Wave induced barrier transparency and melting of quasi-crystalline structures in twodimensional plasma turbulenceAmita Das Citation: Physics of Plasmas (1994-present) 14, 042307 (2007); doi: 10.1063/1.2718927 View online: http://dx.doi.org/10.1063/1.2718927 View Table of Contents: http://scitation.aip.org/content/aip/journal/pop/14/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in A link between nonlinear self-organization and dissipation in drift-wave turbulence Phys. Plasmas 19, 082318 (2012); 10.1063/1.4748143 Two-dimensional bispectral analysis of drift wave turbulence in a cylindrical plasma Phys. Plasmas 17, 052313 (2010); 10.1063/1.3429674 Coherent structure of zonal flow and onset of turbulent transport Phys. Plasmas 12, 062303 (2005); 10.1063/1.1922788 Vorticity probes and the characterization of vortices in the Kelvin–Helmholtz instability in the large plasma deviceexperiment Phys. Plasmas 12, 022303 (2005); 10.1063/1.1830489 On the sink/source effects in two-dimensional plasma turbulence Phys. Plasmas 8, 5091 (2001); 10.1063/1.1415750

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Wave induced barrier transparency and melting of quasi-crystallinestructures in two dimensional plasma turbulence

Amita Dasa�

Institute for Plasma Research, Bhat, Gandhinagar-382428, India

�Received 24 January 2007; accepted 2 March 2007; published online 27 April 2007�

The conservation of energy and enstrophy in two dimensional inviscid hydrodynamics leads to dualcascade behavior. The energy cascades towards long scales and the enstrophy is transferred toshorter scales. The interplay of these dynamical processes leads to self organization and formationof coherent patterns in the two dimensional hydrodynamic turbulence. It was shown by Kukharkinet al. �Phys. Rev. Lett. 25, 2486 �1995�� that this process of self organization occurs in an even moreinteresting fashion in the Hasegawa Mima �HM� equation �Phys. Fluids 21, 21 �1978�� Thisequation is a generalization of the two dimensional Navier Stokes hydrodynamics model in whichthere is a characteristic natural scale in the system �e.g., Larmor radius in the drift wave context�.Kukharkin et al. observed that this scale acts as a barrier in the energy cascade, such that the cascaderate at the longer wavelength side of the barrier is smaller. This work has also shown that theaccumulation of energy around the intrinsic scale leads to the formation of quasi-crystalline patterns.In the present paper it has been demonstrated that the presence of wave excitations leads to anincreased cascade towards longer scales past the natural length scale barrier. It has also beendemonstrated that wave excitations lead to the melting of quasi-crystalline structures. Anotherintriguing but interesting observation is that even though the faster cascade is induced by wavesarising through an anisotropic inhomogeneity in one of the plasma parameters, the spectrum of thefluctuations continues to remain predominantly isotropic. A physical understanding of theobservations is provided by illustrating a close connection between the Kelvin–Helmholtzdestabilization of shear flows and the phenomenon of inverse cascade in 2D fluid flows.© 2007 American Institute of Physics. �DOI: 10.1063/1.2718927�

I. INTRODUCTION

The formation, dynamics and the survival of coherentpatterns in fluid flows have always attracted a great deal ofinterest. In the two dimensional hydrodynamic fluid flowgoverned by the Navier Stokes �NS� equation, there are noinherent natural length scales and the tendency towards selforganization occurs due to the dynamical processes govern-ing the dual cascade behavior. For this system the non-dissipative limit shows conservation of two square integralinvariants, viz., energy and enstrophy. The existence of theseinvariants conspire in such a fashion that energy suffers in-verse cascade towards long scales. The enstrophy on theother hand cascades to shorter scales where it is trappeddominantly in a few short scale patterns. The HasegawaMima2 equation, representing a generalization of the two di-mensional Navier Stokes equation contains a characteristicnatural length scale. It was shown by Kukharkin et al.1 thatfor this model, the process of self organization is relativelymore interesting. This model also conserves energy and en-strophy like two mean square invariants in the non dissipa-tive limit and the dual cascade behavior is also exhibited bythis model. However, due to the existence of the inherentlength scale it displays certain interesting and unique fea-tures. The cascade rate on the two sides of the intrinsic scaleare different causing it to act as a barrier for the inverse

energy cascade. The accumulation of energy around thislength scale then results in the formation of a quasi-crystalline pattern of vortices.

