BIB-VERSION:: CS-TR-v2.0
ID:: CORNELLCS//TR92-1291
ENTRY:: 1993-10-14
ORGANIZATION:: Cornell University, Computer Science Department
LANGUAGE:: English
TITLE:: Pseudospectra of the Linear Navier-Stokes Evolution Operator and 
        Instability of Plane Poiseuille and Couette Flows: (preliminary report)
AUTHOR:: Trefethen, Lloyd N.
AUTHOR:: Trefethen, Anne E. 
AUTHOR:: Reddy, Satish C.   
DATE:: June 1992
PAGES:: 30
ABSTRACT:: 
This is a rough, interim report on some new results concerning the stability 
of plane Poiseuille and Couette fluid flows, following upon recent work by 
Henningson and Reddy, Butler and Farrell, Gustavsson and others. We emphasize 
that the conclusions proposed here have not yet been checked carefully and 
are subject to change.

Our principal results are as follows:

1. Plots of the spectra of the ``full'' Navier-Stokes operator for 
Poiseuille and Couette flows, i.e., without restriction to a wave number pair 
($\alpha, \beta$) or to even or odd modes ($\S\S$4,5).

2. Analogous plots for the pseudospectra of this operator. Comparison 
of the pseudospectra with the spectra gives a new interpretation of why the 
physics of these linear flow problems is not controlled by the location of 
the most unstable eigenvalue ($\S\S$4,5).

3. Demonstration that these pseudospectra predict the Butler-Farrell 
``optimal'' transient energy growth ratios to within a factor of about 2 
($\S$6).

4. Demonstration that about 90% of the Butler-Farrell growth can be 
achieved by a 3$\times$3 linear model obtained by projecting the Navier-Stokes 
problem onto the space spanned by three dominant eigenmodes, for Couette flow, 
or four in the case of Poiseuille flow ($\S$8).

5. Demonstration that although 1 Orr-Sommerfeld mode and 3 Squire modes 
suffice for the 4$\times$4 model in the Poiseuille case, in keeping with a 
recent result of Gustavsson, one can do equally well with 2 modes of each kind 
or with 3 Orr-Sommerfeld modes and 1 Squire mode ($\S$8).

6. Demonstration that the minimal operator perturbation required to 
destabilize a stable flow has norm of order $R^-$$^2$, where $R$ is 
the Reynolds number, though the distance of the least stable eigenvalue from 
the real axis is $O (R^-$$^1$) ($\S$7).

7. Presentation of a 2$\times$2 model illustrating that if the linear 
problems described above are capable of transient energy growth of order $M$ 
(e.g., $M\approx$1000 according to Butler and Farrell), a weak and 
intrinsically energy-conserving nonlinear term can ``bootstrap'' that growth 
to a higher order such as $M^2$. This supports the view that although 
nonlinear terms are of course essential to the subcritical instability of 
fluid flows, the detailed nature of the nonlinear interactions may sometimes 
be relatively unimportant ($\S$2).

8. Adaptation of this ``bootstrapping'' idea to the fluid flows 
considered earlier, particularly the 3$\times$3 approximation for Couette 
flow with $R$=1000.
END:: CORNELLCS//TR92-1291
BODY::
Pseudospectra of the Linear Navier-Stokes
Evolution Operator and Instability of Plane
Poiseullie and Couette Flows:
(preliminary report)
Lloyd N. Trefethen*
Anne E. Trefethen
Satish 0. Reddy
TR 92-1291
June 1992
Department of Computer Science
Cornell University
Ithaca, NY 14853-7501
*Supported by NSF Grant DMS-91 16110.
Pseudospectra of the linear Navier-Stokes evolution operator
and instability of plane Poiseuille and Couette flows:
preliminary report
Lloyd N. Trefethen*
Department of Comp'iter Science
Cornell University
Ithaca, NY 14853
lntC?cs.cornell.edn
Anne E. Trefetben
Cornell Theory Center
Cornell University
Ithaca, NY 14853
aet(Actctc.corneli.edn
Satish C. Iteddy
Conrant Institnte of A4athematjcal Sciences
251 Mercer St.
New York, NY 10012
wddyC?cims.nyu.edn
*Supported by NSF Grant DMS-911611o.
Table of Contents
t. Introduction
2. A 2 x 2 model
3. The linear Navier-Stokes evolution operator
4. Spectra and pseudospectra for plane Conette flow
5. Spectra and pseudospectra for plane Poisenille flow
6. l??ansient energy growth estimates based on pseudospectra
7. Minimal destabilizing perturbations
8. 3 x 3 and 4 x 4 approximations with 90% of the l3utler-i?arrAl growth
9. The 3 x 3 model with nonlinearity
kelerences
t
Abstract
This is a rough, interirn report Ori some new results concerning tlie stability of
plane Poiseuille arid Couette fluid flows, following upon recent work by Henningson
and Reddy, Butler and Farrell, Gustavsson, and others. We emphasize that the
conclusions proposed here have not yet been checked carefully arid are subject to
change.
Our principal results are as follows:
1. Plots of the spectra of the "full" Navier-Stokes operator for Poisenille arid
Couette flows, i.e., without restriction to a wave riumber pair (o,P) or to even or
odd modes (4,5).
2. Analogous plots for the pseudospectra of this operator. Comparison of the
pseudospectra with the spectra gives a new interpretation of why the physics of
these linear flow problems is not controlled by the location of the rnost unstable
eigenvalue (4,5).
3. Demonstration that these pseudospectra predict the Butler-Farrell "opti-
mal" transient energy growth ratios to within a fador of about 2 (6).
4. Demonstration that about 90% of the Butler-Farrell growth can be achieved
by a 3 x 3 linear model obtained by projecting the Navier-Stokes problem onto the
space spanned by three dominant eigenmodes, for Couette ?ow. or four in the case
of Poisenille flow (8).
5. Demonstration that although 1 Orr-Sonimerfeld mode arid 3 Squire modes
suffice for the 4 x 4 model in the Poiseuille case, in keeping with a recent result
of Gustavsson, one cari do equally well with 2 modes of each kind or with 3 Orr-
Somnierfeld modes and 1 Squire mode (8).
