Hydrodynamic Turbulence in Accretion Disks

0.34 MB·English

by  Banibrata Mukhopadhyay

22nd Texas Symposium on Relativistic Astrophysics, Stanford University, December 13-17, 2004 1
Hydrodynamic Turbulence in Accretion Disks
BanibrataMukhopadhyay,NiayeshAfshordi,RameshNarayan
Harvard-SmithsonianCenterforAstrophysics,60GardenStreet,MA02138,USA
Turbulent viscosity in cold accretion disks is likely to be hydrodynamic in origin. We investigate the growth
ofhydrodynamicperturbations inasmallregionofadisk,whichwemodelasalinearshearflowwithCoriolis
force, between two parallel walls. Although there are no exponentially growing eigenmodes in this system,
becauseofthenon-normalnatureofthemodes,itispossibletohavealargetransientgrowthintheenergyof
certainperturbations. Foraconstantangularmomentumdisk,theenergygrowsbymorethanafactorof1000
for a Reynolds number of only 1000, and so turbulence is easily excited. For a Keplerian disk, the growth is
moremodest,andenergygrowthbyafactorof1000requiresaReynoldsnumberofnearlyamillion. Accretion
disks have even larger Reynolds numbers than this. Therefore, transient growth of perturbations could seed
5 turbulenceinsuchdisks.
0
0
2 1. INTRODUCTION quently they again showed[7] that the magnetic field

n dies out when the tidal and Coriolisforces are turned
a off, keeping the Lorentz forces. In two subsequent
The origin of hydrodynamic turbulence is still not
J
papers [8, 9], it was shown through numerical simu-
completely understood. Many efforts have been de-
1
lations that, whereas pure hydrodynamic turbulence
voted to solve this problem, beginning with the work
2
is easily triggeredin plane Couette flow and in a con-
of Kelvin, Rayleigh, and Reynolds at the end of the
stant angular momentum disk, turbulence does not
1 nineteenth century, to more recent work. In most
v cases,a significant mismatch has been found between develop in an unmagnetized Keplerian disk even in
8 the results based on linear theory and experimen- the presence of large initial perturbations. The au-
6 thors argued on this basis that hydrodynamic turbu-
tal (observational) data. In the laboratory, plane
4 lencecannotcontributetoviscosityinaccretiondisks.
Poiseuille and plane Couette flows become turbulent
1
0 at Reynolds numbers, R ≥ Rcexp ∼ 1000, 350, re- Despite the above results, it is probably prema-
5 spectively, while theoretically, plane Couette flow is turetoruleouthydrodynamicturbulenceinrotation-
0 linearly stable at all values of R and plane Poiseuille ally supported disks. Many astrophysicalsystems are
/ is linearly unstable only for R ≥ Rthe = 5772. What known, e.g., star-forming disks, cataclysmic variables
h c
p isthe reasonforthis mismatch? Itwouldappearthat in quiescence, outer regions of AGN disks, etc., in
- for R values in between R = Rexp and R = Rthe, whichthegasiscoldandneutralsothatthereisnegli-
c c
o some new effect allows the system to make a subcriti- gible coupling between the magnetic field andthe gas
r
t cal transition to turbulence in the absence of a linear (seee.g.,[10,11,12]). Inthesesystems,theMagneto-
s instability. Rotational-Instability(MRI) cannotplay arole. How
a
: Inthefieldofastrophysics,hydrodynamicsandtur- do these systems sustain mass transfer in the absence
v
bulencefindextensiveapplications. Accretionflowsin of the MRI? What drives their turbulence? It is pos-
i
X binary stars, young stellar objects and active galactic sible that the turbulence is purely hydrodynamic in
r nuclei must be turbulent in order for the gas to ac- origin. Even in the absence of exponentially growing
a crete. However the actual origin of the turbulence is modes,planeCouette flowisknownto exhibitalarge
unclear and still under debate. Inward mass accre- transient growth in energy for certain initial condi-
tion in a disk occurs as a result of the transfer of an- tions. Isthistransientgrowthresponsibleforthesub-
gular momentum outward by means of some viscous critical turbulence that occurs in this flow, and could
torque in the system. More than three decades ago, a similar phenomenon be responsible for generating
Shakura& Sunyaev[1]andLynden-Bell&Pringle[2] hydrodynamicturbulenceincoldaccretiondisks? Re-
proposed that turbulent viscosity provides the neces- cent laboratoryexperiments on rotating Couette flow
sary viscous torque to cause inward mass transport. inthe narrowgaplimitwithlinearlystablerotational
However, the physical origin of the turbulence in an angular velocity profiles (similar to Keplerian disks)
accretion disk was unclear until the work of Balbus seem to indicate that turbulence does manage to de-
& Hawley [3] who showedthatmagnetizeddisks have velopinsuchflows[13]. Longaretti[14]pointsoutthat
a Magneto-Rotational-Instability (MRI) [4, 5] in the theabsenceofturbulenceinthesimulations[8,9]may
presence of a weak magnetic field. This instability be because of their small effective Reynolds number.
