lyapunov exponents and particle dispersion in drift wave turbulence
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Lyapunov exponents and particle dispersion in drift wave turbulenceThomas Sunn Pedersen, Poul K. Michelsen, and Jens Juul Rasmussen Citation: Physics of Plasmas (1994-present) 3, 2939 (1996); doi: 10.1063/1.871636 View online: http://dx.doi.org/10.1063/1.871636 View Table of Contents: http://scitation.aip.org/content/aip/journal/pop/3/8?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Enhancement of Lyapunov exponents in onedimensional, randomly phased waves Phys. Plasmas 1, 1817 (1994); 10.1063/1.870635 Investigations of attractors arising from the interaction of drift waves and potential relaxation instabilities Phys. Fluids B 4, 3990 (1992); 10.1063/1.860303 Trapped structures in drift wave turbulence Phys. Fluids B 4, 2854 (1992); 10.1063/1.860160 Observation of fast stochastic ion heating by drift waves Phys. Fluids B 3, 3363 (1991); 10.1063/1.859768 Point vortex description of drift wave vortices: Dynamics and transport Phys. Fluids B 3, 3255 (1991); 10.1063/1.859756
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Lyapunov exponents and particle dispersion in drift wave turbulenceThomas Sunn Pedersen, Poul K. Michelsen, and Jens Juul RasmussenAssociation EURATOM-Riso” National Laboratory, Optics and Fluid Dynamics Department,P.O. Box 49, Riso”, DK-4000 Roskilde, Denmark
~Received 14 March 1996; accepted 16 May 1996!
The Hasegawa-Wakatani model equations for resistive drift waves are solved numerically for arange of values of the coupling due to the parallel electron motion. The largest Lyapunov exponent,l1 , is calculated to quantify the unpredictability of the turbulent flow and compared to othercharacteristic inverse time scales of the turbulence such as the linear growth rate and Lagrangianinverse time scales obtained by tracking virtual fluid particles. The results show a correlationbetweenl1 and the relative dispersion exponent,lp , as well as to the inverse Lagrangian integraltime scalet i
21 . A decomposition of the flow into two distinct regions with different relativedispersion is recognized as the Weiss decomposition@J. Weiss, Physica D48, 273 ~1991!#. Theregions in the turbulent flow which contribute tol1 are found not to coincide with the regions whichcontribute most to the relative dispersion of particles. ©1996 American Institute of Physics.@S1070-664X~96!03808-6#
I. INTRODUCTION
The anomalous transport in toroidal devices is com-monly thought to be caused by low frequency electrostaticfluctuations, in particular in the edge region of the plasma.1,2
These fluctuations are believed to be of the drift wave typewhich always appear self-excited at the edge of magnetizedplasmas due to the pressure gradients. A simplified two-dimensional slab model for non-linear drift waves was pro-posed by Hasegawa and Wakatani.3,4 The model includes theeffects of a background density gradient in the direction per-pendicular to the magnetic fieldB, and a generalized Ohm’slaw for the parallel electron motion. In the model, the driftwaves are linearly unstable but after a transitional period ofexponential growth, a non-linear quasi-stationary turbulentstate is achieved. This turbulent state has been studied theo-retically and numerically in a large number of publications~see, e.g., Refs. 5–8 and references therein!. These worksshow that the self-excited quasi-stationary turbulence isstrongly dependent on the assumption about the parallelwave number, but it appears to be independent of the initialconditions. This indicates that the turbulence can be de-scribed in the context of chaos analysis as an attractor in thephase space of the Hasegawa-Wakatani model with a largebasin of attraction, and that the character of the attractorstrongly depends on the model for the parallel wavenumber.8
The quasi-stationary turbulence may be characterized ina number of ways. Previously the effects of the size of therelative density gradient on the perpendicular transport havebeen studied.3,4 The spectral properties such as the spectralindex have been determined,6,7 and the existence of coherentstructures and their influence on the transport have also beeninvestigated.5–7 The presence of intermittent coherent struc-tures complicates the statistical theory based on the spectralproperties of the turbulence, and one may therefore look forother ways to describe turbulence, when coherent structuresare dominant.
In the present contribution, the degree of unpredictabilityor chaoticity is determined by the largest Lyapunov expo-
nent, l1 , for a range of values of the parameters in theHasegawa-Wakatani model. A key issue in the investigationis to clarify to what extentl1 is correlated with other char-acteristic inverse time scales. If there does exist such a cor-relation, the Lyapunov exponents may prove to be usefuldiagnostic tools for plasma turbulence, since they can, atleast in principle, be determined from time series of, e.g.,reflectometer or probe measurements, using a phase spacereconstruction technique.9 Other characteristic inverse timescales are the growth rate of the most unstable mode, thenormalized transport and inverse Lagrangian time scales asdescribed in the following.
The turbulence is analyzed by determining itsLagrangian properties, obtained by tracking virtual fluid par-ticles, passively convected by the velocity field. We shallconcentrate on the relative~or two-particle! dispersion in thelimit of small interparticle distances, i.e., how two nearbyparticles are diverging from each other in the turbulence. Weshow that a linearized theory predicts the decomposition ofthe turbulence into two essentially different regions, one withexponential growth of the particle separation and one with anoscillatory behavior of the particle separation. This decom-position turns out to be similar to the so-called Weissdecomposition10 derived to describe enstrophy transfer intwo-dimensional hydrodynamic turbulence. The linearizedtheory is compared with numerically obtained results for therelative dispersion. An exponential separation of nearby par-ticles is conceptually similar to chaos in a dynamical system,which implies exponential separation of nearby orbits inphase space. Therefore a comparison between the particleseparation exponent,lp , and the largest Lyapunov exponentis made. The Lagrangian velocity auto-correlation functionL(t) is also determined, and two characteristic time scalesdefined byL(t), the microscale and the Lagrangian integraltime scale, are calculated. The results show thatl1 seems tobe correlated with several of the inverse time scales investi-gated, in particular the particle separation exponentlp . Theregions of the turbulence that disperse particles fast are com-pared with the regions of the turbulence that are most unpre-
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dictable to see whether there is also a spatial correlation be-tweenl1 andlp .