The HM model equation is applicable for a variety ofscenarios. The intrinsic scale depends on the associatedphysics in each case. For instance, this scale is the Larmorradius when HM equations are used for the drift wave modelin plasmas. The other situation in which the same model isextensively employed is for geostrophic flow in planetaryatmosphere,3 where the Rossby radius is the intrinsic naturalscale. The two dimensional Electron Magnetohydrodynamic�EMHD� model4 equations in the absence of in-plane mag-netic field also reduce to the HM model, with the electronskin depth c /�pe being the intrinsic length scale of thesystem.

Kukharkin et al.1 performed a numerical simulation forthe HM equation, with a white noise forcing at scales shorterthan the intrinsic scale �say Larmor radii for the drift wavesystem�. The box size of the simulation was chosen so as tocover an entire range of scales spanning from shorter tolonger than Larmor radii. The energy fed at scales shorterthan Larmor radii cascades towards longer scales. It first hitsthe Larmor scale before reaching the box dimension. It wasshown by the simulations of Kukharkin et al.1 that the dif-ference in cascade rates of energy at the two sides of theLarmor scale �faster at shorter and slower at the longer� re-sults in the accumulation of power at the Larmor lengthscale. This produces quasi-crystalline patterns.a�Electronic address: amita@ipr.res.in

PHYSICS OF PLASMAS 14, 042307 �2007�

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In this work we demonstrate that the presence of waveexcitations in the medium causes the barrier at the intrinsicscale to permit increased energy transfer across it resulting inthe melting of the quasi-crystalline pattern. This study there-fore clearly demonstrates that waves play an important rolein spectral power cascade in nonlinear systems. The questionwhether waves influence cascade mechanism has been ad-dressed in the context of magnetohydrodynamics �MHD�.Extensive high resolution simulations have been carried outto discern the difference between the Kolmogorov energyspectrum scaling, viz., k−5/3 and Iroshnikov and Kraichnan5

scaling, e.g., k−3/2, which takes into account the influence ofAlfven waves. The difference in the spectral indices beingtoo small to easily resolve in the numerical simulations, thedebate on the validity of the two scalings continues. On theother hand our simulations give a clear demonstration of theinfluence of waves on the energy cascade behavior of theHM system by exhibiting qualitatively new macroscopic fea-tures of the turbulent state.

Another intriguing but interesting observation of ourstudy is that even though the cascade barrier at the intrinsicscale breaks due to the faster cascade induced by waveswhose origin is connected to the anisotropic inhomogeneityin plasma parameters, the spectrum of the fluctuations con-tinues to remain predominantly isotropic. Although a weaktransient anisotropy in the spectrum predominantly at longerscales is observed, overall the spectra continues to remainisotropic. The physical mechanism behind the process ofisotropization of the spectrum has also been outlined in themanuscript.

The paper has been organized as follows. In Sec. II webriefly introduce the model equation which has been used forthe numerical simulations. Some salient details about thesimulation has also been provided in this particular section.In Sec. III we present the results of our simulations whichshow that the presence of waves induces barrier transparencyprovided the strength of wave excitations exceeds a certainthreshold. The increased transparency of the barrier thenleads to the melting of the quasi-crystalline patterns. This hasbeen illustrated by studying the behavior of the spectrumbefore and after the transition. The results clearly demon-strate an increased cascade towards longer scales. In Sec. IVwe provide a physical interpretation of the observations. Sec-tion V contains the summary of our work.

II. MODEL EQUATION AND NUMERICAL SIMULATION

We demonstrate the influence of wave induced barriertransparency and the melting of the quasi-crystalline struc-ture by considering the driven dissipative Hasegawa Mimaevolution equation2

�

�t��2� − �2�� + z � �� · ��2� + Vn

��

�y= D + F . �1�

In this equation � represents the wave number correspondingto the characteristic natural scale of the system. For �=0 theequation reduces to the two dimensional Navier Stokes equa-tion for incompressible neutral fluid with the variable � rep-resenting the velocity potential. Here, D in Eq. �1� represents

the dissipation and F is the forcing. We have chosen hyper-viscous damping, i.e., D= �−1��p+1���2p��2�� to confine dis-sipation to very short scales. Equation �1� with D=F=0 hasbeen extensively studied in several contexts. It has been em-ployed as a paradigm for drift wave turbulence, where theequation represents the evolution of the normalized scalarpotential e� /Te=�. The length and time are normalized by�s=cs /�ci and �ci

