a ¼ 0:2.
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Fig. 9.—Same as Fig. 8, but for the antisymmetric mode.
Fig. 10.—Same as Fig. 8, but for the m ¼ 2 antisymmetric mode.
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Fig. 11.—Perturbation horizontal velocities, magnetic fields, and pressures for m ¼ 1 antisymmetric unstable modes, for toroidal bands placed at 75 , 60 , 45 , and 30 latitude. Rigid top and bottom boundaries, overshoot (G ¼ 0:1) tachocline stratification. Peak field is a ¼ 0:2 here.
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Fig. 12.—Same as Fig. 11, but for m ¼ 1 antisymmetric mode for radiative (G ¼ 104 ) tachocline stratification.
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boundary conditions, most easily realized in nonlinear simulations. All the unstable mode structures for this case are shown in Figures 8Y10 (respectively for modes m ¼ 1, S; 1, A; 2, A). The disturbance planforms for the no boundary condition case are generally very similar, particularly in the neighborhood of the band. In each figure, the perturbation velocities are shown on the left, perturbation magnetic fields on the right, with the contours of total pressure perturbation overlaid on both. We see in Figure 8 that both radiative and overshoot tachocline disturbances are organized around the location of the toroidal band. The MHD perturbations are confined to this neighborhood. But in the G ¼ 0:1 case, there is also a very high latitude disturbance, which shows up principally in the pressure field. This disturbance must be purely hydrodynamic, since there is no significant magnetic field there in the unperturbed state. We have not studied this instability in detail, but it could play some role in the production of kinetic helicity at high latitudes that might contribute to the workings of the solar dynamo. The disturbance structure in the neighborhood of the toroidal band shows that the flow is nearly magnetoheliostrophic; that is, the flow is clockwise around the high total pressure (red contours) and counterclockwise around the low pressure (blue contours). The total perturbation pressure gradient and the perturbation Coriolis force must be nearly in balance for this to occur. These patterns occur separately on both flanks of the Gaussian profiled toroidal band. The peaks of the highs and lows occur at singular points in the system, where the Doppler shifted phase speed equals the local Alfve´n speed associated with the toroidal field. For banded profiles, there are always two of these points in each hemisphere, since the longitudinal phase speed of the unstable mode is close to the local rotation of the latitude of the peak of the toroidal band. The pattern of perturbation magnetic fields shown indicates that the total field (reference state plus perturbation of, say, amplitude 30% of the undisturbed toroidal field) will have a highpressure point on its equatorward edge at longitudes where the total field is displaced toward the pole, and low pressure on that edge when the total field is displaced toward the equator. In both cases, this means that the local latitudinal total pressure gradient and the local magnetic curvature stress oppose each other. The balance is nearly completed by the perturbation Coriolis force associated with the pattern of velocities shown in the left column. The shallow water system analyzed in Dikpati et al. (2003) shows very similar structure with the perturbation shell thickness playing the role of the pressure perturbation. A substantial difference between the high and low G cases seen in Figure 8 is the latitude extent of the disturbance both poleward and equatorward of the toroidal band—it is much larger in the G ¼ 10 4 case. The reason for this is that with the more subadiabatic stratification, the perturbations are inhibited from extending in the vertical, and so they spread more in latitude. This effect is even more pronounced in Figure 9, for the m ¼ 1, A mode. Figure 9 also shows clearly the effect of the opposite symmetry conditions at the equator, with a peak horizontal velocity there instead of a node, and, for the G ¼ 104 case, a closed heliostrophic circulation about the highs and lows equatorward of the toroidal band. In the neighborhood of the band, the flow and magnetic patterns show the same magnetoheliostrophic relation to the perturbation total pressure field as in Figure 8 for the symmetric mode. Figure 10 displays the m ¼ 2 antisymmetric modes. All of the properties described for m ¼ 1 modes are found here, too, but for disturbances with half the wavelength in longitude. The m ¼ 2, S mode is not shown, because for G ¼ 104, it does not exist and
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Fig. 13.—Latitude of Doppler, Alfve´nic, and stratification singular points for disturbances on a toroidal band that peaks at 45 latitude, as a function of toroidal field strength parameter a, for G ¼ 0:1 (a), 1.0 (b), and 10.0 (c).
