Pulses in Shear Flows_sazonov

CHAPTER 4

PULSES IN SHEAR FLOWS

4.1. Dispersion of a Wave Packet in a Shear Flow

Wave motions in fluids may appear as a result of circumstances other than com-pressibility (acoustic waves) or stratification (internal waves). Waves in shear flowsattract particular interest in hydrodynamics. They appear in the framework of themodel of an incompressible fluid with constant density. Here we mention but a fewof the large number of references concerned with the waves in jets, boundary lay-ers, mixing layers and other shear flows (Landau and Lifshitz, 1989), (Lin, 1955),(Maslowe, 1981), (Timofeev, 1970), (Miles, 1961). Their study was initiated by LordRayleigh in connection with the problem of ‘singing flames’ described in his famousbook (Rayleigh, 1894). Here he used model velocity profiles for the explicit calcula-tion of dispersive curves. He initiated the interest to the problem of hydrodynamicalstability, a problem continues to attract attention today. The core of the theory isthe existence of unstable modes. Nevertheless the study of stable modes (decreasingor neutral) also is of interest for hydrodynamicists. In particular, it is relevant fora better understanding of non-linear wave interactions, the dynamics of turbulentboundary layer, etc.

Hydrodynamical waves differ drastically from acoustic ones: here the velocityof the wave is of the same order as the velocity of the medium. The character ofthe perturbation depends on the velocity profiles in a neighbourhood of the criticallayers, where the velocity of the wave is equal to that of medium. Therefore, severaleffects appear which are unusual for classical acoustics: the continuous spectrum,quasi-eigen modes, etc. These effects can be clearly demonstrated in initial valueproblems. This approach gives a way to eliminate some of the paradoxes that appearwhen considering purely harmonic waves in shear flows.

The traditional way to resolve these paradoxes, i.e. by introducing a small vis-cosity, is unwieldy and cumbersome from an analytical perspective, especially whencompared with the direct solutions of the initial value problems. The problem of theswitching on the source of waves and the asymptotic study of its solution will bepresented below.

129

130 CHAPTER 4

Fig. 45. A plane-parallel flow with the velocity profile U(z); u and w are the componentsof perturbations of the velocity profile, Ωy = ∂zU is vorticity of the main flow and η is itsperturbation.

4.1.1. Linearized equation for small perturbations in a plane-

parallel shear flow

Following the tradition of hydrodynamic stability theory we consider an ideal fluidplane-parallel with velocity profile U(z) and put the x-axis in the direction of theflow (see Fig. 45). Denote the components of velocity perturbation by u, v and w.At the moment we study two-dimensional perturbations only, hence the y-coordinatebecomes irrelevant. Linearizing the Euler equations and using the incompressibilitycondition we obtain the system of the form

∂tu + U∂xu + U ′w + ρ−1∂xp = 0

∂tw + U∂xw + ρ−1∂zp = 0

∂xu + ∂zw = 0.

(4.1)

Here p stands for pressure and ρ denotes the density perturbation. Introducing thestream function φ(t, x, z, ):

u = −∂zφ, w = ∂xφ

one can easily reduce (4.1) to one equation with respect to φ:

L(∂t, ∂x, ∂z; U)φ = 0,

L(∂t, ∂x, ∂z; U) = (∂t + U(z)∂x)(∂2x + ∂2

z )− U ′′(z)∂x.(4.2)

For harmonic in the x-direction perturbations: φ(z, t) exp(ikx), the equation (4.2)takes the form

(∂t + ikU)(∂2z−k2)φ−ikU ′′φ = 0. (4.3)

PULSES IN SHEAR FLOWS 131

Finally, this equation is reduced to the famous Rayleigh equation in the case ofperturbations φ(z) exp(ikx− iωt) harmonic both in x and t

(c− U)(∂2z − k2)φ + U ′′φ = 0 (4.4)

here c = ω/k is the phase velocity of perturbation. If k and ω are both real wesay that φ is a neutral solution. In the case Im ω > 0 or Im ω < 0 the functionφ is an increasing or decreasing solution (in time), respectively. Note that in thecase of neutral perturbations the Rayleigh equation possesses a singularity at thecritical layer, i.e. U(zc) = c for some z = zc. Hence, the coefficient in front of thesecond derivative vanishes for some z = zc. To eliminate this singularities one usuallyconstructs a solution of the Rayleigh equation by means of a limiting procedure fromthe model of viscous fluid when the viscosity tends to zero: ν → 0. The basic equationfor the case of viscous fluid has the form

Lν(∂t, ∂x, ∂z; U)φ = 0

Lν(∂t, ∂x, ∂z; U) = L(∂t, ∂x, ∂z, U)− ν(∂2x + ∂2

z )2.

(4.5)

For harmonic in x and t perturbations it turns into another famous equation namedby Orr and Sommerfeld.

Lν(−iω, ik, ∂z; U)φ = 0 ⇐⇒⇐⇒ (c− U)(∂2

z − k2)φ + U ′′φ− iνk−1(∂2z − k2)2φ = 0.

(4.6)

We shall refer to (4.3) and (4.5) as non-stationary Rayleigh’s and Orr-Sommerfield’sequations, respectively.

Another approach to eliminate singularities is to consider evolutionary problems,i.e. to solve nonstationary Rayleigh equation with an initial condition. Then weobtain pulse-like problems and benefit from the conventional asymptotic methods toinvestigate them in the limit t → 0.

We shall clearly demonstrate that nonstationary approach is less cumbersomethan ‘viscous’ one although to use it requires more careful physical treatment.

4.1.2. Evolution of the 2D perturbations in the Couette flow

The plane-parallel Couette flow between two parallel plates z = −H, H with linearunperturbed velocity profile U = γz provides the simplest example of a shear flow.The study of evolution of initial perturbations in the Couette flow was initiatedby Case (1960), he interpreted the observable decrease of perturbation as a wavedispersion effect, i.e. the dispersion of a wave packet formed by continuous spectrumwaves (see below).

Now we start the detailed exposition. In the case of linear unperturbed velocityprofile the equation (4.2) can be treated as the transport equation for the vorticity

132 CHAPTER 4

perturbation η (only its y-component is not zero identically in a plane-parallel flow):

η = (∂2x + ∂2

z )φ.

Indeed, (4.2) can be written in the form

(∂t + U(z)∂x)η = 0.

This fact implies that vorticity in any layer z of the Couette flow is an integral ofmotion, as a function of horizontal variable x its moves with the velocity of the flowU(z):

η(x, z, t) = η0(x− U(z)t, z)

whereη0(x, z) = (∂2

x + ∂2z )φ0, φ0(x, z) = φ(x, z, 0)

are the initial vorticity and stream function, respectively.This simple remark gives the key for an explicit description of the evolution of an

initial perturbation φ0(x, z) for the Couette flow between parallel platelets z = ±H

with boundary conditions

φ(x,±H, t) = φ0(x,±H) = 0. (4.7)

We have immediately obtained

φ = (∂2x + ∂2

z )−1η0(x− U(z)t, z) (4.8)

where (∂2x +∂2

z )−1 is the resolvent of the Laplace operator incorporating the boundary

conditions (4.7).Now we specify the relation (4.8) for the case of a harmonic along the x-axis initial

perturbationη0(x, z) = η0(z) exp(ikx).

We denote by G(z | h, k) the Green’s function, i.e. the solution of

(∂2z − k2)G = δ(z − h)

satisfying boundary conditions (4.7). A simple computation gives

G(z |h, k) =

sinh k(h + H) sinh k(z −H)

2 cosh 2kH, z ≥ h

sinh k(z + H) sinh k(h−H)

2 cosh 2kH, z ≤ h.

(4.9)

Therefore

φ(t, x, z) =

H∫

−H

G(z |h, k)η0(h) exp[ikx− ikU(h)t]dh. (4.10)


Then we apply the saddle point technique to study the asymptote of (4.10) whent →∞. It is convenient to split the integral in (4.10) into the sum of I1 and I2 (fromz to H and from −H to z, respectively). The contribution from the boundary pointh = z gives the leading term of expansion in both cases. Therefore

φ(t, z) ≈ −η0(z) exp[−ikU(z)t](kγt)−2 + O(t−3) (4.11)

(remember γ = U ′) and any harmonica in the x-direction of initial perturbation(with only exception of the term with k = 0, i.e. the main flow) in the Couetteflow asymptotically decreases as t−2. This due to the fact that the leading terms ofexpansions for I1 and I2 cancel each other and the second terms describe the decreaseof perturbation.

Now we concentrate on the case when the vorticity of initial perturbation is locatedin a thin layer of the width 2d (kd ¿ 1) at the level zc. In this case (4.10) takes theform

φ ≈ eikx

zc+d∫

zc−d

G(z |h, k)η0(h) exp [−i(kγt)h] dh. (4.12)

The interpretation of (4.12) is straightforward. The behaviour of the field dependscrucially on the parameter κ = kγt: in the case κ À 1 formula (4.11) is applicableand the perturbation decreases as t−2. In the opposite case κ ∼< 1 the integrand is aslowly changing function outside the layer (i.e. when |z − zc| > d). Taking advantageof this fact we can use outside the layer the following expression

φ(t, x, z) ≈ A exp [ik (x− U(zc)t)] G(z | zc, k) (4.13)

where

A =

zc+d∫

zc−d

η0(h)dh.

Solution (4.13) represents a harmonic wave propagating along the flow with ve-locity U(zc). In the layer (zc − d, zc + d) solution (4.13) fails. It is convenient tointroduce the characteristic time

td =1

kγd. (4.14)

In the case t ¿ td the perturbation nearly represents a harmonic wave, in theopposite case t À td the perturbation field decreases as t−2 for vertical velocity andas t−1 for horizontal one:

w(t, z) = ikφ(t, z) ∝ t−2

u(t, z) = −∂zφ(t, z) ≈ −iη0(z) exp(−ikU(z)t)k−1γ−2t−1.

134 CHAPTER 4

Fig. 46. Vertical profiles of the velocity components u and w, pressure p and vorticity ηfor a solitary continuous spectrum wave in the Couette flow.

The characteristic time td tends to infinity as d → 0. If we rescale the initialperturbation η0(z) so that A → const as d → 0 (i.e. η0 → δ(z − zc)) the solutiontends to a harmonic wave described by (4.13) (see Fig. 46) being of special interest.

4.1.3. The concept of a CS-Mode

Consider in more details this limiting solution (4.13). It satisfies Rayleigh’s equationwith c = U(zc) anywhere but for the critical layer z = zc, it satisfies the boundarycondition (4.7) as well. It satisfies also the glueing conditions at the critical levelz = zc: the vertical velocity and pressure p = ik−1ρ[(∂t + ikU)∂zφ − ikU ′φ] arecontinuous at this level.

For the solution under study the vorticity is concentrated in an infinitesimallythin layer, more precisely η0 ∼ δ(z − zc). We can treat it as a generalized solutionof Rayleigh’s equation (4.13), i.e. a distribution belonging D′. One can check that itsatisfies Rayleigh’s equation by substitution taking into account that (z − zc)δ(z −zc) ≡ 0 in D′.

We mention that the limiting solution (4.13) fits the model of an ideal fluid becausethis model permits tangential discontinuity (jump of velocity) where vorticity hasthe singularity of the δ-type. Moreover, this solution describes a plane vortex sheetflow with harmonical vortex distribution located at the plane z = zc. In the linearapproximation this sheet does not affect itself. Hence the vortex sheet remains to beplane, conserves the distribution of vortex intensity and drifts along the x-axis withvelocity U(zc).


Thus, in analogy with the conventional waves, the solution (4.13) can be treatedas the eigen-mode of the Couette flow. At the same time this solution is the Green’sfunction of the non-stationary Rayleigh equation for the Couette flow, i.e. the solutionof the problem

L(∂t, ik, ∂z; U)G = δ(t)δ(z − zc). (4.15)

Consider the possibility for generation of such a mode. The Green’s function G

in (4.13) can be treated as the perturbation in the flow had emerged as a result ofvertical shock action fz in this thin layer

fz = δ(t)δ(z − zc)eikx. (4.16)

Let us specify an imaginary experiment for generation of the mode (4.13) in aflow. One creates a harmonic in the x-direction shock action by pushing sharply aslightly corrugated inextensible membrane. This idea is practically feasible in someexperiments with electromagnetic pulses acting on the boundary between two dif-ferent dielectric fluids. The most important source of such mode in the nature is astreamlined profile oscillating in the flow (see Sec. 4.5). Here the exciting force isharmonic in the x-direction due to the periodic changing of angle of attack.

Generation of these waves is due to the adding of supplementary vorticity intothe fluid. It might be the result of an excitement by external volume forces (withthe exception of sources of a volume velocity, they can generate the forced oscillationonly that would disappear when the same sources switched off).

For fixed k the phase velocity of the mode varies in the interval [Umin = U(−H),Umax = U(H)] depending on the parameter zc ∈ [−H, H], i.e. the z-coordinate ofthe layer the initial vorticity generated into. Thus, the mode (4.13) belongs to thecontinuous spectrum (unlike the sound waves in the channels where the phase velocityof every mode can have discrete values for any k).

We can interpret the expression (4.10) as a decomposition of an arbitrary pertur-bation φ(t, z)eikx into superposition of the continuous spectrum modes:

φ(t, z) =

H∫

−H

η0(zc)φCS(z | zc, k)dzc

φCS(z | zc, k) = G(z | zc, k) exp[ikx− ikU(zc)t]

(4.17)

where η0(zc) = (∂2zc− k2)φ0(zc) is a weight factor. Clearly, any harmonic in the

x-direction perturbation φ satisfying the boundary conditions (4.11) can be repre-sented in the form (4.10). Moreover, an arbitrary two-dimensional perturbation inthe Couette flow can be represented as a wave packet of continuous spectrum modesas follows

φ(x, z, t) =1

2π

+H∫

−H

+∞∫

−∞η0(zc, k)φCS(z | zc; k)eikxdkdzc.

136 CHAPTER 4

The weight factor η0(zc, k) is the Fourier transform of η0(x, z) = (∂2x +∂2

z )φ0(x, z).Any mode of the continuous spectrum φCS propagates with its own velocity U(zc)

depending on the z-coordinate of the critical layer (but does not depend on k). Asa result, the wave packet has to be dispersed in time due to the difference in theharmonic velocities. Thus, the perturbation decreases in time asymptotically.

4.1.4. Historical remark

Initially these facts were described briefly in the report (Eliassen et al., 1953) by Nor-wegian researchers, with similar perturbations being studied more thoroughly by vanKampen in plasmas (1955). Continuous spectrum waves attract attention for hydro-dynamic applications after the paper by K. M. Case (1960) who investigated themin the Couette flow. The late references can be found in the reviews (Maslowe, 1981)and (Timofeev, 1970) where the term van Kampen-Case’s wave was introduced. Weshall use the term Case’s wave or CS-mode being only interested in hydrodynamicaluse. It is interesting to mention that abbreviation ‘CS’ may mean that of ‘K. M. Case’and of ‘Continuous Spectrum’ simultaneously.

We stress here that the decrease of the perturbation in a shear flow does notcontradict the energy conservation law. Indeed, the operator

LR = U(z)− U ′′(z)(∂2

z − k2)−1

of the Rayleigh equation described in the form of eigen-value problem for this oper-ator:

LRφ = cφ

is not self-adjoint and allows the energy transference from the perturbation to themain flow. The energy of initial perturbation will eventually be transported into thatof the main flow with the initial flow profile changing very little (Sazonov, 1989). Thecontinuous spectrum solution can be explicitly described for piece-wise linear velocityprofiles which are of often used for the approximation of real ones. Unlike discretespectrum solutions, the perturbation φ has a break at the critical layer (see Sec. 4.5).The existence of continuous spectrum for Rayleigh’s operator L(−iω, ik, ∂z; U) forarbitrary smooth function U(z) was proved by Dickey (1976).

4.1.5. Temporary growth of perturbations in the Couette flow

Thus, any free (i.e. not supported by an external action) two-dimensional perturba-tion can be represented as a wavepacket of CS-modes. The formula (4.10) impliesthat any smooth packet must eventually decrease.

However, this asymptotic decrease does not prevent temporary amplification ofinitial disturbances. A simple example is presented below.


Fig. 47. Temporary growth of the energy of certain perturbation in a shear flow (bottomplot). The perturbation is represented as a superposition of CS-modes having the samewavenumbers, but different critical layers (top plots). Mutual phases of CS-modes changedue to the shear of the flow.

Notice that we can inverse time (t → −t) in (4.10). This procedure is equivalentto the inversion of the velocity gradient (γ → −γ). Simultaneous inversion of t andγ does not affect the integral (4.10) in any way. Thus, we may consider the processin the flow with the inverted velocity gradient as a process with inverted time andvice versa.

Now consider a rather strong initial perturbation φ0 at a time-moment t0 and waituntil time t1 when it practically vanishes (φ1 ≡ φ(t1) ¿ φ0). Then consider a smallperturbation φ1 at a time-moment t1 as initial one in the flow with inverted velocitygradient passing to the reversed evolution of perturbation. Thus, in the time-interval∆t = t1 − t0 we return to the strong perturbation φ0 again, and it is during thisinterval that one can observe the temporal growth of perturbation.

The perturbation should eventually disappear in a large time scale. Nevertheless,the existence of the increasing perturbation (although temporary) is unusual for astable flow such as the Couette flow.

This phenomenon can be very important for problems of 2D-turbulence and flowsstability and has been described in a series of papers (Farrell and Ioannou, 1993),(Shepherd, 1985).

Notice that theoretically any widely dispersed pulse may be relocalized if thephase velocities of all its harmonics are reversed. One observes a similar picture oftemporary growth of perturbation in the shear flow.

Now we present a simple explanation of this phenomena using the concept of CS-modes and the analogy with the pulse dispersion. Perturbation φ0 is a superpositionof CS-modes with nearly coinciding phases (see Fig. 47). They are disphased duringthe evolution due to the drift by the main flow with different velocities. At time t1they have been disphased, as shown in Fig. 47a. Hence, at time t1 they form a small

138 CHAPTER 4

perturbation described by the integral (4.10) of fast oscillatory function.However, these initially strongly disphased CS-modes will have near coinciding

phases at a certain moment in time during the drift (see Fig. 47b). Then they forma rather strong perturbation. After this moment CS-modes continue their drift andlose the coherence thus forming a decreasing perturbation.

This is the first situation of growing perturbations in flows where exponentiallyincreasing modes are absent. In the present work we shall come across anotherexamples of similar phenomena (cf. Sec. 4.2.6, 4.3 and 4.4.).

4.2. Structural Stability of the CS-Mode

Summarizing the results of the previous section, we note that any small free 2D-per-turbation in the Couette flow is nothing but the packet of CS-modes. The decreaseof the perturbation is owing to the dispersion of the packet. Solitary CS-mode is ageneralized solution of the Rayleigh equation, it fits the model of an ideal fluid.

The concept of CS-mode can be simply generalized on flows with piece-wise ve-locity profiles where U ′′(zc) = 0. Then this mode has the same δ-shaped singularitiesof vorticity at the critical level. This case is analyzed in Sec. 4.4.

From mathematical standpoint the existence of CS-modes is due to the singularityof the Rayleigh equation. Thus, one expects the similar modes in other flows wherethe small perturbations are described by an equation with a similar singularity, e.g.in flows with curved velocity profiles, in density stratified fluids, etc. The existenceof CS-modes in such flows is demonstrated in this Section. Also we show that thesolitary CS-mode possesses the more complicated singularity at its critical layer thanthat in the Couette flow. We discuss also the physical meaning of the solitary CS-mode being not so transparent as that for the Couette flow.

Beginner should not be ‘puzzled’ by the singularity of the solitary wave. It is not‘dangerous’ if a superposition of such waves forms a smooth perturbation.