−1, respectively, in the equation. Here, thecharacteristic natural scale of the system is �s=�−1=1 andVn=�s /Ln �Ln being the scale length of plasma density inho-mogeneity along the direction of x�. A linearization of theequation yields a dispersion relation for drift waves �e.g.,�=kyVn / �1+k2��. The equation has also been employed forthe study of geostrophic flows3 where the the Rossby radiusis the natural scale �−1 of the system. The waves in this caseare known as the Rossby waves. The third example is that ofthe two dimensional Electron Magnetohydrodynamics�EMHD� system in the absence of in-plane component ofmagnetic field. In this case the collisionless electron skindepth c /�pe=de=�−1 is the natural length scale of the systemand the presence of an equilibrium density gradient along xis the source for the additional term Vn�� /�y and yields theelectron gradient wave mode.4,6 Clearly, Eq. �1� represents awide class of physical phenomena.

For our simulations we study the driven dissipative casewith both F and D finite. The system is forced through F ina narrow range of wave numbers �k around the wave num-ber kf. In order to study the behavior of the system as theenergy cascades towards longer scales, the system is forcedaround a scale, viz., kf

−1, which is shorter than the naturallength scale �−1 of the system. We use a pseudospectral codewith a square domain of Lx=Ly =20 and a resolution up to512�512 to simulate the evolution of � through Eq. �1�. Aninitial condition with random low energy perturbation in �was chosen and has been depicted in Fig. 1. Note that theinitial configuration for � has no semblance to any quasic-rystalline vortex pattern. We observe that such a pattern de-velops during the course of evolution and is in no way de-pendent on our choice of the initial configuration as well asthe forcing function �as long as it is in the band of wavenumber above ��. Choosing �=1.0 for our simulations, theforcing is carried over the set of wave numbers Kinj lyingbetween kf1=3.0 to kf2=3.5. The choice of forcing functionis similar to the form described by Grossman et al.,7 viz.,

F�p,t� =inj��p,t�

�q�Kinj

���q,t��2; p � Kinj ,

F�p,t� = 0, otherwise.

Here F�p , t� is the pth Fourier mode of the forcing function Fat time t. The above form of forcing injects energy in thesystem at a rate of inj. In Fig. 2 we exhibit the temporalevolution of the total energy in the system for various injec-tion rates inj. The magnitude of the total energy at any timeafter an initial transient phase is in direct proportionality tothe chosen energy injection rate. The saturation would resultwhen the injection rate of energy matches the rate of dissi-

042307-2 Amita Das Phys. Plasmas 14, 042307 �2007�

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FIG. 1. �Color online� The contourplot of initial condition of the field �in the two-dimensional x-y plane.

FIG. 2. The upper subplot shows thetemporal evolution of the energy in thesystem at various injection rates inj.The value of inj are 0.01 �thick dots�,0.03 �dashed curve�, 0.05 �dotted line�,0.08 �solid line�, and 0.1 �uppermostdot-dashed curve�. The lower subplotshows energy at t=200 as a functionof inj.

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pation. For a higher rate of injection the field amplitudewould saturate at a higher level to attain a balance of energyinjection rate with the dissipative term in the equation. Since,the energy cascades towards longer scales and the dissipationhas been confined to shorter scales �specially with the choiceof hyperviscosity� it is difficult to saturate the energy duringany reasonable simulation time.

III. WAVE INDUCED BARRIER TRANSPARENCY

The field � in Eq. �1� is first evolved with a choice ofVn=0, which excludes wave excitations from the system.The initial energy in the system is small and increases withtime with input due to forcing as depicted in Fig. 2. Equation�1� supports two inviscid square integral invariants, viz., en-ergy E=�������2+�2�2dxdx and enstrophy V=�����2��2

+�2����2dxdx in the nondissipative and undriven limit, i.e.,when D=F=0. As in the Navier Stokes system here too,energy E suffers an inverse cascade towards longer scalelengths. We are forcing the system at scales much below thecharacteristics natural scale �−1, hence, as the energy cas-cades towards longer scales, it first hits the scale length at�−1 before reaching the system size. It was shown byKukharkin et al.1 that in such a situation the fluctuationscondense to form quasi-crystalline vortex patterns aroundthis scale. We depict such a condensation in Fig. 3 with thehelp of a two dimensional contour plot of potential �, whichshows localized distinct spatial structures scattered aroundthe two dimensional plane. To quantitatively ascertain thatthe patterns indeed form around the natural scale length �−1

FIG. 3. �Color online� The contourplot of � at t=175 showing the forma-tion of quasi crystalline structures.

