Observation of solitons on vortex filament bush

8 June 1998

PHYSICS LETTERS A

EIXMER

Physics Letters A 242 (1998)

319-325

Observation of solitons on vortex filament bush S. Lugomer Received 17 November

1997: revised manuscript received 2 February 1998; accepted for publication 2 March 1998 Communicated by A.P. Fordy

We show the vortex filnmcnt gcncration by laser-metal scale. The interaction of a laser pulse with a nonuniform

interaction in the surface shear layer on the nanosecond time surface generates a rippled shock wave that oscillates as the

shock propagates. An oscillating shock wave causes vortex cascade angle-locking quasi-two-dimensional frozen pcrmancntly, the alllpliticntion on different

and three-dimensional

thus enabling one to make an a posteriori study. Numerical

of solitons with right-hnndcd h&city

lilamcnts follow dynamical

splittings into a bush-like stiructurc, and

solitons on the filaments. Bccausc of ultra-fast cooling these stticturcs

stay

filtering of the micrographs has revealed

and dumping of the left-handed ones. It has also shown ‘that solitons

I;IWS of a different hierarchy Icvcl, acting simultaneously

on the hr;lnchcs of the

vortex filament bush. @ I908 Elscvicr Science B.V.

as in laser fusion targets [ 6.7 1.

1. Introduction

Difficulties The nonlinear

and noncquilibrium dynamics of the

to gcncratc them on a laboratory scale

for expcrimcntal study arc well known. Ih liquid hc-

fluid shear Iaycr gcneratcs a rich spectrum of turhu-

lium where lhcir size goes down to atom Bcales, they

lcnl structures; among them the vorlcx filaments. Vor-

simply cannot bc observed [ 8 1. In anothcp type of ex-

ICX lilamcnts may appear as closed-loop or open-loop

pcrimcnt special proccdurcs and cquipmcnt are needed

entilics in different physical syslcms: hydrodynamic

like schliercn photography, fluorescent dyes, seeding

thcrmohydrodynamic

of flow by powder etc., to make them visible.

and magnetohydrodynamic,

as

well as in chemical and biological ones [ i -41. They

However, we have found that tiny vortex filament

rcprcscnt objects “embcddcd in an infinite domain” known lo appear on spatial scales ranging from micro-

structures may appear spontaneously in thd shear layer of liquid metal in laser-matter interactions [!I-121,

to-mcso and to mega-scales. In the physical systems

After

the vortex filament scale length ranges from IO-” mIO-” m for superfluid vortices: to 10-* m-10’ m for

turcs stay frozen-in permancnlly because :of ultra-fast

termination

of a 10 ns laser pulse, ithese slruc-

fluid cddics, vortex filaments and tornadoes, and lhcy

cooling (dT/dr sz 3000 K/5 x 1O-‘O Y) z lo9 PY IO” K/s; dt z i2 pulse duration. for a rdugh estima-

may reach IO” m-10” m as in astrophysical systems. They arc known 10 appear in the shock wave intcr-

Con), thus enabling one to make an a pojteriori analysis.

actions with matter conncctcd to the evolution of the nonlinear Raylcigh-Taylor instability (in the spike re-

A scrics of cxpcrimcnts was performed in order to gcncralc micron-scale vortex filaments, 1~ study their spatial organization. their interaction with chock waves

gions) [ 51, which is prcscnt in star collapses as well 0375-9601/98/519.00

@

P//SO.175-9601(98)00175-3

1998 Elsevier Science R.V. All rights rcscrvcd.

320

S. Lu~omer/Phwics

and their dynamics. A Q-switched

Nd:YAP

Letters A 242 (1998)

3119-325

laser of

(2)

E = 0.3 J. of a pulse radiation of 10 ns, was used to irradiate the target. The pulse power density Qp was varied below the volume boiling threshold of IO* W/cm’

It should be noted that the top stream is always the

by variation of the spot size, until the filament struc-

faster one. so that r < I. while s and m are allowed

tures started to appear in the surface morphology. For

to assume any nonnegative value. The shear layer

the target material various refractory metals have been

may also be defined in a different way, for example

used. but the best results were obtained with tantalum

through the characteristic thickness. The most com-

plates of I x I x 0.5 cm in size. The plates were me-

monly used are the vorticity thickness &J and the mo-

chanically polished (so that micron-scale ripples were

mentum thickne:ss 6 [I l,l2.14].