6. Demonstration that the minimal operator perturbation required to destabi-
lize a stable ?ow has norm of order J?--H2, where 1? is the RQynolds number, though
the distance of the least stable eigerivalue from t1ie real axis is o(i?--H1) (7)
t. Presentation of a 2 x 2 model illustrating that if the linear problems de-
scribed above are capable of transient energy growth of order A? (e.g., Al 1000
according to Butler and Farrell), a weak and intrinsically energy-conserving non-
linear term can "bootstrap" that growth to a higher order such as Al2. This
supports the view that although nonlinear terms are of course essential to the sub-
critical instability of fluid flows, the detailed nature of the rionlinear interactions
may sometimes be relatively unimportant (2).
8. Adaptation of this "bootstrapping" idea to the fluid flows considered ear-
lier, particularly the 3 x 3 approximation for Conette flow witli 1? 1000.
2
1. Introduction
A change of paradigm is taking place in the field of hydrodynamic stability.
The traditional paradigm is eigenvalue analysis, which can be summanzed by the
following sequence:
(1)
(2)
Linearize about the laminar solution;
Look for unstable eigenvalues of the linearized problem.
An "unstable eigenvalue" means an eigenvalue in the upper half-plane, thanks to
the convention of introducing a factor --Hi in the linearization. For planar shear
flows, one of the central results of eigenvaliie analysis is Squire's theorem of 1933,
which asserts that in the study of the most unstable eigenvaliies of a flow (i.e.,
eigenvalues with largest imaginary part), one may always restrict attention to 2D
perturbations of the flow field [2].
For some problems (e.g. B6.nard convection, Taylor-Conette flow) the predic-
tions made by eigenvalue analysis match laboratory experiments well. These are
the cases whew the operator involved in the linearized model is normal (i.e., has or-
thogonal eigenfunctions). For other problems (e.g. plane Poiseiiille, plane Conette,
pipe Poiseuille flow), the predictions made by eigenvaliie analysis match labora-
tory experiments poorly. In these problems the operator involved in the linearized
model is highly non-normal, with the basis of eigenfunctions typically having a
condition number that grows exponentially as a flinction ?f J?i/2 as 1? oc (J? is
the Reynolds number) [14,itH]
For example, eigenvalue analysis predicts a critical Reynolds number 1?
5772 for plane Poisenille flow, but in the laboratory, transition to turbulence is
usually observed at R?vnolds numbers closer to 1000. For plane Coiiette flow
eigenvalue analysis predicts linear stability for all Reynolds numbers, but transition
to turbulence is observed at Reynolds numbers as low as 350. Clearly there is
a mismatch in these problems between experimental observations and predictions
based on eigenvaliies. This has been recognized for many years, and the explanation
has usually been attributed to step (1) above. If linear analysis has failed, the
reasoning goes, one must look more closely at the nonlinear terms.
Recently it has emerged, however, that the failure of eigenvalue analysis may
lie principally with step (2), not step (t). Even if all of tbe eigenvalues of a
linearized flow problem lie in the stable half-plane, solutions to that linear problem
may still exhibit transient energy growth, thanks to the non-orthogonality of the
eigenfunctions. If attention is restricted to 2D perturbations, in keeping with the
spirit of Squire's theorem, the following transient energy growth factors are possible
[4,14]:
2D Conette, 1? = 1000: 13.0
2D Poiseuille, 1? = 5000: 45.7.
however, since such transient growth is not a modal phenomenon, there is no solid
3
reason to restrict attention to 2D perturbations. In the past two years it has been
discovered that much larger transierit energy growth factors arc possible if one
allows 3D perturbations [BF,RH,6]:
3D Conette, R--H--Hl000: 1185.
3D Poisenille, R 5000: 4897.
These numbers indicate that even a linearized flow that is asymptotically stable
may include a mechanism of great energy growth. Moreover, the 3D (pseudo)modes
that exhibit this transierit growth have the forrri of streamwise vortices, which have
been observed for many years as a conspiciious feature in laboratory experiments.
In summary, in the past year it has begun to appear that the essential mccli-
anism of energy growth of perturbations of plane Poiscuille arid Couette flows may
be linear, three-dimensional, and nurdated to cigenmodes. This emphasis on linear
but non-modal effects is the emerging riew paradigm in hydrodynamic stability.
The central people involved in these developments have been Farrell arid But-
icr (Harvard); llenningson, Schinid, and R?eddy (MIT, MIT, Courant Institute);
and Gustavsson (Lulca). In particular, two key papers on 3D transient energy
growth have beeri written recently, which we shall refer to as [BF] arid [RH]:
[BF] K. M. Butler arid B. V. Farrell, "Three-dimensional optimal pertur-
bations in viscous shear flow," to appear in Physics of Flrnds, 1992;
[R?II] 5. C. Reddy and D. 5. Henningson, "En&gy growth in viscous channel
flows," submitted to Journal ofiWuid Alechanics, 1992.
The longstanding figure in espousing the new point of view has been Brian Farrell,
who in a succession of papers since 1982 has described the possibilities of non-model
transient effects in fluid flows, both in meteorology and classical fluid mechanics
[?,?1 The discovery of 3D transient eriergy growth is due to Farrell arid his student
Natliryn Butler in 1991, based partly on discussions with lleriningsori; shortly
before, growth factors about 90% as large as the optimal ones had been discovered
independently by Giistavsson [6]. The numbers listed above are taken from [BFj.
This paper is a rough compilation of some computational results that we have
obtained in recent weeks with the aid of Matlab and Fortrari codes adapted from
those used previously by Reddy and Henningson [RllJ. These computations have
led us in a number of directioris, as summarized in the Abstract. We have decided
to record them here in rough form before preparing one or more papers for publi-
cation that will, inevitably, cover only a subset of these issues and appear several
years herice. This paper is incomplete in two respects. First, the computational
results are tentative, and we will undoubtedly find that certain things have to be
changed as we re-examine and extend the calculations in the upcoming months.
Second, we give none of the mathematical or physical details, and few of the nec-
essary references, but assume that the reader is already familiar with the field of
hydrodynarnic stability and with [BF] and [RIl].
4
2. A 2x2 model
Subcritical transition to turbulence depends upon nonlinearity, yet we have
claimed that the energy growth mechanism is essentially linear. In this section
we explain what we mean by this by describing a 2 x 2 nonlinear model problem
which, we believe, may capture some of the process of transition to turbulence. In
particular this model illustrates how a linear growth mechanism of order 1? may
be "bootstrapped" to a nonlinear growth mechanism of order IQ2 or larger. We are
grateful to Philip Holmes for suggesting to us the study of 2 x 2 models.
