Magnetic Buoyancy and Rotational Instabilities in the Tachocline

Peter A. Gilman High Altitude Observatory, National Center for Atmospheric Research, 3080 Center Green, Boulder, CO 80307-3000, USA; [email protected] Received 2017 September 7; revised 2017 November 27; accepted 2018 January 1; published 2018 January 24

Abstract We present results from an analytical model for magnetic buoyancy and rotational instabilities in a full spherical shell tachocline that includes rotation, differential rotation close to that observed helioseismically, and toroidal field. Perturbation solutions are found for the limit of large latitudinal wave number, a limit commonly used to maximize instability due to magnetic buoyancy. We find that at all middle and high latitudes vigorous rotational instability is induced by weak toroidal fields, particularly for high longitudinal wave number, even when the vertical rotation gradient is marginally stable without toroidal field. We infer that this instability will prevent much storage of toroidal fields in the tachocline at these latitudes, but could be responsible for the appearance of ephemeral active regions there. By contrast, the low-latitude vertical rotation gradient, opposite in sign to that at high latitudes, is not only stable itself but also prevents magnetic buoyancy instability until the peak toroidal field is raised above a threshold of about 9 kG at the equator, declining to zero where the vertical rotation gradient changes sign, at 32.3 in our model. Thus this rotation gradient provides a previously unnoticed mechanism for storage of toroidal fields until they amplify by dynamo action to order 10 kG, whereupon they can overcome the rotation gradient to emerge as sunspots. These results provide a new explanation for why sunspots are seen only at low latitudes. The purely rotational instability at latitudes above 50°, even without toroidal fields, also suggests that the high-latitude tachocline should be much thicker, due to HD turbulence, than has been inferred for lower latitudes from helioseismic measurements. Key words: stars: activity – stars: rotation – Sun: interior – Sun: magnetic fields – Sun: rotation – sunspots

1. Introduction dynamo models and simulations (Parker 1975, 1977; Cline et al. 2003). Limits to magnetic buoyancy instability in dynamos Parker (1955) introduced the concept of magnetic buoyancy because of j×B feedbacks on shear flow, representing to explain how magnetic flux generated by the solar dynamo differential rotation, have also been simulated (Vasil & can come to the photosphere and be seen as sunspots. Magnetic Brummell 2008, 2009). Hughes (2007) has recently reviewed buoyancy occurs in a flux tube when the tube is in total (gas ) the status of buoyancy instabilities in the tachocline. plus magnetic pressure equilibrium with its surroundings, One feature common to almost all the work cited above is coupled with the tendency for the temperature inside and that it was done in Cartesian geometry, in order to focus outside the tube to equilibrate due to radiative diffusion. This more on the physical processes involved. Furthermore, all condition is always met if the cross-section of the tube is small the early studies were done before the existence of the solar enough in at least one dimension perpendicular to the local axis tachocline was discovered, so differential rotation in radius of the tube. ’ similar to the tachocline was not considered. Our results for Since Parker s original work, magnetic buoyancy has been instability of combinations of magnetic fields and differential invoked in many contexts. It has been applied both to tubes rotation are done in spherical geometry, and for tachocline fi embedded in a plasma with much weaker magnetic elds differential rotation that is set by helioseismic observations. exterior to it, and to magnetic layers from which tubes might But to get solutions we return to the traditional eigenvalue arise. The study of magnetic buoyancy as a mechanism for instability problem, approximations for which are discussed fi producing instability of a magnetic layer was rst done in Gilman (1970) and more recently in Mizerski et al. (2013). ( ) systematically by Gilman 1970 . The study of magnetic One study that was done including spherical geometry buoyancy instability of magnetic layers has continued ever and differential rotation was that of Acheson (1978), since (Acheson 1978; Acheson & Gibbons 1978; Hughes which did use local analysis in latitude and radius as we 1985a, 1985b;Cattaneoetal.1990;Thelen2000; Tobias & do, but it contains very few numerical results and did not Hughes 2004; Kersale et al. 2007; Davies & Hughes 2011; have the benefit of knowledge of the existence of the Barker et al. 2012;Mizerskietal.2013), as has the study of tachocline. It also placed heavy emphasis on results with magnetic buoyancy instability of isolated toroidal flux tubes finite diffusion. Therefore, unfortunately there are few or rings (Schüssler 1979, 1980; Spruit & van Ballegooijen 1982; detailed points of comparison that can be made with the van Ballegooijen 1982; Moreno Insertis et al. 1992; Ferriz-Mas results presented here. & Schüssler 1993). Magnetic buoyancy has been modeled as a There is also a relevant literature on the destabilization of component of 3D magnetohydrodynamic (MHD) simulations of solar and stellar tachocline rotation gradients, particularly in magneto-convection applied to the solar convection zone the vertical, by weak toroidal magnetic fields (Balbus 1995; (Fan 2001, 2008; Nelson et al. 2014;Weber&Fan2015) and Menou et al. 2004; Parfrey & Menou 2007; Kagan & studied as a form of double-diffusive instability (Schmitt & Wheeler 2014), sometimes referred to as magneto-rotational Rosner 1983; Hughes & Weiss 1995; Skinner & Silvers 2013). instability. It appears that magnetic buoyancy effects are not Magnetic buoyancy effects have been modeled explicitly in solar included in any of these studies, so it is not possible to

1 The Astrophysical Journal, 853:65 (18pp), 2018 January 20 Gilman examine the interplay between magnetic buoyancy and the instability, so the difference might not be easily detected rotational instabilities in these results. We can see the in observations, but it can have profound implications for the absence of magnetic buoyancy effects in the eigenvalue solar dynamo and for the location of emergence of magnetic equations for unstable modes in these studies; they contain flux to the surface (Dikpati & Gilman 2005). no terms involving the vertical gradient of toroidal fields, The same rationale can be used to justify local instability which is essential for magnetic buoyancy instability to exist studies of the radial gradient of rotation, as we do below. In this (Gilman 1970). In addition, only Kagan & Wheeler (2014) case, the starting point for instability analysis is the observation allow for non-axisymmetric instability modes, which are of the radial rotation gradient via helioseismic methods. But known to be important for magnetic buoyancy instability this is difficult to do at high latitudes; in fact we do not (Gilman 1970), particularly when rotation is present. currently know very well what the tachcocline thickness is The approach taken in Kagan & Wheeler (2014) was to there. We only have an estimate of the total change in rotation include full diffusive effects in the stability analysis and across the tachocline, not the radial rotation gradient itself. We differentiate domains in latitude and radius according to the will see that whether the high latitude local radial rotation type of instability found there; we will take the opposite gradient inferred from helioseismic measurements is unstable approach, keeping our model as simple as possible and to local perturbations is crucially dependent on this thickness. building up results first from physically meaningful special Therefore, in this case the instability calculation acquires a cases and limits, which form a basis for interpreting our full different role, namely leading to questioning assumptions made results for magnetic buoyancy and rotational instabilities in a about tachocline thickness at high latitudes. In addition, the differentially rotating spherical shell that represents the timescale for growth of rotational instability may not be very tachocline. different from the timescale to re-establish the radial rotation — Before presenting the details of our analysis of magnetic gradient by angular momentum transport from above unlike buoyancy and rotational instability, some more general the case of coronal mass ejections in the solar atmosphere. discussion is appropriate. Instability calculations can have Only turbulence calculations beyond the scope of this paper different purposes and goals, depending on the physics thought may be able to address the nonlinear competition between to be relevant and the observations that are available. For the rotational instability and restoration of the radial rotation Sun, there are examples of instability in action and observed, gradient. such as the sudden change in magnetic field in the photosphere, By contrast, no reliable direct observations of toroidal magnetic fields inside the Sun have ever been possible, but chromosphere, and corona that quickly leads to eruption of the existence of erupting solar activity and Hale’s magnetic flares and coronal mass ejections. This is a case where the polarity laws tell us such fields must be there, of large evolution of the magnetic field to a point that it becomes enough amplitude to be unstable and to emerge. In this case unstable is on a much longer timescale than the growth of the the profile of the toroidal field we can take to find instability instability itself. And the new magnetic structure can differ is essentially unconstrained. But the growth rate and form substantially from the configuration that led to the instability. fl fi instability takes may be strongly in uenced by the presence Here the reference state that is perturbed is as well de ned as it of rotation, differential rotation, and the latitude location can be observed. where the instability analysis is done, since even with no However, not all instabilities occurring on or in the Sun differential rotation, Coriolis forces are strong functions of are like this. In the convection zone, convective instability is latitude, as is curvature of the toroidal field. With all these always working, but there is no steady reference state that fi — ingredients present, we should expect signi cantly different can be observed only, at most, the average state produced results for low, middle, and high latitudes, and we have by the convective turbulence. This same convection, found these. fl ’ ’ in uenced by the Sun s rotation, generates the Sun s Given the reasoning above, we feel we are justified in putting differential rotation observed in the photosphere and in the rotational and magnetic buoyancy instability of tachocline interior, as is inferred from helioseismic measurements. Thus differential rotation and toroidal fields on an equal footing, and convection leads to the existence of a tachocline, in which we study the interplay between them in full spherical geometry. differential rotation must be imposed from above. It is We well recognize that what we find cannot be the whole entirely possible that the differential rotation can be unstable answer, because instabilities lead to finite amplitude and in both the convection zone and the tachocline, despite the turbulent processes of various kinds, but we judge that the convective turbulence in situ or nearby, because other types model and analyses presented here are an appropriate place to of modes, not convective, can be generated. We may or may start. not be able to measure the changes in differential rotation caused by this instability. This is the basis for doing global instability studies of 2. Model Development differential rotation with latitude and radius in the tachocline, some of which are referenced later in this paper. In this case 2.1. Initial Approximations there is likely an interplay between the growth of unstable We will start from governing equations to which several modes, which modify the differential rotation, and the approximations have been made, in order to simplify the convection zone re-establishing the differential rotation by problem and focus on magnetic buoyancy and rotational momentum transfer from above. As seen in Dikpati (2012), effects in spherical geometry. The easiest to justify are to global instability of differential rotation in the tachocline include the centrifugal force of rotation into local gravity, need not modify the “reference state” very much to bound and ignore the small oblateness of the Sun. Next we assume

