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Endwall effects on the transition between Taylor vortices and spiral vortices
Sebastian Altmeyer1, Christian Hoffmann1, Matti Heise2, Alexander Pinter1, Manfred L¨ ucke1, and Gerd Pfister2
1Institut f¨ ur Theoretische Physik, Universit¨ at des Saarlandes, D66123 Saarbr¨ ucken, Germany
2Institut f¨ ur Experimentelle und Angewandte Physik, Universit¨ at Kiel, D24098 Kiel, Germany
We present numerical simulations as well as experimental results concerning transitions between
Taylor vortices and spiral vortices in the TaylorCouette system with rigid, nonrotating endwalls
in axial direction. As in the axial periodic case, these transitions are performed by wavy structures
appearing via a secondary bifurcation out of Taylor vortices and spirals, respectively. But in the
presence of rigid lids, pure spiral solutions do not occur but are substituted by primary bifurcating,
stable wavy spiral structures (wSPI). Similarly to the periodic system, we found a transition from
Taylor vortices to wSPI mediated by so called wavy Taylor vortices (wTVF) and, on the other hand,
a transition from wSPI to TVF triggered by a propagating defect. We furthermore observed and
investigated the primary bifurcation of wSPI out of basic Ekman flow.
PACS numbers:
Keywords:
47.20.Ky, 47.32.cf, 47.54.r
Taylor vortices, Spiral vortices, Wavy structures, Secondary bifurcations
I.INTRODUCTION
The interaction between Taylor vortex flow (TVF), spi
rals (SPI), and a variety of different wavy solutions was
investigated in numerous publications [1, 2, 3, 4, 5, 6, 7,
8, 9].
Under periodic boundary conditions (pbc), toroidally
closed TVF appears via a primary stationary bifurcation
out of the rotationally symmetric, axially homogeneous
basic circular Couette flow (CCF). Also the two axially
symmetry degenerated, oscillatory SPI states appear via
primary bifurcations out of CCF in a symmetric Hopf
bifurcation together with the ribbon state. The latter
is typically unstable close to onset and can be seen as
a nonlinear superposition of the two oppositely propa
gating spirals to an axially standing wave. The stability
of TVF and SPI at onset is regulated by the order of
their appearance upon increasing the inner cylinder’s ro
tation rate: the first (second) solution to bifurcate out of
CCF is stable (unstable). However, the second unstable
solution becomes stable at larger inner cylinder rotation.
Which state bifurcates first depends on the outer cylinder
rotation rate [10].
Besides parameter regions with monostability of TVF
and SPI, one also observes regions with bistability of both
RIB
wSPI
TVF
SPI
wTVF
FIG. 1:
a suitably chosen control parameter. Stable (unstable) solu
tions are displayed as solid (dashed) lines. Thin arrows in
dicate the transients corresponding to the ’jump’ bifurcation
mentioned in [1].
(Color online) Schematic bifurcation diagram for
states [11]. When moving a control parameter out of this
region, one solution loses its stability and the flow un
dergoes a transition to the remaining stable state, i.e.,
from TVF to SPI or vice versa [11]. Bifurcation the
oretical considerations and symmetry arguments [1, 2]
as well as amplitude expansion techniques in [3] and nu
merical simulations in [9] show that the stability is trans
fered from TVF to SPI via secondarily bifurcating wavy
Taylor vortices (wTVF) [3, 4] and vice versa from SPI
to TVF via secondarily bifurcating wavy spirals (wSPI).
The solution branch of stable TVF (SPI) is connected to
unstable ribbons [12] via stable wTVF (wSPI). Then, a
’jump’ bifurcation [1] from the end of the stable wTVF
(wSPI) branch leads to the stable SPI (TVF) branch.
This bifurcation behavior is schematically illustrated in
Fig. 1.
Note that in the majority of publications, the wTVF
solution branch has been seen to return to the TVF
branch or to undergo higher order bifurcations [4, 6, 7,
8, 13, 14] at larger driving.
