Astron. Astrophys. 356, 1089-1111 (2000)

## 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.

First, we introduce a cartesian system of coordinates and write the shear under the form

where is a constant ( for the Keplerian shear considered in Sect. 2.2). The associated vorticity is .

Inside the vortex, the velocity field can be written

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

Inside the vortex, we find that

where we have taken, by convention, at the centrer of the vortex. On the vortex boundary, the streamfunction is constant with value:

Outside the vortex, we have to solve the Poisson equation

with boundary conditions at infinity

These boundary conditions insure a continuous matching with the shear (A.1)(A.2) at large distances. Introducing the decomposition

the problem (A.8)(A.9) is equivalent to solving the Laplace equation

with boundary conditions at infinity

At this stage, we find it convenient to use elliptic coordinates

with and . The vortex boundary is an ellipse with parameter satisfying

This relation determines the semi-axis a and b of the ellipse in terms of and c:

Alternatively, we have

In terms of elliptic coordinates the Laplace Eq. (A.11) has the simple form

This equation, with the boundary condition (A.12), is solved easily and we obtain outside the vortex

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

In addition, the continuity of on the vortex boundary requires that and B be related by

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:

Then, after some algebra, we find that the continuity of the velocity is satisfied provided that

and

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.

Regrouping all the results, we find that the streamfunction can be expressed inside the vortex () by:

and outside the vortex () by:

## 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 analysis is easiest if we work in a rotating frame of reference. The equations of the problem are provided by the equation of continuity, the equation of motion and the Poisson equation:

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.

We assume that the disk is turbulent and write

where , and are certain constants. With the further assumption that the disk is barotropic, i.e. , we have

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

the equations of the problem become

From the continuity Eq. (B.8), we readily establish that

where and are the values of the surface density in and respectively. We introduce the correlation function of the density fluctuations, defined by:

In homogeneous and isotropic turbulence is a scalar function depending, apart from time, only on the relative distance between the points under consideration. Similarly, we define the quantity

and set

Remembering that , we can rewrite Eq. (B.11) in the form

An equation of motion for can be derived in the following manner. From Eqs. (B.8) and (B.9), it is easy to establish that

or, with more convenient notations

where we have written

The correlation function is related to C by the Poisson equation

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:

Substituting the foregoing expression for the fourth-order correlation functions in Eq. (B.17), we obtain

For sufficiently large values of , the terms in the velocity correlations in Eq. (B.22) will become negligible and Eq. (B.22) will tend to

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.

These equations admit sound waves of the form

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:

Hence , or

Substituting for C, and from Eqs. (B.24)(B.25) (B.26) in Eqs. (B.15)(B.23), we obtain

where are polar coordinates. Eliminating between (B.30) and (B.31) yields

Substituting the foregoing expression for in Eq. (B.29), we arrive at

Using the expressions (B.24) (B.26) (B.28) for C and and remembering that is an eigenfunction of the two-dimensional Laplacian operator with eigenvalue , we obtain the dispersion relation

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).

The function is quadratic in k with a minimum at . The disk will be unstable to some wavelengh provided that , i.e.:

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