Yuan Xiao-Ping et al- Spiral Wave Generation in a Vortex Electric Field

8/3/2019 Yuan Xiao-Ping et al- Spiral Wave Generation in a Vortex Electric Field

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CHIN. PHYS. LETT. Vol.28, No.10 (2011) 100505

Spiral Wave Generation in a Vortex Electric Field *

YUAN Xiao-Ping()1, CHEN Jiang-Xing()2, ZHAO Ye-Hua()2**, LOU Qin()2,WANG Lu-Lu()2, SHEN Qian()2

1Inormation Engineering School, Hangzhou Dianzi University, Hangzhou 3100182

School o Science, Hangzhou Dianzi University, Hangzhou 310018

(Received 22 July 2011)

The efect o a vortical electric eld on nonlinear patterns in excitable media is studied. When an appropriate

vortex electric eld is applied, the system exhibits pattern transition rom chemical turbulence to spiral waves,

which possess the same chirality as the vortex electric eld. The underlying mechanism o this is discussed. We

also show the meandering behavior o a spiral under the taming o a vortex electric eld. The results obtained

here may contribute to control strategies o patterns on surace reaction.

PACS: 05.10.a, 05.45.a, 82.40.CK DOI:10.1088/0256-307X/28/10/100505

Spiral waves are one o the most common andwidely studied patterns in nature. They appear in

hydrodynamic systems, chemical reactions and alarge variety o biological, chemical and physicalsystems.[15] Much attention has been paid to theirrich nonlinear dynamics, as well as potential appli-cations in various biological or physiological systems,since the emergence and instability o spirals usu-ally lead to abnormal states, or example in cardiacarrythmia[6,7] and epilepsy.[8] Much research has beencarried out in studying pattern ormations in cat-alytic CO oxidation on Pt(110),[911] because theyprovide practical utilization in industry. A rich va-riety o spatiotemporal patterns, including travelling

pulses, standing waves, target patterns, spiral wavesand chemical turbulence have been observed in thissystem.[1216]

(a) (b)

(c) (d)

Fig. 1. The evolution o spiral waves induced by clockwise-rotating VEF with = 2.6, = 0.05, and = 1.2: (a) = 0; (b) = 200; (c) = 400; (d) = 1000.

Under the orcing o external eld, the pattern

tends to organize itsel to the pattern with the samesymmetry o the applied eld.[1720] For example, in

time and space symmetry[21,22] the spiral[23] and Tur-ing strips[24] evolve into hexagonal patterns that arecloser to the rotation symmetry o the imposed circu-larly polarized electric eld. In this Letter, we inves-tigate the evolution and transition o spatiotemporalchaos to spirals resulting rom the vortex ow, by im-posing a vortical electric eld (VEF). The meanderingo spiral waves under the inuence o VEF will alsobe studied. The underlying mechanism regarding thetransition o patterns will also be discussed. It will beshown that the VEF is a prominent method that canbe used to tame patterns in surace reactions.

We consider the efect o the vortical electric eldon an excitable media, which is described by a mod-ied FitzHughNagumo model (the Br model).[12]

The two variable reaction-difusion model, with ad-ditional electric term , is given by

= (, ) + 2 + ,

= (, ). (1)

The two-dimensional Laplace operator 2 = 2/2+2/2, and the variables and can be viewed as the ast and slow variables. In this model, (, ) =1

(1 )[ ( + )/], and (, ) describe a de-layed production o inhibitors with (, ) = or0 < 1/3, (, ) = 1 6.75( 1)2 or1/3 < 1, and (, ) = 1 , or > 1. is theratio o their temporal scales, which characterizes theexcitability o the medium. The parameters = 0.84and = 0.07 are xed.