On the other hand, rigid nonrotating lids at the axial
ends (rigid boundary conditions, rbc) induce rotational
symmetric Ekman vortices even for subcritical driving
[10]. This modifies the structure, stability and bifurca
tion behavior of the different states.
One important difference between pbc and rbc is the
absence of pure SPI solutions for rbc, in particular, for
moderate outer cylinder rotation, i.e. within the left part
of Fig. 1. The rotationalsymmetric Ekman modes inter
act with the spiral modes leading to wSPI which bifurcate
primarily out of the basic Ekman state and play a similar
role as the SPI under pbc [9].
Furthermore, under rbc, wSPI loses its stability for
stronger inner cylinder rotation and we found traveling
defects which trigger the transition from wSPI to stable
TVF.
In [15], we described a situation where a similar defect
which separates domains of oppositely traveling spiral
waves propagates through the system and trigger a tran
sition from righthanded to lefthanded spirals or vice
arXiv:0909.1143v1 [nlin.PS] 7 Sep 2009
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versa. Thus, the transition from wSPI to TVF is a fur
ther example for a transition which is triggered by a prop
agating defect.
This paper elucidates how TVF are transformed into
SPI and vice versa under the presence of Ekman induced
disturbances, how stability is transferred between the
branches, and where and what kind of transients occur
under rbc. It is roughly divided into two parts corre
sponding to the transition from TVF to wSPI via wTVF
and the transition from wSPI to TVF via a propagat
ing defect. Structural dynamics, frequencies and wave
number selection are discussed and a comparison of the
results obtained for rbc and pbc is made.
II.SYSTEM
The TaylorCouette system consists of a fluid of kine
matic viscosity ν in the gap between two concentric, in
dependently rotating cylinders (inner, outer radius r1,2;
angular velocities Ω1,2; fixed radius ratio η = r1/r2= 0.5
and fixed length Γ = 12 in units of the gap width
d = r2− r1) and nonrotating, rigid lids at the axial
ends.
Cylindrical coordinates r,ϕ,z are used to decompose
the velocity field into a radial component u, an azimuthal
one v, and an axial one w
u = uer+ v eϕ+ wez.
(2.1)
The system is governed by the NavierStokes equations
∂tu = ∇2u − (u · ∇)u − ∇p.
Here, lengths are scaled by the gap width d and times
by the radial diffusion time d2/ν for momentum across
the gap and the pressure p by ρν2/d2. The Reynolds
numbers
(2.2)
R1= r1Ω1d/ν,R2= r2Ω2d/ν
(2.3)
enter into the boundary conditions for v. R1 and R2
are just the reduced azimuthal velocities of the fluid at
the cylinder surfaces. Within this paper, we hold fixed
R2= −100.
For numerical simulations, we used the G1D3 code de
scribed in [11, 16], i.e. a combination of a finite dif
ferences method in radial r and axial z direction and a
Galerkin expansion in ϕ direction:
f(r,ϕ,z,t) =
?
f ∈ {u,v,w,p}
m
fm(r,z,t)eimϕ,
(2.4)
In the experimental setup, the inner cylinder (ri =
(12.50±0.01) mm) is machined from stainless steel, while
the outer cylinder (r2= (25.00±0.01) mm) is made from
optically polished glass. As fluid thermostatically con
trolled silicone oil with a kinematic viscosity ν = 10.6 cS
is used. At top and bottom the flow is confined by mas
sive endwalls with a till better than 0.03 mm at the outer
diameter. The flow is visualized by elliptical aluminum
particles having a length of 80 µm. Flow visualization
measurements are performed by monitoring the system
with a CCDcamera in front of the cylinder recording
the luminosity along a narrow axial stripe. The spatio
temporal behavior of the flow is then represented by suc
cessive stripes for each time step at a constant ϕ position
leading to continuous space time plots.