The paper is organized as follows: In Sec. II the modelequations are presented and discussed. Section III containsinformation about the simulation and the energy and enstro-phy of the system. In Sec. IV the Lyapunov exponent isintroduced, it is discussed how to calculate it and some re-sults are presented. Lagrangian statistics is computed in Sec.V by using particle tracking and the relation to the Weissdecomposition is demonstrated. Comparison between chao-ticity and particle motion is performed in Sec. VI and con-clusions are given in Sec. VII.
II. THE HASEGAWA-WAKATANI SYSTEM
A. Model equations
We consider the Hasegawa-Wakatani~HW! model3,4 forresistive drift wave turbulence and the basic equationscoupling the densityn and potentialf fluctuations read3,7
S ]
] t1 vE–¹'D ~¹'
2 f !5C ~f2n!1nD2p12f, ~1!
S ]
] t1 vE–¹'D n1
]f
] y5C ~f2n!1nD2pn, ~2!
v[¹'2 f, ~3!
C[2Te
n0e2hvci
Lnrs
]2
]z2, ~4!
D2p[~21!p11¹'2p , ~5!
and
vE[ z3¹f, ~6!
where the normalized variables are defined by
x5x
rs, y5
y
rs, t5
tvcirsLn
, f5ef
Te
Lnrs, n5
n1n0
Lnrs.
~7!
Here,
rs5ATemi
eB0
and
vci5eB0mi
.
D2p is the hyperviscous/hyperdiffusive operator which is in-cluded mainly as a numerical filter.vE is the(E3B)/B2-drift velocity normalized according to the trans-formation to dimensionless variables, Eq.~7!, first intro-duced in Ref. 5 andv(5(¹3vE)• z) is the vorticity.Te isthe electron temperature,h is the parallel resistivity,n0 is thebackground density.Ln is the characteristic length scale forthe relative background density gradient, which is assumedconstant,“ ln n0521/Ln , implying an exponential back-ground density.mi is the ion mass,n1 is the density pertur-bation, andvci is the ion cyclotron frequency.rs is a char-
acteristic scale length. For notational convenience, the‘‘ ; ’s’’ and the' ’s will be omitted in the following.
The equations may also be expressed in terms off andthe non-adiabatic responser5f2n
S ]
]t1vE–¹D ~¹2f!5C r1nD2p12f, ~8!
S ]
]t1vE–¹D r52C ~r1¹22r !1nD2pr1
]f
]y
2¹22@~vE–¹!¹2f#, ~9!
where¹22 is the inverse Laplacian, defined in Fourier spaceas2k2252(kx
21ky2)21. This formulation of the equations
may be used for studying the effect of the coupling termC (f2n) and has numerical advantages in the adiabaticlimit as described in Ref. 8.
B. C and the parallel wave number
From the definition ofC it is seen that in order to reducethe HW model from a fully three-dimensional model to aquasi-two-dimensional model, an assumption has to be madeabout the parallel wave numberkz . Essentially two differentassumptions have been studied numerically in previous pub-lications. The most recently studied assumption5–7 has beenthat there is only one modekz in the parallel direction, sothat C is the same for each perpendicular mode vector(kx ,ky). This constant, which shall be referred to asC 1 , canbe chosen in the range@0;`@ to reflect a specific choice ofvalues of the parallel resistivityh, the parallel wave numberkz , etc.
In the first numerical studies3,4 the assumption was thatfor each value of the perpendicular wave vectork'(kx ,ky),the plasma will choose the parallel wave number that maxi-mizes the growth rate of the linear instability. A simplifieddispersion relation leads to the following result:4,8
C54kyk
2
~11k2!2[C 2 . ~10!
This coupling is strongly anisotropic, being zero fork'5(kx,0)-modes and non-zero for other values ofk' . Weshall mainly consider the caseC5C 1 , however, some caseswith C5C 2 are included for comparison~see also Ref. 8!.
C. Linear growth rate
The linear growth rate for each mode (kx ,ky) in thelocal approximation is expressed as7
g52l
21
l0
2A2AA11
16s2
l04 11, ~11!
where
l5l01l15C 1~11k2!
k212nk2p,
s5C 1kyk2
.
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The growth rate of the most unstable mode as a function ofC 1 is plotted in Fig. 1a. The growth rate has a maximumaroundC 1'0.3 and goes to zero in both limitsC 1→` andC 1→0.
III. SIMULATION DETAILS
The HW equations@~1!, ~2! or ~8!, ~9!# are solved in adoubly periodic domainV, using a de-aliased Fourier spec-tral method in space and the third order semi-implicit stifflystable scheme for the time integration.11 The coupling termsare treated fully implicitly in the regime of strong coupling.The domain size is chosen to beLx5Ly550, and a hyper-viscosity operator of the typeD65¹6 is used. Usually,1282 or 2562 modes are used in the FFT’s~Fast FourierTransforms!, corresponding after de-aliasing to 862 and1722 active modes, respectively.
