The role of coherent vorticity in turbulent transport in resistive drift-wave turbulence
Wouter J. T. Bos, Shinpei Futatani, Sadruddin Benkadda, Marie Farge, Kai Schneider
aa r X i v : . [ phy s i c s . p l a s m - ph ] N ov The role of coherent vorticity in turbulent transport in resistivedrift-wave turbulence
W.J.T. Bos , , S. Futatani , , S. Benkadda , M. Farge and K. Schneider M2P2 – CNRS & CMI, Universit´e de Provence,39, rue Joliot-Curie 13453 Marseille Cedex 13, France LMFA - UMR 5509, - CNRS - Ecole Centrale de Lyon - Universit´eClaude Bernard Lyon 1 - INSA de Lyon, 69134, Ecully Cedex, France France-Japan Magnetic Fusion Laboratory LIA 336 CNRS / UMR 6633,CNRS-Universit´e de Provence. Case 321, 13397 Marseille Cedex 20, France Graduate School of Energy Science, Kyoto University, Japan LMD–CNRS, Ecole Normale Sup´erieure Paris,24 rue Lhomond, 75231 Paris cedex 05, France (Dated: November 7, 2018)
Abstract
The coherent vortex extraction method, a wavelet technique for extracting coherent vorticesout of turbulent flows, is applied to simulations of resistive drift-wave turbulence in magnetizedplasma (Hasegawa-Wakatani system). The aim is to retain only the essential degrees of freedom,responsible for the transport. It is shown that the radial density flux is carried by these coherentmodes. In the quasi-hydrodynamic regime, coherent vortices exhibit depletion of the polarization-drift nonlinearity and vorticity strongly dominates strain, in contrast to the quasi-adiabatic regime.
PACS numbers: 52.55.Fa, 52.35.Ra, 52.25.Fi . INTRODUCTION AND GOVERNING EQUATIONS One important issue in fusion research is the understanding and control of turbulent ra-dial flux of particles and heat in magnetized plasmas, in order to improve the confinementproperties of fusion devices . Indeed turbulence enhances the radial diffusion dramaticallycompared to neo-classical estimations. A long standing question has been : what is therole of coherent structures in this radial transport? The answer to this question requiresextracting and characterizing coherent structures. A particularly appropriate framework toidentify coherent structures is the wavelet representation, where wavelets are basis functionswell localized in both physical and Fourier space . It has already been used to identifycoherent structures in fluid turbulence and to distinguish them from background incoherentnoise . These methods have recently been applied to experimental signals of ion densityin the tokamak scrape-off layer , separating coherent bursts from incoherent noise. In thepresent work these methods are applied to assess the role of coherent vorticity structuresin anomalous radial transport in two-dimensional numerical simulations of drift-wave tur-bulence. Drift waves are now generally considered to play a key role in the dynamics andtransport properties of tokamak edge turbulence (e.g. [10] and references therein). At theedge, the plasma temperature is low and the collision rate relatively large, therefore theresistivity is potentially important. The Hasegawa-Wakatani model is a two-field modelwhich includes the main features of turbulent transport by resistive drift waves.In the present work the two-dimensional slab geometry-version of this model is chosen asa paradigm for drift-wave turbulence in the plasma-edge region. In dimensionless form theHasegawa-Wakatani model reads (cid:18) ∂∂t − D ∇ (cid:19) n + κ ∂φ∂y + c ( n − φ ) = [ n, φ ] , (1) (cid:18) ∂∂t − ν ∇ (cid:19) ∇ φ + c ( n − φ ) = (cid:2) ∇ φ, φ (cid:3) , (2)with n the plasma density fluctuation and φ the electrostatic potential fluctuation. D and ν are the cross-field diffusion of plasma density fluctuations and kinematic viscosity,respectively. The Poisson brackets are defined as[ a, b ] = ∂a∂x ∂b∂y − ∂a∂y ∂b∂x . (3)We identify the x -coordinate with the radial direction and the y -coordinate with the poloidaldirection. The equilibrium density n is non-uniform, with a density gradient dn /dx in the2egative x -direction, such that the equilibrium density scale L n = n / ( dn /dx ) is constantand the value of κ is one. The plasma density fluctuations n are normalized by n , therefore n/n → n , the electrostatic potential is normalized as eφ/T e → φ , the space as x/ρ s → x and the time as ω ci t → t , where e is the electron charge, T e the electron temperature, ω ci the ion cyclotron frequency and ρ s = ( m i T e ) / / ( eB ) is the ion integral lengthscale. B is thestrength of the equilibrium magnetic field in the z -direction and m i is the ion mass. The keyparameter in this model is the adiabaticity c , which represents the strength of the parallelelectron resistivity. It is defined as c = T e k k e n ηω ci , (4)with k k the effective parallel wavenumber and η the electron resistivity.The vorticity ω is related to the electrostatic potential φ by ∇ φ = ω. (5)Note that for c = 0, equation (1) corresponds to the advection-diffusion of a passive scalar inthe presence of a (unity) mean scalar gradient in the x -direction. Equation (2) corresponds inthis case to the vorticity equation. For c → ∞ the Hasegawa-Mima one field approximationis approached, which ignores all resistive effects. For c → c = 0 .