2 The Astrophysical Journal, 853:65 (18pp), 2018 January 20 Gilman the solar tachocline is thin enough that we can ignore the ¶v ¶r 1 ¶pu⎛ ⎞ r + v =- -W+r⎜2 ⎟ sin fu divergence of radii and replace radius with a local vertical ⎝ ⎠ coordinate z. Since the tachocline is a small fraction of the ¶t ¶tR¶f R cos f solar radius at tachocline depths, we make an error of no rvw ⎛ 1 ¶ 1 ¶ --⎜ ()ruv + (rfv2 cos ) more than a few percent, with no change in physics. Even RR⎝ cos fl¶ R cos ff¶ with this approximation, we retain the radial variation of the ¶ ⎞ 1 ¶ ()abc222++ spherically symmetric reference state density; the full + ()rvw ⎟ - equations will be non-Boussinesq. Unlike for studies of ¶z ⎠ R ¶f 2 global HD and MHD instability of the differentially rotating 1 ¶ 1 ¶ tachocline, we do not assume the full governing equations + ()ab + (b2 cos f ) are hydrostatic in the vertical; magnetic buoyancy effects are R cos fl¶ R cos ff¶ certain to occur only in systems where the perturbations are ¶ a2 tan f bc + ()bc ++, non-hydrostatic. By far the most important physical assump- ¶z R R tion we will make is that the thermal relaxation time is very ()2 short for perturbations of interest for magnetic buoyancy, so ¶w ¶r ¶p in the MHD case there will be no temperature perturbations r + w =- - rg and the thermodynamic energy equation can be omitted, as ¶t ¶t ¶z discussed in Gilman (1970) and Mizerski et al. (2013).The ⎛ ⎞ 2 ⎜⎟u rv result is that the equation of state relates perturbation density +Wrf⎝2cos +⎠u + directly to perturbation pressure and magnetic pressure, with R R only reference state temperature present. Since molecular ⎛ 1 ¶ 1 ¶ fi - ⎜ ()ruw + (rfvw cos ) diffusivities of momentum and magnetic elds are much ⎝ R cos fl¶ R cos ff¶ smaller at tachocline depths than is radiative diffusivity, ¶ ⎞ ¶ ()abc222++ we omit these effects for the perturbations we will study. + ()rw2 ⎟ - We will revisit these last two assumptions after obtaining ¶z ⎠ ¶z 2 results. 1 ¶ 1 ¶ + ()ac + (bc cos f ) R cos fl¶ R cos ff¶ ¶ ()ab22+ 2.2. Nonlinear Governing Equations for Velocities and + ()c2 - ,3() Magnetic Fields ¶z R In many magnetic buoyancy instability studies, the equation ¶a ¶ 1 = ()()uc-- wa uc - wa development starts from the governing equations in compact ¶tz¶ R vector calculus notation. This is quite straightforward when 1 ¶ tan f using Cartesian geometry, which most such analyses use. But - ((cos f va-- ub ))(va - ub )(),4 in spherical geometry the situation is more complicated, with R cos ff¶ R curvature terms; in addition, we wish to present the initial ¶b ¶ “fl ” =- ()wb- vc equations in ux divergence form, which does not lend itself ¶tz¶ as well to compact vector notation. Hence we have written the 11¶ equations in component form initially. We note that this is +-+()wb vc ()()va- ub ,5 commonly done for other MHD instability studies, such as R R cos fl¶ fi baroclinic instability in a toroidal magnetic eld, for example in ¶c 1 ¶ Gilman (2017). = ((cos f wb- vc )) With the assumptions described in the previous subsection, ¶tRcos ff¶ the full MHD equations for a rotating thin spherical shell, in 1 ¶ - ()uc- wa ,6 () rotating coordinates, can be written as R cos fl¶ ¶r 1 ¶ 1 + ()ru + ¶u ¶r 1 ¶pu⎛ ⎞ ¶tRcos fl¶ R cos f r + u =- +W+r⎜2 ⎟ ¶t ¶tRcos fl¶ ⎝ R cos f ⎠ ¶ ¶ ´ ()()cosfrv + rw = 0, () 7 ⎛ ⎞ ¶f ¶z ⎜⎟u ´-W+sinfrv ⎝ 2 cos f ⎠ R 1 ¶a 1 ¶ ¶c + ()b cosf + = 0, () 8 ⎛ 1 ¶ 1 ¶ R cos fl¶ R cos ff¶ ¶z - ⎜ ()ru2 + (rfuv cos ) ⎝ R cos fl¶ R cos ff¶ in which uvw,, and a,,bcare respectively velocity and ¶ ⎞ 1 ¶ ()abc222++ magnetic field components in the l,,f z (longitude, latitude, + ()ruw ⎟ - ¶z ⎠ R cosfl¶ 2 vertical) directions; t is time, R the shell radius, p the gas ρ fl Ω ( ) 1 ¶a2 1 ¶ pressure and the uid density. is the constant rotation rate + + ()ab cos f of the coordinate system which we identify for the Sun with the R cos fl¶ R cos ff¶ rate of the interior just below the tachocline. ¶ a + ()ac +- (cbtanf ) , () 1 It is important to stress that the approximations that follow ¶z R are not new, and are in fact common, even standard, in fluid

3 The Astrophysical Journal, 853:65 (18pp), 2018 January 20 Gilman fl dynamics and MHD of compressible uids. This is true even Then if we introduce a new function rs =+1 rrs 00, for approximations specifically designed to bring out magnetic by inspection the hydrostatic balance of the reference state is buoyancy instability, which has a long history of study. What is given by new, and should be our primary focus, is the combination of ¶ps rotation, differential rotation, and spherical geometry, together =-rgs .11() with toroidal magnetic fields containing vertical gradients. ¶z Therefore our description and application of the approxima- It is also useful to recognize that each of the three pressures tions needed to solve the system is relatively brief, distributed, fi and informal. de ned above is related to the corresponding densities by the pRTpRTTp==+rr = We will use the equation of state pRT= r to relate perfect gas law, so 00 00 **00,,s s () 00 s ¢ fi* pressure and density when the temperature T is a xed function r¢RT*()00 + Ts . The quantity R*()TTgH00 +=s , in which fi of z only, given we are assuming an in nitely fast thermal H =+RT*()00 Ts gis the scale height. We will use these relaxation time. The next step in governing equation develop- relations at later points in the equation development. ment is to expand out the total density and total pressure into If we now expand Equations (1)–(3) and (7) in terms of the three parts: static reference state and departures from it (in this expansion Equations (4)–(6) and (8) are symbolically unaffected), and rr=+00 rs()z + r¢,9 () subtract off the reference state, these equations become, in the same order pp=+00 pzps () +¢,10 ( ) ¶u 1 ¶pu¢ ⎛ ⎞ in which r00, p00 are the average density and pressure of the =- +W+⎜2 ⎟ ¶tRrcos fls ¶ ⎝ R cos f⎠ layer, rs ()zpz, s ()are the spherically symmetric reference state departures from the average values, and r¢()lf,,,,zt ⎛ u ⎞ ⎛ 1 ¶ ´-W+-sinffv ⎜⎟ 2 cos w ⎜ ()u2 pzt()lf,,, represent all departures from the time-indepen- ⎝ R⎠ ⎝ R cos fl¶ dent reference state. Because of the assumption of fast thermal 1 ¶ 1 ¶ ⎞ relaxation, there is no T¢, only TT00, s. Then we assume + ()uv cos f + ()ruw ⎟ s ⎠ r¢  rr00 + s, and we ignore r¢ when it is coupled with R cos ff¶ rzs ¶ fi velocities or magnetic elds in the equations of motion. Then 1 ¶ ()abc222++ 11⎛ ¶a 2 we scale all magnetic fields by r12and the gas pressure by r - + ⎜ 00 00 Rrcosfl¶ 2 rR⎝ cos fl¶ so they have units of velocity and velocity squared, ss ⎞ respectively. Then if time frequencies as high as characteristic 1 ¶ ¶ a + ()()(ab cosff+ ac +-cb tan )⎟ , sound waves are ignored, then time derivatives of r¢ can be R cos ff¶ ¶z R ⎠ dropped in the equations of motion and the mass continuity ()12 equation. The HD parts of these equations represent the now standard “anelastic” equations originally developed in geophy- ¶v 1 ¶pu¢ ⎛ ⎞ vw fl =- -W+⎜2 ⎟ sin fu - sical uid dynamics. The approximations used here are not ¶tR¶ff⎝ R cos ⎠ R new, but have a history over the past fifty years in various ⎛ 1 ¶ 1 ¶ contexts. The addition of MHD effects involves no further - ⎜ ()uv + (v2 cos f ) approximations in terms (or asymptotic analysis to derive ⎝ R cos fl¶ R cos ff¶ ) fi fi them , unless magnetic elds are so large that they signi cantly 11¶ ⎞ ¶ ()abc222++ alter the hydrostatics of the reference state (much larger than + ()rvws ⎟ - rz¶ ⎠ Rr ¶f 2 “equipartition” fields), which for the solar tachocline would s s fi fi 11⎛ ¶ 1 ¶ imply a toroidal eld larger than 100 kG, well above elds any + ⎜ ()ab + (b2 cos f ) solar dynamo is likely to generate. Except for the absence of rRs ⎝ cos fl¶ R cos ff¶ diffusion, and the way the equations are scaled, the MHD ¶ a2 tan f bc ⎞ + ()bc ++⎟,13 () version is essentially the same as derived by perturbation ⎠ expansion for global simulations of solar and stellar convection ¶z R R and convectively driven dynamos, as discussed initially in ¶w 1 ¶p¢ r¢g ⎛ u ⎞ v2 Gilman & Glatzmaier (1981), and used in the well known ASH =- - +W⎜⎟2cosf +u + ¶tr¶z r r ⎝ R⎠ R code models, whose roots can be traced back to Gilman & ss00 ⎛ Glatzmaier (1981) and others. There are other even more 1 ¶ 1 ¶ - ⎜ ()uw + (vw cos f ) reduced equations, such as the so-called magneto-Boussinesq ⎝ R cos fl¶ R cos ff¶ equations, as considered in Corfield (1984) and Bowker et al. 11¶ ⎞ ¶ ()abc222++ (2014), but we have not found it necessary to approximate the + ()rw2 ⎟ - s ⎠ equations this much to get significant results, although there is rzs ¶ rzs ¶ 2 little doubt the same asymptotic methods could be used on our ⎛ 11⎜ ¶ 1 ¶ equations to find this simpler system, with spherical geometry, + ()ac + (bc cos f ) rRs ⎝ cos fl¶ R cos ff¶ rotation, and differential rotation. In our system, we have not ¶ ()ab22+ ⎞ approximated terms involving magnetic fields at all, except for + ()c2 - ⎟,14() setting the tachocline radius equal to a constant value. ¶z R ⎠