for G ¼ 0:1, it looks similar to Figure 8, but with half the wavelength in longitudes and without the high-latitude hydrodynamic disturbance. The changes in mode structure with latitude placement of the toroidal bands is shown in Figure 11 for m ¼ 1 modes in the G ¼ 0:1 case, and Figure 12 for the G ¼ 104 case. Figures 11 and 12 show that for all latitude placements, the disturbances remain relatively local in latitude in the neighborhood of the toroidal
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Fig. 14.—Disturbance planforms for G ¼ 0:1 for toroidal bands of various peak strengths that peak at 45 latitude. Top panel is for a ¼ 0:2, middle panel for a ¼ 0:7, bottom panel for a ¼ 1:0.
band. The influence of the equatorial symmetry condition fades with higher latitude placement, such that even for 45 placement the mode structures for the two symmetries differ by little more than their (arbitrary) phase in longitude. For each latitude placement, all the other relationships among perturbation velocities, magnetic fields, and total pressure seen in Figures 8Y10 for modes with 30 band placement hold here, too. The strongest contrast occurs between low and high G modes. In Figure 12, we see that even when the toroidal band is placed at 75 latitude, there is a strong velocity disturbance that reaches all the way to the equator, confirming that the latitudinal transmission of the disturbance is much stronger when the stratification prevents vertical motion that can rearrange the pressure to provide latitude pressure gradients to limit the latitude extent of the velocity fields. Thus, if this global MHD instability occurs with banded toroidal fields in the radiative tachocline, north and south hemispheres will communicate with each other even if the toroidal bands are at extremely high latitudes. By contrast, instability of toroidal bands at high latitudes in the overshoot layer may occur independently in north and south hemispheres, with rearrangement of the thermodynamic structure with latitude leading to
latitudinal forces in the neighborhood of the bands that prevents communication toward and across the equator. We have checked the effects of increasing the stratification parameter S from zero to a high of 0.33 (1/3 of a pressure scale height in the domain for thermodynamic effects) on the mode structure that the velocity and magnetic perturbations in the neighborhood of the toroidal band is virtually unaffected by increasing S . The growth rates change some, but this is not reflected in the perturbation structure. There are differences in the perturbation pressure seen in the very high-latitude hydrodynamic disturbance; we do not pursue these differences further here. Thus, the 3D MHD instability of this system is virtually unaffected by the thickness of the tachocline shell relative to a pressure scale height, at least within the range of parameters we studied. 4.1.3. Role of Singular Points in Instability
As pointed out by many authors (see Lindzen 1988, and references therein), the presence of singular points in a shear instability problem can often give us insight about the properties of the instability and the disturbance structures. Dikpati et al. (2003) discuss the role of singular points of the instability equations in
Fig. 15.—Growth rates and phase velocities for unstable modes of m ¼ 1 as functions of effective gravity G, for broad toroidal field profiles with peak field a ¼ 0:2, 0.5, and 0.8 for vertical wavenumbers n ¼ 1, 3, and 10. Shallow water (SW ) mode results also plotted for comparison.
INSTABILITIES IN A 3D TACHOCLINE determining the unstable mode structure in the shallow water system for banded toroidal fields. Because of the close relation of the equations we have solved here to the shallow water system, discussed above in x 2.3, we have been able to find similar behavior in the 3D case, particularly when S ¼ 0. As defined in Gilman & Dikpati (2002), equations (15), these singular points occur where combinations of two or more speeds in the problem cancel each other out. Thus, there is a singular point at the zeros of !0 c, or where the Doppler shifted phase speed of a mode is zero; there are up to two where the absolute value of the Doppler shifted phase speed equals the local Alfve´n speed, or at the zeros of (!0 c) 2 02 ; and one or two at the zeros of (1 2 )½(!0 c) 2 02 G, or where the local gravity wave speed cancels a latitude-weighted difference between the Doppler-shifted phase velocity and the local Alfve´n speed. Here we will call these singular points respectively Doppler, Alfve´nic, and stratification singular points. Since the modes we are studying are unstable, and therefore c is complex, these are not true singular points, but rather points where their real parts change sign. The smaller is their growth rate, the more nearly