- On hydrodynamic shear turbulence in Keplerian disks: Via transient growth to bypass transition
- 1 Introduction
- 2 From spectral decay to transient growth and bypass transition
- 3 The similarity between Keplerian flow and plane parallel shear flow
- 4 Transient growth in 2D
- 5 Discussion and conclusions
- Appendix A: Bypass scenario of the transition to turbulence
- References

A&A 402, 401-407 (2003)

DOI: 10.1051/0004-6361:20030269

**G. D. Chagelishvili ^{1} - J.-P. Zahn ^{2} -
A. G. Tevzadze ^{1} -
J. G. Lominadze ^{1}**

1 - Center for Plasma Astrophysics, Abastumani
Astrophysical Observatory, Kazbegi 2a, 380060 Tbilisi, Georgia

2 -
LUTH, Observatoire de Paris, 92190 Meudon, France

Received 11 December 2002 / Accepted 6 February 2003

**Abstract**

This paper deals with the problem of hydrodynamic shear
turbulence in non-magnetized Keplerian disks. Several papers have
appeared recently on the subject, on possible linear instabilities
which may be due to the presence of a stable stratification, or
caused by deviations from cylindrical rotation. Here we wish to
draw attention to another route to hydrodynamic turbulence, which
seems to be little known by the astrophysical community, but which
has been intensively discussed among fluid dynamicists during the
past decade. In this so-called *bypass* concept for the onset
of turbulence, perturbations undergo transient growth and if
they have initially a finite amplitude they may reach an amplitude
that is sufficiently large to allow positive feedback through
nonlinear interactions. This transient growth is linear in
nature, and thus it differs in principle from the well-known
nonlinear instability. We describe the type of perturbations that
according to this process are the most likely to lead to
turbulence, namely non-axisymmetric vortex mode perturbations in
the two dimensional limit. We show that the apparently inhibiting
action of the Coriolis force on the dynamics of such vortical
perturbations is substantially diminished due to the pressure
perturbations, contrary to current opinion. We stress the
similarity of the turbulent processes in Keplerian disks and in
Cartesian flows and conclude that the prevalent skepticism of the
astrophysical community about the occurrence of hydrodynamic shear
turbulence in such disks is not founded.

**Key words: **accretion,
accretion disks
- hydrodynamics - instabilities - turbulence

The emergence of X-ray astronomy exposed the importance of
accretion phenomena in many astrophysical systems, such as binary
stars, quasars and active galactic nuclei. The possible
mechanisms governing accretion have been schematically understood
during the very first years of investigations (Shakura & Sunyaev
1973; Pringle 1981): the inward transport of matter and the
outward transport of angular momentum in accretion disks was
ascribed to a turbulent (anomalous) viscosity. The consequent
development in understanding the physics of this phenomenon and
the causes of that turbulence has been irregular and has taken
considerable time. Substantial progress has been achieved in
the nineties with the discovery of a linear instability in
magnetized disks (Balbus & Hawley 1991, 1992, 1998; Hawley &
Balbus 1991, 1992; Hawley et al. 1995; Stone et al. 1996). In
contrast, the solution of the accretion problem in the
non-magnetized case has not yet reached sufficient maturity.
Moreover, the very occurrence of turbulence in non-magnetized
disks has been questioned by Balbus et al. (1996) and Balbus &
Hawley (1998). The reason for this situation is that cylindrical
flows with Keplerian profile belong to the class of smooth shear
flows, i.e. which present no inflection point; it is well
known that these flows are spectrally stable, although they may
become turbulent in the laboratory. The explanation of this
behavior remains one of the fundamental problems of fluid
mechanics. However the situation has changed in recent years, with the
recognition that the so-called non-modal approach provides an
adequate, and even an optimal mathematical formalism to describe
the dynamics of perturbations in plane-parallel shear flows. This
led to the emergence of the *bypass concept* for the onset of
turbulence in spectrally stable flows. This concept has triggered
much interest among fluid dynamicists, but it had little impact so
far on the astrophysical community. For this reason we feel it necessary
to present the salient results of that novel approach in a journal
which is read by astrophysicists, to demonstrate its relevance
for the dynamics of Keplerian disks, and to draw attention
to the bibliography on the subject.