4 The Astrophysical Journal, 853:65 (18pp), 2018 January 20 Gilman 1 ¶ 1 ¶ 1 ¶ 1 ¶ 1 ¶ 1 ¶ u + ()cos fv + ()rws = 0 uv+ ()cos f + ()()rws = 0, 19 R cos fl¶ R cos ff¶ rzs ¶ cos fl¶ cos ff¶ rzs ¶ ()15 ¶a ¶ 1 = ()()uc-- wad uc -- wa The next step in equation development is to make the ¶tz¶ cos f governing equations dimensionless. This is not for the purpose ¶ of further approximations but to make them more universal. ´ ((cosffva-- ub ))( tan va - ub ) , () 20 ¶f Since velocities, magnetic fields, and pressures in the dimensional nonlinear equations above have units of velocity ¶b ¶ =- ()()wb-+ vcd wb - vc or velocity squared, it is straightforward to scale all variables in ¶tz¶ “ ” terms of external parameters that do not depend on the details 1 ¶ of dimensional amplitudes. First we scale horizontal lengths by + ()va- ub ,21 () R, vertical lengths by D, the thickness of the shell, and time by cos fl¶ -1 W . We specify uvab,,, scaled by WR, w, c by dWR,in ¶c 1 ¶ which d = D R, and pressure by W22R . The variable r¢ r is = ((cos f wb- vc )) 00 ¶t ff¶ already dimensionless so no further scaling is required. Another cos 1 ¶ dimensionless parameter we will use is ds = DH. With these - ()uc- wa ,22 () scalings, the full nonlinear governing equations become cos fl¶ ⎛ ⎞ ¶u 1 ¶p¢ u 1 ¶a 1 ¶ ¶c =- ++⎜ ⎟ + ()b cosf + = 0 () 23 2 cos fl¶ cos ff¶ ¶z ¶trcos fls ¶ ⎝ cos f⎠ ⎛ ( )–( ) 1 ¶ Equations 16 23 are then the equations we will perturb to ´-sinfdvuwu() 2 cos f +-⎜ 2 find instabilities. ⎝ cos fl¶ ⎞ 1 ¶ 1 ¶ 2.3. Equations for Calculating Instabilities + ()uv cos f + ()ruws ⎟ cos ff¶ rzs ¶ ⎠ We already defined a spherically symmetric hydrostatic 1 ¶ ()ab2222++d c reference state that was subtracted off from the full nonlinear - governing equations, resulting in the system in Equations cosflrs ¶ 2 (16)–(23). We now separate this system into one representing ⎛ 2 an unperturbed state of rotation and toroidal fields that can be 11⎜ ¶a 1 ¶ + + ()ab cos f functions of f, z (but not longitude or time), which we will rs ⎝ cos fl¶ cos ff¶ denote by subscripts “0,” and linear perturbations on that state ¶ ⎞ + ()ac+- a (df c b tan )⎟ , ()16 that are in general functions of l,,,f z t.Wedenotethese ¶z ⎠ perturbations with subscripts “1.” ⎛ ⎞ ¶v ¶p¢ u 2.3.1. Unperturbed State Equations =- -+⎜2 ⎟ sin fduvw - ⎝ ⎠ ¶t ¶ffcos Our unperturbed state contains only differential rotation and ⎛ 1 ¶ 1 ¶ toroidal field, together with gas pressure. Axisymmetric - ⎜ ()uv+ ( v2 cos f ) velocities and magnetic fields are set to zero, resulting in a ⎝ cos fl¶ cos ff¶ time-independent unperturbed state, found from Equations (17) ⎞ 2222 ( ) 11¶ ⎟ ¶ ()ab++d c and 18 , for latitudinal and vertical force balances. We also + ()rvws - fi fl rz¶ ⎠ r ¶f 2 nd it convenient to introduce angular measures of zonal ow s s fi ⎛ and toroidal eld, such that ua00==wfcos , 00 a cos f.We 11¶ 1 ¶ 2 fi 2 2 + ⎜ ()ab+ ( b cos f ) also de ne a total pressure pa0 =+p0 0 cos f. Then the force rs ⎝ cos fl¶ cos ff¶ balance for the unperturbed state is given by ¶ ⎞ + ()bc++ a2 tanfd bc⎟ , ()17 ¶p0 2 ⎠ - aff0 sin cos ¶z ¶f ¶w 1 ¶p¢ d ++rs()2sincos0,24ww00 f f = () =- -+s puuv¢ 2cosf ++22 22 ¶trd s ¶zrd s ¶p0 2 2 ++dacos f dsp ⎛ ¶z 0 0 ⎜ 1 ¶ 1 ¶ - ()uw+ ( vw cos f ) 2 ⎝ cos fl¶ cos ff¶ -+dwwfrs()200 cos = 0. () 25 ⎞ 2222 By cross-differentiation followed by subtraction of 11¶ 2 ¶ ()ab++d c + ()rws ⎟ - ( ) ( ) rz¶ ⎠ d2rz¶ 2 Equation 25 from Equation 24 we can eliminate the total s s pressure and find the MHD version of the “thermal wind” for ⎛ 11¶ 1 ¶ this system, which relates the latitudinal gradient of fluid + ⎜ ()ac+ ( bc cos f ) rs ⎝ cos fl¶ cos ff¶ density to latitude and vertical gradients of rotation and toroidal field. This relation is of central importance for studying global ¶ ()ab22+ ⎞ + c2 - ⎟, ()18 instability of such gradients, but we do not have to solve ¶z d ⎠ Equations (24) and (25) in the current study. But

5 The Astrophysical Journal, 853:65 (18pp), 2018 January 20 Gilman ( ) ( ) Equations 24 and 25 must be subtracted from the total ¶b1 ¶vb1 ¶ 1 ( )–( ) = a0 - w0 ,31() Equations 16 23 to get the perturbation equations for ¶t ¶l ¶l instability. Equations (24) and (25) contain all the parameters and ¶c1 ¶wc1 ¶ 1 = a0 - w0 ,32() functions that define the reference state that we will perturb for ¶t ¶l ¶l the instability calculations. We are allowed to specify values for δ and ds that are characteristic of the tachocline, as well as 1 ¶a1 1 ¶ ¶c1 fi + ()b1 cosf + = 0, () 33 the function rs which describes the density strati cation within cos fl¶ cos ff¶ ¶z the tachocline. These three quantities are not independent of each other, since they all involve the tachocline thickness, so In Equations (26) and (30) for time changes in ua11, they must be varied together in a consistent way. The quantities respectively, we have used Equations (29) and (33) to rewrite fi w0 and a0, which can be speci ed independently of d,,dssr , some terms into forms more convenient for instability can be functions of both f and z. Once these quantities are eigenvalue problems. specified, we can calculate p0 and p0, if we need them, from Equations (24) and (25). In all instability calculations below, 2.4. Eigenvalue Equations we do not actually use either p0 or p0, because perturbation terms in the equations for unstable modes are everywhere linear 2.4.1. General Case in these variables. Also, in practice we will vary only the In the next step we derive the perturbation eigenvalue amplitude of a0, through a range of values thought to be equations for unstable disturbances from Equations possible in the tachocline, while keeping w0 to a single function (26)–(33) by assuming all independent variables are propor- of latitude and height, which is chosen with guidance from tional to eim()lft+- n t,inwhichm is the longitudinal wave helioseismic observations. number, n the latitudinal wave number, and τ is the complex eigenvalue (ti is the growth rate, tr is proportional to the 2.3.2. Linear Perturbation Equations phase speed). Then we will obtain equations for an as yet After subtracting out the unperturbed state from undetermined dependence of all perturbation variables on z. ( ) ( ) fi Since the coefficients of variables in Equations (26)–(33) are Equations 16 and 17 , and de ning paf1 =+pa1 01cos we get the following equations for the perturbations: functions of latitude, our analysis is essentially local in that coordinate. ¶u1 ¶u111 ¶p The eigenvalue equations are then given by =-w0 - ¶t ¶lfcos rs ¶l im im()wt-=-++ u pwf ()2sinv ++21()wfdwfvw sin -+ () 2 cos 00 01 0 1 cos frs ¶w0 ¶wa0 01¶a ⎛ ⎞ - cosf v1 - cos f w1 + ¶w00¶w ¶f ¶z r ¶l -+dw()2coscos0 wv f - f⎜ + w⎟ s ⎝ ¶f ¶z ⎠ ⎛ 1 ¶a0 ¶a0 ⎛ ⎛ ⎞ + ⎜cosf b1 + cos f c1 1 ¶a00¶a r ⎝ ¶f ¶z ++⎜imaf0 acos ⎜ b + c⎟ s r ⎝ ⎝ ¶f ¶z ⎠ ⎞ s +-dfacoscb 2 sin fa⎟ ,() 26 ⎞ 01 0 1⎠ ⎟ -+2sinfa00bc d cos fa ⎠ ,() 34

¶v111 ¶p in =- -+()2sinwf01u im()wt-=--+ v p ()2sin wu f ¶ ¶f 00 trs rs ¶v1010a ¶b 2sinaf im 2sina0 - w0 + + a1,27() ++a0b a,35() ¶l rss¶l r rssr ¶w 12cos¶pd f 1 ¶pds 1 =- 1 -+s pu()1 +w im()wt0 -=- w - p 2 2 1 01 d22rz¶ d r ¶trzrd s ¶ d s d s s ¶w a ¶c 2cosaf 2 imafa002cos 1010 ++()1coswf0 u +c - a,36 () - w0 + - a1,28() d r dr ¶l rss¶l dr ss im 1 ¶ 1 ¶u1 1 ¶ 1 ¶ uinv+-tan f v + ()rws = 0, () 37 + ()cos fv11+ ()()rws = 0, 29 cos f rzs ¶ cos fl¶ cos ff¶ rzs ¶ af0 cos ¶rs im()wt00-= a im a u + w ¶a1 ¶au1 ¶ 1 rs ¶z =-w0 + a0 ¶t ¶l ¶l ⎛¶w ¶a ¶w ⎛ + cos f⎜ 000bv- + c aw0 ¶rs ¶ 0 ¶a0 ⎝ ¶f ¶f ¶z + cos f⎜ wbv1 + 1 - 1 ⎝ rs ¶z ¶f ¶f ⎞ ¶a0 ⎟ ⎞ - wcw--dw()00 a ,38 () ¶wa0 ¶ 0 ¶z ⎠ + c1 - wcw10101--dw()() a ⎟,30 ¶z ¶z ⎠ im()wt00-= b im a v,39 ()