singular they are, as the true singular point moves closer and closer to the real axis. The closer to the real axis, the more profound their influence locally on the mode structure, since they represent a place where the phase of the mode with longitude changes rapidly with latitude, which is what is needed generally to produce mode structures with ‘‘stresses’’ that extract energy from the unperturbed state. In most cases we have studied, the toroidal field amplitudes are large enough that the Alfve´nic and stratification singular points have the most influence on the mode structure. In addition, the stratification singular point comes into play only for low G. At high G, there are no zeros of this function in the physical domain. Figure 13 displays an example of the location of the various singular points for unstable modes acting on a toroidal band at 45 latitude, for G ¼ 0:1, 1.0, 10.0, which illustrates the evolution from low to high G. At all toroidal fields, the Alfve´nic singular points are prominent, one each on the equatorward and poleward sides of the toroidal field peak. In addition, for G ¼ 0:1, the stratification singular point is also present, and it migrates from the equator to merge with the equatorward Alfve´nic singular point by a ¼ 0:8. By contrast, for higher G, this singular point is always absent. The Doppler singular point is always present, falling in between the Alfve´nic points when the toroidal field is below a certain value, but migrating toward the pole rapidly with increasing a. This is due to the increasingly negative mode phase velocity we have found for the modes with higher growth rates. Figure 14 shows disturbance planforms for three cases, with a ¼ 0:2, 0.7, and 1.0, and with G ¼ 0:1. We can see that the disturbances are in all cases confined to the neighborhood of the toroidal band, but the differences are profound. For low a, the flow is organized around the two Alfve´nic singular points, with clockwise flow around each high-pressure spot (centered on the singular point latitudes) and counterclockwise flow around each spot of low pressure (also centered on the singular point latitudes). Here the horizontal flow has relatively little divergence or convergence. For this a value the perturbation fields are not organized in this way, but instead are displaced from the velocity patterns in longitude, pointing from low to high pressure on the poleward side, and from high to low pressure on the equatorward side. At this low a, from Figure 13 the stratification singular point is still very near the equator where the disturbance amplitudes are very small, and so it has very little influence on the dominant mode structure near the toroidal band.
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Fig. 16.—Same as Fig. 15 except for peak toroidal field parameter a ¼ 0:95, 1.00, and 1.05. (Set of curves for n ¼ 3 have been omitted for clarity.)
But by a ¼ 1:0 this singular point has migrated poleward and merged with the equatorward Alfve´nic point, in effect creating a double singular point at about 40 latitude. This leads to the mode structure seen in the bottom frame of Figure 14, which is very asymmetric about the axis of the toroidal ring. The dominant pressure patterns are on the equatorward side, and the perturbation magnetic fields tend to follow the total pressure contours, with now counterclockwise orientation about the highs, and clockwise around the lows. This represents a near force balance between the total pressure perturbation (dominated at high a by the perturbation magnetic pressure) and perturbation magnetic curvature terms, mathematically like Coriolis forces, but with opposite sign, hence the opposite orientation. By contrast the perturbation velocities show little tendency to follow the pressure contours, and have strong divergence from or convergence to the longitudes of small pressure perturbations. These patterns also imply that there is significant vertical motion in these disturbances. The middle frame in Figure 14 displays an intermediate case at a ¼ 0:7, which shows both low a and high a features. From Figure 13, the stratification singular point is within 10 latitude of the equatorward Alfve´nic point, so its influence is being felt, but is not dominating. What appears to be happening here as a is increased is that we are seeing a reduced form of the polar kink or Tayler instability, which is confined to the neighborhood of the toroidal band. The higher the latitude of the band, the higher its growth rate, and the more like the polar kink instability it becomes. As with the polar kink modes, the phase velocity relative to the rotating frame becomes very negative, moving the Doppler singular point out of
Fig. 17.—Same as Fig. 15, except for various values of S , and with no top and bottom boundary conditions applied.
