Controlling the stability transfer between oppositely traveling waves and standing waves by inversion-symmetry-breaking perturbations.

Abstract

The effect of an externally applied flow on symmetry degenerated waves propagating into opposite directions and standing waves that exchange stability with the traveling waves via mixed states is analyzed. Wave structures that consist of spiral vortices in the counter rotating Taylor-Couette system are investigated by full numerical simulations and explained quantitatively by amplitude equations containing quintic coupling terms. The latter are appropriate to describe the influence of inversion-symmetry-breaking perturbations on many oscillatory instabilities with O(2) symmetry.

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arXiv:physics/0611132v2 [physics.flu-dyn] 6 Jul 2007
Controlling the stability transfer between oppositely traveling
waves and standing waves by inversion-symmetry-breaking
perturbations
A. Pinter∗, M. L¨ ucke, and Ch. Hoffmann
Institut f¨ ur Theoretische Physik,
Universit¨ at des Saarlandes,
Postfach 151150,
D-66041 Saarbr¨ ucken, Germany
(Dated: February 2, 2008)
Abstract
The effect of an externally applied flow on symmetry degenerated waves propagating into opposite
directions and standing waves that exchange stability with the traveling waves via mixed states is
analyzed. Wave structures that consist of spiral vortices in the counter rotating Taylor-Couette
system are investigated by full numerical simulations and explained quantitatively by amplitude
equations containing quintic coupling terms. The latter are appropriate to describe the influence of
inversion symmetry breaking perturbations on many oscillatory instabilities with O(2) symmetry.
PACS numbers: 47.20.Ky, 47.54.-r, 47.32.-y
∗Electronic address: kontakt@alexander-pinter.de
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Many nonlinear structure forming systems that are driven out of equilibrium show a
transition to traveling waves (TWs) as a result of an oscillatory instability [1]. In the
presence of inversion symmetry in one or two spatial directions also a standing wave (SW)
solution bifurcates in addition to the symmetry degenerate, oppositely propagating TWs
at the same threshold. Moreover, depending on the parameters one can have a stability
exchange between TWs and SWs as the driving rate varies. The stability transfer is mediated
by mixed patterns that establish in the solution space a connection between a pure TW and
a pure SW.
Here we investigate how externally tunable symmetry breaking perturbations change
stability, bifurcation properties, and the spatiotemporal behavior of the afore mentioned
structures. To be concrete we investigate vortex waves in the annular gap between counter
rotating concentric cylinders of the Taylor-Couette system [2, 3]. To that end we performed
full numerical simulations of the Navier-Stokes equations with methods described in [4]. We
elucidate that and how these results can be explained quantitatively by amplitude equations
which contain only quintic order terms.
The perturbation is realized in our system by an externally enforced axial through-flow
that can easily be controlled experimentally. However, our results concerning the influence
of inversion symmetry breaking perturbations on TWs, SWs, and on the mixed states are
more general: the quintic order amplitude equations with small symmetry breaking terms
apply to all kinds of O(2) symmetric oscillatory instabilities in the presence of inversion
symmetry breaking perturbations.
Structures – Without symmetry breaking through-flow the following oscillatory vortex
structures occur at small driving [5, 6]: (i) Forward bifurcating TWs consisting of left
handed spiral vortices (L-SPI) or of right handed spiral vortices (R-SPI) that are mirror
images of each other. L-SPI (R-SPI) travel in the annulus between the two cylinders into
(opposite to) the direction of the rotation frequency vector of the inner one, i.e., in our
notation upwards (downwards) [4]. (ii) Forward bifurcating SWs that consist of an equal-
amplitude nonlinear combination of L-SPI and R-SPI that are called ribbons (RIBs) in the
Taylor-Couette literature [7, 8]. (iii) So-called cross-spirals (CR-SPI), i.e., combinations of
L-SPI and R-SPI with different amplitudes. They provide a stability transferring connection
between TW and SW solution branches [3, 6]. The vortex structures (i)-(iii) are axially and
azimuthally periodic with wave numbers k = 2π/λ and M, respectively, with λ = 1.3 and

