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

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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 [FORMULA] embedded in a simple shear [FORMULA]. 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

[EQUATION]

[EQUATION]

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

Inside the vortex, the velocity field can be written

[EQUATION]

[EQUATION]

where [FORMULA] is the aspect ratio of the elliptic patch (a and b denote the semi-axis in the x and y directions respectively) and [FORMULA] the vorticity. We can check that the fluid particles move at constant angular velocity [FORMULA] along concentric ellipses with aspect ratio q.

For an incompressible two-dimensional flow, it is convenient to introduce a streamfunction [FORMULA] defined by [FORMULA] (where [FORMULA] is a unit vector normal to the flow). For a given vorticity field, the streamfunction is solution of the Poisson equation

[EQUATION]

Inside the vortex, we find that

[EQUATION]

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

[EQUATION]

Outside the vortex, we have to solve the Poisson equation

[EQUATION]

with boundary conditions at infinity

[EQUATION]

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

[EQUATION]

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

[EQUATION]

with boundary conditions at infinity

[EQUATION]

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

[EQUATION]

[EQUATION]

with [FORMULA] and [FORMULA]. The vortex boundary is an ellipse with parameter [FORMULA] satisfying

[EQUATION]

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

[EQUATION]

Alternatively, we have

[EQUATION]

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

[EQUATION]

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

[EQUATION]

where B, [FORMULA] and [FORMULA] are some constants. At large distances, [FORMULA] and [FORMULA] (where r and [FORMULA] 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

[EQUATION]

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

[EQUATION]

We must also satisfy the continuity of the tangential velocity [FORMULA] 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:

[EQUATION]

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

[EQUATION]

and

[EQUATION]

The relations (A.24) and (A.21) just determine the constants [FORMULA] 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 ([FORMULA]) by:

[EQUATION]

and outside the vortex ([FORMULA]) by:

[EQUATION]

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 [FORMULA] and constant angular velocity [FORMULA]. 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:

[EQUATION]

[EQUATION]

[EQUATION]

The symbols have their usual meaning and [FORMULA] is the vector [FORMULA] rotated by [FORMULA]. The velocity field [FORMULA] is purely two-dimensional and, in Eqs. (B.1)(B.2), there is summation over repeated indices. In Eq. (B.3), the [FORMULA]-function insures that the disk is infinitely thin.

We assume that the disk is turbulent and write

[EQUATION]

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

[EQUATION]

where [FORMULA] 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

[EQUATION]

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, [FORMULA] 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

[EQUATION]

the equations of the problem become

[EQUATION]

[EQUATION]

[EQUATION]

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

[EQUATION]

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

[EQUATION]

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

[EQUATION]

and set

[EQUATION]

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

[EQUATION]

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

[EQUATION]

or, with more convenient notations

[EQUATION]

where we have written

[EQUATION]

The correlation function [FORMULA] is related to C by the Poisson equation

[EQUATION]

Eqs. (B.15) (B.17) and (B.19) are perfectly general but the system is not closed since the evolution of [FORMULA] involves the fourth-order correlation functions [FORMULA] and [FORMULA] 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:

[EQUATION]

[EQUATION]

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

[EQUATION]

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

[EQUATION]

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

[EQUATION]

[EQUATION]

[EQUATION]

where [FORMULA] is the Bessel function of order zero. The solution (B.26) satisfies the Laplace equation [FORMULA] for [FORMULA] [see Eq. (B.19)]. In the plane [FORMULA], we have [FORMULA]. Eq. (B.19) implies that [FORMULA] must be related to [FORMULA] in a special manner. To see that, we integrate the Poisson equation from [FORMULA] to [FORMULA] (where [FORMULA] is a positive constant) and let [FORMULA]. Since [FORMULA] and [FORMULA] are continuous at [FORMULA], but [FORMULA] is not, we have:

[EQUATION]

Hence [FORMULA], or

[EQUATION]

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

[EQUATION]

[EQUATION]

[EQUATION]

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

[EQUATION]

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

[EQUATION]

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

[EQUATION]

This result differs from the usual dispersion relation (77) in the occurence of [FORMULA] in place of [FORMULA] 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 [FORMULA] is quadratic in k with a minimum at [FORMULA]. The disk will be unstable to some wavelengh [FORMULA] provided that [FORMULA], i.e.:

[EQUATION]

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, [FORMULA] should be replaced by [FORMULA] since small-scale fluctuations are basically three-dimensional. When the turbulent dispersion [FORMULA] dominates over the sound velocity like in our study, we obtain

[EQUATION]

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.

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Online publication: April 17, 2000
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