III.TRANSITIONS BETWEEN TVF AND WSPI
As in the periodic system [9], we also found transitions
between the two primary bifurcating structures TVF and
(w)SPI in the finite length system. The main difference
between both is that we do not observe (neither in sim
ulations nor in experiments) pure spirals for Γ = 12 sys
tems, i.e. helical structures with the continuous symme
try given by f(r,ϕ,z,t) = f(r,kz+Mϕ−ωt) for fixed M,
k, and ω (c.f. [11]). In finite systems with nonrotating
lids generating Ekman vortices with an exponentially de
caying amplitude in axial direction, pure SPI are replaced
by wSPI with a more complex mode spectrum as de
scribed in [9]. This is due to the interaction between the
spiral and the Ekman modes. This section describes first
the bifurcation from TVF to wSPI via wTVF and after
that the transition from wSPI to TVF.
A. Bifurcation from TVF to wSPI
Fig. 2 depicts the bifurcation branches for the interest
ing states (w)SPI, (w)TVF, and ribbons. The different
structures are distinguished by symbols and line colors
and characterized by amplitudes and frequencies of their
significant Fourier modes (m,k) determined by the az
imuthal wave number m and the axial wave number k of
the complete structure. The latter is included in the leg
end box. Solid (dashed) lines with filled (open) symbols
represent stable (unstable) solutions. Fig. 2(a) and (b)
present mode amplitudes um,k of the radial velocity field
u in the finite case (a) and the periodic case (b), respec
tively. (c) and (d) show the corresponding frequencies
ωm,k.
With our way of characterizing the flow by the com
bination of azimuthal and axial Fourier modes, with the
latter being obtained over the full axial extension of the
system, we do not capture, e.g. the Ekman induced axial
variation of m = 0 modes.
We start our discussion of the bifurcation diagram in
Fig. 2(a) in region E’ with a stable k = 4.85 TVF state
which loses its stability in region F’ and C’. This is ex
actly the same behavior as under pbc (b), except that
the stability thresholds are slightly shifted, that the un
stable TVF branch bifurcates out of the k = 3.95 Ekman
state in (a) instead of the CCF as in (b), and – more
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3
0
0.5
1
1.5
2
um,k
(m=0,k=4.85)
(1,3.95)L
(1,3.95)R
(0,4.85)
A’
(a)
(1,4.85)L&(1,4.85)R
F’ E’
(0,3.95)
EkmanC’
(0,4.85)
TVF (k=4.85)
SPI (k=4.85)
SPI (k=3.95)
wSPI (k=3.95)
wTVF (k=4.85)
RIB (k=4.85)
0
1
2
3
4
um,k
(b)
(1,4.85)R
(1,4.85)L&(1,4.85)R
(0,4.85)
(0,4.85)
(1,4.85)L&(1,4.85)R
EF
(1,3.95)R
CCCFA
105110
115
120
R1
25
30
35
ωm,k
(1,3.95)R
(1,4.85)R
(1,4.85)L & (1,4.85)R
(1,4.85)L & (1,4.85)R
FE
(d)
A CCFC
28
30
32
34
ωm,k
(c)
E’ F’
(1,4.85)L & (1,4.85)R
(1,3.95)R
A’EkmanC’
FIG. 2: (Color online) Numerically obtained bifurcation di
agrams for different vortex structures TVF (blue, circles),
SPI (red, triangles), wTVF (black, squares), wSPI (gray, di
amonds), and ribbons (RIB, green, diamonds) versus R1 for
rbc (a,c) as well as for pbc (b,d). Curves with the same color,
symbol, and linewidth represent different modes of the same
solution. Solid (dashed) lines with filled (open) symbols refer
to stable (unstable) states. Shown are the significant radial
flow field amplitude modes um,k at midgap (a,b) and the
corresponding frequencies ωm,k (c,d). The indices R and L
correspond to right and lefthanded spiral modes. The short
arrows pointing to the abscissa in (a) denote the R1 values
of the snapshots in Fig. 3, the long arrow in (a) indicates the
direction for the transition TVF −→ wTVF −→ wSPI. The
sections labelled with letters correspond to different stability
regions as listed in the table (c.f. Fig. 3 in [9]):
region A A’ C C’ F F’ E E’
TVF
SPI
wSPI
wTVF 
RIB

s

 u u u u s s
ss  s
ss s 
s s 
 u  u  u 

s

u
stable (s)
unstable (u)
nonexistent ()
importantly – that the final state is a wSPI and not a
pure SPI. We omit the Ekman branch itself in the figure
due to visibility reasons.
