Journal of Modern Physics, 2013, 4, 18-22
doi:10.4236/jmp.2013.45B004 Published Online May 2013 (http://www.scirp.org/journal/jmp)
Stability and Vorticity Production in Stratified
E. S. Uchava1,2,3, A. G. Tevzadze1, G. D. Chagelishvili1,2,3
1Department of Physics, Faculty of Exact and Natural Sciences, Tbilisi State University, Tbilisi, Georgia
2E. Kharadze Abastumani Astrophysical Observatory, Ilia State University, Tbilisi, Georgia
3M. Nodia Institute of Geophysics, Tbilisi State University, Tbilisi, Georgia
We study local linear non-axisymmetric perturbations in fully stratified 3D astrophysical disks. Radial stratification is
set to be described by power law, while vertical stratification is set to be exponential. We analyze the linear perturba-
tions in local shearing sheet frame and derive WKB dispersion equation. We show that stratification laws of the disk
matter define not only the thermal stability of the disk, but also the efficiency of the potential vorticity production by
rotationg convective turbulence in astrophysical disks. Taken developed convective turbulence we assume nonlinear
tendencies set by linear spectrum and show that vortices are unlikely to be generated in rigid rotatio n flows. In con trast,
differential rotation yields much higher vortex production rate that depends on the disk thickness, distance from the
central object and the spectral characteristics of the developed thermal turbulence. It seems that measurements of the
temperature and density distribution in accretion disks may indicate the efficiency of the turbulence development and
largely define the luminosity characteristic of accreting flows.
Keywords: Accretion Disks; Protoplanetary Disks; Turbulence
Stability and turbu lence in hydrodynamic accretion disks
are often considered as a key phenomena in defining the
anomalous viscosity and corresponding accretion rate in
high energy accretion disks [1,2], as well as in the proc-
ess of planet formation in the core accretion model in
protoplanet ary disks [3- 6] .
In presented research note we give non-axisymmetric
local linear WKB stability analysis of differentially ro-
tating fully stratified astro physical disks and describe the
possibility of po tential vorticity g eneration. We speculate
on the nonlinear developments of the convective turbu-
lence and what will be the consequencies for the vortex
production in such flows.
The horizontal component of the potential vorticity is
thought to be most important in protoplanetary disks for
core accretion modes. In this respect we show that hori-
zontal vortices can only be generated from vertical vor-
ticity if no shear is present. We anticipate the spectral
characteristic of the convective turbulence in differenc-
tially rotating flows base on the growth rates of thermal
nstability. Thus, we are able to estimate the polarity of
the vortices produced in stratified astrophysical flows.
In Sec. 2 we give mathematical formalism of the con-
sidered physical model. We introduce linear perturba-
tions and derive rigid rotation as well as WKB spectum
for differentially rotating flows. We demonstrate th e pos-
sibility of vortex production in stratified flows. The
findings are shortly discussed and the paper is summa-
rized in Sec. 3.
2. Physical Model
Let us consider the Boussinesq flow rotating around a
central gravitating object under the influence of radial
and vertical stratification in cylindrical co-ordinates:
Copyright © 2013 SciRes. JMP
E. S. UCHAVA ET AL. 19
We consider equilibrium state of the disk with radial
power law and vertical exponential stratification:
S indices describe the radial structure
of the disk, H is the vertical stratification length-scale,
while parameter describes the vertical structure. Note
that here differs from standard adiabatic index
can have values less than unity. In this case the direction
of the entropy stratification is reversed. We keep the
value of general aiming to mimic the different heating
and cooling properties of the disk (not explicitly consid-
ered in this paper).
In the thin disk approximation we consider radial and
vertical gravitational acceleration constant over disk height:
where G is the gravitational constant and M is the mass
of central gravitating object. In the equilibrium we con-
sider azimuthal flow V0 = (0, r , 0). Hence, introduc-
ing Keplerian angular velocity:
we may derive the system parameters in equilibrium:
with quasi-Keplerian angular velocity:
Eqs. (7)- (9) set an equilibrium disk model that is baro-
clinic in nature: 0P
. In the case of proto-
planetary disks when pressure normally decreases with
P > 0 ) Equation (13) indicates sub-Keplerian
equilibrium flow ().
2.1. Linear Perturbations
We employ Boussinesq approximation where the change
of density is due to thermal effects and compressibility is
neglected. Equations governing the dynamics of linear
perturbations are inhomogeneous in space due to several
distinctive factors. Among these we can distinguish ra-
dial and vertical stratification of the disk matter, radial
inhomogeneity of the angular velocity and global curva-
ture of the flow. In order to simplify linear analysis we
can deal with these complications separately. We use
local shearing sheet analysis, which is designed to deal
with the latter factor: flow curvature. To deal with back-
ground inhomogeneities due to fully stratified state we
re-scale linear perturbations in global frame in such a
way to remove explicit coordinate dependence in local
frame (see the 2D analog in ). Hence, we split physi-
cal variables into the background components and linear
perturbations as follows:
St SrzS t
Note the specific scalling factors for the perturbations
of velocity and entropy. Using local shearing sheet frame
that co-rotates with the disk flow at r = r0 radius:
we neglect the flow curvature and study effects of the
differential rotation in the form of th e uniform shear flow.
Hence, the angular velocity of th e rotation is reduced to
where the Oort's parameters are defined as follows:
Now we can employ Fourier expansion of linear per-
turbations in space with time varying phase:
where: ()(0) 2
and the following characteristic wavenumbers are intro-
, , , .