Our approach parallels that of Longaretti (2002), who analyzed the existing experimental and numerical results, and who also insisted, as we shall do here, on the similarity of subcritical shear turbulence in plane and rotating flows.

Our paper is organized as follows. In Sect. 2 we outline the fundamental problems of fluid mechanics bearing on the description of plane-parallel smooth shear flows. We present the recent developments in shear flow analysis made by the hydrodynamic community, and the novel bypass concept for the onset of turbulence in shear flows. In Sect. 3 we analyze the quadratic forms of the dynamical equations and discuss the essential role of pressure perturbations, which counteract the effect of the Coriolis force in 2D flows. In Sect. 4 we describe the transient growth of two dimensional perturbations. This will allow us to stress the similarity between the behavior of Keplerian and Cartesian flows, and the negligible role of the Coriolis force in the 2D perturbation kinematics and energetics. Finally we conclude in Sect. 5 that the bypass concept provides a plausible scenario leading to turbulence in astrophysical disks. In the Appendix we present a simple sketch of the bypass scenario.

From the theoretical viewpoint there are flows that are spectrally stable at all Reynolds numbers (e.g. plane Couette or pipe Hagen-Poiseuille flows), while some others become spectrally unstable at high enough Reynolds numbers (e.g. plane Poiseuille or Blasius flows). In the latter case the flow is characterized by the critical Reynolds number that is the marginal value of this parameter above which the spectral instability occurs. In this sense the critical Reynolds number of the spectrally stable flow is infinity.

Here we shall consider smooth shear flows (without inflection point in the velocity profile), like in Keplerian disks. These are linearly stable according to classical fluid mechanics. As shown by Rayleigh (1880), the existence of an inflection point (more precisely of a vorticity extremum, as pointed out by Fjørtøft 1950) in the equilibrium velocity profile is a necessary condition for the occurrence of a linear (spectral) instability in hydrodynamic flows. Thus smooth shear flows (flows without a vorticity extremum) are "relaxed'' in this context: they are spectrally stable, meaning that exponentially growing solutions are absent. However, it is well known from laboratory experiments and from numerical simulations that finite amplitude perturbations may cause a transition from laminar to turbulent state at moderate, less then critical Reynolds number.

This has led to the development of the concept of *nonlinear
instability* in hydrodynamics (cf. Bayly 1986; Bayly et al. 1988;
Herbert 1988; Orszag & Kells 1980; Orszag & Patera 1980, 1983).
Until about ten years ago, the predominant view of this
laminar-turbulent transition was centered around the slow linear
amplification of exponentially growing perturbations (the familiar
T-S waves), which modify the flow profile and thereby allow a
secondary instability, further nonlinearity and finally a
breakdown to turbulent flow. According to this concept of
nonlinear instability, the perturbations (and the turbulent state
itself) are energetically sustained by nonlinear processes. This
concept of nonlinear instability has been borrowed by
astrophysicists, who still use it to explain turbulent processes
in smooth astrophysical flows, where no spectrally unstable
solution is known, and in particular in Keplerian disks flows (see
Balbus & Hawley 1998). However, there are subcritical transition
phenomena that cannot be attributed to the nonlinear
instability.

During the last decade of the 20th century, another viewpoint
emerged in the hydrodynamic community on the understanding of the onset
of turbulence in spectrally stable shear flows, called *bypass transition* (cf. Boberg & Brosa 1988; Butler & Farrell
1992; Reddy & Henningson 1993; Trefethen et al. 1993;
Morkovin 1993; Gebhardt & Grossmann 1994; Henningson & Reddy
1994; Baggett et al. 1995; Waleffe 1997; Grossmann 2000;
Reshotko 2001; Chagelishvili et al. 2002; Chapman 2002;
Rempfer 2003). Although the bypass transition scenario involves
nonlinear interactions - which intervene once the perturbations
have reached finite amplitude - the dominant mechanism leading to
these large amplitudes appears to be linear. This concept of the
onset of turbulence is based on the *linear transient growth
of vortex mode (aperiodic) perturbations*. The potential for
transient growth has been recognized for more then a century
(see Kelvin 1987; Orr 1907a,b). However, only recently has
the importance of the phenomenon
been better understood (cf. Moffatt 1967; Marcus & Press 1977;
Gustavsson & Hultgren 1980; Craik & Criminale 1986; Farrell &
Ioannou 1993; Reddy & Henningson 1993; Chagelishvili et al.
1997).