In F’ (and also in F), TVF becomes unstable against
wTVF maintaining the same wave number (k = 4.85).
Note that in (b), the wave number is determined by the
predefined periodicity length. At the right border of C
(C’), wTVF undergoes a transition to the remaining sta
ble SPI (wSPI) solution with k = 4.85 (k = 3.95), i.e., the
wave number changes during this transition – c.f. [9] for a
detailed description of the stability properties, the bifur
cation behavior, and the structure of wTVF and wSPI.
We added the corresponding k = 3.95 SPI solution
branch in (b) in order to emphasize the identical onsets
of k = 3.95 SPI (b) and k = 3.95 wSPI (a). As the SPI
solution in (b), also the wSPI solution in (a) is stable
within the whole parameter range displayed here. The
transition from wTVF to SPI (wSPI) includes an unsta
ble transient ribbon state, but we did not try to stabilize
this state for rbc.
Generally speaking, there are three major aspects con
cerning the finite and the periodic system: (i) wSPI in
(a) play quite the same role as the pure SPI in (b) – we
indicated this by the prime at the labels A,C,E,F distin
guishing the different stability regions (c.f. [9, 17]). (ii)
for transition TVF −→ wSPI, the finite Γ = 12 system
selects the same wave number k = 4.85 for all toroidally
closed structures (TVF, wTVF) and k = 3.95 for the
helical solution (wSPI). Thus, transitions from TVF to
wSPI are generally accompanied by a change in the wave
number. (iii) finite boundary conditions superimpose ro
tational symmetric disturbances. Therefore, all thresh
olds (dotted vertical lines in Fig. 2(a,b)) of solutions with
rotational or toroidal symmetry are shifted towards lower
R1compared to the respective thresholds in the periodic
system. The wTVF onsets EF and E’F’ coincide very
well in both cases.
In Fig. 2(a), the dominant mode (0,4.85) of the un
stable k = 4.85 TVF branch ends up in the (0,4.85) Ek
man mode at A’C’ which is a subdominant mode in the
k = 3.95 Ekman state. Therefore, we included the mode
(0,4.85) which is a higher Fourier mode of the k = 3.95
wSPI state.
Note that due to the absence of symmetry breaking
effects like axial throughflow, righthanded and left
handed spiral solutions are equivalent [11, 17] and there
fore simply indicated by (w)SPI.
1.Frequencies
Fig. 2(c) and (d) provide the frequencies ωm,k of the
corresponding mode amplitudes in (a) and (b). We omit
those frequencies of TVF and wTVF which are zero.
SPI – for pbc (d), spirals and ribbons grow via a pri
mary Hopf bifurcation with a common frequency out of
CCF. The difference between the spiral frequencies for
pbc (d) and rbc (c) is a consequence of the Reynolds
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stress driven (intrinsic) axial net flow which is directed
oppositely to the spiral propagation. In finite systems,
this net flow is suppressed by impermeable lids which
leads to a shift in the axial phase velocity and thereby
also in the frequency [11]. This effect can also be seen for
the wSPI in (c) and the SPI in (d): whereas the frequen
cies of both spirals (k = 4.85 and k = 3.95) are nearly
identical at EF, the frequencies of the k = 3.95 SPI at
EF and the k = 3.95 wSPI at E’F’ differ.
wTVF – On the other hand, the wTVF frequencies
at the bifurcation thresholds EF and E’F’ are almost
identical. Here, the intrinsic net flow of the (1,4.85)L
mode is compensated by that of the (1,4.85)Rmode in
both cases (c) and (d). Moreover, the variation of wTVF
and SPI frequencies in region F differs significantly.