7, and a quasi-hydrodynamic case with c = 0 .
01. The case c = 0 . c ( n − φ ). The influenceof this term on the density field can however be considered to be negligible in the quasi-hydrodynamic case .The quantity of interest, the radial particle density flux, is the correlation between theradial velocity u r = − ∂φ/∂y and the particle density,Γ r = h nu r i , (6)where the brackets denote an average over both time and space. The question we address inthis paper is how coherent structures contribute to this flux. To investigate this, direct nu-merical simulations of the Hasegawa-Wakatani system are performed on a periodic domain3
14 0 14 0−12 12
FIG. 1: One realization of the vorticity field for the quasi-hydrodynamic case (left) and for thequasi-adiabatic case (right). The abscissa corresponds to the radial position. The ordinate indicatesthe poloidal position. Both range from 0 to 64 ρ s . The white frames indicate the dipoles we haveselected in both cases. discretized with N = 512 gridpoints. The length of the domain is 64 ρ s . A finite differencemethod is used in which the nonlinear terms are computed using a method developed byArakawa . The time stepping is performed using a predictor-corrector scheme. The plasmadensity diffusion D and viscosity ν are set to 0 .
01 in normalized units. Computations areperformed up to t = 612. At t ≈
100 the kinetic energy saturates and a statistically station-ary state is reached, independent of the (random) initial conditions. Typical realizationsof the vorticity field are shown in figure 1, where one observes coherent structures for bothcases. In each case we select a dipolar structure that we indicate by a white frame. Thequasi-hydrodynamic case exhibits coherent vortices of very different sizes and intensities, incontrast to the quasi-adiabatic case where the coherent structures are more similar in sizeand intensity.
II. COHERENT VORTEX EXTRACTION (CVE)A. Method
Definitions and details on the orthogonal wavelet transform and its extension to higherdimensions can be found, e.g., in [7,17]. In the following we fix the notation for the orthogonalwavelet decomposition of a two–dimensional scalar valued field. The wavelet transformunfolds the field into scales, positions and directions using a set of dilated, translated and4otated functions, called wavelets. Each wavelet is well-localized in space, oscillating ( i.e., it has at least a vanishing mean, or better its first m moments vanish), and smooth ( i.e., itsFourier transform exhibits fast decay for wavenumbers tending to infinity). We here applythe coherent vortex extraction (CVE) algorithm using orthogonal wavelets. In dimensiontwo, orthogonal wavelets span three directions (horizontal, vertical and diagonal), due tothe tensor product construction. To go from one scale to the next, wavelets are dilatedby a factor two and the translation step doubles accordingly. Wavelet coefficients are thusrepresented on a dyadic grid .We apply the CVE algorithm to the vorticity fields ω of both the quasi-hydrodynamic andthe quasi-adiabatic regime. The extraction is performed from the vorticity since enstrophyis an inviscid invariant in the hydrodynamic limit. Moreover, vorticity is Galilean invariantin contrast to velocity and streamfunction. We consider the quasi-stationary state of thesimulations, i.e., when a saturated regime is reached, and we decompose the vorticity field,given at resolution N = 2 J , into an orthogonal wavelet series ω ( x, y ) = X λ ∈ Λ e ω λ ψ λ ( x, y ) , (7)where the multi–index λ = ( j, i x , i y , d ) denotes the scale j the position i =( i x , i y ) and the three directions d = 1 , ,
