Effect of the dynamical phases on the nonlinear amplitudes' evolution

0.49 MB·English

by  Miguel D. Bustamante

Effect of the dynamical phases on the nonlinear amplitudes’ evolution
Miguel D. Bustamante† and Elena Kartashova∗
† School of Mathematical Sciences, University College Dublin, Belfield, Dublin 4, Ireland, EU
∗ RISC, J.Kepler University, Linz 4040, Austria,EU
InthisLetterweshowhowthenonlinearevolutionofaresonanttriaddependsonthespecialcom-
binationofthemodes’phaseschosenaccordingtotheresonanceconditions. Thisphasecombination
is called dynamical phase. Its evolution is studied for two integrable cases: a triad and a cluster
formed by two connected triads, using a numerical method which is fully validated by monitoring
the conserved quantities known analytically. We show that dynamical phases, usually regarded as
equal to zero or constants, play a substantial role in the dynamics of the clusters. Indeed, some
effects are (i) to diminish the period of energy exchange τ within a cluster by 20% and more; (ii)
to diminish, at time scale τ, the variability of wave energies by 25% and more; (iii) to generate
9 a new time scale, T >> τ, in which we observe considerable energy exchange within a cluster, as
0 well as a periodic behaviour (with period T) in the variability of modes’ energies. These findings
0 can be applied, for example, to the control of energy input,exchange and output in Tokamaks; for
2 explanationofsomeexperimentalresults; toguideandimprovetheperformanceofexperiments;to
interpret theresults of numerical simulations, etc.
n
a
J PACSnumbers: 47.27.Ak,47.27.ed,52.25.Fi

5
1
1. Introduction. Nonlinear resonances are ubiqui- sortofgeneralmisunderstandingpersists,concerningthe

] tous in physics. Euler equations, regarded with various relevanceofϕ forthe generaldynamicsofthe system. It
I
boundary conditions and specific values of some param- is a common belief that for an exact resonance to occur,
S
. eters, describe an enormous number of nonlinear disper- itisnecessarythatthephaseϕiseitherzero([10],p.132,
n
sive wave systems (capillary waves, surface water waves, Eq.(6.7);[11], p.156,Eq.(3.26.19),etc.) orconstant(e.g.
i
l atmosphericplanetarywaves,driftwavesinplasma,etc.) [6]).
n
[ all possessing nonlinear resonances [1]. Nonlinear reso- The main goal in this Letter is to show that in the case
nances appear in mechanics [2], astronomy [3], medicine of generic initial conditions, the phase ϕ which we call
1
[4], etc., etc. In this Letter we will regard the simplest from here on dynamical phase affects the evolution of
v
nonlinear resonant systems corresponding to the 3-wave the amplitudes and therefore,has a directimpact onthe
4
5 resonance conditions. Examples of these nonlinear reso- behaviour of any physical system governed by a triad
3 nantsystemsare,inorderofsimplicity,triads(whichare as well as small clusters of resonant triads. As it was
2 integrable), and small groups of connected triads which shownin[12], some of these clusters are describedby in-
.
1 are known to be important for various physical applica- tegrable systems and for them a complete set of conser-
0 tions (large-scale motions in the Earth’s atmosphere [5], vation laws (CLs) was given explicitly, showing that dy-
9
laboratory experiments with gravity-capillary waves [6], namicalphasesarerelevantinthedeterminationofthese
0
etc.). Dynamical system for a triad will be regarded in CLs. In this paper we present differential equations for
:
v the standard Manley-Rowe form: the two independent dynamical phases appearing in the
Xi butterfly, a resonance cluster formed by two triads con-
B˙1 =ZB2∗B3, B˙2 =ZB1∗B3, B˙3 = ZB1B2, (1) nected via one mode. We investigate the integrable case
r −
a ofbutterflybysolvingnumericallythereducedevolution-
where (B1,B2,B3) are complex amplitudes of 3 reso-
arydifferentialequationsshownin[12]forthephasesand
nantly interacting modes B expi(k x ω(k )t), while
j j· − j amplitudes. The numerical integration is fully validated
the corresponding resonance conditions are
by monitoring the conservationlaws, knownanalytically
ω(k1)+ω(k2) ω(k3)=0, k1+k2 k3 =0, (2) from our previous work [12]. We study the effects of the
− − phases onthe modes’ amplitudes, andsome physicalim-
where ω(k) is the dispersion relation and k is the plications are briefly discussed.
wavevector. 2. Triad with complex amplitudes. TheCLsforthe
Sys. (1) has been studied both in its real and com- Sys.(1) have the form
plex form by numerous researchers (e.g., [7], [8], etc.).
