|
Items in cart [4] |
Questions? Call 888-624-8373 |
| Copyright © 2009. National Academy of Sciences. All rights reserved. Terms of Use and Privacy Statement |
Below are the first 10 and last 10 pages of uncorrected machine-read text (when available) of this chapter, followed by the top 30 algorithmically extracted key phrases from the chapter as a whole.
Intended to provide our own search engines and external engines with highly rich, chapter-representative searchable text on the opening pages of each chapter.
Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.
Do not use for reproduction, copying, pasting, or reading; exclusively for search engines.
OCR for page 55
Turbulization of Shear Flows in Astrophysics
G.D. CHAGELISHVILT, R.G. CHANISHVILT, AND J.G. LOMINADZE
Abastumani Astrophysical Obsenatory
INTRODUCTION
According to a most widely spread representation (Shakura and Sun-
yaev 1973; Pringle 1981; Liang and Nolan 19843, the disk accretion phe-
nomenon is based on the anomalous transport of the accreting matter
angular momentum outward due to turbulent viscosity. It is also the
turbulence which provides such an accretion law that a bulk of released
gravitational energy of the accreting matter in the disk thermalizes, and as
a result, is emitted and provides the observed X-ray spectrum for some bi-
na~y sources. Consequently, the investigation of the turbulence emergence
in accretion disks is a problem of paramount importance (of course, to
elucidate the possibility of turbuli7~tion of other shear flows in astrophysics
is also important).
Turbulence can be created in accretion disks by a superdiabatic pres-
sure gradient across a disk (in the regions where such a gradient exists) and
by differential rotation of the matter, i.e. by a shear of angular velocity In
the Keplerian rotation of the matter, Q ~ r -3/2 (Shakura et al. 1978~. The
superadiabatic pressure gradient resulting in thermal convection is charac-
tenstic of only some parts of the accretion disk (Taylor 198Ct, Lominadze
and Chagelishvili 1984; Chagelishvili et al. 19861. Therefore, in Me other
parts of the disk the presence of the turbulence can be ensured only by
the shear instability. The problem of instability in shear flows is treated
in quite a number of works (Narayan et al. 1987). They give us a global
analysis of irrotational (potential) modes in a two-dimensional compress-
ible shear flow, implying that, in the presence of reDect~g boundanes,
those modes can increase and that the characteristic time of the increase
55
OCR for page 56
56
AMERICAN AND SOVIET PERSPECTIVES
Is several orders as high as the dynamic time in the medium. Due to the
slowness of the increase of irrotational modes and to the problematical
character of the theories applicability as presented in Narayan e! al. (19g7)
for thin accretion disks, we have tried to develop an alternative theory of
turbulization shear Dows. It is quite contrary to the above theory. Instead
of the global analysis, we make a local one, far from the flow boundaries.
Instead of the irrotational perturbations, we examine vertical ones. Lastly,
we consider the medium to be incompressible. To gain our aim, the Dow
in the accretion disk can be modeled by a plane Couette flow, without
regard for boundary conditions: it can be modelled by a plane shear Dow
in infinite incompressible fluid. According to the present scenano, there
is a critical perturbation level in the free shear flow. If the level of initial
perturbations exceeds the critical one, the flow is turbulized.