3, corresponding to horizontal, verti-cal and diagonal wavelets respectively. The corresponding index set Λ is Λ = { λ = ( j, i x , i y , d ) , j = 0 , ..., J − i x , i y = 0 ... j − , d = 1 , , } . Due to orthogonality thewavelet coefficients are given by e ω λ = h ω, ψ λ i , where h· , ·i denotes the L -inner productdefined as h f, g i = R f ( x, y ) g ( x, y ) dxdy . The wavelet coefficients measure fluctuations of ω at scale 2 − j around the position i , in one of the three directions d . Here a Coifman 30wavelet is used, which is orthogonal and has 10 vanishing moments ( R x n ψ ( x ) dx = 0 for n = 0 , ... • Decomposition: compute the wavelet coefficients e ω λ using the fast wavelet transform . • Thresholding: apply the thresholding function ρ ε to the wavelet coefficients e ω λ , thusdiscarding the coefficients with absolute values smaller than the threshold ε . • Reconstruction: reconstruct the coherent vorticity field ω C from the thresholded5 ABLE I: Compression rate (% of coefficients retained), retained energy E = (cid:10) φ ∇ φ (cid:11) ,enstrophy Z = (cid:10) ω (cid:11) , and radial flux Γ r , after applying the CVE filter to the vorticity field ofthe quasi-hydrodynamic and quasi-adiabatic 2D drift-wave turbulence simulations.Compr. (%) E (%) Z (%) Γ r (%)Quasi-hydrodynamic ( c =0.01) 1.3 99.9 97 99Quasi-adiabatic ( c =0.7) 1.8 99.0 93 98 wavelet coefficients using the fast inverse wavelet transform.The incoherent vorticity field is obtained by simple subtraction, i.e., ω I = ω − ω C .The thresholding function is given by ρ ε ( a ) = n a if | a | >ε | a |≤ ε , (8)where ε denotes the threshold, ε = √ Z ln N , (9)where Z = h ω, ω i is the enstrophy (which corresponds to half of the variance of the vorticityfluctuations) and N the resolution. This threshold value allows for optimal denoising in aminmax sense, assuming the noise to be additive, Gaussian and white .In summary, this decomposition yields ω = ω C + ω I . Due to orthogonality we have h ω C , ω I i = 0 and hence it follows that enstrophy is conserved, i.e., Z = Z C + Z I . Let usmention that the computational cost of the Fast Wavelet Transform (FWT) is of O ( N ) . B. Compression rates
The results of the extraction are displayed in table I. The compression rate is in bothcases very significant: for the quasi-hydrodynamic case, 1 .
3% of the modes retain more than99 .
9% of the energy and 97% of the enstrophy. For the quasi-adiabatic case, 1 .
8% of themodes retain 99 .
0% of the energy and 93% of the enstrophy. The contribution of the coherentvorticity to the radial flux is also given in table I. The coherent modes, which contain mostof the energy and enstrophy, are responsible for 99% of the radial particle density flux Γ r inthe quasi-hydrodynamic case, and for 98% of Γ r in the quasi-adiabatic case. In other words,Γ r is almost exclusively carried by the coherent structures.6 -5 -4 -3 -2 -1 -25 -20 -15 -10 -5 0 5 10 15 20 P ( ω ) ω tot.coh.inc. 10 -5 -4 -3 -2 -1 -20 -15 -10 -5 0 5 10 15 20 P ( ω ) ω tot.coh.inc10 -7 -6 -5 -4 -3 -2 -1
1 10 100 Z ( k ) k k -2.5 k tot.coh.inc. 10 -6 -5 -4 -3 -2 -1
1 10 100 Z ( k ) k k -2 k tot.coh.inc. FIG. 2: Top: PDF of the vorticity. Bottom: Fourier spectrum of the enstrophy versus wavenumber.Left: quasi-hydrodynamic case. Right: quasi-adiabatic case. Dashed line: total field, solid line:coherent part, dotted line: incoherent part. Note that the coherent contribution (solid) superposesthe total field (dashed), which is thus hidden under the solid line in all four figures. The straightlines indicating power laws are plotted for reference.
C. Wavenumber spectra and probability density functions
Spectra and probability density functions (PDF), averaged over 512 realizations duringthe time interval 100 < t ≤ IG. 3: (Color online) Scatter-plot of vorticity against electrostatic potential for the coherent part(top) and incoherent part (bottom). Left quasi-hydrodynamic case, right quasi-adiabatic case. Thelight grey (red online) dots correspond to the total field, the dark grey (blue online) dots to thedipoles we have selected in Fig. 1.
97% of the total enstrophy Z , with Z = 1 .
4. For the quasi-adiabatic case, the coherent partretains 93% of the variance of the vorticity fluctuations and hence 93% Z , with Z = 3 .