Generally speaking, we investigate in this section the effects of different factors(velocity profile curvature, fluid viscosity, density stratification, etc.) on the solitaryCS-mode, i.e. the topic concerns its structural stability.

One should remember that the Rayleigh equation may have parasitic solutionsthat cannot be obtained as limits when ν → 0 from solutions of the Orr-Sommerfieldequation (Landau and Lifshitz, 1989). Therefore the problem of structural stabilityof Case’s wave is of great importance.

For this aim we study the evolution of the following perturbation

φ(t = 0) = G(z | zc, k) exp(ikx)

in different flows. We demonstrate that in the limits U ′′ → 0, ν → 0, N → 0 (whereN is the buoyancy or Brunt-Vaisala frequency) or A → 0 (A is the amplitude of


Case’s wave), respectively, the solution of the Cauchy problem with initial conditionstends to (4.13) on any finite interval.

We consider now the case of an unbounded in the z-direction flow (H → ∞)to avoid unwieldy expressions. This assumption does not affect the solution if thedistance between the layer zc where the external force of the type (4.15 ) is appliedand boundary is large when compared with the wavelength λ = 2π/k. Consideringthe limit H →∞ in (4.9) we obtain the expression for Case’s wave:

φCS = A exp [−k |z − zc|+ ik (x− U(zc)t)] . (4.18)

4.2.1. The effect of the curvature of velocity profile

The existence of continuous spectrum for the Rayleigh equation has been proved byL. Dickey (1976). Here we obtain an explicit expression for the CS-mode using thesmallness of velocity profile curvature as a limit of an initial value problem.

Consider the non-stationary non-homogeneous Rayleigh equation

(∂t + ikU)(∂2z − k2)φ− ikU ′′φ = −2kδ(t)δ(z − zc) (4.19)

with boundary conditionsφ(z → ±∞) → 0 (4.20)

and the following conditionφ(t < 0) ≡ 0 (4.21)

implied by the causality principle. The factor 2k is introduced for convenience. Afterthis normalization the Case wave with amplitude A = 1 satisfies (4.19) for the Couetteflow. Indeed, perform the one-sided Fourier transform

φ(c, z) =

∞∫

0

φ(t, z) exp(ikct)dt.

Obviously the spectrum φ obeys the non-homogeneous Rayleigh equation

(c− U)(∂2z − k2)φ + U ′′φ = −2iδ(z − zc).

If U ′′ ≡ 0 the spectrum φ has the pole c = U(zc) as the only singularity:

φ(c, z) = ik−1(c− U(zc))−1 exp(−k | z − zc |).

The residual at the pole gives the Case wave

φ = exp [−k |z − zc|+ ik (x− U(zc)t)] .

140 CHAPTER 4

We mention that for the distributed in the z-direction external forcefz(z)δ(t) exp(ikx) the spectrum φ(c, z) is represented as follows

φ(c, z) =∫

fz(h)(c− U(h))−1 exp(−k |z − h|)dh. (4.22)

The character of singularity φ depends on the analytical properties of fz(z). As anexample, if the external force vanishes outside the thin layer of the size 2d (kd ¿ 1)and constant (f0) inside it, the ‘distributed’ pole in (4.22) reduces to the couple oflogarithmic branch points:

φ ≈ f0 exp(−k |z|)d∫

−d

(c− γh)−1dh = f0 exp(−k |z|) logc− γd

c + γd.

Turning back to propagation of this perturbation after initiation in the flow withthe curvature of velocity profile, we use the dimensionless variables z = k(z − zc),t = U ′

ct (U ′c = U ′(zc)) to simplify the notations. Then (4.19) takes the form

(∂t + iU)(∂2z − 1)φ− iU ′′φ = −2δ(t)δ(z). (4.23)

Here U(z) = k[U(z/k) − Uc]/U′c is the dimensionless velocity of the flow. Using

the Fourier transform with respect to dimensionless time φ(ω, z), we obtain for thespectrum

(ω − U)(∂2z − 1)φ + U ′′φ = −2iδ(z).

Note that the dimensionless frequency ω coincides with the dimensionless phasevelocity c. The solution of (4.23) can be written in the form

φ =

2iW (ω)−1φ2(ω, z)φ1(ω, 0), z > 0

2iW (ω)−1φ1(ω, z)φ2(ω, 0), z < 0(4.24)

where W (ω) = φ2(ω, z)∂zφ1(ω, z) − φ1(ω, z)∂zφ2(ω, z) is the Wronskian of the par-ticular solutions φ1 and φ2 of the Rayleigh equation vanishing when z → −∞, andz → +∞, respectively.

Suppose that the velocity profile U(z) is nearly linear, moreover it has no pointsof inflection and can be represented in the form

U(z) = z[1 + σz/2 + γ2(σz)2 + γ3(σz)3 + · · ·

](4.25)

where σ = U ′′(0) is a small parameter. It specifies the curvature of the velocity profilein a neighbourhood of the level of localization zc of the external force fz. For thisvelocity profile the solution of the Rayleigh equation can be approximated by

φ1,2 = exp(± |z|)[1∓ σ

2F (±ω ∓ z) + O(σ2)

], |z| ¿ |σ|−1 (4.26)


φ1,2 = exp

±z ± 1

2

z∫

0

U ′′(u)du

U(u)− ω

(−2ω)−σ/2, |z| À 1. (4.27)

The formula (4.26) is obtained by the successive approximations method usingthe decomposition in a σ-series and (4.27) is obtained by the JWKB method. Bothexpressions fit the glueing condition in the intermediate domain 1 ¿ |z| ¿ |σ|−1.

Substitute (4.26), (4.27) in (4.24) and perform the inverse Fourier transform.Then we obtain

φ(t, z) =1

2π

∫

Γω

φ(ω, z) exp (−iωt) dω (4.28)

where the contour Γω passes in the upper half-plane above all singularities of theintegrand φ(ω). The list of these singularities includes the logarithmic branch pointω = 0 of the functions φ1(ω, 0) (z > 0) or φ2(ω, 0) (z < 0) (it reduces to a singlepole when σ = 0); the ‘moving’ (along the horizontal axis with change of parameterz) logarithmic branch point ω = z (|z| ¿ |σ|−1) or ω ≈ U (|z| ¿ 1) (these formulaeconform each other because U ≈ z + σz2/2 + O(σ2) when z → 0). The movingbranch point responds to c = U(z), i.e. the velocity of the flow on the z-horizon indimensional variables. These branch points disappear in the limit σ → 0. We presentan approximate expression for the spectrum φ(ω, z) in a neighbourhood of the branchpoints. When ω → 0

φ(ω, z) ≈

φ2(0, z)(i/ω)φ0(ω), z > 0

φ1(0, z)(i/ω)φ0(−ω), z < 0(4.29)

where φ0 = 1 + (σ/2)(−C + πi − 2ω log 2ω) + O(σ2) + O(ω) + O(σω log ω), C =0, 5772156649 . . . is the Eiler constant. When ω → z

φ(ω, z) ≈

φ1(0, z)(i/z)e−zφz(ω − z), z > 0

φ2(0, z)(i/z)e−zφz(z − ω), z < 0(4.30)

where φz(ξ) = 1− (σ/2)(C− πi−2ξ log(−2ξ) + O(σ2) + O(ξ) + O(σξ log ξ)).We put cuts emanating from the branch points as extending vertically down-

wards and specify the branches of multi-valued functions log and Ei according to thecausality principle (it implies that φ(t, z) ≡ 0, t < 0). For this aim log and Ei has totake real values on the positive and negative real half-axis, respectively. We deformthe initial contour Γω into two contours Γ0 and Γz passing around the branch pointsω = 0 and ω = z, respectively (see Fig. 48). Denoting these integrals as φ0 and φz:

φ(t, z) = φ0(t, z) + φz(t, z) (4.31)

we study their asymptotes as t → ∞. If t > 0 the integrand in (4.28) decreasesexponentially in the lower half-plane. Therefore a neighbourhood of branch points

142 CHAPTER 4

Fig. 48. Contour of integration in the analysis of Case’s wave in a flow with a smallcurvature of the velocity profile.

provides the main contributions to these integrals. Using this approximation andnoting that the branches of log differ by the additive constant 2πi we obtain

φ0 =

φ2(0, z)1 + (σ/2)(πi−C + 2i/t), z > 0

φ1(0, z)1 + (σ/2)(πi−C−2i/t), z < 0(4.32)

φz =

(σ/(2zt2))φ1(0, z) exp(−z + izt), z > 0

−(σ/(2zt2))φ2(0, z) exp(z + izt), z < 0.(4.33)

Formulae (4.32), (4.33) fail in a neighbourhood of the critical layer z ≈ 0 wherethe mutual interconnection of the branch points has to be taken into account. In thedomain |z| ¿ 1, | ω |¿ 1 we obtain the following approximation for the spectrumφ(ω, z):

φ ≈

i

ω

[(1 + z +

σπi

2− σC

)− σ log 2ω + σ(1− z

ω) log(2z − 2ω)

], z > 0

i

ω

[(1− z +

σπi

2− σC

)+ σ log(−2ω) + σ(

z

ω− 1) log(2ω − 2z)

], z < 0

Then

φ ≈

1− z + σπi/2− σC + (σi/t)(e−izt − 1) + σz[Ei(−izt)− log(2z)], z > 0,

1 + z + σπi/2− σC− (σi/t)(e−izt − 1)− σz[Ei(−izt)− log(−2z)], z < 0,(4.34)

Thus, formulae (4.32)–(4.34) describe the asymptotic behaviour of the Green’sfunction for a small perturbation in the shear flow with a nearly linear profile. Incontrast with the case of the Couette flow, the initially harmonic in time perturbation


(4.16) produces a non-harmonic motion. Nevertheless, in the limit t → ∞ it tendsto a harmonic wave

φ(t, x, z) → Φ[k(z − zc)] exp[ik(x− U(zc)t)] (4.35)

where

Φ(z) = limt→∞φ(t, z) =

φ2(0, z)[1 + (σ/2)(πi−C)], z > 0

φ1(0, z)[1 + (σ/2)(πi−C)], z < 0.

Neglecting the factor [1 + (σ/2)(πi−C)] which is nearly 1, we obtain a represen-tation for Φ(z):

Φ(z) =

exp(− |z|)1 + j(σ/2)[Ei(−2 |z|)− log(2 |z|)], |z| ¿ |σ|−1

exp− |z| − (j/2)∫ z

j/2(U ′′(h)/U(h))dh, |z| À 1

(4.36)

where j = signz.The field (4.35) keeps the singularity in the critical layer being a discontinuity of

the first derivative for the term of the order of σ0 and logarithmic singularity z log |z|for the term of the order of σ1.Therefore Φ′′(z) treated as a distribution belongingD′ can be represented in the neighbourhood of the origin in the form

Φ′′(z) = −2δ(z)− σP 1

zexp(− |z|) + regular terms

where P(1/z) = (log |z|)′ is the principal value of (1/z) (see, e.g. (Vladimirov, 1971)).The field (4.35) is to be treated as an eigenmode of continuous spectrum for the

nearly linear profile because it satisfies both the Rayleigh equation (when c = U(zc))as a generalized solution and the causality principle. The last one is fulfilled because(4.35) is the limit of an evolutionary problem with initial perturbation (4.16). Thesmall curvature of the profile in a neighbourhood of the critical layer changes thecharacter of the singularity but affects the vertical velocities field negligibly in quan-titative aspect (a term of the order of σ1 appears). At the same time the horizontalvelocity tends to infinity near the critical layer as log |z|:

u = −∂zφ ≈ sign z + σ log |2z| , |z| ¿ 1.

However, the formulae (4.33) implies that the stabilization time of the large hor-izontal velocities grows near the critical layer. Indeed, the horizontal velocities fieldtakes the form

u(t, z) = −∂zφ(t, z) = sign z + σ log(t/2) ∝ log t

in the domain |z| ¿ 1, |zt| ¿ 1.

144 CHAPTER 4

The vorticity of the Case wave in the curved velocity profile flow is not local-ized entirely in the critical layer, it may be described by the following distributionbelonging D′

ω ∼ 2δ(z) + σP 1

zexp(− |z|), |z| ¿ 1. (4.37)

Because U ′′ 6= 0 the vorticity of the main flow changes from layer to layer. Thisvariability creates its vertical transport. To keep the stationarity of the picture thevorticity of the critical layer would create a vertical velocities field compensating thistransport. Actually, one can check that the perturbation (4.36) is an equilibrium oneto stabilize the vorticity across the flow.

Now we demonstrate that an arbitrary (smooth) perturbation in the flow (4.25)can be decomposed into the waves of continuous spectrum just as in the Couetteflow:

φ(t, z)eikx =∫

G(z | h, k) exp[ik(x− U(h)t)]g(h)dh. (4.38)

Here G(z | h, k) = Φ (k(z − h)) and g(h) is an amplitude of the Case wave withcritical layer z = h.

The proof is easy. Indeed, we can calculate the amplitude g(h) for an arbitraryinitial perturbation of the form φ(0, x, z) = φ0(z)eikx. For this aim we set t = 0 in(4.38) and act on both sides by the linear operator (∂2

z − k2). In this way we obtainthe following relation for η

η0(z) =∫

ηCS(z − h)g(h)dh

and ηCS(z) = (∂2z − k2)Φ(kz). Calculating ηCS and using dimensionless variables

ηCS(z) = −2δ(z) + σ exp(− |z|)P 1

z+ O(σ2)

we reduce (4.38) to the following integral equation

η0(z) = −2g(z) + σ∫

exp(− | z − ξ |)P 1

z − ξg(ξ)dξ + O(σ2).

Now we use the Fourier transform with respect to z:

η(κ) =∫

η(z) exp (iκz) dz. (4.39)

Taking into account the expression of the Fourier transform of the functionexp(−|z|)P(1/z) as log[(1− iκ)/(1 + iκ)] we obtain:

η0(κ) = −2g(κ) + σ log1− iκ

1 + iκg(κ) ⇐⇒


⇐⇒ g(κ) = −1

2η0(κ)− σ

4η0(κ) log

1− iκ

1 + iκ+ O(σ2).

Performing the inverse Fourier transform we finally have the relation

g(z) = −1

2η0(z)− σ

4

∫exp(−|z − ξ|)(z − ξ)−1η0(ξ)dξ + O(σ2).

In particular, if initial vorticity is δ-shaped η0(z) = δ(z), the amplitude of the Casewaves is described by the formula (4.37). We can treat the formulae obtained as thedispersion of the Case waves packet being generated by the initial excitation (4.16).We note that the time of coherence decreases with the increase of distance betweenthe critical layers and that of initial excitation. The contribution of Case’s waves withcritical layers around zc is essential only in the vanishing domain |z| ∼ σt−2 becauseof the nearness of phase velocities. In the limit t →∞ all these waves eliminate eachother with the only exception of the central Case wave. The same phenomena takesplace for the initial perturbation of vorticity of the type (4.36) in the Couette flow.

4.2.2. The effects of viscosity

First, we note that the continuous spectrum waves described above must disappear inviscous flows1. In the case of an ideal fluid their existence is related to the singularityof the Rayleigh equation lacking in the Orr-Sommerfeld equation. Therefore spectralproblems for the Rayleigh and the Orr-Sommerfeld’s equations are drastically differ-ent. However, the influence of viscosity becomes essential in evolutionary problemsonly in large time scale. Now we demonstrate the effects of viscosity in the problemof evolution of the initial perturbation (4.16) which is governed by the equation

(∂t + iU)(∂2z − 1)φ− iU ′′φ−R−1(∂2

z − 1)φ = −2δ(z)δ(t) (4.40)

with initial conditions (4.20)–(4.21). Here we use dimensionless coordinates z and t

and R = U ′c/(νk2) stands for the Reynolds number.

For simplicity we start with the case of linear velocity profile (σ = 0, U ≡ z), anduse the Fourier transform with respect to z (instead of t) to present the solution inan explicit form. Using (4.40) we obtain for the spectrum φ(t, κ) the equation

(∂t − ∂κ)(κ2 + 1)φ + R−1(κ2 + 1)2φ = −2δ(t)

that yields an explicit solution for the spatial spectrum

φ =θ(t)

π

+∞∫

−∞

dκ

1 + κ2eiκz exp

(− t3

12R− t

R− t2κ

R+

tκ2

R

).

1In viscous flows in boundary layers continuous spectrum waves of other types might exist aswell, in this case the amplitude of perturbation tends to a constant value far from the boundary(Grosch and Salwen, 1978).

146 CHAPTER 4

Performing the inverse Fourier transform we express the solution in terms of theauxiliary error function:

φ(t, z) =

1

2exp

(− t3

3R− it2

R− z

)erfc

−1

2

√R

tz +

(1− it

2

) √R

t

+

+1

2exp

(− t3

3R+

it2

R+ z

)erfc

+

1

2

√R

tz +

(1 +

it

2

) √R

t

θ(t).

(4.41)

If any of inequalities t À R−1/3 or |z| À√

t/R is fulfilled we can simplify (4.41)using the asymptote of erfc for large arguments:

φ(t, z) = exp

(− |z| − t3

3R− it2

Rsign z

)+

+ 2

√t

πR

t

R− 1

4

z

√R

t+ it

√t

R

2−1

exp

(−Rz

4t− izt

2− t3

12R− t

R

).

(4.42)

Note that the first addend prevails in the domain t ¿ R−1/3 and |z| À√

t/R,therefore the solution of the evolutionary problem can be written in the form

φ(t, z) ≈ exp(− |z|)[1 + O(t3/R)].

For t fixed and R →∞ it tends to the solution (4.18) of an inviscid evolutionaryproblem.

When t À R−1/3 the second addend prevails for all z (to be more specific the firstaddend ∼ exp(−t3/3R) and the second one ∼ exp(−t3/12R)). Using the dimensionalvariables we conclude that the solutions of viscous and inviscid evolutionary problemsmatch in the domain

| z − zc |À zν ≈ (tν)1/2

t ¿ tν ≈ ν−1/3(kU ′c)−2/3.

(4.43)

When t ¿ tν the influence of viscosity is essential only in a neighbourhood of thehorizon where initial perturbation was localized, the thickness of the viscous layerzν ∼ t1/2. We shall now look at the problem (4.40) in the layer −H ≤ z ≤ H withsticking boundary conditions when z = ±H. We can easily demonstrate that theinfluence of the viscosity is essential in the boundary layers of the size zν also. Ifthe external force acts on the layer of a finite thickness d, the effects of the viscositybecome essential for large time when zν > d, i.e. t > tdν = d2/ν. This fact can beeasily obtained from the study of convolution φ ∗ fz where φ is the solution (4.41)and fz is the distribution of the initial perturbation of the z-direction.

The physical interpretation of the solution (4.41) was presented by Timofeev(1970): the estimation (4.43) was obtained from quantitative considerations of vor-ticity ‘diffusion’. As a matter of the fact, the diffusion equation

dtω = ν∆ω


where dt = ∂t + U∂x, ∆ = ∂2x + ∂2

z governs the propagation of vorticity in a viscousfluid. It means that a solitary Case wave (i.e. an infinitesimally thin layer of vorticity)turns into a packet of the size zν(t). In a shear flow the dispersion time of the packetis of the scale

t ∼ 1

zν(t)kU ′ ,

this estimate leads to the expression (4.43) for the viscous time tν .We mention here an analogy with some spectral problems in the theory of thin

shells. In the framework of momentless theory eigenmodes satisfy an equation ofthe second order containing a singularity. In some cases (e.g. for conical or nearlyconical shells) it admits continuous spectrum. After taking into consideration bendingmoments it is to be replaced by an operator ∆2 of the order four. Thus, the intervalsof continuous spectrum turn into ‘clots’ of discrete spectrum lines (Aslanyan andLidsky, 1974). Note that operator ∆2 appears in the theory without the factor i incontrast with the Orr-Sommerfeld equation.