We must particularly emphasize that the flow can only be turbulized by
non-potential (vortex) perturbations. The fact that the turbulization comes
at finite initial perturbations is explained by Me non-ordinary nature of the
temporal evolution of the vortex perturbations at the linear stage. Such a
non-ordinarily for some astrophysical phenomena was found for the first
time in the work of Lominadze et al. (1988~. The distinguishing feature
of their analysis, as well as of Goldreich and Linden-Bell (1965), is that
the linear equations are integrated in a co-moving coordinate system with
moving axes (X~OY~. That is to say that two local coordinate systems
are used to analyze equations (see Figure 1~: a Cartesian one (XOY) with
the Y-ans pointing the velocity direction and X-a~ns directed orthogonally,
along the shear of the velocity, and another system, with moving axes
(X~OY'). The Y~-axis is parallel to the Y-ams, and the X~-ams moves
together with the unperturbed flow. (Lee relation between (x,y) and (x:,y~ ~
coordinates is given by equation (4~. The idea of the paper is as follows:
equations are projected onto the Cartesian coordinate system axes. Then
we substitute the variables using equation (4) so as not to change the
sense of the projection of the physical quantities. That is, we do not make
a physical transition from the XOY frame to the X:OY~, but a formal
substitution of variables that makes the analysis easier. In fact, the linear
theory equations (6~-~83 written in the terms of the new variables x and
y: are uniform in regard to them Nut they are already not uniform in
regard to time). So, expanding a perturbance into Fourier modes in regard
to x: and ye, one can follow separately the temporal evolution of every
Fourier mode. After returning to the former coordinates, we can see
that besides the time variation of every Fourier mode amplitude, there
is variation of its wave number along the X-a~ns in the direction of the
velocity shear of the main flow (see equation (19~. There is a pecularity
In the temporal evolution of the Fourier modes: the evolution law is not
exponential, but power one (see equation (18~. The growth can take place
OCR for page 57
HIGH-ENERGY ASTROPHYS CS
AX ~ X'
_ _4
_ ~
l
57
Y Y4
FIGURE 1 The local Cartesian coordinate system XOY and the system with moving axes
XlOY1 are presented. The dashed arrows show the direction of the main flow velocity Uo
at different distances from the Yeats. Uo(O,Uo3~0~; Uo' = Ax; A > 0. X~-ams moves
together with the shear flow.
when I*-k=(t)/kyA > 0, and where knots and kit, are wave numbers of
the Fourier modes under consideration along the velocity shear and the
main stream velocity, respectively, but in the XOY coordinate system. A is
a characteristic parameter of the velocity shear ~Jo, = Ax). When writing
the wave number k=(t) we especially emphasize the time dependence to
stress that the spatial scale of Fourier modes varies in time along the X-axis.
We must also stress that the Fourier expansion was done in terms of the
variables x, and ye (see equation (93), k= (t) and ky being the wave numbers
of those Fourier modes in the XOY system.
From the condition k~(t)/~3,A > 0 and equation (183 one can see
that, according to linear theory, the growth time of every Fourier mode
is limited. At first a Fourier mode draws on energy from the main Dow
and grows; after some time k~(t) changes its sign, and We growth of the
Fourier modes turns into weakening: they "return" the energy back to the
main flow.
It follows from the above that the possibility of turbulization of free
OCR for page 58
~8
AMERICAN AND SOVIET PE~PECT~S
shear cows is ambiguous. Indeed, if you start from infinitesimal perturba-
tions, you may assume that because of a limited growth time of Fourier
harmonics they will not grow up to non-linearity, that is, up to such values
where non-linear effects come into play. So the evolution of the perturba-
tion will be purely linear, and it will disappear '~without a trace:" with no
turbulization in the media. But if the level of initial perturbations in the
media is not too small, then at a certain stage of increase of the Fourier
harmonics, non-linear (cascade) processes come into play that can provide
turbulization of the flow.
LINEAR THEORY OF PERTURBATION
Let us consider the linear theory of the temporal evolution of two-
dimensional perturbances irk a plane, free-shear flow of incompressible
fluid. 1b get the growth effect alone, let us first consider a nonviscous
fluid. Allowance for viscosity won't be too difficult, so our result will be
generalized for the case of a viscous fluid (see below3.
Let us direct the Y-a~ns of the Cartesian coordinate system along the
main how velocity UO (O,Uoy, O), and the X-axis along the shear of the
flow velocity Uoy = Ax (see Figure 1~. Considenng the problem to be
two-dimensional, we write the equations of continuity and motion for the
perturbed quantities:
ou=+Buy=O,
fix by
{~3~+Ax~ }u==-NIP,
{it +A~;~3 }u y+Au~
(1)
(2)
UP
(3)
where us and U3, are the components of the perturbance velocity in the
Cartesian coordinate system, P is the pressure perturbance normalized by
the density of the matter p.