Let I? be a large parameter and consider the matrix*
= An,			(2.!)
The 2 x 2 linear problem ut = An has solution n(1) = etAu(O), and IletAll is the norm
of the solution operator. Figure 2.1 plots this norm as a function of I for R = 1,
10, 102, io3, and i0?. Throughout this paper, is the usual 2-norm induced by
the inner product (n, v) = fn*v.
Figure 2.1 reveals the following features of the curves lictAll. The initial slope
(determined by t1ie lield of values of A) is 1?/2. Tlie asymptotic slope as I
(determined by the spectrum of A) is --H1. In between these extremes, for finite I,
there is a large hump, of height Al iQ/4. This corresponds to transient amplitude
growth of order 1?, in analogy to the transient growth described in [6j, [131'], and
Now let us add a nonlinear term of llamiltonian form, i.e., one that transfers
energy from the decaying modes of the linear problem to the growing modet but
does not in itself add energy. The simplest problem of this kind is
nt = AntinliRn,
I?			(2.2)
for the same matrix A. The matrix B has imaginary eigenvalues, and t1ie nonlinear
term does not add energy into the system directly, as is shown by t1ie vanishing of
B from the formula
dm112 __
di			--H 2(u,nt) = 2(n,An)t2IInJI(n,Bu)
--H 2(n,Ati) = (n,(AtAT)n).
(2.3)
*The results would be essentially the same if we took both diagonal entries equal to --H1. However we
have used the less elegant example (2.1) to makc sure there is no misunderstanding that the points we
are making do not depend on nondiagonaflzability.
t The phrase "growing mode' here and below is cardess, since the eigenmodes of this problem all decay.
We are speaking of components that experience transient growth, not asymptotic.
5
10?
II0?AII
102
100
10-2
0			2			4			6			8			10
Figure 2.1. II ciAll vs. I for the linear model problem (2.1).
102
100
IIu(t)!i
10-2
10?
10-
v
IIu(O)II
10--H2
0			4			6			8			10
Figure 2.2. IIu(t)II vs. I f??r solutions to the nonlinear model problim
(2.2) with J?=25 and various initial vectors u(0)(0,coust)T.
1?
linear
growth
factor
Al
12.5			3.18
25			6.28
50			12.5
100			25.0
threshold
ampliti?de
Af2c
.043			.433
.0105			.414
.0026			.406
.00065			.406
Table 2.1. The "bootstrappiug" of the linear growth factor M causes
the threshold amplitude to be of order M--H2.
6
Figure 2.2 shows the effect of the nonlinear term on the solutions u(1), with
I? = 25. The figure shows the norms IIu(t)II for solutions starting frorn six different
initial vectors of the form u(0) (0, 11?I(0)11)T. When IIu(O)II is small1 well below
10--H?, the curves of Figure 2.2 match those of Figure 2.1, indicating that the evo-
lution is essentially linear. For larger IIu(O)II, larger than a threshold amplitude
10--H2, the nonlinearity has a pronounced effect, and the threshold amplitude is
about 10--H2. Initial vectors larger than this do riot decay to zero but blow up and
are eventually attracted to a large-amplitude critical point of (2.2).
What is important about the behavior of Figure 2.2 is that the amplitude
amplification exhibited is far greater thari that of the purely linear problem. Ta-
ble 2.1 lists the linear growth factor Al and the threshold amphtude c- (for this
particular initial vector) for four values of R. One sees that ( is of order M--H21
riot Al--H' as one might suppose from the idea that the growth mechanism is "only
linear." This bootstrapping phenomenon can be explained as follows. Suppose
the solution at time 0 consists of a vector of amplitude ? in the direction of the
growing rnode of (2.1). At later time the solution has grown to order Al? by the
linear gwwth mechanism but moved into the decaying mode. Meanwhile, however,
the nonlinear term in (2.2) has had the effect of transferring some of this energy
back to the growing mode of (2.1), with amplitude (M?)2 since the nonlinearity is
quadratic. If (Al?)2 is of order less than ?, the process is not self-sustaining and
energy dies awa??. On the other hand if (M?)2 is of order greater than ?, then
there is more energy than was present at the start and feedback occiirs, leading to
self-sustained amplitude growth and to an eventual amplitude, as it happens, on
the order of 1?. Thus the threshold amplitude is c =
Figure 2.3 gives another look at this model problem by depicting several
trajectories iii the phase plane (u,1u2)T for 1? 5. The linear term (A) forces tra-
jectories to the right in the upper half-plane arid to the left in the lower halfplane.
For very small initial arnplitudes the trajectories then turn a corner and head back
to the stable critical point at the origin. The nonlinear term (B), however, si'-
perirnposes ori this linear portrait a counterclockwise rotation, transferring energy
frorn the decaying to the growing mode of the linear term. For initial airiplitudes
larger than order J?--H2 the process is self-sustaining and the trajectory spirals up
to the nonlinear critical point.
The growth process described here is "essentially linear" in the following re-
spects. First, eq. (2.3) shows that the nonlinear term does not add energy to the
system in a direct sense. (The same situation holds in hydrodynamics, as em-
phasized by Reddy and llenningson [RllJ.) Second, the growth process we have
described is not specific to (2.3) but would apply to many different nonlinear
terms. Any quadratic nonlinear term that transfers energy from decaying to grow-
ing modes is likely to generate a bootstrapping effect with threshold amplitude
= O(M--H2) much as in the experiment presented here. A random perturbation,
for example, will often suffice. (Not all norilinear terms are suitable; for example,
2
A
-4			-2			0			2			4
Figure 2.3. Trajedories in the phase plane for (2.2) with R 5. The
basin of attraction of the stable critical point at the origin is of width
o(i?-2).
replacing B by its transpose transfers energy in the wrong direction.) higher-order
nonlinearities lead to similar effects, again independently of details; for example
with a cubic nonlinear term one expects ? o(Ai?3/2). Of course the critical
points of the nonlinear system such as those of Figure 2.3 are entirely dependent
on the nonlinear details, but that is the problem of turbnlence, not instability.
From amplitude O(?) to 0(t) the nonlinear terms are small compared with the
linear ones and the critical points are far away.