The bypass concept implies that the perturbation energy extracted from the basic flow by linear transient mechanisms causes the increase of the total perturbation energy during the transition process. The nonlinear terms are conservative and only redistribute the energy produced by the linear mechanisms. (A simple sketch of the bypass scenario in a Keplerian flow is given in the Appendix.)

Thus, according to this concept, the transient growth of
perturbations (i.e. the linear process) is the key element
in the transition to turbulence in spectrally stable flows. The
importance of this linear process has been stressed in the titles
of several seminal papers: Henningson & Reddy (1994) (*"On
the role of linear mechanisms in transition to turbulence''*);
Baggett et al. (1995) (*"A mostly linear model of transition
to turbulence''*) and Reshotko (2001) (*"Transient growth: A
factor in bypass transition''*).

The scheme of investigation of shear flow dynamics implies the
following steps: introduction of perturbations into a mean flow,
linearization of the governing equations, and description of the
dynamics of perturbations and flow using the solutions of the
initial value problem (i.e. following temporal balances in
the flow). In principle this can be done, but in practice it is a
formidable task. Therefore, the mathematical approach was changed.
This was done by assuming that the solution is separable in
eigenmodes, and then establishing the existence of at least one
unstable eigensolution. This approach became canonical in time,
and resulted in a shift of attention, which became directed to the
asymptotic stability of the flow, while *no attention* was paid
to any particular initial value or to the finite time period of
the dynamics. Indeed, this phase of the evolution was not thought
to have any significance - it was left to speculation. But
recently the early transient period of the perturbations has been
shown to reveal "rich'' and complicated behavior leading to
different consequences.

It was found in the 1990s that smooth shear flows are crowded by
intense processes of mean flow energy extraction by perturbations,
energy exchange between perturbations, etc. even in the linear
approximation, while following the classical theory they are
spectrally stable and consequently "relaxed''. It has
been especially shown that a superposition of decaying normal modes may grow
initially, but will eventually decay as time goes on - a new *linear* transient channel of energy exchange between the mean flow
and perturbations has appeared. Moreover, it has been shown that
transient growth can be significant even for subcritical values of
the Reynolds number and that its interplay with nonlinear
processes can result in the transition to turbulence without any
"nonlinear instability'' of the flows.

In fact an exact resonance is not necessary to obtain the
transient growth of perturbations. This is the consequence of the
non-normal character of operators that describe the linear
dynamics of perturbations in flows (see, e.g. Reddy et al.
1993). The fact that the eigenfunctions of the linearized
Navier-Stokes equations are not orthogonal (i.e. the
operator is non-normal) is enough to allow for solutions that
exhibit transient growth, depending on the initial conditions,
before finally decaying (see Criminale & Drazin 1990; Trefethen
et al. 1993). The mechanism of transient growth is
essentially inviscid - the operators are highly non-normal for
large Reynolds numbers and the transient growth is asymptotically
large in *Re*. (This fact is extremely important in the case of
Keplerian accretion disks, where the Reynolds number is literally
astronomical:
*Re*>10^{10} !)

These developments provoked a change of paradigm in the study of
linear processes in the considered flows. It was the so-called
*nonmodal approach* that became extensively used and even
canonized in the 1990s on these grounds. As a result, substantial
progress in the understanding of the shear flow phenomena has been
achieved. The nonmodal analysis - some modification of the
initial value problem - implicates the change of independent
variables from a laboratory to a moving frame and the study of
temporal evolution of spatial Fourier harmonics (SFH) of
perturbations without any spectral expansion in time (see Sect. 4).

Since our aim is to elucidate the basic similarity of the dynamics of plane shear flows and of Keplerian disks, we approximate the Keplerian flow in the two dimensional (2D) limit by its tangent plane parallel flow, while retaining the effect of rotation through the Coriolis force. This model, which ignores the purely geometrical complications, is known as the shearing sheet model; it has been has been used for the study of the linear dynamics of both wave and vortex mode 2D perturbations (cf. Goldreich & Lynden-Bell 1965; Goldreich & Tremaine 1978; Drury 1980; Nakagawa & Sekiya 1992 for wave mode perturbations, and Lominadze et al. 1988; Fridman 1989 and Ioannou & Kakouris 2001 for vortex mode perturbations).