4.2.3. The effects of stratification

We are interested in the propagation of the initial perturbation (4.16) in an exponen-tially stratified inviscid flow. Therefore we study the non-stationary Taylor-Goldsteinequation with r.h.s. and conditions (4.20), (4.21):

(i∂t − U)2(∂2z − 1)φ− i(i∂t − U)U ′′φ + Iφ = −2(i∂t − U)δ(z)δ(t).

Here I = (N/U ′)2 stands for the Richardson number and N stands for the Brunt-Vaisala frequency. Now we concentrate on the case of the linear velocity profile:

U ′′ ≡ 0, U ≡ z.

Calculating the Fourier transform in t we easily obtain for φ the stationary Taylor-Goldstein equation

(ω − z)2(∂2z − 1)φ + Iφ = −2i(ω − z)δ(z). (4.44)

We write down solutions of homogeneous equation when z → ∞ and z → −∞,

respectively:φε(±(z − ω))

here φε(ξ) = (2ξ/π)1/2K1/2−ε(ξ), ε = 1/2 −√

1/4− I (ε ≈ I + O(I2) when I ¿ 1),Kν denotes the Mcdonald function. Their Wronskian W = 2. Now the solution ofthe problem (4.44) takes the form

φ =

(i/ω)φε(+ω)φε(z − ω), z < 0

(i/ω)φε(−ω)φε(ω − z), z > 0.

148 CHAPTER 4

This function has two branch points ω = 0 and ω = z. Using asymptote of theMcdonald function we specify the asymptote φ in a neighborhood of the branchpoints. When ω ≈ 0

φ =

21−εφε(z)fo(ω), z < 0

(−2)1−εφε(−z)fo(ω), z > 0.

where f0(ω) = ωε−1(1 + O(ω2) + O(ω−ε)) and when ω ≈ z

φ =

(−2)−εφε(z)fz(ω − z), z < 0

2−εφε(−z)fz(ω − z), z > 0.

where fz(ω) = ωε(1 + O(ω2) + O(ω1−ε)). For small z the branch points are close toeach other:

φ ≈ πi sign z

2 cos2 επ

e−πiε21−2ε

Γ2(1/2 + ε)ωε−1(ω − z)ε + · · · (4.45)

Performing the inverse Fourier transform and integrating along a neighbourhoodof the branch points we obtain

φ = φ0 + φz

φ0 =

√π

2

i21−εφε(|z|) sign z

cos επ Γ(1/2 + ε) Γ(1− ε)t−ε + O(t−ε−2) + O(t−1+ε)

φz =

√π

2

i21−εφε(|z|) sign z

cos επ Γ(1/2 + ε) Γ(−ε)e−iztt−1−ε + O(t−ε−3) + O(t−2+ε).

When z → 0 formula (4.45) implies

φ ≈ Γ(ε)Γ(1 + ε)

Γ(1 + 2ε)Γ2(1/2 + ε)

πi21−2εe−πiεsign z

2 cos2 επz2εΦ(ε, 1 + 2ε,−izt)

where Φ is the confluent hupergeometrical function of the first type:

Φ(a, c, x) =Γ(c)

Γ(a)Γ(c− a)

1∫

0

exuua−1(1− u)c−a−1du.

One can easily check using the formulae above that for fixed t and I → ∞ thesolution φ0 tends to exp(− |z|), i.e. it turns into a solitary Case wave in the Couetteflow of nonstratified fluid. Taking into account a small (but finite) stratification leadsto the decreasing of perturbation as t−ε when t → ∞. Moreover, the character ofsingularity at the critical layer will change: for positive I the perturbation of thevertical velocity tends to zero as z2ε when z → 0.


Fig. 49. Profiles of vertical velocity of a continuous spectrum wave: (a) in a flow with asmall curvature of velocity profile, (b) in a flow of a stratified fluid.

Note that the decrease of perturbation as t−ε when t →∞ can be treated as thedispersion of the packet formed by continuous spectrum waves in the stratified flow.It has no connection with radiation of energy and its transport by internal waves.

In fact, continuous spectrum waves with fixed phase velocity c (γzc = c) take theform (see Fig. 49)

φ±CS = φ[±k(z − zc)] exp[ik(x− ct)]

where φ(z) =√

2z/πK1/2−ε(z)θ(z) are generalized solutions of the Taylor-Goldsteinequation

z(φ′′ − φ)− ε(1− ε)z−1φ = 0. (4.46)

(for convenience we use dimensionless variables here). Checking it up one should usethe following relation

φ′′(z) ∼ δ(ε)(z)+ regular terms, here δ(ε) ∈ D′ is a distribution described below.The function φ(z) fits equation ( 4.46) in the conventional sense outside the criticallayer z = 0 matching conditions in this layer.

We decompose the initial perturbation φ0 = exp(− |z|) with respect to the func-tion φ(z)

exp(− |z|) =

∞∫

0

φ(h)g+(z − h)dh +

0∫

∞φ(−h)g−(z + h)dh

The weight functions g+ and g− are to be calculated below. The Fourier transformof (4.39) with respect to z yields

[(κ− i)−1 − (κ + i)−1]/2i = φ(κ)g+(κ)− φ(−κ)g−(−κ).

Assuming the symmetry of weight functions g+(κ) = g−(κ) we obtain

g+(κ) = [2(1 + iκ)φ(κ)]−1.

150 CHAPTER 4

Now we express the spectrum of solution in terms of the hypergeometric function

F (a, b; c; z) =Γ(c)

Γ(b)Γ(c− b)

1∫

0

tb−1(1− t)c−b−1(1− tz)−adt.

The results above lead to the following relation

φ(κ) = 21−ε(1 + iκ)ε−2 ε(1− ε)π

sin επF [2− ε, 1− ε; 2;

iκ− 1

iκ + 1].

that implies for g(z) the integral representation

g+(z) =1

8π

2ε sin επ

επ(1− ε)

∞∫

−∞

(1 + iκ)1−εeiκzdκ

F [2− ε, 1− ε; 2; iκ−1iκ+1

]. (4.47)

Calculating asymptotes g(z) when z →∞ and z → 0, we obtain

g+(z →∞) ∼ εz−2+ε, g+(z → 0) ∼ ε−1z−εθ(z).

The behaviour of the wave packet depends critically on the character of the weightfunction singularity. In particular, the packet of Case’s waves decays as t−2 in thecase of a continuous function g+(z) (see Sec. 4.1), it decays as t−1 in the case of astep function g+(z) and tends to a stationary wave if g+ contains δ-singularity (seeSec. 4.1). In the case under consideration the singularity is higher than θ-function butlower than δ-function. As a result, the packet in the Couette flow with distributionof amplitudes of the form (4.47) decays as t−ε, 0 < ε < 1.

For convenience we list here some properties of distributions δ(ε)(z) ∈ D′ definedby the relation

δ(z)(ε) ∗ φ(z) ≡∞∫

−∞δ(ε)(h)φ(z − h) dh =

=1

2π

∞+i0∫

−∞+i0

(iκ)εeiκz( ∞∫

−∞φ(h)e−iκhdh

)dκ

where φ(z) ∈ D, (iκ)ε > 0 when iκ > 0. This definition directly implies the followingproperties

a. δ0(z) = δ(z) if ε = 0 (usual δ-function);b. δn(z) = dn

z δ(z) when n = 1, 2, 3, . . .(n-th derivative of δ-function);c. δ(−n)(z) =

∫ z−∞ · · ·

∫ zn−1−∞ δ(zn)dzn · · · dz1, (δ(−1)(z) = θ(z)) (n-th primitive of δ-

function);d. δ(ε)(z) = z−1−εθ(z)/Γ(−ε) when ε < 0 and ε 6= n (δ(ε)(z)φ(z) ∈ L1, φ(z) ∈ D);


e. δ(ε)(z) = z−1−ε/Γ(−ε) when z > 0 and ε 6= ±n (non-local distribution when ε isa fraction);

f. δ(ε)(z) = (iκ)ε (Fourier transform of the distribution).

We come across the distribution δ(ε)(z) in the following examples:

1. The differentiation of a function with a power singularity. Let

f(z) = z−εθ(z)e−αz ∈ L1

where 0 < ε < 1, α > 0 (the exponential function is introduced for the sake ofconvergency at infinity). Then

f ′(z) = Γ(1− ε)δ(ε)(z)e−αz − αz−εθ(z)e−αz ∈ D′

2. A wave field of the 2D acoustical dipole (ν = 2) could be described by distribu-tion δ(1/2):

E2(r, t) =θ(ct− r)√c2t2 − r2

,∂

∂xE2(r, t) = cos ϕ

∂

∂rE2(r, t) =

= cos ϕ

√π

r + ctδ(1/2)(r − ct)− 1

2

1

r + ct

θ(ct− r)√c2t2 − r2

∈ D′

x = r cos ϕ.

4.2.4. The non-linear effects

Speaking intuitively, the vorticity imposed in a fluid is ‘frozen’ and transported to-gether with fluid particles. This idea is formalized by the well-known Thomson the-orem on conservation of the velocuty circulation (see (Lamb, 1895)). The externalforce fz of the type (4.16) creates a vortex sheet at a layer z = 0, drifted by the mainflow and disturbed by its own velocity field. In flows with constant vorticity (theCouette flow or a homogeneous flow), the nonlinear evolution of such vortex sheetcan be calculated by the contour dynamics method, as well as by that of an approx-imation of the sheet by a chain of point-vortices (the so-called Helmholtz vortices).Both of these methods demonstrate that eventually the initially plane sheet turnsinto wave-like movements. Its shape then becomes similar to the Riemann wave. Fi-nally, it should rolled up to form the chain of localized vortices (see Fig. 50). After afinite time, the curvature of the sheet tends to infinity at some selected points, therebeing the centers of formation of the localized vortices. In the absence of the shear,these vortices have the same intensity and the alternating directions of rotations. Inthe Couette flow that coinciding with the vorticity of the main flow prevails. Thus,

152 CHAPTER 4

Fig. 50. Deformation of CS-mode before rolling-up (top) and after rolling-up (bottom).

a solitary CS-mode being considered for enough long time is a strong non-linearstructure.

In analogy to the viscous case, we can evaluate the time interval where the gradu-ally occurring nonlinear distortion of the CS-modes can be neglected. We can roughlyestimate this time interval through the parameters of the mode.

Denote by ε an amplitude of the velocity field (break of the tangential componentof velocity) at the critical layer:

u0 =ε

2exp(−k |z − zc|)sign(z − zc), φ0 =

ε

2kexp(−k |z − zc|)

w0 =iε

2exp(−k |z − zc|), η0 = −εδ(z − zc).

The vertical deformation of the sheet ζ would be of the order of ζ ≈ |wt| ∼ εt attime t. When this layer fails to keep its flatness, the velocity shear would producehorizontal distortions of the order of ∆x ∼ ∫ t

0 ζ(t)U ′dt ∼ εU ′t2. Non-linear effects canbe neglected while these distortions are small when compared with the wavelengthof the perturbation λ = 2π/k, i.e.

t ¿ min(tz, tx), tz ≈ (kε)−1, tx ≈ (kεU ′)−1/2. (4.48)

Comparing the times tz and tx one can easily check that the change of the velocityon the characteristic scale of perturbation is of the order of ∆U ' U ′/k. In the case


of a large shear (ε ¿ ∆U) horizontal perturbations are essential, in the opposite caseof a small shear and a short-wave perturbation (ε À ∆U) the vertical distortions ofthe vortex sheet become more important before the horizontal ones. When conditions(4.48) are fulfilled, the vorticity layer stands flat and the linear theory may be applied.

The estimations (4.48) are in reasonable accordance with the numerical compu-tations of vortex sheet evolutions.

Here we add the following note. Although the nonlinear evolution of a solitaryCS-mode is rather intuitive, that of a smooth wavepacket is not so transparent. Thereare no effective numerical or analytical methods to calculate it for a large period oftime. A thin wavepacket (distributed vortex sheet with the typical width d) kd ¿ 1can be treated as a solitary CS-mode until its radius of curvature exceeds the widthd. However, a detailed hydrodynamic picture after these events is still unknown.

The problem of nonlinear evolution of 2D-perturbations (the CS-modes wave-packets) is of primary interest in hydrodynamics. The dynamics of a finite sizevortex in shear flows is a particular case of this problem which has attracted variousnumerical approaches. Note that while the evolution of a solitary CS-mode of a finiteamplitude and that of a wide packet with infinitesimally small amplitude (see Sec. 4.1)are two limiting cases of this problem, the evolution of a finite vortex presents anintermediate and the most unwieldy case.

Our experience of numerical computations in these problems leads to the followinghypothesis. It is probable that a very thin wave-packet kd ¿ 1, ε/d À γ will roll-up to create a chain of localized vortices (may be they exist for a limited time). Avery wide packet kd ∼> 1 with a small amplitude ε/d ¿ γ should vanish becauseof the CS-mode disphasing before the nonlinear interaction. In the intermediatecase, the total velocity field will split into localized perturbations with closed streamlines. In other places, the velocity field will eventually vanish in accordance withthe linear theory. This picture is somewhat similar to one-dimensional situationsdescribed by the KdV-equation and other equations for solitons (e.g. (Leibovich andSeebass, 1974)). Initial perturbations split into localized structures, i.e. solitons, withother parts of the field eventually disappearing due to dispersion. We emphasis herethat this is only a hypothesis, present state-of-the-art technology is not sufficient toconfirm or reject it.

4.2.5. The 3D effects

Now we consider the 3D-problem in the simplest case when the force of the type (4.16)is acting on the Couette flow but the wave-vector k deviates from the direction ofthe main flow:

fz = δ(t)δ(z) exp(ikx cos ϕ + iky sin ϕ).

Here ϕ is an angle between the k-direction and the x-axis. Solving the linearized 3D

154 CHAPTER 4

system of the Eiler equations we obtain the following expression for the field velocityevolution

w = exp(−k |z|), u′ = exp(−k |z|)sign(z)

v′ = exp(−k |z|)1− exp(ikzγt cos ϕ)

kztan ϕ.

(4.49)

Here u′ = u cos ϕ+v sin ϕ is the velocity component in the k-direction, v′ = −u sin ϕ+v cos ϕ is normal both to k and the z-axis component of velocity. Note that the veloc-ity field in the plane k, z is just the same as in the 2D-case (cf. Fig. 46). However,the third velocity component appears which tends to infinity in the vicinity of thecritical layer as t → ∞. The energy of perturbation (4.49) also grows indefinitelywith the time

E =ρ

2

+∞∫

−∞〈|u|2 + |v|2 + |w|2〉dz =

=ρ

8k

1 + 2 tan2 ϕ

[t arctan(

t

2)− log

(1 + (

t

2)2

)]

where t = tγ cos ϕ is a dimensionless time, 〈·〉 denotes averaging over the spacialperiod.

This is the second example of the growing perturbation in the Couette flow.Actually, the energy of such growth is taken from the main flow. Hence, its profileshould slightly change. Analysis shows that we can represent the perturbation (4.49)as an evolution of a wave-packet formed by the singular CS-modes of two types. Thefirst type is a generalization of the 2D CS-mode to 3D flows: this mode tends to 2Dmode as ϕ → 0:

wCS = exp(−k |z|), u′CS = exp(−k |z|)sign(z)

v′CS = exp(−k |z|)P 1

zsin ϕ.

(4.50)

The second type of continuous spectrum modes has a clear physical meaning ofan infinitesimally thin jet normal to the k-wavenumber

w2 = 0, u′2 = 0, v′2 = δ(z). (4.51)

Both waves are drifted by the main flow with the velocity of their critical layerz = 0. Thus, the phase velocity in k-direction equals to cph = U(zc)/ cos ϕ. Bothwaves possess the indefinite energy (unlike the analogous 2D waves). But energy ofa superposition of several CS-modes differs from the sum of their energies (the samesituation emerges in the 2D-case as well). As a result, the energy of a smooth wave-packet is finite at any finite time. Both waves (4.50) and (4.51 ) provide generalized


solutions of the 3D Rayleigh equation. In fact the perturbation (4.49) can be treatedas disphasing of CS-modes of two kinds, i.e. it can be described as

w =∫

(A(h)wCS(z|h) + B(h)w2(z|h))dh.

Here A(h) and B(h) are weight functions. We demonstrate this using the Fouriertransform with respect to the z-coordinate. Then we obtain the following spectra forthe perturbation (4.49):

w =k

k2 + κ2, u′ =

κ

k2 + κ2

v′ =

[arctan

(κ

k

)− arctan

(κ− kγt cos ϕ

k

)]sin ϕ

(4.52)

and two type of CS-modes:

wCS =k

k2 + κ2, u′CS =

κ

k2 + κ2v′CS = arctan

(κ

k

)sin ϕ

u′2 = 0, v′2 = 1, w2 = 0.(4.53)

Comparing (4.52) and (4.53) we obtain the spectra of weight functions A(h) andB(h) and the inverse Fourier transform provides explicit expressions

A(h) = δ(h), B(h) = P 1

htan ϕ.

Thus, perturbation (4.49) is a solitary wave-packet of both type of modes. It isa superposition of a solitary CS-mode of the first type and continuum of the secondtype modes whereas its evolution can be explained as the disphasing of the CS-modesof the second type owing to the shear of velocity.

Three-dimensional wave packets in the Couette flow of viscid fluid are consideredin (Sazonov, 1996) where the temporal growth of perturbation energy is analyzed.The typical energy dependence over time is plotted in Fig. 51 (here a is typical widthof the packet in z-direction; the case a = 0 corresponds to generation of perturbationby δ(z)-shaped force, in analogy to the case considered in this Section for inviscidfluid). Remark that energy of initial perturbation may increase several tens andhundreds times over, but eventually it should vanish. Similar plots can be found inmany papers concerning numerical computation of flow transition to turbulence (e.g.(O’Sullivan and Breuer, 1994), (Gustavsson, 1991), (Bergstrom, 1992), (Bergstrom,1993) and (Butler and Farrell, 1992))

156 CHAPTER 4

Fig. 51. Time dependence of energy for 3D wave-packets in the Couette flow: width ofa packet ka is indicated near the corresponding curve. Here the angle between flow andwave: ϕ = π/4 and Reynolds number Re = U ′/(k2ν) = 1000.

4.3. Quasi-Eigen (QE) Modes in Ideal Fluid Flows

4.3.1. Rayleigh’s theorem and the problem of decaying eigenmodes

existence in a flow without points of inflection of velocity profile

In Sec. 4.1 and 4.2 we treated some evolutionary problems in flows with linear ornearly linear profiles. The waves of continuous spectrum only can exist in theseproblems. Here we select the problem of pulse propagation in the flows with piece-wise linear profiles as a model. In this case neutral modes of discrete spectrum shouldbe incorporated as well. For a convex profile (U ′′ > 0) being a small perturbationof a piece-wise linear one both neutral or decaying mode of discrete spectrum fail toexist. Nevertheless, some specific wave packets of continuous spectrum waves wouldbe studied carefully. In some applications (e.g. in studying of non-linear interactions)they behave like weakly decaying discrete spectrum modes. Therefore, the problemunder study is closely connected with the famous Rayleigh theorem which statesthat in the flows without points of inflection growing (or decaying) modes of discretespectrum cannot exist.

The proof of Rayleigh’s theorem is quite simple. Actually, one multiplies theRayleigh equation written in the form

(∂2z − k2)w + U ′′(c− U)−1w = 0

by the conjugate solution w∗ and integrates from −H to H with respect to z. Finally,separate the imaginary part of the equation at hand

H∫

−H

ciU′′(z) |w(z)|2

(cr − U(z))2 + c2i

dz = 0. (4.54)


This equality implies that U ′′(z) has to change the sign in the interval (−H, H)(because of the positivity of all other factors in the integrand).