8
3. The linear Navier-Stokes evolution operator
Now we want to move from 2 x 2 matrices closer to fluid mechanics. Let
u0 = u0(x,y,z) be a time-independent solution to the Navier-Stokes equations in a
geometry of interest. For plane Poisenille and Couette flows the geometry is the
infinite region bounded by two fixed plates at y =0 and y = 1, and u0 corresponds
to the usual laminar flow solution between these boundaries a parabolic velocity
profile for Poiseuille flow, and a linear profile for Couette flow.
Let uotu(I)=u0(x,y,z)tu(1,x,y1z) be a slightly perturbed solution to the
Navier-Stokes equations. If we take ti to be infinitesimal, then it satisfies an an-
tonomous linear equation of the form
du
di
where i = ?ffl?? and  is a linear operator. (The factor i is purely a matter of
convention.) The precise choice of variables in the formulation of u is not essential
to the idea of (3.1), but a convenient choice is to take u to be a vector consisting
of the normal velocity and the normal vorticity, as descnbed in [6], [BF] and [RH]
(`?normal" means in the y direction, normal to the solid boundaries). We will
call the operator  the linear Navier-Stokes evolulion operator, or the Navier-
Stokes operator for short. `!?he fundamental problem of linear stability theory
is to analyze the evolution of solutions to this operator, or formally speaking, the
operator exponential e?itfl [15,20]. In typical numerical calculations this translates
to a matrix exponential for a matrix of dimension in the range 20--H200.
The operator  is rarely mentioned ill the literature of hydrodynamic stability.
Instead, several reductions are made at the start of the discussion. First, one
simplifies the problem by Fourier transforming in the unbounded variables x and
z, introducing two real wavenumbers er and ,3, respectively. Second, one simplifies
the problem further, in the Poisenille case, by considenug separately the evolution
of even or odd perturbations (with respect to the centerhue in y). Third, one
assumes an exponential dependence with respect to I also, permitting a complex
frequency .? It is in this step that the evolution problem (3.1) is replaced by an
eigenvallie problem.
Because of Squire's theorem, a fourth simplification is also usually introduced,
at least until the literature of the last few years. One sets 13 = 0, permitting only
2D perturbations.
The product of all four of these siniplifications is the Orr-Sommerfeld equa-
tion: an eigenvalue equation in one spatial variable (y) in which the x wavenumber
er appears as a parameter [2]. The analogous eigenvalue problem corresponding to
3D instead of 2D flow perturbations can be formulated in terms of coupled "Orr-
Sommerfeld modes" and ?Squire modes" [BF,Rll], but seems to have no single
*Alternativdy, one may take w to be real and allow a to be complex. We do not consider th? and other
variations in this paper, but they are important.
9
name other than these. The 3D evolution problem (3.1), i.e., without eigenvalue
analysis, seems to have no agreed-upon name at all, an indication of how prevalent
the tendency in the hydrodynamic stability literature has been to equate linear
analysis with eigenvalue analysis.*
The first two of these simplifications Fourier transformation and even/odd
separation are physically reasonable, because the modes that are being separated
are orthogonal. The situation is analogous to that of a block diagonal matrix,
with the blocks corresponding to various choices of ?, P, and even/odd; since the
couplings between the blocks are zero, nothing is lost by analyzing them separately
(at least for the linear problem). On the other hand the third simplification,
looking at eigenvalues, is physically unreasonable when the eigenmodes are far
from orthogonal, and this is analogous to the case of a triangular matrix in which
the off-diagonal terms are far from zero. Though the off-diagonal terms in such a
matrix have no effect on the eigenvalues, they may have a controlling effect on the
actual behavior of the matrix in an evolution process [18].
The papers [BF] and [ItH] are the first to analyze (3.1) systematically without
making the physically unreasonable third and fourth steps just described. However,
both of these papers do make the first and second simplifications involving a1 ?,
and even or odd modes. Thus the spectra and pseudospectra that are discussed
in those papers correspond to fixed choices of a, ?, and even/odd. For example,
Figure 5 of [ItH] depicts the spectra and pseudospectra for Poisenille ?ow with
a = 1 and p 0 and 1? 3000. The spectrum takes the form of an asymmetric Y,
a shape that has been familiar now for about 20 years [12,14]. Figure 7 of [RH]
treats the same problem but with a = 0 and P =2, leading to a symmetric picturc
and a pure imaginary spectrum. This corresponds approximately to the "optimal"
growth-m&ximizing mode of [BF], a streamwise vortex, which is now recognized as
more important physically but is a less familiar picture.
The question with which we began this project was1 what do tbe spectra and
pseudospectra of the full operator  look like, i.e., with no restrictions at all?
These sets are defined as follows:
spectrum: A() = [z E C: II(zI--H )?1II =
&pseudospectrum: A?() = fzcC: II(z1?)?'II >?--H`J
=[zEC:zEA(tS)forsome?with??J??J.
For a discussion of the idea of pseudospectra with computed matnx examples, see
[18]; a comprehensive work is in preparation [19]. Because of the orthogonality
described above, one has
= Ua,?+A(op,?),			A?() = Uap?A?(?p+)			(3.2)
*See [8J for a discussion of an analogous situation in the literature of the numerical solution of ODEs.
10
where I represents the even/odd choice in the Poisenille case. Thus in principle, to
find A() or one must look at the corresponding sets as computed by Reddy
and llenningson or Butler and Farrell for all parameters ?, P, I. in practice one
performs a continuous optimization over 0 and P, once for the even modes and once
for the odd modes. We have done this in Matlab on Sparc workstations, taking
advantage of the codes of Reddy and iienningson based on Chebyshev spectral
discretizations; we omit the details.
Unlike the spectrum, the pseudospectra A() depend on the choice of norm
II II.? For the results to be physically meaningful (and for (3.2) to be valid), one
must pick a norm that corresponds to the square root of energy:
Energy--H lull2.			(3.3)
If u consists of a velocity coupled with a vorticity, then the appropriate norm
is the weighted 2-norm obtained after a diagonal scaling of the vorticity terms by
the wave number amplitudes; see [6], [Rll], and [BF] for details.
In summary, the problem of optimal amplitude growth for ttie 3D linearized
Navier-Stokes problem is the problem of the operator norms
Amplitude: e?it II
as a function of t > 0. For optimal energy growth we square this quantity and
consider
Energy: llc?????ll2.
*As must any quantity that is physically important.