Now we introduce a coordinate system with moving axes XENON. Its
ongm and the Y~-a~s coincide with the respective characteristics of the
XOY system; and the X:-a~s moves together with the undisturbed Dow
(see Figure i). This is equivalent to the change in variables
Hi =~; ye =y-Ant; to =t, (4)
or
OCR for page 59
HIGH-ENER~Y ASTROPHYSICS
_ = _-AX1-
With the new vanables, equations (1)-(3) take on the form
(A _ At1 A ) Uz + A Y = 0,
A Z =-{ A -Atl by }P.'
At + Auk =-{ A }P
59
(5)
(6)
(7)
(8)
As it was already noted in the introduction, the substitution of the
variables (4) is not a physical transition to the new frame, because in
equations (63-~8),aswellasin equations (1~-~3), the quantifies us end Al are
components of the perturbation velocity in a Cartesian coordinate system.
The coefficients of the initial linear equation system (1~-~3) depended on
the space coordinate x. After our transformations, this inhomogeneity was
changed into a time inl~omogeneity. So we can perform a Fourier analysis
of equations (6~-~8) in respect to the variables x~ and yl:
( us ~ += { u=(k=~' Kim, t) ~
By ~ = /j ~k=~dly~ ~ uy~k=~'ky~'t) ie~(ik=~xi +ily~yi) (9)
~ P ) -= ~ p~k2~,ky~,t) J
Substituting expansion (9) into equations (6~-~8), we obtain
Okra-key Ati fun ~ by: By = 0
=-take-key Ati jP,
0tY + Au2 =-ily~P,
Solving equation system (10~-~12), say, for velocity, we get
u2(k k ti) = Uzbeks ky o) (k~l/kyt) + ~
U (k k t -U (k k o)[(k~l/kYl}201](1-Atlkyl/k2l)
Y =1' Y1' 1 - y 21, Y1'
(10)
(11)
(12)
(13)
(kxl/ky1-Atl)2 + ~, (14)
OCR for page 60
60
AMERICAN AND SOVIET PERSPECTIVES
u(kxl,kyllil) = {u2 + uy}2 =
[{k~l/kY1-Atl } + 1]
.
(15)
As one can see from these formulae, only those Fourier harmonics of the
initial perturbation grow the wave numbers of which satisfier the inequality
k
HA > 0'
and the growth takes place in time t < t*, where
t* _ key /kyl A
(16)
(17)
It must be stressed that the solutions obtained here exist when the initial
vorticity Is non-zero.
Using notation (17), we may write solution (13) in the form
UX(k2~, kin, ti) = u=(k2t, ky:, O) A2(~* t)2 + 1
(18)
As one can see from this formula, for I* ~ 1 at first (as long as Aft*- t) >
1) the growth of a Fourier harmonic is rather explosive: the growth rate of
the perturbations greatly increases in time. However, it is only towards the
end (when Aft*-t) ~ 1 that the growth stops abruptly and (when t > t*)
a weakening follows (see Figure 2a).
We have performed Founer analysis of perturbances for the variables
x1 and ye and followed the time evolution of Founer harmonic amplitudes.
What happens to Founer harmonics in an ordinary space, be., in the
Cartesian coordinate system (XOY)? Using equations (43 and (9) one can
introduce a parameter that determines the characteristic linear dimension
of each of the Founer harmonics along the awns X and Y at every particular
instant
knits = k2'-ky:Ati, by = Aye.
(19)
We can see from these formulae that the space scale along the X-axis of
every Founer harmonics under examination varies in time.
Viscosity plays an important role in many hydrodynamic flows, He
more so In turbulization phenomena. For some astrophysical flows, the
viscosity was considered in terms of Lominadze et al. (1988) in the work
of Fridman (1989~. Using the results of the latter work, we can easily
OCR for page 61
(t)
kx
'1
HIGH-ENERGY ASTROPHYSICS
O (kX1, k%2, t)
-
Jim
.
l
\!