If regarded in the amplitude-phase representation Bj = I13 =|B1|2+|B3|2, I23 =|B2|2+|B3|2, (3)
|Bj|expiθj, Sys.(1) is equivalent to a system for the 3 (IT =Im(B1B2B3∗),
real amplitudes B and the phase combination ϕ =
j
| |
θ1+θ2 θ3,theindividualphasesθj beingslavevariables and their knowledge is enough for the explicit solution
−
and can be obtained by quadratures [9]. The dynamical to be constructed. Here conservedquantities I13 andI23
equation for ϕ is also known ([8], p.43, Eq.(28)). Still, a are not energy and enstrophy anymore but their linear
2
Π Π
1 Π 3 3
€€€€€€
0.5 4 2.5 €3€€€4€Π€€€€€€ 2.5 €3€€€4€Π€€€€€€
2 2
Π Π
0 0 1.5 €€2€€€€ 1.5 €€2€€€€
-0.5 1 €Π€€€€€ 1 €Π€€€€€
-€Π€€€€€ 0.5 4 0.5 4
-1 4
0 2 4 6 8 10 0 2 4 6 8 100 0 2 4 6 8 100
Π Π Π
3 3 3
2.5 €3€€€€Π€€€€€€ 2.5 €3€€€€Π€€€€€€ 2.5 €3€€€€Π€€€€€€
4 4 4
2 2 2
Π Π Π
€€€€€€ €€€€€€ €€€€€€
1.5 2 1.5 2 1.5 2
1 Π 1 Π 1 Π
€€€€€€ €€€€€€ €€€€€€
4 4 4
0.5 0.5 0.5
0 0 0
0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10
FIG.1: Coloronline. Plotsofthemodes’amplitudesanddynamicalphaseasfunctionsoftime,foratriadwithZ =1. Foreach
frame, ϕ(t) is (red) solid, C1(t) is (purple) dotted, C2(t) is (blue) dash-dotted,C3(t) is (green) dashed. Upper panel. Initial
condition α=0.7 andconservedquantitiesI23 =1,I13 =1.1 forallframes. Initialconditionsforthephaseandcorresponding
cubic conserved quantity: Left: ϕ = 0,IT = 0. Middle: ϕ = 0.1,IT = 0.041. Right: ϕ = π/2,IT = 0.408. Lower panel:
ConservedquantityI13=1.1forallframes. Initialconditionsandcorrespondingconservedquantities: Left: α=1.56,ϕ=0.01
and I23 = 1,IT = 3.2×10−5. Middle: α = 0.7,ϕ = 0.01 and I23 = 0.1,IT = 5.1×10−4. Right: α = 0.7,ϕ = π/2 and
I23 = 0.1,IT = 5.1×10−2. Here, horizontal axe denotes non-dimensional time; vertical left and right axes denote amplitude
and phase correspondingly.
combinations [5]. The analytical solution of Sys.(1) can ofthecorrespondingHamiltonians,andsimplifytheform
be found in [8], as well as the equations on two of the ofthe dynamicalsystemsandconservationlaws(see [12]
three phases θ (the standardamplitude-phase represen- for more details).