We have

produced), and washed in alcohol. The characteristics of the surface morphology have

SW = (U, - C/z)

(3)

been studied by the scanning electron microscope. To make topological details visible, the micrographs have been numerically

and

filtered. Image analysis of the mi-

II/Z

J

crographs was done by using the Photo-Styler pro-

t?=

gramme.

2. Generation

IJ*

.

u, -

r/, dv u2

(4)

.’

assumptions have been made for p(y)

A short and powerful laser pulse of a Gaussian proents of pressure and tcmpcralurc. The vertical thermal gradient VT,

u, -

where II is the thickness of the fluid layer. Various

of the shear layer

file gcncratcs a molten laycr with characrcristic gradi-

lations. taking into account low and medium Reynolds numbers. The Reynolds number Rc is delincd by

increases the

of the upper surface layer that is in

as well as for

SW and 0 by various authors in their numerical simu-

Re

tempcralurc

Ul - CL(Y) CL(y)-

-- u‘h -

v

(5)

’

contact with the ionized vapour IO the boiling tem-

= T,j ), while the lower surface layer is at Tf, > T 2 T,,, the melting temperature.

perature, Tn (T

The surface layer bchavcs as a bilaycr charactcrised by T~.pl,~l,...,

L/I ,...

and

T~.~,Kz

with T, > Tz;p~ < pz;~~ x K~,...,U, . . . , UI >> i/z, where p is the density,

,...,

&

K

,...

or is the thcr>

r/2

whcrc U = (&

- UI )/2. and v is the kinematic vis-

cosily. Such conditions lcad to the vortex filament formation as a kind of Kelvin-Hclmholtz

instability in

the shear layer with the Re numhcr in the range 01 lo*-IOI.

Cerremfiorl of rhe oscillatory

shock. On the unpol-

mal conductivity and U the lluid velocity. Evidently.

ished or mechanically

the upper layer (index

turgct, the ablation pressure gcncrales a rippled shock

I)

is a high momentum. tn.

polished, irregular (rippled)

while the lower one is the low

wave in accordance with the surface topology. We have

momentum layer with rn2 = pz(/tz. Thus, a shear layer

obscrvcd this in a number of experiments in which var-

is formed that establishes mixing of both layers and

ious hydrodynamic formalions on the ablation surface

layer with ml = PII/:.

have been generated [ 9- I2 J. Endo ct al. [ i 5 ] and rc-

generates the vorticity at their interface [ I 1,121. The shear flow can be described by the dimcnsionless paramctcrs, namely the velocity and density ratios r and 5, respectively, defined as [ I l-13 J

u,

ccntly Wouchuk and Nishihara [ l6J and lshizaki and Nishihara

[ 17) have discussed the propagation of a

rippled shock wave driven by nonuniform laser ablation that develops on the surface with an initial rough-

(1)

ness. given by aoexp( ix-y). where a~, and k are the surface amplitude and the perturbation wave number,

The third paramctcr. called the momentum ratio m,

respectively. These nonuniformities induce pcrturbalions in the shock-comprcsscd. the ablation, and the isothermal rarefaction regions [ 17 1.

r=

-.

UI

s=

P? -. PI

can hc dcfincd according to Ref. [ 131 as

S. Lugomer/Ph.vsics

Once a rippled bation

Laws

shock is launched a pressure pertur-

is induced

by the lateral

fluid motion

the shock. The pressure perturbation

behind

causes the ripple

on the shock front to be reversed and subsequently cillate.

as the pressure perturbation

formation

of the ablation

The amplitude

os-

increases the de-

front monotonically

[ 171.

of the shock ripple decays as the shock

propagates. Since the pressure perturbation

at the abla-

tion front also decays with time, the deformation

of the

ablation

[ 171.

front approaches

In the weak shock limit

an asymptotic

value

(as we assume to be the case

for the shock generated

at Q,, <

10’ W/cm’),

and

for the shock Mach number M, < I. one has for the shock font ripple n,/uo [ 171. Furthermore,

A 242 (1998)

which show many overcrossings Reconstruction acteristic following

(1, ( t )

-

a,

= Jo(r,)

procedure

parameter.

the bifurcation

of the ripple

value of the bifurcation

lilamcnts

is rippled.