6 The Astrophysical Journal, 853:65 (18pp), 2018 January 20 Gilman

im()wt00-= c im a w,40 () 2.4.3. Reduction to a Single Eigenvalue Equation im ¶c Equations (42)–(45) can be reduced to a single polynomial ainbtanb+-f += 0,() 41 by the standard technique of requiring the determinant of the cos f ¶z coefﬁcients of the variables uwa,,,c to vanish to yield non- trivial solutions. The resulting equation is a quartic equation of 2.4.2. Limiting Case: n ¥ the form

4 2 Consistent with the assumption that in the tachocline the ()()()mambmcwt0 -+qqq wt0 -+ wt0 -+=0, radiative diffusivity is much larger than either viscous or ()46 magnetic diffusion, as used in Gilman (1970) and Mizerski et al. (2013) and other magnetic buoyancy instability studies, in which we now greatly simplify the eigenvalue problem by letting the m2a2 2 ⎛ ⎞ latitudinal wave number n ¥, but keep the products 2 0 af0 cos ⎜⎟ds fi aq =- -⎝ -2⎠ nvnbn,,p nite. The variables vb,,p  0imply they drop rrssd d out of Equations (34)–(36) and (38). Equation (39) becomes ⎛ ⎛ ⎞ ⎞ ¶a0 1 ¶rs 0 = 0. In addition, since pfa=+pacos 0 , we can substitute ´ ⎜ - ⎜ + da⎟ ⎟ ⎝ ⎠ 0 for p in Equation (36) to form the magnetic buoyancy force ⎝ ¶zrs ¶z ⎠ there. In this limit the equations separate into a system of four 2 f ⎛ ¶w ⎞ ( ) 2cos ⎜ 0 ⎟ equations for the four variables uwa,,,c, from Equations 34 , -+++()()ffwd00⎝ 2,47 w ⎠ () (36), (38), and (40), and three separate equations, from d ¶z Equations (35), (37), and (41), that can be used to find 2 2 2cosmffa0()+ w0 np,,nv nb respectively, if needed, once uwa,,,c are known. bq =- The reduced equations to solve for the eigenvalue τ and the rs variables uwa,,,c are given by ⎛d 1 ¶r ⎞ ´-⎜ s 21f () +-d s ⎟,48 () ⎝ ⎠ d rs ¶z ¶w0 im()()wt00-=-+ u d2coscos w w f - fw ¶z m4aaf4 m2 2 cos2 ⎛⎛d ⎞ a c =+0 0 ⎜⎜⎟s -2 0 imafa a cos ⎛ ¶ ⎞ q 2 ⎝⎝ ⎠ 00⎜ ⎟ r drrs d s ++⎝ + da0⎠c,42() s rrzss¶ ⎛ ⎞ ⎛ ⎞⎞ ¶a0 ¶w0 ´ ⎜ +++da00⎟ 21f () w ⎜ - dw0⎟⎟ . () 49 ⎛ ⎞ ⎝ ¶z ⎠ ⎝ ¶z ⎠⎠ cos fa0 ⎜⎟ds im()wt0 -= w -2 a dr ⎝ d ⎠ s In the definitions of aqqq,,bcwe have added a “flag,” 2cosf ima0 denoted by f, which marks the places in the equations of the +++()1,43w0 u c () d rs dimensionless rotation of the system. When rotation is included, f=1; when it is absent, f=0. This allows us to ⎛⎛ ⎛ 1 ¶r ⎞ use the same formulas for the case with no rotation, to calculate im()wt-= a im a u +cos f⎜⎜⎜ s + da⎟ magnetic buoyancy instability from the same formulas in the 00⎝⎝ ⎠ 0 ⎝ rs ¶z non-rotating case (obviously the differential rotation can be ) ¶aw⎞ ⎛¶ ⎞ ⎞ switched off by setting w0 = 0 everywhere . - 00⎟w + ⎜ - dw ⎟c⎟,44() ⎠ ⎝ 0⎠ ⎠ Two limiting cases are important: when there is no rotation ¶z ¶z and no differential rotation, and when there is no toroidal field. With no rotation, Equation (49) becomes quadratic in t2; when im()wt00-= c im a w.45 () 2 there is no toroidal field, one (mwt0 - ) factors out, and the We see that in this limit that there are no longer any vertical rest yields a quadratic equation for (mwt0 - ). This is because derivatives of perturbation variables; the perturbation equations in both cases bq=0. Without toroidal field, we are solving just are now algebraic and in principle apply independently to each reduced forms of Equations (42) and (43). As checks, we have elevation and each latitude in the domain. This is analogous to done the algebra both ways in both cases and obtained identical what was found in Cartesian geometry in Gilman (1970) and answers. Mizerski et al. (2013). The latter showed how this system can It is well known that analytical solutions exist for quartic be reconciled with solutions of the two-point boundary value equations. We use the forms described in the Handbook of problem for the same system. Ultimately that should be done Mathematical Tables and Formulas by Burington (1953),an for the system derived here, but we leave that to future studies. admittedly ancient, but reliable, source. These solutions work We show below that much can be learned about the MHD of even in the special cases of zero rotation or zero toroidal field, the tachocline without addressing that issue. It should also be yielding answers very close to those from the corresponding noted that, in principle, the Cartesian eigenvalue equations of quadratic forms in these cases, a useful check on the general Gilman (1970) can be recovered from Equations (42)–(45),if forms and our FORTRAN codes, each of which was written one focuses on the equator and discards all curvature terms. independently for the general and special cases. As a further The numerical results in Gilman (1970) are different because check, for the general case we also found sample solutions there the focus was on the very special case of an isothermal from the general Equation (49) by minimizing its left-hand side atmosphere, for analytical convenience, which is significantly by scanning through eigenvalue space, using as guidance for different from the tachocline conditions assumed here. Also, of the scan the analytical solutions found. Agreement was very course, in 1970 the tachocline was not known to exist. good in all cases tried.

7 The Astrophysical Journal, 853:65 (18pp), 2018 January 20 Gilman

In brief, as documented in Burington (1953), quartics are parameter rs is absent, so these results are independent of solved by finding the roots of the “resolvent cubic,” whose density effects, therefore the same in Boussinesq and non- coefficients are related algabraically to the quartic coefficients. Boussinesq regimes. The reduced equation is a simple The roots of the quartic are sums and differences of square quadratic, given by roots of the resolvent cubic’s roots. From the details of the 22 analytical solutions in Burington’s tables, we can see that ()mwt0 -=2cos f () 1 + w0 unstable modes are guaranteed only when the quantity ⎛ ⎞ ⎜ 1 ¶w0 ⎟ H =+ba23427is positive. The quantities a , b are ´++2 w0 .52() rr rr ⎝ d ¶z ⎠ related to aqqq,,bc in Equations (46)–(49) by ar = 2 22 -acq 48 - q 4 and br =-()aacbq 108 +qq 3 -q 8 8. Clearly we get a complex eigenvalue, signifying instability, All unstable modes we find have H > 0. In principle, we can only when the last factor on the right-hand side of fi nd the boundaries of unstable domains in our parameter space Equation (52) is negative. This occurs only poleward of a = by solving the equation H 0. Given the complexity of the certain latitude, which is itself a weak linear function of z. With algebra involved, this is impractical; instead we find the d = 0.04, or tachocline thickness of 4% of the tachocline stability boundaries, where they exist, by approaching them from an adjacent unstable domain. radius, or about 3% of the solar radius, we get an instability It is important to recognize that Equations (46)–(49) are boundary latitude of 49.93 at the bottom, and 49.3 4 at the top. valid for all latitudes and vertical elevations, and for all profiles Since the whole right-hand side is zero at the pole, there is of differential rotation and toroidal field, and for all stratifica- instability between about 50° and the pole. The actual τ tions, as represented by rs. They also apply for a wide range of eigenvalue is given by tachocline thicknesses as measured by d, d . Within the s 1 approximations we have made, and in the limit n ¥, d, d ⎛ ⎛ ⎞⎞ s ⎜ 1 ¶w0 ⎟ 2 can even vary with latitude. tw=m 000cos f⎜ 2() 1 + w 2 ++ w ⎟ .53 () ⎝ ⎝ d ¶z ⎠⎠

3. Results From Equation (53) we see that, as usual in dissipation-free The results we now present will be for specified profiles of systems, we get one growing and one decaying mode of equal differential rotation and toroidal field. In the absence of more growth and decay rates poleward of the boundary latitude, both detailed guidance from helioseismic observations, we will of which are advected passively by the local rotation relative to assume differential rotation is linear in z, in the form the rotating coordinate, for that latitude and elevation. The 2 4 growth rate is the same for all m, including m=0. Instability wff002=+()()sssin + s4 sin z , 50 virtually always occurs at all depths, with a maximum at the top in which we will usually take ss02==-=-0.05, 0.14, s 4 .14, of the shell, where w0 is most negative. Equatorward of the so the differential rotation ranges from zero at the bottom of the boundary we get a pair of neutral oscillation modes, one faster tachocline at z=0, to its maximum at the top at z=1. and the other slower than the local rotation rate by the same We will also do calculations for constant rotation ( f = 1) amounts. These departures from the passive propagation rate where we set s024,,ss= 0, to see the effect of constant are a measure of the strength of the restoring force that makes rotation at different latitudes on magnetic buoyancy the oscillation. Here, too, the stabilizing force is independent of instability. m, including m=0. This measure of restoring force is an For the toroidal field we will consider a quadratic profile of indication of how much destabilizing force would be needed to the form generate instability by some other means, in our case by

aa000=-zz()1, () 51 magnetic buoyancy. In Figure 1 we plot the growth rate of the unstable mode at high latitudes, together with the restoring which peaks at mid-depth in the tachocline and is zero at force, which has the same dimensionless units, for all low fi its bottom and top. Obviously other pro les are possible, latitudes. We see the peak growth rate occurs between 60° and ( ) including linear combinations of that in Equation 51 and a 65°. In our scaling, a growth rate of 0.01 corresponds to an fi constant eld, but we feel we capture the essential behavior e-folding growth time of about a year. Thus the rotational with these two cases. We have also done a few calculations instability we find is quite vigorous, with growth times of a fi with constant toroidal eld to assess the effect of a uniform week or less. Therefore it must be taken into account in fi eld on instability of the differential rotation. consideration of the HD and MHD of the tachocline. But the In all the results that follow, unless indicated otherwise, the strength of the restoring force at low latitudes is about three growth rates plotted are the maximum growth rate for that case, times greater than the destabilizing force at high latitudes. found by scanning all the local growth rates in the elevation The vigor and even the existence of this high-latitude range z = 0.,1.. instability in our model for the tachocline is a direct consequence of the assumption of infinitely fast radiative 3.1. Rotation and Differential Rotation Only diffusion compared to viscous or magnetic diffusion. For In our system, even if there is no toroidal field, there can still adiabatic perturbations in the stably stratified tachocline the be instability because there is differential rotation. We can find instability would be governed by a Richardson number-type the eigenvalues for neutral and unstable modes either from condition, modified by the presence of rotation. But a restoring Equations (42) and (43) directly, or from Equations (46)–(49), force from ordinary (as opposed to magnetic) buoyancy is in the limit of a00  0. Either way, the reference state density absent in our system. Without magnetic fields, in fact, all