The dynamical equations are written in the local co-moving Cartesian co-ordinate system:

where () are standard cylindrical co-ordinates and is the local angular velocity at

is the shear parameter (in Keplerian disks ).

For the present purpose, we shall consider only 2D perturbations,
independent of the axial coordinate *z*. The resulting dynamical
linear equations for the perturbations of the radial velocity
(), the azimuthal velocity ()
and the pressure (*p*)
take the form:

The incompressible limit is taken here to leave out wave mode perturbations and to keep only the vortex mode perturbations, which are the basic ingredient of the bypass scenario. In this case the energy density of the perturbation depends only on its kinematic characteristics and not on the thermodynamics, such as pressure.

Pressure terms are also absent when the dynamical equations are written in vorticity form (for ), taking the curl of Eqs. (3, 4). This is probably the reason why the pressure perturbations often have been ignored when discussing the dynamics of shear flows. However they play a very important role, even in the incompressible case, because they are the mediators of momentum exchange between fluid particles, which results in the transient growth of the vortex mode. (The physics of the "mediator activity" of the pressure perturbations is described in detail in Chagelishvili et al. 1993, 1996.) The importance of the pressure terms is clearly seen in the following analysis.

We multiply Eqs. (3) and (4) respectively by and ,
in order to put them in kinetic energy form, and
average them over a domain which is symmetrical in *x*. This
procedure is similar to that performed in papers by Balbus et al.
(1996) and Balbus & Hawley (1998).

The source of instability is represented by the shear term . However, the Coriolis term is explicitly present too, and in Eq. (7) it overbalances the source term, since in the Keplerian disk. It thus would seem at first sight that the Coriolis force has a profound influence on the shear flow stability, as has been argued by Balbus & Hawley (1996), but this conclusion is contradicted by the fact that the Coriolis terms disappear when summing Eqs. (6) and (7), meaning that the growth rate of the total kinetic energy is independent of the Coriolis force.

A more detailed analysis will show that the Coriolis terms may
be eliminated also on the level of the dynamical equations,
by a suitable renormalization of the pressure perturbation.
In this two-dimensional and incompressible case, the
perturbation velocity field derives from the stream
function :

(8) |

Renormalizing the pressure perturbation

(9) |

Eqs. (3-7) can be rewritten as follows:

(10) |

(11) |

(12) |

(13) |

(14) |

In Sect. 2 we outlined the bypass concept in Cartesian/planar shear flows (a simple sketch of the bypass scenario is presented in the Appendix). Since we wish to apply a similar scenario to differentially rotating disks, we shall write the Cartesian counterparts of the averaged equations for comparison. If

where is the shear parameter of the Cartesian flow ;

These equations are identical to the normalized equations derived above for the Keplerian case, which stresses the similarity of these two flows. But they deal with spatially averaged quantities, and one may wish to establish that the similarity is even more profound, namely that the local kinematics and energy of 2D vortex perturbations of plane and disk flows are the same in time and space, starting from identical initial conditions.

Mainly for the purpose of illustrating the bypass mechanism,
we shall describe now the evolution in time of an
initial perturbation, in two dimensions. We perform
a spatial Fourier transform of all relevant variables, as
shown here for the pressure fluctuation:

In a shearing flow, the perturbations cannot keep the form of a simple wave, since the wave-number of each spatial Fourier harmonics (SFH) depends on time (see Criminale & Drazin 1990 for a rigorous mathematical interpretation). In our geometry

thus the wave-number of each SFH varies in time along the flow shear: it "drifts'' in -space. Therefore

Equations (5), (6) and (7) take the
following form in Fourier space:

For simplicity, we ignored here the viscous forces; their action may be easily incorporated in the analysis, as we shall see below.

One readily shows that these equations
possess a time invariant

(22) |

which expresses the conservation of vorticity in Fourier space. Making use of this invariant, we may write the solution of the system (19-21) as follows:

(23) |

(24) |

(25) |

We see that the perturbed quantities vary aperiodically in time (as is natural of vortex mode perturbations), and that they undergo transient amplification. Energy is exchanged between the perturbations and the background flow, and that represents the basis of the bypass transition scenario. It is important to note that the kinematic characteristics of the perturbations do not depend on the rotation rate - they are identical to those of Cartesian shear flows. Only the pressure perturbation depends on . Consequently, the energetics of the SFH is identical to that of the Cartesian case:

We see that the kinetic energy reaches maximum amplitude for

= | |||

= | (27) |

This expression is identical to that describing the evolution of the pressure perturbation in a Cartesian flow, which proves the similarity of transient growth in both types of flows, in two dimensions.