j
tation Bj = Cjexp(iθj) is used). Below we use slightly As it is shown in Fig.1, when initial dynamical phase is
different notations, introduced in [12], because they are zero it will remain zero at all times, but the amplitudes
more convenient for further studies of bigger groups of will change sign periodically (upper-left panel). When
connected triads. The equation for the dynamical phase dynamical phase is initially very small but non-zero, the
can be easily deduced and reads as amplitudes become purely positive and dynamical phase
ϕ˙ = I (C−2+C−2 C−2). (4) will have abrupt jumps, at those times when the ampli-
− T 1 2 − 3 tudes used to change sign (lower-left panel). Physically,
Combining (3) and (4), it is easy to see that the con- in terms of squares of amplitudes, the dynamics in both
straint I = 0 implies that dynamical phase vanishes, cases is quite the same (figure not shown). However,the
T
i.e. ϕ=0. phase’s dynamics, with its periodic motion, is revealed
IfweputI =0,thesolutionforamplitudestakesavery in the non-zero case. As it is shown in upper panel (left,
T
simple and familiar form: middle and right), non-zero dynamical phase influences
the evolutionofamplitudes,sothatasinitialϕincreases
C1(t)=dn(( t+t0) z√I13,I23)√I13 from0toπ/2,therangeofamplitudevariationsdecreases
− I13
C2(t)=cn((−t+t0) z√I13,II1233)√I23 (5) ffrroomm 51ttoo30..1 and the period of the motions decreases
C3(t)=sn(( t+t0) z√I13,I23)√I23
− I13 InFig.1,lowerpanel,weshowthatthenotionofA-mode
wheret0 is defined by initial conditions and can also be (active) and P-mode (passive) introduced in [13] (com-
written out explicitly. paretostabilitycriterion[14])isusefulalsointhecaseof
3. Triad, case I =0. Nowwepresentsomeresultsfor non-zero dynamical phase ϕ. A-mode is the mode with
T
6
the cases when I =0 but otherwise the resonance con- thehighestfrequency,andtwoothermodesarecalledP-
T
6
ditions in the standard form (2) are satisfied. In Fig.1, modes. Inthe pictures,C1 is P-modeandC3 is A-mode.
the characteristic evolution of amplitudes is shown, de- Lower-middle picture: when initial value of amplitude
pending on the value of IT. To characterize the initial C1 >> C3,C2 (4 times on figure), the P-mode C1 keeps
conditions,thevariableα=arctan(C3/C2)hasbeencho- itsenergyandtheA-modeC3 interactsstronglywiththe
sen for a triad and variables αa = arctan(C3a/C2a) and remaining P-mode C2. If C2 >> C3,C1 then the situ-
αb = arctan(C3b/C2b) have been chosen for a butterfly. ation will be qualitatively the same, with the P-mode
These variables appear naturally from the explicit form C2 keeping the energy. Lower-left figure: on the other
3
Π €Π€€€€€ Π
2
€Π€2€€€€ -€5€€€2€Π€€€€€€
0 -€1€€€5€2€€€€Π€€€€€ €Π€2€€€€
-€2€€€1€€€€€Π€€€€€
-€Π€2€€€€ -€2€€€7€2€€€€Π€€€€€
2
-Π 0 2 4 6 8 10 12 14 0 20 40 60 80 100 120 0 0 20 40 60 80 100 120
1 1 0.9
0.8
0.8
0.5
0.7
0.6 0.6
0
0.4 0.5
-0.5 0.4
0.2
0.3
-1 0 0.2
0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14
2.5 2.5 2.5
2 2 2
1.5 1.5 1.5
1 1 1
0.5 0.5 0.5
0 0 0
0 20 40 60 80 100 120 0 20 40 60 80 100 120 0 20 40 60 80 100 120
FIG. 2: Color online. Upper panel. Plots of dynamical phases as functions of time, for a butterfly with Za = Zb = 1. For
each frame, ϕa(t) is (red) solid and ϕb(t) is (black) dashed. Initial conditions αa = 0.3,αb = 0.7; and conserved quantities
Iab = 1.1,I23a = I23b = 0.5 for all frames. Initial conditions for the phases and corresponding cubic conserved quantities:
Left: ϕa = ϕb = 0,IB = 0. Middle: ϕa =0,ϕb = π/2,IB = 0.13. Right: ϕa =ϕb = π/2,IB = 0.36. Middle panel. Plots
of modes’ amplitudes of a butterfly as functions of time, same initial conditions and parameters as in the corresponding left,
middle and right frames of upperpanel. Connecting mode C1 is (blue) dotted, C2a is (green) long-dash-dotted,C3a is (green)
long-dashed, C2b is (purple) short-dash-dotted, C3b is (purple) short-dashed. Lower panel. Plots of amplitudes squared, as
functions of time, same initial conditions and parameters as in the corresponding left, middle and right frames of upper and
middle panels. Colors as above. To facilitate view, C22b,C32b are shifted upwards by the value 1, and C12 is shifted upwards by
the value 1.75. Here, in all three panels, the horizontal axe denotes non-dimensional time; the vertical axe denotes phase in
theupperpanel and amplitudes in themiddle and lower panels.