Clearly, the sign of ci is not relevant. Therefore both increasing and decreasingmodes cannot exist in flows without points of inflection. Due to the integral formof the equality (4.54) the Rayleigh theorem is applicable to the non-analytic profiles(in particular, to profiles with breaks where U ′′ ≈ ∆U ′δ(z)). The neutral modes ofdiscrete spectrum can exist in the case of zero curvature of the velocity profile intheir critical layers only. This fact can be easily derived from (4.54) in the limitingcase ci → 0. On the other hand this relation does not imply any restriction on theneutral modes of continuous spectrum because they cannot be treated as the limitsof decreasing or increasing waves.

In spite of the simplicity of the Rayleigh theorem, one can point out severalobservations that seem to contradict it. Firstly, the numerical studies of eigenmodesin boundary layer of viscous fluid having profiles without points of inflection revealan existence of modes with the following properties: their complex phase velocitydoes not depend on viscosity for large Reynolds numbers (see (Landahl, 1967)).Secondly, the neutral modes fail to keep the structural stability with respect to smalldeformations of velocity profile U(z). More precisely, consider a flow admitting aneutral mode with U ′′ ≥ 0 everywhere but for its critical layer where U ′′ ≡ 0. Thenconsider a small deformations of the profile in a neighbourhood of the critical layer,such that U ′′ < 0 in its smaller neighbourhood. Hence, at least two points of inflectionhave to appear. Using the small perturbations technique one can show that this modehas to turn into an increasing mode. In the opposite case, when U ′′ > 0 everywhereas a result of the small deformation, this mode has to disappear due to the Rayleightheorem. This result seems to contradict common sense. One can hardly imaginean oscillator with the following properties: neutral oscillations when friction is zero;small negative friction leading to increasing eigenmodes and for any small positivefriction, decreasing oscillations failing to exist.

Keeping in mind that the problems of wave propagation in acoustical and electro-magnetic stratified waveguides lead to the equation of the form φ′′+f(z, ω)φ = 0 withhomogeneous boundary conditions, we note that eigenmodes respond to the resid-uals in the poles of solutions (see (Borovikov and Molotkov, 1988), (Brekhovskikhand Lysanov, 1991)). The Rayleigh equation has the same form if we put f ≡U ′′(z)/(c−U(z))−k2. Therefore, one can suppose that a pole will disappear under asmall perturbation of profile (see (Timofeev, 1970)). However, we shall demonstratethat the disappearance of an eigenmode in this problem does not necessarily respondto the disappearance of a pole.

All of these contradictions can be resolved if one takes into account the singularityof the stationary Rayleigh equation (4.4), and the consistent analysis of the problemcan be obtained in the framework of the non-stationary Rayleigh equation (4.3).Therefore we shall study the evolutionary problems in shear flows.

158 CHAPTER 4

Fig. 52. A family of model profiles: (a) σ > 0, (b) σ = 0, (c) σ < 0.

4.3.2. Evolutionary problems

The simplest flow possessing a discrete spectrum mode (moreover, it is unique in thiscase) is the piece-wise linear profile with one point of break. Let us consider a smallperturbation of this profile having a small curvature at the critical layer (see Fig. 52)using the small perturbation technique.

Consider an unbounded in the z-direction velocity profile U(z) = 0 when z < 0and a small perturbation of linear profile when z > 0 without points of inflection,which increases monotonically. Using dimensionless coordinates z = kz, t = U ′(+0)t,we write down

z(U) = kU(z/k)/U ′(+0) =

= θ(z)z[1 + σz/2 + γ2(σγ)2 + · · ·

] (4.55)

where σ characterizes the curvature of velocity profile above the break:

U ′′(z) = δ(z) + θ(z)[σ + O(zσ2)]

If σ > 0 the curvature is positive everywhere, if σ < 0 it changes the sign. In thecase of the piece-wise linear profile (σ = 0) there is unique mode of discrete spectrum.It is a neutral (ωi = 0) mode of the form

w(z, t) = exp(− |z| − iω0t), (4.56)

Its phase velocity ω0 = 1/2 (for dimensionless variables it coincides with fre-quency). Unlike the continuous spectrum waves the perturbation (4.56) has no sin-gularity at the critical layer zc = 1/2 + O(σ)(U(zc) = ω0). Using the successiveapproximation technique one can check that this mode becomes an exponentially


increasing one when σ < 0: ωi = O(σ) > 0 (see below). This mode would disappearfor an infinitesimally small σ > 0.

Let us concentrate on the precise formulation now: we seek the solution of thenon-stationary Rayleigh equation:

(i∂t − U)(∂2z − 1)w + U ′′w = −2δ(t)δ(z) (4.57)

w ≡ 0 when t < 0, w → 0 when |z| → ∞ (4.58)

where U is defined by (4.55). We note that the mode of discrete spectrum of theform (4.56) (multiplied by θ(t)) is the solution of the problem in the particular caseσ = 0.

It is convenient to solve the problem in the domains z > 0 and z < 0 separatelyusing the glueing conditions at the boundary z = 0: the continuity of the verticalvelocity and the pressure jump at the moment t = 0:

[w]0 = 0, (∂t + iU)[w′]0 + [U ′]0w = δ(t), (4.59)

here [f(x)]h is the jump of a function f(z) at the layer z = h. Using the one-sidedFourier transform with respect to t we obtain the stationary Rayleigh equation forthe spectrum in the domain z > 0. The function U(z) in this equation can be treatedagain as a perturbed linear profile. We choose its particular solution which decreaseswhen z → +∞:

w1(ω, z) = e−z(1 + (σ/2)(e−2δEi(2δ)− log(−2δ)) + O(σ2))

δ = ω − z, 0 < z ¿ σ−1.

On the other hand the solution decreasing when z → −∞ takes the form w2 = ez.Using the glueing conditions in the spectral form

[w]0 = 0, ω[∂zw]0 + w(0) = −2

and calculating the inverse Fourier transform we obtain the following integral repre-sentation for the velocity field in the upper half-space:

w(z, t) =1

2π

∫

Γω

w1(z, ω)

w1(0, ω)

e−iωt

F (ω)dω, z > 0 (4.60)

here the contour Γω passes in the upper half-plane Im ω > 0 is parallel to the axisRe ω and

F (ω) = i[(1/2) + (ω/2)∂zw1(0, ω)/w1(0, ω)− ∂zw2(0, ω)/w2(0, ω)] ≈≈ i[(1/2)− ω(1 + (σ/2)e−2ωEi(2ω))] + O(σ2).

The integrand in (4.60) has the following singularities:

160 CHAPTER 4

Fig. 53. Analysis of a decaying quasi-eigen mode in a flow with velocity profile withoutpoints of inflection.

a) a logarithmic branch point ω = 0 is a singularity of w1(0, ω) and ∂zw1(0, ω)b) a logarithmic branch point ω = z is a singularity of w1(z, ω)c) a pole ωp = ωr + iωi is a zero of function F (ω).

It is convenient to put the cuts vertically downwards from the branch points andfix the branches of multi-valued function using the condition: w(z, ω) tends to zeroas |ω| → ∞, Im ω > 0. This condition implies the causality principle (i.e. integral(4.60) vanishes when t < 0).

It is easy to calculate ωp by successive approximations technique:

ωp = 1/2 + (σ/4)e−1Ei(1) + O(σ2) (4.61)

taking into account the proper choice of the branch of the function Ei we obtain

ωr = 1/2− (σ/4)e−1i(1) + O(σ2),

ωi = −(σπ/4)e−1 + O(σ2)

(here i(ω) = Re Ei(ω)).When t > 0 one deforms the contour Γω in the lower half-plane downwards to

the singular points of the integrand. It splits into three parts: the contour Γp passesaround the pole and contours Γ0 and Γz pass along cuts and around branch pointsω = 0 and ω = z, respectively (see Fig. 53). We denote the respective integrals bywp, w0 and wz. In the domain 0 < z ¿| σ |−1 the integral wp can be treated as adecreasing (if σ > 0) or increasing (if σ < 0) harmonical perturbation:

wp = exp(−z − iωpt)[1 + (σ/2)e2z−1Ei(1− 2z)− log(2z − 1)−(e2z−1 − 1)πiθ(z − 1/2)+ O(σ2)].

(4.62)


Thus, the velocity field wp with σ < 0 satisfies the Rayleigh equation (4.57) andauxiliary condition (4.58). Hence, it can be treated as an increasing eigen mode. Thefield wp can be generated by means of vertical force fz ∼ δ(t)∂zwp. In the case σ > 0the pole ωp lies in the lower half-plane. Therefore the cut associated with the movingbranch point ω = z intersects the pole when z = zc = 1/2. As a result, it appears atthe other side of the cut. Thus, the field wp (i.e. residual in the pole ωp) has a breakin the term of the order of σ and discontinuity in the term of the order of σ2 at thecritical layer:

[∂zwp]zc= σπi exp(−zc − iωpt)

[wp]zc= −σπiωi exp(−zc − iωpt) = (σ2π2i/4e) exp(−zc − iωpt).

(4.63)

In the case σ > 0 the field wp is not an eigenmode in the proper sense because itfails to satisfy the Rayleigh equation at the critical layer. It can be generated by apermanent external force only.

Unexpectedly the total field turns to be an analytical function: non-analyticity ofthe field wp at the critical layer when σ > 0 is compensated by the same singularityof the field wz. Both perturbations wz, w0 are of the order of σ when σ → 0, theydecrease in inverse proportion for large time (t À 1):

w0 ≈ −2σt−2 exp(− |z|)(1 + θ(z))

wz ≈

2σt−2e−zθ(z)(1− 2z)e−izt, t |z − zc| À 1

σ[it−1e−izt + (zc − z) exp(−iωpτ)Ei(it(zc − z))], t |z − zc| ∼< 1.

(4.64)

In spite of the fact that the perturbations wz and w0 are negligible when comparedwith wp on the small time scale (just after the initial excitation (4.16)), they decreaseslower when σ > 0. Hence they become of the order of wp at the time of the order oftσ ∼ σ−1 (using dimensional variables, of the order of tσ ∼ k/U ′′

c ). When t À τσ theperturbations wz + w0 prevails.

Thus, in the case σ > 0 and t ¿ tσ the total perturbation can be approximatedby a time-harmonic wave with time-dependence exp(−iωpt), when t À tσ it decreasesas a degree of t. When σ > 0 the term wz cannot be eliminated: if it is dropped theterm wp satisfying the Rayleigh equation and one boundary condition (e.g. z → −∞)would fail to satisfy the second boundary condition (when z → +∞). One can easilycheck this fact using (4.62) and the following observation: an analytical continuationwp from lower half-plane (z < zc) to the upper one (z > zc) differs from wp bythe addend (σ/2) exp(z − 1− iωpt)2πi emerging from the analytical continuation ofEi(1− 2z).

Obviously, the partition of the total field into harmonic part wp and non-harmonicaddend wz is conventional, e.g. one can vary the angle of the cut that begins at thebranch point ω = U(z). This variation moves the point of intersection of the pole

162 CHAPTER 4

trajectory and the cut from z = ωp. However, in a sense the vertical cut is an optimalone: the smaller the angle between the cut and the axis Re ω, the greater t would beto use the asymptote (4.64).

Rigorously speaking, the harmonic part wp (i.e. the residual at the pole ωp) cannotbe treated as an eigenmode. Nevertheless, it prevails in a neighbourhood of thecritical layer when the curvature of the velocity profile is small enough. Presumably,in some cases (e.g. in nonlinear interactions (Goldstein, 1961), (Terent’ev, 1981),(Terent’ev, 1984), or in excitation by an external force (Voronovich and Rybak, 1978),(Ostrovsky et al., 1986)) it would reveal some properties of an eigenmode. Thereforethe harmonic part wp will be referred to as a ‘quasi-eigen mode’, and its existencedoes not contradict the Rayleigh theorem.

The results above were obtained for the velocity field in the model profile (4.55).We shall not approach here the more sophisticated problem of transfer to the realshear flows: boundary layers, jets, tracks, etc. We believe that the qualitative answershould be the same in spite of complexity of precise calculation of the poles. Inreality, for any real velocity profile U(z) the spectrum of the perturbation contains amoving branch point c = U(z), which passes along the axis Re c with variation of z.The singularity at this point does not depend on the velocity profile when U ′′ 6= 0

U ′′(z) (c− U(z))

(U ′(z))2log

[k(c− U(z))

U ′(z)

].

The cut starting vertically downwards from point c = U(z) can intersect somepoles of the spectrum in the lower half-plane c for specific values of z. Hence, theresiduals at the poles would be discontinuous for these specific values of z and wouldnot be eigenvalues in the proper sense (Landahl, 1967). Thus, the Rayleigh theoremwhich states the non-existence of eigenmodes cannot be extended for non-existenceof poles. If the curvature in a neighbourhood of the critical layer is small enough, thepole lies near the axis Re c. Hence, the residual wp behaves like a weakly decreasingeigenmode in some resonance effects.

In order for numerical computation of these eigenmodes using an appropriateboundary values problem one should consider an analytic continuation of the profileU(z) to complex z-plane in a neighbourhood of the critical layer. One chooses thebranch of U(z) after the continuation along the round-about the critical level zc:U(zc) = cp using the Landau rule, see (Lifshitz and Pitaevsky, 1979).

In particular, for any profile U(z) changing between Umin and Umax (Umin =−∞, Umax = ∞ for profile of the form (4.55)) the branch points in the c-plane c = Umin

and c = Umax emerge. Usually one use the cut joining these points (see Fig. 54) andspecifies the so-called physical branch of the function U(z) on the c-plane. If for agiven k the profile admits an unstable mode, one can distinguish a pole of the physi-cal branch in the upper half-plane (lying in the so-called Howard circle (LeBlone andMysak, 1978)) (see Fig. 54), and corresponding complex-conjugate pole c∗p (lying in


Fig. 54. Contours of integration with cut located on a physical sheet.

Fig. 55. Contours of integration for cut deformed into the lower half-plane (A part ofnon-physical sheet is dashed).

the lower half-plane of the physical sheet, and describing a decreasing mode). A sta-ble velocity profile cannot have any poles of the physical branch due to the Rayleightheorem.

We deform the initial contour of integration passing in the upper half-plane intosmall circles around the poles (of any pole exists) and a contour around the cut (seeFig. 55). It gives us a way to split the total field into a sum of eigenmodes (if anyof them exists) and integral of continuous spectrum waves (speaking intuitively, onecan associate with the cut a ‘distributed pole’, see Sec. 4.2.1). Generally, one cannotcalculate the integral along the cut explicitly. Hence one concentrates on computationof its asymptotic behaviour when t →∞ using the vertically downwards cuts (in thedirection of the steepest descent of the integrand). Using this approach one shouldstudy a part of the ‘non-physical’ sheet as well and calculate residuals at its poles.On the other hand one should omit the poles of the type c∗p1, they are conjugated to

164 CHAPTER 4

Fig. 56. Cuts are deformed vertically downwards (steepest descent paths of integration).

the poles lying in the upper half-plane. Besides, the new branch point covered bythe original cut should appear. The cut started in the branch point would intersectthe poles in the lower half-plane. Due to this fact the residuals at the poles of the‘non-physical’ branch cannot be treated as the eigenmodes in the proper sense (seeFig. 56). Thus, the disappearance of the modes is not related with the movement ofsome poles to the ‘non-physical’ sheet. Instead, the correct explanation is related tothe moving branch point.

The phenomenon of quasi-eigen modes is of great importance for nonlinear prob-lems because these modes can resonate like the usual modes. As a rule, the in-vestigation of many nonlinear phenomena: i.e. three-waves interactions, explosiveinstability, etc., is based on the preliminary solution of the respective linear problemand the analysis of dispersion curves (see, e.g. (Zakharov, 1974), (Craik and Adams,1979), (Craik, 1985), (Voronovich and Rybak, 1978)), and (Romanova, 1994). Dueto analytical obstacles in the investigation of flows with arbitrary smooth profilesand numerical difficulties in the analysis of quasi-eigen modes, one approximates thereal profiles by piece-wise linear profiles for which it is easy to obtain the dispersivecurves. But dispersive curves of real flows (including those for the quasi-eigen modes)can be drastically different from that for piece-wise linear flows.

Consider, e.g. the simplest mixing-layer described by the hyperbolic tangentU(z) = U∞ tanh(z/H) (Fig. 57) and draw the ‘spacial’ dispersion curves (Re c(k),Im c(k)) for smooth and piece-wise linear profiles (Fig. 58). For a small k the disper-sion curves for increasing modes display similar behaviour. But in the vicinity of thecritical value of the wavenumber their behaviour crucially diverges. Two branch ofdispersion curves (for increasing and decreasing modes) merge at kcr. When k > kcr

two neural curves appear. They have infinitely large derivatives at the merging point(and infinitely high group velocities). For a real profile, we should consider only one


Fig. 57. Velocity profile shaped as ‘tanh’ (thick curve). This model is popular fordescription of mixing layers. U∞ and H are parameters of the flow. Piece-wise linearprofile with the same parameters (thin line).

Fig. 58. ‘Spatial’ dispersion curves for smooth (thick curve) and piece-wise linear (thincurve) profiles.

branch for this profile (the second pole never contributes to an initial value problem).When k = kcr2 the curve intersects the plane Re c = 0 and further on it describes aquasi-eigen mode. It intersects the plane at an angle rather than orthogonal as inthe case of a piece-wise linear profile. Thus, in a real profile, effects such as neutralmodes, infinitely high group velocity, parts of dispersion curves of negative energy,etc., disappear. Generally, we might expect several dispersive curves, another be-haviour in the vicinity of the critical wavenumber and slightly decaying waves (inaccordance to the so-called Landau attenuation (Lifshitz and Pitaevsky, 1979)) in-stead of neutral waves. Hence, it is necessary to verify many nonlinear effects forstructural stability in the smoothing of real velocity and density profiles.

We mention the following examples. If the vortex is described by a piece-wise

166 CHAPTER 4

profile the hydrodynamical neutral mode can propagate around it (neutral DS-mode).If one incorporates a small compressibility of medium, one obtains acoustic radiationinstability which leads to a vortex collapse (e.g. (Kop’yev and Leont’yev, 1988),(Gryanik, 1988)). However, this phenomena is structurally unstable in the smoothingof the velocity profile of the vortex. In the smoothed vortex, we have a decaying quasi-eigen mode (instead of a neutral one) which decays in the real vortex much fasterthen the development of acoustic instability (Danilov, 1989).

Another application is described by Shrira (1989) where nonlinear interaction ofQE-modes with the surface sea waves leads to a change in the spatial spectrum of thelatter. For more details about the so-called subsurface QE-mode, see in Sec. 4.4.5.

4.4. The Green’s Function of the Rayleigh Equation for a Flow with aDiscrete Spectrum Mode

Some interesting physical phenomena can be distinguished in the study of evolution-ary problems when the initial vorticity is concentrated not at the layer of the velocitybreak (as in Sec. 4.3) but at an arbitrary layer. Consider a flow with a piece-wiselinear profile (or its small perturbation) admitting the modes both of discrete andcontinuous spectra. The resonance interaction of these modes is especially effectivein the case when a source of perturbation is located not far from the critical layerof eigenmode or quasi-eigen mode. It leads to an unbounded growth of the initialperturbation as a power of t. This effect is known as an algebraic instability, it canbe studied in boundary layers as well as in flows of stratified fluids (Chimonas, 1979),(Landahl, 1980).

