Astrophysics, Vol. 58, No. 1, March , 2015
VORTICES IN ROTATING GRAVITATING GAS DISKS
M. G. Abrahamyan
Linear and nonlinear vortex perturbations of a gravitating gaseous disk are examined in the geostrophic and post-geostrophic approximations. The structures of the isolated monopole and dipole vortex (modons) solutions of these equations are studied. Two types of mass distributions in dipole vortices are found. The first type of modon is characterized by an asymmetrically positioned single circular densification and one rarefaction. The second type is characterized by two asymmetrically positioned densifications and two rarefactions, where the second densification-rarefaction pair is crescent shaped. The constant density contours of a dipole vortex in a light gas disk coincide with the streamlines of the vortex; in a selfgravitating disk the constant density contours in the vortex do not coincide with streamlines. Possible manifestations of monopole and dipole vortices in astrophysical objects are discussed. Keywords: vortex; monopole; dipole; structure: gravitating disk
1. Introduction
The nonlinear equations for the dynamics of two-dimensional vortices are very important in the physics of the ocean and atmosphere, in plasma physics, and in astrophysics. All of these vortical structures are described by nonlinear equations of the same type. In hydrodynamics, this is the well-known equation [1]
Erevan State University, Armenia; e-mail:
[email protected]
Original article submitted November 13, 2014; accepted for publication December 5, 2014. Translated from Astrofizika, Vol. 58, No. 1, pp. 105-119 (February 2015). 0571-7256/15/5801-0089 ©2015 Springer Science+Business Media New York
89
∂ (1 − ∆)ψ − ν 0 ∂ψ − (e z × ∇ψ )∇∆ψ , ∂t ∂y
(1)
which describes nonlinear Rossby waves in the atmosphere [2] and drift nonlinear waves in plasmas [3]. Here ψ (x, y, z ) is the stream function, which is related to the velocity by v = e z × ∇ψ . In the case of plasmas, ψ is also the electrostatic potential; the constant ν 0 is determined by the gradient of the density of the equilibrium state. An exact solution of this equation has been obtained [4] for a stationary isolated dipole vortex (modon) moving along the Y axis on rotating shallow water. A solution of the same form was obtained later for a large number of similar equations [5-10].
More complicated vortex structures have been studied by numerical methods and
laboratory experiments in a number of papers [11-18]. The equilibrium of astrophysical disks is ensured by a gravitational field (external or intrinsic). Nonlinear vortex perturbations of uniformly rotating gravitating masses with a barotropic equation of state have been examined [9]. In the limits of random, short-wavelength (much shorter than the Jeans wavelength, i.e., λ > λ J ) perturbations, these nonlinear equations transform [9] into Eq. (1) [1]. In this paper, linear and nonlinear equations for the perturbations of a gravitating gas disk with differential rotation are examined in the geostrophic and post-geostrophic approximations. The structures of isolated monopole and dipole vortices are obtained and studied for the case of weakly differential or uniform rotation. The form of the distortions of the distribution of mass in the vortical regions of a disk are studied. Two types of mass distributions in modons are obtained. It is shown, in particular, that the constant density contours coincide with the streamlines of a dipole vortex in the case of short-wavelength perturbations, but not in the long-wavelength limit.
2. Model and basic equations for linear perturbations We consider a gravitating gas disk with a volume mass density ρ(r ) rotating at an angular velocity of Ω (r ) about the Z axis. We study two dimensional perturbations in the plane of the disk, neglecting its vertical structure, and take any perturbed function f to have the form f → f 0 (r )+ f (r , ϕ, t ), where f0(r) describes the equilibrium state and f (r , ϕ, t ) is a small, but finite, perturbation.
We assume that the perturbations are isentropic, i.e., S = const, so that the enthalpy H (S , P ) = H (P ), where
P is the pressure and dH = ρ −1 dP = cs2 ρ −1 d ρ ,
(2)
where cs is the speed of sound. Equation (2) is evidently the equation of state for the gas in the disk.
