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The response of convectively unstable equations to a localized oscillatory forcing (i.e. the ‘signalling problem’) is studied. The full mathematical structure of this class of problems is elucidated by examining partial differential equations of second-order (the linear Ginzburg–Landau equation) and fourth-order (the linear Kuramoto–Sivashinsky equation) in space. The long-time asymptotic behaviours of the Fourier–Laplace integral solutions are obtained via contour integration and the method of steepest descent. In the process, a general algorithm is developed to extract the important physical characteristics of such problems. The algorithm allows one to determine the velocities that bound the transient and spatially growing portions of the response, as well as a closed-form transfer function that relates the oscillatory disturbance amplitude to that of the spatially growing solution. A new velocity is identified that provides the most meaningful demarcation of the two regions. The algorithm also provides a straightforward criterion for identifying ‘contributing’ saddles that determine the long-time asymptotic behaviour and ‘non-contributing’ saddles that give errant solutions. Lastly, a discontinuity that arises in the long-time asymptotic solution, identified in prior studies, is resolved.
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Algorithm for spatio-temporal analysis of the
signaling problem
Nathaniel S. Barlow1, Steven J. Weinstein2, and Brian T. Helenbrook3.
1School of Mathematical Sciences
Rochester Institute of Technology, Rochester, NY 14623, USA
2Department of Chemical Engineering
Rochester Institute of Technology, Rochester, NY 14623, USA
3Department of Mechanical and Aeronautical Engineering
Clarkson University, Potsdam, NY, 13699, USA
This article has been accepted for publication in the IMA Journal of
Applied Mathematics Published by Oxford University Press. The final pub-
lished version can be found here: http://imamat.oxfordjournals.org/
cgi/content/abstract/hxv040?ijkey=eszq3GhUiiUKtTA&keytype=ref
Abstract
The response of convectively unstable equations to a localized oscil-
latory forcing (i.e. the “signaling problem”) is studied. The full math-
ematical structure of this class of problems is elucidated by examining
partial differential equations of second order (the linear Ginzburg-
Landau equation) and fourth order (the linear Kuramoto-Sivashinsky
equation) in space. The long-time asymptotic behaviors of the Fourier-
Laplace integral solutions are obtained via contour integration and
the method of steepest descent. In the process, a general algorithm
is developed to extract the important physical characteristics of such
problems. The algorithm allows one to determine the velocities that
bound the transient and spatially growing portions of the response, as
well as a closed-form transfer function that relates the oscillatory dis-
turbance amplitude to that of the spatially growing solution. A new
velocity is identified that provides the most meaningful demarcation
1
of the two regions. The algorithm also provides a straightforward cri-
terion for identifying “contributing” saddles that determine the long-
time asymptotic behavior and “non-contributing” saddles that give
errant solutions. Lastly, a discontinuity that arises in the long-time
asymptotic solution, identified in prior studies, is resolved. signaling
problem; convective instability; linear Ginzburg-Landau equation; lin-
ear Kuramoto-Sivashinsky equation; long-time asymptotics; method
of steepest descent; residue theorem
1 Introduction
Fluid flows are often subject to localized ambient disturbances such as ma-
chine vibrations, flow pulsations, and air disturbances. Perhaps the simplest
problem that models the response to these disturbances is the so-called “sig-
naling” problem. For small 1-D disturbances propagating in the x-direction
of the domain x(−∞,) and time t, a mathematical model of the re-
sponse to forcing can be constructed with the linearized governing operator,
L, of the response, h(x, t), as:
Lh =(x)eft(1a)
with constraints:
h(x, 0) = 0 for all x, h 0 as x→ ±∞ for all t, (1b)
where Ais the forcing amplitude, δ(x) is the Dirac delta function, and ωfis
the real-valued forcing frequency. The smoothness of himplies that the above
boundary conditions in xalso apply to all spatial derivatives of has x→ ±∞.
Note that additional initial conditions are necessary to achieve a well-posed
problem if the time derivatives in operator Lare 2nd order or higher. Fig-
ure 1 provides a schematic of the solution structure that arises in signaling
problems. If Lis convectively unstable, time-growing wave disturbances
propagate out of a domain of finite length leaving behind a spatially growing
or decaying wave train that oscillates in time. Alternatively, if a flow is abso-
lutely unstable, initiated disturbances grow and spread in both the upstream
and downstream direction and ultimately infect the entire fluid domain; in
such cases, a process described by (1) will not be sustainable even if initiat-
ing disturbances are infinitesimal. The various velocities shown in the figure
identify solution features of either mathematical or physical significance that
2
will be discussed in the paper. V,Vmax, and V+are established velocities
from general spatio-temporal stability analysis [Huerre & Rossi, 1998]. The
other velocites are specific to the signaling problem. These will all be dis-
cussed in the mathematical analysis, including the new velocity VEwhich we
introduce.
(a)
Oscill ating sou rce
Sig naling
regi on
x
VE
V+
Transi ent re gion
VI
Vmax
V
(b)
Oscill ating
source
x
Sig naling
regi on
V+
VI
Vmax
Transi ent region
V
VE
Figure 1: A schematic of signaling behavior at a fixed time for (a) spatially
unstable and (b) spatially stable forcing. Key regions of the response are
delimited by velocities V(with various subscripts) that move downstream
with the wave packet.
The signaling problem is of importance in many manufacturing processes.
For example, in coating processes the goal is typically to create thin films,
often on the order of microns thick, on a substrate that is moving at high
speeds. In a key portion of coating processes, wide and thin liquid layers flow
simultaneously down the inclined planar surfaces of a coating die (referred
to as a “slide”) and are subsequently deposited onto a moving substrate to
form a multiple layer liquid structure; this structure is subsequently solidified
to create a multiple-layer product [Weinstein & Ruschak, 2004]. In the slide
3
flow, it is often impossible to avoid convectively unstable conditions, i.e. the
usual approach of operating below a neutral stability threshold is not feasible.
If the disturbance growth rate is large, small disturbances can be magnified
to the degree that they lead to unsalable products. Thus, the management of
convectively unstable fluid flows is a practical matter, and because thickness
variations in micron-sized layers on the order of 1% may exceed product
specifications, linearized analysis is appropriate.
To manage such flows, one must understand their response to distur-
bances. It is commonplace for two classes of disturbances to arise in such
processes. The first can be modeled as a transient perturbation due to some
non-repetitive disturbance, such as when a coating die is repositioned sud-
denly over a moving substrate during film formation and coating. Since this
repositioning is time-limited, the problem can be formulated as a response to
an initial perturbation. An alternative type of disturbance arises, for exam-
ple, when a fan turns on near a coating die and the die begins to vibrate at a
set frequency. The signaling problem provides a framework for understanding
the response to such oscillatory disturbances.
Although numerical solution of the linearized system (1) can in gen-
eral be obtained, such solutions often hide the relatively simple underly-
ing structure described in Figure 1. Of particular interest is the identifi-
cation of the bounding velocities, and ultimately, the relationship between
the amplitude of a vibration, A, and the amplitude of the spatially grow-
ing wave-trains in the signaling region shown in Figure 1. As the tran-
sient portion of the solution ultimately exits a domain of finite length, it is
this continually forced spatially growing wave-train that remains. If the
amplitude of this wave-train at the end of the physical domain is small
enough, it may fall within the operating tolerance of a process - for ex-
ample, an acceptable level of film distortion may be achieved after coat-
ing [Weinstein et al., 1993, Weinstein & Ruschak, 2004]. Thus, the transfer
function between the disturbance amplitude and the spatially growing wave-
train amplitude is of significant interest. In many cases, the features of the
solution just described may be all that are practically necessary, and can be
used in lieu of a full solution of the governing system (1).
There are several prior works that have analyzed the signaling problem
(see [Gordillo & erez-Saborid, 2002] and references therein). The main goal
of this paper is to provide a simple algorithm to extract the key features of
the signaling problem. Note that certain preliminary features of the algo-
rithm have been presented in [Barlow et al., 2012], and here we improve and
4
clarify it: Saddle points that would lead to errant asymptotic results are
identified (and dealt with), and a new velocity, VE, is introduced which is
easily determined and has more significance to the signaling problem than the
velocities typically identified. It is also shown how the various integral terms
of a stability analysis combine to give a physically meaningful prediction of
the signaling response.
The paper is organized as follows. In Section 2, we solve a relatively sim-
ple signaling problem – the linear real-coefficient Ginzburg-Landau equation
with a forcing function of the type given in (1a): the preliminary asymptotic
analysis and instability classification is done in 2.1 and the signaling analysis
begins in 2.2. All features of this solution are found analytically, enabling the
propagation features of the solution and the intricacies incurred from forcing
to be examined completely. This examination culminates in an algorithm
in Section 3, used to extract critical velocities and a transfer function for
signaling problems in general. To illustrate the application of the method
towards a problem with additional structure, in Section 4 we apply the algo-
rithm to an oscillatory-forced Kuramoto-Sivashinsky operator, and compare
the results with a full solution of the system. In Appendix A, we provide a full
asymptotic expansion and continued fraction solution of the problem given
in Section 2, and use these to examine the discontinuity at the velocity VI,
which is the limiting velocity introduced by [Gordillo & erez-Saborid, 2002].