provides a natural means to generate turbulence and Transientgrowthoccurs in a linear system because
transport angular momentum outward. Later, Haw- of the non-normal nature of the associated operator
ley,Gammie &Balbus[6]showedthatthe turbulence [15, 16, 17]. How, and if, such growth might fi-
diesoutiftheLorentzforcesareturnedoffinamagne- nally cause turbulence is not well understood. One
tohydrodynamically turbulent Kepleriandisk. Subse- possibility is that the growth causes fluctuations to
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2 22nd Texas Symposium on Relativistic Astrophysics, Stanford University, December 13-17, 2004
acquire large amplitudes, and non-linear interactions ζˆ(x)eiσtei(kyy+kzz), where σ = σR +iσI is an eigen-
then push the system into self-sustained chaotic tur- value of the problem. After substituting these per-
bulence. turbations in (1) and (2), we solve the combinedOrr-
We present here some results on shearing rotating SommerfeldandSquireequationstoobtainthe eigen-
flowsfollowingthemethodsthathavebeendeveloped system {σ,uˆ,ζˆ}. The detailed set of equations and
to study the evolution of non-normal modes in plane its reduction to an eigenvalue equation form and the
Couette flow. We employ a local approximation, and solution procedure are described in [18].
focus mainly on an eigenvalue analysis of perturba- Asmentionedabove,eveninthe absenceofanyex-
tionsinEuleriancoordinates(fordetails,see[18],and ponentially growing mode in the system, plane Cou-
also[19, 20] for a similaranalysisin Lagrangiancoor- ette flow can exhibit a large transient growth in the
dinates). In the next section, we briefly describe the energy of certain perturbations [15, 17]. This growth
basic model and write down the set of equations to occurs in the absence of non-linear effects and is be-
be solved. Then in §3, we present a few important lieved to facilitate the transitionfrom laminar to tur-
results. We conclude with a discussion in §4. bulent flow. The maximum growth in the perturbed
energy can be expressed as
E
2. MODEL G (t)=optimum t , (3)
K
E
(cid:18) 0(cid:19)
We consider a small patch of an accretion disk where
whose local geometry is described in terms of Carte-
sian coordinates, x = (r −r0), y = r0(φ−φ0). We E = 1 1 k2uˆ†uˆ+ ∂uˆ†∂uˆ +ζˆ†ζˆ dx (4)
assume that the flow is incompressible and extends t 8k2 ∂x ∂x
from x=−1 to +1, between two rigid walls with no- Z−1" #
slip boundary conditions. The y and z directions are
andE istheinitialenergyoftheperturbationattime
0
unbounded with periodic boundary conditions. This
t = 0. By “optimum” we mean that we consider all
flow behaves like rotating Couette flow in the narrow
possibleinitialperturbationsandchoosethatfunction
gaplimit, whenthe unperturbedvelocitycorresponds
that maximizes the growth of energy at time t. To
toalinearshear,U~ =(0,−x,0),andtheCoriolisforce understand the technical details of how to compute
is described by the angular frequency ~ω = (0,0,1/q): the growth, see [18].
Ω = Ω (r /r)q. Here all the coordinates and quanti-
0 0
ties are expressedin terms of dimensionless variables.
Note, q = 3/2 corresponds to a Keplerian disk, q = 1
3. RESULTS
and 2 to a flat rotation curve and a constant angular
momentum disk, respectively, and q = 0 corresponds
The eigenvalue analysis shows that, for plane Cou-
to rigid body rotation. A schematic diagram of the
ette flow and for a rotating flow with q ≤2, there are
flow in the local coordinates is shown in Fig. 1.