The accuracy of the code is checked by following thetime derivatives of the energyE and the generalized enstro-phyW defined as
E5Ekin1Epot51
2E EV
~¹f!2dx dy1E EVn2dx dy,
~12!
W 51
2E EV
~n2v!2dx dy. ~13!
E andW are not conserved quantities, but follow the equa-tions
dE
dt5E E
V2n
]f
]ydx dy2C E E
V~f2n!2dx dy
2nE EV
fD pv2nD pn dx dy
[G f1GC1Gn,E , ~14!
dW
dt5E E
V2n
]f
]ydx dy2nE E
V~n2v!
3~D pn2D pv!dx dy
[G f1Gn,W . ~15!
From these the accuracy of the code can be estimated fromthe following expressions:
eE~ t 
One of the main purposes of our investigations is toanalyze whether there is a correlation betweenl1 and otherproperties of the flow such as the perpendicular transportG f or the particle dispersion properties discussed in Sec. V.Any correlation is potentially interesting becausel1 can inprinciple be measured for edge turbulence in a real plasmaexperiment.l1 may be calculated from a time series of, e.g.,reflectometer measurements by the use of a phase space re-construction technique.9 Thus if a correlation betweenl1
and other properties of the flow does exist then informationabout these properties may be obtained indirectly by measur-ing l1 . We perform such an analysis by varying the opera-tors C andn of the HW system.
A. Numerical implementation
A standard way of determining the Lyapunov exponentsis to linearize the system at each point on an orbit in phasespace. The Jacobian is determined and the perturbation isevolved in time by multiplying it with the Jacobian. Unfor-tunately, the Jacobian is aD3D matrix. Since in our caseD52MN we get forM5N5172 ~corresponding to 2562
modes in the FFT’s! that the Jacobian hasD2
5 (2•172•172)2'3.5•109 elements, which is totally incom-prehensible. Instead one may follow two separate flows inthe simulations, which initially are in almost the same state,and observe how the difference between the two flows growsexponentially in time. However, there are several problemswith this approach. First of all, the difference between theflows does not stay infinitesimal, and after a finite time, thetwo flows will be essentially uncorrelated and there is nolonger exponential growth of the difference between theflows. Saturation of the perturbation is observed for this rea-son. We have chosen a more proper approach, similar to theone employed in Ref. 13. We perturb a given flow and derivethe linearized equations for the perturbation. Then the per-turbation will stay infinitesimal in the mathematical sense,and no saturation of the perturbation occurs. The linearizedequations for the perturbation are derived as follows: Firstassume infinitesimal perturbations (df,dn). The linearizedequations fordf,dn are found by subtracting the equationfor f, n in the ‘‘unperturbed’’ flow from the equation forf1df, n1dn in the ‘‘perturbed’’ flow. Then neglecting thesecond order terms yields the following equations for theperturbations (df,dn):
]dv
]t1vE–¹dv1dvE–¹v5C ~df2dn!1D2p~dv!, ~18!
]dn
]t1vE–¹dn1dvE–¹n
5C ~df2dn!2]df
]y1D2p~dn!. ~19!
Similarly, we obtain the linearized equations in ther -formulation
]dv
]t1vE–¹dv1dvE–¹v5C dr1D2p~dv!, ~20!
]dr
]t1vE–¹dr1dvE–¹r
52C ~dr1¹22dr !1]df
]y
1D2p~dr !2¹22~vE–¹dv1dvE–¹v!. ~21!
The HW equations and the equations for the perturbation areevolved in time simultaneously. The largest Lyapunov expo-nent l1 is obtained as the growth rate of the perturbationafter some initial time when it has reached a constant value~see Appendix A!. The following norm was chosen for thecalculation:
u~f,n!u5AE EV
f21n2dx dy. ~22!
With this approach the relative error inl1 due to intrinsicerrors in the time stepping is of the ordereE.
B. Results
We have determinedl1 for different values ofC 1 in therange 0.01<C 1<10 and forC5C 2 for different values ofthe viscosityn. The results are shown in Table I. In Fig. 1bwe have plottedl1 as a function ofC 1 for a specific choiceof the viscosity. There seems to be a maximum forl1 closeto C 150.1, i.e., the system appears to be most chaotic in thisintermediate regime. It is observed thatl1 decreases, i.e., thechaoticity diminishes asC 1 increases. Also in the hydrody-namic limit C 1→0 there is a tendency for a decreasingl1
but it is much weaker. We may conclude from these resultsthat the largest Lyapunov exponent shows a clear tendency togo to zero asC 1 increases and that it shows a tendency todecrease asC 1→0, but not necessarily to zero.
TABLE I. Lyapunov exponents.
n M5N C 1510 C 152 C 151 C 150.1 C 150.02 C 150.01 C5C 2
5•1024 128 ;0 * 0.332 0.350 * 0.238 0.1441024 128 0.0112 0.253 0.339 0.388 0.327 0.313 0.2545•1026 256 * * 0.388 0.478 * * *
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Comparingl1 in Fig. 1b with the linear growth rategfor different values ofC 1 ~Fig. 1a! we observe some corre-spondence between the shape of the two graphs. This dem-onstrates that the stronger linear instability the more chaos inthe system. Figure 2 shows that viscosity has the effect ofsuppressing the chaoticity. This indicates that the smallscales are active in the generation of chaos in the HW sys-tem. We note that the maximum linear growth rate is virtu-ally unaffected by hyperviscosity of the size considered here.This is so because the maximum linear growth rate occursfor intermediate values ofk that are relatively unaffected bythe hyperviscosity. On small scales, the nonlinear convectiveterms dominate the HW system together with the viscosity.Our findings suggest that the convective term, in combina-tion with the linear instability, are the sources of chaos in theHW system, whereas viscosity tends to damp the chaos. Thisalso seems intuitively reasonable.