Copyright © 2013 SciRes. JMP
E. S. UCHAVA ET AL.
Hence, the system governing the linear dynamics of
perturbations in the limit 22
kk is reduced to the
ktu kuku (23)
Similar system in Boussinesq limit has been already
derived for Lagnrangian perturbations to study the mo-
mentum transport by linear perturbations in fully strati-
fied disks . In present note we focus on the vorticity
production and stratistical properties of nonlinear turbu-
lent state developed due to thermal instability.
The linear perturbations of potential vorticity that can
be also derived from Ertel's theorem is the following:
In the considered stratified flow potential vorticity
production is described by the following linear equation:
dW tkks t
Dynamics of vorticity is defined by potential vorticity
perturbations, that is system invariant in barotropic flows
(kP = kS = 0).
2.2. Rigid Rotation Spectrum
Dispersion equation can be derived straightforwardly in
the case of rigid body rotation, i.e. zero shear limit in
local frame (A=0, B = -0). In this case dispersion equa-
tion is derived using Fourier expansion of spatial har-
monics in time (see Eqs. (19) - (23). Hence, the disper-
sion equation reads as fo llows:
is the frequency of density-spiral waves in stably strati-
fied flows ,
Stability of the linear perturbations in this case is de-
fined by parameter
and is consistent with the axisym-
metric stability criterion derived in :
= 0. Otherwise,
instability is set by
> 0 and exponential damping by
2.3. Non-axisymmetric WKB Spectrum
Within WKB approximation we assume that
Applying ansatz (30) into Eqs. (19-23) we derive the
WKB local dispersion equation of linear perturbations in
the stratified differentially rotating disk flows:
Figrue 1 shows numerical values of the WKB growth
rates in fully stratified accretion disk flow with sub-Ke-
plerian differential rotation, weak radial and unstable
(< 1) vertical stratification laws. Figure shows asym-
metry of the instability, when strongest growth occurs for
linear modes with .
3. Discussion and Summary
We have studied 3D stratified hydrodynamics accretion
disks in local shearing sh eet approx imation. We drive the
stability characteristics, as well as conditions for the
generation of linear perturbations of potential vorticity.
Thermal instability that developes into the convective
/buoyant turbulence tend s to modify the stratification law
of thermal instability in a way to minimize its linear
growth rate. In our case rigid rotation spectrumwill yield:
=0 (see Eq. (26)). In this limit perturbations of the
potential vorticity can not be generated (see Equation
(27) (,) .Wtconstk
On the other hand the vertical and horizontal compo-
Copyright © 2013 SciRes. JMP
E. S. UCHAVA ET AL. 21
nents of the vorticity are coupled (see Eqs. 24). Hence,
horizontal vorticity can be generated only by redistribu-
tion of vertical vorticity and no net production occurs.
The differential rotation introduces modification to
growth rates, and thus turbulence does not tend to 0
0 state any more. In a crude approximation we may esti-
mate that in this case flow with 1
=0 sets up.
Hence, using Eqs. (25,28) and (32) the spectrally inte-
grated vortex production function in time can be de-
dtk k k
Using rough estimate for Oort constant ,
and weak radial stratification we get: 0
Figure 1. Local linear WKB growth rate of convective in-
stability in fully stratified disk with differential rotation.
Here = 0.98, kSH = 0.01, kPH = 0.01. Top graph shows ( kx,
ky) plane, while the bottom panel shows two isocounturs
given for of 0.25 and 0.5 maximal growth rates for the con-
sidered setup. Profound assymetry of the instability is re-
vealed in kxky > 0 area, while the spectrum is symmetric
with respect to kz.
It seems that statistical properties of the turbulence
(sign of the spectrally averaged stochastic entropy per-
turbation) defines the polarity of the generated vortices:
cyclonic or anticyclonic. On the other hand, vortices are
produced most effectively in thicker disks (H) at inner
radia, where the angular velocity is highest.
Mean flow vorticity is negative, thus W < 0 corre-
sponds to the cyclonic circulation and W > 0 to anti-
cyclonic one. Taking into account the spectral asymme-
try of the convective turbulence that most likely will be
similar to the linear growth rates in stratified differen-
tially rotating flows (more spectral power at kxky > 0 and
symmetry with respect to kz) we conclude that the sign of
produced ponetial vorticity is defined by the sign of en-
tropy perturbations. Thus, cooler areas should produce
cyclonic, while the hotter areas anticyclonic vortices,
respectively. We have presented qualitative description
of the vorticity generation mechanism in fully stratified
astrophysical disks. In our analysis we employ both, lin-
ear spectral analysis, as well as nonlinear estimates to
render outcome of developed turbulence. Naturally, deci-
sive conclusion on the possibility and efficiency of vor-
tex stearing by thermal forces in differentially rotating
disks should be given by numerical simulations. Inter-
estingly, it has been already shown that vertical convec-
tion can transport angular momentum outward and in
some cases be self-sustained state . Realistic numeri-
cal simulations of fully stratified disks will need global
high resolution method to properly descrithe effects of
vertical as well as radial stratification.
Presented analysis can be applied to the protoplanetary
disks, where vorticity productio n is essential for the core
accretion model (see e.g. ). Consider early stages of
the disk flattering, when shock waves are produced by
infalling matter. Shock waves produce positive entropy
perturbations, thus yielding the excitation of anticyclonic
vortices. On the other hand, it is belived that long-lived
anticyclonic vortices are most important for early stages
of planet formation, where they can trap dust particles
and give rise to rapid formation of planetesimals. In
highly opaque (thick) accretion disks of compact objects
vorticity can steer turbulence. Thus, different thermody-
namic stratification profiles can yield different anoma-
lous viscosity and luminosity functions of observed ob-
The research was supported by Georgian National Sci-
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