4.4.1. Piece-wise linear profile

Let the source of the type (4.16) located at the layer z = h > 0 acts on the flow withthe velocity profile

U(z) = γzθ(z). (4.65)

The response on this perturbation can be described by the Green’s functionG(t, z | h, k), i.e. by the solution of (4.19) with auxiliary conditions (4.20) and (4.21).Using dimensionless coordinates z = kz, t = γt we write its Fourier transform withrespect to t in the form

G(ω, z | h) = i(ω − h)−1

[exp(−|z − h|+ exp(−h− | z |)

ω/ω0 − 1

]

where ω0 = 1/2 is the dimensionless frequency of the discrete spectrum mode, h = kh.Hereafter we shall omit the sign ‘bar’ for dimensionless variables. The residual at thepole ω = h gives a continuous spectrum wave GCS with the critical layer z = h, on


the other hand the residual at the pole ω = ω0 can be treated as a discrete spectrumwave GDS with the critical layer z = ω0:

G = GCS + GDS (4.66)

GCS =

[exp(−| z − h |) +

exp(−h− | z |)h/ω0 − 1

]exp(−iht)

GDS = −exp(−h− | z |)h/ω0 − 1

exp(−iω0t).

Phase velocities of these waves expressed in dimensional coordinates are

cCS = U(h), cDS = γ/2k.

One can easily check that the amplitude of the perturbation increases when thedistance between the critical layer of the discrete mode zc = (2k)−1 and the layerof external force decreases because the poles of the Fourier transform G(ω) nearlycoincide. When h = zc they coincide exactly and the pole of the second order appears.It contributes by a perturbation increasing linearly with time:

w = [−itω0 exp(−h− | z |) + exp(−| z − h |)] exp(−iω0t).

Thus, in the flow (4.65) an exponentially increasing mode cannot exist becauseof the constant sign of the curvature. Nevertheless, this profile admits perturbationsincreasing as a power of time.

For physical insight into this effect we calculate the vorticity η = (∂2z − k2)w =

ikΩy of the perturbation (4.65):

η = η(1) + η(2),

η(1) = −2δ(z − h) exp(−iht)θ(t)

η(2) =

−2δ(z)e−h exp(−iht)− exp(−iω0t)

h/ω0 − 1θ(t), h 6= ω0

2itω0δ(z)e−h exp(−iω0τ)θ(t), h = ω0.

(4.67)

The vorticity is located into two layers z = h and z = 0. In the first layer itsamplitude is constant in time, in the second one it vanishes when t = 0 and oscillatesafterwards. Thus, the initial excitation (4.16) at the moment t = 0 creates thevorticity level η(1) at the horizon z = h moving with the velocity U(h). In turnthe vorticity level creates hydrodynamical perturbations exponentially decreasing farfrom it:

w(1) = η(1) ∗ exp(−| z |)/(−2), u(1) = i∂zw(1) (4.68)

168 CHAPTER 4

here exp(−| z |)/(−2) stands for the solution of (∂2z − 1)w = δ(z) with the auxiliary

condition (4.20). The vertical velocity w(1) leads to the distortion of the boundary ofthe main flow. In turn this effect leads to the appearance of the secondary vorticitylayer. The layer η(2) creates its own periodical velocity field that is additive to thefield (4.68)

w(2) = η(2) ∗ exp(−| z |)/(−2), u(2) = i∂zw(2). (4.69)

The field (u(2), w(2)) propagates with the velocity cDS (or ω0 in the dimensionlesscoordinates). If the speed of the initial vorticity layer U(h) 6= ω0, the influenceof initial perturbation sometimes acts in phase and sometimes acts in anti-phasewith that of the secondary perturbation changing periodically. It is apparent if onetransforms the expression for η(2) (4.67) when h 6= ω0 to the form

η(2) = 4iδ(z)(h/ω0 − 1)−1 exp[−h− it(h + ω0)/2] sin[t(h− ω0)/2]θ(t).

Clearly, this is a pulsation with the frequency (h− ω0)/2 and amplitude (h/ω0 −1)−1e−h increasing when h → ω0. The equality h = ω0 (i.e. U(h) = cDS) leads to theresonance: the influence of the initial perturbation acts always in phase with thatof the secondary perturbation. Thus, the amplitude of the secondary perturbationalways grows. The main difference from the case of usual instability with the expo-nential growth of the perturbation is the absence of any feedback. The secondaryperturbation η(2) fails to influence the primary one η(1) (in the framework of the linearscheme). Therefore, the amplitude of η(1) does not increase.

The existence time of a solitary vorticity layer is obviously restricted in real flowsdue to different facts: one should take into account its finite width, viscosity, etc.Thus, the time of the growth of final amplitude of perturbation is restricted also.Estimations show that for an initial perturbation with amplitude w0:

w(0, k, z) = w0 exp(−k | z − zc |)

(in dimensional variables) the maximal amplitude of perturbation at the layer z = 0

wmax ∼ tmkcDSw0 (4.70)

where tm ≈ min(tν , td) and tν is the viscous time (see (4.43)), td stands for the time ofcoherence loss for Case’s waves in a vorticity layer of width 2d (kd ¿ 1) (see (4.14)).Thus, due to purely linear effects the critical perturbation in the flow (4.65) increasesin a finite number of times. The energy for this growth is taken from the energy ofthe main flow (Sazonov, 1989). In contrast to the conventional instability when anyinitial perturbation initiates the growth of amplitude restricted by non-linear effectsonly and therefore, leads to a non-reversible change of the flow, in the case of algebraicinstability the final amplitude depends on the amplitude of the initial perturbationand other conditions. Too small perturbations can be restricted in growth without


reaching the non-linear values. Therefore, they cannot change the structure of theflow. Non-linear effects come into play when the initial amplitude is greater thansome threshold. Usually this threshold is small enough and lies in the linear scale.Here we evaluate the amplitude of initial perturbation for a vorticity layer of thewidth 2d (kd ¿ 1) lying near the critical layer of a discrete spectrum mode. In orderfor the final displacement to be of the order of k−1, the initial velocity perturbationshould be greater than wth

w0 > wth ∼ U ′d.

We omit simple computation leading to this estimation.

4.4.2. A velocity profile with a small curvature

We trace here an analogy between some resonance effects in an oscillator with a smallfriction and velocity behaviour in a profile with a small curvature in a neighbourhoodof the critical layer. After that, we shall demonstrate that some resonance phenomenaare possible when no discrete spectrum mode in the proper sense can exist in the flow,only the existence of a slowly decreasing quasi-eigen mode is supposed. Denote byG(t, z | h) the solution of (4.57) with the velocity profile of the type (4.65) fitting theauxiliary conditions (4.58). (We continue to use dimensionless variables). Its Fouriertransform G satisfies the Rayleigh equation with r.h.s. We seek for its solution of theform

G =

Aexp(z), z < 0

Bw1(ω, z) + Cw2(ω, z), 0 < z < h

(C + D)w2(ω, z), z > h

where w1,2 = exp(±z)[1 ∓ (σ/2)F (∓ω ± z) + O(σ2)], are particular solutions of theRayleigh equation with slightly curved velocity profile and F (ξ) = exp(2ξ)Ei(−2ξ)−log(2ξ). Remember that we use dimensionless scales 1/k and 1/U ′(+0). We shallfind the amplitudes A,B,C, D using glueing conditions at the levels z = 0 and z = h.They include the continuity of w and pressure p at the level z = 0, continuity w andcondition (4.58) on the pressure jump at the layer z = h where the external forceacts. Cumbersome computation leads to the expressions

A = 2e−h[1 + F (h− ω)]ω(1− σπi)/∆

B = e−h[1 + F (h− ω)]/∆

C = e−h[1 + F (h− ω)][1− F (ω) + ω(F (ω) + F1(ω))]/∆

D = −e−h[1− F (ω − h)]/∆1

170 CHAPTER 4

Fig. 59. Contours of integration in the analysis of the Green’s function in a flow with abreak in the velocity profile.

where∆ = i(1− σπi)(ω − h)

∆1 = −2[1 + (F (ω)− F1(ω))/2](ω − ωp)∆

F (ξ) = (σ/2)[exp(2ξ)Ei(−2ξ)− log(2ξ)]

F1(ξ) = (σ/2)[exp(2ξ)Ei(−2ξ) + log(2ξ)].

The Fourier transform G has the following singularities: the pole ωp defined by(4.59) and logarithmic branch points ω = h, ω = z and ω = 0. Calculating theinverse Fourier transform we deform the contour of integration downwards and splitit into contours Γp, Γh, Γz, Γ0 around the respected singularities (see Fig. 59). Denotethe integrals along these contours by Gp, Gh, Gz, G0, respectively. One can easilycheck that when t ∼< 1 the pulses Gz and G0 are of the order of σ, they decreaseas t−2 as t tends to infinity. The residual at the pole ωp has a singularity whenz = zc(U(zc) = Re cp), it can be treated as a slowly decreasing quasi-eigen mode

Gp ≈ wp/(ωp − h)

where wp is defined by (4.62). Clearly, its amplitude increases when h → zc but itsgrowth is restricted from above by a threshold of the order of σ−1 in contrast to thecase σ = 0:

Gp,max ≈ wp/ωi.

When σ 6= 0 the singularities ω = h and ω = zc cannot merge exactly becauseω = h moves along the axis Re ω with change of h and the pole ω = zc lies below it.

Finally, the integral Gh tends to a solitary wave of continuous spectrum with thecritical layer z = h when t →∞.

Gh(t, z | h) = GCS(z | h) exp(−iht) + O(t−2) (4.71)


GCS(z | h) =

e−h−z(h− ωp)−1, z < 0

−e−h(w1(h, z) + w2(h, z)(h− ωp)−1)/2, 0 < z < h

−e−h(1 + (h− ωp)−1)w1(h, z), z > h.

Here GCS is a generalized solution of the Rayleigh equation (i.e. the classical solutionwhen z 6= h satisfying the glueing conditions at the level z = h). When t < tσ theamplitude of Gh is of the same order as that of the quasi-eigen mode of discretespectrum. However, the quasi-eigen mode slowly decays. Thus, the neutral mode ofdiscrete spectrum prevails eventually when t →∞. Its amplitude can be rather largeif the critical layer of Case’s wave lies not far from that of the quasi-eigen mode.

Hence, a flow with curved velocity profile holds the resonance mechanism of am-plification of initial perturbation. In the linear approximation the amplitude reachesits maximal values given in (4.70) at time t ≈ min(tν , t∆, tσ), tσ ∼ k(U ′′)−1. Themechanism of this amplification is basically the same as in the case of a piece-wiselinear profile. However, its detailed picture is more complicated because when z > 0the vorticity of the main flow does not conserve anymore. Calculating its perturba-tions by the primary and secondary vorticity layers, we obtain addends Gz, G0 andthe third term of the order of O(t−1) in (4.71).

Algebraic instability can be discovered in more sophisticated flows as well, if theyadmit neutral or slowly decreasing eigenmodes or quasi-eigen modes with criticallayers in the flow. Then the vorticity of the initial perturbation localized in a neigh-borhood of the critical layer leads to the resonance growth of perturbations in thestratified flows (see Sec. 4.5). Finally, note that a natural source of the periodic ini-tial conditions can be realized by different periodic (or near periodic) structures. Asa few examples, we mention a chain of singular vortices, the Karman vortex street,etc. Their only property of importance is to move with the phase velocity of one ofdiscrete spectrum modes. One can remember the well-known Phillips mechanism ofthe amplification of wind waves on the sea surface (Phillips, 1969). Here the role ofthe initial periodic perturbation is played by turbulent vortices in the air flow movingwith the phase velocity of the surface waves.

4.4.3. A Long-wavelengths approximation for the Green’s function

So far we treated some model problems to get insight into some interesting physi-cal phenomena. Among them we concentrated on the quasi-eigen modes, algebraicinstability, etc. to reveal their relation with the singularities of the spectrum of therespected solutions. This study facilitates an approach to the investigation of realflows by means of a long-wavelengths approximation. This technique of constructionof particular solutions for the Rayleigh equation as a power series with respect tok was pioneered by Heisenberg (Heisenberg, 1924). It provides a powerful tool forstudying of boundary layer flows (Schlichting, 1959), near-surface currents in oceans

172 CHAPTER 4

Fig. 60. Profile of a subsurface flow (a). Movement of pole c on c-plane. Its intersectionwith the segment [U2, U1] at c∗ (b). Movement of its image on z-plane and variation of theintegration path when z crosses the real axis (c).

(Shrira, 1989), etc.We start with a plane-parallel flow in the upper half-space z > 0 with a monotonic

velocity profile U(z)(U ′(z) ≤ 0). To simplify the calculations we suppose that U(z)vanishes outside the layer of width H, moreover it has a finite derivative at thelevel z = 0 and vanishes at the level z = −H (see Fig. 60a). Consider the Cauchyproblem for the non-stationary Rayleigh equation with an initial data harmonic inthe x-direction

w(0, x, z) = w0(z)eikx, w0(0) = 0.

One can easily obtain the solution using the Green’s function

w(t, x, z) = w(t, x)eikx = eikx∫

η0(h)G(t, z | h, k)dh.

Here η0(z) = (∂2z − k2)w0(z) and G is the solution of (4.19) fitting conditions (4.21)

and boundary conditions

G(t, 0 | h, k) = 0 (4.72)

(∂z + k)G(t, z | h, k) |z=H= 0. (4.73)

Performing the one-sided Fourier transform

G(c, z | h, k) =

∞∫

0

G(t, z | h, k)eikctdt.


we obtain for the spectrum G the stationary Rayleigh equation with r.h.s.

(∂2z − k2)G− V ′′

VG =

i

k

δ(z − h)

V (h)(4.74)

where V (z) = U(z)− c.Denote by φ1,2(z) the particular solutions of the homogeneous Rayleigh equation

V (∂2z − k2)φ1,2 − V ′′φ1,2 = 0.

They are uniquely defined by the following conditions: φ1 fits ( 4.73) and φ2 fits(4.74). In terms of these solutions the Green’s function takes the form

G(c, z | h, k) =

iφ1(z)φ2(h)

kV (h)W (c), z < h

iφ2(z)φ1(h)

kV (h)W (c), z > h.

Here W (c) = φ′1 φ2− φ1 φ′2 stands for the Wronskian of the particular solutions φ1,2.In case of a small k one can decompose φ1,2 into a series following the formalism

introduced by Heisenberg (1924). It is convenient to express the Rayleigh equationin terms of the new variables

y =

y1

y2

=

w/V

ik−1(V w′ − V ′w)

.

Physical meaning of components of the vector y is as follows: y1/(ik) is a verticaldisplacement, y1 is a slope of the displacement of a liquid particle, y2 = p/ρ is thepressure divided by the density.

Hence, the equation takes the form of the system of the first order ODE with thefactor k:

y′ = ikAy, A =

0 −V 2

V −2 0

. (4.75)

One can easily transform this system into the system of integral equations

y(z) = y(z0) + ik

z∫

z0

A(z1)y(z1)dz1.

Here y(z0) = y0 is an initial condition for equation (4.75). Clearly, we are faced nowby the problem of obtaining the fundamental matrix F, i.e. the solution

F(z0 | z) = E + ik

z∫

z0

A(z1)F(z0 | z1)dz1. (4.76)

174 CHAPTER 4

where E stands for the unit-matrix because of the relation

y(z) = F(z0 | z)y(z0), F =

f11 f12

f21 f22

.

Iterating the equation (4.76) with F(0) = E one obtains its solution in the formof an absolutely convergent series with respect to k:

F = E + ik

z∫

z0

Adz1 − k2

z∫

z0

A

z1∫

z0

Adz2dz1 − ik3

z∫

z0

A

z1∫

z0

A

z2∫

z0

Adz3dz2dz1 + · · ·

Calculate φ1,2 in terms of the components of the fundamental matrix F

G(c, k, z | h) =

−∆−1V (z)f12(0 | z)[f11(H | h)− iV 2(H)f12(H | h)], z < h,

−∆−1V (z)f12(0 | h)[f11(H | z)− iV 2(H)f12(H | z)], z > h.

G(c, k, z | h) = −∆−1V (z)f12(0 | z)[f11(H | h)− iV 2(H)f12(H | h)], z < h

G(c, k, z | h) = −∆−1V (z)f12(0 | h)[f11(H | z)− iV 2(H)f12(H | z)], z > h

∆ = k2f22(0 | H) + iV 2(H)f12(0 | H).

Neglecting of the terms of the order of k2 (and higher) leads to an approximateexpression for G:

G ≈

iV (z)∆−11 I(c; 0 | z)[1 + kV 2(H)I(c; h | H)], z < h,

iV (z)∆−11 I(c; 0 | h)[1 + kV 2(H)I(c; z | H)], z > h,

(4.77)

G ≈ iV (z)∆−11 I(c; 0 | z)[1 + kV 2(H)I(c; h | H)], z < h

G ≈ iV (z)∆−11 I(c; 0 | h)[1 + kV 2(H)I(c; z | H)], z > h,

where ∆1 = k[1 + kV 2(H)I(c; 0 | H)] and

I(c; z1 | z2) =

z2∫

z1

(c− U(z))−2dz, 0 ≤ z1,2 ≤ H. (4.78)

We study now the inverse Fourier transform

G(t, k, z | h) =k

2π

∫

Γc

G(c, k, z | h)e−ikctdc.

The contour Γc passes above all the singularities of the function G(c). First, weconcentrate on the singularities of the function I(c) defined in (4.73). The functionI(c) is uniquely defined on the Riemann surface with the cut [U1, U2] (U1,2 = U(z1,2)),


the branch of the function I(c) under study is denoted as L0. Consider an analyticalcontinuation of the function I(c) to be specified as follows. Suppose that the param-eter c moves from the upper half-plane and intersects the cut at the point c∗ = U(z∗).Thus, the function I(c) changes into the next branch to be denoted as L1. Supposethat U(z) is an analytic function in a neighbourhood of z∗ and there exists the locallyone-to-one correspondence c → z: c = U(z). Denote by zc the image of c: U(zc) = c

and use the relation

c− c∗ ≈ U ′(z∗)(z − z∗) + o(z − z∗). (4.79)

Because of the inequality U ′(z∗) < 0, the image zc of c moves from the lower half-plane to the upper one when c intersects the cut downwards. After this variation ofparameter pushing zc up the cut, we deform the contour of integration to go aroundzc upwards (see Fig. 60 b,c). The integral along the deformed contour defines ananalytical continuation of I(c) into the branch L1. Note that the correspondencec → z fails to be one-to-one at the point where U ′(z) = 0 (or z′(U) = ∞) and atbranch points of U(z). We suppose that the segment [z1, z2] is free from the pointsof both mentioned types.

Change the variable z → U in (4.78) and integrate by parts to calculate the limitc → U1 (or c → U2). Adding and subtracting the term

z′′(c)U2∫

U1

(U − c)−1dU

we obtain

I(c; z1 | z2) = z(U1 − c)−1 − z(U2 − c)−1z′′(c) log[(c− U1)/(c− U2)]+

+

U2∫

U1

[z′′(U)− z′′(c)]/(U − c)dU. (4.80)

Here z1,2 = (dz/dU) |U=U1,2= [U ′(z)]−1, z′′(U) = −U ′′/(U ′(z))3. The last addend in(4.80) is an analytic function in a neighbourhood of the segment [U1, U2] when z′′(U)has the same property. The first and second addends have the poles c = U1 andc = U2, respectively. The same points plays the roles of the logarithmic branch forthe third addend in (4.80).

One can similarly check that for profile U(z) containing the jump of the n-thderivative at the point z3 ∈ (z1, z2)

I(c) ≈

a1(U3 − c)−1 + a2 log(U3 − c), n = 1

a3(U3 − c)n−2 log(U3 − c), n ≥ 2

176 CHAPTER 4

in a neighbourhood of the point c = U3 = U(z3).The case when U ′ → 0 when z → z1 or z → z2 turns out to be more complicated,

e.g. when z → z1 and

U ≈ a1(z − z1)n + a2(z − z1)

n+1 + · · · , n > 1

the inverse function has the following representation in a neighbourhood of z1

z − z1 ≈ (U/a1)ν − νa2(U/a1)

(1+ν)/(1+ν) + · · ·

where ν = 1/n (0 < ν < 1). As a result, the leading term of asymptote for I(c) whenc → 0 has the form

I(c) ≈ λ1cν−2[1 + λ2c

(1+ν2)/(1−ν)] + terms of higher order.