FIG. 4. The evolution of spectral average kx2� and ky

2�, respectively areshown in the upper and lower subplots for �=0 �thick solid line� and �=1�dotted line�. The thin solid horizontal line has been drawn at unit abscissafor the sake of comparison.

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we show the evolution of the mean square average value ofthe wave vectors in Fig. 4 for �=0 �solid line� and for �=1.0 �dotted line�. The mean square average wave vectorsare defined as

kx2� =

��kx2���kx,ky��2dkxdky

�����kx,ky��2dkxdky;

ky2� =

��ky2���kx,ky��2dkxdky

�����kx,ky��2dkxdky.

It should be noted from the figure that kx2� and ky

2� aresimilar, as expected from isotropy in the x−y plane. Theinitial average values of these wave vectors are high, this isbecause the initial spectrum was chosen to be concentratedaround the higher wave numbers and the energy injectionthrough forcing is also carried around the high wave numberband of 3.0�kf �3.5. However, kx

2� and ky2� are seen to

diminish with time. For �=0 the average values kx2� and ky

2�asymptote around a few times of �kmin

2 � where kmin2

= �2 /L�2=0.01. Here L=20 is the simulation box size.This shows that for �=0 the inverse cascade of spectra con-tinues up to the simulations box size. However, for �=1 themean square average asymptotes at a comparatively higher

value around unity. This has been depicted by plotting ahorizontal line with abscissa at unity in the plot for compari-son. This gives a clear indication that the power accumula-tion occurs around the natural scale length inherent in thissystem.

We now investigate how the presence of wave excita-tions impact this scale length barrier. For this purpose thelinear term in Eq. �1�, viz., Vn�� /�y is switched on duringthe evolution. We evolve the system with Vn=0 up to t=200, the time around which the system reaches a statewhere the mean square wave numbers have achieved asteady value, and quasi-crystalline structures do get formed.For these simulations we have chosen �=1.0. In Fig. 5 wecompare the evolution of mean square wave numbers �kx

2�and ky

2� in subplots �a� and �b�, respectively� for variousdistinct cases. The thick solid line illustrates the case forwhich Vn remains zero throughout. However, in this case theenergy injection has been switched off beyond t=200. Thusthe simulation beyond t=200 depicts a decaying run. Forcomparison we have plotted a case �dotted line� again withVn=0 throughout and for which the forcing continues to bepresent. It is clear from the comparison of these two runs thatfor the decaying case �solid line beyond t=200� there is a

FIG. 5. Evolution of mean square av-erage kx

2� and ky2� are shown in the

subplots �a� and �b�, respectively, forthree distinct cases. �1� Dotted lineshowing the evolution for Vn=0 andinj=0.01 for all times. �2� Solid linerepresents the case for which Vn=0 forall times, however, inj has beenswitched off from the value of 0.01 tozero at t=200. �3� The dot-dashed lineshows similar run as in �2� but forwhich Vn=0.8 after t=200. In subplots�c� and �d� evolution of �i� total energyEtot=�0

kmaxE�k�dk �here E�k� depictsenergy in wave number k� with solidline, �ii� the energy contained inmodes with scale lengths longer thanthe intrinsic scale, i.e., El=�0

�E�k�dkby dashed line, and �iii� the energy inmodes with scale lengths shorter thanthe intrinsic scale Es=��

kmaxE�k�dkhave been plotted for the simulationcases �2� and �3�, respectively.

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slight decrease in the value of mean square wave numbers incomparison to the forced case �dotted line�. This can happeneither due to the damping at higher wave numbers by hyper-viscosity or due to the fact that the barrier at the intrinsicnatural scale is leaky and continues to permit the inverseenergy cascade. To ascertain this we have in subplot �c� plot-ted the evolution of �i� total energy with solid line represent-ing Etot=�0

kmaxE�k�dk �here E�k� depicts energy in wave num-ber k�, �ii� the energy contained in modes with scale lengthslonger than the intrinsic scale, i.e., �El=�0

�E�k�dk� by dashedline, and �iii� the energy in modes with scale lengths shorterthan the intrinsic scale �Es=��

kmaxE�k�dk� plotted by dottedline. The plot clearly shows that while Etot and Es decreaseonce the forcing has been switched off, however, the energycontained in scales longer than the intrinsic length scale El

continues to increase. This clearly indicates that the barrier atthe intrinsic scale � is slightly leaky. A small amount ofenergy continues to cascade past the natural scale determinedby the slower cascade rate at the longer length scale side ofthe barrier.