2 Signaling in 1-D: a model problem
To elucidate key features of the signaling problem, we begin with a model
of instability in a convection-diffusion process; we examine the forced linear
real-coefficient Ginzburg-Landau equation in h(x, t),
∂h
∂t +ah
∂x bh c2h
∂x2=(x)eft, t 0
h(x, 0) = h0δ(x) for all x, h 0 as x→ ±∞ for all t, (2)
where his the system response (this can be viewed, for example, as the
amplitude of a small disturbance along the interface of a film); a,b, and care
real-valued parameters; Ais the forcing amplitude; and h0is the amplitude
of a delta function perturbation of the initial condition. In contrast to the
system (1) provided in the introduction, we impose a nonhomogenous initial
condition; this will allow us to examine the similarities between the signaling
5
response and the perturbation response. Note, that the homogenous initial
condition in (1) suffices to fully characterize the structure of the signaling
problem including transients, as will become apparent at the end of this
section. The stability analysis of the unforced version of (2) can be found
in [Hunt & Crighton, 1991] and [Huerre & Rossi, 1998]. The simplicity of (2)
allows us to not only highlight techniques for extracting features relevant to
signaling, but also to justify these techniques through their connection to an
integral solution that can be evaluated analytically.
Following the procedure given in [Barlow et al., 2010], the integral so-
lution of (2) is obtained by first taking the successive Fourier and Laplace
transforms such that the partial differential equation in h(x, t) becomes an
algebraic equation in H(k, s). It is implicit in this notation that xtransforms
to k,ttransforms to s, and htransforms in aggregate to H. After solving
for the doubly transformed variable H, one may then take the subsequent
inverse Fourier and Laplace transforms to recover h(x, t). Using the stan-
dard convention in wave problems of rotating the Laplace contour such that
s=iω, the integral solution of (2) becomes
h(x, t) = 1
4π2Z
−∞ Z+0
−∞+0
1
D(k, ω)iA
(ωωf)+h0ei(kxωt)dk, (3)
where
D(k, ω) = iω+ ika b+ck2(4a)
and D(k, ω)=0 is the dispersion relation, written as
ω=ka + ibick2(4b)
which describes the relationship between the complex wavenumber k=
kr+ikiand complex frequency ω=ωr+iωiin the flow governed by (2). In
the above and all of the following, the subscripts rand iindicate the real and
imaginary components, respectively. Note that D(k, ω)=0 is easily obtained
by substituting h= ei(kxωt)into the homogeneous form of the governing
equation (here (2)) with no constraints applied. Essentially, all key features
of the solution can be obtained by examining the properties of (4b), as shall
be demonstrated in what follows. The above form of the dispersion rela-
tion allows ωto be determined easily if one is given k. It is convenient to
rewrite (4a) as
D(k, ω) = c[kk1(ω)] [kk2(ω)] (4c)
6
where k1and k2are the k-roots of (4b). The parameter τ0in (3) is chosen
such that the horizontal integration path in the ωr-ωiplane passes above
all singularities in the integrand; this restriction is embedded in the inverse
Laplace transform [Morse & Feshbach, 1953], and assures that causality (no
disturbance can arise prior to the time t=0) is satisfied. The path of the
Fourier inversion is taken to be along the kr-axis in (3), although any path
in the domain of convergence of the transform may be chosen. The kr-axis
is guaranteed to be a path of convergence based on its relationship to the
discrete Fourier series that arises from a self-adjoint eigenvalue problem on
a finite domain, where the eigenvalues are real [Weinberger, 1965].
The double integral in (3) is evaluated through one integration in ω
(Laplace integral) and one in k(Fourier integral), which may be done in
any order. As is usually the case, one order is arguably easier than another,
but the answers will be equivalent. The first integration (in ωor k) is car-
ried out by applying the residue theorem, which leads to the substitution of
either k(ω) roots or ω(k) roots of D(k, ω)=0 in to the argument of the expo-
nential in (3). Although both approaches have been taken in the literature
to analyze the response to an initial disturbance [Ashpis & Reshotko, 1990,
Huerre, 2000, Schmid & Henningson, 2001], performing the Laplace integral
first is generally easier [Barlow et al., 2010, Barlow et al., 2011] and this sim-
plicity is even more apparent in the signaling problem [Gordillo & erez-Saborid, 2002,
Barlow et al., 2012].
Evaluating the inner (Laplace) integral of (3) by the method of residues
leaves us with the Fourier integral solution,
h(x, t) = A
2πZ
−∞ "ei[kxω(k)t]
[ω(k)ωf]∂D
∂ω ω(k)
+ei(kxωft)
D(k, ωf)#dkh0
2πiZ
−∞
ei[kxω(k)t]
∂D
∂ω ω(k)
dk
(5)
where ω(k) is given by (4b). The first integral is due to the forcing and
the second is due to the initial condition, as evidenced by their respective
coefficients. The denominators of both forcing terms in (5) can be placed in
a simplified form that explicitly shows the location of their zeros. Using (4),
7
the denominators of (5) may be simplified such that
h(x, t) = A
2πc Z
−∞
ei[kxω(k)t]
(kkf1)(kkf2)dk
| {z }
hA(x,t)
+Aeiωft
2πc Z
−∞
eikx
(kkf1)(kkf2)dk
| {z }
hB(x,t)
+h0
2πZ
−∞
ei[kxω(k)t]dk
| {z }
hC(x,t)
.(6)
where
kf1,2k1,2(ωf) = ai±a2+ 4 b c + 4 c ωfi
2c.(7)
The result (6) is the exact solution of the system (2), and as indicated,
three distinct portions of the solution may be identified. The signaling prob-
lem incurs the first 2 integrals (hAand hB) and involves poles that incorporate
the forcing frequency. The last integral, hC, incorporates the effect of the ini-
tial condition and is independent of the oscillatory forcing. In what follows,
we examine the structure of these integrals more closely and the response
contained therein. In general, the analysis of the integral solution begins by
first determining if the system is stable, convectively unstable, or absolutely
unstable for a prescribed set up parameter values (in the current problem
these are a,b,c). Then, one may discontinue any effort to solve for the
response to oscillatory forcing for situations that lead to absolute instability
from the outset, since forcing becomes irrelevant in such cases (the transient
quickly overtakes any effect due to forcing). If the system is stable, the tran-
sient will eventually become sub-dominant to forcing at a given xlocation.
In such cases, it is still of general interest to examine the system response, as
the relationship between initiating amplitude and downstream disturbances
may be practically relevant; even damped waves, if large enough, can be
problematic. As will be seen, the approach taken in this paper is general
enough to handle all of these cases, although we focus most attention on
convectively unstable system responses. If the choice of parameter values
leads to convective instability, the next step is to examine the long-time be-
havior of the convectively unstable transient. The final step is to examine
the effect of forcing on the wave-train that trails the convectively unstable
transient. In what follows, Section 2.1 and Section 2.2 describe how each
of the above steps are handled as part of the overall integral solution (6).
8
Section 2.3 generalizes key features of the disturbance response for signaling
problems of the form (1).
2.1 Features of transient solution and stability classi-
fication
This section provides a spatio-temporal stability analysis of the problem,
which is done prior to analyzing the effect of signaling. Note that there are
several approaches given in the literature to perform spatio-temporal stabil-
ity analysis [Sturrock, 1958, Gaster, 1968, Briggs, 1964, Kupfer et al., 1987,
Lingwood, 1997, Fokas & Papageorgiou, 2005, Barlow et al., 2012]. For pur-
poses of clarity, we first examine the asymptotic behavior of hC(6), which
has an exponential argument identical to that of hA, yet the integrand is
devoid of any poles. All the steps needed to evaluate the long-time asympt-
potic behavior of hCare also needed to determine the asymptotic behavior
of hA; these steps are tied to the method of steepest descent that uses saddle
points of the identical exponential argument in these integrals. Of course,
the complication of poles needs to be incorporated to evaluate hA, but the
presentation is simplified by first considering the evaluation of hC, which in-
cludes some established techniques (contained in the above-cited literature)
that are detailed to facilitate the analysis of hAthat follow.
Noting that kis real in the integrals of (6), we observe that growth of
the integrands, and hence the solution, depends strictly on the generally
complex function ω(k) given in (4b). If ω(k) has a positive imaginary part
for a given real wavenumber then the solution will grow exponentially in
time. This justifies the use of classical temporal analysis to determine the
stability of the problem. The maximum of the amplification curve (i.e ωi
vs. kr) bounds all values of ωifor all real kand thus serves as a single
number to characterize stability; if this maximum has a value of ωi>0, the
flow is unstable. For later reference, the wavenumber corresponding to this
maximum is denoted as kmax; for completeness, the value of kmax is obtained
from ∂ωi
∂kr=0, ωi>0, and 2ωi
∂k2
r<0. A temporal stability plot corresponding
to the dispersion relation (4) with k=kris shown in Figure 2. Applying the
temporal instability criteria (i.e. ωi(kmax)>0) to (4b), it can be shown that
a flow governed by (2) is unstable if b > 0.
Assuming parameters are such that the problem is unstable, the next step
is to determine whether the problem is absolutely or convectively unstable.
9
−0.6 −0.4 −0.2 0 0.2 0.4
−0.25
−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
kr
ωi
kmax= 0
Figure 2: Temporal stability plot for the dispersion relation (4) with a=3,
b=0.1, and c=1.
