BAROCLINIC INSTABILITY IN THE SOLAR TACHOCLINE

Transcription

1 C 214. The American Astronomical Society. All rights reserved. Printed in the U.S.A. doi:1.188/4-637x/787/1/6 BAROCLINIC INSTABILITY IN THE SOLAR TACHOCLINE Peter Gilman and Mausumi Dikpati High Altitude Observatory, National Center for Atmospheric Research, 38 Center Green, Boulder, CO 837-3, USA; Received 213 May 31; accepted 214 April 4; published 214 May 5 ABSTRACT The solar tachocline is likely to be close to a geostrophic thermal wind, for which the Coriolis force associated with differential rotation is closely balanced by a latitudinal pressure gradient, leading to a tight relation between the vertical gradient of rotation and the latitudinal entropy gradient. Using a hydrostatic but nongeostrophic spherical shell model, we examine baroclinic instability of the tachocline thermal wind. We find that both the overshoot and radiative parts of the tachocline should be baroclinicly unstable at most latitudes. Growth rates are roughly five times higher in middle and high latitudes compared to low latitudes, and much higher in the overshoot than in the radiative tachocline. They range in e-folding amplification from 1 days in the high latitude overshoot tachocline, down to 2 yr for the low latitude radiative tachocline. In the radiative tachocline only, longitudinal wavenumbers m = 1, 2 are unstable, while in the overshoot tachocline a much broader range of m are unstable. At all latitudes and with all stratifications, the longitudinal scale of the most unstable mode is comparable to the Rossby deformation radius, while the growth rate is set by the local latitudinal entropy gradient. Baroclinic instability in the tachocline competing with instability of the latitude rotation gradient established in earlier studies should be important for the workings of the solar dynamo and should be expected to be found in most stars that contain an interface between radiative and convective domains. Key words: dynamo instabilities stars: rotation Sun: interior Sun: magnetic fields Sun: rotation 1. INTRODUCTION There has been extensive analysis of instabilities that could occur in the solar tachocline (Parker 1979; Knobloch & Spruit 1982; Spruit & van Ballegooijen 1982; van Ballegooijen 1983; Schmitt & Rosner 1983; Spruit & Knobloch 1984; Cattaneo & Hughes 1988; Moreno Insertis et al. 1992; Balbus & Hawley 1994; Gilman & Fox 1997; Schatzman et al. 2; Arltetal. 25; Arlt & Sule 27; Kitchatinov & Rüdiger 29). The work of Acheson (1978) and Zhang et al. (23) is also quite relevant to solar and stellar tachoclines. Instability can arise in the tachocline because there is energy stored in the latitudinal and radial gradients of the rotation that must be present, as well as well as in latitudinal and radial gradients of toroidal field that should be present due to the action of the solar dynamo. The HD instability problem for latitudinal differential rotation (commonly referred to as barotropic instability, because no thermodynamics is involved) has been analyzed by Charbonneau et al. (1999), Dikpati & Gilman (21), and Garaud (21), who found that latitudinal differential rotation in the tachocline could be unstable to two-dimensional (2D) disturbances if the differential rotation there includes a term proportional to the fourth power of the sine of the latitude, or if shallow water type radial fluid displacements are allowed. Dikpati (212) has carried this problem into the nonlinear regime, and shown that for solar conditions a whole spectrum of global modes, peaking for longitudinal wavenumbers m = 1, 2 can be sustained, for which there is a nonlinear oscillation of kinetic energy between these modes and the differential rotation. Much attention has been paid to the case with gradients in latitude of both rotation and toroidal field present (Gilman & Fox1997; Dikpati & Gilman 1999; Gilman 2; Dikpati et al. 23; Dikpati 24; Dikpati et al. 24; Miesch et al. 27). A primary result of that work is that toroidal fields can destabilize latitudinal differential rotation that is stable in the absence of fields. Disturbances with longitudinal wavenumbers m = 1, 2 result from broad toroidal fields, while narrow bands of toroidal field can lead to instabilities with m as high as 7. These instabilities may play a role in producing active longitudes (Dikpati & Gilman 25). These instability studies have been done for 2D (longitudelatitude), shallow water, and 3D MHD systems, in all cases with rotation and toroidal field independent of radius. Very little analysis has been done for the case for which there are radial gradients in toroidal field and rotation. Solving the instability problem when both latitudinal and radial gradients are present is much more difficult (McIntyre 197). Here we examine the instabilities likely to be present due to radial gradients in the absence of latitudinal gradients. The differential rotation in the tachocline is probably at least in part a thermal wind, in which the radial gradient of rotation is related to the latitudinal gradient of temperature or specific entropy, a consequence of geostrophic balance in latitude and hydrostatic balance in radius within the tachocline. With toroidal fields present, the thermal wind itself can be hydromagnetic (Gedzelman 1975). In the hydrodynamic case, instability of thermal winds is called baroclinic instability, first studied in dynamical meteorology. We will use the same terminology here in what follows. In the meteorological and oceanographic context, the literature on baroclinic instability is vast. Some of the classic work on this topic has been summarized for the astrophysical community in, for example, Knobloch & Spruit (1982), Spruit & Knobloch (1984), Knobloch & Spruit (1985), and Tassoul (2). In the early studies of this instability, it was usually assumed that the perturbations were quasi-geostrophic, that is, the Coriolis and horizontal pressure gradient forces were nearly in balance. The perturbation equations describe how the quasigeostrophic system evolved with time. Later studies relaxed this assumption to find unstable modes in the nongeostrophic (Stone 1966; Derome & Dolph 197; Knobloch & Spruit 1985), and even the nonhydrostatic case (Stone 1971). Both of these approximations and the departures from them are relevant to the 1