One can write down an explicit (but unwieldy) expression for coefficients λ1, λ2

(cf. (Gilmore, 1981)).Summing up, we list the singularities of the function G described by the approx-

imate formula (4.77):

i) the pole cp where the term ∆1 vanishes (see (4.77 ))ii) the logarithmic branch points c = Uh, c = U0 and c = Uz;iii) a branch point near c = 0 located at a distance of the order of c(1+ν2)/(1+ν).

Here Uz = U(z), Uh = U(h). The pole cp has been obtained by the iterationtechnique:

cp = cr + ici

cr = U0 + kU0/U′0 + O(k2)

ci = −k2U ′′0 (U0/U

′0)

4π + O(k3).

Calculating the inverse Fourier transform we deform the contour downwards andsplit it into five circles around each singularity (see Fig. 61). Denote these integralsas Gp, Gh, Gz, G0, GH and consider them separately.

The residual at the pole cp can be simplified in the parameter domains h À zc ≈k(U0/U

′0)

2 and h ∼< zc. In the first case it does not depend essentially on h:

Gp ≈ (U ′0)−1[∆Uz0 − U ′′

0 (U ′0)−2(Uz − cp)(U0 − cp) log((Uz − cp)/(U0 − cp))]×× exp(−ikcpt).

In the second case h ∼< zc the residual Gp gains its maximum when h = zc

Gp ≈ ∆Uz0(U′0)−1[1− kU2

0 (U ′0(Uh − cp))

−1+

+kU ′′0 U2

0 (U ′0)−3 ln((Uz − cp)/(U0 − cp))] exp(−ikcpt)

(4.81)


Fig. 61. Contours of integrations for the Green’s function of a subsurface flow.

max Gp = ∆Uz0(U′0)

2(kU ′′0 U2

0 πi)−1 exp(−ikcpt).

Here ∆Uz0 = Uz − U0. We stress again that the branches of log in (4.81) are to bedifferent on both sides of the critical layer to fit the boundary conditions (4.72) and(4.73). (In fact, the analytic continuation of Gp satisfying (4.73) at the point z = H

takes the value Gp(0) ≈ 2πik2U ′′0 U4

0 (U ′0)−5 (h À zc) on the boundary z = 0, thus it

cannot fit both boundary conditions simultaneously). As a result, we can calculatethe jump of Gp at the layer z = zc

[G]zc≈ k3(U ′′

0 )2U60 (U ′

0)−8

to be compensated by the non-harmonic perturbation Uz.The perturbation Gh tends to a harmonic Case’s wave as t →∞

Gh = GCS[1 + iU ′′h (U ′

h)−2(kt)−1] exp(−ikpUht)

GCS =

∆U/U ′, z > h

kI(Uh; 0 | z)∆Uzh/U′h, z < h, h, z À zc

∆U0h∆Uzh (U ′0(Uh − cp))

−1, h, z ∼ zc

where ∆Uzh = Uz − Uh, ∆U0h = U0 − Uh.Finally, other addends Gz, G0, GH turn out to be negligible when compared with

the terms Gp and Gh above. The role of the term Gz is to compensate the jump ofthe quasi-eigen mode at the critical layer. The term GH depends on the singularityat the layer z = H where the shear flow meets the homogeneous one, it becomessmaller when the break at the layer decreases.

Summing up, we note that an external action on a near surface sublayer whichis localized in the z-direction and harmonic in the x-direction, generates both a CS-mode with the same critical layer as the layer of the force and a weakly decaying

178 CHAPTER 4

QE-mode. The action of a force with a smooth z- profile can be described by theconvolution of this profile and the Green’s function. Avoiding these unwieldy compu-tations, Shrira (1989) presents a qualitative description of the perturbation evolutionand correct evaluation of its time scale. He demonstrates that the dominant contri-bution of the QE-mode at the time scale

1

k Im ci

≈ π

k3U ′′0

(U ′

0

U0

)4

can be treated as an intermediate asymptote of the perturbation due to the disphasingof the CS-mode. In the special case when the external force acts in vicinity of thecritical layer of the QE-mode, it generates the temporal growth of the perturbation(i.e. the algebraic instability) described in details in Sec. 4.4.1 and 4.4.2 for modelprofiles.

4.5. Localized Source for CS-Mode

Previously we considered a localized in time, but distributed in space (along the flow),source (excitation). Now we consider a more realistic case. Let an oscillating profile(e.g. a wing) is located along the y-axis under a small angle of attack streamlinedby the flow. It is well known (see (Goldstein, 1961)) that the flow separation at theback edge of the profile leads to an appearance of the vertical bulk force localized ina small neighbourhood of separation point. It is given by the relation

fz ≈ U2ρπd sin α.

Here U is velocity of the flow, ρ is fluid density, d is cross-section size of the profileand α is angle of attack. If α changes with the time, fz changes as well. Thus, theoscillating profile creates an oscillating external force fz. Hence, an oscillating stream-lined profile provides a natural source of hydrodynamical perturbations including thesolitary Case wave (see Sec. 4.5.1).

First, we consider a purely harmonic source

fz = δ(x)δ(z − h) exp(−iΩt), (4.82)

afterwards we shall study an evolutionary problem of its switching on.

4.5.1. The Couette flow

We shall demonstrate now that in the Couette flow a source of the type (4.82) gen-erates the Case wave in a shear flow and a non-wave perturbation as well. The Casewave propagates along the flow with the velocity U(h). Thus, we are interested insolutions of the basic equation of the form

L(∂t, ∂x, ∂z; U)w = ∂x(∂zfx − ∂xfz)


Fig. 62. Contours of integration in the analysis of the action of a localized periodic forcein the Couette flow.

describing an action of the external force f(t, x, z) = (fx, 0, fz) on the shear flow withthe velocity profile U(z). In particular, when fx ≡ 0 and fz has the form (4.82) weobtain the problem

L(−iΩ, ∂x, ∂z; U)w = δ′′(x)δ(z − h) (4.83)

where w has to fit the auxiliary conditions (4.20).We approach the problem (4.83), (4.20) using the Fourier transform with respect

to x. The spectrum w obeys the equation

[−iΩ + ikU(z)](∂2z − k2)w = −k2δ(z).

Choosing h = 0, U = U0 + γz we obtain the sought-for solution

w = iκ[2U0(k − k0)]−1 exp[−κ |z| − iΩt] (4.84)

where k0 = Ω/U0, κ(k) = k sign Re k. Note that κ(k) = limα→0

√k2 − α2 and the

function√

k2 − α2 is defined on the Riemann surface with cuts (α + i0, α + i∞) and(−α − i0,−α − i∞). In other words, the solution (4.84) is defined on the surfacewith cuts (i0, i∞), (−i0,−i∞). One choose the branch of the multi-valued functionthat is positive when k > α. Therefore, the spectrum w decreases when |z| → ∞ (inthe case of a compressible fluid, α would be finite, see (Mironov, 1975)).

Calculating the inverse Fourier transform we choose the direction of the patharound the pole k = k0 using the causality principle. More precisely, consider acomplex frequency with an infinitesimal imaginary part Ω = Ω0 + i0. In otherwords, we treat the harmonic oscillation of the source (4.84) as a limit of increasingoscillations exp(−iΩ0t + εt) when ε → 0. Thus, the pole k0 = Ω0/U0 + i0 is movinginto the upper half-plane and contour of integration passes below it.

180 CHAPTER 4

Let us deform the initial contour of integration Γ putting it below the cut (Γ−)when x < 0 and above the cut (Γ+) when x > 0. When x > 0 the residual at the polek = k0 has to be taken into account. As a result of a straightforward calculation, weobtain the following expression

w = (4πU0)−1[2κ(κ2 + z2)−1 + ik0 exp(k0 |z|+ ik0x)Ei(−k0 |z| − ik0x)+

ik0 exp(−k0 |z|+ ik0x)Ei(k0 |z| − ik0x)] + (4.85)

+ (Ω/2U20 ) exp(−k0 |z|+ ik0x− iΩt)θ(x).

One chooses the branch of the integral exponent Ei(u) in (4.85) that is real whenu < 0, this multi-valued function is defined on the surface with the cut (0, +∞). Itis interesting to note that (4.85) represents a continuous function. Indeed, the jumpsof the functions θ(x) and Ei(k0 |z| − ik0x) at x = 0 compensate each other. Theformula (4.85) can be simplified when | k0x |À 1:

w ≈ i(2πΩx2)−1 exp(−iΩt) + (Ω/2U20 ) exp(−k0 |z|+ ik0x− iΩt)θ(x).

The first addend in (4.85) emerging as a result of integration along the cut de-creases far from the source. The flow at hand is potential with the only exception ofthe point x = 0, z = 0 where the source is located. The second addend emerging asthe residual at the pole k = k0 describes the Case wave propagating with the velocityU0. This flow is potential but for the critical layer z = 0, it has the δ -singularity atthis layer.

Finally, we note that one cannot generate the Case wave by a source of volumevelocity at the point x = 0, z = 0. This source creates a potential flow when xz 6= 0.This flow decreases far from the source and disappears immediately when the sourceis switched off.

4.5.2. Slightly curved profile

Now we consider the problem (4.83), (4.20) for the flow with the curved velocityprofile (4.25) and demonstrate that the solution can be approximated by the solitaryCase wave in the limit x → ∞. However, for any fixed x > 0 the singularity of thesolution at the critical layer has the type

w∼ΩU ′′0

U0U ′0

(z |z|) log |z| . (4.86)

Hence, it is weaker when compared with the singularity of Case’s wave:

W ∼ U ′′0

U ′0

z log |z| .


In the particular domain |k0z| ¿ |σ|−1 we can express the solution using the smallperturbation technique with respect to the small parameter σ = (U ′′

0 U0/ΩU ′0):

w = w0 + w1 + O(σ2).

Here w0 stands for the solution (4.85) for a shear flow with a linear velocity profile.Using again the Fourier transform with respect to x we obtain the equation

[Ω− k(U0 + U ′0z)](∂2

z − k2)w1 = −kU ′′0 w0 + kU ′′

0 (z2/2)(∂2z − k2)w0. (4.87)

Here w1 stands for the correction of the order of σ in the Fourier transform of thesought-for solution. We can integrate (4.87) explicitly

w = G ∗ w0

−kU ′′0 [Ω− k(U0 + U ′

0z)]=

=iU ′′

0

4U0U ′0

1

k − k0

+∞∫

−∞

exp(−κ |z| − κ |z − z1| − iΩt)

z − (k0/k)z1 + z1

dz

by means of the Green’s function G(z) = − exp(−κ |z|)/2 of the equation

(∂2z − k2)G = δ(z).

Here z1 = U0/U′0 denotes the distance between the layer of the source location and the

layer where the velocity of the flow coincides with the velocity of the source (in otherwords, it equals zero in the coordinate system of the source). Integrating explicitly,we obtain

w1 = iU ′′0 (4U0U

′0)−1[Y1(k)− Y2(k)] exp(−iΩt), (4.88)

Y1 = j(k − k0)−1[exp(y1)Ei(y2)− exp(−y3) log y2]

Y2 = j(k − k0)−1[exp(−y1)Ei(y4)− exp(−y3) log(−y2)]

wherey1 = 2sjk0z1 − sjk(2z1 + z), y2 = 2sj(k − k0)

y3 = sjkz, y4 = 2sj(k0z1 − kz − kz1)

j = sign z, s = sign Re k = κ(k)/k.

The function Y1 is defined on the surface with the additional cut (k0, k0 + i∞),one chooses the branches of Ei and log using the following condition: Ei(y2) has tobe real when y2 < 0 and log has to be real when y2 > 0. The function Y2 is definedon the surface with the additional cut (k2, k2 + ∞), where k2 = kz1/(z + z1), onechooses the branch of the function that is real when y4 < 0.

Calculating the inverse Fourier transform of the function (4.88) we deform thecontour of integration as we did previously, see Sec. 4.4.1. Dealing with the function

182 CHAPTER 4

Fig. 63. Contours of integration in the analysis of perturbations in a flow with a smallcurvature of the velocity profile.

Y1(k) when x > 0 we should add the integral along the additional cut (the contourΓ1 in Fig. 63a) to the integral along the cut (+i0, +i∞) (the contour Γ+ in Fig. 63a).In a similar way, dealing with the function Y2(k) when x > 0 we should add theintegral along the additional cut (the contour Γ0 in Fig. 63) to integral around thepole k = k0 and along the cut (the contour Γ2 in Fig. 63b).

We cannot calculate the integrals along Γ+ and Γ− in an explicit form. However,we are able to calculate their asymptotes when k0x À 1 using an expansion of theintegrand with respect to k:

∫

Γ+,Γ−

· · · dk ≈ i |z|U ′′0

4πΩx2.

All the other integrals can be calculated explicitly, one should keep in mind thatthe branches of Ei and log differ by the additive constant 2πi above and below thecut.

Now we present the result neglecting the terms vanishing when x →∞:

w ≈ w01 + U ′′0 U0(2ΩU ′

0)−1[j exp(2k0 |z|)×

×Ei(−2k0 |z|)− j log(2k0 |z|)− jC + I]. (4.89)

Here I stands for the integral along the contour Γ2. This term is essential at thelayer | k0xz/z1 |∼< 1 only. In this domain for I we obtain an approximate expression

I ≈ −(iπk0z/2)[Ei(−ik0zx/z1) + (iz1/k0zx) exp(−ik0zx/z1)]. (4.90)

If the term I is neglected, the expression (4.89) coincides with the solution (4.35).Hence it expresses the solitary Case wave in the slightly curved velocity profile.


Both sources (4.82) and (4.16) generate a packet of the Case waves in the shearflow with non-linear velocity profile. Each harmonica of the packet propagates with itsown phase velocity. As a result of their interference, all waves compensate each otherwith the only exception of the central harmonica. The addend I in (4.89) describesthe contribution of the Case waves with critical layers near z = 0; the width of thelayer where this contributions is essential decreases as x−1. One can easily checkusing (4.89) and (4.90) that the leading terms of asymptote when z → 0 of thesolitary Case wave and the addend I cancel each other. Therefore, the singularity ofthe vertical velocity w at the critical layer turns out to be more sophisticated, see(4.86). As a result, the horizontal velocity u does not tend to infinity as log(1/z),but remains finite in a neighbourhood of the critical layer.

4.5.3. A stratified shear flow

Now we study an action of the source (4.82) on a shear flow admitting discretespectrum modes side by side with the modes of continuous spectrum. As a model,we consider an unbounded in the z-direction flow with the velocity profile

U =

U0 + γ1z, z > 0

U0 + γ2z, z < 0(4.91)

and piece-wise constant stratification

ρ =

ρ1, z > 0

ρ2, z < 0.(4.92)

The dispersion relation for discrete spectrum modes takes the form

Z(k, ω) ≡ s(ω − kU0)2 − (ω − kU0)ω0 − kδ = 0

where

ω0 =ρ1γ1 − ρ2γ2

ρ1 + ρ2

, δ = −gρ1 − ρ2

ρ1 + ρ2

> 0, s = sign Re k.

The qualitative behaviour of dispersion curves for different U0 when ω0 > 0 isillustrated in Fig. 64. The dispersion curves for ω0 < 0 can be obtain by the reflectionwith respect to the axis ω. In the same Fig. 64 the dispersion curves of discrete modeswhen ω0 > 0 are presented for the case of a homogeneous fluid (δ = 0) with the breakof velocity profile 2ω0 = γ1 − γ2. The dispersion curves when ω < 0 can be obtainedby the reflection with respect to the origin.

The characteristic behaviour of the discrete spectrum modes in the flow (4.78)–(4.79) is due to the stratification, as well to the break of velocity profile. When U → 0

184 CHAPTER 4

Fig. 64. Dispersion curves for different values of parameters: a–e — for stratified flows;f–h — for nonstratified flows (δ = 0); i — map for regions with different numbers of rootsk(ω).

they tend to the usual internal waves on thermocline with dispersion relation of theform

ksδ = ω2.

When the stratification vanishes they tend to the hydrodynamical mode (4.56)


and to an additional mode ω = 0.As usual, we analyze this problem using the Fourier transform with respect to x.

Glueing conditions for this transform w at layers z = 0 and z = h take the form

[w]0 = 0, [w]h = 0,

(ω/k − U0) [ρ∂zw]0 + [ρU ′]0 w(0) + [ρ]0 g(ω/k − U0)−1w = 0,

(ω/k − U0) [ρ∂zw]h = −ikρ−11 .

We seek the solution in the form (cf. (4.76))

w = A[exp(−κ | z − h |) + B exp(−κ |z|)] exp(−iΩt)

A(k) = κ[2ρ1(kU0 − Ω)]−1, B(k) = [ω0(Ω− kU0) + kδ](Z(Ω, k))−1e−iκh

Uh = U(h) = U0 + γ1h.

Calculating the inverse Fourier transform we concentrate the attention on theterms that do not vanish when |x| → ∞. These terms are due to the residuals atpoles of functions A(k) and B(k). In particular, the residual at the pole kh = Ω/Uh

of the function A(k) gives Case’s wave propagating along the flow

wCS = iΩ(2ρ1Uh)−1 exp[−iΩ(t− x/Uh)]exp(−kh | z − h |)+

+[ω0(Ω− khU0) + khδ](Z(Ω, kh))−1 exp(−khh− kh |z|)θ(x).

(4.93)

The residuals at the poles of the function B(k) (zeros of Z(Ω, k)) represent theinternal waves generated by the source. For fixed ω there exist up to four differentdiscrete spectrum modes with different wavelengths (see Fig. 64a, ω0 < ω < ω∗).

Denote the poles with positive real part by k+1,2 and the poles with negative part

by k−1,2. Using (4.93) we obtain

k+1,2 =

1

2U0

[2Ω− ω0 +

δ

U0

±√

ω20 −

2ω0δ

U0

+δ2

U20

+ 4Ωδ

U0

]

k−1,2 =1

2U0

[2Ω + ω0 − δ

U0

±√

ω20 −

2ω0δ

U0

+δ2

U20

− 4Ωδ

U0

].

After some variation of parameters the real part of k+1,2 can become negative or

the real part of k−1,2 can become positive. In this case the respective pole passes underthe cuts (i0, i∞), (−i0,−i∞) and fail to contribute into the far field of the resolution,e.g. if ω0 > 0, U0 > −δ/ω0, Ω < ω0, then Re k+

2 < 0 (see Fig. 64a–d). Therefore thepole produces a decreasing wave and changes the near field only.

When Ω tends to Ω∗:

Ω∗ = ±U0(ω0 − δ/U0)2/4δ

186 CHAPTER 4

the roots k−1 and k−2 (or k+1 and k+

2 ) coincide and equal to

k∗ =| ω20 − 2δ2/U2

0 | /4δ.

They are complex for real Ω in the domain Ω > Ω∗ and represent the exponentiallydecreasing waves. More precisely, these waves decrease in the horizontal direction asexp(− |x Im k|)).

Selection of poles that are relevant for the far field behaviour is ruled by thecausality principle. When x > 0 one should check whether the pole lies in the upperhalf-plane for complex Ω (Ω = Ω0 + i0). There are just the poles to be intersected bythe contour Γ+ during its deformation from the initial position along the axis Re k.On the other hand, when x < 0 the poles lying in the lower half-plane for Ω = Ω0 + i0are relevant only. Using the relation

k(Ω + iε) ≈ k(Ω) + iεk′(Ω) = k(Ω) + iε/Cg

where Cg = Ω′(k) is the group velocity of the discrete spectrum mode, we obtain theexpression

k(Ω) = k(Ω0) + i0 sign (Cg).