A vortex electric eld could be realized inexperiments.[19] For example, a long and straight

*Supported by the National Natural Science Foundation o China under Grant Nos 10747120 and 11005026, and the Natural

Science Foundation rom the Educational Commission o Zhejiang Province (GK100801067).**To whom correspondence should be addressed. Email: [email protected] 2011 Chinese Physical Society and IOP Publishing Ltd

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electric solenoid will generate a VEF by inducing achangeable magnetic eld. Thereore, we could dis-cuss using the vortex electric current, which induces,or example, transition between spiral waves and otherstates, the homogeneous sates or turbulence, etc. Its

intensity is described by

=

, (2)

where is the intensity o the magnetic eld, and theunction / describes the variation o the externalmagnetic eld. Following Eq. (2), we obtain the ol-lowing expression when we consider that the area othe magnetic eld is small,

= 2

2

=

2

2

, (3)

where is the radius o the solenoid. is externalelectric current that is input to the solenoid; is aproportional coecient describing and is in directproportion to ; =

( 0)2 + ( 0)2, 0 and

0 represent the coordination origin, and (, ) denesthe site position o the eld. In this case, > , i.e. thestudied point is outside the solenoid, and an electriceld = 2/2 = / (the electric current/ = ) is investigated. Clearly, by taking into ac-count the efect o a counterclockwise VEF, the vor-tex electric eld can be divided into two directionsalong the - and -axes: = sin = /2, = cos = /2. Here is the angle o theeld point with respect to the positive -axes. Thusthe polarized medium is described by

= (, ) + 2 +

+

,

= (, ). (4)

Inside the solenoid, we set = = 0. For simulationsin 2D, zero-ux conditions have been employed in boththe medium and solenoid interace boundaries. A spa-

tial discretization o = = 0.3906 on a 256256array with xed time step = 0.02 has been used inan explicit Euler scheme.

Now we study the inuences o the VEF on the dy-namical behaviors o nonlinear patterns. In Fig. 1(a)we select a chaotic state as an initial state. Then theVEF (with an intensity o = 2.6), induced by asolenoid located on the center o the medium, is im-posed on. With the evolution o time, rom Fig. 1(b),one can see that a small spiral is generated in the mid-dle o the disk at about = 200. Gradually, the smallspiral wipes away rom the turbulence and simultane-

ously grows into the outer region, which can be seenin Fig. 1(c). Eventually, the medium is occupied by awell developed single spiral (see Fig. 1(d)). Thereore,

the VEF can develop spirals to suppress the surround-ing turbulence, which may act as an ecient controlmethod to terminate the chaotic state.

0.4 0.8 1.2 1.6 2.0 2.4

20

40

60

80

100

D

(a) (b) (c)

(d) (e) (f)

(g)

Fig. 2. (a)() Patterns induced by diferent intensities oVEF with = 0.05 and = 1.2 at = 500. (a) = 0; (b) = 0.6; (c) = 0.8; (d) = 1.2; (e) = 1.6; () = 2.8.(g) The diameter o the developed spiral as a unction othe amplitude o the VEF.

0 . 2 5 5 0 . 2 6 0 0 . 2 6 5 0 . 2 7 0 0 . 2 7 5 0 . 2 8 0

2 0 0 0

4 0 0 0

6 0 0 0

8 0 0 0

1 0 0 0 0

1 2 0 0 0

1 4 0 0 0

1 6 0 0 0 = 3 . 2

= 3 . 0

P

o

w

e

r

F r e q u e n c y

Fig. 3. The power spectra obtained rom the FFT methodat diferent VEF amplitudes. The other parameters arethe same as those in Fig. 1.

In the VEF there is only one parameter, that isthe intensity, which can be tuned. From this point,the control method is very simple. In simulation it isound that there is a critical value o VEF intensity,below which the spiral can not be generated. In Fig.2(a)(), we present the states ater long-time evolu-tion at = 500. It is evident that the stronger the im-

posed VEF, the larger the size o the developed spiral.In Fig. 2(g), we show the dependence o the diametero the developed spiral on the amplitude o the VEF.

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Then, one can use the VEF to suppress turbulenceand develop a spiral with the required sizes.

(a) (b) (c)

Fig. 4. Pattern evolution induced by a counterclockwise-rotating VEF with = 2.8, = 0.05: (a) = 0; (b) = 300; (c) = 1000.

(a) (b)

Fig. 5. A multi-arm spiral developed by the vortex eldwith a large radius. = 5.0, = 0.05. (a) = 7.8; (b) = 11.7.