2 solar tachocline. So too are effects of compressibility (Gross 1997). Long before the existence of the solar tachocline was known, Gilman (1967a, 1967b, 1967c, 1967d, 1969a) and Gedzelman (1972) considered baroclinic instability in the presence of a toroidal field, with application to the Sun in mind. Baroclinic instability appears now to be much more relevant to the Sun, in particular the solar tachocline, than it was before the tachocline had been discovered. Gilman (1967a) discussed mechanisms for generating domains within the convection zone where the average thermodynamic structure can be subadiabatic, and that is still thought to be a possibility (Rempel 25). But the obvious location for baroclinic instability to occur is the tachocline. It was recognized by Knobloch & Spruit (1982), Spruit & Knobloch (1984) and Knobloch & Spruit (1985) that in slowly rotating stars like the Sun, baroclinic instability is likely to occur only near the boundary between the convection zone and the radiative interior below. Recently Balbus et al. (212) and Balbus & Schaan (212) have discussed how the thermal wind needed for baroclinic instability might be established in the solar tachocline and convection zone. Why might baroclinic instability in the solar tachocline be important? Just as it does in the Earth s atmosphere, it can alter the mean zonal winds, i.e., the differential rotation. It can contribute to latitudinal mixing in the tachocline, in competition with global instability of the latitudinal gradient studied in various references cited above. It can be a driver of meridional circulation. It can produce an α-effect that could help generate poloidal field in the tachocline from toroidal field there, and it could limit the growth of toroidal field by disrupting its amplification there due to shearing by differential rotation. It can also establish preferred longitudinal wavenumbers in the flow there, which could create magnetic patterns that might propagate to the surface to be seen in synoptic maps of the surface fields, such as active longitudes. It is even possible that baroclinic instability in the tachocline can drive a dynamo itself. Baroclinic wave dynamos have been studied by Gilman (1969b, 1969c), Braginsky & Roberts (1975), among others. How important these possible effects are will depend on how fast the growth rates of unstable modes are, which, from results already known, will in turn depend on the degree of subadiabaticity of the stratification in the tachocline, as well as the strength of the thermal wind there. Therefore, to answer these questions we need to analyze quantitatively baroclinic instability for parameter values that are plausible for the tachocline. This paper will focus on the purely hydrodynamic problem; later papers will include the effects of magnetic fields. 2. DEVELOPMENT OF THEORY The basic development of the equations to solve is determined by what assumptions are made for the system. Two possible assumptions are particularly important. First, do we take the system to be in hydrostatic balance, or do we allow departures from hydrostatic? Second, do we assume the system is in geostrophic balance, that is, to a first approximation, do we assume that horizontal pressure gradients are balanced by Coriolis forces? In the early studies of baroclinic instability, both assumptions were made, because observations were consistent with them, and because the equations became much simpler to solve. More recently, first the geostrophic and then the hydrostatic assumptions have been dropped, to achieve greater realism. In the present work, we will also assume hydrostatics but not geostrophy. The Rossby number of this system should be substantially less than unity, so traditional baroclinic instability theory should apply. Almost all studies of baroclinic instability assume an ideal fluid, i.e., no diffusion of momentum or heat. We make the same assumption. Because the tachocline is so thin, we also assume the system is Boussinesq, that is, variations in fluid density with radius or height are ignored in all inertial terms, including Coriolis force terms. All these assumptions are the same as made in Knobloch & Spruit (1985), as well as, more recently, in Miesch & Gilman (24) and Gilman et al. (27), from which we will take the equations we solve. Before giving the detailed mathematical formulation, it is instructive to understand intuitively the physics of baroclinic instability. This is discussed using astrophysical terminology in Spruit & Knobloch (1984). The basic idea is that in a stably stratified fluid system in thermal wind balance (hydrostatic with latitudinal pressure gradients balancing Coriolis forces from differential rotation), all thermodynamic variables in the unperturbed state are functions of latitude. Fluid displacements whose trajectories in latitude have slopes that are shallower than the slopes of the temperature or specific entropy surfaces of the reference state can extract the available potential energy from the system to drive growing perturbations. These perturbations transport thermal energy from latitudes where it is higher to latitudes where it is lower (baroclinic instability in the Earth s atmosphere transports heat from warm low latitudes to colder high latitudes). How effective this process is depends on, among other things, how subadiabatic the stratification is for a specified latitude gradient of potential temperature. The more subadiabatic, the more shallow the fluid trajectories must be, which generally means the further the fluid element must travel in longitude and/or latitude to release a given amount of energy. This requires that the spacial scale of the disturbance must be larger with more subadiabatic stratification. We will clearly see this effect in the solutions we present below. In a finite container, such as a spherical shell like the tachocline, there are limits to how large this scale can be; this can bound the amplitude and indeed the occurrence of instability. Also, the Coriolis force from the component of rotation in the radial direction, which in hydrostatic shallow fluids is the only component that is active in the dynamics, varies with latitude, from zero at the equator to a maximum at the poles, which causes baroclinic instability to have different properties at different latitudes. In the results reported below, we will see all of these effects playing roles in determining growth rates and horizontal scales of unstable modes. 3. MATHEMATICAL FORMULATION In developing the equations for baroclinic instability for the solar tachocline, it is convenient to begin with the so-called hydrostatic primitive equations (HPEs) of Miesch & Gilman (24), subsequently developed for instability calculations in Gilman et al. (27). Both of these formulations are more general than we will use here, in that they include MHD effects within the same approximations. In the HPEs, the core assumption is that the fluid shell is thin compared to its radius, and that the disturbances being modeled are large in horizontal scale compared to the shell thickness, which assumption leads directly to the disturbances being hydrostatic, so that the vertical equation of motion contains no time derivative and reduces to an equation for hydrostatic balance for the perturbations. But for this system 2