Since wTVF is a timeperiodic rotating state that does
not propagate axially, all mode frequencies are either zero
(ω0,4.85= 0) or multiples of ω1,4.85. So, the dynamics of
wTVF is rather simple while the spatial structure is more
complex.
2.Spatiotemporal behavior
In order to elucidate the different states arising dur
ing the transition TVF (snapshot #5) −→ wTVF (#4,
#3) −→ wSPI (#2, #1), Fig. 3(a) gives isosurfaces of
the azimuthal vorticity ∂zu − ∂rw of stationary states
at different R1values which are marked by short arrows
pointing to the abscissa of Fig. 2(a).
The pure k = 4.85 TVF state (#5) becomes unstable
against toroidally closed but axially modulated k = 4.85
wTVF (#4). Obviously, the modulation strength in
creases towards midsystem where the Ekman influence
is minimal. As the m ?= 0 mode contributions grow, the
formerly rotational symmetric structure becomes more
and more deformed and the vorticity tubes narrow at a
certain ϕ position (#4). This means that the maximal
vorticity within the (r,z) plane at this ϕ position de
creases with R1– the vortex ’intensity’ becomes weaker
there. Note that this indentation of the vortex tubes as
well as the defect rotate with the whole structure.
Finally, the isosurfaces are completely constricted and
separated (#3). After displacing the ends of the tubes,
new connections are established and the vorticity in
creases now to the final distribution in the k = 3.95 wSPI
(#2).
The last snapshot, (#1), depicts a situation in which
the system is very weakly supercritical and therefore, the
Ekman vortices remain the dominant structure.
Fig. 3(b) presents the experimentally obtained spatio
temporal behavior describing the dynamics of the differ
ent states after an initial jump from R1= 115 to 109 at
the right border of the plot beginning with a pure TVF
state in (#5) with k ≈ 4.8 which then undergoes (begin
ning at midheight) a transition to wTVF with the same
axial wave number (#4). After a transient (#3) which
corresponds to the ’jump’ bifurcation described above,
FIG. 3: (Color online) (a) Numerical simulations: snapshots
of isosurfaces of the azimuthal vorticity ∂zu−∂rw = ±40 (red:
+40, green: 40) at five different R1 values marked by arrows
in Fig. 2(a) during the transition TVF −→ wTVF −→ wSPI
(from right to left). Red (green) coloring on the additional
ϕ =const. plane denotes positive (negative) vorticity. We use
4π cylinders in (a) in order to present the whole structure in
one single 3dimensional plot. (b) Experimentally obtained
spatiotemporal flow visualization of the transition which was
triggered by an initial jump from R1 = 115 to 109. The plots
cover the complete system length of Γ = 12.
wSPI with k = 3.6 are finally established (#2). Note
that we also verified in further experiments the stabil
ity and stationarity of the TVF (#5), wTVF (#4), and
wSPI (#2) states for suitable fixed control parameters
R1.
3. Wave number selection
Due to the finite boundary conditions, the toroidally
closed structures (TVF, wTVF) can occur with discretely
different axial wave numbers depending on the initial con
ditions. We found at least three stable TVF states with
7 (k = 3.83), 8 (k = 4.85), and 9 (k = 5.81) vortex
pairs in region E’ (only the 8 vortex pairs TVF state is
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100105 110
115
R1
120125130
0
0.5
1
1.5
2
um,k
(m=0,k=3.83)
(1,3.95)R
(1,3.95)L
Ekman
(0,3.95)
P1
P2
TVF
wSPI
FIG. 4:
tion there. Short arrows pointing to the abscissa identify the
snapshots in Fig. 5(a). The long arrow indicates the direction
of the transition wSPI −→ TVF.