Focusing on the evaluation of the integral hCin (6), it is convenient to rewrite
the exponential in the integrand as eφ(k)twhere
ϕ(k) = i[kV ω(k)],and V=x/t. (8)
With ω(k) given by (4b), the integral hCmay be evaluated analytically.
However, this is not typically the case and so here we discuss a general ap-
proach - one which is also appropriate for the later evaluation of hA. For
either integral, the sign of ϕ(k) cannot readily be determined, and so one
cannot close the contour in any judicious way to enable a simple evaluation
via Cauchy’s theorem. Instead, we look for an integration path that enables
asymptotic analysis as t→ ∞ via the method of steepest descent. In the
steepest descent method, a path is chosen that passes through saddle points
of the argument of the exponential in such a way that the imaginary part
of (8) is constant and the real part passes through a maximum at the sad-
dle(s). An asymptotic expansion about the saddle point may then be used to
deduce the dominant exponential behavior as tbecomes large using Laplace’s
method [Bender & Orszag, 1978]. The location of the maximum of Re[ϕ(k)]
(Re[] denotes the real part of the argument) is known as a saddle point and
satisfies the condition
dk ks
=V(9a)
10
or equivalently, the conditions
∂ωr
∂krks
=V(9b)
and ∂ωi
∂krks
= 0,(9c)
where ksis the complex wavenumber associated with the saddle. The com-
plex k-plane contains an infinite number of saddle points, each corresponding
to some real-valued velocity Vx/t. As shall be demonstrated, it is im-
portant to recognize that the location k=kmax + 0i found from temporal
analysis (see Figure 2 and surrounding discussion) is a saddle point because
(∂ωi/∂kr)|kmax = 0 and we can associate the velocity Vmax = (∂ωr/∂kr)|kmax
with this point.
Through application of the saddle point definitions (9) to the dispersion
relation (4b), the locus of all saddle points of the integrals hCand hAin (6)
is given by
ks=i
2c[Va].(10)
Note that it is not always possible to find an explicit expression for the saddle
points. In such cases, it is helpful to first determine the saddle point kmax
from classical temporal analysis (i.e. Figure 2), and then identify the contour
(∂ωi/∂kr)|ks= 0 (in the kr-kiplane) that passes through kmax. Figure 3 shows
the locus of saddles. The specific saddles labeled in the figure are defined in
a later part of this section.
The steepest paths through a saddle point in the kr-kiplane are (in
general) given by the contours
Im [ϕ(k)] = Im [ϕ(ks)] (11)
where Im denotes the imaginary part of the bracketed argument. Substitut-
ing (10) and (4b) into (11) leads to a family of curves describing the steepest
paths, given by
kr(V2ckia) = 0.(12)
By construction, curves described by (12) will pass through the saddle points (10),
each associated with a given value of V. The path of steepest descent passes
perpendicular to isocontours of Re[ϕ] that decrease away from the saddle;
11
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
kr
ki
locus of sadd les
k0
(ks)
(ks)E
(ks)+
kmax
Figure 3: The locus of kmax and nearby saddles for the integrals hAand hC
in (6) with a=3, b=0.1, and c=1. (ks)Eis determined using ωf=0.65.
this decreases the magnitude of the exponential argument in hCfor a given
time. For large times, the main contribution then comes from the saddle
point, which is a maximum along this path, and allows for the asymptotic
behavior to be deduced.
To determine whether a problem is absolutely or convectively unstable, it
is necessary to examine the velocity Vx/t = 0, which is synominous with
a zero group velocity
dk = 0 (i.e. ∂ωr
∂kr=ωi
∂kr=0). The saddle point(s), (k0, ω0)
(for complex kand ω) of D(k, ω)=0 that are paired with a velocity V= 0
determines the nature of instability for the medium; in the characterization, it
is easiest to view V=0 as the initial disturbance location x=0. The “absolute
growth rate” Im[ω0] is the exponential growth rate in time of waves traveling
at a zero velocity. If the flow is unstable (again, which can be assessed by
classical temporal stability theory), yet Im[ω0] is negative, a disturbance at
x=0 will decay and this indicates that the disturbance convects as it grows,
i.e. the instability is convective. If the flow is unstable and Im[ω0] is positive,
a disturbance at x=0 will continue to grow with time, and the instability is
characterized as absolute. According to these arguments, using equation (10)
with V= 0 and equation (4b) the system (2) exhibits absolute instabilities
if b>a2/(4c)>0 and convectively instabilities if 0 < b < a2/(4c). These
conclusions are guaranteed to be correct only if the integral hCin (6) can be
12
−2 −1 0 1 2
−3.5
−3
−2.5
−2
−1.5
−1
−0.5
0
0.5
−8
−6
−6
−4
−4
−4
−4
−2
−2
−2
−2
−2
−2
0
0
0
0
2
2
4
4
kr
ki
SA
k0
SD
CR
CL
Figure 4: Steepest descent (SD) and ascent (SA) paths through the saddle
k0() for the integral solution of (2) with a=3, b=0.1, and c=1. Parameters
are chosen such that the flow is convectively unstable. The SD path is closed
back to the kr-axis through vertical links CRand CL. Isocontours are shown
for values of Re[ϕ].
evaluated using the steepest descent path through the identified saddle point
(k0, ω0); this is not always the case, so care needs to be taken in making this
assessment as follows.
For our problem, with steepest paths given by (12), one may verify that
Re[ϕ] reaches a maximum at ksalong the path V2ckia= 0 and a
minimum at ksalong the path kr=0. Hence, the possible steepest descent
paths for (2) are given by
ki=Va
2c,(13)
which describes a family of horizontal lines (for each Vx/t) in the kr-ki
plane that move through the corresponding saddle point in (10).
For a saddle point to contribute to the integral solution, one must be able
to connect its steepest descent path to original path of integration (the kr-
axis) while preserving Re[ϕ(ks)] as a maximum along the total path, which is
a closed contour. An illustration of the steepest descent and steepest ascent
paths is provided in Figure 4 for k0(the saddle associated with V=0), where
a closed integration path is used to relate the original Fourier integration
path (the kr-axis) with the path of steepest descent via Cauchy’s theorem.
The vertical path (that of ascent) is not useful for the asymptotic analy-
sis, since the magnitude of the exponential argument increases away from
13
the saddle and leads to a non-converging integral [Morse & Feshbach, 1953].
Along the closed integration path, Re[ϕ(ks)] is a maximum, enabling asymp-
totic analysis (discussed below). In Section 4 we provide an example where
this is not the case, leading to non-contributing saddles. Although the above
criteria may be applied to any saddle, for k0it is equivalent to Briggs’ cri-
teria [Briggs, 1964], used to classify unstable systems as absolutely or con-
vectively unstable. Additional structure of the transient solution may be
identified as follows.
Following notation similar to that used in [Monkewitz, 1990, Huerre, 2000,
Barlow et al., 2012], we label the exponent at the saddle for each velocity V
as
σ=ϕ(ks) = i[ksVω(ks)] .(14)
In accordance with (10), σis strictly real in the current problem and is given
by
σ=b1
4c[Va]2σr,(15)
where σrdescribes the exponential growth in time for a given velocity, V;
this is plotted in Figure 5 for a convectively unstable flow. Note that the
velocity V=0 has a negative growth rate associated with it, as expected
for a convective instability. If the flow were absolutely unstable, growth
for the velocity V=0 would instead be positive. For contributing saddles,
this is another place in the analysis where one could choose to classify the
instability; given that σr=Im[ω0] for V=0, this classification method is, in
fact, identical to that described earlier.
The long-time asymptotic structure presented above can be characterized
by “limiting velocities”. The limiting velocities that bound the front and
back of the time-growing wave packet are defined by the velocity-saddle pairs
where σr= 0. These are V+(and saddle (ks)+), which bounds the ‘front’
of the transient (the leading edge) and V(and saddle (ks)), which bounds
the trailing edge of the transient; these velocities are labeled in the response
shown in Figure 1 and in the growth plot of Figure 5. In the current problem,
one may obtain these velocities explicitly using (15); they are V±=a±2bc,
derived previously in [Hunt & Crighton, 1991]. In the signaling problem,
there is an interplay between spatial and temporal growth that gives rise to
other important limiting velocities (shown in Figures 1 and 5b); these are
discussed in Section 2.2.
In Figure 5 note that the downward parabola indicates that growth has a
14
(a)
−4 −2 0 2 4 6 8
−14
−12
−10
−8
−6
−4
−2
0
2
σr
V
V+
V
V= 0
(b)
2.2 2.4 2.6 2.8 3 3.2 3.4 3.6
−0.04
−0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
V
σr
Vmax
VE
VV+
˜σr
σr
Figure 5: Temporal growth rates σr=Re[ϕ(ks)] for velocities Vx/t in the
integral solution of (2) with a=3, b=0.1, and c=1. Parameters are chosen
such that the flow is convectively unstable. ˜σris determined using ωf=0.65.
Subfigure (b) is an enlarged view of the boxed region shown in subfigure (a).
maximum at V=Vmax. In general, the saddle kmax will always contribute to
the long-time behavior since kmax lies on the kr-axis, and is thus oriented (in
the kr-kiplane) in such a way that its steepest descent path may always be
closed back to the kr-axis; a proof is given in Appendix A. By extension, the
locus of saddles that contains the saddle kmax will be composed of saddles
that also contribute to the growing wave-packet. Non-contributing saddles
typically lie along σrvs. Vcurves that have infinite growth rates at large V;
such cases are encountered in Section 4. To confirm that the saddles (ks)+,
(ks), and kmax contribute to the solution, one may draw the steepest descent
curves through the saddles, as shown in Figure 6, where it is apparent that a
closed path may be drawn back to the kr-axis with Re(ϕ) being maximum at
the saddle point. In each subfigure of Figure 6, the steepest descent path is
closed back to the original path of integration (the kr-axis) through vertical
lines, denoted as CRand CL. Although these links do not need to lie at ±∞
for their contribution to be subdominant, doing so leads them to contribute
nothing to the final solution.
