This is the thermal wind relation, which relates the z dependence of the angular velocity — to the baroclinicity of the system (seeEq. [2.83]). By making use of Eqs. (3.82)-(3.85), we thus have
so that the rotation $ ^ $0 is always opposite to the rotation ^ ^ Equation (3.87) is also the condition that makes the tensor M symmetric. Indeed, since the sets ($0, ^0) and ($, can be interchanged, it follows at once that
and the curves £ ■ M£ = constant represent a family of concentric conics. At each point of the configuration, one can thus find their two (orthogonal) principal axes (x, y) so that
where fx and are the components of £ along the principal axes. Because the trace and the determinant of M are invariant with respect to a rotation of the axes, we also have
From Eqs. (3.90)-(3.92) we observe that £ ■ M£ is positive definite if and only if «x and «y are both positive. Hence, the conditions of stability are trace M > 0 and det M > 0, (3.93)
or, returning to the original variables,
dm dz dz dm
Equations (3.94) and (3.95) are often known as the Solberg-H0iland conditions for dynamical stability.
Now, as was shown by Holmboe (1948), the equation governing small axisymmetric oscillations can be brought to the form d 2£
This equation gives the meridional acceleration in the perturbed motion as a result of two forces. The first term on the right-hand side of Eq. (3.96) represents the centrifugal buoyancy. It is directed opposite to the unit vector \m, and it has the magnitude ($ ■ £). Since the vector $ is perpendicular to the surfaces j = constant, it follows at once that only the component of £ perpendicular to these surfaces is active in the generation of centrifugal buoyancy. The second term represents the gravitational buoyancy. It is in the same direction as the effective gravity g, and it has the magnitude |g|(^ ■ $). Thus, only the component of £ perpendicular to the surfaces S = constant contributes to the gravitational buoyancy. The stability of the system depends on the direction of the resultant buoyancy with reference to all permissible displacements £.
In the limit j = 0, stability conditions (3.94) and (3.95) reduce to the single inequality
which is the condition for the temperature lapse rate to be subadiabatic throughout the configuration (see Eq. [2.136]). Not unexpectedly, the solution of Eq. (3.96) then reduces to stable buoyancy oscillations.
In the limit S = constant, the configuration degenerates into a barotrope. In this case, the stability condition (3.94) becomes dj
d m with the solution of Eq. (3.96) being stable inertial oscillations. Note that criterion (3.98) generalizes to homentropic fluids the well-known Rayleigh criterion for an incompressible fluid.*
Given these results, one would be tempted to conclude that, in the general case of a baroclinic star, stability conditions (3.94) and (3.95) are equivalent to conditions (3.97) and (3.98) simultaneously. This is not quite true, as will become apparent from the following discussion.
Since we are mainly interested in the radiative regions of a rotating barocline, let us restrict our discussion to the case for which trace M > 0 (see Eq. [3.91]). Figures 3.1 and 3.2 depict, at any given point, two plausible orientations of the basic vectors. In Figure 3.1 the vector products $0 x and $ x ^ both point along the same direction, so that the determinant of M is positive (see Eq. [3.92]). This implies stability. On the
* See, e.g.,Chandrasekhar, S.,Hydrodynamic andHydromagneticStability, Section 66, Oxford: Clarendon Press, 1961 (New York: Dover Publications, 1981).
contrary, in Figure 3.2 the vector products $0 x and $ x ^ have opposite signs. Their scalar product is therefore negative, and the determinant of M if negative. This implies instability. By virtue of Eqs. (3.82) and (3.84), this determinant identically vanishes when the surfaces j = constant and S = constant coincide. This limiting case corresponds to a neutral state of equilibrium.
In summary, in this section we have considered the dynamical stability of a baro-clinic star with respect to axially symmetric motions. Restricting our analysis to short-wavelength disturbances, we have shown that the radiative zone of a baroclinic star is stable with respect to these motions if and only if, on each surface S = constant, the angular momentum per unit mass Qm2 increases as we move from the poles to the equator. In other words, if the specific angular momentum decreases radially outward on the surfaces S = constant, there exist unstable motions. In geophysics, this form of instability is called symmetric instability.
Not unexpectedly, in the radiative regions of a barotropic stellar model, this instability occurs whenever Qm2 decreases with increasing distance from the rotation axis. In the case of a stably stratified baroclinic star, however, Figure 3.2 shows that the configuration may become unstable with respect to axially symmetric motions (i.e., trace M > 0 and det M < 0) even when N2 > 0 and d (Qm 2)/dm > 0. This is clear proof that stability conditions (3.97) and (3.98) are not, in general, equivalent to the Solberg-H0iland conditions (Eqs. [3.94] and [3.95]).
What is the exact link between the simple model presented in Section 2.7 and the more elaborate discussion made in this section? It is a simple matter to show that these two models are strictly equivalent. Indeed, as was pointed out by Ooyama (1966), the tensor M that corresponds to a rotating fluid layer in the f plane approximation is given by
(see Eqs. [2.135] and [2.136]). Accordingly, we can write
so that the condition det M < 0 implies Ri < 1, and conversely (see Eq. [2.137]). Condition (2.151) is therefore equivalent to the Solberg-H0iland conditions for dynamical instability.
To conclude, let us mention the work of Lorimer and Monaghan (1980), who have made a preliminary numerical investigation of the symmetric instability in differentially rotating polytropes. Following these authors, this instability is a violent one in the sense that, given an initially unstable j-distribution, a slowly rotating barotrope will at once generate meridional currents and nonaxisymmetric motions in the nonlinear regime, where the resulting flow becomes chaotic with a very slow trend to equilibrium. Further studies along these lines would be most welcome.