3 we do not assume that the perturbations are quasi-geostrophic, that is, that to lowest order that they contain a balance between Coriolis forces and horizontal pressure gradients. Therefore, the so-called Rossby number, which estimates closeness to geostrophic balance, need not be much smaller than unity, and no expansion in Rossby number as a small parameter is invoked. This approach means that the equations used can be in the inertial reference frame rather than in a rotating frame (Coriolis forces can still be easily identified in the inertial frame when it is useful to do so). The HPEs can be thought of as a generalization of the socalled shallow water equations (Dikpati & Gilman 21), which are also hydrostatic and not geostrophic, which, in simplest onelayer form, contain only specified vertical coordinate (z) dependence (horizontal velocities independent of z vertical coordinate, vertical velocity a linear function of z). Multilayer shallow water models are possible, as discussed for example in Pedlosky (1987), in which reference state and perturbation variables (including layer thickness) are different in the different layers. For studies of instability of latitude gradients of rotation, such as in Charbonneau et al. (1999), Dikpati & Gilman (21), and Gilman et al. (27), all the geometrical factors that vary between equator and pole in a spherical shell were included. In Gilman et al. (27), the differential rotation was taken to be independent of z while the disturbances could be functions of z. Therefore the solutions still separated in latitude and z, and all latitudinally varying quantities in the reference state could be retained. Here, to isolate effects of vertical gradients in the reference state, it is necessary to confine the equations solved to a fairly narrow band of latitudes (as has commonly been done for baroclinic instability applied to planetary atmospheres and oceans), within which all functions of latitude are taken to be constant in latitude. Therefore, our instability analysis is, in effect, local in latitude. The reference state we perturb will be independent of longitude, but full longitudinal variation in the perturbations is retained in the normal mode formulation. With these restrictions, the perturbation functional forms separate in longitude and latitude (as well as time), leaving a system of ordinary differential equations in the vertical coordinate. We start from Equations (1) (1) of Gilman et al. (27) with all the magnetic terms omitted and follow the same sequence of developments that follow there to derive the perturbation equations. ( t + ( t + u cos φ u cos φ 3.1. Nonlinear HD Equations u λ + w (v cos φ) + cos φ φ z λ + v φ + w z =, (1) ) u+ uv tan φ = 1 p cos φ λ, (2) λ + v φ + w ) v + u 2 tan φ = p z φ, (3) p z = G1/2 θ, (4) ( t + u cos φ λ + v φ + w ) θ + G 1/2 w =, (5) z in which λ, φ, and z are the longitude, latitude, and vertical coordinates; u, v, and w are the corresponding velocities; p is the gas pressure; and θ is the so-called potential temperature. In dimensional terms, potential temperature (commonly used in atmospheric dynamics) is related to the astrophysically familiar specific entropy s by the relation s = c p ln θ. G 1/2 = Nd/Rω c, N being the Brunt Väisälä frequency, d the thickness of the tachocline, R the radius of the Sun at the depth of the tachocline, and ω c the rotation rate of the solar interior. Roughly speaking, G 1 corresponds to the radiative, bottom part of the tachocline, and G 1 to the overshoot, top part of the tachocline (Dikpati & Gilman 21). We will obtain instability results for all G for which instability is found. In the subsections that follow, we will derive equations that apply to the unperturbed reference state that contains only longitude-independent differential rotation, potential temperature and pressure, and linear perturbations about that reference state. Formally, these are found by representing each variable as the sum of an axisymmetric part, denoted by subscript zero, and a perturbation part, denoted by a subscript one. For the reference state differential rotation, we introduce the angular velocity ω, so that we take u = ω cos φ Reference State Equations The zeroth-order equations for the unperturbed reference state are found from Equations (3) and (4) by collecting terms containing only variables with subscript zero. There will be only a differential rotation angular velocity ω, a pressure p, and a potential temperature θ. In this state there is only centrifugal balance in latitude (in the inertial frame; geostrophic balance in the rotating frame), and hydrostatic balance in the vertical, ω 2 cos φ sin φ = p φ and p z = G1/2 θ. (7) From Equations (6) and (7), we can relate the latitudinal gradient in potential temperature to the vertical gradient of rotation, as G 1/2 θ φ = cos φ sin φ z ω2. (8) This single relationship defines the thermal wind in the system we are using. It says that for every reference state rotation that varies with z, there is a corresponding latitudinal gradient of potential temperature or specific entropy required for latitudinal and vertical force balance to be maintained. In this system, the effective gravity G can also be a function of z, soitispossible to model a tachocline that contains both radiative and overshoot parts. If we specify the differential rotation in Equation (8) as a function of both latitude and z, we can determine the latitudinal potential temperature gradient everywhere. Since baroclinic instability is really driven by the latitudinal potential temperature gradient, we can infer some local but general properties of the instability from Equation (8). It says that the latitude gradient has zeroes at the equator, poles, and where the vertical rotation changes sign. Our quantitative instability results below agree with these inferences. The equatorial zero occurs because the Coriolis force in the local horizontal plane vanishes there, and the polar zero comes from the shrinking of latitude circles to zero, so there is no surface across which to transport heat and release energy. (6) 3

