7.5: Rotational Stability and Mixing

A complete discussion of the stability of a rotating star is quite complicated and beyond the scope of this book. However, we consider some of the important effects on the stability of rotating stars. The usual approach to the subject of stability involves finding the spectrum of perturbations for which the equations of motion are stable (i.e., the perturbations will damp out with time). A related approach is to use the Virial theorem⁷, which after all is just a spatial moment of the equations of motion. Various physical processes may occur and give rise to an instability:

1. Buoyancy forces that result from thermal stratification
2. Perturbations that may grow in the presence of an angular momentum gradient
3. Instabilities in the presence of a magnetic field
4. Shear instabilities producing flows both parallel and perpendicular to the local gravity field
5. Failure of the equipotential surfaces being surfaces of constant temperature and pressure [that is, ω ≠ ω(s)]
6. Development of a molecular weight gradient as a result of nuclear evolution
7. Diffusion of heat, angular momentum, and the mean molecular weight

Of all these effects, probably the most important for the stability of rotating stars is the various shear instabilities.

a. Shear Instabilities

The existence of a velocity gradient implies the presence of particle interactions resulting from changes in the macroscopic velocity field. These interactions result from the collisions that are the product of the differential stream motion of the gas, and the severity of these collisions is usually characterized by the viscosity n of the material. The viscosity will try to remove the velocity gradient. However, if the shear is too great, the velocity field will break up into turbulent flow. The conditions of the flow can be characterized by a dimensionless number known as the Reynolds number R_e, which for rotating stars is \[R_e \approx \frac{\omega R_*^2}{v}\label{7.4.1}\]

Should this number exceed a critical value, known as the critical Reynolds number, which is about 10³, the flow will break up into turbulent eddies and the smooth macroscopic motion will become chaotic.

It is useful to break the notion of shear motion into two limiting cases. Motion along the equipotential surfaces will be unopposed by gravity and any of the phenomena that arise from the gravity field. Thus, perturbations that produce horizontal shear can grow unopposed except by the dissipative forces that arise from the viscosity of the gas. However, shear instabilities that arise from motions perpendicular to the equipotential surfaces must overcome forces caused by the temperature and perhaps molecular weight gradients. Thus, the star will be much more stable against vertical shear instabilities, and the time scales for their respective growths will be quite different. For the horizontal shear instabilities the time scale is dominated by the viscosity, while for vertical shear instabilities the time scale for development is essentially the thermal, or Kelvin-Helmholtz, time scale. Thus, \[t_h \approx \frac{R_*^2}{v} \quad t_v \approx t_{\mathrm{K}-\mathrm{H}}\label{7.4.2}\]

The nature of the viscosity of stellar material has long been a subject of heated debate. If one calculates the viscosity simply on the basis of the collisional interaction of the atoms of the gas, one will obtain an extremely small number and an associated growth time scale which is long compared to the nuclear time scale for the star. However, if the flow becomes turbulent, then the dominant collisions occur, not between atoms, but between turbulent elements, giving rise to a "turbulent viscosity" which is many orders of magnitude greater than the kinematic viscosity of the atoms themselves. Unfortunately, the theory of turbulent flow is not sufficiently developed to yield reliable values for the turbulent viscosity, so we must rely on empirical values for systems with dimensions vastly smaller than those of stars. Nevertheless, the prevailing opinion seems to be that turbulent viscosity will be many orders of magnitude greater than kinematic viscosity and so shear instabilities will be of considerable importance in bringing about the redistribution of angular momentum within the star.

From the arguments in Section 7.3 [equation \ref{7.3.40}] it seemed likely that the Eddington-Sweet circulation currents could redistribute material and angular momentum on a time scale comparable to the Kelvin-Helmholtz time for a rapidly rotating star. This is the same order of magnitude as the time scale for the development of the vertical shear instabilities. However, it is rather greater than the time scale for the horizontal shear instabilities should they result from turbulent flow. So these horizontal shear instabilities would appear to be the dominant phenomenon that redistributes the material and angular momentum in the most rapidly rotating stars. This would lead to a steady-state rotation law where the angular velocity was constant on equipotential surfaces and had a condition for stability of the form \[\frac{\partial^2\left(\omega s^2\right)^2}{\partial(\operatorname{Cos} \theta)^2} \neq 0\label{7.4.3}\]