8 The Astrophysical Journal, 853:65 (18pp), 2018 January 20 Gilman

Figure 1. Dimensionless growth rates (positive values) and restoring force amplitudes (negative values) of linear modes in the spherical shell with rotation and differential rotation but with no toroidal fields. The right-hand scale is Figure 2. Tachocline thickness (curved line including dashed part) required to dimensional e-folding growth time. make rotational instability growth rates go to zero. The flat solid line depicts the tachocline thickness used for all latitudes equatorward of 50°. pressure perturbations are zero. Only Coriolis and inertial forces are acting. about how well the solar dynamo will be able to generate and The physical interpretation of this instability is as follows. store well organized toroidal fields at these depths and latitudes Moving fluid elements are always conserving their total angular in the presence of strong hydrodynamic turbulence. One momentum, even if they are non-axisymmetric (m ¹ 0) possible consequence is that the tachocline at high latitudes because there are no longitudinal pressure forces. At low may be much thicker than usually assumed. In line with basic latitudes, where the vertical Coriolis forces are strong and fluid dynamics in fluid boundary layers, such as at the surface angular velocity increases with z, an upward moving element of the Earth, it is possible that the rotational turbulence expands arrives at its new location with smaller angular velocity than its the tachocline to a thickness for which the system is marginally surroundings, and therefore falls back. At high latitudes, the stable. We can use Equations (52) or (53) to calculate the Coriolis force is weaker, so it is less restoring, but an upward thickness, as measured by δ, that the tachocline would have to fl moving uid element arrives at its new location with higher be as a function of latitude. Figure 2 shows the result. We see angular velocity than its surroundings, since the ambient that δ rises from 0.04 at around 50° to almost 0.13 at the pole, a angular velocity decreases with height there. This pushes the thickening of a factor of more than three. It is well known that particle upward further, leading to instability. At intermediate inferences of internal properties of the Sun by helioseismic latitudes there is a near balance between Coriolis and inertial techniques at high latitudes is very uncertain due to effects; they exactly balance at the instability boundary near foreshortening, so there are not many real data to compare 50°. The energy source for unstable modes is clearly the with. Our result can be considered to be a prediction for differential rotation itself. helioseismologists to test. It could be that a lower vertical Some may question the validity of this instability result in rotation gradient over a deeper layer would be easier to real systems, given the assumptions made. But we see no estimate. Antia & Basu (2011) do find helioseismic evidence reason to doubt it for radiative zones of stars, since there the that the tachocline is thicker at high latitudes than at the assumption of fast radiative diffusivity is reasonable. We see equator, but they also see evidence that the tachocline is nothing to prevent the growth of perturbations that have an prolate, so its structure variations with latitude may be more extremely small scale in latitude. complex. Our result has important implications for our understanding It is equally important to recognize that the strong restoring of the HD and MHD of the tachocline, including how turbulent force from Coriolis and inertial forces at low latitudes seen in it is. In the HD realm, these latitudinally thin disturbances Figure 1 implies that tachocline rotationally driven HD should create rotationally driven turbulence at all high latitudes turbulence is likely to be absent at these latitudes. This could where instability occurs. In the MHD realm it raises questions still be true even when toroidal fields are present, opening up

9 The Astrophysical Journal, 853:65 (18pp), 2018 January 20 Gilman the possibility of an additional toroidal ﬁeld storage mechanism, beyond the often invoked negative buoyancy of the sub- adiabatic stratiﬁcation. Indeed, that mechanism is negated for disturbances in a plasma in which radiative diffusion dominates over adiabatic effects. We explore these issues in detail in later sections. In closing, we comment that this local rotational instability is distinct from the global rotational instability studied by Watson (1981), who did not consider radial or vertical rotation gradients. In the global instability, pressure forces play a key role, and unstable disturbances are like quasi-geostrophic Rossby waves in which there is nearly, but not quite, a horizontal force balance between Coriolis forces and pressure gradients. By contrast, in our local instability, as stated above, pressure forces are zero everywhere. Only Coriolis and inertial forces are acting. Baroclinic instability of the vertical rotation gradient in the form of a “thermal wind” should also be present in the tachocline (Gilman & Dikpati 2014; Gilman 2015, 2016, 2017), but this instability is completely different in that here too horizontal pressure gradients play a key role, the disturbances are hydrostatic, and the system has adiabatic perturbations in the sub- adiabatic layer rather than isothermal as assumed here.

3.2. Toroidal Field Only The other special case worth studying using our system is the case with neither rotation nor differential rotation ( f ,0w0 = ). In this case the coefficient bq in Equation (46) is again zero; this time the equation becomes a quadratic equation in t2, and so is easily solved. In addition to magnetic buoyancy, the only Figure 3. Dimensionless growth rates of modes with m =-015 for selected perturbation forces remaining in the system are magnetic latitudes for a toroidal field of 10 kG with neither rotation nor differential stresses, proportional to m, and “curvature” stresses arising rotation present. Instability is purely from magnetic buoyancy. The right-hand from the spherical geometry. For the case of constant a ,or scale gives dimensional e-folding growth times in days. Growth rate curves for 0 fi constant toroidal field, there are no unstable modes, because all other toroidal eld choices scale linearly, so these curves are universal. there is no energy source for instability. For the quadratic profile of a0 defined by Equation (51), there is instability for all lower latitudes. For m=0 modes, the so-called curvature a00 and all latitudes. This instability is always confined to the stress is not stabilizing at all, at least for modes very narrow in upper half of the shell, where the toroidal field is declining in latitude, which are what we have focused on here. amplitude with height. Figure 3 displays growth rates as a We see that magnetic buoyancy instability is also very function of longitudinal wave number m between 0 and 15 for vigorous, with e-folding times for all latitudes being slightly disturbances at selected latitudes between equator and pole for less than two days. So even a 1 kG peak toroidal field yields a10kGfield. These growth rate curves are self-similar, so they e-folding times of significantly less than a month. For the fi have the same shape for all other peak toroidal eld choices. To m=0 modes the growth rate peak, its elevation z= 0.815, and fi get the correct growth rates for any other peak toroidal eld, the elevation range z = 0.53, 1.00 over which there is simply scale the vertical scales in Figure 3 by the ratio of the instability are all independent of latitude. For higher m modes, chosen field to 10 kG. = the higher the latitude and the higher the m, the closer to the top We see that, for each latitude, the m 0 mode is most is the peak growth rate, and the higher the elevation of the unstable, but only very slightly so near the equator; as the instability boundary. It is easy to prove algebraically that latitude is increased, growth rates fall off faster and faster as m unstable modes always have zero phase velocity in longitude is increased. For latitudes for which the highest m plotted with (t = ) growth rate is less than 15, the next higher m mode is not r 0 , so in no sense do the unstable modes have properties of Alfvén waves. The other, neutral, eigenvalues do character- unstable. The higher the latitude, the fewer the m that are fi unstable at all. Near the equator curvature is very weak just as it ize Alfvén waves in a spherical shell system, modi ed by is in the Cartesian limit. Here weak magnetic stresses at low m magnetic buoyancy. We have not studied these oscillations are virtually balanced by enhanced magnetic buoyancy from in detail. finite m. As the latitude is increased, the same m corresponds to If we compare the purely rotational instability growth rates a shorter longitudinal distance, leading to increased magnetic and restoring force in Figure 1 with the magnetic buoyancy stress, since it is proportional to m. But magnetic buoyancy is growth rates in Figure 3, we see that in the general case, when not enhanced further, so for high enough m, stabilizing both toroidal field and differential rotation are present, in a magnetic stresses overpower the magnetic buoyancy and the qualitative sense we should expect the two instabilities to add at high m modes decline in growth rate, ultimately to zero. For high latitudes, making those latitudes even more unstable, m=0 this stabilizing magnetic stress is zero even very close to while at low latitudes magnetic buoyancy will have to be large the pole, so the growth rate for that mode is the same as at enough to overcome the rotational restoring force there. In

10 The Astrophysical Journal, 853:65 (18pp), 2018 January 20 Gilman addition, even a constant rotation will affect magnetic buoyancy instability, and constant toroidal field can change rotational stability and instability. Understanding all these possibilities requires that we solve the full quartic Equation (46), which we give results for next. As we shall see, the presence of even a very weak toroidal field, whether constant or varying with z, can destabilize an otherwise stable differential rotation profile.