4 With specified differential rotation, from Equations (7) and (8), we are able to calculate the pressure and potential temperature of the reference state implied by those choices. To be consistent, we need only assume that θ / φ = at the equator and poles, to match with the right hand side of Equation (8). In the development of instability equations that follows, we will assume that ω is a function of z only. In that case, the partial derivative in z in Equation (8) can be replaced by the ordinary derivative. In addition, in this case an expression for the quantity dθ /dz can be found by integrating Equation (8) once with respect to φ and setting the integration constant (which can at most be a function of z only) equal to zero, since the spherically symmetric part of the reference state is already accounted for by G. The resulting form is dθ dz = sin2 φ 2G 1/2 d 2 dz 2 ω2. (9) 3.3. Boundary Conditions To solve the first-order perturbation equations, we must apply boundary conditions to close the problem. In this first study of tachocline instability from vertical variations in rotation, we confine the analysis to the tachocline itself by taking w = at the top and bottom boundary at all latitudes. These conditions apply rather well to the bottom of the tachocline, since the domain below is very subadiabatic, allowing little penetration of the flow, and has little turbulence. By contrast, flow and field should penetrate the upper boundary; later studies will address this penetration with a multilayer shallow water model as an extension of that discussed in Dikpati & Gilman (21) with guidance from Pedlosky (1987). Since we will be applying the governing equations to a fixed latitude, rather than seeking solutions that are global in latitude, we are effectively treating either an infinite plane tangent to the sphere at the chosen latitude, or a narrow band in latitude that has sidewalls. Neither is that realistic for the Sun, but both make the problem tractable as a 2D (longitude-radius) problem. For most purposes, we will use the infinite plane tangent to the spherical shell at a specified latitude, and assume solutions that are formally periodic in latitude. The wavenumber of these disturbances would tell us the latitudinal scale. In practice, in this paper, we will solve the instability problem assuming the latitudinal wavenumber is zero, so all disturbances are independent of latitude in that plane. In contrast to the case of solutions that are global in latitude, because we are evaluating the reference state to be perturbed at a fixed latitude, there is no longer a pole problem to deal with First-order Perturbation Equations We first give the most general form of the perturbation equations, for which no further assumptions about dependence on latitude, longitude, or the vertical coordinate z are made. In this form, we retain all axisymmetric variables, including v,w, which for the reference state we have chosen would be zero initially. All first-order perturbation variables are given a subscript 1. These perturbation equations are given by u 1 λ + φ (v 1 cos φ) + cos φ w 1 =, (1) z ( ) t + ω λ + v φ + w u 1 + v 1 z φ (ω cos φ) + w 1 z (ω cos φ) (ω cos φv 1 + v u 1 )tanφ = cos φ p 1 λ, (11) ( ) t + ω λ + v φ + w v v 1 + v 1 z φ + w v 1 z +2sinφω u 1 = p 1 φ, (12) p 1 z = G1/2 θ 1. (13) In principle, once we have found the first-order linear perturbation equations, we can also find second-order equations that calculate how the reference state is initially altered by the growing perturbations. Formally we can do this by subtracting these equations from the original nonlinear set, Equations (1) (5), with all variables split into their axisymmetric and perturbation parts, then averaging over longitude to eliminate all terms that are linear in perturbation quantities and retaining all products of perturbation variables. These equations would tell us how the reference state changes initially in response to unstable firstorder perturbations (Dikpati & Gilman 21). In this paper, we will focus on the instability itself, so we omit further consideration of the second-order equations. To illustrate the energy exchanges among the different types more clearly, we could also derive energy equations from the second-order system Eigenvalue Equations Here we make additional simplifications and approximations to reduce the problem to an eigenvalue problem for a complex eigenvalue τ, the real part of which is the oscillation frequency and the imaginary part the growth rate of the unstable mode. We assume that the reference state, represented by ω, θ φ are independent of φ, the other axisymmetric variables v,w are initially zero, and all other functions are evaluated at a fixed latitude. In that case, we can represent all the perturbation variables as functions of z multiplied by the form e i(mλ+nφ τt). With this representation, τ r /m represents the phase speed in longitude, and τ r /n the phase speed in latitude. With these choices, the resulting eigenvalue equations from Equations (1) (13) are imu 1 + incos φ v 1 + cos φ dw 1 dz =, (14) dω i(mω τ)u 1 2sinφ ω v 1 + cos φ dz w 1 = im p 1, cos φ (15) i(mω τ)v 1 +2sinφ ω u 1 = inp 1, (16) i(mω τ)θ 1 + θ φ v 1 + dp 1 dz = G1/2 θ 1, (17) ( G 1/2 + dθ ) w 1 =. (18) dz 4

5 3.6. Reference States We choose reference states to perturb that are fairly simple, but still plausible for the tachocline. We also expect latitude variations in these states, even though, since we are concentrating here on the vertical variations, we will not include such variations explicitly in the equations we are solving. However, we will solve the equations for particular latitudes, and we want the reference state to be locally appropriate for those latitudes. We will take the vertical gradient in rotation to be independent of z at each latitude, but use a realistic profile in latitude from which to calculate local rotation rates for each latitude. The bottom of the shell will be in uniform rotation with latitude, much as is the bottom of the solar tachocline. This implies that the thermal wind or vertical gradient of rotation changes sign at an intermediate latitude, which, for our choice of parameters, is at The high latitude vertical gradient will be significantly larger than the low latitude gradient, as is observed. Therefore, we write the reference state differential rotation we are perturbing as ω = ω c +(r r 2 sin 2 φ r 4 sin 4 φ )z. (19) Obviously, we could choose many other profiles of rotation with depth within the solar tachocline, but we have no observational information about the actual profiles, so we stick with the simplest case, from which much can be learned. In the context of the Earth s atmosphere, the instability of this linear profile is known as the Eady problem, after the original analysis for baroclinic instability by Eady (1949) Relative Merits of Vertically Continuous versus Layer Models The classical treatment of the non-geostrophic version of the Eady problem is that of Stone (1966, 1971). The latter paper is non-hydrostatic as well. Stone showed that there are three instabilities contained in this system (Equations (14) (18)). These include baroclinic instability, the so-called symmetric instability, and Kelvin Helmholtz (KH) instability. Using expansions in appropriate small parameters and asymptotic analysis, Stone developed a map in parameter space for where each of these instabilities is likely to be most important. Due to limitations of such expansions and asymptotic analysis, this map has gaps. Singularities in the equations, where the phase velocity of the mode equals the local flow speed of the unperturbed state, also add complications and limitations. Stone showed clearly that symmetric modes (m = modes in our