V. LAGRANGIAN STATISTICS
The dispersion properties of the turbulent field can beexpressed via the Lagrangian statistical properties of theflow, see e.g., Ref. 14. In order to derive the Lagrangianstatistics of the drift wave turbulence in the HW model wetrack virtual particles. That is, passively convected, masslessparticles which do not affect the flow in any way. They sim-ply follow the fluid locally, i.e., their velocity equals the driftvelocity vE5 z3¹f. For details about the numerical imple-mentation of particle tracking see Appendix B. The equationof motion for a particlep with position r p5(x,y) becomes:
dr p
dt5vE~r p~ t !,t ![vp~ t !⇔ ~23!
F dx
dt
dy
dt
G p
5F 2]f
]y~r p,t !
]f
]x~r p,t !
G . ~24!
The explicit time dependence ofvE is kept to show that theflow velocity changes in time following the HW equations.The velocity vE is divergence free and strictly two-dimensional so the particles should not be expected to con-centrate in regions with high densityn since the growth ofthe density perturbation is an effect of a combination of theparallel electron dynamics and the background density gra-dient. Since the two-dimensional drift velocity flow is diver-gence free, the flow, in which the particles move, is incom-pressible.
A. Relative dispersion
The distance between a particle and a neighbor particleis defined as
d rel~r1 ,r2!5ur1~ t !2r2~ t !u.
If, for each particlep in the ensemble, a neighbor particlep is defined, then the ensemble averaged relative divergencecan be defined as
d rel~t!5^ur p~t!2r p~t!u&p . ~25!
If the neighbor particles are chosen so thatur p2r pu'kmax21
wherekmax21 relates to the minimum wavelength present in the
flow, then the particles and their neighbors will initially sepa-rate slowly since the velocity field varies only slightly overspatial distances smaller thanlmin52p/kmax. As the par-ticles separate, their velocities may become increasingly dif-ferent which makes the particles separate faster so that theirvelocities diverge faster, etc.15 A simple, intuitive model ofthis could be obtained by assuming that the rate of diver-gence~the difference in velocities! is proportional to the dis-tance between the particles, which would lead to the follow-ing:
d
dtd rel5lpd rel⇒d rel5d rel,0e
lpt.
If this is the case, then the particle motion can be said to bechaotic in the sense that there issensitive dependence on theinitial conditions and that two neighbor particles separateexponentially. However, since the equation of motion@Eq.~23!# does not define a dynamical system with a two-dimensional phase space (x,y) then lp is not a Lyapunovexponent in the strict mathematical sense defined in Appen-dix A. On the other hand, it may still be interpreted as aninverse time scale that quantifies the chaoticity and unpre-dictability of particle motion in the turbulence, just as thelargest Lyapunov exponent of a dynamical system,l1 .Therefore a comparison betweenl1 andlp is appropriate. Inthe following we show that a simplified and linearized theorypredicts exponential divergence, but only in some regions ofthe flow.
Previously, in theoretical studies, it has been found thatthe shape of the kinetic energy spectrum is important for therelative dispersion. The relative distance grows exponentiallyfor a spectral index ofa53 but not for smaller values of thespectral index.15–17In the following we consider a linearizedtheory for the relative dispersion of particles. We show that
FIG. 2. The Lyapunov exponentl1 for C 151 (h) and forC 150.1 (s),the particle exponentlp for C 151 (n) and forC 150.1 (1), and 1/2t i forC 150.1 (3) as a function ofn.
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this leads to a decomposition of the flow into two regionscorresponding to exponential growth and oscillatory behav-ior of the relative particle distance.
Consider a particle and a neighboring particle with aninfinitesimal relative distancedr moving in the fluctuatingvelocity field. Assume for simplicity that the velocity field is‘‘frozen,’’ i.e., that it is constant in time, and that the field isinfinitely differentiable. The equations of motion for the twoparticles read
d
dtr5v~r !5FuvG
d
dt~r1dr !5v~r1dr !
J⇒ d
dtdr5F ]u
]x
]u
]y
]v]x
]v]y
G dr1o~ udr u2!.
Since the distance is assumed infinitesimal the last term canbe omitted. This leaves a linear system of first order ordinarydifferential equations. We insert u52]f/]y andv5]f/]x and find the eigenvalues of the system matrix
U2]2f
]x]y2l 2
]2f
]y2
]2f
]x2]2f
]x]y2lU50⇔
~26!
l25S ]2f
]x]yD2
2]2f
]x2]2f
]y2[Q .
Thus, for Q,0 there are two imaginary eigenvaluesl56 iA2Q and the distance in the two eigendirections willbe purely oscillatory whereas forQ.0 there will be two realeigenvaluesl56AQ , which means that the distance willshrink exponentially in one eigendirection and grow expo-nentially in the other eigendirection. We should then expectto measure exponential growth of the relative distance inQ.0 regions since the exponential growth in the directioncorresponding to the positive eigenvalue will dominate overthe exponential shrinking in the other eigendirection, and ofcourse the growth should be faster in regions with high val-ues ofQ . In regions withQ,0 we should expect a slow oreven no growth of the relative distance for small interparticledistances since this first order approximation predicts no av-erage growth. It should be emphasized that the argumentsleading to Eq.~26! are based on linearized calculations andnaturally only apply in the initial case.