To give some insight into the mechanism o thesuppression o turbulence by a developing spiral wave,we present the power spectra obtained rom the FFTmethod at diferent VEF amplitudes in Fig. 3. Thedata is plotted at the same spatial point. WithoutVEF, the turbulence has a background requency o

0 = 0.238. Ater a VEF is applied on the medium,a spiral is developed with a requency that dependson the amplitude o the VEF. I the VEF is strongenough to generate a spiral with a requency higherthan 0, the induced spiral will wipe away rom thesurrounding turbulence. Figure 3 veries this point.One can nd that the spiral requency is higher than0 and increases with amplitude, rom = 0.261( = 2.4), = 0.268 ( = 3.0), to = 0.271 ( = 3.2).I the VEF is so weak that its corresponding spiral re-quency is smaller than 0, no spiral will grow. This isthe reason why there is a critical value o to suppress

turbulence.The amplitude o the VEF is decreased along theradius to the outside. Due to the act that the re-quency o the spiral shows VEF amplitude depen-dence, its value decreases accordingly. Thus, the re-quency o the spiral is inhomogeneous, that is, roma higher value to 0. When the requency o the spi-ral in the edge region is decreased to 0, the spiralcan no longer grow again, and a state described byspiral-turbulence coexistence is observed. Thus it isclear that the stronger VEF leads to higher requencyand larger sizes o developed spiral, which can be seen

in Fig. 2.It is known that the VEF has its chirality: a clock-wise or counterclockwise vortex eld. From the simula-

tion, it is always shown that the process is the growtho a single spiral wave. Thus, we pay attention to theinitial state when the VEF is imposed. Many small spi-rals orm the background chaotic state. Generally, onesmall spiral has two ends with opposite chirality. One

pattern tends to organize itsel to the pattern with thesame symmetry o the applied eld. Once the VEF isswitched on, one end o the spiral has the same chiral-ity as the developed VEF. In Fig. 4, we show the resultunder the control o the counterclockwise VEF. Thecounterclockwise rotating spiral veries our point.

I the radius o the solenoid is increased, comparedto the small radius cases, there are many small spiralsavailable that can be taken as seeds. Consequently,based on the mechanism discussed above, multi-armspirals will be developed, and this is conrmed inFig. 5. It is observed that the larger the radius, thegreater the number o spiral arms. For example, ave-arm spiral is shown in Fig. 5(a) when is 7.8(20 grids), while an eight-arm spiral is observed inFig. 5(b) as is up to 11.7 (30 grids).

As a less-investigated external eld, it is interest-ing to study the tip motion under its inuences.[25,26]

In Fig. 6, we present two types o motion. We selectan outward meandering spiral as the initial state. Thespiral rotates clockwise with two requencies that canbe indicated by the compound motion o the spiraltip. In Fig. 6(a), the clockwise trajectory (see the ar-row ) showed by 1 and 2 is the initial state withoutcontrol. The counterclockwise VEF will introduce thethird requency, which gives another counterclockwisemotion (see the arrow ) indicated by the trajectorywith radius 3. With stronger amplitude, VEF reversesthe second motion by increasing the second requency,which leads to the changes rom outward meanderingto inward meandering, and clockwise to counterclock-wise rotation. This result is shown in Fig. 6(b).

105100

105

110

115

120

125

130

r

r

r

(b)(a)

115 120 125 130

105

110

115

120

125

x

ba

y

x

115 125 135

Fig. 6. (a) The trajectories o the motion o the spiral tip.The clockwise trajectory (see the arrow ) showed by 1and 2 is the initial state without control. Another, coun-terclockwise, motion (see the arrow ) indicated by thetrajectory with radius 3 is induced by the counterclock-wise VEF. (b) The inward-meandering trajectory o thesecond motion modulated by the VEF.

In conclusion, we suggested a control method tosuppress spatial-temporal chaos in excitable media.The size and requency o the developed spirals are

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ound to increase with the intensity o the imposedVEF. When the radius o the solenoid is increased,multi-arm spirals can be generated. It is ound thatthe chirality o the generated spiral is always the sameas the imposed VEF. The VEF can also be utilized to

tune the dynamics o meandering spirals. One can eas-ily develop spiral waves o the required size by modu-lating the intensity o the VEF. It is expected that ourresults will contribute to the understanding o chemi-cal reaction-difusion systems, especially in CO oxida-tion on a Pt surace.

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Yuan Xiao-Ping et al- Spiral Wave Generation in a Vortex Electric Field