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the domain entirely, so that the instability and its mode structure is entirely of MHD origin. This transition occurs by a ¼ 0:85, from Figure 13a. 4.2. Broad Toroidal Field Profiles 4.2.1. Growth Rates and Phase Velocities
Figures 15Y18 display disturbance growth rates and phase velocities for instability of broad toroidal field profiles. For such profiles, only m ¼ 1 modes are unstable. Figure 15 shows both these quantities as functions of G, for toroidal field parameters a ¼ 0:2, 0.5, 0.8, rigid top and bottom, and S ¼ 0. For each a, only one mode symmetry is unstable, symmetric when a ¼ 0:2, antisymmetric when a ¼ 0:5 and 0.8. This antisymmetric mode leads to the ‘‘clamshell’’ mode of instability first discussed in Cally (2001) and seen in the nonlinear simulations in MGD07. We see in Figure 15 the same self-similar behavior of growth rates and phase velocity with vertical wavenumber n. We also see that the shallow water mode instability always extends to a lower G, and as a is increased, even n ¼ 1 modes are not unstable at all for overshoot G values. Thus, we should expect the Sun to pick the shallow water mode (allowing deformation of the top boundary of the shell) over modes with more vertical structure, even in the ideal MHD case. But absent at low G are the much higher growth rate modes we saw with no boundary conditions in the banded profile case. We see again that for high G, the modes with vertical structure and the shallow water modes asymptote to the same growth rates and phase velocities. We also see that for low G at all a, the phase velocities of the modes of given n acquire a very slow phase velocity relative to their shallow water counterparts, the amount of this slowness increasing with a. Unlike the results for banded toroidal fields described above, this occurs even though the growth rates remain relatively small. The instability properties change radically if we increase the toroidal field parameter a still further. Above a ¼ 0:94, the high growth rate modes at low G suddenly appear again. We see this in Figure 16, which displays growth rates and phase velocities as functions of G for different n and a ¼ 0:95, 1.0, and 1.05. Here, the higher the n, the higher the G at which the high growth rate modes appear. Clearly in these cases the vertical velocity plays an important role in the instability. When the growth rate levels out at e-folding times of several days, rather than the 1 year times at high G, the phase velocities have dropped to near 1 in the rotating frame, or zero in an inertial frame. So these high growth rate modes behave as if the system is not rotating at all. As in the banded toroidal field case, these are forms of Tayler instability (Tayler 1973). The high toroidal field amplitude is needed to overcome rotational effects. But choosing high vertical wavenumber n negates the stabilizing effect of the subadiabatic stratification. Figures 17 and 18 demonstrate that whether we apply boundary conditions at top and bottom of the shell or not, and whether we assume S ¼ 0 or not, makes a large difference in the occurrence or absence of high growth rate modes at low G. Figure 17 shows results when we apply no top or bottom boundary conditions, with n ¼ 1, and we increase S from zero. For S ¼ 0, the no boundary condition case differs from the rigid top and bottom case only by allowing low growth rate instability for somewhat lower G, making it somewhat more competitive with the shallow water mode. At high G, the asymptotes for rigid top and bottom and no boundary condition cases are essentially the same. But when S is increased, the instability at low G reappears, the more so for higher a. The phase velocities of these unstable modes are
Fig. 18.—Same as Fig. 15, except for n ¼ 1 only, for various S values between 0 and 0.5.
increasingly negative in the rotating frame, but almost independent of S . Finally, in Figure 18, we show that with rigid top and bottom, the growth rates depend only slightly on S for all a, this dependence being discernible on the plot only for intermediate G values. Consistent with the results in Figure 16, only the a ¼ 1 case shows high growth rates modes at low G. Note also that the phase velocities of low G unstable modes drop sharply as a is increased from 0.8 to 1.0. To interpret physically the results shown in Figures 15Y18, we need to examine the structure of unstable modes, including the role of singular points of the equations. 4.2.2. Disturbance Structures and Role of Singular Points
The principal differences in disturbance planforms for broad toroidal field profiles occur between cases with relatively high peak toroidal field together with low effective gravity G, and cases with high G together with low toroidal field strength. Figure 19 shows an example of these differences. The top panels are for a ¼ 1 and G ¼ 0:1, 1, and 10 respectively, the bottom panels for a ¼ 0:2 and G ¼ 10. Rigid top and bottom boundary conditions have been applied, and S ¼ 0. The patterns shown change hardly at all for the no boundary condition case, or for S > 0. The G ¼ 10 cases show a pattern very similar to that seen in Dikpati et al. (2003) for the shallow water case, with most pronounced perturbations in the neighborhood of the singular point where the Doppler-shifted phase velocity equals the local Alfve´n speed; with a ¼ 0:2 this occurs at between 15 and 20 latitude, with a ¼ 1 at 5 latitude. We can see in these cases that the disturbance in both the magnetic and velocity fields is global (unlike the perturbation fields in the banded toroidal field case), with both flow and field tending to follow the pressure contours.
Fig. 19.—Sample disturbance planforms for cases with high a and several G values, and low a and high G, for a broad toroidal field.