B. Discussion of the decomposition
The criterion for decomposing the flow into regimeswith Q.0 andQ,0, discussed above was derived previ-ously in the context of enstrophy transfer in two-dimensionalturbulence by Weiss,10 and is known as the Weiss decompo-sition. The present derivation shows the direct applicabilityof the decomposition to the relative dispersion of particles in
the turbulence. In Weiss’ formulation the definition ofQ ,which is mathematically equivalent to Eq.~26!, reads
Q5 14~s22v2!,
where
s25S ]u
]x2
]v]y D
2
1S ]u
]y1
]v]xD
2
is the square of the strain~or deformation! andv is the curl~or rotation! of the velocity field, i.e., the vorticity, as definedin Eq. ~3!. From Weiss’ expression it can be seen thatQ,0 regions areelliptic regions where rotation dominatesover strain, andQ.0 regions are strain-dominatedhyper-bolic regions. The Weiss decomposition has previously beenemployed in connection with the absolute dispersion of par-ticles in two-dimensional~2-D! incompressible Navier-Stokes turbulence, showing that the dispersion laws are sig-nificantly different in the two different regimes.18
From the derivation presented here it is expected thatalso the relative particle dispersion is different in the hyper-bolic and elliptic regions. Therefore, the particles are dividedinto two ensembles, according to the sign ofQ , and the
FIG. 3. Contour plot of the vorticityv in the saturated state forC5C150.1, n51024, M5N5128 and Contour spacing 4. Dashed linescorrespond to negative values, solid lines correspond to positive values.
FIG. 4. Contour plot of the Weiss fieldQ in the saturated state for the samesituation as in Fig. 3. Contour spacing 20.
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Lagrangian statistics is calculated for the two ensemblesseparately as well as for the ensemble consisting of all par-ticles. For the calculation of the ensemble and time averagedrelative dispersion the particles are divided into four en-sembles corresponding to strongly elliptic (Q,24), weaklyelliptic (24,Q,0), weakly hyperbolic (0,Q,4), andstrongly hyperbolic (Q.4) flow regions. This is done inorder to facilitate a detailed comparison to the predictions ofthe linearized theory. The values64 are chosen on the basisof preliminary numerical simulations.
The Weiss field for each particle is obtained by a purelyspectral interpolation of the Weiss field in mode space. Thecalculation of the Weiss field in mode space is done using theFFT algorithm and the 2/3 de-aliasing scheme since theWeiss field is a nonlinear quadratic function off. Comparedto the spatial interpolation of the velocity field, the low speedof the spectral interpolation is not a major problem for theWeiss field since it is calculated only when the statistics arecalculated which is every;20 time steps.
Contour plots of the vorticityv and Weiss fieldQ areshown in Figs. 3 and 4. It can be verified that the stronglyelliptic regions coincide with vortex structures in the vortic-ity. Strongly hyperbolic regions are found just outside thevortices, especially for the strong monopoles and the dipoles.From this, it seems that the Weiss decomposition can behelpful in diagnosing coherent structures. However, it is notclear whether the Weiss field directly contains informationabout the lifetimes of these structures.
C. Results for the relative dispersion
In Fig. 5 the relative dispersion is shown for differentregions of the flow as defined by the Weiss field. The ten-dency is clear. Particles in the hyperbolic regions of the flowspread faster than particles in elliptic regions and the ten-dency is stronger in strongly hyperbolic and elliptic areas.This result is in good agreement with the theoretical predic-tion leading to the Weiss criterion. The Weiss decompositionis thus useful when studying relative dispersion of particles.
In Fig. 6 the relative dispersion exponent, averaged overall particles, is shown for different values ofC 1 and forC5C 2 . The parametern is constant,n51024. It is seenthat the exponent is not constant in time, but that is has adecreasing trend after an initial ‘‘overshoot.’’ Therefore theaverage relative particle divergence is not purely exponentialin time. However, it is a reasonable approximation for4,t,10 for all values ofC 1 andC5C 2 , so for the sake ofobtaining a time scale for the relative particle dispersion westill assume exponential divergence. In this spirit, the particleexponents,lp , are calculated as averages over the time in-terval where they are approximately constant, typically4,t,10. This is done for different values of the couplingparameterC . Three different values of the parametern wereused to investigate the effects of the hyperviscous/hyperdiffusive damping. The results are displayed in TableII. As also evident on Fig. 6,lp seems to have a maximumaround some intermediate value ofC 1 , betweenC 150.1andC 151.
D. The Lagrangian autocorrelation function
We determine the Lagrangian velocity autocorrelationfunction defined as:
L~t,t !5^vp~ t !–vp~ t1t!&p
^uvp~ t !u2&p. ~27!
One may perform a time average ofL(t,t) to get
L~t!5^L~t,t !& t5 limt→`
S 1t Es50
t
L~t,s!dsD . ~28!
FIG. 5. Comparison between the relative dispersion in different areas of theflow for C 150.1, n51024. ‘‘All’’ is the average of all particles and ‘‘Half’’is the average of half the particles chosen randomly.
FIG. 6. Comparison between the relative dispersion exponent for differentchoices of the coupling parameterC for n51024. The exponents are aver-aged over all particles.
TABLE II. Relative particle dispersion exponents.
n M5N C 1510 C 151 C 150.1 C 150.01 C5C 2
5•1024 128 0.063 0.25 0.24 0.17 0.141024 128 0.070 0.25 0.27 0.19 0.165•1026 256 * 0.25 0.29 * *
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It is readily seen thatL(t50)51. The Lagrangian integraltime scalet i is defined as
t i5Es50
`
L~s!ds. ~29!