11
4, Spectra and pseudospectra for plane Couette flow
Figures 4.t--H4.4 depict the spectra and pseudospectra for Couette flows with
I? 250, 500, 1000, and 2000. Note that the real and imaginary axes in these plots
are not equally scaled. The various parts of these plots can be described as follows.
The shaded region in each plot is the spectrum A(). This was computed
along the lines suggested in (3.2), by compiiting A(a?p) for values of a and (3on
a grid (there is no even/odd choice iii the Couette case). The result is a set of dots
in the complex plane that fill out a region that we take to be the spectrum and
shade accordingly.
The three curves in each plot are the boundanes of the &pseudospedra A?()
for 6 10--H2, 10--H2?5, l0--H? (frorri upper to lower). These clirves are obtained by
cornputirig II(zJ--H)--H111 at each point on a grid in tlie -plane and then sending
this grid data to a contour plotter. Each grid value requires an optimization over
a and t3, and thus these are large computations.
Finally, the asterisks along the imaginary axis represent the (discrete) eigeri-
values for a case of particular interest: a 0 and ? 1.6, approximately the
"optimal" mode of Butler arid Farrell.
The plots in this and the next section are rough; so far, oiir data come pri-
marily from Matlab computations with fairly small inatrices. In particular, the
boundary of the smallest ?-pseudospedriim iii the plots is irregular in many cases,
and sometimes appears to ciit through the spectrrini. This is impossible, and is an
indication that our results are inaccurate. Such matters will be clearcd lip before
final publication.
Let us consider what is revealed in Figures 4.1 4.4. First of all, the spectrum
of IC is always in the lower half-plane. This is well known, arid corresponds to
the fact that plane Coiiette flow is stable for all Reynolds numbers according to
eigenvalue analysis.
The pseudospectra of , on the other hand, protnide far into the ripper half-
plane. If  were a normal operator, the boundary curve of Af() f??r each 6 would
lie exactly at a distance 6 from A(). It is evident iri the figures that the curves
lie much further frorri  than this. In maiiy respects, for a finite time at least, IC
will behave as if it has eigerivalues in the upper half-plane. In the later sections
we will see what quantitative conclusions can be drawri frorn these pseudospectra.
One cannot see in Figures 4.1--H4.4 what values of a and ? couespond to various
points in the spectrum and pseudospectra. ?Ve have not investigated this point in
detail, but roughly speaking, the situation is as follows. The upper boundary of the
spectrum corresponds to small values of p (less thari 1) arid values of a that grow
in proportion to Rez. The values of a in the pseudospectra are corriparable, biit
the values of ? are considerably larger. This is a?other reflection of the fact that
although domiriant eigenvalues can be taken to be two-dirnensiorial, the nori-modal
aspects of this operator are strongly three-dimensional.
12
0.1--
-1			0
Figure 4.1. Spectrum (shaded) aiid c--pseudospec?ra (c.--H--H 10--H2, 10--H2.5,
to--H3) for Couetle flow wi?h 1?--H250.
n
0
-1
0
Figure 4.2. Couette flow, 1? =500.
-0.1
13
0.1
--H1
0
Figure 4.3. Coiie??e fl()w, 1?--H--H 1000.
0.1
0
0
Figure 4.4. Couette flow, 1? 2000.
14
-1
5. Spectra and pseudospectra for plane Poiseujile flow
Figures 5.1--H5.4 are analogous to Figures 4.1--H4.4, but for Poisenille instead of
Conette flow. The Reynolds numbers are R = 1000, 3000, 5772, and 8000. The
asterisks show eigenvalues corresponding to a = 1 and t3 = 2, and the circles show
eigenvalues corresponding to a =0 and P =0 (the Y shape).
The geometry is more complicated here than in the Conette case. First of all
there is the famous difference that in this case, for 1?> 5772, there is an eigenvaliie
in the upper half-plane. This dominant eigenvalue corresponds to the "mogul"
that is visible in the four figures, which begins quite small at = 1000, becomes
more prominent at fi = 3000, hits the real axis at 1? = 5772, arid then crosses into
the upper half-plane for 1? = 8000. The modes that make up the mogul correspond
to small values of ?pj (typically ? 2) and values of a that grow approximately in
proportion to R'ez. In particular, the most unstable point at the top of the mogul
corresponds approximately to ? = 0 (Squire's theorem) and a 1.
A second and quite distinct family of modes that delineates the spectrum is
the ?Mt. Fuji" branch, the steady slope that forms the boundary of the shaded
region to the outside of the mogul and is continued as a dashed lirie underneath.
This family corresponds to larger values of ??, typically > 2, and perturbations
that are more highly three-dimensional.
The regions we have shaded are bounded by just these two families of eigenval-
lies, and represent a correct depiction of the spectrum A() if attention is restricted
to even modes. However, there is a further part of the spectrum corresponding to
odd modes, also shaped like Mt. Fuji b?t with a higher imaginary part. This addi-
tional branch is marked by the highest of the dashed curves in our plots. Properly,
this upper dashed curve should contribute to the boundary of the shaded region,
biit we have not yet carried out complete enough calculations for odd modes to
draw this part of the picture properly. hence our compromise: Figures 5.1 5.4
depict spectra and pseudospectra for even modes oilly, but the boundary of the
odd mode spectrum is superimposed as an extra dashed line.
As in Couette flow, the pseudospectra tell a very different story from the
spectra. The striking feature is that the mogul has negligible effect on the pseu-
dospectra, indicating that it probably corresponds to modes at a substantial angle
from the other eigenmodes of the system.* The action is in the pseudospectra,
not the spectra, and near the imaginary axis, riot near the mogul.
Fortunately, though these figures represent imperfectly the effects of odd
modes near and below the uppermost dashed curve, the pseudospectral contours
in the upper half-plane are probably correct for the full operator  with no restric-
tion, for preliminary computations indicate that the even-mode resolvent norm is
larger than the odd-mode resolvent norm ill the upper half-plane.
*As mentioned in the last section, the cases where pseudospectral boundaries intersect the shaded region
represent errors in our computations. So far as we know these are local errors, with little impact on the
rest of the plot.
15
0.1
0
-1			0			1
Figure 5.1. Spectrum (shaded) aud &pseudospectra ((?= 10--H2, 10--H2.5,
i0--H3, t0--H???) for Poiseullieflow with I? = 1000. See the text for comments
about odd vs. even modes.