notation) are most unstable for very large n and therefore are very small scale in latitude, close to the thickness of the layer, the tachocline for our problem; they also occur in only a limited range of the so-called Richardson number, which measures the ratio of Brunt Väisälä frequency to the vertical shear in linear rotational velocity. Similarly, KH instability modes are most unstable for very large m, forwhicheven Coriolis forces are not significant, corresponding to wavelengths in longitude also comparable to the tachocline thickness and even shorter. Including departures from hydrostatic balance in the perturbations reduces the growth rate of these small scale instabilities relative to that of baroclinic modes, making them still less important. Stone showed how to isolate the baroclinic modes that will be global in scale by picking the right longitudinal and latitudinal scales in a plausible way. It is possible to show that all of these instabilities are still present in even a two-layer model, and the same limits as considered by Stone can still be taken to isolate and emphasize global scale baroclinic instability. Phillips (1954) was the first to develop a two-layer model for baroclinic instability, and many other studies have followed using this class of model. Therefore, we can have confidence that, in terms of the physics included, a two-layer model is not too simple to be appropriate for application to the solar tachocline. The extensive use of layer models in the literature and described for example in Pedlosky (1987) also supports that perspective. And unlike models with continuous rotation profiles with height, singularities are generally not a problem for the analysis, and all the gaps in parameter space that limited the results in Stone (1966, 1971) can be filled in using just algebra. Finally, it is much easier to generalize a two-layer model to include toroidal fields in the reference state. Unlike in the geostrophic Eady problem with toroidal field (Gilman 1969a), it appears to be impossible to reduce the perturbation equations to a single differential equation of modest order that can be solved with techniques similar to those used by Stone (1966) and Gilman (1969a), even if the toroidal field is independent of height. With the two-layer model, the problem still is reduced to algebra for the eigenvalue, in general, an eighth-order polynomial. For all the reasons given above, henceforth in this paper we use a two-layer model for studying baroclinic instability in the solar tachocline. It is worth noting that, since the observed tachocline straddles the depth where the stratification changes from a nearly adiabatically stratified overshoot layer, to the much more subadiabatic radiative domain just below it, the actual tachocline can be usefully characterized as having two layers with different stratification parameters. The one-layer shallow water models referenced in the introduction can be generalized to HD and MHD two-layer versions that contain disturbances that are global in both longitude and latitude; the results we obtain below for baroclinic instability in a two-layer tachocline can inform that generalization. 4. TWO-LAYER MODEL 4.1. Levels for Variables and Boundary Conditions In layer models, the variables to be solved for are evaluated at a minimum number of levels, with variables staggered in the vertical coordinate to maximize accuracy and facilitate application of boundary conditions. A two-layer model actually has five levels, when one counts the upper and lower boundaries and the interface between the two layers. Horizontal velocities u, v and pressure p are evaluated at levels 1 and 3, and vertical velocity w and potential temperature θ are evaluated at level 2, the interface between the two layers. Boundary conditions that w vanishes at top and bottom are applied at levels zero and 4. The second subscript on ω denotes the level at which it is evaluated using Equation (19). Pedlosky (1987) contains extensive discussion of the relationships between multilayer and multi-level models. There is a close relationship between, for example, shallow water models that have two layers each of variable thickness, and the simplest two-layer or five-level approximation to our perturbation equations Two-layer Eigenvalue Equations Starting from Equations (14) (18), the two-layer eigenvalue equations are given by imu 1 + incos φ v 1 + cos φ Δz w 2 =, (2) 5

6 imu 3 + incos φ v 3 cos φ Δz w 2 =, (21) i(mω 1 τ)u 1 2sinφ ω 1 v 1 + cos φ 2 dω dz w im cos φ p 1 =, (22) i(mω 3 τ)u 3 2sinφ ω 3 v 3 + cos φ 2 dω dz w im p 3 =, cos φ (23) i(mω 1 τ)v 1 +2sinφ ω 1 u 1 + inp 1 =, (24) i(mω 3 τ)v 3 +2sinφ ω 3 u 3 + inp 3 =, (25) p 3 p 1 G 1/2 θ 2 =, (26) Δz i(mω 2 τ)θ 2 + θ ( v 1 + v 3 φ + G 1/2 + dθ ) 2 2 dz w 2 =. 2 (27) In Equations (2), (21), and (26), Δz is the (equal) thickness of each layer, which, in this approximation, now appears explicitly in the perturbation equations. It also is the vertical spacing between adjacent levels Unstable and Oscillatory Modes in Two-layer System The homogeneous equation set (2) (27) has unique solutions provided that the 8 8 determinant of the coefficients of the left-hand sides vanishes. In general, this results in a fifthorder polynomial for the complex eigenvalue τ. There are no analytical forms of solutions for polynomials of this or higher order, but solutions can be found iteratively from sensible initial guesses. Since there is no diffusion in the system, all unstable modes will come as complex conjugate pairs, one mode growing, the other decaying at the same rate. At most, there can be two of these pairs so there must always be at least one purely oscillatory mode. In parts of the parameter space where no instability is found, there will in general be five oscillatory modes, involving different combinations of restoring forces associated with Coriolis forces, hydrostatic pressure gradients, and advection by the differential rotation. Parameter domains may also exist where there is just one unstable pair plus three oscillatory modes. A complete analysis would look at all of these cases, but our interest is primarily in baroclinic instabilities, because these are most likely to manifest themselves in global patterns, so below we take certain parameter limits within the 8 8 determinant to isolate baroclinic modes Reduced Eigenvalue Equation We wish to focus here on baroclinic instability rather than on symmetric or KH instability, for the reasons given in the previous section. If we take n = we can filter out the symmetric instability, as was done in Stone (1966). KH instability can be avoided provided m is not very large compared to unity. Since from Stone (1966) we know that these two instabilities have maximum growth rates respectively for n or m approaching infinity, what we are omitting modes of instability that are not global, and