Thcrcforc.

(gcncratcd

oscillating

r. This makes the

parameter

to the well-known

problem

formblation

the amplitude

as the shock propa-

in addition

to the vortex

hy the shear flow),

shock wave,

one has the

that is traveling

by

[ 19.201,

equation

< rs,,

ifs,,

is in [0, I],

which gives for r > 3 a “time dcpcndcnt” front

tends to oscillate

gates [ IS- I7].

Thus, every curve connecting

(7)

of order 0

and 2. respcctivcly. the shock

(gener-

of r, where r is the

points of the same level hasi the same

x,+1 = rx,( I -x,)

Once

of the space.

shows the number of filaments

the difference

where Jo and Jz arc the Bcsscl functions

regularity

in the paramdter

ated in the cascades) as a function bifurcation

and the

the points of the same cascade

curves, a beautiful

process is obtained

Fig. 2 (right)

itmto chaos.

distributed

is needed to make thils similar-

ity real. Connecting branching

process char-

bifurcation

points are not regularly

level by irregular

in Fig. 2 sug-

with the branching

for period doubling

Bifurcation

and undercrossings.

of the bush shown

gests the similarity

connection 2M,?+2 + 3M5 + I J:(r,),

311

319-325

in a radial

splution

of

period 2. That is. the equation _r,,+r = x,, dcvkzlops two real roots in addition

to the steady states. Thi: solution for r ranging from r t r? = 3.0

is stable and attracting

to r = rj z 3.45. At rz a period 4 solution, dcvclops and the period 2 solution

loses its stability

I 191. Bc-

direction.

twccn r3 and roe z 3.56YY8, stahlc solutionslof

Splirtirrg of the vortex filmml by the oscillfltory shock. The shock wave causes an axial compression

2k arc succcssivcly supplanted by the stahlclsoIutions of period 2k + I. The lower period solutionk becomc

on the filament. that may lead to filament bending. or hrcaking or to splitting, as dcscrihcd by Erlcbachcr ct

rcminisccnl

al. [ I8 ] Consequently,

the oscillating

by the single laser pulse). vortex filament splittings. A micrograph microscope

shock (driven

causes the cascade of the

obtained

by the scanning

on a Ta surface

after

electron

irradiation oricntcd

with a cascade of angle-locking

by a vortex

bifurcations;

see Fig. I. The angle between the splittcd vortices is 0 = 57’-60’ in all bifurcation stages. The filament scgmcnrs arc of variable position is ahout (length

of the splitting 120 pm

length with a rather irregular points. The filament

to I50 pm

to thickness ratio)

The hchavior

while

thickness

the aspect ratio

was in the range from 5.68

to 8.8 for the shortest filaments.

and from 16.4 to 35,

for the largest ones. Filaments with the aspect ratio approaching 100 wcrc obscrvcd but they arc not common [ I?]. The cascade of vortex filaments splitting gcncratcs a three-dimensional

bush, the branches of

of the itcratcs of dq, (I)

of LildilU’S

infinite

Lions in that the bifurcations riodicitics,

to higher and digher pcbehakiors,

incrcascs in r. At rd

thcrc arc no stahlc periodic

is

scqucncc of transi-

and thus to more complex

cur with even smaller

IO ns laser pulse shows a radially filament

unstahlc.

period

solutions.

oc-

x 3.57

The odcillations

all scttlc down to what appears to be pcriocil 2 [ 191. However,

the period

number changes, depc’nding

on

how many digits in rare taken into account. This indicates that at rm the system is no more scnsitjive to the initial

state. and cntcrs in the chaotic

Since the illustration (Fig.

2 right)

behav$or

[ 191.

in the phase space of the bush

is typical

for period doublin$

bifurca-

tion to chaos, WC attrihutc the above values to the bifurcation paramctcrs as given in this diagrarb. Reconstruction of the hush rcvcals the exiistcncc ot solitons on the largest filaments, which arc three dimensional in the central zone of the bush, $nd thcrc arc quasi-two-dimensional ones on its sides, This indicatcs that filaments arc subjected to torsi$n which

S. &ujpncr/Phwics

Lcrrers A 242 (19981 319-325

/ Fig.