The microscalem is defined by the curvature ofL(t) at theorigin t50
d2L~t!
dt2 Ut50
522
m2 . ~30!
The integral scalet i can be estimated numerically fromthe numerical arrayL( iDt) wherei denotes a non-negativeinteger running from 0 to some maximum numberimax andDt is the interval between each calculation ofL. If Dt! t iandimaxDt @ t i then
t i'(i50
imax
L~ iDt!Dt.
This rather crude integral estimate is sufficient for the pur-poses in the present contribution.
Employing the Taylor expansion aroundt50 ofL(t)512t2/m2 the microscalem is determined by fitting aparabola toL. It can be seen from Table III thatm is smallerfor particles in elliptic regions of the flow than for particlesin hyperbolic regions. We estimatem;0.3 in elliptic regionsandm;0.6 in hyperbolic regions. This may seem counter-intuitive but reasoning in the following way provides an ex-planation: InL(t), the scalar productv(t)•v(t1t) be-comes zero if the particle has zero velocity~which does notoccur often in a turbulent flow! or if the velocity att1t isperpendicular to the velocity att. A 690° rotation of thevelocity is necessary for this to occur. Since elliptic regionsare rotation-dominated whereas the hyperbolic regions aredominated by strain~or deformation!, one may expect thatL initially decreases faster in elliptic~rotation dominated!regions. This tendency might not hold for longer times sinceparticles which are rotational will return to their original ve-locity after a rotation of 2p and this will have the oppositeeffect i.e., it will tend to makeL larger. In Fig. 7L(t) isshown for 0<t<6, a time span longer than the integral timescalet i which is 0.67 for this case. An oscillatory behavior isobserved, particularly for particles in elliptic regions, as ex-pected.
An important issue in the present contribution is the pos-sible correlation between the largest Lyapunov exponentl1 , as defined in Sec. IV, and other inverse time scales of theturbulence. The two time scalest i andm are candidates for a
comparison withl1 . In Table IV the values oft i are pre-sented for different choices of the parametersC andn. Thevery large values oft i for C5C 2 can be understood fromthe results of Ref. 8. The dominatingk''(0.3,0) modemakes the fluid flow in they-direction and this motion maygo on for a very long time, transporting fluid particles with-out changing their velocities significantly. This does not pre-vent particles from separating exponentially~or semi-exponentially! although the separation is somewhat slowerthan for the intermediate regime ofC 1 values. This mayappear contradictory but it should be born in mind that thesmall scales~high wave numbers! of f dominate in the rela-tive dispersion process whereas the intermediate and largescales off dominate the velocities of particles, and thereforethe shape ofL(t). This is particularly the case forC5C 2
due to the strong (kx,0) mode.
VI. COMPARISON BETWEEN CHAOTICITY ANDPARTICLE MOTION
A. Inverse time scales
One of the most important reasons for studying electro-static drift wave turbulence is to understand the basic mecha-nisms of the strong transport of particles across the magneticfield lines. Therefore, a comparison betweenl1 and the non-linear particle fluxG f is interesting. However, sinceG f , inthe saturated state, becomes increasingly larger asC 1→0~see Ref. 8! whereasl1 peaks for an intermediate value ofC 1 , there seems to be no simple correlation between chao-ticity and anomalous transport.
Other possible correlations could be betweenl1 and thetwo characteristic Lagrangian inverse time scaleslp and
TABLE III. The Lagrangian microscalem.
Ensemble n M5N C 1510 C 151 C 150.1 C 150.01 C5C 2
All 5 •1024 128 4.2 0.84 0.63 0.30 *Q.0 5•1024 128 4.7 1.05 0.81 0.50 *Q,0 5•1024 128 3.7 0.68 0.48 0.20 *All 1024 128 4.0 0.83 0.44 0.29 3.0Q.0 1024 128 4.4 1.05 0.56 0.35 3.6Q,0 1024 128 3.6 0.67 0.33 0.20 2.5
FIG. 7. The Lagrangian auto-correlation for a time span longer than theintegral timet i for different particle ensembles.C5C 150.1, n51024.
TABLE IV. The Lagrangian integral timet i .
n M5N C 1510 C 151 C 150.1 C 150.01 C5C 2
5•1024 128 2.55 0.68 0.74 0.83 71.31024 128 2.66 0.65 0.67 0.81 69.45•1026 256 * 0.78 0.61 * *
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1/t i found in Sec. V. These inverse time scales are shown inthe Figs. 1b and 2 together withl1 as functions ofC 1 andn. It is clear that the time scales are not identical but theyseem to be correlated.l1 and lp are of the same orderwhereas 1/t i is somewhat larger thanl1 and lp . For thesake of comparison we have plotted the quantity 1/2t i in-stead of 1/t i . As previously mentioned, this correlation isinteresting sincel1 is in principle measurable in real plasmaexperiments. The Lagrangian inverse time scalest i
21 andlp are not directly obtainable from plasma experiments sinceit is not possible to track individual particles in a real plasma.For lp the correlation may be extended even to include thestrongly anisotropic turbulence obtained forC5C 2 whereasit is clear that this can not be done for 1/t i . It still remains anopen question if the correlation found here is specific to theHW model or if it may be observed in other models as well.
B. Spatial dependencies
As seen from Fig. 4 and also noted previously in Ref. 18vortex cores coincide with strongly elliptic regions whereasthe edge regions of vortical structures are strongly hyper-
bolic. Since it has been confirmed from the numerical simu-lations that the relative dispersion is much stronger in hyper-bolic regions, we conclude that the hyperbolic regions,particularly vortex edge regions, are the main contributors tothe overall relative particle dispersion.