We next consider the evolution of wave numbers in sub-plots �a� and �b� of Fig. 5 depicted by the dot-dashed line.This also corresponds to a decaying run beyond t=200, how-ever, here Vn has been switched on to 0.8 at t=200. Thisthick dot-dashed line shows a more drastic reduction in theaverage values of the wave numbers. The evolution of vari-ous energies corresponding to this particular simulation hasbeen shown in the subplot �d� of the figure. Here too, whileEtot and Es decrease, El keeps increasing even after t=200.The reduction in Es and the enhancement of El is much moredrastic in this case than compared to those depicted in sub-plot �c�, where Vn=0. This clearly demonstrates that in thepresence of a finite Vn �which permits linear wave excitationsin the system� the barrier at the intrinsic scale length of thesystem gets increasingly transparent. It is interesting to com-pare the spatial structures at a later time say t=400 for thetwo cases depicted by solid and dot-dashed lines. This hasbeen shown in the contour plots of Fig. 6. While the simu-lation with Vn=0 continues to exhibit localized vortex pat-terns at various spatial locations, the case for which Vn wasswitched on clearly shows the melting of vortices. The gen-eration of longer scale excitations can be distinctly observedin this case.

Our simulations have also shown that this barrier trans-parency gets induced only when Vn exceeds a certain thresh-old. Typically, this occurs when the linear and the nonlinearterms of the evolution equations become comparable. Thevalue of Vn required to induce barrier transparency for thosecases when the saturated energy is higher in the system �e.g.,when it is stirred at a higher energy injection rate� is found tobe high. This threshold on Vn is found to scale with thesquare root of the total energy in the system.

Kukharkin et al. had illustrated the formation of quasi-crystalline patterns with the help of the plot of structurefunctions. In particular, the plot of eighth order structurefunction was the most prominent depiction of the existenceof long range order in the quasi-crystalline phase. In Fig. 7we illustrate the melting transition for the quasi-crystalline

pattern by plotting the eighth order structure functions S8�r�as a function of r, defined by

S8�r� =�����r�1 + r�� − ��r�1��8d2r�1

��d2r�1

�2�

at three different times. Here ��r�=�2�−�. The solid linecorresponds to t=175 when the quasi crystalline pattern for-

FIG. 6. �Color online� Contours for constant � are shown at t=400 for cases�ii� and �iii� of Fig. 5 in upper and lower subplots, respectively.

FIG. 7. The plot of 8th order structure function for case �iii� of Fig. 5 at threedistinct times, viz., t=175 �solid line�, t=500 �dot-dashed line� and t=1000 �dotted line�.

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mation has already taken place. The amplitude of the func-tion shows recurrent oscillations with r. These oscillationsare quasi periodic in space. This is due to the quasi-crystalline pattern formation in this stage. The maximumvalue of the ordinate r in the plot corresponds to half thesimulation box size. At later time when the wave excitationsare present �i.e., after Vn has been switched on�, the functiontries to become periodic �dot-dashed line of the figure corre-sponding to t=500�. Such periodicity is expected in the pres-ence of waves. The spatial periodicity length is seen to in-crease and the amplitude of the oscillations are observed toreduce with time. This can be discerned from the comparisonof plots for t=500 �dot-dashed line� and for t=1000 �dottedline�. The oscillation length scale increases and there is alsoa decay in the amplitude of the oscillations. This suggeststhat the presence of waves induces a melting transition of thequasi-crystalline patterns whence they slowly try to approachthe characteristics of the liquid state.