hand, if C3 >> C1,C2, then a completely different time describing evolution of a PP-butterfly has the form
evolution is observed and all modes interact.
B˙1 =ZaB2∗aB3a+ZbB2∗bB3b,
One more important general feature of the dynami- B˙2a =ZaB1∗B3a, B˙2b =ZbB1∗B3b, (6)
cal phase is shown on the upper- and lower-rightpanels.
Indeed, independently of details of the initial values of B˙3a =−ZaB1B2a, B˙3b =−ZbB1B2b .
mC1iz,eCd2,wCh3e,nththeevianriitaitailocnornadnitgieonoffothrethaemdpylintaumdeicsailspmhiansie- wherenotation B1 = B1a = B1b is chosen for the
amplitude of the mode common for both triads while
ϕ is equal to π/2. This can be used in real physical sys-
tems in order to control the exchange of energy between B2a,B3a,B2b,B3b are other four modes of the butterfly
cluster. The set of constructed conservation laws reads
resonant modes, at no energy cost: the choice of initial
as
dynamical phase does not change the energy of the sys-
tem, which is a sum of squares of amplitudes, obviously I23a = B2a 2+ B3a 2, I23b = B2b 2+ B3b 2,
independent of the dynamical phase (and of any phase, | | | | | | | |
for that matter). Iab =|B1|2+|B3a|2+|B3b|2, (7)
IB =Im(ZaB1B2aB3∗a+ZbB1B2bB3∗b)
4. Butterfly with complex amplitudes. As it was
shown [13], clusters formed by two triads a and b con- Similar to the triad, the use of standard representation
nected via one mode canhaveone of three types accord- B = C exp(iθ ) shows that the Sys.(7) has effectively
j j j
inglytothetypesofconnectingmodeineachtriad: PP-, 3 degrees of freedom and two dynamical phases are im-
AP- and AA-butterfly. In this Letter, a PP-butterfly is portant: ϕa = θ1a +θ2a θ3a, ϕb = θ1b +θ2b θ3b,
− −
taken as a representative example. Dynamical system withthe requirementθ1a =θ1b whichcorrespondsto the
4
choice of connecting mode. Accordingly, equations on 10 times and more for the same technical equipment.
the dynamical phases take form To gain more insight into the phenomenon of zonal
•
flows in plasmas which are now regarded as the main
ϕ˙a =−IB(C1−2+C2−a2−C3−a2), (8) component in all regimes of drift wave turbulence. “The
(ϕ˙b =−IB(C1−2+C2−b2−C3−b2). progressofplasmaphysicsinducedaparadigmshiftfrom
the previous ‘linear, localand deterministic’ view of tur-
Inordertostudytheeffectofnon-zerodynamicalphases bulent transport to the new ‘nonlinear, nonlocal (both
forbutterfly,weperformednumericalsimulationsforthe in real and wave number space), statistical’ view of tur-
integrable case Z = Z . This allows us to compare the bulent transport. Physics of the drift wave-zonal flow
a b
results with the triad, which is just a particular case of system is a prototypicalexample of this evolutionin un-
the integrable butterfly. derstanding the turbulence and structure formation in
In Fig.2, left column: we show phases, amplitudes plasmas”([17]). In [18], a modulational instability of
and amplitudes squared (energies) for initial conditions Rossby/plasma drift waves leads to generation of zonal
ϕ =ϕ =0. The dynamics is quite similar to the triad jets through a process in which the wave amplitudes are
a b
withinitialconditionϕ=0showninFig.1upperleft. In initially well-approximated by a kite, a cluster consist-
Fig.2, middle column: phases, amplitudes and energies ing of two triads connected via two modes. The study
are shown for initial conditions ϕ = π/2,ϕ = 0, while of the behaviour of associated dynamical phases could
a b
in the right column the same data are presented, for ini- lead to a deeper understanding of zonal jet formation.