no exponentially growing modes in the system, i.e.,
Writing the perturbations as U (x,y,z,t) →
x for no choice of the parameters is there an eigenvalue
u(x,y,z,t), U (x,y,z,t) → U (x) + v(x,y,z,t),
y y with σ > 0. However, for plane Couette flow, previ-
I
U (x,y,z,t)→w(x,y,z,t), P/ρ→P/ρ+p(x,y,z,t),
z ous workers (e.g. [15, 16]) have shown that there is
and combining the linearized Navier-Stokes and con-
a large transient growth even for R = 1000. The in-
tinuity equations, we obtain
terestingpointisthatthemaximumgrowthinenergy
forplaneCouetteflowandaconstantangularmomen-
∂ ∂ ∂2U ∂u 2∂ζ 1
+U ∇2u− + = ∇4u, (1) tum disk (q =2) are very similar. Figure 2 shows the
∂t ∂y ∂x2 ∂y q∂z R
(cid:18) (cid:19) contoursofconstantmaximumgrowth(Gmax)andthe
∂ ∂ ∂U ∂u 2∂u 1 timeatwhichthegrowthismaximized(t )forboth
+U ζ− − = ∇2ζ, (2) max
∂t ∂y ∂x ∂z q ∂z R plane Couette and q = 2. Whereas for q = 2, Gmax
(cid:18) (cid:19)
occurs exactly on the k axis, for plane Couette flow
z
where ζ = ∂w/∂y − ∂v/∂z. Equations (1) and (2) it is slightly off the axis. This is the only difference
are the standard Orr-Sommerfeld and Squire equa- between the two cases.
tions, respectively, except that they now have addi-
tional terms due to the presence of rotation in the
system. We have to solve equations (1) and (2) with
the boundary conditions: u = ∂u/∂x = ζ = 0, at
x=±1 (no slip).
Due to translational invariance in the y and
z directions, the perturbations can be chosen as
u(x,y,z,t) = uˆ(x)eiσtei(kyy+kzz) and ζ(x,y,z,t) =
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22nd Texas Symposium on Relativistic Astrophysics, Stanford University, December 13-17, 2004 3
Figure 1: Unperturbedflow in thelocal shearing box. The arrows represent thevelocity field.
2 2
1.5 1.5
1 1
0.5 0.5
0 0
0 0.1 0.2 0.3 0 0.1 0.2 0.3
3 3
2 2
1 1
0 0
0 0.1 0.2 0.3 0 0.1 0.2 0.3
Figure 2: Contours of (a) Gmax for planeCouette flow, where dashed contours correspond to 10,30,...90; solid to
150,200,...600; dotted to 700,800,...1100; dot-dashed to 1150,1160,...1200; (b) tmax for plane Couette flow, where
dot-dashed contours correspond to 20,25,...45; solid to 50,55,...100; dotted to 110,120,...180; (c) Gmax for q=2 flow,
where contour labels are thesame as in (a); (d) tmax for q=2 flow, where contour labels are thesame as in (b). All
calculations are for R=1000.
Table 1
Figure 3 shows how G and t vary as a func-
max max
Maximum Growth for Various q and R=1000 tion of Reynolds number, R, for plane Couette flow
and q = 2 flow. We see that G scales as R2 and
max
t as R in both cases, again highlighting the sim-
max
q ky kz Gmax tmax ilarity of the two flows. The physical reason is that
1.5 1.21 0 13.04 8.8 a constant angularmomentum disk has zero epicyclic
frequency. Thus, the effect of rotation is essentially
1.7 1.06 0.74 13.4 9
cancelled out, making the basic structure of the sys-
1.9 0.54 1.44 23.75 12.4
tem very similar to that of plane Couette flow.
1.99 0.04 1.8 122.8 27.3
For a given R, when q slightly deviates from 2,
2 0 1.66 1165.9 138.5 growth immediately reduces dramatically due to the
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4 22nd Texas Symposium on Relativistic Astrophysics, Stanford University, December 13-17, 2004
Figure 3: Solid and dotted lines correspond to plane Couette and a q=2 disk respectively.
introduction of a small epicyclic frequency (= κ = turbation at t = 0 is a leading wave with negative
2(2−q)Ω). This indicates the dominanteffect that x-wavevector kx and with |kx| ≫ ky. With time, the
κ has on the fluid dynamics. Table 1 and Fig. 4a wavefrontsare straightenedout by the shear, until at
p
showhowGmax andtmax changeasq decreasesbelow t=tmax,thewavefrontsarealmostradialandkx ∼0.
2. It is interesting to note that, for a q = 2 flow the Then, at yet later times, the wave becomes trailing
growth is maximum for k =0, while for a Keplerian and the energy also decreases. The perturbations are
y
disk (q = 1.5) the growth is maximum for k = 0. verysimilarto the growingperturbationdescribedby
z
Therefore, the location of G moves in the k −k other authors [21, 22, 23].
max y z
planesystematicallyfromoneaxistotheother,asqis
varied. Thus, fora constantangularmomentumdisk,
we need to include verticalstructure in the perturba-
4. DISCUSSION
tions to understand the energy growth, whereas for a
Keplerian disk a 2-dimensional analysis is sufficient.