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M = 2 throughout this paper. They rotate with characteristic constant angular velocities
as a whole into the same direction as the inner cylinder [4]. Thereby they are forced to
propagate axially with the exception of RIB vortices which rotate only but do not propagate.
Order parameters – Figures 1-2 show bifurcation diagrams of SPI, CR-SPI, and RIB
solutions in a system of radius ratio η = 1/2 for fixed R1= 240 versus the reduced distance
µ = (R2− R0
2)/|R0
2| from the spiral onset R0
2in the absence of through-flow, Re = 0. Here,
R1and R2are the Reynolds numbers defined by the rotational velocities of the inner and
outer cylinder, respectively, and Re is the Reynolds number of the imposed axial through-
flow. In Figs. 1 and 2, the influence of a small through-flow (Re = 0.02) is compared with
the situation without through-flow. Order parameters in Fig. 1 are the squared amplitudes
|A|2,|B|2of the dominant critical modes
u2,1(t) = |A|e−iωAt,u2,−1(t) = |B|e−iωBt
(1)
in the double Fourier decomposition of the radial velocity u at mid-gap in azimuthal and
axial direction. The indices 2 and ±1 identify azimuthal and axial modes, respectively. The
linear stability analysis of the basic state consisting of a superposition of circular Couette
flow and of annular Poiseuille flow in axial direction shows that the growth rates of these
modes become positive at the respective bifurcation thresholds µAand µBof M = 2 L-SPI
and R-SPI, respectively. Note also that for all relaxed vortex structures investigated here
the moduli and frequencies of u2,±1in Eq. (1) are constant.
In Fig. 2 we show in addition the bifurcation diagrams of the combinations
S =?|A|2+ |B|2?/2,D =?|A|2− |B|2?/2(2)
since they are convenient to describe in particular CR-SPI.
Bifurcation scenario for zero through-flow – In the symmetry degenerate case without
through-flow L-SPI (A ?= 0 = B,D > 0) and R-SPI (A = 0 ?= B,D < 0) have identical
bifurcation properties. They are stable close to onset whereas RIB (A = B,D = 0) are
initially unstable. In the driving range shown in figures 1 and 2 the squared amplitudes of
these two states grow practically linearly with the reduced distance µ from the common onset
at µ = 0, albeit with different slopes. Then, there appear in a finite supercritical driving
interval stable CR-SPI solutions which transfer stability from SPI to RIB. The solution
which bifurcates with B = 0 out of the L-SPI is identified as a L-CR-SPI with |A| > |B|,