One may identify the regions that are main contributorsto chaos in the turbulence by plotting the perturbation(df,dn) together with the reference field (f,n). This isdone in Figs. 8–11 forC 150.1. dn seems to give rise to themain contribution, especially in regions with strong gradientsin the density. It also seems that vortical structures do notcontribute significantly toudnu. We therefore conclude thatalthough the averaged values ofl1 andlp are rather similar,the spatial distribution of the sources may be different.
VII. CONCLUSION
From the presented results, the largest Lyapunov expo-nent,l1 , seems to be correlated to other characteristic in-verse time scales of the turbulence and this indicates thatl1 may be a valuable diagnostic tool. However, the actualphysical regions of the turbulence active in the generation ofchaos seem to be quite different from those active in therelative dispersion process. Thus, there are fundamental dif-
FIG. 8. Contour plot off for the same situation as in Fig. 3. Contourspacing 4.0.
FIG. 9. Contour plot ofdf in the saturated state for the same situation as inFig. 3. Contour spacing 6.0.
FIG. 10. Contour plot ofn for the same situation as in Fig. 3. Contourspacing 4.0.
FIG. 11. Contour plot ofdn in the saturated state for the same situation asin Fig. 3. Contour spacing 6.0.
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ferences between the exponential separation of nearby par-ticles and exponential separation of two almost identical tur-bulent states.
The Weiss decomposition has been derived in the con-text of relative dispersion. This rather simple derivationshows together with numerical results its direct applicabilityto the relative dispersion processes. The results for theLagrangian autocorrelation function show that the micros-cale is significantly smaller in elliptic regions than in hyper-bolic regions. Since vortex cores are strongly elliptic regions,we conclude that particles inside vortices have a shorter mi-croscale than particles outside vortices due to the fact that thedirection of velocity changes rapidly in the quasi-circularmotion inside vortices. We find that the Weiss fieldQ is auseful tool for identification of vortex structures.
ACKNOWLEDGMENTS
Discussions with J. S. Hesthaven, J. P. Lynov and A. H.Nielsen on various aspects of the numerical methods andwith H. L. Pecseli on the aspects of particle dispersion aregratefully acknowledged. One of the authors~T.S.P! is in-debted to P. E. Bak for introducing the chaos theory.
Computer resources were supported by the Danish Re-search Council.
APPENDIX A: LYAPUNOV EXPONENTS
The HW system is a dynamical system described bymode coupling equations which may be written in the form
dr
dt5F~r !,
where
r5~r 1 ,r 2 ,r 3 , . . . ,r i , . . . !,
F5~F1 ,F2 ,F3 , . . . ,Fi , . . . !.
Eachr specifies a state of the system and it may be repre-sented as a point in a suitably chosen vector space called thephase space. Solution curves in phase space are calledorbits.The dimension of the phase space is in principle infinitesince the HW system is a system of coupled partial differen-tial equations~PDE’s!, and the discrete numbering of thecoordinatesi should be replaced by a continuous variable tosignify that the infinite dimension is non-countable. How-ever, in the numerical solution, periodic boundary conditionsare applied and the Fourier series is truncated at some maxi-mum mode numberM in the x-direction and some maximummode numberN in the y-direction. Therefore, the system hasa finite, but very high dimension, of the order 104–105, de-pending on the spatial resolution.
In dissipative systems such as the HW system, the orbitsoften contract to a subset of the phase space ast→` and thissubset usually has a lower dimension than the phase spaceitself. Such a subset is called anattractor. As previouslymentioned, the numerical simulations strongly indicate thatthere exists an attractor for the~truncated! HW system. In
chaos analysis attractors are often characterized by their di-mension~which may be defined in a number of ways! and bythe Lyapunov exponents.
In the phase space, one may define a Lyapunov exponentfor each positionr0 on the attractor and an infinitesimal dis-placementd0 as
l~r0 ,d0!5 limt→`
1
tlnS ud~ t !u
ud0uD .
We may choose the infinitesimal displacement inD mutuallyorthogonal directions (D denotes the phase space dimen-sion!. In general, the set ofD Lyapunov exponents found bychoosing different orthogonal directions ofd0 will be inde-pendent of the choice ofr0 . Therefore, ther0 dependence ofl may be omitted. We may write the set of Lyapunov expo-nents asl1>l2> . . .>lD ~assuming a finite dimensionDof the phase space!. For a finite dimensional phase space thenorm u* u may be any norm defined in this phase space. Sincethere is a Lyapunov exponent for each dimension in phasespace it follows that for a dynamical system of PDE’s thereis an infinity of Lyapunov exponents. In the truncated, dou-bly periodic model, there are 2MN Lyapunov exponents.However, the largest Lyapunov exponentl1 contains the in-formation about the unpredictability and chaoticity of thesystem. Therefore we will concentrate onl1 only.
The largest Lyapunov exponentl1 is defined as
l15 limt→`
1
t2t0lnS udr ~ t !u
udr ~ t0!uD ,
where dr (t) measures a perturbation of the flow with theinitial perturbationdr (t0) chosen randomly. It will eventu-ally point in the direction that is most unstable since it willhave a component in the eigenvector direction correspondingto the largest eigenvalue of the Jacobian. Ifl1 is positivethen this means that the perturbation, on average, grows asudr (t)u;el1t. In other words, two almost identical flows willdiverge exponentially in phase space as time passes and thisessentially makes long term prediction of the flow impos-sible. The system is then said to bechaoticand the conditionl1.0 is a commonly accepted definition of deterministicchaos, provided that the state of the system is confined to anattractor fort→`.