1. Vortex fil:uucnt bush gtmmted

by the single pulse of a Q-switched Nd:YAG

laser on Ta surhco in the qinr of nonstationary The lam energy density, Q = df
vaporization I hst

T - 5000 K. when: M N 200x.

results were ohtuined with tantalum plates of I x I x 0.05 cm in siyx. The platss wcrc nrch;micully

polished and w;lshcd in alcohol.

The characteristics of the surface morphology have bcvn studied by the scanning electron cuicrosco~. To ~uakc topological dctails visible. the micrographs huvc bcx!n numerically filtered. by using a Hcwkt-Packard compared with those ohtaincd by numerical simulations.

The surfxc

sc;mnc’r and the Adobe Photoshop progmnunc. and the objects

shrm

loycr pncr;rtcs

the vortex tilarncnts.

shock. generated in the regime of nonstationary ablution. CBUWS the c~~adc of vortex tilarnent splittings structure. Torsion

of the filaments gcncratcs the solitons on likunrnts

upper side) and quasi-two dinunsional

while thr oscillatory

and the appcxmcc of a bush-like

which arc three dimensional in the central zone of the bush (ccntcr.

in the side zones of the bush (MI

and right side).

gcncra~cssolitons and puts them into motion. Solirons in the central zone of the bush arc threedimensional multisolitons that travel in a helical way. while the soldons in the side zones of the bush arc

vanishes (7 = 0) in the side zones. Two-dimensional solitons arc described by the modilicd Kortcwegdc Vrics equation (mKdV) with the parameter K

quasi-two-dimensional single solitons that travel by simple “slipping” along the filamcn~s. This indicates

described by the nonlinear Schrtiingcr equation with the complex parameter P [ 21-231. The solitons on the filaments generated in laser experiments arc fast

that the torsion T is not constant in all the branches of the bush but has a maximum in the central zone and

(curvalurcf,

while the three-dimensional

ones arc

and moving without friction along the filament; they

S. Lugcmcr/Ph.vsics

30

Letters A 242 (1998)

319-325

SOLITONS

T--

-___ FIRST

CHAOTIC REGION

FILAMENT POPULATION

Fig. 2. Kcconswuction of the vor~cx filanlen~ bush from Fig. I. Solitono in the central zone have maximal torsion (T = T,,,J, ); 111cside solitons hwc

npproximatcly

r = 0. Th~~~forc. the tonion

I is not constant but has a distribution over the brnnchcs of tM_ hush. The

illuslr;~tion on the right side rWc;ds the period doubling transition to chaos in the branching proccss of the vorlcn lil;mlrnt. (:Sw ICXI. )

reach a velocity

hc~wccn

IO’

m/s to lo” m/s for

an asymmetry of the stability charactoristic6 bctwccn

two-dimensional-like solitons, and an about 3-4 times

Icft- and right-handed hcliccs [ 25 I. If the 8nagnitutlc

larger v&city

of the axial flow is small, than it tends to stnbilisc Icft-

for 3D ones. Thus, 3D [ I I, I2 1 solitons

in the central xonc of the bush arc cxposcd to larger

handed hcliccs and dcstabilisc the right-hahtlcd ones

torsion and thcrcforc they travel faster than quasi-2D

for positive hclicity. For ncgativc hclicity thi: opposite

ones. in agrccmcnt with a thcorctical prediction of

is true. If the magnitude of the axial flow is Ilargcr than

Kicca, that the torsion 7 may considerably influcncc

some value it tends to suppress both type of hcliccs.

the propagation velocity of twisted filaments (24).

The suppression of the long-wave mode is &tablishcd

The filament thinness parameter is playing an impor-

at modcratc values of (W/VI

tant role in the validity of the Iocaliscd approximation

swirl velocity), the critical value of which is smaller

that guides thin vortex filament motion and soliton

for Icft-handed hcliccs if the hclicity is posiltivc [ 25 1.

formation.