We find that significanttransientgrowthofpertur-
Even though the growth decreases dramatically as
bations is possible in a shear flow with Coriolis force
q falls below 2, nevertheless, at a given q, if the
between walls (see also [24]). This system is an ideal-
ReynoldsnumberRincreasestoalargeenoughvalue,
ized local analog of an accretion disk. Although the
the growth can become significant. In Fig. 4b, we
systemdoesnothaveanyunstableeigenmodes,never-
show how the peak value of growthincreases with in-
theless,becauseofthenon-normalnatureoftheeigen-
creasing R for Keplerian disks. It is found that G
max modesa significantleveloftransientenergygrowthis
scalesasR2/3andt asR1/3forlargeR. Therefore,
max
possible for appropriate choice of initial conditions.
while the presence of a finite epicyclic frequency has
If the maximum growth exceeds the threshold for in-
a strong dynamical effect on growth, it does not rule
ducingturbulence,itisplausiblethatthismechanism
outalargegrowth. Onejustneedsmuchlargervalues
could drive the system to a turbulent state. Presum-
ofR. ColdastrophysicalaccretiondiskscanhaveRas
ably, once the system becomes turbulent it can re-
high as 1010 or more, so large growth should in prin-
main turbulent as a result of nonlinear interactions
ciple be easily achievedin such disks. Also at smaller
and feedback among the perturbations.
k , G scales as k2/3, while at larger k , G de-
y max y y max In this mechanismof turbulence, the maximum en-
creasesas ky−4/3. At large t, tmax scales as ky−2/3. For ergy growth and the time needed for this growth are
adetailedderivationofthesescalingrelations,see[18]. probably the main factors that control the transition
For a Keplerian flow the maximum growth occurs to hydrodynamic turbulence. As mentioned in §1, for
for k ∼1.2, k =0. Figure 5 shows the development plane Couette flow Rexp ∼ 350 and according to our
y z c
withtimeoftheperturbedvelocitycomponentu(x,y) analysis, for R = 350, G = 145, and the corre-
max
correspondingtoR=1000. Weshowsnapshotscorre- sponding t = 42.3. Since a q = 2 disk is very
max
sponding to four times: t = 0, t /2, t , 3t /2. similar to plane Couette flow, the critical Reynolds
max max max
The perturbations are seen to resemble plane waves number for turbulence in this case is also likely to be
that are frozen in the shearing flow. The initial per- R ∼350. For this R, G =143.5 and t =48.3.
c max max
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22nd Texas Symposium on Relativistic Astrophysics, Stanford University, December 13-17, 2004 5
Figure 4: Growth curvesfor (a) q=1.5 (solid), 1.7 (dotted),1.9 (dashed),1.99 (dot-dashed);other parameters are
given in Table 1. (b) Growth curves for q=1.5 when {R=4000,ky =1.2} (solid), {R=1000,ky =1.21} (dotted),
{R=500,ky =1.29} (dashed); kz =0.
Figure 5: Development of theperturbed x-component of thevelocity u(x,y) as a function of time for theoptimal
perturbation with the maximum growth of energy in a Keplerian flow with R=1000. The perturbation has ky =1.21,
kz =0, tmax =8.8. Panels show the perturbation at (a) t=0, (b) t=tmax/2=4.4, (c) t=tmax =8.8, (d)
t=3tmax/2=13.2. Solid and dotted contours correspond to positive and negative valuesof u respectively.
Based on the above idea, we make the plausible of 100 greater than in the case of plane Couette flow.
assumption that the threshold energy growth factor The corresponding t = 28.3, which is comparable
max
needed for transition to turbulence in a shear flow to that in plane Couette flow, and is not too large
is E ∼ 145. Applying this prescription to the opti- compared to the accretion time-scale of a geometri-
c
maltwo-dimensionalperturbationsofaKepleriandisk cally thin disk.
(§§2,3), we estimate the critical Reynolds number for
Instead of taking R ∼ 350, we might wish to be
c
a Keplerian flow to be Rc ∼ 3.4×104, i.e., a factor conservative and take Rc ∼ 1000 for plane Couette
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6 22nd Texas Symposium on Relativistic Astrophysics, Stanford University, December 13-17, 2004
flow and q = 2 flow. In this case, G ∼ 1200 and turbed shear flows. It remains to be seen if these in-
max
t ∼ 120−140. Applying the requirement of E ∼ stabilitiesarepresentinperturbedflowssuchasthose
max c
1200toaKeplerianflow,wefindR ∼106andt ∼ shown in Figure 5.
c max
100. Now the critical Reynolds number is a factor of Acknowledgments
1000 greater than in the case of plane Couette flow.
An interesting result is that epicyclic motions in
This work was supported in part by NASA grant
a differentially-rotating disk kill the growth dramat-
NAG5-10780and NSF grant AST 0307433.
ically and as a result the critical Reynolds number
R becomes higher. This also changes the optimum
c
wavevector {k ,k } of the perturbations needed to
y z
produce energy growth. For a constant angular mo- References
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