therefore likely to be of interest primarily for local dynamics of the tachocline. We further assume in this analysis that the subadiabatic stratification of the system is determined locally in latitude by convective and radiative processes not explicitly included in our system, so in the thermodynamic equation (29), we set dθ /dz evaluated at level 2 equal to zero. This simply means the subadiabatic stratification associated with overshooting convection in the tachocline is not modified much by the presence of a thermal wind. With the above simplifications, and using the two-layer version of the thermal wind equation (8), the determinant of the coefficients of the homogeneous equation set (2) (27) reduces to a single polynomial (quartic) equation for the eigenvalue τ, given by m 2 Δz 2 (mω 3 τ)(mω 1 τ)g +2msin 2 φ cos 2 φ ω 2 (ω 3 ω 1 ) 2 τ cos 2 φ (mω 2 τ) ((mω 1 τ)((mω 3 τ) 2 4ω3 2 sin2 φ ) +(mω 3 τ)((mω 1 τ) 2 4ω1 2 sin2 φ )) =. (28) There exist closed form solutions for quartics, but it is easier to solve Equation (28) forτ by trial and error. We do this also so that we can use the same approach for more general cases that come from the fifth-order polynomial system, as well as when we generalize our system to include MHD perturbations about a toroidal field. In this case, which is also amenable to two-layer models, the polynomial for the eigenvalue becomes seventh order. Since τ is complex, we must scan through a complex space. But we are helped greatly by the realization that in baroclinic instability problems of this type, experience has shown that the phase velocity c r of unstable modes, which is related to τ by τ r /m, closely follows the rotation speed at mid-depth at the latitude for which the equations are being solved. This provides a very good initial guess for τ r, from which we can converge rapidly on τ i and get the whole eigenvalue with accuracy of 1 4 or better in just one or two iterations. Our results below verify that c r = ω 2 typically to an accuracy of 1 3 or even better. This property extends to marginally unstable modes with zero growth rate, so it is possible to use this property to estimate the boundary of instability as a function of G and m for all latitudes. For a given latitude and m instability is possible for values of G below this boundary value. In other words, instability should occur if the stratification is sufficiently close to adiabatic in the tachocline Instability Boundary In solving all eigenvalue equations for unstable modes, it is very helpful to map out in parameter space the boundary for instability, where the growth rate τ i approaches zero. This is not always possible in closed algebraic form, but here it is (another advantage of the two-layer model). For all latitudes for which there is a vertical rotation gradient, so ω 1,ω 2,ω 3 are not equal, we can solve Equation (28) forg, assuming τ i =, c r = ω 2, which gives G = 2sin2 φ cos 2 φ ω2 2 (ω 3 ω 1 ) 2 m 2 Δz 2 (ω 3 ω 2 )(ω 2 ω 1 ). (29) Equation (29) says G = at the equator and pole, so there should be no instability there for any m for positive G (subadiabatic stratification). We form detailed solutions that this is indeed the case. At the latitude ( 32. 3) where the vertical 6

7 Figure 1. Upper limit in stratification parameter G of instability as a function of longitudinal wavenumber m for low latitudes. Figure 2. Upper limit in stratification parameter G of instability as a function of longitudinal wavenumber m for high latitudes. rotation gradient is zero, Equation (29) cannot be used because it contains division by zero. G at the instability boundary is formally indeterminate there, but we should expect no instability because at that latitude there is no thermal wind to drive it. We verify below that this is the case. For all latitudes between equator and pole, and away from the latitude where the radial rotation gradient changes sign, we find from Equation (29) the upper limit in G for which there can be instability for each longitudinal wavenumber m. The results are shown in Figures 1 (low latitudes) and 2 (high latitudes). Low latitude G values are plotted as diamonds in Figure 1,high latitude G values as triangles in Figure 2. Since m is discrete, only the points represent real data; the solid curves are an aid to the eye to compare G values for different m at each latitude. In Figure 1, results are for latitudes between 5 and 3 in increments of 5 ; in Figure 2, between 35 and 85 with the same increment. In Figure 1, The boundary in G for a given m increases monotonically as the latitude increases; the curve with lowest G is the 5 curve. The next curve above is for 1, etc. In Figure 2, the lowest curve is for latitude 85, rising monotonically in G for the same m. We see from Figures 1 and 2 that the G value below which there can be instability is a monotonically declining function of m. This means in effect that as the longitudinal scale of the disturbance gets shorter, for it to be unstable requires that Figure 3. Growth rate of unstable modes for G 1 for all latitudes for which there is instability. Only m = 1 is unstable for G 3. For G = 1, m = 1,2 are unstable. The gap between low and high latitude instability is where growth rates approach zero at the latitude where the radial differential rotation changes sign. the subadiabatic stratification be closer to the adiabatic value. This is because, the higher the m, from continuity of mass, the higher is the vertical motion for a given longitudinal motion, and therefore the steeper the slopes of the fluid particle trajectories. More work must be done by the system against the negative buoyancy force that tends to suppress vertical motion. With smaller G, this negative buoyancy force is smaller. It is perhaps surprising that the instability boundary rises in G as midlatitudes are approached from either lower or higher latitudes, since the thermal wind or vertical gradient in differential rotation, which we shall see drives the instability, peaks at latitudes intermediate between either equator and pole, and the latitude where it changes sign. But we must keep in mind that Figures 1 and 2 show only the upper limit to G for instability; they do not indicate the growth rate itself, which we will find does peak not very far in latitude from the peak radial differential rotation, in both low and high latitudes. Similarly, we might infer from Figures 1 and 2 that m = 1 would always be the most unstable mode, since the upper bound in G declines monotonically with m at all latitudes. But again, here, the upper bound in G does not tell us the growth rate, and we show below that, if modes with a range of m values are unstable, the most unstable mode will fall within that range, not at m = 1. On the other hand, for larger and larger G, fewer and fewer m s can be unstable, with m = 1 always the last to be rendered stable. This suggests that the range of unstable m always starts with m = 1; we verify this inference below with detailed calculations of growth rates of unstable modes Growth Rates and Phase Velocities It is clear from Figures 1 and 2 that the larger is the stratification parameter G (the more subadiabatic the stratification is), the narrower is the range of m that is unstable. It is convenient to start searching for unstable modes for the highest G possible, which would apply to the radiative part of the solar tachocline. Figure 3 shows the results of this search for G 1. We find that for G = 1, modes m = 1, 2 are unstable at both low and high latitudes, while for higher G, only m = 1 is unstable. The instability cuts off entirely for G 8. In this case the fluid particle trajectories in longitude and latitude have to be so nearly horizontal as to be unable to fit into the spherical shell and still be long enough to release enough energy to drive the instability. 7