(hc

ratio of axial and

The overall view of hclicity on the bush rcvcals

A detailed study of the micrograph by numerical

a great regularity: a Icft-handed h&city

otl quasi-2D

filtration has shown an even more complex structure

solitons on the right side of the bush, and then follows

of the solitons on the “branches” of the vortex fila-

a right-handed helicity of 3D solitons. thcb the Icft-

ment bush. There is discrimination

in the picture in-

banded helicity of 3D sojilons,

and finally, a right-

tensity of left-handed solitons and right-handed ones.

handed helicity of quasi-2D solitons (on the left side

The Icft-handed solitons on the filaments (both their

of the bush). Thcrcfore, the angular distribution of the

three-dimensional ones in the central zone and two-

hclicity of solitons on the vortex filament bush is a

dimensional ones on the periphery of the bush) are

periodic one. It is possible that such a distliibution of

more intensive than the right-handed ones; see Fig. 3.

right- and left-handed hclicity is rcsponsihle for the

According to the Fukumoto-Miyazaki

highly symmetric bush gcomclry, in a fan sihapc.

theory (which

gives a gcncratization of the localiscd induced approximation) this discrimination is based on the prcsencc of the axial flow within the core which brings about

Solitons. or the solitary waves that propagate along a thin vortex filament, can transport physical quantitics such as mass, linear and angular momentum,

I

LEFT- HANDED HELlCiTY

IDAMPED

Fig. 3. Magnification

and numerical

SOLITONS

I I ;

RIGHT - HANDED HELIC fTY AMPLIFED

liltration of dclails in the micrograph

left-handed helicity. We have M N 1500x.

(a) Three-dimensional

while those with left-handed helicity are strongly damaged (left

IW~

SOLITONS

very

importanl details. relating IO the right- or the

rolitons with right hcliciry are amplified (right side of the micrograph).

sideof the micrognph).Notice

that the “border” between them is actually

;1 “double helix,*’ i.e. IWO helically paired filaments are there with Icft- and another one with right-huldcd

helicity.

lo

In particular. with respect to the case studied one

another one (261. Rciations bctwccn these quantities

can say that the “line slructurc” of the laser generated

and the polynomial invariants Fl, F2, F3 . . . F,, (Fl is the kinetic cncrgy, F2 the hclicity. F3 the Lagrangian

pattern reveals the prcscncc of quasi-two-dimensional solitons which can bc described by mKdV equation;

of the system) wcrc found by Ricca [ 2 I j. Hc has shown that the dynamical laws describing this par-

that thcrc arc 3D solitons on filaments without axial flow (probably with a very small one), which can be

ricular type of llow directly dcpcnd on the number of invariants Fl, F2. . . (family of invariants) prcscnt. Thus, the “family of invariants” corresponds 10 the

dcscribcd by the nonlinear Schriidinger equation; and

and energy, from a particular region of the fluid

type of llow [ 26 1.

finally, thcrc arc solitons with the presence of a strong axial flow, appearing as Icft- and right-handed ones, which can bc dcscribcd by the Fukumoto-Miyazaki

S. L.ugnmer/Ph_vsics

Lrrers

A 242 (1998)

325

319-325

interactions, one has the possibility to grasp the universal mechanisms which govern the nonlinear spatiotemporal dynamics in various systems, the relation of these mechanisms to the underlying space-tiime characteristics, and their possible universality.

Acknowledgement The author is thankful to Dr. Renzo L. RicEa, Dept. of Appl. Mathematics and Theoretical

Physics, Uni-

versity College. London, for very useful comments and a critical reading of the manuscript. Fig. 3. (b) A quasi-two-dimensional

soliton on a filament fomied

in the right side of the bush in Fig. filtration. We have rCI w 1200x. because the Icft-handed

I

obtained by numerical

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Notice that it is scarcely visible.

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diffcrcnt branches of the same vortex filament bush. This can bc understood as a consequence of axial flow and of the inhomogcncous spatial distribution of two crucial paramctcrs: the torsion 7, and the curvature

K,

over the branches of the bush; conscqucntly. they gcncrate solitons the motion of which follows dynamical laws at diffcrcnt hierarchical lcvcls. In addition, the viscous intcruction of filaments with the background fluid may cause a change of the aspcct ratio by the core spreading, which explains the existcncc of a broad range of aspect ratio values. The viscous

interaction

is expected lo incrcasc

(i)

toward

the pcriphcry region of the bush, and (ii) at the second

half of the pulse when its intensity goes down. This may strongly affect those filaments which arc formed last in the cascade splitting. Such filaments arc cxpcctcd to bc rclativcly thick and short. Conscqucntly. they have no time to grow up to the stage where they satisfy the conditions for soliton formation. Therefore, one has the situation - the most complex one - whcrc the conditions of the filament branching and of the soliton formation change in space-time. By examining

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