3.3. Toroidal Fields with Rotation but No Differential Rotation What happens to magnetic buoyancy instability in a spherical shell tachocline when uniform rotation is present? To answer this question, we must solve the full quartic Equation (46) with the switch f=1, which captures the Coriolis force from the rotation of the system, but with w0 = 0 everywhere (no rotation relative to the coordinate system rate). For simplicity, we keep d = 0.04 at all latitudes, the results in Figures 1 and 2 notwithstanding. The growth rates for the most unstable m at each latitude selected for a particular choice of toroidal field are shown in Figure 4. The dimensional e-folding times are shown on the right-hand axis, ranging from 100y down to 0.01y. The range of toroidal field goes from 0.5 kG, for which rotation has a very strong influence, up to 20 kG, for which rotation has very little influence. The integers along each growth rate curve denote the most unstable m at that latitude for that peak field. We see several important effects of rotation, which vary strongly with latitude. Growth rates for low and moderate toroidal field are most strongly depressed at the equator, and Figure 4. Growth rates of most unstable m (integers near continuous curves) fi least at the pole. In fact, the growth rates close to the pole are for selected latitudes and toroidal elds with rotation but not differential rotation present. Growth rates at 90° are the same as without rotation. Note that identical to those without rotation. This is because in the limit n growth rates for the range of toroidal field chosen cover almost 4 orders of very large, the Coriolis force is proportional to cos f, which is magnitude. Dimensional e-folding growth times are shown on the right-hand maximum at the equator, zero at the pole. Particularly for low scale. toroidal fields, the growth rates rise roughly as 1cos2 f with increasing latitude. This is because in the eigenvalue Equation (46), rotational terms are everywhere proportional indicating that the influence of rotation at all latitudes is weak to cos2 f; both remaining equations of motion, in longitude and compared to the magnetic effects. z, contain only Coriolis forces proportional to cos f.As Since at very high latitudes rotation has very little influence, discussed earlier, in the large-n limit with no magnetic field there the growth rate increases linearly with toroidal field, as in present, fluid particles in unstable disturbances of all m the non-rotating case. By contrast, at low latitudes, as toroidal conserve angular momentum which, with no differential field is increased the growth rate rise is proportional to the rotation, is always inhibiting to growing disturbances. When square of the toroidal field. This is because, for weak fields, a toroidal field is added, angular momentum is no longer terms that produce magnetic buoyancy (all terms containing conserved because magnetic stresses exert longitudinal torques. vertical gradients of a0) and magnetic stresses are proportional These stresses are proportional to m and are therefore absent for to the square of the peak field in the coefficient aq, while the m=0. This means that, unless the toroidal field is large Coriolis terms there are independent of toroidal field. Similarly, (∼20 kG in our case), m=0 modes are no longer unstable. We in cq the magnetic terms are proportional to the fourth power of find that a broad range of m > 0 modes are unstable, starting the peak field, while the stabilizing Coriolis terms are with m=1. Each higher m is more unstable than the previous proportional to the field squared, as they are in bq. The one, until the m of maximum growth rate is reached, denoted in stabilizing effect of magnetic stresses is also evident in these Figure 4 by the integers along each growth rate curve. Beyond same coefficients, since the ratio of magnetic stress terms to that m, the growth rates decline again, as the stabilizing effect magnetic buoyancy terms is always proportional to m2, but of magnetic stresses increasingly negates the magnetic buoy- independent of the peak toroidal field, as set by a00. ancy. The higher the latitude, the weaker the Coriolis force, It follows from the same reasoning, verified by calculations thus the lower the m for maximum instability. The drop in m of not shown, that at all latitudes there is instability for all toroidal maximum growth rate accelerates with latitude, because the field amplitudes, no matter how low, for a wide range of non- higher the latitude the shorter are the physical distances in zero m. They just have very low growth rates, of little relevance longitude for the same m, since these are proportional to cos f, to solar cycles, because their e-folding growth times are so long amplifying the magnetic stress for the same m. Not surpris- compared to a solar cycle. By contrast, for toroidal fields of ingly, the latitude for which the most unstable m declines to 2 kG and above, the e-folding times at all latitudes are zero declines with increasing toroidal field. For field of 20 kG substantially shorter than one year; for fields greater than and above, m=0 is the most unstable mode for all latitudes, 4 kG, they are shorter than a month, making them of interest for

11 The Astrophysical Journal, 853:65 (18pp), 2018 January 20 Gilman the appearance of individual active regions at the surface. However, such e-folding times do not indicate how fast a toroidal flux tube with this peak field amplitude takes to traverse the convection zone, or what latitude it will appear at, at the surface. We will see in the next subsection that when differential rotation is added, for low latitudes toroidal fields below a certain value are no longer unstable at all. As shown in Section 3.2, in the absence of rotation, unstable modes driven by magnetic buoyancy have no phase velocity in longitude. When we add rotation, we find that the most unstable modes, depicted in Figure 4, do propagate prograde relative to the rotating frame, at a rate that is a function of latitude and increases rapidly with peak field strength for weak fields, to a maximum of roughly 1.5% of the rotating frame rate when the most unstable m declines toward zero. We have not explored this propagation in detail, but it appears to be a “slow” magneto-rotational oscillation characteristic of global MHD systems in which rotational effects are stronger than magnetic effects. Mathematically, this coupling between rotation and toroidal field comes from the coefficient bq in Equation (46), which is non-zero even if there is no magnetic buoyancy or differential rotation in the system.

3.4. Toroidal Fields with Rotation and Differential Rotation Here we find solutions for the more general case with both toroidal field and differential rotation present. Our results above for differential rotation with no toroidal field indicate that we may find very different results for low, and middle and high latitudes. That is what we find. We first look at low latitudes. Figure 5. Instability domains in low and middle latitudes when toroidal field, fi 3.4.1. Low Latitudes rotation, and differential rotation are all present, showing the toroidal eld threshhold for instability for latitudes lower than 32°.3. Upper curves depict the After extensive surveying of solutions for eigenvalues from toroidal field for which e-folding growth times are respectively five weeks and ° the full quartic Equation (46),wefind that for all latitudes for one week, together with the most unstable m. Note that above latitude 32.3 all m are unstable, details for which are shown in later figures. which the angular velocity in the tachocline increases with z,so for all latitudes f  32.3,afinite toroidal field is needed to produce instability, below which all eigenvalues are real, are times short enough to be relevant to the appearance of new representing neutral waves. Figure 5 lays out this result. We see active regions on the solar surface. We see that the one week e- that the minimum peak toroidal field increases as the equator is folding time requires a toroidal field of almost 12 kG at the approached, reaching ∼8.7 kG at the equator. This result equator, declining only to 11 kG by 15° and about 8 kG at 30°. implies that dynamo processes must amplify the toroidal field Thus to get a flux tube to enter the convection zone above to values above the stability boundary to have a chance to quickly in sunspot latitudes requires at least an 8 kG field. This escape the tachocline and rise to the surface. Thus we have time is much shorter than the timescale for amplification of the found an additional, apparently not previously noticed, toroidal tachocline toroidal field by dynamo processes, particularly the flux storage mechanism, to allow build-up of field amplitudes in situ shearing of poloidal field by the differential rotation. to the point that they may be large enough to produce sunspots The details of unstable mode growth rates for a sample when they rise by magnetic buoyancy or by advection by latitude, 15°, are shown in Figure 6, which displays the growth convection to emerge to the photosphere. The overlap in rate of each unstable m for m up to 15, for toroidal fields latitude between this stable domain and the latitude domain of between 7.52 kG for instability onset, up to 20 kG, for which observed sunspots seems very unlikely to be a coincidence, so the influence of both rotation and differential rotation are weak. this new storage mechanism needs to be considered in addition For each toroidal field choice, the most unstable m is labeled by to, or perhaps even in place of, storage created by negative its integer value. We find at all low latitudes, the first unstable buoyancy of the sub-adiabatic tachocline. We say “in place of,” mode is always m=1. As the toroidal field is increased, the because in the large-n limit we have taken, ordinary negative range of unstable modes increases rapidly and the m for the buoyancy effects are wiped out by the rapid approach to most unstable mode rises. In this example, the highest m=12 temperature equilibrium within a latitudinally very thin tube. that is most unstable occurs for toroidal fields of 10–12 kG, We also show in Figure 5 curves for e-folding growth times above which the most unstable m declines rapidly and the of five weeks and one week (chosen to be short compared to growth rate curve flattens. The m=0 mode is unstable only for the time it takes for dynamo action to replace the toroidal field, toroidal fields 14 kG. Physically what is happening is that, to but close to that estimated for the rise time of a buoyant flux overcome the stabilizing effect of conservation of angular loop through the convection zone), together with the typical momentum, m > 0 must be selected. At instability onset it is longitudinal wave number m of the most unstable mode. These always m=1 because that is always the mode with weakest,

12 The Astrophysical Journal, 853:65 (18pp), 2018 January 20 Gilman

Figure 6. Growth rates of unstable modes at latitude 15° for wave numbers m = 015– , for selected toroidal ﬁelds above the instability boundary. Integers label the most unstable m for each toroidal ﬁeld.

Figure 7. Growth rates of most unstable modes m from Figure 6 as a function of elevation z, for the toroidal fields shown in Figure 6. The peak growth rates but still non-zero, magnetic stress, that can break the angular always occur near z=0.8. Vertical scales cover the same ranges as in Figure 6. momentum conservation. As the toroidal field is increased, magnetic buoyancy rises quickly and can overcome a larger magnetic stress characteristic of higher m modes, so the most typically a few percent higher than that from surface Doppler unstable m rises. For large enough toroidal field, all modes, methods (see, e.g., Figure3 of Howard et al. 1984). This including m=0, are almost equally unstable because the difference has usually been explained as being caused by influence of rotation through Coriolis forces, and differential transmission of the rotation rate at the “anchoring” depth of a rotation, becomes relatively weak. It is also worth noting how rising toroidal loop (Gilman & Foukal 1979), but the inherent powerful magnetic buoyancy is in causing instability with super-rotation we have found may also play a role in rotation and differential rotation. When the peak toroidal field determining the spot rotation speed. is raised from about 1% above critical for instability, to about It is instructive to look also at the profile of growth rates with twice critical, the growth rate is increased by factor of elevation z for the most unstable m as toroidal field is increased. about 100. Figure 7 shows the result. We see that instability occurs for When we look at the longitudinal phase velocities of the lowest toroidal field near z=0.8, which is very close to the most unstable modes as a function of toroidal field peak above maximum of aa00¶¶z, which is the source of magnetic the threshold for instability, we find that just above critical (at buoyancy in Equation (46). As the toroidal field is increased, 7.52 kG) the speed of the modes is about 0.5% above that of the the domain of instability in z expands to both higher and lower rotation rate at the elevation of the peak growth rate for that z, until by 20 kG it nearly fills the range 0.5<

13 The Astrophysical Journal, 853:65 (18pp), 2018 January 20 Gilman arrive at their new elevation with substantially smaller angular momentum relative to their surroundings than they would experience in a layer with no differential rotation. Therefore the downward force combating magnetic buoyancy is much larger. Above 32.3 this gradient reverses, so the radial rotation gradient can reinforce magnetic buoyancy, which there has only Coriolis forces to overcome. And the Coriolis forces decline with increasing latitude, at a rate proportional to cos f, further enhancing instability. As we show next, we get instability for all toroidal ﬁeld amplitudes at these higher latitudes.