The above analysis has been done in the inviscid case. Accounting
for the viscous forces is a straightforward procedure (see Fridman
1989): it may be done by multiplying the obtained equations by the
factor
.
For instance, Eqs. (19) and (26) will read as follows:

(28) |

(29) |

To estimate the maximum amplification which is achieved in thistransient growth, let us express

(30) |

which can reach a huge value in astrophysical disks.

We have seen that the concept of nonlinear instability in smooth
spectrally stable shear flows has undergone substantial revision
in the hydrodynamics community. Nowadays the concept of *the
bypass transition to turbulence* has become a favorite and is under
intensive development.

Let us summarize the main features of this concept:

- The onset of turbulence
and the turbulent state itself in smooth spectrally stable shear
flows is supported energetically by *the linear transient
growth of vortex mode perturbations* - the key ingredient of the
turbulence is the vortex mode (eddy) perturbation, and the key
phenomenon is the linear transient growth of the perturbation.

- Nonlinear processes *do not contribute to any energy
growth*, but regenerate vortex mode perturbations that are able to
extract shear flow energy. Doing so, nonlinear processes only *indirectly* favor the energy extraction by the vortex mode
perturbations.

- The non-orthogonal nature of the linearized
Navier-Stokes equations is the formal basis of the transient
growth.

- The non-orthogonal nature increases with
increasing Reynolds number; thus the operators are highly
non-normal for the huge Reynolds numbers of Keplerian disks
(*Re*>10^{10}) and the transient growth is asymptotically large
in *Re*.

In this paper we wanted to contribute to the revival of hydrodynamic shear turbulence as a possible explanation for the "anomalous'' viscosity in non-magnetized Keplerian accretion disks. Specifically, we wished to draw the attention of astrophysicists to the bypass concept.

We showed that in disk flows there exist vortex mode perturbations which are similar to those that are held responsible for the onset of turbulence in the Cartesian shear flow. The key point of our analysis is the interpretation of the important role of the pressure perturbations in the dynamical processes.

In fact, the kinematics and energetics of the vortex mode perturbations are identical in the rotating disk and the Cartesian shear flows in the 2D case. The Coriolis force only causes deviation of the pressure perturbations from the Cartesian case. By focusing on the epicyclic motions, and underestimating the action of the pressure terms, one is led to the false conclusion that the Coriolis force suppresses hydrodynamic turbulence in Keplerian flows (Balbus et al. 1996; Balbus & Hawley 1998).

This property of the pressure perturbations has been established here in the 2D case. In the more general case of 3D perturbations, the dynamics of vortex mode perturbations is somewhat more complicated, as it will be shown in a forthcoming paper (Tevzadze et al.). However, the role of the pressure perturbations is still to counteract the Coriolis force. In the 3D disk case, the wave-number domain where the perturbation undergoes transient growth is smaller in comparison to 3D Cartesian flows, but this is compensated for by the very large Reynolds number characterizing astrophysical disks.

Other scenarios which may lead to hydrodynamic turbulence in astrophysical disks have been presented recently: they invoke a linear (spectral) instability arising from the stratification perpendicular to the disk (Dubrulle et al. 2002) or due to deviations from cylindrical rotation (Urpin 2002; Klahr & Bodenheimer 2002). Such instabilities may well compete with the bypass mechanism presented here, and it is not possible to conclude presently which is the best candidate to render astrophysical disks turbulent.

In laboratory experiments the field narrows, because there is no stratification in the fluid (other than imposed on purpose), and there the bypass mechanism provides an attractive explanation for the turbulence detected in flows that are linearly stable (angular momentum increasing outwards). We refer to Couette-Taylor experiments performed by Wendt (1933), Taylor (1936), Coles (1950) and Van Atta (1966). Very recently such turbulence has been observed also in rotation profiles which share the properties of Keplerian disks, namely with their angular velocity decreasing outwards (Richard 2001).