0.j
-1
0
Figure 5.2. Poiseujile flow, 1? =3000.
16
0.1
-0.1
0
Figure 5.3. Poise?i11cflow, 1? 5772.
0
-1
0
Figure 5.4. Poiseujileflow, 1? 8000.
17
6, ?ansient energy growth estimates based on pseudospectra
The Butler-Farrell optimal perturbations are functions that achieve the max-
imum possible transient energy growth in a linear Navier-Stokes evolution process
that has all its eigenvalues in the stable half-plane. It is interesting that by ex-
amining the pseudospectra of these operators in the unstable half-plane without
knowing anything about the problem they come from??ne can predict that tran-
sient growth like this must occur and estimate its magnitude to within a fador of
about 2. Bounds of this kind have been considered previously in [Rll] and [14].
The estimate that makes this possible is a trivial one that might be calkM
the `easy half of the Kreiss matrix theorem for exponentials" [11,13]. Suppose
I gir1j < Al for some constant Al and all I > 0. Then by the Laplace translorm, for
any with Itez > 0 we have
II(zI--H)?1JI			f0?e--HSzetflds ? Al' f0?e?szds --H M
Contrapositively, if II(zI--H)--H'II > Al/Izi for some A'l and some with R.ez>0,
then we must have IIetII > Al for some I > 0. Thus we have the bound
s?u>i?) IIe?II > RSJzlP>o IzIII(zJ--H )--H`II.
In particular, taking z to be a real number y> 0, and rotating by 900 in keeping
with the usual convention of hydrodynamic stability gives
sup 1e?itfl11 > sup I?] I (?yJ--H )--H1 (6.1)
t>o			y>O
We can state this bound in the language of pseudospectra if we define the c-
pseudospeciral abscissa of  by
The estimate becomes
slip Imz.			(6.2)
s?>p0 e?itflII > snp-?'??.			(6.3)
From (6.3) and the pictures of the last two sections, with the aid of a ruler,
one can estimate transient growth ratios. Consider for example Poisenille flow with
1? 3000. From Figure 5.2 one sees that the boundary of the ?-pseudospectrum
with c- = 10--H??? cuts the imaginary axis at a height of about 0.00S. Thus we have
a10?35()?0.008 and therefore
sup 16--Hitfl			0 008			25.3,
t>o			>
18
or, squaring to measure energy,
supIIc?itII2 > 640.
t>o
The actual supremum in (6.3) for this problem is 29.3 ? (determined with
a computer instead of a ruler), attained with ? 3.346 x io--H5. Now the actual
optimal growth for this problem is 42.0 #m63. Thus (6.3) falls short of the
maximum by only about a factor of 2.
Tables 6.1 and 6.2 list the estimates (6.3) for a range of Reynolds numbers for
both Poisenille and Conette flow (compare Tables 1,11, III of [BPi). The numbers
reveal that in both the Poisenille and Conette cases, the lower bound (6.3) captures
about 70% of the optimal amplitude growth and thus about 50% of the optimal
energy growth. Moreover, the wavenumber /3 that gives the optimal estimate
(6.3) is virtually identical to the wavenumber of the true optimal. Clearly for this
problem, the essential features of the growth mechanism are well represented in
the pseudospectra of .
amplitude			energy
growth			growth
factor			factor
M			Al2
I?
1000
3000
5000
9.78/14.0			95.6/196.
29.3/42.0			858./1763.
48.73/69.98			2375./4898.
(ft/102.6)/(R/71.45) (R/102.6)2/(R/7l.45)2
0/
0/
0/
0/0
2.05/2.05
2.05/
/2.044
/2.04
Table 6.1. Transient growth for Poisenille Jiow at various Reynolds numbers.
In eacli xx/yy pair, xx corresponds to the estimate (6.3) based on pseudospe&
tra and yy to the actual Butler-Farrell optimal.
amplitude			energy
growth			growth
factor			factor
___Al___			M2___			p
250			5.76/8.597			33.2/73.9			0/0.12			1.54/1.61
500			11.9/17.20			141.6/296.0			0/0.067			1.63/1.60
1000			23.5/34.42			552.3/1185.			0/0.035			1.63/1.60
2000			47.0/68.84			22t1./4739.			0/0.018			1.62/1.60
oo (R/42.6)/(J?/29.05) (R/42.6)2/(1?/29.05)2			0/0			1.62/1.60
Table 6.2. Same as Table 6.1 but for Couette Ilow.
1?
19
7. Minimal destabilizing perturbations
One method of coming to grips with the energy growth mechanism in lin-
earized hydrodynamics is to study the Butler-Farrell "optimal," the initial function
that achieves maximum possible transient energy growth before eventually decay-
ing. In the language of linear algebra, this optimal is the dominant right singular
vector for the operator etC with t chosen to maximize 1etC1 [5]. Another approach,
considered in the last section, is to find the extremum of the pseudospectral esti-
mate (6.3). This can be described as the dominant right singular vector for the
operator (iyI?)--H' with y chosen to maximize ll(?yI--H)--H'll. From the close
agreement of the & and ? parameters listed in Tables 6.1 and 6.2, it appears likely
that both of these methods identify mudi the same streamwise vortex. We have
not confirmed this in detail.
In this section we consider a third approach to understanding the destabilizing
effects of non-normality. We ask: what is the minimal-norm perturbation ? sucli
that  t ? is unstable, i.e., has eigenvalues in the unstable half-plane? We call
such an ?, or its norm, a minimal destabilizing perturbation. Fquivalently, what
is the smallest e for which A?() touches the real axis? Also equivalently, what is
the inverse of the largest value of ll(?' )--H`II for z C IR?
From the pseudospectral plots of Sections 4 and 5 it is cl??ar that tlie de-
termining point in the complex plane is z 0 (except iii Poiseuille flow with 1?
greater than a number very slightly below 5772). Thns the question of minimal
destabilizing perturbations reduces to the question, how large is l(zI--H)?'ll for
0? The corresponding function is the dominant right singular vector of --H?
itself, that is, the function that  most nearly annihilates. Questions like this of
??nearness to instability" have received attention in the past decade in the litera-
tures of numerical linear algebra and control theory. See for example [1], which
contains a plot of a computed pseudospectrum, [9], and [10], where the minimal
norm ll?ll is called the stability radius for . We are grateful to kidgwa? Scott for
suggesting that we apply these ideas to the Navier-Stokes operator.