3.4.2. Middle and High Latitudes To calculate unstable modes for middle and high latitudes, we need to return to the issue of the thickness of the tachocline. From results in Figure 1 we know that, without toroidal fields but specifying a tachocline thickness such that d = 0.04, there is vigorous, purely rotational instability at latitudes higher than about 50°. We show in Figure 2 that this instability can be neutralized if we assume that the tachocline thickness, as measured by δ, increases with latitude. In order to isolate the effects of toroidal fields, we adopt the profile of δ shown in Figure 2 for all latitudes for which it exceeds 0.04. For latitudes lower than that we retain d = 0.04, since we know of no observational evidence that the mid- or low-latitude tachocline is significantly thinner than given by d = 0.04. The parameter rs is also adjusted to reflect the greater density variation across the thicker tachocline at high latitudes. Anticipating the results presented below, we will find instability for all middle and high latitudes for all toroidal field amplitudes, even very weak ones. Keeping d = 0.04 at middle and high latitudes will only make Figure 8. Growth rates of unstable modes at selected middle and high latitudes the instability even more vigorous. as a function of longitudinal wave number m, for a 30 G peak toroidal field. Our instability results for middle and high latitudes are This represents instability of the vertical rotation gradient induced by the presence of a weak toroidal field. shown in Figures 8–12. In Figure 8 we present growth rates at several latitudes for a relatively weak toroidal field of 30 G, as a function of longitudinal wave number m.Wefind that growth latitude and elevation of the growth rate. Taken all together, rates for a given m increase strongly with latitude; growth rates this means that the growing disturbances are completely for the same m are about four times larger at 45° than they are passive, being just swept along by the local rotation rate at 35°, and another factor of four larger at 55°. All growth rates without reacting back on it. There is no evidence of “super- for a given latitude increase linearly with m up to m=100 and rotation” of any of these modes for weak toroidal fields, from higher. For 35°, m=100 modes have e-folding growth times zero up to toroidal fields approaching 1 kG and m of order 100. of about one year, and for 45° and 55° only of order one month. These properties indicate that the instability is of the Figures 9–12 respectively show the full variation with toroidal differential rotation destabilized by the presence of a toroidal field peak amplitude of growth rates and e-folding times for field. We have checked and found that even if the toroidal field each latitude presented in Figure 8. These figures show that at is independent of z we get instability that differs only all latitudes the growth rate increases linearly with peak quantitatively with z but not much with latitude. These results toroidal field as well as m, so at all latitudes shown for all weak can be understood by considering the action of the vertical toroidal fields (1kG) the growth rates give e-folding times rotation gradient on a perturbation of the toroidal field. Such a much less than a year. It is clear from Figures 9–12 that to get perturbation of wave number m is sheared by the differential any moderation of the growth rates we must go to both higher rotation in such a way as to create a local Maxwell stress, peak toroidal fields and higher m. proportional to both m and the local toroidal field, that extracts If we look at the profiles of growth rate as a function of energy from the differential rotation, growing the perturbation. elevation z we find that for all m and all toroidal fields shown In the absence of any toroidal field, a hydrodynamic the system is unstable at all elevations from bottom to top, with perturbation would be sheared in the same way but would the growth rate profile (not shown) rather closely following the create a Reynolds stress of the opposite sign to the Maxwell profile of toroidal field with z itself. This is very different from stress, which would add energy to, not subtract energy from, what we found for instability at low latitudes, for example, as the differential rotation, so the perturbation would be stable, not shown in Figure 7. Furthermore, a check of the longitudinal unstable. In the MHD case the presence of perturbed toroidal phase velocities from tr shows that the phase speed of the field negates angular momentum conservation in the perturba- unstable modes is essentially independent of both m and tions. We conclude that these results show a form of magneto- toroidal field peak, but depends on both latitude and elevation, rotational instability in the mid- and high-latitude tachocline, with values that closely approximate the rotation w0 at the qualitatively similar to that found by Kagan & Wheeler (2014).

14 The Astrophysical Journal, 853:65 (18pp), 2018 January 20 Gilman

Figure 9. Growth rates of rotational instability induced by selected peak Figure 10. Same as Figure 9, but for latitude 45°. toroidal fields at latitude 35°, for the elevation of peak growth rate, which is always very close to the elevation of the peak in toroidal field, defined by Equation (51). rotation created by even quite weak fields, particularly for latitudes poleward of 45°, imply that these weak fields should As stated earlier, magnetic buoyancy was not included in their be injected quickly into the turbulent convection zone. What model, so they could not assess the relative importance of form they have when they are carried to the solar surface we magneto-rotational and magnetic buoyancy instabilities in the cannot answer with our model. The implication is that most of same model. the amplification of toroidal field must occur at low latitudes If we had kept the tachocline thickness fixed at d = 0.04, where they can be held at tachocline depth by the radial high latitudes would have been even more unstable than shown gradient of rotation found at low latitudes. Menou et al. (2004) – in Figures 8 12. It is clear that for these latitudes magnetic reached a similar conclusion, but their result applied only to fi buoyancy plays no role for weak elds in producing instability. m=0 modes. fi Nevertheless the instability we nd is just as vigorous in terms We can also ask whether including more physics in our of growth rates as for magnetic buoyancy instability at low instability model, such as diffusion of momentum and/or latitudes above the threshold for instability shown in Figures 5 magnetic fields, would fundamentally alter our conclusions. and 6. Magnetic buoyancy effects can be found at high But, as discussed in Gilman (1970) and more recent latitudes when the peak toroidal field is large enough, at least publications, radiative diffusivity χ is orders of magnitude 10 kG, but these are not very relevant because the magneto- larger than either that of momentum ν (the smallest) or of rotational instability will break up any dynamo-generated fi η toroidal field at these latitudes before it can reach such large magnetic elds . This is the physical basis of our original fi assumption that temperature equilibrium occurs much faster in amplitudes. The departures of growth rates from linear pro les fi with m for 1 kG peak toroidal fields are the first sign of perturbations than does diffusion of elds or velocities. From ( ) 21- 321- reactions of the magnetic stresses to start to inhibit rotational Gilman 1970 we take n = 20 cm s , h =´210cms, 721- instability. Pure magnetic buoyancy instability becomes and c =´210cms. In our model latitude and longitude important only at even much higher toroidal field strengths. linear distance is scaled by the tachocline radius R, but we can The magnitude and omnipresence of instability we have scale more conservatively by the tachocline thickness dR. Then found for middle and high latitudes has profound implications the momentum diffusion time for the depth of the tachocline for for the possibility of storage of magnetic flux at tachocline high m modes is roughly (dnR)22m . Then for m = 103, we get depth until it get large enough to generate sunspots at the a timescale of 2 ´ 109 s, or 63 years! The magnetic diffusion surface. The results say that, for the assumptions we have time is 0.63 years, which is starting to be significant for made, such storage is impossible. The instability of differential m = 103 modes for the growth rates shown in Figures 8–12.

15 The Astrophysical Journal, 853:65 (18pp), 2018 January 20 Gilman

Figure 11. Same as Figure 9, but for latitude 55°. Figure 12. Same as Figure 9, but for latitude 75°.

By contrast, the radiative diffusion time is less than one hour. 4. Implications for the Solar Tachocline Similar arguments can be made for the diffusion in latitude for and the Solar Dynamo large n. What all this says is that our results assuming infinitely We have shown above that instability of differential rotation fast thermal diffusion but no viscous or ohmic diffusion should and toroidal fields occurs in very different forms at latitudes m = 3 – be valid out to 10 , as displayed in Figures 8 12.Ifwe equatorward and poleward of the latitude where the vertical had used R instead of dR for our length scale, modes of the ° 3 rotation gradient changes sign, in our model at 32.3. At low same m or n would have diffusion times 10 times longer; this latitudes, no instability due to either magnetic buoyancy or long time would make diffusion in latitude and longitude vertical rotation gradients occurs until the toroidal field is raised irrelevant compared to diffusion in radius. Similarly, if we above a threshold value as large as nearly 9 kG near the assumed modes experience diffusion in z independent of their equator, above which magnetic buoyancy drives the instability. wave number, then the momentum diffusion timescale For latitudes poleward of 32°.3, the vertical rotation gradient is 2 becomes just (dR) n, leading to a value longer by a factor unstable for any non-zero toroidal field, even if that gradient is 2 6 3 m , so a factor 10 for modes with m = 10 ! stable when toroidal field is absent. For large enough ( ) Both Parfrey & Menou 2007 and Kagan & Wheeler longitudinal wave number m, the disturbance growth rates are ( ) 2014 found evidence of magneto-rotational instability at so short that little build-up or storage of toroidal field is middle and high latitudes but absent at low latitudes, as we possible. It has been commonly argued that in the tachocline have found, but in Parfrey & Menou (2007) it was only for toroidal field can be stored for extended periods because it has axisymmetric (m = 0) modes; they claimed magnetic buoy- sub-adiabatic stratification. This mechanism has been invoked ancy instability becomes important only if toroidal fields are numerous times to say the tachocline plays a key role in the of order 100 kG, which is an order of magnitude higher than solar dynamo, by allowing amplification of toroidal fields we have found at low latitudes here. Furthermore, in the limit enough to produce sunspots at the surface at low latitudes. But we have taken, the m=0 mode shows no magneto- the reasoning assumes implicitly or explicitly that the rotational instability but, as seen in Figures 8–12,this perturbations are adiabatic or close to it, rather than being instability grows linearly with m for fixed weak toroidal field isothermal as we have assumed. That implies a lower limit to to very high values (short e-folding times). Since neither the latitudinal scale of disturbances, which disturbances are free Parfrey & Menou (2007) nor Kagan & Wheeler (2014) to go below. Therefore if our physical assumptions are correct, allowed for magnetic buoyancy, neither could capture the magnetic field storage by sub-adiabatic stratification is threshold in toroidal field for magnetic buoyancy instability prevented. On the other hand, we have found a previously to occur at low latitudes. apparently overlooked mechanism to create toroidal field