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i.e., D > 0. The symmetry degenerate R-CR-SPI (|B| > |A|,D < 0) bifurcates with A = 0
out of the R-SPI. In the former |B| grows and |A| decreases – and vice versa in the latter
– until the CR-SPI branches end with A = B,D = 0 in the RIB solution. The amplitude
variations of the CR-SPI solutions, however, are such that the sum S remains practically
constant, cf. Fig. 2(a). The RIB state loses stability outside the plot range of figures 1 and
2 to another type of amplitude-modulated CR-SPI that are not discussed here.
Through-flow induced changes – The axial through-flow significantly perturbs and changes
structure, dynamics, and bifurcation behavior of the SPI vortex solutions discussed so far
[4, 9, 10, 11, 12]: RIBs cease to exist in the strict sense, L- and R-SPI are no longer
mirror images of each other, and also L-CR-SPI are no longer related to R-CR-SPI by this
symmetry operation. However, spirals retain their spatiotemporal structure in the through-
flow, i.e., they still do not depend on ϕ,z,t separately but only on the phase combination
φA= Mϕ + kz − ωAt or φB= Mϕ − kz − ωBt, respectively. L-SPI (red color in Figs. 1-2)
bifurcate for small Re > 0 at a threshold value µA< 0 prior to R-SPI (orange color) which
bifurcate at 0 < µB≈ −µAout of the basic state.
L-SPI are again stable at onset, but then lose stability to L-CR-SPI (violet color) which
remain stable in the plotted parameter regime. The L-CR-SPI solution approaches the
(Re = 0) RIB state with increasing µ, but retains with A ?= B a finite distance D > 0.
On the other hand, R-SPI are unstable for small and large µ, but stable for intermediate µ.
Stability is exchanged with a R-CR-SPI (magenta color ) which has a stable as well as an
unstable branch resulting from a saddle-node bifurcation at µSin Figs. 1, 2. For small µ the
unstable R-CR-SPI lies close to the (Re = 0) RIB solution and bifurcates with finite D < 0
out of the R-SPI slightly above µB. Note also that the sum of the squared amplitudes S of
CR-SPI is no longer constant as for the case of Re = 0.
Amplitude equations – The changes in spatiotemporal, bifurcation, and stability behavior
of SPI, CR-SPI, and RIB states by a small through-flow can be explained and described
close to onset quantitatively within an amplitude-equation approach. To demonstrate that
we focus here on the bifurcation properties of the moduli |A|,|B| of the critical modes.
In order to reproduce the bifurcation and stability behavior of the aforementioned vortex
states including the CR-SPI one needs coupled equations for A and B of at least quintic
order. Higher-order terms that are suggested in [3] are not necessary to ensure the existence
of CR-SPI solutions. Symmetry arguments [3, 6, 9] restrict the form of the equations for

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the moduli to
τAd|A|
dt
= |A|?(µ − µA) + bA|A|2+ cA|B|2+ eA
?|A|2|B|2− |B|4??,(3a)
τBd|B|
dt
= |B|?(µ − µB) + bB|B|2+ cB|A|2+ eB
?|B|2|A|2− |A|4??
(3b)
with real coefficients that depend in general on Re. Here we have discarded the quintic
terms |A|5and |B|5in view of the linear variation of the squared SPI moduli with µ, cf.
Fig. 1. Furthermore, we made a special choice for the coefficients of the last terms |A||B|4
and |B||A|4in Eq. (3) that is motivated by the linear variation of the squared RIB moduli
with µ for Re = 0 and that suffices to describe the behavior for small Re as well.
Coefficients – As a result of the inversion symmetry under z ↔ −z which includes revert-
ing the through-flow the coupled equations are invariant under the operation (A,B,Re) ↔
(B,A,−Re) so that the coefficients in Eq. (3) obey relations like, e.g., cA(Re) = cB(−Re).
To reproduce the bifurcation properties of the moduli for small Re as in Figs. 1, 2 it suffices
to incorporate the Re dependence to linear order in the coefficients
µA(Re) = −µ(1)Re,µB(Re) = µ(1)Re(4a)
cA(Re) = c + c(1)Re,cB(Re) = c − c(1)Re(4b)
only and to ignore any Re dependence of the others by setting
bA= bB= b,eA= eB= e.(4c)
The choice bA= bB= b reflects the fact that the linear growth of the squared SPI moduli
with the distance from their respective thresholds at µAand µBis unchanged by the through-
flow. On the other hand, the downwards shift of the L-SPI onset being for small through-
flow of equal magnitude as the upwards shift of the R-SPI onset is reflected by µA(Re) =
−µB(Re) = −µ(1)Re with positive µ(1). The coupling constant eA= eB = e ensures the
existence of CR-SPI solutions [6]. The flow induced changes of the coupling constants cA
and cBreflect the perturbation and destruction of the RIB states and ensures for positive
Re L-CR-SPI solutions with D > 0 when µ is large. The values of the coefficients [13] were
obtained from linear stability analyses and by fits to the full numerical nonlinear results.
Fixed points – With the coefficients (4) it is straightforward to derive from (3) the following