(a)
(b)
t
61
FIGURE 2 (a) Time evolution of the amplitude of a perturbation velocity Fourier
ilaI-mOIliG The graph is plotted in accordance tenth equation (15~. (by lime dependence of
the spatial scale along the X-a~ns of Fourier harmonics increasing at initial time ~2~0~A
> 0~. The graph is plotted in accordance with equation (19~.
generalize our solutions to a viscous fluid too. For instance, ~ the case of
a viscous fluid equation (13) takes the form
u~(k=t, kyt,ti) = u2(k=t, key 0) kin' + k2 em
{_u ~ [k2(t) ~ ky2] dti,
(20)
where z, = pip is the kinematic viscosity. Naturally, osmosis impedes
perturbation grown. For every specific z, one can find Me minimum
characteristic dimension of perturbations (maximal k-~/~) at which
Fourier harmonics can increase.
OCR for page 62
62
AMERICAW AND SOVIET PERSPECTIVES
QUALITATIVE SCENARIO OF TURBULIZATION
Let us now describe the processes which can lead, we presume, to
turbulization of shear flows. Of course, such a process represents an
aggregate of linear and non-linear phenomena. Let us assume that the
linear phenomena proceed in accordance to the theory exposed in the
previous section. Then any initial perturbation can be presented as a set
of Fourier harmonics. For the following discussion it is convenient to
introduce a notion of a k-plane (kooky). Without a loss of generality,
we shall also hold that A > 0. Let us follow linear dynamics of Fourier
harmonics. In addition to amplitude variations of Fourier harmonics, their
characteristic scales vary in the direction of the velocity shear: along the
X-axis (see equation (19~. This variation on the kooky plane means a
drift of Fourier harmonics in the direction shown by the arrows in Figure
3. The Fourier harmonics that satisfy condition k=(O)/ky < 0 (k=~/ky <
O), weaken. The Fourier harmonics that satisfy the condition k2 (0~/ky >
O at first draw on energy from the main flow and increase, while skim
decreases. Each Fourier harmonics grows till k=(t) becomes zero (see
Figure 2a and 2b). When it becomes zero, knots reverses sign, as it follows
from the linear theory, and the growth changes to weakening: the Fourier
harmonic returns the energy back to the main flow. Hence it is clear that
if non-linear phenomena do not come to play, any initial perturbations will
disappear without leaving a trace.
A turbulence cannot exist without a permanent energy input from
the main flow to perturbations. According to equation (16), when A >
O. this takes place if there are Fourier harmonics with wave numbers
corresponding to the first and third quarters of the k=Ok3, plane (see
Figure 3) which we shall refer to as a "growth area." But because of the
drift of Fourier harmonics from the "growth area," the linear mechanism
cannot provide a permanent input of energy from the main flow to the
perturbation. Actually, if at me zero time (t = 03, in addition to w~ken~g
Fourier harmonics, we have a packet of increasing Fourier harmonics (the
packet that fills the cross-hatched area in Figure 3), the packet will drift
in the k=Ok3, plane (! ,11 ,111), so that ~ due time (t = to) it will come
to be outside the "growth area." Thus, later on the "growth area" will be
empty and there will be nothing to draw energy with from the main flow.
What can the non-linear phenomena result in? They take no part in
energy exchange with We main Dow (Joseph 1976~. But they redistn~ute
energy among the Fourier harmonics. So the non-linear phenomena may
return a part of the energy of the above packet back to the "growth area,"
as a result of the decay processes (k' ~ k" = k, see Figure 3~. Then Fourier
harmonics which are able to take energy from the main flow reappear and
drift in the kooky plane. It will be easily understood that if there is a
OCR for page 63
HIGH-ENERGY ASTROPHYSICS
t_~`'
hi\!
I,,
,
, ~ 1
, , l,
t t t\
1 1 1 ~ 1
1
1 ~ 1
It I ~ ~ ~
, Al
-
'
rig ~ ~I ~I ~
I/ ~ 1\
t = 0
/l ~ l\\~ ~t=t'i
/} ~ ~ T\~l 1
~ ~1
A> 0
63
t =r
FIGURE 3 The short arrows show the directions of the drift of Fourier harmonics
described in the linear theory in the k space (when A > 0~. A drift of Fourier harmonic
packet filling cross-hatched area I in the k space ~ HI ~III) is shown. After some time
(t = tat, the packet comes to be outside of the "growth area" (k~(t"^A < 0~. An
_ _ _
example of the decay process k' + k" = k which can return a part of the perturbation
energy back to the "growth area" (k~(t)/KyA > 0) is given.