8 Figure 4. Growth rate of the most unstable longitudinal wavenumber for longitudes between 5 and 85, for selected G between 3 and 1 2. Both low and high latitude growth rates approach zero between 3 and 35 as we should expect, since the radial differential rotation, and therefore the latitudinal entropy gradient, change sign in between those latitudes. Not surprisingly, as G is increased, the range of latitudes that is unstable goes down, both shrinking toward midlatitudes, as suggested by the results in Figures 1 and 2. Growth rates are substantially higher for high latitude compared to low latitude modes. This is because both the latitudinal entropy gradient and Coriolis forces are larger in high latitudes. The amplitudes of the growth rates are plotted in dimensionless units. As in earlier instability work (Gilman & Fox 1997; Dikpati & Gilman 21), a growth rate of 1 2 corresponds in the Sun to an e-folding rise time of about 1 yr. Therefore, for these high G values, low latitude modes grow quite slowly, taking two years or longer to grow by a factor e. ForG as high as four, this growth takes a whole sunspot cycle. By contrast, in high latitudes for G = 1thee-folding time can be as short as about four months. From the results in Figure 3, we can infer that in the radiative tachocline baroclinic instability should be occurring at both low and high latitudes, with growth rates short compared to a sunspot cycle, but long compared to the time for formation of a new active region in the photosphere. It is tempting to speculate that an MHD manifestation of these unstable modes could be related to persistent global magnetic patterns identified as active longitudes. The results for high G show that only the very lowest longitudinal wavenumbers are unstable. How does this change as the stratification parameter G is decreased? Figure 4 depicts the growth rates of the most unstable modes at each latitude. We see that at all latitudes, growth rates roughly double for each decline in G by about a factor of three. At the same time, the high latitude growth rates are always about five times greater in high latitudes than in low. The smaller is G, the easier it is for the sloped fluid particle trajectories to convert potential energy of the latitudinal entropy gradient into kinetic energy of the growing disturbances. Furthermore, the latitude range within which instability occurs widens toward the pole and the equator as G is decreased, leading to instability at all latitudes sampled for G.1. The extension to lower latitudes is possible because the greater ability of the growing disturbances to convert potential energy overcomes the weaker Coriolis force as the equator is approached. The poleward extension happens because with steeper slopes in Figure 5. Growth rate of most unstable longitudinal wavenumber for low and high latitude domains, as a function of G. The lower curve is for low latitudes, the upper curve for high latitudes. The wavenumber of each mode is given inside the square boxes, and the latitude where the instability is strongest is indicated to the left of each box. particle trajectories allowed, the unstable modes can be physically smaller in longitude and latitude dimensions. As a result of the spreading in latitude of the unstable domain with declining G, the latitude of peak growth rate migrates away from the latitude where the radial rotation gradient changes sign. This is illustrated in Figure 5, where we have plotted the maximum growth rate found separately in low and high latitudes, together with the latitude where it occurs and the wavenumber for which it occurs. We see that for a wide range of stratifications identified with the overshoot tachocline, the strongest instability occurs near 2 and 65. The low latitude instability branch peaks in the middle of the domain of sunspots; the high latitude branch peaks where spots are never seen. This suggests that baroclinic instability may play quite different roles with respect to low and high latitude toroidal fields occurring in the tachocline. We see in Figure 5 that as G decreases to values within the overshoot tachocline, the maximum growth rate in both low and high latitudes asymptotes to a value of roughly G 1/2.High latitude maximum growth rates are about a factor of five larger than those for low latitudes for the same G. Remarkably, the most unstable wavenumber is virtually the same in low and high latitudes, steadily increasing as G gets smaller. Thus, for all G, we can expect that in a nonlinear system the wavenumber or wavenumbers with largest amplitude will be the same or nearly the same at all latitudes. Again, this possibility needs to be tested with actual nonlinear calculations. Having the same dominant m at low and high latitudes says that the angular dimension of the disturbances are about the same while their linear dimension in longitude is smaller and smaller as the pole is approached. It is also clear from Figures 4 and 5 that for G, characteristic of the overshoot tachocline, the growth rates become quite large, leading to e-folding times of just four months in low latitudes with G =.3 and less than 1 month in high latitudes. These high rates are therefore of interest for understanding the origins and evolution of individual active regions, and their recurrence at certain longitudes over several months to years. Particularly in high latitudes, baroclinic instability may have strong effects on toroidal fields produced there by the solar dynamo. For these lower G values, or more nearly adiabatic stratification that is characteristic of the overshoot part of the tachocline, we can infer that the overshoot tachocline should be 8