(Color online) Extension of Fig. 2(a) – see descrip
presented in Fig. 2 and 3). All of them undergo a transi
tion to wTVF in region F’ for specific R1values without
changing their respective wave number k. Finally, all
wTVF states ’jump’ (accompanied by a change in k) to
the k = 3.95 wSPI solution.
The experimentally (k = 4.53 for (w)TVF and k =
4.03 for wSPI) and the numerically (k = 4.85 for (w)TVF
and k = 3.95 for wSPI) obtained axial wave numbers
differ slightly. However, numerical simulations as well as
experimental results exhibit the same jumps in k during
the transition wTVF −→ wSPI.
B.Transition from wSPI to TVF
Under pbc, one finds the bifurcation sequence SPI −→
wSPI −→ TVF. While the transition from wSPI to TVF
is mediated by a ’jump’ bifurcation [1, 9], pure SPI, on
the other hand, could not be observed neither in rbc sim
ulations nor in experiments.
As described above for rbc, wSPI bifurcates primar
ily out of the basic Ekman flow as a stable solution for
stronger counterrotation. However, we found wSPI to
become unstable against TVF via an other kind of tran
sition taking place beyond region E’ of Fig. 2(a). This
transition is mediated by a propagating defect which sep
arates the wSPI from a wTVF regime pushing wSPI out
and pulling wTVF through the bulk. Once the defect
crossed the whole bulk, the modulation amplitude of the
wTVF vanishes and pure TVF remains.
Fig. 4 gives a enhanced version of the bifurcation di
agram in Fig. 2(a) with a slightly extended R1 range
covering the transient wSPI −→ TVF transition in the
gray marked region P2. The short arrows mark the R1
values for which isovorticity snapshots are presented in
Fig. 5(a).
Starting with a pure wSPI solution at small R1, an ad
ditional defect is generated near the upper Ekmanspiral
defect after increasing R1beyond the left border of region
P1and remains at its axial position for any R1within the
whole region P1. At the right border of P1, the defect
begins to propagate towards the other axial end. This is
a transient state which ends up in a pure k = 3.83 TVF
solution after the annihilation of the propagating spiral
spiral defect at the lower Ekmanspiral defect. The gray
marked region P2 gives the behavior of the amplitudes
during this transient from wSPI to TVF which occurs
within the range 120 < R1 < 122. Note that the fi
nal TVF (k = 3.83) is different from that discussed in
Fig. 2(a) which has k = 4.85. This TVF state is one of
several stable TVF states with different wave numbers
(c.f. Sec. IIIA3).
This transition sequence agrees with experimental re
sults as presented in Fig. 5(b) showing a spatiotemporal
flow visualization of the transition even after an initial
jump from R1= 107 to 120 at the left border of the plot.
1.Spatiotemporal behavior
The arrows in Fig. 4 mark the R1 values of the ten
snapshots of Fig. 5 depicting the isosurfaces of the az
imuthal vorticity (a) as well as a spatiotemporal plot of
the experimentally obtained velocity field (b). Both se
quences illustrate the structural changes during the tran
sition wSPI −→ TVF mediated by a propagating defect.
Starting with a wSPI state (#1) in Fig. 5 and increas
ing R1, the second upper closed vortex becomes wavylike
deformed in axial direction (#2#5) while the deforma
tion rotates with the whole structure. The modulation
becomes stronger with increasing R1until a second defect
evolves out of the phase generating Ekman spiral defect
(#2).
In (#5), this defect detaches from the upper Ekman
vortex and begins to propagate (#6)(#9), pushing the
wSPI and pulling a wTVF state through the bulk. There
fore, the localized wSPI region shrinks and the localized
wTVF domain grows. During this process, neither the
wave number of wSPI nor that of wTVF change signif
icantly, because new wTVF vortices are generated di
rectly behind the propagating defect: comparing (#7)
and (#8), one observes that the vortex tubes become
constricted and separated and after displacing the ends
of the tubes, new connections are established and a new
wTVF vortex is generated. Finally, the defect reaches
the bottom end and merges with the lower Ekman vor
tex (#10) leaving behind a pure TVF state.