The minimal destabilizing perturbation is a very diff&ent quantity from the
the distance of the spectnim from the unstable half-plane. When a matnx or op-
erator is non-normal, the minimal destabilizing perturbation may be far smaller
than the location of the spectrum might suggest, and this turns out to be true
for the Navier-Stokes problem. Tables 7.1 and 7.2 show that whereas the eigen-
value distance scales as R--H1 (for 1? < 5772, in the Poisenille case), the minimal
destabilizing perturbation scales as 1?--H2
For Poiseuille flow with R = 1000, for example, which is roughly the Iteynolds
number at which transition to turbulence is observed in the laboratory, Table 7.1
indicates that a perturbation ? with ll?ll 3 x io--H4 should be enough to make t?
unstable. We emphasize that this is not a perturbation of the data, such as might
be introduced by imperfections in the experimental apparatus, but a perturbation
of the operator. The relevance of this to the evolution of fluid flow is that the
20
distance
of spectrum
from unstable
hatfplane
1?			___			___			__			p
minimal
destabilizing
perturbation
E
1.201 x to--H3
3.010 x to--H?
3.346 x 10--H?
1.205x 10--H?
(R/17.36)--H2
500			4.93 x 10--H3			0			1.62
1000			2.47x10--H3			0			1.62
3000			8.22 x to--H?			0			1.62
5000			4.93x10--H4			0			1.62
(R/2.467)--H'			0			1.62
Table 7.1. MiMmal destabilizing perturbations for Poiseiiille flow at various
Reynolds numbers. The last line represents convenient summary formulas; for
R> 5772 in actuality a different eigenvalue from the one reflected bere crosses
into the upper half-plane.
distance			minimal
of spectrum destabilizing
from unstable pertii rbation
half-plane			E
I?
250			9.87 x to--H?			1.051 x to--H?			0
500			4.93 x to--H?			2.633 x to--H?			0
1000			2.47 x to--H?			6.587 x to--H?			0
2000			1.23x10--H3			1.647x10--H5			0
oo
(J?/2.467)--H'			(ft/8.117)--H2			0
Table 7.2. Same as Table 7.1 but for Couette flow.
1.18
1.18
1.18
1.18
1.18
nonlinear terms can be thought of, locally, as introduci?g approximately such a
perturbation, with the scale determined by the nature of the nonlinearities and
the amplitude of the data (which in turn depends on those imperfections in the
laboratory apparatus). ?re do not know if these ideas can be made precise enough
to lead to a prediction of threshold amplitudes for transition to turbulence.
A comparison of the second columns of Tables 7.1 reveals that the distance
of the spectrum from the upper half-plane is the same for Poisenille and Conette
flows (until I? gets close to 5772 in the former case). The dominant eigenvalues that
account for this in the Poisenille case correspond to odd modes. If one restricts
attention to even modes, the modes which dominate the pseudospedra in the
upper half-plane, the distance of the spectrum changes to (!?/9.270)--H'
21
8. 3x3 and 4x4 approximations with 90% of the Butler-Farrell growth
The transient energy growth that is the subject of this paper and of [BF] and
[Ri'] is not a phenomenon of individual eigenmodes. However, we have found that
by projecting the linear Navier-Stokes operator  onto a 3-dimensional subspace in
the Conette case, or a 4-dimensional subspace in the Poisenille case, we can retain
on the order of 90% of the energy growth of the full operator. In other words,
much of the physics of transient energy growth can be captured by a 3 x 3 or 4 x 4
matrix, though not a 1 x 1 matrix.
The mechanics of such a projection are a matter of straightforward numencal
linear algebra [5,14]. Having reduced  to an N x AT matrix L by discretization,
suppose we wish to project L onto the column space of an N x k matnx Q with
orthonormal columns. The appropriate projected version of L is Q*LQ. Snppose
in particular that we started with an i? x k matrix V whose columns are selected
eigenvectors of L, satisfying LV = VD for some k x k diagonal eigenvalue matrix D.
If V = QR is a QR (Gram-Schmidt) decomposition* of V, with Q of dimensions
N x k and 1? of dimensions k x k, then we have Q*V = 1? and Q = VI?--H1 and
therefore
Q*LQ = Q*LVI?-1 = Q*VDR-' = I? D?IQ-1
(8.1)
Thus I?D!?--H1 (upper-triangular) is the matrix representation of the projection of
L onto the space spanned by selected eigenvectors, and in Matlab its computation
is a matter of a few lines.
The operator  and hence the matnx L are block 2 x 2 and block triangular;
the diagonal blocks are the Orr-Sommerfeld and Squire operators described in
[BF] and [R?H]. It is convenient to label the various eigenvalues of  as OS or
Squire eigenvalues, and number them by decreasing imaginary part. We have
experimented with projections onto the spaces spanned by various subsets of the
eigenmodes, and have reached the following conclusions. For Conette ?ow, most
of the transient energy growth can be captured by projection onto a 3-dimensional
subspace:
Couette: OS modes 1,3 and Squire mode 1.
For Poiseuille flow, most of the transient energy growth can be captured by pro-
jection onto any of three different 4-dimensional subspaces:
Poisenille:
OS mode 1 and Squire modes 1,2,3, or
OS modes 1,2 and Squire modes 1,2, or
OS modes 1,2,3 and Squire mode 1.
Figure 8.1 illustrates the Conette case for the fixed approximately optimal
parameters ? = 0, P = 1.6. The solid curves are the norms Iletnil as fundions of 1.
* Caution we are using R for both the Reynolds number and a triangular matrix.
22
80
IIetII
40
0			400
800 ?
Figure 8.1. Transient amplitude growth for Cou4te flow (solid) and its
3 x 3 projection onto OS modes 1,3 and Squire mode 4 (dashed).
We plot amplitudes; for energies the uumb&s should be squared. The dashed
curves are the corresponding norms after projection onto the space spanned by OS
modes 1,3 and Squire mode 1.
Table 8.1 presents these results numericatly, listing the fractions of the total
amplitude and energies (squared amplitudes) obtained by these projections. For
It ? 1000 the 3 x 3 approximation evidently captures more than 86% of the energy
growth.