In a cylindrical coordinate system rotating with angular velocity Ω 0 ≡ Ω(r0 ) (see Fig. 1) the perturbed state
of the disk is described by the two-dimensional hydrodynamic equations:1 1
Here and in the following a prime denotes differentiation of equilibrium parameters of the disk with respect to the radial coordinate r.
90
y
R
x a
u O
r r0 ϕ
Fig. 1. The local coordinate system.
d v dt + 2 Ω 0 e z × v + e ϕvr r Ω ′ + ∇Φ = 0 ,
(3)
d ρ dt + ρ∇v = 0 ,
(4)
where the full velocity is given by V = V0 + v ,
V0 ≡ e ϕ (Ω − Ω 0 )r ,
(2')
Φ is the sum of the perturbations in the gravitational potential U, and the enthalpy Φ ≡ U+ H ,
(5)
d dt = ∂ ∂ t + V0 ∂ r ∂ϕ + v∇ ;
(6)
∆U = 4π G ρ .
(7)
with
and the Poisson equation
In the equilibrium state the disk is axially symmetric and has no radial flow. In Eq. (3) we have used the notation (5) and the condition for radial equilibrium of the disk, Ω 2 r = d Φ 0 dr .
(8)
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With Eq. (2) the continuity equation (4) can be rewritten in terms of enthalpy as
dH dt + cs2 ∇v = 0 .
(9)
Taking the curl (rot) of Eq. (3) and then combining the result with the continuity equation and carrying out some simple transformations, we obtain d dt {[rot z v + 2 Ω + r Ω′] ρ}= 0 .
(10)
A similar formula for vortical perturbations of a collisional light stellar disk has been obtained before [19], but, because of the pressure anisotropy, its right hand side is nonzero. The expression in curly brackets in Eq. (10) is referred to as the generalized vorticity. This equation shows that for two-dimensional isoentropic perturbations the generalized vorticity is conserved along the streamlines.
Thus, for stationary perturbations the generalized
vorticity is the stream function ψ:
(rot z v + 2Ω + r Ω′) ρ = B(ψ ).
(11)
It has been shown [20] that it is impossible to obtain a localized vortex using the linear function B (ψ ) with
differential rotation. By suitable choice of B (ψ ) in the form of a quadratic function (in the isotropic case in the framework of the geostrophic approximation), a stationary monopole localized non-drifting vortex has been obtained [19] for a differentially rotating stellar disk with a nonmonotonic rotation curve.
Non-drifting stationary vortex solutions can be obtained in a uniformly rotating (Ω = const) gravitating disk even without specifying the function B (ψ ) , since then Eq. (11) can be written in the form of a Jacobian
J {ψ, (rot z v + 2Ω ) ρ}= 0 that is satisfied by arbitrary circularly symmetric vortical perturbations around a point O.
3. Vortices in the geostrophic approximation
In this approximation it is assumed that the gradient pressure force is balanced by Coriolis and gravitational forces. This corresponds to neglecting the inertial term d v dt in the equation of motion (3); i.e., d Ω dt > U corresponds to smallscale perturbations: k 2 k J2 >> 1 or λ λ J . Then ψ = U 2Ω 0 ≡ φ and Eq. (18') transforms to
{(
)
}
B (φ) = ρ 0−1 1 − κ 20 ω2J ∆φ + x β .
(21')
By choosing the function B it is now possible to study the stationary vortical solutions of Eqs. (21) and (21'). Here we shall examine the simplest case of a uniformly rotating disk that is uniform in density: β = 0 and κ 20
= 4Ω 20 , when the generalized vorticity is constant and equal to Γ ρ0 π a 2 , where Γ is the velocity circulation
(integral). We shall assume that the circulation Γ is nonzero only within a circle of radius a ( 0
Γ a there is no density perturbation, so that the vortex is separated from the disk by a thin transition layer of size δ R