,
permanent non-linear return of a part of perturbations to the "growth
region," the now will be turb~li~ed.
Thus the comparative analysis of linear and non-linear mechanisms
gives us a general idea of the dynamics of Fourier harmonics and energy
redistribution among them. What can be observed at small initial perturba-
tions? Because of the linear drift, the Fourier harmonics will be driven out
of the "growth area," and because of the smallness of the amplitude, the
decay processes will be weak and will not be able to resist the permanent
linear theory drift. As a result, the perturbation will be damped without
inducing turbulization in the media. The higher the amplitude of initial
perturbations, the stronger the effects of non-linearity. Finally, at a cer-
tain amplitude (which, of course, must depend on the initial perturbation
spectrum and the value of the viscosity coefficient), the decay processes
will be able to compensate the linear drift of the Fourier harmonics and
ensure a permanent "drawing" of energy from the main flow and lead to
the turbulization of the media. So, according to the present scenario, Were
OCR for page 64
64
AMERICAN AND SOVIET PERSPECTIVES
is a critical level for initial perturbations in the free shear flow. If the level
of initial perturbations exceeds the critical one, the flow gets turbulized.
The local theory of turbulization of a plane Couette flow presented
here can be applied not only to accretion disks, but to other shear flows in
astrophysics, such as galactic disks, planetary rings, protostellar nebulae and
possibly, to some rotating stars as well We think that beyond astrophysical
applications, this theory is a ray of hope for explanations of turbulization
of some "earthly" hydrodynamic flows which are steady to infinitesimal
perturbations but are turbulized by finite ones. Of course, our theory is
qualitative and needs more rigorous mathematical formulation in order
to be quite reliable: a theory of weak and strong turbulence is to be
developed. We have developed a theory of weak turbulence, but the extent
of calculations does not allow us to include it into the present paper. Here
we only give ache final equation of the theory of weak turbulence:
Seek _k Amok _ 2k~kyA id, ~ 7,{~2 ~ E21~ =
k2 ~ k2~,tT',`~24~y,~,¢-A
~ T y ~
/
dk'dk"~(k-k'-kin.
(kit_ k=) k k ky2ky'2{[t~b(k"'ky''t)-¢(k',ky,t)]
{Ex,Ex,, - 2k,2 k92 [~(k",ky/,t)-~b(k=,ky,t)]ExEx,'}. (21)
Here Ek is the energy density of a Fourier mode of a perturbation with
fixed k, and
?{ = k2 arctg 11 + k~/ky(`ks/ky - At)
RE~:RENCES
Chagelishvili, G.D, RG. Chanishvili, and J.G. Lommadze. 1986. Proceedings of the Joint
Varenna-Abastumani Intemational School and Workshop. Sukhumi, USSR, (ESA
SP-251), p. 563.
Fridman, NM. 1989. P~s'ma Astran. Zb., in press.
Goldreich, P., and D. Lynden-Bell. 1965. MNRAS 130:125.
Joseph, D.D. 1976. Stability of Fluid Motions. Springer Verlag, New YorL
I iang, E.P., and P.L. Nolan. 1984. Space Sal. Rev. 38:353.
Lominadze, J.G., and G.D. Chagelishvili. 1984. Astron. Zh. 61:290.
Lominadze, J.G, G.D. Chagelishvili, and RG. Chanishvili. 1988. Pis'ma
Astron. Zh. 14:856.
Narayan, R., and P. Goldreich. 1987. MNRAS 288:1.
Mingle, J.E. 1981. Ann. Rev. Astron. and Astrophys. 19:137.
Shakura, N.I. and RA Sunyaev. 1973. Astron. and Astrophy~ 24:337.
Shakura, N.I., RN Sunyaev, and S.S. Zilitinkevich.1978. Astron. and Astrophys 62:179.
Representative terms from entire chapter:
main flow