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Fig. 21.—Energetics for a nonlinear simulation (see text). The volume-integrated magnetic energy contained in the mean toroidal field ( TFME; solid line) and in the non-axisymmetric (m > 1) component of the magnetic field ( NAME; dotted line) are shown as a function of time. Also shown are the integrated kinetic energy in the differential rotation ( DRKE; dashed line) and in the non-axisymmetric velocity field ( NAKE; dash-dotted line), again integrated over the volume of the layer. Vertical dashed lines indicate the times shown in Fig. 22. Fig. 20.—Latitude of Doppler, Alfve´nic, and stratification singular points for disturbances on a broad toroidal field, as a function of toroidal field strength parameter a, for G ¼ 0:1 (red ), 1.0 (black), and 10.0 (blue).
Higher a leads to flow and field following these contours even more closely. This is characteristic of disturbances containing rather little vertical motion, and influenced significantly by rotation. By contrast, the top frame for G ¼ 0:1 and a ¼ 1 shows a completely different planform, one that is concentrated near the poles, and one that, from the pattern of arrows, must contain large amounts of both vertical flow and field, to satisfy mass and field continuity. This is a form of the polar kink instability found by Cally (2003). Its rapid growth means that rotational effects on the disturbance structure are very small, leading to strong departures from patterns in with the perturbation flow and field tend to follow the pressure contours. An intermediate case with G ¼ 1:0 (second frame from the top) shows both a broader high-latitude structure plus a low-latitude structure. The disturbance structures seen in Figure 19 are all consistent with the location of the three types of singular points, shown as functions of peak toroidal field in Figure 20. For G ¼ 10 cases, for both low and high a, there is no stratification singular point present between equator and pole, so the mode structure is governed almost entirely by the location of the Alfve´nic singular point, which for a ¼ 0:2 is at about 15 latitude, and for a ¼ 1:0 at about 5 latitude, both consistent with the latitude of sharp structures in the bottom two panels of Figure 19. At the other limit, for G ¼ 0:1, we see in Figure 20 that the Alfve´nic and stratification singular points essentially coincide at a latitude near 60 , right near the equatorward edge of the dominant polar disturbance structure in the top frame of Figure 19. For the intermediate case (Fig. 19, second panel from the top), we get both a high-latitude structure that starts around 40 latitude, and a sharp low-latitude structure near 20 latitude. The latter is consistent with the position of the Alfve´nic singular point for a ¼ 1:0 in Figure 20, which also must locally overpower the effects of the even lower latitude stratification singular point, needed to produce the high-latitude structure. 4.2.3. Competition between Polar Kinks and Clam-Shell Instabilities
Even although this polar kink mode has a much higher growth rate than modes that predominate at low latitudes, there is no reason why both cannot coexist in a full spherical shell, especially after enough time has elapsed that both can reach finite ampli-
tude. We have done an experiment with the nonlinear system of MGD07, which shows that in fact the clam-shell instability eventually overtakes the polar kink instability. Details are as follows. Figures 21Y22 shows results from a nonlinear simulation with G ¼ 104 , S ¼ 0:28, a ¼ 1, r ¼ 0:044, and s ¼ 0:18. The numerical model is described in detail by MGD07. The computational domain is a layer with impenetrable, stress-free, perfectly conducting boundaries (z ¼ 0, 1), with a constant radiative flux through the shell. The simulation was initiated with a depthindependent equilibrium state as described in xx 2.2Y2.3 and was allowed to evolve freely, with no external forcing. Initially the perturbation magnetic and kinetic energy, NAME and NAKE, grow rapidly until t 20 when they become comparable with the kinetic energy contained in the differential rotation, DRKE (Fig. 21). At this stage, NAME, NAKE, and DRKE are all much less than the magnetic energy contained in the mean toroidal field, TFME, which has not changed significantly from its initial value. Between t 20 Y80, NAME and NAKE decrease but subsequently they increase again, with a slower growth rate. After t 170 the NAME dominates over TFME, and all energy components slowly decay thereafter. This corresponds to the opening up and saturation of the clam-shell instability as illustrated in Figure 22. The top row in Figure 22 shows the scalar magnetic potential J, defined in MGD07 such that B ¼ : < ð J zˆ Þ þ : < : < ðCˆzÞ ¼ Bt þ Bp ;
ð46Þ
where Bt ¼ : < ð J zˆ Þ ¼ zˆ < :J is the toroidal field component. Contours of J are thus parallel to Bt . At t ¼ 10 ( Fig. 22a), the J contours are nearly indistinguishable from the initial condition, corresponding to 0 ¼ . However, the vertical field c (Fig. 22c) exhibits structure near the poles with longitudinal wavenumbers m ¼ 1 and 2. This is a manifestation of the polar kink instability. The polar kink instability saturates by modifying the differential rotation and the magnetic configuration near the poles, leaving lower latitudes undisturbed. Thus, the integrated energies TFME and DRKE shown in Figure 21 remain largely unchanged. As the polar kink subsides, the clam-shell instability proceeds, ultimately giving rise to a dramatic restructuring of the magnetic field (Figs. 22b and 22d ) and the differential rotation.