The dispersion relation of the waves excited in the pres-ence of finite Vn is �k=kyVn / �1+k2�, which clearly is aniso-tropic in the x-y plane. However, an interesting observationis that even though the melting transition is induced by theseanisotropic waves the spectral property of the system contin-ues to remain isotropic. This has been clearly demonstratedin Fig. 8 where a comparison of kx

2� and ky2� has been made

for two different values of Vn. The solid and dashed linecorrespond to kx

2� for Vn=0.8 and Vn=0.2, respectively. Thedots hovering around each of these line are the values of ky

2�for the respective cases. Note, that the difference in kx

2� andky

2� is very mild and is of oscillatory character. It shows thatthere is no preferential enhancement or decrement in thevalue of ky

2� as compared to kx2�. Basically, the fluctuating

spectrum continues to remain isotropic even after Vn isswitched on permitting waves with anisotropic dispersion re-lation. However, since the evaluation of the mean square

average values of the wave numbers k2� is biased towardsthe shorter wavelength side of the spectrum, we have alsocompared the average values of inverse of the wave numbersquares, viz., 1/kx

2� and 1/ky2�. This accentuates the pres-

ence of any anisotropy in the longer wavelength regime ofthe spectrum. Such plots show a distinct difference in thevalues of 1/kx

2� and 1/ky2�, but the difference oscillates in

time such that on an average the spectrum can be consideredto be isotropic. This implies that there is no preferential for-mation of scales with a particular symmetry in time.

We now demonstrate that barrier transparency leading tomelting occurs only for dispersive waves. The denominator,viz., �1+k2� appearing in the dispersion relation is crucial forthe melting transition to occur in the presence of waves. Fora Doppler shifted real frequency these effects are not ob-served. We simulate the following evolution equation todemonstrate this

�

�t��2� − �2�� + z � �� · ��2� + Vn

��2� − �2�

�y

= D + F . �3�

Here the linear dispersion relation is merely a Dopplershifted real frequency, viz., �k=kyVn. The plot of meansquare wave numbers with time in Fig. 9 shows no suddendrop at the time t=200 when the value of Vn was switchedon to 0.8. There is no difference between the two cases ofVn=0 �dashed line� and the Vn=0.8 �dots� in the plot for thesimulation of Eq. �3�. The contour plots of the field � alsoshow no sign of melting transition when Vn is switched on.As expected, a translational invariant system such as thisdoes not induce any barrier transparency and/or melting tran-sition of the quasi-crystalline vortex patterns. The dispersivenature of the of the normal modes are essential for this tooccur.

IV. INTERPRETATION OF NUMERICALOBSERVATIONS

The section is devoted to the interpretation of the nu-merical observations of the 2D fluid flow systems. In particu-lar we illustrate that the process of inverse cascade andisotropization of the spectrum associated with the nonlinear-ity z��� ·��2� of the 2D incompressible fluid flow systemis inherently associated with the underlying physics of theKelvin–Helmholtz �KH� destabilization of shear flows.8 Thissimilarity provides the explanation for the numerical obser-vations associated with the role of intrinsic natural lengthand time scale on energy cascade behavior outlined in theprevious sections. The accumulation of power at the naturallength scale can be understood by realizing that the growthrate of the Kelvin–Helmholtz destabilization of the shearflow pattern depends on whether the scale length of the ex-cited mode is smaller or longer than the natural length scaleof the system. Moreover, the observed wave induced trans-parency beyond a certain threshold value of Vn can also beunderstood by analyzing the behavior of the growth rate ofKH mode in the presence of Vn.

The simulation was initiated by injecting power at veryshort scales. This is tantamount to exciting sheared flows in

FIG. 8. This figure shows the evolution of mean square wave numbers forthe case �iii� �mentioned in Fig. 5� simulation The evolution of kx

2� for Vn

=0.2 �dashed line� and Vn=0.8 �solid line� has been shown in the figure. Thedotted line hovering around the two curves depict the evolution of ky

2�. Notethat the difference in kx

2� and ky2� is very mild and of oscillatory nature,

indicating transient weak anisotropy in the spectrum.

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the system at short scales. For the sake of simplicity anddefiniteness let us concentrate on the flow due to one particu-lar scale of wave number say k0x. The fluid being incom-pressible the fluid flow is orthogonal to the x direction, i.e., itis along y for this mode. The shear scale length associatedwith this flow is �k0x

−1. A linearization of the nonlinear term�z��� ·��2�� in Eq. �1�, treating this flow as an equilib-rium would produce the requisite differential equation for theevaluation of the eigenvalues �growth rate� and the eigen-mode structure of the KH destabilization associated with thissheared flow pattern. It is well known that typically thegrowth rate of the shear flow having a shear width of , isfinite only for those wave numbers say ky �orthogonal to theshear direction� for which ky 1 and maximizes when ky�O�0.5� O�1�. For ky�1 and beyond the growth ratevanishes. Thus, as a result of KH destabilization of the origi-nal shear flow, a mode with shear in the orthogonal directionand a shear width almost twice as long as gets excitedpreferentially. This excited mode also constitutes a shearedflow, and hence is itself KH unstable. Thus the mechanism ofKH destabilization continues; at each stage producing longerand longer scales. The KH destabilization thus provides theunderlying mechanism for the inverse cascade in 2D fluidflows. It should also be noted that at every stage, the KHdestabilization produces shear scales orthogonal to the origi-nal flow direction. This particular aspect of the KH instabil-ity is responsible for the isotropization of the spectrum.