R
amplitude
amplitude			energy
percentage			energy			perce?tage
250			8.416			99.24			70.82			98.49
500			16.58			97.88			274.9			95.80
1000			31.69			92.88			1004.			86.77
2000			53.76			78.69			2890.			61.92
Table 8.1. Percentage of the optimal amplitude and energy maxima
achieved by the dashed curves of Figure 8.1 (Conette flow, a =0, P = 1.6).
23
80
IIetcII
40\\
0			400
800
Figure 8.2. Transient amplitude growth for Poiscuille flow (solid) and
its 4 x 4 projection onto 2 OS modes and 2 Squire modes (dashed).
Figure 8.2 and Table 8.2 are the the analogues of ?igure 8.1 a?id Table 8.1 for
a 4 x 4 approximation to the Poiseuille problem based on 2 OS modes and 2 Squire
modes, with c? = 0 and ? = 2.04. As in the Conette case, the low-dimensional
approximation is obviously a good one, capturing more than 76% of the total
energy growth for the Reynolds numbers shown.
1?
1000
2000
3000
4000
5000
amplitude
12.75
24.85
37.02
49.11
61.13
amplitude			energy
percentage			energy			percentage
91.17			162.7			83.13
88.87			617.4			78.98
88.20			1370.			77.79
87.72			2411.			76.94
87.35			3737.			76.30
Table 8.2. Percentage of the optimal amplitude and energy inaxima
achieved by the dashed curves of Figure 8.2 (Poisenille flow, a = 0, P =
2.04).
It is interesting to examine, for this Poisenille problem, what combinations
of modes suffice to capture a substantial fraction of the energy growth of the full
24
operator  (still with a? = 0, P = 2.04). Table 8.3 lists results for various numbers
of OS and Squire modes, for I? = 5000:
no. of
OS
modes
2
3
4
t
2
3
4
5
--H0.00503			0.08533			--H0.60812
0			--H0.00948			--H0.10894
0			0			--H0.02477
A			ix			(8.3)
no. of
Squire
modes
2
3
4
amplitude			energy
percentage			percentage
8.09			0.69
87.35			76.30
97.61			95.27
99.40			98.81
2			50.86			25.87
3			88.80			78.85
4			94.40			89.11
5			95.07			90.37
1			43.55			18.97
1			85.10			72.43
1			94.93			90.11
1			96.35			92.84
2			3			89.16			79.49
3			2			88.30			77.96
Table 8.3. Dependence of transient growth on the choice of cigen-
modes included in the low-dimensional approximation (Poisenille flow,
!?=500Q &=0, ?=2.04).
The niiinbers in this table are closely related to the recent results of Gustavsson
[6]. For a slightly different choice of ? and /3, Gustavsson showed that a huge
amount of transient energy growth can be achieved by a model consisting of one
OS mode and many Squire modes. The independent results of [BF] then showed
that Ciustavsson's model had captured about 90% of the optimal energy growth,
a conclusion that is reproduced in the figures above. The figures 5110w that the
combination of one OS mode with many Squire modes is not the only one that
works; one can do just as well, for example, by taking I Squire mode and many
OS modes.
Here are some 3 x 3 matrices for Conette flow. For ? = 500:
--H0.01005			0.16652			--H0.62491
A = i x			0			--H0.01896			--H0.11473
0			0			--H0.04953
(8.2)
For 1? 1000:
25
The scaling of these matrices as a function of 1? appears to be
0(1)
A			J?-1			0(1)
0(1)			0(R)
0(1)
which is not far from the 2 x 2 model of Section 2.
(8.4)
26
9. The 3x3 model with nonlinearity
In Section 2 we presented a 2 x 2 model problem iii which a locally energy-
conserving nonlinear term had the effect of bootstrapping a linear growth mecha-
nism into a nonlinear instability. In the last section we showed that although this
2 x 2 matrix does not seem to arise in the equations of linearized hydrodynamics,
analogous matrices do seem to arise, to good approximation, of dimensions 3 x 3
and 4 x 4. To conclude this paper we will now close the loop by illustrating that
the bootstrapping effect applies also to these more realistically obtained matri-
ces. In particular we consider the 3 x 3 matrix corresponding to Conette ?ow with
1? 1000.
In analogy to (2.2), consider
011
B= -1			0			1
1			--H1			0
tit = AtitIItitIvBti,			(9.1)
with A defined by (8.3). The parameter v is 1 for a quadratic w?nlineanty, 2 for
a cubic nonlinearity, and so on. Figures 9.1 and 9.2 are analogues of Figures 2.2
with i' = 1 and v = 2, respectively, based on initial vectors of the form ti(0) =
(0,0, 11ti(O)11)T. Clearly the behavior is much the same as iii Figure 2.2. Below
a threshold initial amplitude, the evolution is approximately linear. Above that
amplitude, there is positive feedback and eventual growth to a large-amplitude
critical point.
The observed threshold amplitudes are
v = 1:			2.3 x 10?6,
v=2:			(?2.5x10?4.
Both of these numbers are far less than the inverse of 31.69, the linear growth
factor listed in Table 8.1. For threshold energies instead of amplitudes, we square
these numbers and find ourselves at the level of 10--H8 or below.
The next step in this analysis should be to replace the arbitrary nonlinear
term in (9.1) by some approximation to the actual nonlinear terms present in the
Navier-Stokes equations [7]. These terms transfer energy not only between modes
for fixed ? and P, but also across modes (see e.g. [7]). One could then attempt
to estimate threshold perturbation amplitudes for transition to turbulence in a
fashion that would be quantitative rather than merely siiggestive.
27
loo
IIu(t)II
lo?
10-			t
0			200			400
Figure 9.1. IIt(t)II vs t for solutions to the model problem (9.1) with
quadratic iionlii?earity (v 1). Compare Figuw. 2.2. The threshold am-
plitude is ? 2.3 x 10--H6
loo
M
10-			t
0			200			400
Figure 9.2. Same as Figure 9.1 but for a cubic ?onhnearity (v 2).
The threshold amplitude is ( 2.5 x io--H4.
28
IIu(t)II
Acknowledgments
This paper would not have been written without the continuing advice over
several years of Dan llenningson. We have also benefited from "non-modal" discus-
sions with Peter Schmid, Brian Farrell, and Kathryn Butler, and from discuss?ons
on various topics with Walter Mascarenhas, Philip Holmes, Ridgway Scott, and
Andr6. Weideman. Finally, our computations have been made possible by a re-
markable software system, Matlab.
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