tial conditions ϕ = ϕ = π/2. We observe from Figs.1 Structure formation (for 3-wave resonance processes) is
a b
and 2 some effects from the dynamical phases ϕ (t) and presented in [19], examples of non-local interactions are
a
ϕ (t) of a butterfly (correspondingly, the phase ϕ(t) of givenin[20]aswellasthe casesof‘weak’locality(waves
b
a triad): to diminish the period of energy exchange τ with wave numbers of order n and n2 can form a reso-
within a cluster by 20% and more; to reduce the vari- nancecluster,nandn3-cannot);effectsofinitialenergy
ability of wave energies by 25% and more; to generate a distribution among the modes of a cluster are studied in
new time scale, T >> τ, in which there is considerable [13]. InthisLetterweidentifydynamicalphaseasanad-
energy exchange within a cluster, as well as a periodic ditionalimportantparameterforanytheoreticalstudyof
behaviour (with period T) in the variability of modes’ nonlinearwavesystems. We wouldlike to pointoutthat
energies. the explicit equation for the dynamical phase of a triad
All computations have been done using Mathematica hasbeenknownformorethan40years(Eq. (3.31),[21])
and we have validated the code by checking the corre- in plasma physics and probably even earlier in nonlin-
sponding conservation laws, particularly those of cubic ear optics. We consider as our material impact in this
and quartic dependence on the amplitudes (introduced metier the detailed study of dynamical phases’ effect on
in [12]). These conservation laws are stably conserved nonlinear evolution of a triad and a butterfly.
during the whole numerical simulation, within a relative To guide and improve the performance and analy-
error of 10−12. sis•of laboratory experiments. We showed that non-zero
A comment regarding the ergodicity of the integrable phases can dramatically reduce the variability of the os-
butterfly. We observe in a parametric plot of cos(ϕ ) cillations (Figs. 1 and 2, left columns). It would then
a
vs. cos(ϕ ) as functions of time (figure not shown), be possible to tune initial conditions and/or forcing (in
b
that the seemingly periodic motions are indeed precess- rotatingwatertanks,forexample)inorderthatthemea-
ing with precession speed depending on the initial con- surementof the resonantmodes’ oscillationsbe less sub-
ditions. This is a generic feature of integrable systems ject to errors. In [6] results of laboratory experiments
which are not superintegrable. with gravity-capillary waves are presented, and corre-
5. Conclusions. Effects of non-zero dynamical phases sponding dynamical system for three connected triads
should be taken into account in the following situations. is written out and solved explicitly. The authors report
Tocontrolenergyinput,exchangeandoutputinlab- qualitative agreement of the observation with the solu-
•
oratory experiments, e.g. in Tokamaks. Indeed, a possi- tions of the dynamical system though the magnitudes of
bilityofconcentratingenergyinasmallsetofdriftwaves observed amplitudes are higher than those theoretically
viasomeinstabilitymechanismshasbeenconjecturedby predicted. For all calculations the dynamical phase of
V.I.Petviashvilisome15yearsago[15]. Assoonasenergy the initially excited triad was set to π which might be
is concentrated in a resonant cluster, mode amplitudes the source of this discrepancy.
can become dangerously large. In [16] it was shown that To interpret the results of numerical simulations.
•
the appearance of resonances can be completely avoided For instance, in [5] a generic model of intra-seasonal os-
by special choice of the form of the laboratory facilities cillations in the Earth’s atmosphere has been presented
which, of course, is too costly a game with Tokamaks. which describes the processes with periods of the order
On the other hand, adjustment of dynamical phases can of 30-90 days. In this model, dynamical phase has not
diminishamplitudesofresonantlyinteractingdriftwaves been taken into account yet. A new time scale, T >>τ,
5
whichisclearlyobservableinFig. 2,corresponds,forthe ReviewoftheArmedForcesMedicalServices(IRAFMS),
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