15
(a)
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
kr
ki
−2
−2
−2
−2
0
0
0
0
0
0
0
2
2
2
22
4
4
(ks)
SA
SD
CR
CL
(b)
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
kr
ki
−2
−2
−2
−2
0
0
0
0
0
0
0
0
2
2
2
22
4
44
SD
SA
0.1 0.1
0.1
kmax
0.1
(c)
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
kr
ki
−2
−2
−2
−2
0
0
0
0
0
0
0
22
2
22
2
44
SA
(ks)+SD
CR
CL
Figure 6: Steepest descent (SD) and ascent (SA) paths through the saddles
(ks),kmax, and (ks)+for the velocities (a) V, (b) Vmax , and (c) V+, superim-
posed onto contours of Re[ϕ] for the integral solution of (2) with a=3, b=0.1,
and c=1. Note that in (b) the SD path is coincident with the kr-axis and so
no contour integration is needed for evaluation. Parameters are chosen such
that the flow is convectively unstable.
16
2.2 Influence of poles due to forcing & the full solution
Now that the flow has been classified as convectively unstable, the effect of
forcing as described by the integrals hAand hBin (6) is obtained. Upon ex-
amination of the exponential terms in (6), one may observe that downstream
(x > 0) spatial growth occurs through a negative imaginary kand spatial
decay (i.e. negative growth) occurs through a positive imaginary k. These
modes are accessed through the poles due to forcing, as described below.
The integral hB(x, t) in (6) can be evaluated via the residue theorem. The
real part of the exponential argument along the closure of the contour must
approach negative infinity as k→ ±∞ to ensure convergence in accordance
with Jordan’s Lemma. When x > 0, one must close the integration path with
a semicircle in the upper half of the kr-kiplane and vice-versa for x < 0. The
two poles of the integrand are at kf1and kf2whose locations are shown in
Figure 7. Depending on the value of ωf, the kf2pole can lie in the upper
half plane, the lower half plane, or along the kr-axis. The pole kf1, however,
remains in the lower half plane for all ωf>0. If x > 0, the contribution to
hBwould be 2πi Res(kf2) if kf2is in the upper half plane or zero otherwise.
If x < 0, the closure is in the lower half plane and the contribution is 2πi
Res(kf1) plus 2πi Res(kf2) if kf2is in the lower half plane. After evaluating
the residue for each of these 3 cases, the expression for hB(x, t) becomes
hB(x, t) = iAeiωft
c(kf2kf1)H(x)H(ωfωf,c)eikf2x+H(x)eikf1xH(ωf,c ωf)e ikf2x
(16)
where ωf,c =apb/c is the value at which kf2crosses the real axis and His the
Heavyside step function. We augment the usual definition of H(x) such that
H(0) = 0.5. This definition incorporates the case for ωf=ωf c, where the kf2
pole lies on the kr-axis and so its principal value contribution ±πi Res(kf2) is
included in (16). The integral hBis neutral in time and represents a spatially
decaying waveform oscillating with the forcing frequency ωf. It is important
to point out that the integral hBcannot be responsible for spatial growth as
the step functions that arise to assure convergence preclude such growth. For
example, here kf1has a negative imaginary part which would give growth
for x > 0 but it is multiplied by H(x) in (16). Since hBis nonzero for all
xand tand thus has no element of transience, physical reasoning suggests
that it must sometimes cancel with components of hA; this is in fact the case,
and is demonstrated in what follows. It is a common misconception that hB
is responsible for growth. This error and its propagation in the literature is
17
ωf, c
ωf= 0
ωf= 0
in cr eas in g ωfin cr easin g ωf
ωf, c kr
ki
kf2(ωf)
kf1(ωf)
Figure 7: Schematic of pole location in hB, given by (7) with the constraint
0<b<a2/(4c) such that the system (2) is convectively unstable. The
real positive forcing frequency increases from ωf=0 in the direction of the
arrows, eventually reaching ωf,c =apb/c where forcing transitions from
spatial growth to spatial decay.
discussed at length in [Barlow et al., 2012].
We now proceed to use the method of steepest descent to evaluate hA
in the limit of large time. Since the argument of the exponential in hAis
identical to that of hC, the integral hAcan be evaluated using the identical
steepest descent paths to those used for hC. The analysis thus proceeds
the same way as for hCexcept that now one must consider the effect of
the poles. Using a (closed) rectangular path of integration (the same as
those shown in Figures 4 and 6), the residue theorem is used to relate the
steepest descent contour to the kr-axis. Any poles lying within the closed
path will contribute a residue to the solution. Depending on the value of the
forcing frequency ωf, the placement of the corresponding poles changes the
evaluation of the contour integration (through the residue theorem). This is
illustrated in Figure 8 which shows a map of several ks-Vpairs (×) and their
corresponding paths of steepest descent along with the pole locations () for
18
ωf< ωf,c. The infinitely long rectangular path of integration, connected
through the vertical links denoted as CRand CL, may or may not enclose
a pole, depending on the value of V. The position of the steepest descent
contour moves upward with increasing Vand downward with decreasing V,
which leads to the following scenario for poles in the lower half plane: The
kf1pole may only be enclosed by a rectangular path back to the kr-axis
when using a steepest descent contour associated with an upstream velocity
V < Vf1<0. At kf1,ω(k) = ωfand the pole contributes a term which
is decaying as determined by the real part of the product iV kf1. Similarly
if kf2has a positive imaginary part (i.e. the pole lies above the kr-axis), it
will be enclosed when V > Vf2>0 but the growth will also be negative.
Spatial growth arises from a residue contribution when the kf2pole has a
negative imaginary part (i.e when ωf< ωf,c as shown in Figure 8) and is
enclosed using steepest descent paths associated with downstream velocities
Vf2> V > 0.
To understand the physical meaning of the above results, we begin by
examining the behavior near the source associated with the pole that causes
spatial growth (i.e. kf2for ωf< ωf,c). One may view the velocity V=0 as
being coincident with the source location x=0 for any t. Equivalently, one
may view V=0 as the limit as t→ ∞ at a fixed x. Using either interpretation,
a signal due to forcing must be encountered in the limit as |V| → 0 (i.e., in
the neighborhood of V=0) because the signal is continuous from its source
at x=0. As Vis reduced from +to V=0 in Figures 8abc, the pole kf2
becomes enclosed (at the velocity labeled Vf2), as shown. Intuition suggest
that the resulting residue contribution is consistent with the effect of the
oscillating source being seen only in the vicinity of the source at large times
for given values of x.
The above argument does not apply to the pole kf1; as Vis increased from
−∞ to V= 0 in Figures 8edc, the pole kf1is enclosed away from the
source (V < Vf1) but is not enclosed as the source is approached. Intuition
suggests that the contribution of the pole is not consistent with a continuous
signal emanating from the source as the source is approached (i.e., x0 for
fixed t). In this case, the contributions from kf1in hAcombine with terms
in hBto give a physically meaningful response. To show this, we continue
with the explicit evaluation of hAin (6). The relation between hAand the
path of steepest descent is obtained using the residue theorem, following the
notation shown in Figure 8. The evaluation of hAfor ωf< ωf,c (pole location
19
(a)
0
0
kr
ki
(SD)V > Vf2
hA
CR
V=0
V < 0
V > 0
hA+RCR+R(SD )V >Vf2
+RCL= 0
ksVf2
Vf1
kf1
kf2
CL
(b)
0
0
kr
ki
hA
(SD)0< V <Vf2V= 0
V > 0
V < 0
hA+RCR+R(SD )0<V <V f2
+RCL
=2πi Res(kf2)
CLCR
Vf2
kf2
ks
Vf1
kf1
(c)
0
0
kr
ki
hA
(SD)V= 0
V=0
V > 0
V < 0
CR
CL
ks
kf2
Vf1
kf1
hA+RCR+R(SD )V=0+RCL
=2πi Res(kf2)
Vf2
(d)
0
0
kr
ki
hA
V=0
V < 0
CL
CR
V > 0
Vf1
(SD)Vf1<V < 0
ks
kf1
kf2
hA+RCR+R(SD )Vf1<V <0
+RCL
=2πiRes(kf2)
Vf2
(e)
0
0
kr
ki
V < 0
(SD)V < Vf1
hA+RCR+R(SD )V <Vf1
+RCL
=2πi Res(kf2)2πi Res(kf1)
CLCR
V > 0V= 0
ks
Vf1
kf2
hA
kf1
Vf2
Figure 8: Paths for evaluating the integral hAgiven in (6), taking into account poles
() from forcing of the type ωf< ωf,c . Paths of steepest descent through saddles (×)
describe the large time behavior for velocities (a) V > Vf2; (b) 0 < V < Vf2; (c) V=0; (d)
Vf1< V < 0; (e) V < Vf1. In constructing the above figure, parameters are chosen such
that (2) describes a convectively unstable flow.