9 experiencing vigorous baroclinic instability at almost all latitudes. This instability occurs for a wide range of longitudinal wavenumbers, which should lead to a spectrum of baroclinic waves, or wave packets, of substantial amplitude. These modes should have the power to substantially modify the reference state differential rotation. Just how this occurs requires a nonlinear model that is beyond the scope of this paper Other Properties of Unstable Modes 1. The range of unstable wavenumbers increases as the stratification parameter G is decreased, as suggested by Figures 1 and 2. ForG =.1, the lowest we considered, the range is m = 1,27. Thus, there should be many more unstable modes in the overshoot tachocline than in the radiative tachocline. This is primarily due to the fact that with smaller G there is less resistance to vertical motion, which, from mass conservation, there is more of for high m with the same longitudinal motion. 2. The phase velocities in longitude of all unstable modes is virtually identical to the rotation velocity in the middle of the layer (level 2) at the latitude of the perturbation. These were computed to an accuracy of 1 4 from an initial guess taken to be the rotation rate at latitude φ at level 2. This is a common property of unstable baroclinic modes. 3. The longitudinal scale ( mr cos φ, in which R is the solar radius at tachocline depth) of the most unstable mode at each latitude declines by a factor of 2 or so between equator and 85, the highest latitude for which calculations were done. But estimated relative to the Rossby radius of deformation (Pedlosky 1987), given by L D = G 1/2 R/2sinφ, which measures the relative importance of Coriolis forces and subadiabatic stratification in baroclinic flow, the longitudinal scale is almost always in the range.5.6, changing very little with latitude, indicating the mechanism for instability is the same at all latitudes, despite the large variation in physical scale in longitude. 4. As expected, all unstable baroclinic modes transport specific entropy and temperature from warm latitudes to cold, to release potential energy to drive the instability 5. CONCLUSIONS: APPLICATION TO THE SOLAR TACHOCLINE We have established that baroclinic instability should be occurring at almost all latitudes in both the overshoot (upper) and radiative (lower) parts of the solar tachocline. The only latitudes where unstable modes are absent are at the poles and at the latitude (32. 3) where the vertical rotation gradient, and consequently the latitudinal entropy gradient, change sign. Since the vertical gradient of rotation in the tachocline is maintained by Reynolds stresses from the convection zone above, this instability should lead to finite amplitude disturbances being present in the tachocline at all times. The range of longitudinal wavenumbers that are unstable is much larger in the overshoot than in the radiative tachocline, the most unstable mode has a much higher wavenumber, and the growth rates are much larger. For both overshoot and radiative tachoclines, growth rates are also about five times larger poleward of the null point in the vertical rotation gradient than equatorward of it, due to the stronger vertical rotation gradient there, and therefore the stronger latitudinal entropy gradient, as well as the stronger Coriolis forces. The difference in disturbance scales between radiative and overshoot tachoclines comes from similar differences in the Rossby radius of deformation. In the overshoot tachocline, high latitude modes amplify by a factor of e in as little as 1 days, and in low latitudes, in 1 2 months. These timescales are short enough to be relevant to the timescale for eruption of new surface magnetic features, and comparable to the convective turnover times of deep convection in the convection zone. Therefore, these modes should be playing a substantial role in the overall dynamics and MHD of the tachocline. By contrast, in the radiative tachocline the e-folding amplification factor is about.5 2 yr in high latitudes and 2 2 yr in low latitudes. Therefore, these modes are probably not connected to flux eruption processes, but could play a role in very persistent magnetic patterns such as active longitudes, as well as dynamic features such as the torsional oscillations. In general, all unstable baroclinic modes contain kinetic helicity, so they can act as sources of the α-effect in the tachocline, often invoked in solar dynamos to produce poloidal from toroidal fields. How does the baroclinic instability we have established for the solar tachocline relate to instability of latitudinal differential rotation thought to occur there that were reviewed in the introduction? The comparison is not precise because the modes of instability to latitudinal differential rotation are global in latitude, while the unstable baroclinic modes found here are local. Ultimately, we need to solve for unstable modes arising from both the vertical and latitudinal rotation gradients, but that is beyond the scope of this paper. In the radiative tachocline, only modes with m = 1, 2are baroclinicly unstable. These same m s are the unstable modes found in 2D (Charbonneau et al. 1999), shallow water (Dikpati & Gilman 21) systems with latitude rotation gradient, and have similar growth rates. Therefore, in the case where both rotation gradients are present, we expect that the unstable modes will still be for m = 1, 2 and have properties of both barotropic and baroclinic instability. These modes will transport both angular momentum and heat in latitude. Since energy will be extracted from both gradients, we would expect that the growth rates would be somewhat larger than for modes driven by just one rotation gradient. Furthermore, barotropic instability is not prevented locally by the sign change in the vertical rotation gradient, so these unstable modes should be truly global, not confined to separate low and high latitude regimes. In the overshoot tachocline, the situation is different. From Dikpati & Gilman (21), we still expect global unstable modes due to the rotation gradient in latitude, still confined to modes with m = 1, 2. However, in this case, the most unstable baroclinic modes occur for much larger m, whose growth rates are also much larger. Therefore, in the overshoot tachocline we expect baroclinic instability to dominate, with perhaps a secondary peak in growth rate in modes with m = 1, 2 that are responding globally to the latitudinal rotation gradient. Only nonlinear calculations in the spherical shell with both rotation gradients present can determine what really happens. The relative importance of barotropic (shallow water) and baroclinic instability may vary with latitude, but both are likely to be more intense in high latitudes, poleward of the latitude where the vertical rotation gradient changes sign. We have shown that growth rates of local disturbances in latitude are much higher in high latitudes than low, because of the stronger vertical gradients there, and the global analysis of instability of the latitude gradient of rotation shows that amplitudes of those 9