APPENDIX B: NUMERICAL IMPLEMENTATION OFPARTICLE TRACKING
The accuracy of the particle tracking is a major concernsince the relative particle dispersion in its nature is very sen-sitive to numerical errors. The results of accuracy tests pre-viously published have been somewhat contradictory. In Ref.19 it is stated that the spatial approximation is the dominat-ing source of numerical error whereas it is concluded in Ref.20 that the time stepping error is the most important factorexcept for high order time stepping schemes. Therefore, ahigh accuracy scheme has been chosen for both the spatialand the temporal approximation. The drawback is that CPUtime considerations limit the number of particles to1022103. Choosing a less accurate but faster spatial interpo-lation scheme such as interpolating splines would allowmore than ten times more particles. This would yield better
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statistics, but it might result in erroneous results, especiallyfor the relative dispersion, since the numerical errors couldbe systematic in the sense that they do not average out.
1. Spatial approximation
The spatial interpolation is purely spectral, i.e., the fullFourier series expansion of thex-componentu52]f/]y ofthe velocity field is used to get the value ofu at the particleposition
u~x,y!5 (m50
M
(n50
N
um,nei ~2pnx/Lx12pmy/Ly!.
A similar expression holds for they-component of thevelocity fieldv5]f/]x. The use of this method means thatthe interpolation error is negligible and the error in the cal-culation of the particle velocity is completely dominated bythe truncation error and the error in the time stepping of theHW equations. The drawback of this method is that it re-quires a lot of floating point operations. The algorithm com-plexity is O(npMN) and sine and cosine function calls arevery time consuming. However, since the sine and cosinevalues needed for the summation are independent in the twodirections then one may calculate and store these first anduse the stored values in the summation. Then only(2npM12npN) trigonometric function calls are necessaryand the code runs 10–20 times faster depending on the ar-chitecture of the computer. The spectral interpolation codehas been checked by comparing the results of a test case withresults from another spectral interpolation code.
2. Temporal approximation
Several different schemes have been investigated. Theschemes have been chosen such that they only require thecalculation of the velocity field once or twice per time step.In addition it should be noted that for time stepping virtualparticles in a doubly periodic domain there is no stabilityproblem, only accuracy is of concern.
The test case used was a ‘‘frozen’’ Lamb dipole21 ~i.e.,no time evolution of the dipole!. A higher order scheme wasused in combination with a very small time step in order toobtain a particle trajectory~forming a closed loop inside avortex! with an accuracy close to machine accuracy. Thendifferent schemes were initialized with values from this tra-jectory in order to make the initialization error negligible.Different values of the time step were tried. The results canbe seen in Fig. 12. Note that the Courant, Friedrichs, Levy~CFL! number in this case is not of direct relevance to theproblem. It was found that a fourth order Adams-Bashforth~AB4! predictor step in combination with a fourth orderAdams-Moulton~AM4! corrector step was superior to theother schemes which were AB3/AM3, AB4, SS3~third orderstiffly stable scheme, with the implicit part set to zero!. Inagreement with Ref. 22, the partially corrected scheme waschosen since it yields the same accuracy as the fully cor-rected scheme and only requires one calculation of the ve-locity per time step. The approximate radius of the closedtrajectory of the particle is 2•1022 so even for the largesttime step~which corresponds to one full revolution of the
closed particle trajectory in 20 time steps! the error is about1% of the radius. The effects of machine accuracy can beseen for the 4th order schemes for the lowest time step. TheAB4/AM4 partially corrected scheme is
r i115r i1Dt
24~55F i259F i21137F i2229F i23!,
F i115vE~ r i11!
~predictor step!,
r i115r i1Dt
24~9F i11119F i25F i211F i22!
~corrector step!.
3. Initialization of particles
The temporal initialization of the AB4/AM4 scheme isperformed by using lower order integration schemes:
t50→t51:Euler step;t51→t52:2nd order AB/AM predictor-corrector step;t52→t53:3rd order AB/AM predictor-corrector step.The accuracy of the initialization is a concern. One
should be aware that the particle trajectories are less accuratefor the first few time steps due to the use of low orderschemes.
The particles are initialized randomly in space with uni-form probability density in the box@0;Lx@3@0;Ly@ using apseudo-random number generator. For each particlep theposition of the neighbour particlep is initialized by adding arandomly oriented displacement to the position of particlep. The displacement has to be small so that exponential di-vergence may be seen. On the other hand, if the displacementis chosen on the order of the grid size or smaller, then thedifference in velocity between a particle and its neighbourwill be due to the very highest modes in the system. Thesemodes are heavily affected by the hyperviscosity and by thetruncation error. An initial displacement somewhat smallerthan the smallest grid size is chosen and the actual growth
FIG. 12. Comparison of the accuracy of different time stepping schemes.AB4/AM4 partially corrected (h), AB4/AM4 fully corrected (s), AB4(n), AB3/AM3 partially corrected~1! and SS3 (3).
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rate calculated when the displacement has grown to approxi-mately the grid size. In this way also the temporal initializa-tion error is suppressed.
4. Convergence check
The convergence of particle ensemble averages ischecked by comparing the statistics of all the particles tostatistics obtained by averaging over only half of the par-ticles. Convergence of the time integration is only checked ina few cases since it is CPU time consuming. The time step islowered to 1/2 of the original time step in the test runs. Sincethe particle time stepping is fourth order accurate, and theflow time stepping is third order accurate, it follows thatparticle statistics for the flow is third order convergent intime. Therefore the test runs should be expected to be eighttimes more accurate, and convergence can be verified.
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