The most plausible rotation law that would satisfy these constraints is rigid rotation, and this may well be the only equilibrium law for rapidly rotating stars. However, many questions must to be answered before it can be determined if this law actually exists in these stars.

b. Chemical Composition Gradient and Suppression of Mixing

composition m did not appear on the right-hand side of equation \ref{7.3.5}. In an evolving star, the chemical composition is continually changing as a result of nuclear processes. Thus, for the early-type stars, we expect the convective core to change its chemical composition on a nuclear time scale, causing m to increase with time. This will lead to a discontinuity in the chemical composition at the core-envelope interface. Now imagine a blob of helium displaced upward by the circulation currents into the less dense hydrogen envelope. The forces of hydrostatic equilibrium will tend to restore the higher-density helium to the core, while the circulation currents will try to mix the helium higher in the hydrogen envelope. Fricke and Kippenhahn⁸ have shown that ratio of the circulation velocity to the restoring velocity induced by hydrostatic equilibrium is given by \[\frac{v_c}{v_\mu} \sim \frac{0.3 w^2}{\Delta \mu / \mu_c}\label{7.4.4}\]

Since the greatest value of w which is allowed is unity, and since a pure helium core will produce \(\Delta\mu/\mu_{\mathrm{c}}.0.5\), we would expect the core-envelope interface to be stable against any vertical motion that would allow mixing. For the typical B star where \(\mathrm{w} \approx 0.4\), a reasonably small gradient in the chemical composition will stabilize the star against rotationally driven mixing, so we may expect the stellar evolution scenarios for the upper main sequence stars described in chapter 5 to remain correct. (Recently some two- and three- dimensional model interior calculations have cast doubt on this conclusion, but the issue is far from definitively resolved).

c. Additional Types of Instabilities

Conditions that can lead to instability in a rotating star seem so numerous that some physicists have despaired from finding any angular momentum distribution that is stable for the lifetime of the star, and it may well be true that no such distribution exists. The number and type of instabilities that can occur are indeed legion. However, what is relevant for the theory of stellar evolution is the time scale for the development of these instabilities and what they do to the star. For rapidly rotating stars, shear instabilities are likely to occur and lead to a rotation law where the angular velocity is constant on equipotential surfaces. There are additional constraints on the rotation law. Should an outward displacement that conserves angular momentum produce a perturbation that has a greater angular velocity than the local velocity field, the perturbation will be dynamically unstable and will grow on the dynamical time scale. This basically geometric instability is sometimes called the Solberg-Hφiland instability, and it constrains the angular momentum per unit mass so that \[\left.\frac{\partial\left(\omega s^2\right)^2}{\partial s}\right|_{\rho=\text { const }}>0\label{7.4.5}\]

Thus, angular velocity laws that decrease faster than s^-2 will be dynamically unstable. A similar criterion holds for the Goldreich-Schubert-Fricke instability. However, the time scale for its development is very much longer because this instability basically arises from the removal of buoyancy stabilization of the temperature gradient by thermal diffusion. If we add to the angular velocity constraints the notion that the rotation law should be derivable from a potential [equation \ref{7.1.13}], then the constraints on the angular velocity distribution become \[\frac{\partial\left(\omega s^2\right)^2}{\partial s}>0 \quad \frac{\partial\left(\omega s^2\right)^2}{\partial z}=0\label{7.4.6}\]

The notion that the rotation law should be conservative is largely based on personal prejudice and will be wrong if dissipative forces like those arising from viscosity are present. Under these conditions the criterion for stability becomes \[\left\{\frac{\partial\left[\ln \left(\omega s^2\right)^2\right]}{\partial z}\right\}^2 \geq \frac{v}{k T} \frac{g_{\mathrm{eff}}}{\omega^2(s)}\label{7.4.7}\]

Since the quantity \(v / \mathrm{kT}\) is usually quite small for stars [i.e., of the order of 10^-6(cgs)], the Goldreich-Schubert-Fricke instability is unimportant except in cases of slow rotation and long nuclear time scales. Thus, this instability has been applied to the sun with some interesting results. However, it can be easily stabilized by a molecular weight gradient such as that described by equation \ref{7.4.4}.

Under conditions of rapid rotation, one might expect non-axis symmetric motions to occur that can separate surfaces of constant pressure from equipotential surfaces. Instabilities resulting from such situations are generally referred to as baroclinic instabilities. These and other types of diffusive instabilities we leave to others to discuss.