Thus, the sign of the imaginary part of the pole depends on the slope of therespective dispersion curve or, equivalently, on the sign of the group velocity Ω′(k).Naturally, the group velocities of all modes in the problem are directed from thesource, e.g. the waves with wavenumbers k+

1 , k+2 , k−2 go to the right (while phase

velocity of the wave k−2 is directed to the source) and the wave with wavenumber k−1goes to the left when 0 < U0 < δ/ω0, ω0 > 0, ω0 < Ω < ω∗ (see Fig. 64b).

In the case when all the poles kh, k±1,2 are different, we obtain for the internal

waves the expression

w±1,2 = ±k±1,2

ω0(Ω− k±1,2U0) + k±1,2δ

2ρ1(k±1,2U0 − Ω)(k±1,2 − k±2,1)

×

×θ(±Re k±1,2)θ(xΩ′(Re k±1,2)) exp[k±1,2(∓h∓ |z|+ ix)− iΩt] (4.94)

As a result of this study, the total field is represented as the sum of five wavesand a near field to be neglected far from the source:

w(t, x, z) = wCS + w+1 + w+

2 + w−1 + w−

2 + O(x−1).

In the case Ω = Ω∗ the wavenumbers of two internal waves coincide (and equalto k∗). As a result, their group velocity vanishes in the coordinate system of thesource and their amplitude tends to infinity. One can say that the harmonic sourcetransmits an infinite energy to the waves during the infinite time span. The energyis mainly located near the source because of the fact that group velocity vanishes.


This effect is well-known for the case of acoustic waveguides (Isakovich, 1973): if thefrequency of a source coincides with the critical frequency of a mode (and, therefore,the group velocity of the mode vanishes), the amplitude of the mode tends to infinity.Note that this effect is still valid if one changes the source of external force fz by asource of volume velocity.

Now we concentrate on the case of coincidence for wave numbers of different types:one of a discrete spectrum mode (say, k+

2 ) and one of a continuous spectrum mode(it means that the source is located at the critical layer of the wave w+

2 ). Moreover,we suppose that the group velocity of the wave w+

2 is negative for given ω0, δ and U0.Then the amplitude of this wave has to be infinite also.

Unlike the previous case, the energy of the perturbation is taken from the mainflow instead of the source and the group velocity vanishes. This effect can be realizedonly in the case of the source of external force which is the only suitable type ofsource to generate Case’s waves. Below we shall study the evolutionary problems indetails to get insight into this effect (see Sec. 4.5.2).

Note that in the case at hand the merging poles lie in different half-planes (Im k >

0 and Im k < 0), hence the contour of integration passes between them.In the case of the coincidence of the wavenumber for Case’s wave and that of

internal waves k+1 or k+

2 , we concentrate on the case of the positive velocity of theinternal wave. Then the merging poles lie on the same side of the contour of inte-gration. If, e.g. kh = k+

1 one can easily study the limits of (4.80) and (4.81) whenk+

1 → kh and obtain that the velocity field is finite at any time. However, it growswith the increase of distance from the source in the direction of the flow:

limk11→kh

(wCS + w+1 ) = iΩ(2ρ1Uh)

−1 exp(ikhx− iΩt)exp(−kh | z − h |)+

+(ix− h− |z|)[ω0(Ω− khU0) + khδ][U20 (kh − k+

2 )]−1 exp(−khh− kh |z|)θ(x).

This is the well-known instability emerged in the case of coincidence of roots ofthe characteristic equation leading to the linear growth of the perturbation far fromthe source (see (Monin, 1986) and (Terent’ev, 1984)).

4.5.4. Nonstationary case

Consider the switching at the moment t = 0 of a harmonic source of perturbation ina shear flow admitting discrete spectrum modes:

fz = δ(x)δ(z − h) exp(−iΩt)θ(x).

We restrict ourselves to the simplest case of a non-stratified flow with the piece-wise-linear velocity profile

U(z) = U0 + γzθ(z). (4.95)

188 CHAPTER 4

Thus, we have to study the non-stationary Rayleigh equation with the velocityprofile (4.95)

L(∂t, ∂x, ∂z; U) = δ′′(x)δ(z − h) exp(−iΩt)θ(t)

and auxiliary conditions (4.20) and (4.21). Using the one-sided Fourier transformwith respect to t and the usual (two-sided) Fourier transform with respect to x weobtain for the spectrum the Rayleigh equation with r.h.s.:

(ω/k − U)(∂2z − k2) w + U ′′ w = −k2(ω − Ω)δ(z − h).

Its solution fitting the condition (4.20) and the glueing conditions at levels z = 0and z = h (see (4.61)) takes the form

w = κ[2(ω − Ω)(ω − kUh)]−1[exp(−κ | z − h |)+

+sω0(ω − kU0 − sω0)−1 exp(−κh− κ |z|)].

Calculate the inverse Fourier transform with respect to t. Taking into accountthe residuals at the poles ω = Ω, ω = kUh and ω = kU0 + sω0 we obtain

w = (iκ/2Uh)(k − kh)−1[exp(−ikUht)− exp(−iΩt)] exp(−κ |z − h|)+

+(iω0kh/2D±)(k − kh)−1[exp(−ikUht)− exp(−iΩt)] exp(−κh− κ |z|)+

+(iω0k±/2D±)(k − k±)−1[exp(−iΩt)− exp(−ikU0t− isω0t)] exp(−κh− κ |z|)++(iω0sks/2D±)(k − sks)

−1[exp(−ikU0t− isω0t)− exp(−ikUht)]×× exp(−κh− κ |z|). (4.96)

Here D± = (Uh − U0)Ω− sω0Uh and k± = (Ω− sω0)/U0 stands for wavenumbersof the discrete spectrum wave with the frequency Ω, ks = ω0(Uh − U0)

−1 = (2h)−1

and ωs = γ/2 + U0ks = γ/2 + U0/2h.Now the inverse Fourier transform with respect to x is given by the cumbersome

expression

w =

iΩ

4πU2h

[F (kh, | z − h |, Uh, Uh, 0) + F (−kh, | z − h |, Uh, Uh, 0)]−

− iΩω0

4πUhD+

F (kh, |z|+ h, Uh, Uh, 0) +iΩω0

4πUhD−F (−kh, |z|+ h, Uh, Uh, 0)+

+iΩk+

4πD+

F (k+, |z|+ h, U+, U0, 0) +iΩk−4πD−

F (−k−, |z|+ h, U−, U0, 0)+

+iΩks

4πD+

F (ks, |z|+ h, Uh, U0, Uh) +iΩks

4πD−F ∗(ks, |z|+ h, Uh, U0, Uh)+


+1

2πUh

[x exp(−iΩt)

(x− h)2 + x2− x− Uht

(z − h)2 + (x− Uht)2

]θ(x)

(4.97)

F (k, z, cph, Cg, cs) = exp[−kz + ik(x− cpht)]××Ei[kz − ik(x− cst)]− Ei[kz − ik(x− Cgt]−

2πiθ(kx)θ(cst− x, x− Cgt)

(4.98)

U± =Ωω0

Ω± ω0

, θ(x, y) = θ(x)θ(y)− θ(−x)θ(−y) =

1, x > 0, y > 0,

−1, x < 0, y < 0,

0, xy < 0.

Here ∗ stands for the complex conjugation. One chooses the branch of Ei(y) on theplane with the cut y > 0 to be positive when y < 0.

Calculate the asymptote of (4.98) when | x |, | x− cst |, | x−Cgt |À 1. If any ofthe conditions

kz > 0, x ∈ (cst, Cgt) (4.99)

fails, the function (4.98) decreases as x−1. In the case when all the condition (4.99)are fulfilled simultaneously, the solution (4.98) represents a waves packet propagatingalong the x-axis with the phase velocity cph. At the moment t it is located in theinterval [cst, Cgt]:

F = ±2πiexp[−kz + ik(x− cpht)] + O(x−1).

The addends in (4.98) containing Ei are essential in domains

| x− cst |, | x− Cgt |∼< k−1,

they describe the transient processes near the points x = cst and x = Cgt.The non-decreasing when x →∞ parts of solution (4.84) attract the main interest.

The addends I–IV describe the Case waves packet propagating along the flow withthe phase velocity Uh. At the moment t it occupies the region [0, Uht) (unlike thepacket treated in Sec. 4.1.1, Case’s waves are distributed in k instead of h). Theaddends V–VI describe the discrete spectrum modes with wavenumbers k+ and k−,phase velocities U+ and U− and frequency Ω, propagating with the velocity U0 (inthis case U0 coincides with the group velocity of the discrete spectrum mode). Thecondition of propagation is Re k+ > 0 for the first wave and Re k− < 0 for the secondone. They are located from the source x = 0 to the point x = U0t.

The addends VII and VIII describe the propagation of another discrete spectrumwave with wavenumber ks and frequency ωs = Uhks = γ/2 + U0/2h (in contrast toall other waves with frequencies equal to the frequency of the source Ω). The phasevelocity of the wave Uh coincides with that of Case’s wave, but its group velocityU0 coincides with that of other discrete spectrum modes for this flow. This wave is

190 CHAPTER 4

generated by the source harmonics with frequency ω = ωs. It has the critical layer z

= h. The front of these waves propagates with the velocity Cg = U0 and the secondboundary of the packet propagates with the velocity cs = Uh coinciding with thefront velocity of Case’s waves (one can imagine that the front of Case’s waves is thesource of these waves). Thus, the perturbation ws occupies the region [Uht, U0t]. Inthe case U0Uh < 0 it eventually covers the whole space when t →∞.

Thus, in the case U0Uh < 0 the solution of non-stationary problem does notconverge to that of the stationary problem treated in Sec. 4.5.3 (we suppose thatδ 6= 0) because of the additional wave ws with the frequency ωs 6= Ω. This statementcontains no contradiction. Indeed, the wave ws is a solution of the homogeneousRayleigh equation, in other words it is a free oscillation. It appears because ofinitial switching on the source and disappears in the limit t →∞ if one includes aninfinitesimal viscosity in the problem. All the other waves would conserve in the limitbeing fed up by the source. Therefore, ws is a natural analogue of a free oscillationemerged after the switching of usual oscillator without dumping. If one includes aninfinitesimal dumping in the model the constrained oscillations with the frequency ofthe external force would survive only.

Now we present a physical interpretation of the solution (4.97)–(4.98). The sourceof the external force creates the vorticity tail at t = 0 moving with the velocity ofthe flow Uh:

η(1) ∼ δ(z − h) exp[−iΩ(t− x/Uh)],

In turn it creates periodical perturbations of the velocity field which is exponen-tially small far from the critical layer.

The vertical velocities of this perturbation disturbs the boundary of vorticity inthe main flow creating the secondary vorticity tail at the layer z = 0. This secondarylayer propagates with phase velocities U+ and U− and group velocity U0. The har-monica with the frequency ωs of the initial vorticity layer generates the wave ws dueto the resonance mechanism (see Sec. 4.4.1).

In the case of coincidence of the phase velocity of the initial vorticity layerwith that of the secondary vorticity tail, the resonance effect becomes crucial (seeSec. 4.4.1). It is convenient to use the coordinate system x1, y1 moving with thegroup velocity of the mode under resonance: x1 = x−U0t, z1 = z. In this coordinatesystem the source moves with velocity −U0 and the velocity of the flow at the horizonof the source equals ∆U = γh (it coincides with the velocity of the generated Casewave). When x is fixed and the vorticity tracks interaction time grows, the localamplitude of secondary perturbations grows as well. The registered picture dependson the relation of velocities U0 and Uh.

1. Let Uh > U0 > 0 (γ > 0). In this case the source moves to the left, theprimary vorticity track propagates from the point x1 = 0 to the left with the velocity−U0 and to the right with the velocity ∆U . The shape of the total perturbation


Fig. 65. Envelops for different shapes of the pulses in a flow with an algebraic instability.The type of instability depends on parameters U0 and Uh: (a) and (b) kinematic instability,(c) and (d) absolute instability. Pictures to the left correspond to the frame moving withthe lower fluid, pictures to the right correspond to the frame sticking to the source.

is shown in Fig. 65a. In the initial coordinate system x, y the amplitude of theperturbation grows with time when U0t < x < Uht and does not depend on the timewhen 0 < x < U0t. Nevertheless, it grows linearly with the increase of the distancefrom the source (see Fig. 64a, right).

2. Let U0 > Uh (γ < 0). The vorticity track moves to the left occupying the region(−U0t, ∆Ut). For any x1 from this region the influence of the primary track startsat the moment −x1/U0 being essential up to the registration time t. If the point ofregistration is located below the track (∆Ut < x1 < 0), this influence continues duringthe time ∆t = t2−t1 = xUh/(U0∆U) from t1 = x1/U0 to t2 = −x1/∆U . Thus, it hadterminated at the moment of registration t and the amplitude of the perturbation is

192 CHAPTER 4

saturated at the point x1 before the moment t. The saturated amplitude grows withthe increase of distance from the point x1 = 0 as is shown in Fig. 65b. Using againthe initial coordinate system one can obtain the shape of the saturated perturbationwhich grows far from the source.

3. Let U0 < 0, Uh > 0(γ > 0). Thus, the velocity of the flow at the level of thesource and the group velocity of the discrete spectrum mode participating in the res-onant interaction are directed to the opposite sides. In the coordinate system x1, y1the source moves to the right with velocity −U0 and the track moves in the samedirection with velocity ∆U > −U0 (see Fig. 65c). The time of interaction at a pointx1 located downstream the tracks ∆t = (−x1/∆U) + (x1/(−U0)) = −x1Uh/(∆UU0).Hence, the saturate amplitude of the perturbation is proportional to x1 as well. Re-turning to the initial coordinate system one obtains the shape of the perturbationat a point x which grows linearly with time and decreases far from the source (seeFig. 65c, right). When t →∞ the amplitude tends to infinity everywhere, the energyof the perturbation is taken from the main flow (Sazonov, 1989). Hence this is thecase of the absolute instability.

4. We consider the special case U0 = 0 separately. In this case the frame x, zcoincides with x1, z1 because of the fact that the group velocity of the discretespectrum mode coincides with the velocity of the source. When Uh > 0 the trackpropagates to the right with velocity Uh, its time of interaction at the point x < Uht:∆t = t − x/Uh (see Fig. 65d). The amplitude of the perturbation tends to infinitywhen t → ∞. In the region lying to the left to the source the amplitude of theperturbation is expressed as the function Ei. Hence, it decreases as C(t)x−1 withC(t) indefinitely increasing when t →∞. As a result, the amplitude of perturbationtends to infinity with time to the left of the source as well without propagationphenomena.

The picture above described at the intuitive level can be rigorously deduced fromthe exact solution (4.97)–(4.98). As the only example, we consider here the casekh → k+. In this resonant case D+ → 0, ks → k+ forcing the addends III, V and VIItend to zero. Expressing Ω = ωs + ∆ one obtains the relation

k+ = kS + ∆/U0, kh = ks + ∆/Uh, D+ = ∆U −∆.

Tending ∆ → 0 in (4.97)–(4.98) one can express the growing part of the solutionoutside the singularities (x = tU0, x = tUh and x = 0) an explicit form:

w =i

2π

(Ω

∆U

)2

exp(iksx− iωst)×

× [(x/Uh − t)θ(x, Uht− x)− (x/U0 − t)θ(x, U0t− x)] .


4.5.5. Remark on the algebraic instability

Here we summarize and discuss the different aspects of the hydrodynamical stabilitytheory that were touched on in this Chapter.

Classical theory of hydrodynamical instability (see (Rayleigh, 1894), (Lamb,1895), (Lin, 1955)) associates the instability with the existence of exponentially grow-ing modes in the flow. It describes the initial exponential growth of the perturbationuntil it reaches non-linear values. We emphasize that the analysis of flow stabilityfor such an approach is based entirely on a spectral problem without detailed studyof the actual evolution of the perturbations.

However, in some recent papers other mechanisms of instability are observed whensolving several evolutionary problems for stable (in the classical sense) flows. It meansthat one can observe growing perturbations in the flow in spite of the absence of anyincreasing eigenmodes for this flow. The growth of perturbation is not exponentialat an initial stage, often it is a polynomial. This gives the name to this type ofinstability: i.e. algebraic.

First, we mention the paper (Chimonas, 1979) where the term ‘algebraic insta-bility’ was introduced. The mechanism of instability described here is owing tosimultaneous effects of buoyancy and shear in the flow.

In papers (Shepherd, 1985), (Farrell and Ioannou, 1993) the increase of perturba-tion energy was thoroughly studied in the Couette flow (which is stable profile fromthe standpoint of the classical theory). It was demonstrated that an initial pertur-bation is equally likely to either decrease or increase (growth is possible only duringa finite time span). Hence, the averaged energy of perturbation excited by randomforces may exceed the work by those forces on the fluid. We emphasize that thismechanism works in the 2D-case for a purely linear model. A possible energy growthin this model is depicted in Fig. 47.

Some other types of algebraic instability based on resonance interactions are de-scribed in Secs. 4.2.5, 4.4 and 4.5. Here we distinguish at least two perturbations andinterpret one of them as an excitation that affects others in a resonant interactionwithout any feedback, e.g. in Sec. 4.4.1 the excitation disturbance is a primary vortexsheet (see (4.68)–(4.69)). The simplest mechanical model of this type is an oscillatorunder the action of an external force, which is an obvious excitation in this case.

By analogy, we can say that a feedback is essential for the development of theusual exponential instability. The greater amplitude of the perturbation, the fasteris its growth. We keep in mind an oscillator with negative friction as the simplestexample. In this case, however small are the initial perturbations, they will eventuallyreach the values when non-linear interactions come into play. They lead to a crucialchange in the structure of the whole flow, i.e. developing of the secondary structures,chaotizations, etc.

Moreover, only two-dimensional perturbations are studied traditionally in the

194 CHAPTER 4

classical theory, because their increment is maximal due to the well-known Squire the-orem. The situation with the algebraic instability is drastically different (e.g. (Gus-tavsson, 1991), (Bergstrom, 1992), (Bergstrom, 1993), (Butler and Farrell, 1992)). Inparticular, in the paper (Sazonov, 1996) and in Sec. 4.2.5 it was demonstrated thatthe growth of the 3D-pertubations may be caused mainly by the velocity componentwhich is orthogonal both to the direction of the gradient and to the wavenumber.

In contrast to the usual exponential instability with the unrestricted growth ofperturbation in linear approximation, the maximal values of amplitude in algebraicinstability may be restricted by purely linear effects (cf. (4.70)).

Thus, a finite amplification of the initial perturbation results from the algebraicinstability. However, the coefficient of the amplification depends on the parametersof the flow and can be considerably greater than 1.

Another important feature of the algebraic instability is the appearance of athreshold: if an initial amplitude of the perturbation exceeds this threshold, theamplified perturbation can reach non-linear levels and inreversible changes in theflow structure occur. On the other hand, the perturbation below this thresholdwould eventually disappear after the stage of initial growth. Thus, they produce onlyreversible changes in the initial flow.

Based on cited papers and results of this Chapter one can conclude that algebraicinstability appears in many hydrodynamical and geophysical situations. Presumably,it is especially important for slightly supercritical hydrodynamic systems where thefinal stages of evolution are crucially dependent on initial excitations.

4.6. Pulse Propagation in Unstable Media

4.6.1. The basic model

Now we consider pulse propagation in a medium with the usual (not algebraic) in-stability admitting exponentially increasing modes. Crighton and Oswell (1991) de-scribed the evolution of perturbations in an uniform inviscid incompressible flow overa flexible elastic plate. A similar, but somewhat more tractable model, was con-sidered by Danilov and Mironov (1993) where, however, the plate is replaced by astretched membrane, whose vibrations are governed by simpler equations. For thesake of the principle of considering every phenomenon using the simplest model, weshall describe here the model with a membrane.