16 The Astrophysical Journal, 853:65 (18pp), 2018 January 20 Gilman storage, for low latitudes, which has its origin in the strong turbulence driven by the vertical rotation gradient imposed on it increase with radius of the low-latitude rotation rate from the by the convection zone above. If so, what are the implications bottom to the top of the low-latitude tachocline. The latitude of this turbulence for the thermal structure of the tachocline at range for which this occurs is very close to that for the middle and high latitudes? Storage of toroidal magnetic fields occurrence of sunspots, which is very suggestive that this so that they can amplify enough to produce sunspots may be mechanism is what allows toroidal fields large enough to possible only at low latitudes, due to the vertical rotation produce sunspots to be produced. gradient there. The negative buoyancy of the tachocline may be The lack of toroidal field storage capability at middle and neutralized for small magnetic scales by the fast radiative high latitudes need not imply that no toroidal field is produced diffusion, so that storage by the sub-adiabatic stratification of by dynamo action there, just that it cannot get very large. What the tachocline would not be possible. The workings of the solar could be happening is that dynamo action is producing toroidal dynamo may be altered by our results, but flux-transport field there all the time, but due to rotational instability in the dynamos can still work to produce the correct butterfly tachocline it is quickly injected into the convection zone where diagrams for sunspot cycles. The “extended solar cycle” may it can rise or be carried to the photosphere in the form of be produced by rotational turbulence at high latitudes that ephemeral active regions. The latitude belt where production of injects low-amplitude magnetic fields into the convection zone, fi toroidal eld is largest can still migrate toward the equator, which can reach the photosphere in the form of ephemeral fl carried there by meridional circulation, as demonstrated in ux active regions. ( ) transport dynamo models Dikpati 2016 . From the results in Within the mathematical context of the instability eigenva- – ° Figures 7 12, we can argue that the closer to 32.3 this peak lues and eigenfunctions themselves, there remains the question fi toroidal eld gets, the larger it can become, because the the of what is the eigenfunction structure with elevation when the rotational instability growth rates continually decline. Once this full two-boundary eigenfunction problem is solved. This was peak crosses the latitude where the vertical rotation gradient addressed in Mizerski et al. (2013) and elsewhere for the changes sign, much greater build-up is allowed because the fi Cartesian problem with no rotation or differential rotation. toroidal eld no longer immediately escapes into the convec- There it was found that if the instability was “direct,” which tion zone. means the eigenvalue (both real and imaginary parts) goes to It should be possible to test the concept for the solar dynamo zero at the instability boundary, then the eigenfunction does described above using currently available flux transport ( ) peak at the elevation z of the maximum in local growth rate. dynamo models, such as described in Dikpati 2016 . This But if the instability is “oscillatory,” meaning a non-zero real might be done by making the “α-quenching” parameter part of the eigenvalue at the instability boundary, then the commonly used in such models a function of latitude, set relationship between the local analysis and the two-point close to zero for latitudes above 32.3 and a much higher value boundary value results is much more complex. What we have for latitudes below that value. Alternatively, a parameter that is found is that without rotation or differential rotation, the set to capture the effect of toroidal flux leaving the tachocline instability is direct, as in the much more extensively studied can itself be a function of latitude. These changes could be significant particularly when modern data assimilation methods Cartesian case. But when rotation or rotation and differential are included in flux transport dynamo models to carry out rotation are added, the situation is less clear. In both cases, observing system simulation experiments or making solar cycle there is a non-zero real part to the eigenvalue as the instability ( boundary is approached but, particularly when there is predictions with such models Dikpati & Anderson 2012; ( ) Dikpati et al. 2014, 2016, 2016). In addition, our results offer longitude dependence m ¹ 0 , the real part can be directly guidance on what amplitude of toroidal field to include in related to the local rotation of the system at the latitude and models for the recently discovered tachocline nonlinear elevation of the local analysis This is precisely true in our fi oscillations (Dikpati et al. 2017). results for the case that nds instability from differential fi Separately from the dynamo, our hydrodynamic instability rotation in the absence of toroidal elds. It is unclear whether fi “ ” fi results suggest the tachocline may get thicker with latitude this result quali es as oscillatory in the de nitions used in ( ) above about 50°. This is something helioseismologists can look Mizerski et al. 2013 and elsewhere. It would be good if the fi for in current or future data. In parallel, very high spatial experts on this particular issue were to sort out this signi cant resolution, very low diffusivity hydrodynamic turbulence point in the rotating case, since we know that the tachocline simulations are needed to test whether rotational instability of rotates, and rotates differentially. the middle- and high-latitude tachocline does thicken it to the There is another consideration, namely what happens when point where the instability is neutralized. Similar simulations numerical simulations of the growth of unstable modes to the that include MHD could test the validity of our assumptions of level nonlinear effects become important. While we cannot temperature equilibrium for perturbations that are at the same know the answer for sure before the simulations are done, it time essentially momentum and magnetic field diffusion-free. seems plausible that finite disturbance amplitude will occupy at How well are these assumptions satisfied in turbulence least the whole elevation range for which local instability was simulations in a rotating spherical shell, as opposed to assumed found, and probably more, due to penetration of the ab initio in an analytical theory of tachocline instabilities? perturbations into the stable domain in the lower half of the domain. All these areas are topics for future exploration. Clearly new 5. Conclusions and Future Studies or revisited helioseismic analyses are needed to determine the If the results we have found are valid for the solar tachocline, depth and structure of the middle- and high-latitude tachocline then it seems clear we need to rethink our view of both and the degree and type of turbulence there. High-resolution, tachocline structure and dynamics. The tachocline may be low-diffusivity simulations of rotationally induced turbulence substantially thicker at high latitudes than elsewhere, due to are needed for the tachocline at all latitudes, which can verify

17 The Astrophysical Journal, 853:65 (18pp), 2018 January 20 Gilman or refute our analytical results in both the hydrodynamic and Bowker, J. A., Hughes, D. W., & Kersale, E. 2014, GAFD, 108, 553 MHD cases. They can also help us better understand the Burington, R. S. 1953, Handbook of Mathematical Tables and Formulas ( ) dynamical and MHD interactions between the tachocline and Sandusky, OH: Handbook Publishers Inc. fl Cattaneo, F., Chiueh, T., & Hughes, D. W. 1990, JFM, 219, 1 convection zone at all latitudes. New ux-transport dynamo Cline, K. S., Brummell, N. H., & Cattaneo, F. 2003, ApJ, 599, 1449 calculations can be done with dynamo parameter modifications Corfield, C. N. 1984, GAFD, 29, 19 to see the effects of the loss of flux storage at middle and high Davies, C. R., & Hughes, D. W. 2011, ApJ, 727, 112 latitudes. Dikpati, M. 2012, ApJ, 745, 128 Dikpati, M. 2016, AsJPh, 25, 341 The results shown above were all found for solar tachocline Dikpati, M., & Anderson, J. L. 2012, ApJ, 756, 20 conditions, but there is every reason to believe they are relevant Dikpati, M., Anderson, J. L., & Mitra, D. 2014, GRL, 41, 5361 for all stars that have tachoclines. They provide an expectation Dikpati, M., Anderson, J. L., & Mitra, D. 2016, ApJ, 828, 91 that stars with equatorial accelerations like the Sun will also Dikpati, M., Cally, P. S., McIntosh, S. W., & Heifetz, E. 2017, Nature Sci. tend to have starspots preferentially at low latitudes, as has Rep., 7, 14750 ( Dikpati, M., & Gilman, P. A. 2005, ApJL, 635, L193 been observed for the active K4V star HAT-P-11 Morris Dikpati, M., Mitra, D., & Anderson, J. L. 2016, AdSpR, 58, 1589 et al. 2017). For stars with equatorial decelerations, we might Fan, Y. 2001, ApJ, 546, 509 expect spot zones to be at middle or high latitudes. The sign Fan, Y. 2008, ApJ, 676, 680 change in the vertical rotation gradient should still occur in the Ferriz-Mas, A., & Schüssler, M. 1993, GAFD, 72, 209 ° Gilman, P. A. 1970, ApJ, 162, 1019 neighborhood of 30 due to a balance of torques between low Gilman, P. A. 2015, ApJ, 801, 22 and high latitudes in the star. But the two cases are not mirror Gilman, P. A. 2016, ApJ, 818, 170 images of each other, because the relevant component of Gilman, P. A. 2017, ApJ, 842, 130 Coriolis force is much weaker at high latitudes than low, no Gilman, P. A., & Dikpati, M. 2014, ApJ, 787, 60 matter the sign of the latitudinal differential rotation. Angular Gilman, P. A., & Foukal, P. V. 1979, ApJ, 229, 1179 Gilman, P. A., & Glatzmaier, G. A. 1981, ApJS, 45, 335 momentum per unit mass is much lower near the poles Howard, R., Gilman, P. A., & Gilman, P. I. 1984, ApJ, 283, 373 compared to low latitudes, and so the mechanism for storing Hughes, D. W. 1985a, GAFD, 32, 273 toroidal flux will also be much weaker. It could be that stars Hughes, D. W. 1985b, GAFD, 34, 99 with “anti-solar” differential rotation show a much weaker Hughes, D. W. 2007, The Solar Tachocline (Cambridge: Cambridge Univ. Press), 275 tendency to concentrate spots in particular latitude bands, or Hughes, D. W., & Weiss, N. O. 1995, JFM, 301, 383 may have few if any spots at all, because the rotational storage Kagan, D., & Wheeler, J. C. 2014, ApJ, 787, 21 mechanism is relatively weak at all latitudes. Kersale, E., Hughes, D. W., & Tobias, S. M. 2007, ApJL, 663, L113 Menou, K., Balbus, S. A., & Spruit, H. C. 2004, ApJ, 607, 564 The National Center for Atmospheric Research is supported Mizerski, K. A., Davies, C. R., & Hughes, D. W. 2013, ApJS, 205, 16 Moreno Insertis, F., Schüssler, M., & Ferriz-Mas, A. 1992, A&A, 264, 686 primarily by the National Science Foundation. The author is Morris, B. M., Hebb, L., Davenport, J. R. A., Rohn, G., & Hawley, S. L. 2017, supported primarily by TIAA. The author thanks Mausumi ApJ, 846, 99 Dikpati for reviewing the manuscript, and an anonymous Nelson, N. J., Brown, B. P., Brun, A. S., Miesch, M. S., & Toomre, J. 2014, referee for helpful comments that led to improvements in the SoPh, 289, 441 Parfrey, K. P., & Menou, K. 2007, ApJL, 667, L207 presentation. Parker, E. N. 1955, ApJ, 121, 491 Parker, E. N. 1975, ApJ, 198, 205 ORCID iDs Parker, E. N. 1977, ApJ, 215, 370 Schmitt, J. H. M. M., & Rosner, R. 1983, ApJ, 265, 901 Peter A. Gilman https://orcid.org/0000-0002-1639-6252 Schüssler, M. 1979, A&A, 71, 79 Schüssler, M. 1980, A&A, 89, 26 Skinner, D. M., & Kilvers, L. J. 2013, MNRAS, 436, 531 References Spruit, H. C., & van Ballegooijen, A. A. 1982, A&A, 106, 58 Thelen, J.-C 2000, MNRAS, 315, 155 Acheson, D. J. 1978, RSPTA, 289, 40 Tobias, S. M., & Hughes, D. W. 2004, ApJ, 603, 785 Acheson, D. J., & Gibbons, M. P. 1978, JFM, 85, 743 van Ballegooijen, A. A. 1982, A&A, 113, 99 Antia, H. M., & Basu, S. 2011, ApJL, 735, L45 Vasil, G. M., & Brummell, N. H. 2008, ApJ, 686, 709 Balbus, S. A. 1995, ApJ, 453, 380 Vasil, G. M., & Brummell, N. H. 2009, ApJ, 690, 783 Barker, A. J., Silvers, L. J., Proctor, M. R. E., & Weiss, N. O. 2012, MNRAS, Watson, M. 1981, GAFD, 16, 285 424, 115 Weber, M. A., & Fan, Y. 2015, SoPh, 290, 1295