20
as shown in Figure 8) is thus
hA=
2πi [Res(kf2) + Res(kf1)] R(SD)V:V < Vf1
2πi Res(kf2)R(SD)V:Vf1< V < Vf2
R(SD)V:V > Vf2.
(17)
For ωf> ωf ,c,kf2lies in the upper half kr-kiplane (see Figure 7) and the
evaluation is:
hA=
2πi Res(kf1)R(SD)V:V < Vf1
R(SD)V:Vf1< V < Vf2
2πi Res(kf2)R(SD)V:V > Vf2
(18)
The CLand CRintegrals have not been included in the above because they
evaluate to 0. R(S D)Vdescribes the path of steepest descent through the
saddle ksassociated with a given velocity V; this integral is evaluated in
Appendix B. The residues in (17) and (18) are obtained from the integrand
of hAin (6) and are given by
Res(kf1,2) = Aei(kf1,2xωft)
2πc (kf2,1kf1,2).(19)
Combining all possible contributions to the integral according to (17)
and (18), hAmay be written compactly as
hA=iAeiωft
c(kf2kf1)Ff2(x, t) + H(x)H(Vf1V) e ikf1xZ(SD)V
,
Ff2(x, t) =
H(x)H(VVf2) e ikf2x:ωf> ωf,c
H(x) + H(x)H(Vf2V) e ikf2x:ωf< ωf,c.
(20)
Note that the cases of V=Vf1and V=Vf2are included in (20) by defining
H(0) = 1/2 to account for the principal value that is needed when the kf1,2
pole lies along the path of steepest descent. Although omitted above, the
case of ωf=ωf,c can similarly be found by taking the principal value of the
kf2pole lying along the kr-axis.
21
The complete physical description of wave propagation is recovered through
the superposition h=hA+hB+hCin accordance with (6) to yield
h(x, t) = iAeiωft
c(kf2kf1)H(x)H(VVf1) e ikf1x+H(x)H(Vf2V) eikf2x+hCZ(S D)V
.
(21)
The contribution to the response from the residues alone (i.e. everything ex-
cept R(SD)V&hC) in (21)) is plotted for the case of spatial growth (ω < ωf,c)
at various times in Figure 9. Note that the signaling region is restricted to lie
between the velocities Vf1and Vf2. Thus, the contradiction that disturbances
due to forcing were observed in both hBand hAat x→ ∞ is eliminated by
the cancellation that occurs between these two integrals.
In the solution method outlined above, the velocity Vf2characterizes a
separation between pure temporal growth and spatial/temporal growth; it is
the velocity of the “limiting ray” first discussed by [Gordillo & erez-Saborid, 2002].
We refer to this velocity as an intermediate velocity VI, as it lies between the
velocities Vand V+, which bound the time-growing transient. As shown in
Figure 8, VI(here, Vf2) is determined by locating the steepest descent path
that passes through the pole causing spatial growth from forcing. While the
ray VIis useful in determining the region unaffected by spatial growth, it
does not provide one with a clear delineation between the regions where spa-
tial or temporal growth is dominant, since temporal growth is also dominant
for nearby velocities V< V < VI(see Figure 1) as t→ ∞. In what follows,
a velocity is introduced that distinctly separates the signaling and transient
regions shown in Figure 1.
To find this velocity, we need to examine the long-time behavior of the
solution (21). The evaluation of the transient (time-growing) portion of (21)
(R(SD)V) requires the evaluation of hAalong the steepest descent path.
Two solution methods for evaluating this integral are presented in Appendix
B: a long-time asymptotic expansion and a continued fraction solution. In
unforced problems, only the first term of the large-time asymptotic expan-
sion of R(SD)V(see Appendix B, eqn. 34 with m=0) is typically required to
sufficiently describe transient wave behavior. Substituting this into (21) and
simplifying for x > 0 (instability is convective and moving downstream), the
large-time asymptotic solution to (2) is then
h(x, t)AiH(Vf2V)
c(kf2kf1)e˜σt 1
2πct A
c[kf2ks] [kf1ks]h0eσt (22)
22
(a)
−100 −50 0 50 100
−3
−2
−1
0
1
2
3
4
x 10−5
x
hhC+R(S D)V
(b)
−100 −50 0 50 100
−3
−2
−1
0
1
2
3
4
x 10−5
x
hhC+R(SD)V
Vf1Vf2
(c)
−100 −50 0 50 100
−3
−2
−1
0
1
2
3
4
x 10−5
x
hhC+R(SD)V
Vf1Vf2
(d)
−100 −50 0 50 100
−3
−2
−1
0
1
2
3
4
x 10−5
x
hhC+R(SD)V
Vf1
Vf2
Figure 9: The response solely due to the residues (i.e. hhC+R(SD)V)
in (21) for (a) t= 0.1, (b) t= 5. (c) t= 10, (d) t= 20. Parameters
(a=3, b=0.1, c=1) chosen in (2) for convective instability and spatial growth
(ωf= 0.65 < ωf,c). Note that only the real components of the signal are
shown above.
23
where ks,σ, and ˜σare functions of Vx/t:ksis given by (10), σis given
by (15), and ˜σ=ϕ(kf2) = i[kf2Vωf]. The real part of ˜σ, ˜σr=Im[kf2]V,
is the spatial growth rate (substituting V t =x), which can also be viewed as
the “effective” temporal growth rate along the constant velocity ray due to
forcing. Note that the asymptotic form of the saddle point contribution to
hAis the same as that of hC, except with a different amplitude. To capture
the interplay between signaling and transient responses, then, h0= 0 may
be taken without loss of generality. In doing, so, the form of (2) reduces to
that of the standard signaling problem (1).
The controllable portion of the flow resides not in the transient, but
instead within the signaling region and is governed by the first term in (22);
hence this is where our main interests lie. This term can be thought of as a
transfer function, relating input amplitude Aand frequency ωfwith output
waveform. The region where this transfer function dominates (as tbecomes
large) can be determined by finding the velocity where the effective growth
due to forcing is equal to the transient growth rate. This velocity of equal
growth VEis defined then by
σr|VE= ˜σr|VE(23)
which can be found analytically or graphically from the intersection of the
two curves (shown in Figure 5b). The transfer function is then defined in it
region of applicability as
h(x, t)V <VE=H(x)iAei(kf2xωft)
c(kf2kf1)(24)
for waves in a medium described by (2) with unstable forcing (i.e. ωf< ωf,c)
and a downstream propagating convectively unstable transient. The transfer
function is compared with the Fourier series solution to (2) in Figure (10),
where it can be seen that VEclearly separates regions of spatial and temporal
growth. The Fourier series is constructed on a periodic domain, following
the approach in [Barlow et al., 2010]. Observe that VEfalls between the
velocities Vand V+, which bound the time-growing packet; this makes sense
for the case of forced spatial growth (ωf< ωf,c). For forced spatial decay
(ωf> ωf,c), the velocity VEinstead lies outside of this region, as illustrated
in Figure 1b. For either case, VEis expected to lie (in a broad sense) between
the source (V=0) and the peak of the convectively unstable transient.
24
0 50 100 150 200 250 300
−1.5
−1
−0.5
0
0.5
1
1.5
2
x 10−3
x
h
V
V+
VE
VI
Figure 10: Fourier series solution (solid line) of (2) (with h0=0) at t=70 using
a=3, b=0.1, c=1, ωf=0.65, and A=5×105. Residue given by (24) (dashed
line), shown beyond its region of applicability (V > VE). Note that only the
real components of the signal are shown above; this is the solution to (2)
with a forcing of (x) cos(ωft).
25
0 50 100 150 200 250 300
−1.5
−1
−0.5
0
0.5
1
1.5
2
x 10−3
x
h
Vf2
Figure 11: Asymptotic Fourier integral solution of (2) as given by (22) with
h0=0 ( - -) compared with a Fourier series solution (––) with a=3, b=0.1,
c=1, ωf=0.65, and A=5×105, shown for t=70. Note that only the real
components of the signal are shown above; this is the solution to (2) with a
forcing of (x) cos(ωft).
The asymptotic solution of the signaling problem described by (2) over
the entire domain (eq. (22) with h0= 0) is compared with a Fourier series
solution (of (2) with h0=0) in Figure 11. As seen in the figure, the asymptotic
solution is not continuous at V=Vf2. [Gordillo & P´erez-Saborid, 2002]
pointed out that misalignment between exact residue contributions (for all
t) and an asymptotic approximation of R(SD)V(for large t) leads to this
discontinuity. If the optimal number of terms are included in the asymptotic
expansion, the effect of the discontinuity weakens for any time t; this is shown
in Appendix B. The discontinuity vanishes completely if enough terms are
used in the continued fraction solution to R(SD)V, also shown in Appendix
B. The above completes the analysis of system (2). In the next subsection,
we generalize the structure of the signaling problem just described to other
signaling problems of the form (1).
26
2.3 Generalizations
Rather than go through an entire analysis of the integrals that arise in the
general signaling problem (1), a simpler approach can be used to deduce the
transfer function. First we observe that the integrals hAwith hBwill always
have the same kfpole locations (see [Barlow et al., 2012] for an example
that is 2nd-order in t). The poles from hBwill be enclosed or not enclosed
depending on whether xis greater or less than zero. For hA, the poles are
either enclosed or not enclosed depending on the location of the steepest
descent curves.