As described above, Fig. 5(b) elucidates the spatio
temporal behavior during the transition after an initial
jump from R1= 107 to 120 at the left border of the plot.
As in the numerical simulations, the experimental setup
realizes a transition from wSPI to TVF via a propagating
defect. Furthermore, the experimental wave numbers for
wSPI k = 4.03 and TVF k = 3.84 agree very well with
the numerical wave numbers for wSPI k = 3.95 and TVF
k = 3.83.
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FIG. 5: (Color online) (a) Numerical simulations: snapshots of isosurfaces of the azimuthal vorticity ∂zu − ∂rw = ±40 (red:
+40, green: 40) of flow states at ten different R1values marked by short arrows pointing to the abscissa in Fig. 4 and visualizing
the transition wSPI −→ TVF while increasing R1. Red (green) coloring on the additional ϕ =const. planes denote positive
(negative) vorticity. 4π cylinders are used in order to present the whole structure in one single 3dimensional plot. Note that
the propagating defect (#5#9) is a transient state. (b) Experimentally obtained spatiotemporal flow visualization of the
transition which was triggered by an initial jump from R1 = 107 to 120. In axial direction, the plots cover the complete system
length of Γ = 12.
We’d like to stress that in the here presented experi
ments, the transitions are initiated by an instantaneous
jump (after preparing the initial state) into the param
eter regime where the final structure was expected to
be stable. The experimentally obtained spatiotemporal
plots disclose a sequence of transient structures which
correspond to stationary structures when quasistatically
driving the system by a R1ramp. We indicated the cor
responding structures by identical numbers of the snap
shots in Fig. 3 and 5.
IV. SUMMARY
We investigated the bifurcation behavior for the tran
sition between Taylor vortices and wavy spirals in a finite
length TaylorCouette system with nonrotating, rigid
lids. In contrast to periodic boundary conditions where
pure SPI solutions exist (even for outer cylinder at rest),
here, helical solutions occur as wavy structures due to the
admixture of Ekman induced m = 0 mode components
in the Fourier spectra.
Under finite system geometry, we found a transition
wTVF(k )
wTVF(k )
defect
0
TVF(k )
1
2
TVF(k )
2
1
RIB
wSPI(k )
FIG. 6:
sen control parameter containing the results of Fig. 2(a) and
Fig. 4 for the finite system (in contrast to Fig. 1 for the pe
riodic system). Stable (unstable) solutions are displayed as
solid (dashed) lines. Thin arrows indicate the transients cor
responding to the ’jump’ bifurcation. We included the ribbon
branch as an intermediate unstable solution.
Schematic bifurcation diagram for a suitable cho
from TVF to wSPI via wTVF which is analogue to
the transition from TVF to SPI via wTVF in the peri
odic system. wSPI (helical open vortices) are selected
with a distinct wave number k0 whereas several bi
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furcation branches corresponding to TVF and wTVF
states (toroidally closed vortices) exist simultaneously
and multistably with different axial wave numbers (e.g.
k1 and k2), i.e.different numbers of vortex pairs (as
schematically depicted in Fig. 6). The transitions from
TVF to wSPI and vice versa are in general accompanied
by a change in k.
An other kind of transition performing the change from
wSPI to TVF is triggered via a propagating defect. This
defect pushes the wSPI out of the system and pulls wTVF
and finally TVF into the bulk.
The coincidence of the wTVF frequency and the SPI
frequency at the bifurcation point disappears in the finite
system.
Acknowledgement
We thank the Deutsche Forschungsgemeinschaft for
support.
[1] M. Golubitsky, I. Stewart, and D. Schaeffer, Singulari
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