The role of intrinsic scale �−1 of Eq. �1�1 and the pres-ence of the wave excitations on the phenomena of inversecascade can also be understood by analyzing their implica-tions on the KH growth rate. For analytical tractability, wechoose for this purpose a step velocity shear profile. Thelinearized Eq. �1� in the presence of a y directed equilibriumflow �yv0�x�= y��0 /�x� can be written as

�

�t��2� − �2�� + v0

�

�y��2� − �2��

−��

�y

�

�x �2�0 −

�2�0

�x2 � + Vn��

�y= 0. �4�

We have used �=�0+� in Eq. �1� and retained the termswhich are linear in the perturbed field �. The sheared flowbeing a step function of the form v0�x�=−V0+2V0��x� weobtain the following equations in the two spatial region I�−��x�0� and region II �0�x��� after Fourier decom-posing in time t and spatial coordinate y

d2�I,II

dx2 − �I,II2 �I,II = 0, �5�

where

�I2 = ��2 + ky

2� −�2kyV0

�� + kyV0�−

kyVn

�� + kyV0�,

�II2 = ��2 + ky

2� +�2kyV0

�� − kyV0�−

kyVn

�� − kyV0�. �6�

Choosing �I and �II as positive square roots we can write thesolutions in the two regions as

�I = A exp��Ix�; �II = C exp��IIx� . �7�

The constants and the dispersion relation has to be deter-mined by the matching conditions which are the continuityof the functions9

FIG. 9. Comparison in the meansquare wave number evolution for asystem having non dispersive Dopplershifted frequency. The two plotsdashed line for Vn=0 and dots corre-sponding to Vn=0.8 exhibit no differ-ence. Like previous cases �Fig. 5� thesystem is forced up to t=200 and rep-resents a decaying run beyond thistime.

042307-8 Amita Das Phys. Plasmas 14, 042307 �2007�

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f1 = �� − kyv0�d�

dx+ ky�

dv0

dx,

f2 =�

�� − kyv0�.

Applying these conditions we get

�� − kyV0��− �IIC� = �� + kyV0���IA� ,

C

�� − kyV0�=

A

�� + kyV0�,

which, upon elimination of A and C, yields the followingdispersion relation

�3�3�2 + 4ky2� − kyVn�3�2 + ky

2V02� + �ky

2V02��2 + 4ky

2� = 0.

�8�

For Vn=0 we have the following expression for the growthrate �= Imag���:

� = kyV0� �2 + 4ky2

3�2 + 4ky2 . �9�

Substituting, �=0 provides the growth rate of the neutralhydrodynamic fluid as �=kyV0. When � is finite the growthrate for those modes with scales 4ky

2�3�2 �i.e., shorter thanthe natural scale length� has the form �kyV0. On the otherhand modes with scale length longer than the intrinsic scalehave a slower growth rate of kyV0 /�3. We believe that incontrast to the behavior of the hydrodynamic Navier stokesfluid, this change in the form of the growth rate on the twosides of the natural scale length �faster for shorter scale ex-citations and slower for longer scale modes� is responsiblefor the barrier formation and accumulation of power aroundthe intrinsic scale length of �−1. This leads to the formationof the quasi crystalline structures.

Let us now try to understand the phenomena of inducedtransparency of the barrier in the presence of wave excita-tions which results in the melting of the quasi crystallinestructures. When Vn is finite the cubic dispersion relationEq. �8� needs to be solved. We do not attempt any perturba-tive solution in Vn here, as the simulations have shown thatthe barrier transparency occurs only when the value of Vn issuch that the linear term associated with it becomes compa-rable to the nonlinear term. Figure 10 shows the plot of thegrowth rate as a function of Vn. The figure clearly shows thatthe growth rate of the KH mode first diminishes and then itsubsequently increases with Vn. At higher values of Vn, thegrowth rate even exceeds the value at Vn=0. It is in thisregime that one observes the wave induced transparency inenergy cascade past the natural length scale barrier and/orthe melting phenomena associated with quasi-crystals.