There are a few possible outcomes for the cancellations that occur between
hAand hB: The most important is when the residue contribution of hAoccurs
only for |V|less than some limiting value (e.g. 0 < V < Vf2for kf2with
ω < ωf,c). This will occur when the pole is between the V= 0 steepest
descent curve and the real axis and is the case that leads to exponential
spatial growth; in this case, the residue contribution is the transfer function
that relates initial amplitude to that of the spatially growing wave-train.
The other cases correspond to residue contributions that have opposite signs
that cancel when |V|exceeds some limiting value (e.g. V < Vf1for kf1), or
complete cancellation of residue contributions (e.g. kf2for V < 0). These
are the only possible outcomes, based on the physical intuition that there
should be no effect of forcing as x→ ∞ for a finite t.
For clarity, we now limit our discussion to spatial growth and decay of the
forced solution that trails a downstream propagating transient. The steepest
descent path for V= 0 passes through the saddle k0which, for a downstream
propagating convective instability, lies below the kr-axis [Briggs, 1964], and
thus lies below the saddle kmax (which lies on the kr-axis). This trend im-
plies that both the saddles and their associated steepest descent paths move
upward in the kr-kiplane with increasing V.
Based on the above discussion, Table 1 shows how to determine the ap-
propriate transfer function that relates the forcing disturbance to signal am-
plitude at large times for signaling problems (as defined in (1)) that are
first-order in t. Note, that in Table 1, the residue contribution is given
in its general form, where f(k, ω) is the numerator in the Fourier-Laplace
integral solution (e.g. iAin (3)) and D
∂k |kff=D
∂ω |ωf
∂ω
∂k |kf. For our prob-
lem, Res(kf) is given by (19). If one considers the integral hAalone, residues
from poles of case (i) in Table 1 lead to upstream decay when one considers
steepest descent paths associated with saddles for V < 0. However, such
27
Table 1 Possible scenarios for poles, kf, for a downstream propagating con-
vective instability.
residue contribution = sgn[Im(kf)] f(kf, ωf) e i(kfxωft)
∂D
∂k |kff
| {z }
2πi Res(kf)
(i) If an hApole lies below the kr-axis and above the steepest descent path
for V=0 such that it is enclosed between them, then the above residue
contribution is the desired transfer function, describing spatial growth
in the downstream direction.
(ii) If an hApole lies above both the kr-axis and the steepest descent path
for V=0, then the residue will interact with hBsuch that the negative
of the above residue contribution is the desired transfer function, de-
scribing spatial decay in the downstream direction.
(iii) If an hApole lies below both the steepest descent path for V=0 and the
kr-axis, then the residue will interact with hBsuch that the negative
of the above residue contribution is the desired transfer function, de-
scribing spatial decay in the upstream direction.
28
residues will completely cancel with the residue for these poles in hB, leading
to a zero contribution; this structure is clearly elucidated in Section 2.2 and
Figure 8. Thus, we see that a given pole can only affect either the upstream
or downstream solution, but not both.
In the case of several poles, it useful to distinguish the dominant pole
relevant to the spatial growth trailing a downstream propagating transient
(i.e. for V > 0) as the one with the most negative imaginary part, but
which also falls into scenario (i) (the only situation where growth occurs) in
Table 1. For spatial decay trailing a downstream propagating transient (e.g.,
ωfin Figure 7 is such that Im(kf2)>0), the dominant pole will be the one
with the least positive imaginary part, and this will fall into scenario (ii) in
Table 1.
The dominant pole is then used in determining the spatial growth function
˜σr=Im[kf]Vthat allows one to determine the velocity VE(i.e. the velocity
that limits the transfer function) from (23). In the previous subsection (23)
is justified based on the large-time asymptotic solution (22) to a specific
signaling problem (2). However, VEmay be determined from (23) for the
general class of signaling problems given by (1) since the arguments in the
exponentials of the integral solution will in general be of the same form.
3 Algorithm for Signaling Analysis
We now summarize the analysis made above into an algorithm that may be
used to extract the limiting velocities where structure in the flow changes,
as well as a transfer function relating input signal to output waveform in
the signaling region of Figure 1. In what follows, we assume knowledge of
the dispersion relation ω(k). The overall procedure is split into two parts,
Algorithms 1 and 2. For impulsively forced flows (or where disturbances are
created through initial conditions along), only algorithm 1 is needed.
29
Algorithm 1 Assessment of system stability and extraction of key elements.
Although the algorithm is written based on our examination of problems that
are first-order in t, the general structure will remain for higher-order problems
(for examples, see [Gordillo & P´erez-Saborid, 2002, Barlow et al., 2012] ).
1. Temporal Stability: Plot ωivs kr. If there is a maximum satisfying ωi>0,
the flow is unstable (e.g. Figure 2). Denote the location of this maximum
as the saddle kmax. This saddle, and its associated growth rate, will appear
in the transient waveform as tbecomes large. Also, determine the velocity
of maximum growth Vmax = (dω/dk)kmax .
2. Convective/Absolute Instability: Find the saddle(s) k0where dω/dk=0 (cor-
responding to V=0). Plot k0in the kr-kiplane along with contours of con-
stant ωi(i.e. Re[ϕ(k)] for V=0). Additionally, draw the lines of constant
ωrthat pass through each k0saddle; these are the steepest paths for V=0
(see Figure 4). Identify the path of steepest descent that travels through
curves of decreasing ωiaway from the saddle. Verify that a closed integration
contour may be formed by connecting the path of steepest descent to the
kr-axis (original path of integration) such that ωi(k0) is a maximum along
the closed path. If this can be done, the point k0is used to classify the in-
stability; if not, the saddle may not be used and is termed non-contributing.
If ωi(k0)<0 and the saddle is contributing, the flow is convectively unsta-
ble and it is worthwhile to complete all steps of the signaling analysis. If
ωi(k0)>0 and the saddle is contributing, the flow is absolutely unstable
and forcing will be subdominant at large times in the signaling region of
Figure 1. In this latter case, if one is interested in only the classification
of the instability, then one would stop here. However, if the breadth of the
absolutely unstable transient region is also of interest, one would continue
on with additional steps.
3. Loci of Saddles: Find all saddles by setting Im[dω/dk]ks= 0. In general,
this will yield a locus of kivs. krvalues. Identify the locus that includes kmax
and the contributing k0(e.g. Figure 3); the saddles along this locus are also
contributing. Note that the knowledge of kmax and/or k0being contributing
saddles may be used as initial guesses in problems where the dispersion
relation is difficult to solve analytically. For problems that are second order
(or higher) in ω, branch points in kmay be present. In such cases, it is useful
to examine loci of saddles on each Riemann surface separately through the
use of complex polar coordinates; examples are given in [Barlow et al., 2012].
4. Growth: For each locus of saddles, find corresponding velocities
V=Re[dω/dk]ksand aggregate growth σr=Re[ϕ(ks)], where ϕ(k)=i[kV
ω(k)]. Plot σrvs. Vand confirm that loci of contributing saddles lead to
bounded growth (e.g. Figure 5) for large |V|. Bounded growth is a self-
consistency check, as it should be a feature of saddles that are contributing.
5. Bounding Velocities: Locate the velocities Vand V+where σr= 0. These
bound temporal growth in the transient solution at its trailing and leading
edges, respectively. Note that the velocity Vmax must lie between these two
values.
30
Algorithm 2 Incorporation of oscillatory forcing. For clarity in the steps be-
low, we focus specifically on forced waves trailing a downstream propagating
transient (i.e. for V > 0).
1. Poles: Plot the locus of forcing pole locations for 0 < ωf<on the
steepest descent plot made for the Convective/Absolute stability analysis
(V=0). Poles that lie between the V= 0 steepest descent curve and the
kr-axis fall into scenario (i) described in Table 1. These are the poles that
cause forced spatial growth. Poles that lie above the kr-axis or below the
V= 0 steepest descent path fall into scenario (ii) and (iii), respectively (see
Table 1). These are the poles that cause forced spatial decay. If there are
multiple forcing poles that fall into any of these cases for the same ωf, the
one with the largest value of ˜σr=Im[kf]Vis the dominant pole.
2. Extent of Spatial Growth: Using the dominant pole obtained above, find
the velocities at which ˜σr=σr. The velocity that satisfies this relation
and lies between V=0 and Vmax (e.g. Figure 5) is the velocity VE, which
separates the region of the solution dominated by spatial growth and the
region dominated by temporal growth as t→ ∞.
3. Transfer Function: The transfer function for spatial growth is then given by
h(x, t)V <VE=f(kf, ωf)ei(kfxωft)
∂D
∂k |kff
,as t→ ∞.(25)
where kfis the dominant pole as defined above. This approximation pro-
vides a simple transfer function between the input forcing and the output
waveform. In the case of a forced but spatially decaying solution to the
signaling problem, one may refer to Table 1 to determine the correct sign
of the transfer function, which has the same magnitude as given in (25). If
one is not concerned with the initial transient, the only information needed
to describe the asymptotic behavior of the solution is provided by (25).