Hence, a membrane with a density per unit area ρ1 and tension per unit width T isimmersed in an ideal incompressible medium of density ρ, which moves with a velocityU on both sides of the membrane. The equation for the transverse displacements(flexion) η(x, t) of the membrane has the form

ρ1∂2t η − T∂2

xη = ∆p + f(x, t)


where f(x, t) is the driving force, and ∆p is the differential pressure across the mem-brane. The perturbation of the fluid beyond the membrane obeys the Laplace equa-tion

(∂2x + ∂2

z )φ = 0

where φ stands for the velocity potential of the fluid, and the pressure is expressedthrough it by the relation

∇p = ρ(∂t + U∂x)∇φ.

Besides, the potential is connected with the membrane displacement as follows

∂zϕ = (∂t + U∂x)η.

Using the length-scale ρ1/ρ it is convenient to pass to dimensionless space andtime coordinates

x = (2ρ/ρ1)x, z = (2ρ/ρ1)z, t = (2ρ/ρ1)t/cm

where cm =√

T/ρ1 is the wave velocity in the membrane without fluid. In these

coordinates U = U/cm is the ratio of flow velocity to the velocity of waves in the freemembrane. Further on we shall omit bars.

4.6.2. An instantaneous source

Consider a source, localized both in space and in time:

f(x, t) = δ(x)δ(t).

Using the usual Fourier transform for x and the one-sided Fourier transform fort, we can write down the solution of the problem as follows

η(x, t) =1

2π

∫

Γω

dω

+∞∫

−∞dk

ks exp(ikx− iωt)

D(ω, k)(4.100)

where

D(ω, k) = ks(ω2 − k2) + (ω − kU)2 (4.101)

s = sign (Re k).

Here D(ω, k) = 0 is the dispersion equation of the model. Solving this equation forthe frequency, we find

ω1,2 =Uk ±

√k4 + k3s(1− U2)

1 + ks(4.102)

196 CHAPTER 4

Fig. 66. Dispersion curves for waves on the membrane. a) 0 < U < 1; b) 1 < U < U∗; c)U > U∗.

The dispersion curves obtained for real values of k are shown in Fig. 66 for positivefrequencies. It follows from (4.102) that the membrane is unstable for U > 1 in therange of wavenumbers 0 < k < U2−1.

The list of singularities of the integrand in (4.100) as a function of k includespoles (zeros of the dispersion equation) and branch points (one of them coincideswith the origin). In analogy with the previous section (see Fig. 62) we draw the cuts(+i0, +i∞) and (−i0,−i), and consider the integrand on the plane with those cuts.For positive x we shall deform the initial contour of integration with respect to k to


the upper half-plane, and downwards for negative x.The integral over k then reduces to the sum of the residuals and an integral along

the edges of the cut I(ω):

η(x, t) =1

2π

∫

Γω

dωe−iωt

[I(ω) +

∑n

k s exp(ikn(ω)x)

∂D(ω, k)/∂k|k=kn

]

Here kn(ω) denotes the n-th root of the dispersion relation in the upper half-plane ofk for ω ∈ Γω. The number of roots depends on the frequency: as the latter is varied,the roots can drop below the cut or rise above it.

To evaluate the integral over the frequency, we deform the contour Γω downwards.The main contribution to the integral in the limit t → ∞ is defined by the pointswhere the integrand has singularities. They include branch points of the dispersionrelation ω = ωb, which are determined by the condition ∂D/∂k = 0 at which tworoots kn(ω) merge. Clearly, ∂D/∂k ∼ (ω − ωb)

1/2 in a neighbourhood of thesepoints, hence each of them contributes a term proportional to t exp[ik(ωb)x − ωbt].However, if any two roots merging at the point ω = ωb are among the roots kn, thenet contribution from them is exactly zero, and the singularity of the integrand isremovable. The contour Γω can be deformed downwards in this case. Otherwise, thecontribution the point ω = ωb to the asymptotic representation as t → ∞ increaseswith time at a fixed point in space if Im ωb > 0. The instability is absolute in thiscase.

Consequently, the ascertainment the type of instability starts with the finding thepoints ωb where Im ωb > 0 , and then we must trace the paths of the roots convergingto these points in the limit Im ω → +∞. If these roots are on opposite sides of thereal axis of k as

Im ω → +∞the instability is absolute. (About the concept of absolute and convective instabilitysee, e.g. (Lifshitz and Pitaevsky, 1979)). This problem is quite tractable in ourcase because the dispersion relation is a cubic polynomial in k. Therefore, usingthe Cardan solutions we reduce the normally troublesome tracing problem to anelementary form.

Differentiating the dispersion relation (4.101) with respect to k, we deduce thefollowing equation for the critical points kcr = k(ωb) from the system of two equationsD(ω, k) = 0 and ∂D(ω, k)/∂k = 0:

4k3cr + (12− 4U2)k2

cr + (9− 14U2 + U4)kcr + 4U4 − 4U2 = 0

Re k > 0.(4.103)

The equation for Re kcr < 0 differs from the above by the sign of even powers ofk. Its analysis is similar to that of equation (4.103). The value of ωb can be easily

198 CHAPTER 4

found from kcr, due to the equation

2ωbU = k3cr + (U2 − 3)k2

cr + 2U2kcr.

The roots of equation (4.103) can be found by the Cardan solutions. However, inthe case of interest U > 1 certain conclusions about the behaviour of the roots kcr canbe inferred without any calculations. In fact, the free term of (4.103) is positive in thiscase, implying that one of the roots of the equation is negative real. This root mustbe discarded, since it does not satisfy the condition Re kcr > 0. The other two rootskcr = k1 and kcr = k2 (see Fig. 66b) are real in the region 1 < U < U∗ = (6

√3−9)1/2.

These roots merge at U = U∗, i.e. the dispersion relation has a triple root. Using theconditions at the triple root

D(ω, k) = 0,∂D(ω, k)

∂k= 0,

∂2D(ω, k)

∂k2= 0

we find U∗ = (6√

3− 9)1/2 ≈ 1.18, kcr = k∗ ≈ 0.464, and ω∗ ≈ 0.316. In so far as theroots kcr and the frequencies ωb are real in the region 1 < U < U∗, this is the regionof convective instability. Two of the roots kb enter the complex plane at U > U∗.The root with a negative imaginary part corresponds to a frequency with a positiveimaginary part. We find that two waves meet at this point, one being observeddownstream (the root is included among the roots kn), and the other upstream (notamong the roots kn). Therefore the system is absolutely unstable for U > U∗. Thelocation of this root contradicts common sense at first glance. But it is certainly truethat

Re kb = Re k∗

for U = U∗ + ε in the limit ε → 0. However, this value of the real part lies to theright of k0 = U2 − 1, where the region of complex frequencies for real values of thewavenumber ends. The root corresponding to the wave observed downstream arrivesfrom the region Im k > 0 as Im ω tends from +∞ to Im ωb < 0. But it cannot crossthe real axis in the limit k → k∗. Invoking the Cardan solitons, we can easily tracethe true paths of the roots. The root in question crosses the real axis at k < k0 andthen, crossing into the lower half-plane of k, arrives at a point with a large real part.The behaviour of the roots remains unchanged with a further increase of the velocity.

Thus, we have shown that a membrane in a flow is convectively unstable for1 < U < U∗ = (6

√3− 9)1/2 - and is absolutely unstable for U > U∗.

4.6.3. Switching on an oscillating source

The next step is to consider the problem of wave generation by a source with areal frequency Ω. In view of the symmetry of the dispersion curve, we limit thediscussion to positive frequencies. We refer to Fig. 66b for an insight into the nature


of the attendant problems. Let Ω < ω2. Then three waves are excited with Re k > 0.We denote their wavenumbers by k+

1 , k+2 , and k+

3 . A wave with Re k < 0 is alsoexcited. We denote its wavenumber by k−3 . We seek to know whether these wavesare observed upstream or downstream. The group velocity is negative for the firstwave, positive for the second, and negative for the third and fourth waves. We takethe direction of the group velocity as the direction in which the wave is observed,and notice that only the k+

2 -wave will be observed downstream. This conclusion iscontradicted by the fact that k+

1 , k+2 and k+

3 are negative energy waves, i.e. waveswhose generation diminishes the energy of the system (the transition from a positiveenergy wave to a negative energy wave takes place at a point where ∂D(ω, k)/∂ω = 0or ω = 0; see (Lifshitz and Pitaevsky, 1979)). The energy flux, which is equal to theproduct of the group velocity and the energy density, is opposite in sign to the groupvelocity. In contradiction to the above, this implies that the k+

1 and k+3 -waves are

observed downstream. But this conclusion is incorrect: actually the k+3 - wave must be

observed wherever k−3 is, because the k−3 branch is essentially the continuation of k+3

to negative frequencies. At large absolute values of the frequency these waves changecontinuously into a wave propagating in the unloaded membrane in the negativedirection.

We find, therefore, that the problem of localization of the regions where wavesof frequency Ω are observed is not trivial, and attempts to answer it by intuitivereasoning can lead to contradictions. These contradictions are associated with thestandard concept that energy propagates away from the source. This is certainlytrue for passive systems. However, in the case of active systems such as that underinvestigation, the energy does not necessarily propagate away from the source; it cancome from the flow. The only criterion in this case is the causality principle. Namely,harmonic waves must be obtained from the solution of the source excitation problemin the limit t →∞.

The behaviour of the waves changes significantly after transition through thepoint U = 1 which separates the stable and the unstable regions. We begin with thisextreme case. The solutions of (4.101) for the case U = 1 have the elementary form

k+1 = ω

k+2 = (1/2)(1− ω)− [(1/4)(1− ω)2 − ω]1/2

k+3 = (1/2)(1− ω) + [(1/4)(1− ω)2 − ω]1/2

k−3 = (1/2)(−1− ω) + [(1/4)(1 + ω)2 + ω]1/2.

Fixing the real part of the frequency and letting the imaginary part tend to +∞,we find that Im k+

1 → +∞. The root k+2 now drops below the cut Re k = 0 for a finite

imaginary part of the frequency, and the root itself has a positive imaginary part.The imaginary parts of the other two roots tend to −∞. Consequently, the k+

1 and

200 CHAPTER 4

k+2 waves will be observed downstream, and the other two upstream. The k+

2 and k+3

waves merge and become complex conjugates at high frequencies. It follows from theabove equations that they represents the waves decaying downstream and upstream,respectively. The k+

2 and k+3 waves have negative energy in the region Ω < ω2. This

is confirmed by the fact that the sum of the kinetic and potential energies of motionof the membrane is of the order of 1/k times smaller than the kinetic energy of thewave-induced motion of the medium in the limit k → 0.

For the lower branch of the dispersion curve the phase velocity is of the orderof k1/2 times smaller than the velocity of the medium in the limit k → 0. Thismeans that the energy of the wave-induced motion of the medium is negative (see,e.g. (Ostrovsky et al., 1986)) and that it specifically determines the wave energy inthe limit k → 0. Since the sign of the energy changes only at points where ω = 0or ∂D(ω, k)/∂ω = 0, the k+

2 and k+3 -waves have negative energy. This means that

the energy fluxes transported by these waves are directed toward the source. Suchwaves are usually called anomalous. To generate these anomalous waves the sourcemust act as a sink for energy extracted from the flow. If a resonator is placed at thesite of the source, the amplitude of its oscillations must increase as a result of theradiation of anomalous waves. This behaviour of a resonator is physically equivalentto the phenomenon of radiative instability of a moving oscillator in a stationarymedium (Abramovich et al., 1986) after transformation to a reference frame tied tothe oscillator.

Next we consider the case in when U is only slightly greater than 1, α = U−1 ¿ 1.This case is intriguing in that the behaviour of the waves can be treated analyticallyby the perturbation method. The k−3 and k+

3 branches are known to remain struc-turally stable in this case and to change very little from the case U = 1. Indeed,they have wave numbers of the order of 1 or higher and can therefore change onlyslightly. As before, they correspond to the waves observed upstream. The k+

3 -wavehas negative energy, transports energy toward the source, and is anomalous. Thek−3 -wave has positive energy and transports energy away from the source. We useasymptotic analysis to learn the fate of the k+

1 and k+2 -waves. It follows from (4.101)

that significant changes from the case U = 1 are possible only in a neighborhoodof the point (ω0, k0). Since ω0, k0 ∼ α, we set ω = αω′, where ω′ = O(1). Writ-ing K = αK0 + αK1 + · · ·, where K0, K1, . . . = O(1), from (4.101) we obtain theexpression for k (which is the solvability condition for K2)

k = αω′ + α2(ω′2 − ω′ ±

√ω′4 − 2K3

0

)+ O(α3) ≈

≈ ω + ω2 − ω(U−1)±√

ω4 − 2ω3(U−1).(4.104)

This equation describes the two waves k+1 and k+

2 . It is valid for |ω| ¿ 1. Atfrequencies large in comparison with α, it yields the following expressions for the


Fig. 67. Paths of the roots on the k-plane when a real part of the frequency is fixedand an imaginary part increases from zero: a) U = 1.2, Ω = 0.3415; b–d) paths for variousfrequencies Ω of the source in the case 1 < U < Ucr.

upper root in equation (4.103):

k+2 = ω + 2ω2 − 2ω(U−1) + O(max[ω3, α2ω]) (4.105)

and for the lower root:

k+1 = ω + O(max[ω3, α3ω]) (4.106)

If Re ω is very small, the root described by (4.105) drops below the cut for

Im ω ≈ (Re ω)1/2

(equation (4.106) is valid in this case), i.e. it behaves like in the case U = 1. Weuse the method of successive approximations instead of asymptotic expansion in theregion |ω| À α, adopting the solutions for U = 1 as the zero-th approximation toformulate solutions for this case. Matching the resulting solutions with those obtainedabove, we verify that the behaviour of the roots for a large positive imaginary partof the frequency is similar to the behaviour of the roots for U = 1.

For a real-valued frequency we infer from (4.103) that two complex-conjugatesolutions with small wavenumbers exist in the region 0 < ω < ω1 ≈ 2(U−1). Thek+

2 -wave decays downstream, whereas the k+1 -wave grows. The paths of the roots on

the k-plane for a fixed real part of the frequency and an imaginary part which startsfrom the origin are shown schematically in Fig. 67b for Re ω ¿ α, and in Fig. 67c for

202 CHAPTER 4

Re ω ≥ α. The roots k+1 and k+

2 merge at the point (ω1, k1) from opposite half-spacesIm k < 0 and Im k > 0 (see Fig. 67). In the region ω1 < ω < ω2 the solution with thesmaller wave number corresponds to the k+

1 -wave, and the solution with the largerwavenumber corresponds to the k+

2 -wave.Thus, the detailed analysis of the case α ¿ 1 leads to the following description of

waves generated by a source of frequency Ω. A pair of waves k+1 and k+

2 and a pairof waves k+

3 and k−3 are generated in the interval 0 < Ω < ω1. Then the first pair isobserved upstream, and the second pair is observed downstream. The k+

1 -wave grows,and the k+

2 -wave decays downstream. The k+3 -wave has a negative group velocity and

negative energy and transports energy toward the source. The k−3 -wave is an ordinarywave, which has a negative group velocity and positive energy and transports energyaway from the source. The k+

1 and k+2 -waves merge at the point Ω = ω1. Then in

the interval ω1 < Ω < ω2 the k+2 -wave has a positive group velocity and negative

energy and transports energy towards the source. It is an anomalous wave as in thecase U = 1. The source must act as an energy sink for this wave. It encounters theanomalous k+

3 -wave at the point Ω = ω2, k = k2. Both waves decay downstream andupstream, respectively, at high frequencies. In the interval ω1 < Ω < ω0 the k+

1 -wavehas a negative group velocity and negative energy, and transports a positive energyflux. It is most instructive to trace the path of the root k+

1 as the imaginary part ofthe frequency increases from zero. Initially it drops down the real axis on the k-plane(since the group velocity is negative) and then ascends into the upper half-plane;the root differs considerably from the others in its local and global behaviour. Thek+

1 -wave changes to an ordinary positive energy wave above the point ω0.An analysis of the paths of the roots by means of the Cardan solutions shows

that they remain unchanged in the entire region 1 < U < U∗. On the other hand,the behaviour of the roots in the stable region U < 1 is similar to the case U = 1.

The situations treated here are typical and occur in the analysis of waves in flow-loaded structures and in the investigation of waves in flows (Briggs, 1964), (Crightonand Oswell, 1991) and (Leib and M. E. Goldstein, 1986). It is necessary to proceedfrom the causality principle in any such case, i.e. to trace the paths of the roots inthe limit of a large positive imaginary part of the frequency. Before this analysis ofan unstable system, however, the type of instability must be ascertained. Positiveand negative energy waves can be generated in flow systems. Negative energy wavescan be ordinary waves, i.e. transport energy away from the source, or they can beanomalous, extracting energy from the flow and delivering it to the source. Theenergy flux and the group velocity have opposite signs for ordinary negative-energywaves. As a result, the path of the root first enters the half-plane determined bythe sign of the group velocity and then transfers to the other half-plane (see, e.g.(Briggs, 1964), (Crighton and Oswell, 1991) and (Leib and M. E. Goldstein, 1986),for examples of the situations described above.)


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212 CHAPTER 4

CONTENTS CONTENTS CONTENTS

4 PULSES IN SHEAR FLOWS 1294.1 Dispersion of a Wave Packet in a Shear Flow . . . . . . . . . . . . . . 129

4.1.1 Linearized equation for small perturbations in a plane- parallelshear flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

4.1.2 Evolution of the 2D perturbations in the Couette flow . . . . . 1314.1.3 The concept of a CS-Mode . . . . . . . . . . . . . . . . . . . . 1344.1.4 Historical remark . . . . . . . . . . . . . . . . . . . . . . . . . 1364.1.5 Temporary growth of perturbations in the Couette flow . . . . 136

4.2 Structural Stability of the CS-Mode . . . . . . . . . . . . . . . . . . . 1384.2.1 The effect of the curvature of velocity profile . . . . . . . . . . 1394.2.2 The effects of viscosity . . . . . . . . . . . . . . . . . . . . . . 1454.2.3 The effects of stratification . . . . . . . . . . . . . . . . . . . . 1474.2.4 The non-linear effects . . . . . . . . . . . . . . . . . . . . . . . 1514.2.5 The 3D effects . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

4.3 Quasi-Eigen (QE) Modes in Ideal Fluid Flows . . . . . . . . . . . . . 1554.3.1 Rayleigh’s theorem and the problem of decaying eigenmodes

existence in a flow without points of inflection of velocity profile 1564.3.2 Evolutionary problems . . . . . . . . . . . . . . . . . . . . . . 157

4.4 The Green’s Function of the Rayleigh Equation for a Flow with aDiscrete Spectrum Mode . . . . . . . . . . . . . . . . . . . . . . . . . 1664.4.1 Piece-wise linear profile . . . . . . . . . . . . . . . . . . . . . . 1664.4.2 A velocity profile with a small curvature . . . . . . . . . . . . 1694.4.3 A Long-wavelengths approximation for the Green’s function . 171

4.5 Localized Source for CS-Mode . . . . . . . . . . . . . . . . . . . . . . 1784.5.1 The Couette flow . . . . . . . . . . . . . . . . . . . . . . . . . 1784.5.2 Slightly curved profile . . . . . . . . . . . . . . . . . . . . . . 1804.5.3 A stratified shear flow . . . . . . . . . . . . . . . . . . . . . . 1834.5.4 Nonstationary case . . . . . . . . . . . . . . . . . . . . . . . . 1874.5.5 Remark on the algebraic instability . . . . . . . . . . . . . . . 192

4.6 Pulse Propagation in Unstable Media . . . . . . . . . . . . . . . . . . 1944.6.1 The basic model . . . . . . . . . . . . . . . . . . . . . . . . . 1944.6.2 An instantaneous source . . . . . . . . . . . . . . . . . . . . . 1954.6.3 Switching on an oscillating source . . . . . . . . . . . . . . . . 198

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Pulses in Shear Flows_sazonov