In Section 2.7 we have considered a basic flow that has a shear in the vertical direction, that is, along the direction of the effective gravity (see Eq. [2.143]). Besides the symmetric instability, this very simple model exhibits two forms of dynamical instability with respect to nonaxisymmetric motions. One of them - the shear-flow instability -occurs when the Richardson number satisfies the condition
(see Eq. [2.137]). In this case, then, instability sets in when the vertical shear is so steep that the destabilizing effect of inertia overwhelms the stabilizing effect of buoyancy. Its maximum growth rates are associated with short-wavelength zonal disturbances. The other one - the baroclinic instability - occurs for almost all positive values of the Richardson number, and it is associated with zonal disturbances of all wavelengths (see Section 2.7.2). These nonaxisymmetric motions may become unstable because the isothermal surfaces and the isobaric surfaces do not coincide in a barocline. Hence, the potential energy of the basic flow can be converted into kinetic energy of baroclinic waves. This is quite different from the shear-flow instability, which is a form of barotropic instability, because it draws its energy mainly from the kinetic energy of the zonal motion.
Not unexpectedly, the case of a rotating star satisfying condition (3.48) is much more complex than the simple problem discussed in Section 2.7. For example, letting Q = Q(m) one can easily see that both vertical and latitudinal shears become possible. In general, for a star rotating with some assigned angular velocity Q = Q(m, z), the stability problem is complicated by the presence of a vertical shear as well as latitudinal variations of both the angular velocity and the temperature over an isobaric surface. In the simple barotropic case, the component of the rotational motion with latitudinal shear will become unstable to disturbances that transfer momentum down the meridional gradient in angular velocity, thus weakening the basic zonal flow. This is the reason why this instability is called a barotropic instability. It disappears only if the surfaces Q = constant and p = constant coincide. (Recall that the shear-flow instability is also a form of barotropic instability, drawing its energy from the component of the rotational motion with vertical shear.) In the general baroclinic case, thus, the basic zonal flow may develop all these instabilities with respect to nonaxisymmetric disturbances:
(a) the barotropic instability, because there is a latitudinal gradient in angular velocity,
(b) the shear-flow instability, because there is a vertical shear in the rotational motion, and (c) the baroclinic instability, because the isothermal surfaces are always inclined to the isobaric surfaces in a barocline.
Although the importance of shear-flow instability has long been recognized, the other two instabilities have received scant attention in the astronomical literature. Important progress has been made by Fujimoto (1987,1988) and Hanawa (1987), who investigated the stability of a baroclinic star with respect to nonaxisymmetric, isentropic disturbances. Their calculations strongly suggest the prevalence of the barotropic and baroclinic instabilities in differentially rotating stars, for all positive values of the Richardson number, at least for short azimuthal wavelengths; the instabilities disappear only if the rotation is strictly uniform at every point. As we shall see in Section 3.6, this is an important result because shear-flow instability generates small-scale eddies wherever condition (3.101) is satisfied. Since there is no reason to expect this inequality to be satisfied at every point in a stellar radiative zone, it is evident that one can hardly justify the presence of turbulence in a rotating star on the basis of shear-flow instability alone.
The stability criterion derived in Section 3.4.2 is based on the assumption that the displaced fluid particles move isentropically (DS/Dt = 0). Whereas viscous effects are negligible in a star, the effects of radiative conductivity may become important, at least for sufficiently small mass elements, because of the smoothing of temperature differences by radiative transfer. To be specific, consider a system that has a subadiabatic temperature gradient so that it is dynamically stable with respect to isentropic motions. If the fluid is unstable without the isentropic constraint due to a slightly adverse angular momentum distribution, the thermal conductivity will thus reduce the restoring force of thermal buoyancy. Hence, it will permit axially symmetric disturbances to grow, with their amplitudes being limited by the thermal conductivity that relaxes the isentropic constraint. As we shall see in this section, two types of thermally unstable motions can occur simultaneously in a baroclinic star.
Consider an axially symmetric star that rotates with the assigned angular velocity Q = Q(m, z). Assume that it satisfies the Solberg-Hailand conditions for dynamical stability with respect to axially symmetric disturbances. We shall consider a simple ideal gas with negligible radiation pressure. However, we shall make allowance for a gradient of chemical composition. By virtue of Eq. (3.8), we thus have p a p T//x, where /x is a function of position and time. Since we are chiefly interested in a dissipation mechanism (i.e., radiative conductivity), we may expect that the most unstable perturbations will be found to have wavelengths that are much smaller than the star's radius. It is therefore expedient to work with a simplified set of equations that approximate the exact equations in a small region of the star. The analysis is restricted to small axisymmetric motions, and we assume that their size is much smaller than any scale height of the equilibrium model. Then, the coefficients in the perturbation equations will be independent of m, z, and t, so that the Eulerian changes may be expanded in plane waves of the form exp[nt + i (km m + kzz)]. (3.102)
Consistent with the above approximations, we may now take S V = 0. We shall also make use of the Boussinesq approximation for compressibility effects; the pressure variations thus contribute little to the density variations.
By virtue of Eq. (3.102), the momentum equation reduces to
pp where £ is a two-dimensional vector with components and £z (see Eq. [3.74]). Note that Eq. (3.103) already incorporates the conservation of angular momentum of each mass element along its path. (This property still holds because we can rightfully neglect viscous friction.) Similarly, by virtue of Eq. (3.3), our approximations lead to the condition k •£ = 0, thus implying that the wave vector k is transverse to the displacement £. Letting next
£ = % a (where a is the unit vector along £) and multiplying Eq. (3.103) by a, we obtain n2% = - S-P (a ■ ^o) - % (a ■ S)(a ■ $o) (3.104)
Outside the central regions where thermonuclear reactions take place, Eq. (3.4) reduces to
The small-perturbation counterpart of this equation can be brought to the form
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