31
4 Application to a problem that is 4th-order
in x
To demonstrate the above procedure on a more elaborate problem, we exam-
ine the signaling response of the following equation in h(x, t) which is fourth
order in space and first order in time:
∂h
∂t +4h
∂x4+ 4α3h
∂x3+6α2+ 2β22h
∂x2+ 4 α2+β2α∂h
∂x
+hγ+α2+β22ih=(x)eft, t 0
h(x, 0) = 0 for all x, h 0 as x→ ±∞ for all t, (26)
where α,β, and γare real parameters. Following the Fourier-Laplace process
described in Section 2.1, the integral solution is given by (3) with
D(k, ω) = iω+γ+ (kiα+β)2(kiαβ)2(27a)
where D(k, ω)=0 is the dispersion relation, written as
ω=iγi (kiα+β)2(kiαβ)2.(27b)
In actuality, (26) was derived from the dispersion relation to generate a prob-
lem with properties that demonstrate the effect of non-contributing saddles
and not the other way around, but this would not normally be the case for
a physical problem. For a connection back to canonical partial differential
equations, one may view (26) as a modified linear Kuramoto-Sivashinsky
equation, such as the one used in [Lin & Kondic, 2010] to model thin film
flow. Further applications built on Kuramoto-Sivashinsky equations are dis-
cussed in [Fokas & Papageorgiou, 2005].
4.1 Assessment of system stability
Following Algorithm 1 given in the previous section, we begin by performing
a classical temporal stability analysis and find kmax such that Im[
dk ]kmax
∂ωi
∂kr|kmax = 0 and ωi(kmax )>0 for purely real k. From the dispersion rela-
tion (27b),
dk = 4 (α+ki) α2+ 2 α k i + β2k2.(28)
32
−1 −0.5 0 0.5 1
−0.6
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
kr
ωi
kmax
kmin
kmax
Figure 12: Temporal stability plot for dispersion relation (27b) with α=
1/2, β=1/5, γ=1/2.
Setting the imaginary part of this expression to 0 while assuming kis real
leads to three k-roots (i.e. saddles along the kr-axis),
0,±p3α2+β2.
To determine if the problem is unstable, these values are inserted into the
dispersion relation (27b). The temporal growth-rates corresponding to the
three possible values are
ωi=γα2+β22,γ+α2(8α2+ 4β2),γ+α2(8α2+ 4β2).
The growth-rate corresponding to k= 0 is less than the other two so this is a
local minimum, kmin, as indicated in Figure 12. The second two roots given
above are local maxima (kmax =±p3α2+β2, also shown in Figure 12) and
indicate that the problem is unstable (i.e. ωi(kmax )>0) if γ < α2(8α2+4β2).
Substituting kmax back into (28), the velocity of maximum growth is then
Vmax
dk kmax =8α(4α2+β2).
We now proceed to the spatial/temporal classification (Algorithm 1: Step
2) and determine the saddle points with zero group velocity by setting
dk = 0
in eq. (28). This gives three saddles, ˆ
k0=αi and k0=αi±β. Figure 13
shows these three saddles as well as contours of Re(ϕ) for V=0. The steepest
descent path is perpendicular to these contours and, for the left and right k0
saddles, this curve is a horizontal line. This path may be closed back to the
33
kr-axis such that the growth rate Re[ϕ] is maximum along the path at each
k0. Although this path also passes through the central saddle ˆ
k0, the value
of ωireaches a local minimum at this point, as indicated by the contours
magnified in Figure 13b. The steepest descent path for the center saddle
is instead a vertical line which may not be closed back to the kr-axis, and
hence this saddle is non-contributing. The growth rate for the contributing
k0saddles may be found analytically using the expressions developed above
and is ωi(k0) = γ. This flow is thus absolutely unstable for γ < 0 and
convectively unstable for 0 < γ < α2(8α2+ 4β2) where the upper bound on
γcomes from the requirement that the flow be temporally unstable.
As all small detour, we note that the above classification may be equiv-
alently made through Briggs’ criteria [Briggs, 1964]. This criteria defines
a “pinch point” as a saddle point for V= 0 where the ωiisocontours (i.e.
Re(ϕ) for V= 0 as shown in Figure 13) approach the saddle point from above
and below the real axis as ωiis decreased from large values (as associated
with the original Laplace contour in (3)) to the value at the saddle. This
ensures that the path of the ωintegral in (3) lies above singularities ω(k),
such that causality is preserved. The criteria states that these “pinch points”
are contributing saddles and can thus be used to classify the instability. If
one applies this criteria to Figure 13, Briggs’ criteria is satisfied for the two
outer saddles, k0, since the value of ωialong contours above and below the
kr-axis is indeed decreasing towards the pinch at k0. If we magnify near the
region of the middle saddle ˆ
k0, as done in Figure 13b, it becomes clear that,
as ωiis decreased, pinching now occurs from the left and right towards the
saddle - a scenario which does not satisfy Briggs criteria.
Proceeding with Step 3 of Algorithm 1, we find all the saddles by setting
Im[dω/dk]ks= 0. This can be done by defining a grid of kvalues, calculat-
ing the imaginary part of
dk on this grid and extracting the zero contours.
Figure 14 shows three saddle curves, each passing through the three ki= 0
saddles which are the extremums of Figure 12. As the saddle ˆ
k0was shown
to be non-contributing, it is expected that the saddles on the middle curve
(locus of saddles #2) will not contribute to the long-term behavior. This
result could also have been deduced by noticing that the saddle kmin lies on
locus #2, and since any minimum from a temporal growth curve such as
Figure 12 will lead to a non-contributing saddle (see Appendix A), all other
saddles on this locus will be non-contributing.
For Step 4, we calculate and plot σrversus Valong the saddle curve,
where Vis given by Re[
dk ks]. This is shown in Figure 15. This figure
34
−1.5 −1 −0.5 0 0.5 1 1.5
−1.4
−1.2
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
kr
ki
−12
−8
−8
−8
−4
−4
−4
−4
0
0
0
0
0
0
0
0
4
4
4
4
−1
−1
−1
−1
−1
−1
−1
−1
−0.5
−0.5
−0.5
−0.5
−0.5
−0.5
−0.5
−0.5
−0.5
−0.5
−0.5
−0.5
ˆ
k0
k0
k0
kf1
(SD)k0(S D )k0
0. 55
0. 55
0. 75
0. 75
(b )
(a)
−0.2 −0.1 0 0.1 0.2
−0.55
−0.5
−0.45
kr
ki
k0k0
0.499
0.499
0.499
0.499
0. 5
0. 5
0. 5
0. 5
0. 5
0. 5
0. 5
0. 5
0.501
0.5005
0.5005 ˆ
k0
0.501
(S D)ˆ
k0
0.5016
0.5016
(S D)ˆ
k0
0.5016
0.5016
0.502
0.502
(b )
Figure 13: Re(ϕ) contours for V=0 (i.e. contours of constant ωi) for (26),
showing contributing (k0) and non-contributing (ˆ
k0) saddles. The steepest
descent (SD) contour for V= 0 (− −) is shown in (a) for k0and (b) for
ˆ
k0. Parameter values of α=1/2, β = 1/5, and γ= 1/2 are chosen for
convective instability.
35
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
kr
ki
kmax
kmax
locu s of saddles #2
locu s of saddles #3
k0k0
ˆ
k0
(ks)+
(ks)+
kmi n
(ks)E
locu s of saddles #1
(ks)
(ks)
(ks)E
Figure 14: Saddle points of (26) with α=1/2, β = 1/5, γ= 1/2. (ks)E
found using ωf=1.
confirms that the locus of saddles #2 (from Figure 14) is non-contributing
because as Vgets large the growth rate σrincreases without bound. The
two velocities corresponding to zero growth (Algorithm 1: Step 5), Vand
V+, are labeled in Figure 15 along the curve of contributing saddles (#1 &
3). Note that if we had not distinguished between sets of contributing and
non-contributing saddles, there would be additional candidates for Vand
V+from the second locus of saddles.
4.2 Incorporation of oscillatory forcing
To determine the forced behavior, the poles associated with the forcing fre-
quency must be examined. For this problem, these poles are the k-roots of
D(k, ωf) = 0 in (27a) and are given by
kf1,2,3,4=αi ±qβ2±pfγ.
The above poles are plotted in Figure 16 for the parameters used in the
previous subsection for convective instability.
36
−4 −2 0 2 4 6 8 10
−0.6
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
V
σr
V+
V
Vmax
σr, sadd le s#1,3
σr, sadd le s #2
˜σrVE
Figure 15: Growth rate of saddles as a function of Vfor (26). Parameter val-
ues of α=1/2, β = 1/5, and γ= 1/2 are chosen for convective instability.
˜σris found using ωf=1.
−1 −0.5 0 0.5 1
−1.5
−1
−0.5
0
0.5
ki
kr
0.6 0.8 1 1.2
−0.02
0
0.02
0.04
0.06
0.08
k0k0(SD)V= 0
kf2
kf4
kf3
ωf= 1
kf1
ωf= 1
ωf= 0.3
ωf= 0.3
ωf= 1
ωf= 0.3
ωf= 1
ωf= 0
ωf= 0
ωf= 0
ωf= 0
ωf= 0.3
Figure 16: Forcing poles of (26) with α=1/2, β = 1/5, and γ= 1/2.
37
As the V= 0 steepest descent curve is a horizontal line at ki=αi, kf1
thus falls into scenario (i) of Table 1 for the frequency range where Im[kf1]<0
(e.g., for ωf=1 in Figure 16 ), and these frequencies cause downstream spatial
growth. All other poles fall into scenarios (ii) and (iii) of Table 1 and lead
to downstream and upstream spatial decay, respectively. Not only is kf1the
only pole that causes growth here, it is also the dominant pole for the case
of spatial decay (for Im[kf1]>0, scenario (ii)), and so we focus our attention
on the behavior induced by this pole for both instances. We first examine
the case of spatial growth.