V. SUMMARY AND DISCUSSION

One of the major distinctions between the neutral hydro-dynamic fluid flow and the behavior of the plasma fluid isthat the plasma fluid is ridden with several characteristicslength and time scales. The role of these characteristicsscales in defining the turbulent state �spectral properties, cas-cade rates etc.� has been an issue of extensive investigation.However, most of the investigations in this regard have beencarried out in the context of Magnetohydrodynamic modeland also largely been focussed at determining the spectralpower law indices and scaling properties of the structurefunctions. For instance, in the context of Magnetohydrody-namic �MHD� fluid, the question whether or not the Alfvenwave influence the cascade properties has been studied bydiscerning the difference between the Kolmogorov energyspectrum scaling, viz., k−5/3 and Iroshnikov and Kraichnan5

scaling, e.g., k−3/2. The difference in the spectral indices be-ing too small cannot be resolved easily in the numericalsimulations. The debate on the validity of the two scalingsthus continues. In this work we have discerned the role ofwaves on the energy cascade behavior of the Hasegawa–Mima �HM� system by observing qualitatively new macro-scopic features of the turbulent state.

The study on 2D plasma turbulence model depicted bythe Hasegawa Mima �HM� system carried out by Kukharkinet al.1 demonstrated that the existence of the intrinsic naturallength scale with the HM set of equations acts as a barrier tothe inverse energy cascade. This leads to the formation ofquasi-crystalline structures around the typical length scale.This was attributed to the difference in the cascade rate at thetwo sides of the characteristic length scale, by Kukharkin etal.1 By considering the presence of dispersive drift waves inthe system arising through the inhomogeneous plasma back-ground �defined by Vn, the normalized inhomogeneity scalelength� we have been able to show that the barrier in theenergy cascade becomes transparent provided Vn exceeds acertain threshold. This results in the melting of the quasi

FIG. 10. Plot showing the growth rate of the Kelvin–Helmholtz instabilityas a function of Vn. Here the other parameter values are as follows V0

=0.1 and ky =1.0.

042307-9 Wave induced barrier transparency and melting… Phys. Plasmas 14, 042307 �2007�

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crystalline patterns. The dispersive nature of the waves isnecessary to induce this transparency. Moreover, it is ob-served that even though the wave dispersion relation is an-isotropic the resultant spectrum continues to remain isotro-pic. There is only a weak anisotropy �oscillatory in time� atlong scales.

All these observations can be understood by realizingthat the phenomenon of inverse cascade in 2D flows is es-sentially a nonlinear manifestation of the Kelvin–Helmholtzinstability associated with the shear flow patterns. It wasshown analytically by considering a simple case of step ve-locity shear profile that the growth rate of the KH instabilityalso exhibits similar features as demonstrated by the inverseenergy cascade rate. More extensive studies need to be doneto make this comparison more succinct and quantitative. Forinstance it would be desirable to seek a definite dependenceof the cascade rate on the KH growth rate of the distinct

shear flow patterns present in the system. Efforts in this di-rection are currently being carried out.

ACKNOWLEDGMENT

The author would like to thank Professor P. K. Kaw forhis keen interest and encouragement in the work.

1N. Kukharkin, S. A. Orszag and V. Yakhot, Phys. Rev. Lett. 75, 2486�1995�.

2A. Hasegawa and K. Mima, Phys. Fluids 21, 87 �1978�.3J. C. Charney, J. Atmos. Sci. 28, 1087 �1971�.4A. S. Kingsep, K. V. Chukbar and V. V. Yankov, in Reviews of PlasmaPhysics �Consultants Bureau, New York, 1990�, Vol. 16, and referencestherein; A. Das and P. H. Diamond, Phys. Plasmas 7, 170 �2000�.

5R. H. Kraichnan, Phys. Fluids 8, 1385 �1985�.6T. M. Abdalla, B. N. Kuvshinov, T. J. Schep and E. Westerhof, Phys.Plasmas 8, 3957 �2001�.

7S. Grossman and D. Lohse, Phys. Fluids 6, 611 �1994�.8P. G. Drazin and W. H. Reid, Hydrodynamic Stability �Cambridge Univer-sity Press, Cambridge, London, 1981�, p 14.

9A. Das and P. Kaw, Phys. Plasmas 8, 4518 �2001�.

042307-10 Amita Das Phys. Plasmas 14, 042307 �2007�

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