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Fig. 22.—Scalar magnetic potential J (aYb; see text) and vertical field c (c, d ) are shown for a nonlinear simulation at z ¼ 0:23. Images are displayed as Molleweide projections, which include all 360 of longitude and in which lines of constant latitude are horizontal. Red and blue tones represent positive and negative values respectively, with black contour lines highlighted in aYb. Left and right columns correspond to t ¼ 10 (a, c) and t ¼ 170 (b, d ).
Thus, although the polar kink instability has a larger growth rate, it is eventually overwhelmed by the clam-shell mode. 5. SUMMARY From all of the above results taken together, we see that the 3D version of the global MHD instability of coexisting toroidal fields and differential rotation in a continuously subadiabatically stratified spherical shell behaves in almost all respects very similarly to the purely 2D MHD instability when G is large, as in the radiative part of the tachocline. When G is of order 1 or smaller, the 3D system behaves rather like the shallow water version of the instability except that unlike the shallow water case, for large enough toroidal field there is no low G cutoff due to the deformation of the top surface of the shell, and there are very high growth rate modes seen. This is true for both banded and broad toroidal field profiles, but occurs at lower toroidal field in the banded toroidal profile case. Furthermore, the results are hardly affected at all at high G by inclusion of the thermodynamic effects of the shell thickness as a finite fraction of a pressure scale height, and only modestly affected at low G. Similarly, different boundary conditions at top and bottom have very little effect at high G (because the disturbances contain very little vertical motion or field as also seen in MGD07), and only modest effect at low G. Generally speaking, the highest growth rates realized for different vertical wavenumbers n as a function of G are about the same for all n. Since this is an ideal MHD result, we should expect that with diffusion added, we will find that the modes of
Blackman, E. G., & Brandenburg, A. 2002, ApJ, 579, 359 Caligari, P., Moreno-Insertis, F., & Schu¨ssler, M. 1995, ApJ, 441, 886 Cally, P. S. 2001, Sol. Phys., 199, 231 ———. 2003, MNRAS, 339, 957 Cally, P. S., Dikpati, M., & Gilman, P. A. 2003, ApJ, 582, 1190 Charbonneau, P., & MacGregor, K. B. 1992, ApJ, 387, 639
lowest n will have the highest growth rates, and therefore be most likely to dominate in nonlinear simulations. These results are consistent with the nonlinear simulations of MGD07, in which initially 3D disturbances evolved toward 2D forms over time, in which the vertical velocities are very small, but horizontal velocities vary with height ( layering). We conclude that the tendency of unstable modes of global MHD instability in the solar tachocline to tend to be 2D (longitudelatitude) in character (excepting only the relatively transient ‘‘polar kink’’ modes) to be quite robust, at least under the assumptions and approximations we have made. But this result in no way precludes other modes of instability being present in the solar tachocline that are much more 3D, such as unstable modes due to magnetic buoyancy (Schu¨ssler et al. 1994; Ferriz-Mas & Schu¨ssler 1994). But other equations, including for example nonhydrostatic effects, must be used to study such modes. There is every reason to believe that different spatial and timescales of instabilities can coexist in the solar tachocline. Simulating both with a common set of governing equations is a much more involved calculation than we have tried here.
We thank Keith MacGregor for a thorough review of the entire manuscript. We extend our thanks also to an anonymous referee for a very careful review and many helpful comments on the earlier version of this paper. This work is partially supported by NASA grants NNH05AB521, NNH06AD51I, W-10,177, and W-10,175, and the NCAR Director’s opportunity fund.
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