To determine the velocity which bounds the spatial growth region (Algo-
rithm 2: Step 2), Figure 15 shows the growth rate ˜σrassociated with the kf1
pole along with the unforced growth rate, σr. Here, ωf=1 has been chosen be-
cause it leads to spatial growth, as detailed above and indicated in Figure 16.
In Figure 15, one can see that between Vand Vmax there is a velocity where
the unforced growth and the forced growth is equal; this is VE. Note from
Figure 15 that, had we not distinguished between sets of contributing and
non-contributing saddles, there would be an additional (incorrect) candidate
for VEfrom the second locus of saddles.
Finally, using the dominant spatial growth pole kf1and (25), the transfer
function describing signaling in problem (26) is given by
h(x, t)|V|<|V|E=iAei(kf1xωft)
4piωfγ(kf1iα).(29)
Both the transfer function and the predicted limiting rays for a spatially
growing wave are validated in Figure 17a where they are superimposed onto
the Fourier series solution of (26). The same steps as above can be applied
for the case of spatial decay. Choosing ωf=0.3, which leads to a spatially
decaying contribution from kf1(see Figure 16), the sign in (29) flips once due
to the sgn[Im(kf)] term in (25) and then once more from applying criteria
(ii) in Table 1. Thus, the transfer function for the forced spatially decaying
wave where ωf=0.3 is, in fact, also given by (29). Both the transfer function
and the predicted limiting rays for a spatially decaying wave are validated
in Figure 17b where they are superimposed onto the Fourier series solution
of (26). Note that, for the spatially decaying case, the VEray (determined
using the same steps as before) now lies upstream of the Vray.
38
0 50 100 150 200 250 300 350 400 450
−10
−8
−6
−4
−2
0
2
4
6
8
10
h
x
(a)
VV+
VE
0 50 100 150 200 250 300 350 400 450
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
h
x
VV+
VE
(b)
Figure 17: Fourier series solution (solid line) of (26) at t=75 using α=
1/2, β = 1/5, and γ= 1/2 for (a) ωf=1 (spatial growth) and (b) ωf=0.3
(spatial decay). Residue given by (25) (dashed line), shown beyond its region
of applicability (V > VE), with the dominant pole kf1from Figure 16. Note
that only the real components of the signal are shown above; this is the
solution to (26) with a forcing of (x) cos(ωft).
39
5 Conclusions
In this work, we have developed an algorithm that allows one to systemati-
cally determine the response of a convectively unstable fluid flow to oscilla-
tory forcing. This algorithm builds upon our previous work [Barlow et al., 2012]
so that the key features of the solution response may be extracted. Addi-
tional structure is provided for identifying saddle points that contribute to
the long-time asymptotic behavior, and those that do not. The theory pre-
sented here is developed via the method of steepest descent and thus may
be applied to saddle points associated with any velocity. The consequence of
forcing is the inclusion of poles whose influence depends on their locations
in the complex plane relative to steepest descent paths. We have established
that these poles do not invalidate conclusions about stability obtained by ex-
amining steepest descent paths alone. Consequently, conclusions drawn for
waves of zero velocity here yields results equivalent to the well-known Briggs’
criteria [Briggs, 1964].
For a simple convection-diffusion signaling problem, we are able to clearly
and analytically show the cancellation that occurs between two pieces that
arise from the Fourier-Laplace integral solution: (i) a spatially decaying
“background term” and (ii) a temporally growing “transient” piece that is
capable of inducing spatial growth and decay. The interaction between these
pieces depends on the forcing frequency of the disturbance and dispersion
relation of the medium, but the mechanisms that provide cancellation are
believed to be general.
In examining the competition between spatial and temporal growth, a
new velocity is introduced that distinctly separates regions dominated by
each growth-type in the limit of large time. This velocity is relatively easy to
deduce for a given dispersion relation. The moving front specified by this ve-
locity sets the extent of the time-bounded spatial growth region downstream
from the source of a disturbance. We provide an analytic closed-form transfer
function that is exact in this “signaling region”, relating forcing frequency
and forcing amplitude with the amplitude and frequency of the resultant
waveform.
The methodology developed here also provides one with limiting ve-
locities important to the transient waveform emphasized in previous tech-
niques [Gordillo & erez-Saborid, 2002, Huerre, 2000]. For the modified Kuramoto-
Shivasinsky equation, the existence of non-contributing saddles requires an
examination of the path of steepest descent, which allows one to distinguish
40
between physical and non-physical wave-numbers, and by extension - veloc-
ities, that limit the extent of spatial and temporal growth.
Although developed here using problems that are first-order in time, the
method may be extended to higher-order problems (see [Barlow et al., 2012]
for example). Problems that are 2nd-order in time or higher incur the com-
plication of branch points, and so care must be taken to ensure that poles
are enclosed on the correct Riemann sheet in complex wave-number space.
Funding
N.S.B. acknowledges partial support from NSF Award No. 1048579.
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A Proof that kmax contributes to the Integral
Solution
In classical temporal stability theory, ω=ω(kr) is assumed. That is, the
wavenumber is assumed to be purely real although ωis generally complex,
and so ωr=ωr(kr) and ωi=ωi(kr). If there is a maximum in ωi(kr) at the
point (kmax,ωi,max), then ks=kmax +0i is a saddle point from definition (9c),
and its velocity is given by (9b), denoted as Vmax .
In this appendix we show that the saddle kmax will always contribute to
the long time behavior of the generalized Fourier integral,
h(x, t) = Z
−∞
F(k)eϕ(k)tdk, (30)
that often arises in the analysis of wave phenomenon, where ϕ(k) is given
by (8). We are interested in the value of Vand ksfor which σr(V)=Re[ϕ(ks(V))]
is maximized. To determine this maximum growth rate (for the asymptotic
long-time solution to (30)) over the full domain of generally complex-valued
saddle points, we solve for values of Vsuch that r/dV =0; using (14) this
becomes:
Re iks+dks
dV V
dk ks= 0.
Noting that the term in the square brackets is zero from the definition of a
saddle (9a), the solution to the above equation is
(ks,i)max = 0.(31)
43
The above result indicates that if a maximum growth rate exists, it occurs
at a saddle point that lies along the kr-axis. This saddle, then, would corre-
spond to the real-valued quantity kmax with associated speed Vmax and growth
ωi,max extracted from temporal analysis as described above. Equation (31) is
necessary, but not sufficient, to assure that maximum growth occurs at kmax
over all values of V.
To move towards a sufficient condition, we need to establish that kmax
in fact contributes to the long time asymptotic solution to (30). To do
so, we must establish that a closed contour can be constructed such that
Re[ϕ] is a maximum at the saddle point. Furthermore, the path through the
saddle point must be chosen such that Im[ϕ] is constant. This is the path of
steepest descent. An example is given in Figure 6b where the kr-axis is the
steepest descent path and thus the above conditions are satisfied for the kmax
saddle. In general, kmax will always be orientated such that these conditions
are satisfied because, although the kr-axis may not be a path of steepest
descent, it will always be a path of descent. Since the definition of a saddle
point implies continuity of ϕin the vicinity of a saddle, one may deform
the integration path from one path of descent (e.g. the kr-axis, for kmax) to
another (e. g. path of steepest descent), such that Re[ϕ] is a maximum at
the saddle point.
At this point, we have established that kmax is an extremum of σr(V)
and, if it is also a maximum, then it contributes to the long-time solution
of (30). This can be confirmed by a simple inspection of a σrversus Vplot
(e.g Figures 5 and 15). This completes the proof that any kmax obtained from
such a plot is a contributing saddle. This assessment reveals that the quantity
kmax will always be in the final waveform, validating its use as a measure for
characterizing the stability of the medium, both from the perspective of an
individual Fourier mode and from the long time behavior.
Note, that a minimum of ωi(kr) is also a saddle point, according to defini-
tion (9) (e.g. kmin in Section 4). For this saddle, the kr-axis is an ascent curve
of Re[ϕ] in the kr-kiplane. This isolates the kr-axis within the neighboring
ascent paths surrounding a simple saddle, making it inaccessible to any path
of descent. This indicates that minimums of ωi(kr) cannot contribute to the
long-time asymptotic behavior.
44
B Full Asymptotic Solution of R(SD)Vin (21)
In this appendix we confirm that the discontinuity that appears in the so-
lution of (2) at V=Vf2(see Figure 11) does indeed vanish (for any t) by
keeping additional terms in the long-time asymptotic expansion of R(SD)V
in (21). Note that R(SD)Vis the contribution to hAin (6) evaluated along the
path of steepest descent. The full long-time asymptotic expansion of R(SD)V
may be obtained by applying Laplace’s method [Bender & Orszag, 1978] to
hA, using an expansion point of k=ksgiven by (10). The expansion of
the argument of the exponential in hAabout k=kstruncates exactly to
σc(kks)2t, where σis given by (15). To simplify further, we recognize
that the (integration) path of steepest descent (as given by (13)) is along the
horizontal line k=kr+ks(where ksis purely imaginary) from to −∞ (see
Figure 8). Upon changing the integration variable to krfor the R0
portion
of the path and kr