Appendix A: elliptic vortex in a uniform shear
In his monograph, Saffman (1992) reports the existence of an exact solution of the incompressible two-dimensional Euler equation consisting of an elliptic patch of uniform vorticity embedded in a simple shear . Since this vortex solution is the starting point of our study, we establish in this appendix the condition (13) for its existence and construct the corresponding streamfunction. These results will be useful for subsequent studies.
where is a constant ( for the Keplerian shear considered in Sect. 2.2). The associated vorticity is .
where is the aspect ratio of the elliptic patch (a and b denote the semi-axis in the x and y directions respectively) and the vorticity. We can check that the fluid particles move at constant angular velocity along concentric ellipses with aspect ratio q.
For an incompressible two-dimensional flow, it is convenient to introduce a streamfunction defined by (where is a unit vector normal to the flow). For a given vorticity field, the streamfunction is solution of the Poisson equation
where B, and are some constants. At large distances, and (where r and are polar coordinates) and we recover the usual multipolar expansion of the streamfunction. The condition that the vortex boundary is a streamline destroys most of the terms in the series (A.19). There remains only
We must also satisfy the continuity of the tangential velocity at the contact with the vortex. To that purpose, we need first to express the streamfunction (A.6) inside the vortex in terms of elliptic coordinates. We find:
The relations (A.24) and (A.21) just determine the constants and B appearing in the expression (A.20) of the streamfunction. In addition, the condition (A.23) must be satisfied for a solution to exist. In a given shear, Eq. (A.23) imposes a relation between the vorticity and the aspect ratio of the vortex.
Appendix B: gravitational instability of a turbulent rotating disk
In this appendix, we derive an instability criterion for a turbulent rotating disk in the approximation where the disk is a sheet of zero thickness with constant mean density and constant angular velocity . Our study is inspired by the work of Chandrasekhar (1951) concerning the stability of an infinite homogeneous turbulent medium. To our knowledge, the results presented in this appendix are new.
The symbols have their usual meaning and is the vector rotated by . The velocity field is purely two-dimensional and, in Eqs. (B.1)(B.2), there is summation over repeated indices. In Eq. (B.3), the -function insures that the disk is infinitely thin.
where denotes the velocity of sound. Then, we invoke Jeans swindle (see, e.g., Binney & Tremaine, 1987) to eliminate the centrifugal term, i.e. we write
This process is permissible if we assume that the centrifugal force is balanced by a gravitational force that is produced by some unspecified mass distribution (recall that for an infinite uniform disk, is necessarily vertical and cannot, by itself, compensate the centrifugal force). In our situation, the external force is provided by the sun's gravity. With the further approximation
Eqs. (B.15) (B.17) and (B.19) are perfectly general but the system is not closed since the evolution of involves the fourth-order correlation functions and which are not known. To go further, we shall suppose that these fourth-order correlations can be expressed in terms of the second-order correlations as in a joint Gaussian distribution. For our purposes, this approximation is reasonable and should provide ample accuracy. Then, by virtue of Wick's theorem, we have:
Eqs. (B.15)(B.23) and (B.19) are mathematically similar to the linearized equations which appear in the usual problem of the Jeans instability for a thin rotating disk (see, e.g., Binney & Tremaine 1987). However, their physical meaning is different since Eqs. (B.15)(B.23) and (B.19) have been derived explicitly from the analysis of the correlations in a turbulent medium.
where is the Bessel function of order zero. The solution (B.26) satisfies the Laplace equation for [see Eq. (B.19)]. In the plane , we have . Eq. (B.19) implies that must be related to in a special manner. To see that, we integrate the Poisson equation from to (where is a positive constant) and let . Since and are continuous at , but is not, we have:
This result differs from the usual dispersion relation (77) in the occurence of in place of and by a factor 2. Similar differences with the usual Jeans dispersion relation occur in the analysis of an infinite homogeneous turbulent medium (Chandrasekhar 1951).
This is the generalisation of the Toomre instability criterion (78) to the case where the disk is turbulent. For a real disk, with finite thickness H, should be replaced by since small-scale fluctuations are basically three-dimensional. When the turbulent dispersion dominates over the sound velocity like in our study, we obtain
This result justifies the procedure used in Sect. 3.4 of replacing the sound velocity occuring in Eq. (78) by the more relevant turbulent dispersion of the particles. However, in more general situations, the turbulent dispersion needs not be large compared to the velocity of sound and the criterion (B.35) should be used.
© European Southern Observatory (ESO) 2000
Online publication: April 17, 2000