Decisive conclusions about the self-sustenance of the turbulent state need to be supported by numerical simulations. To our knowledge, the simulations reported so far have failed to detect hydrodynamic shear turbulence in rotating flows with angular momentum increasing outwards. In a recent paper, Longaretti (2002) has discussed the possible explanations for this negative result. The main reason is probably the lack of spatial resolution, which still prevents us from reaching even the Reynolds numbers at which turbulence is detected in the laboratory. At this stage, we hope that the analysis presented here will help to build a fruitful background for the nonlinear numerical stability analysis of rotating astrophysical shear flows. A definite answer will result from the convergence of theoretical, numerical and experimental work.

This work is supported by the International Science and Technology Center grant G-553. G.D.C. would like to acknowledge the hospitality of Observatoire de Paris (LUTH). The authors would like to thank the referee for helpful comments on the early version of the paper.

We present here a simple sketch of the bypass scenario applied to
Keplerian disk flow, in the wave-number plane
(*k*_{x}, *k*_{y}) (see
Fig. A.1). We shall define as the "active domain'' the region
where viscous dissipation may be neglected, i.e. where
,
with
.

The linear dynamics of the perturbation may be described by
following each of its spatial Fourier harmonics (SFH) in this
wave-number plane. We single out one that is located initially at
some point 1 in the amplification part of the "active domain'',
which meets the condition
*k*_{x}/*k*_{y} < 0. According to Eq. (18), as *k*_{x}(*t*) varies in time, the SFH drifts in
the direction marked by the arrows, since *A*<0. (We present the
drift of the SFH only in the upper half-plane; since the
perturbation is real, there is a counterpart in the lower
half-plane.) Initially, as |*k*_{x}(*t*)| decreases, the energy of
the SFH grows. This growth lasts until the wave vector crosses the
line *k*_{x}=0 (point 2). Then, while undergoing attenuation, the
SFH continues its drift until it reaches point 3, where it is
dissipated through viscous friction. The same will occur with all
other Fourier harmonics. Consequently, if the nonlinear
interaction between different Fourier harmonics is inefficient,
the perturbation disappears eventually.
Permanent extraction of
shear energy by the perturbations is necessary for their
maintenance, which is possible when quadrants II and IV where
*k*_{x}/*k*_{y} < 0 are being repopulated through nonlinear interactions
between Fourier harmonics of quadrants I and III that have reached
sufficient amplitude. This may be achieved through three-wave
processes
;
four wave
processes
;
five wave processes, etc. In the figure we present an example of
three wave process
that contributes to the regeneration of an SFH in the
amplification area, transferring perturbation energy to it from
the attenuation areas. The bypass scenario implies the dominance
of this regeneration tendency of nonlinear processes, i.e.
predominant transfer of perturbation energy from quadrants I and III to quadrants II and IV, in other words *nonlinear positive
feedback*. To what extent the reproduction of the Fourier
harmonics in quadrants II and IV is sustained, even in the case of
positive feedback, depends both on the amplitude and on the
spectrum of the initial perturbation. Nonlinear decay processes
are weak at low amplitudes and are not able to compensate the
linear drift of SFH in **k**-plane. As a result, weak
perturbations are damped without any trace, and without inducing
transition to turbulence. The higher the amplitude of initial
perturbation the stronger are the nonlinear effects. At a
certain amplitude (which, of course, depends on the initial vortex
perturbation spectrum in **k**-plane and on the Reynolds
number), the nonlinear processes are able to counterbalance the action
of the linear drift, thus ensuring permanent return of SFH to the
amplification areas (quadrants II and IV). This eventually ensures
a permanent extraction of energy from the background flow and
maintenance of the perturbations, and thus of turbulence.

Therefore the bypass scenario can be realized only in the case of finite amplitude perturbations and in each case it has a threshold that depends on the perturbation spectrum and the Reynolds number.

Let us summarize. According to the bypass concept, vortex mode
(eddy) perturbations are the basic ingredient of hydrodynamic
shear turbulence. The bypass scenario involves the interplay of
four basic phenomena:

- the linear "drift'' of SFH in the **k**-plane;

- the transient growth of SFH;

- the usual viscous
dissipation;

- nonlinear processes that close the feedback loop
of the transition by mixing - by the angular redistribution of SFH in **k**-plane.

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