4. Asimilar result is observed in the enstrophy spectrum computed from the Fourier transform ofthe vorticity field, averaged over wavenumber shells of radius | k | , the wavenumber. The totaland coherent enstrophy are the same all over the inertial range and at the highest wavenum-bers, in the dissipation range, the incoherent part contributes to the spectral enstrophydensity. Both coherent and incoherent contributions are spread all over the spectral range,but they present different spectral slopes in the inertial range and therefore different spatialcorrelations. From the integral wavenumber to the dissipation wavenumber, a negative slopefor the coherent contribution, corresponding to long range spatial correlations, is observed.8he incoherent part shows a positive slope with a power-law dependence close to k in theinertial range. This corresponds to an equipartition of kinetic energy in two dimensions. Asimilar result was obtained in three-dimensional isotropic Navier-Stokes turbulence . D. Scatter-plots
We show in figure 3 scatter-plots of the vorticity versus the electrostatic potential corre-sponding to the fields in figure 1. Both the total part and the incoherent part are shown.Since the coherent part is almost identical to the total part, it has been omitted. Also shown,superposed on the same figures, is the scatter-plot corresponding to the zoom on the dipo-lar structures indicated by a white frame in figure 1. In the freely decaying hydrodynamiccase, c = 0, Joyce and Montgomery showed that a functional relation φ ( ω ) = α sinh( βω )should be expected, corresponding to a final state of decay depleted from nonlinearity. Theparameters α and β are Lagrangian multipliers, necessary for maximizing the entropy underconstraints. The value 1 /β can be associated with a (negative) temperature . Depletionfrom nonlinearity corresponds to steady solutions of the Euler equation, [ ω, φ ] = 0, impliedby the existence of a functional relation φ ( ω ). Indeed drift-wave turbulence contains aninternal instability which prevents the flow from decaying. This forcing is present in bothcases considered here and a sinh-Poisson relation cannot be expected a priori for the globalflows. Moreover, the two-field model [equations (1) and (2)] contains two nonlinearities, firstthe polarization-drift nonlinearity in the vorticity equation, second the E × B nonlinearityin the density equation. The latter disappears in the adiabatic limit as n and φ are in phase,which corresponds to a linear functional relationship. In figure 3, a local depletion of thepolarization-drift nonlinearity is seen for the quasi-hydrodynamic case. The scatter-plot of φ − ω , corresponding to the dipolar structure, that is indicated by a white frame in figure1 (left), is close to a sinh-Poisson relation (solid black curve) in spite of the presence ofthe forcing term. In the quasi-adiabatic case the dipolar structure, that is indicated by awhite frame in figure 1 (right), does not exhibit such a functional relation. In the incoher-ent parts (Figure 3, bottom) no functional relation can be distinguished, which confirmsthat the incoherent part does not contain any structure, for both quasi-hydrodynamic andquasi-adiabatic cases. 9 . Strain versus vorticity A question is now how to quantitatively distinguish between the structures in both cases.Intuitively it can be inferred that different regions of high vorticity in the quasi-adiabaticcase involve strong mutual shearing which strongly limits their lifetime and the chance toreach a functional relation φ ( ω ). Koniges et al. determined the lifetime of individual eddiescompared to the eddy-turnover time τ over , i.e. the time it takes for a fluid element in an eddyto make a 2 π rotation. They estimated the lifetime of the quasi-hydrodynamic eddies to beapproximately 10 τ over , and the lifetime of the adiabatic eddies (for c = 2 .
0) approximately τ over . As mentioned in their paper, this measure is quite subjective and very time-consuming,especially if a full PDF of the lifetimes is to be obtained. Here we propose a simpler way todistinguish the coherent structures for the different regimes.In fluid turbulence the Weiss criterion Q is a local measure of the strain compared tothe vorticity for a 2D velocity field. The Weiss field is defined as: Q = 14 (cid:0) σ − ω (cid:1) , (10)with σ = (cid:18) ∂u∂x − ∂v∂y (cid:19) + (cid:18) ∂u∂y + ∂v∂x (cid:19) . (11) u and v are two orthogonal components of the velocity vector. The Weiss criterion wasproposed to identify coherent structures, but it may lead to ambiguous results becausethe underlying assumption that the velocity gradient varies slowly with respect to the vor-ticity gradient is not generally valid . We here apply the same criterion to drift-waveturbulence but not to identify coherent structures (this being done by the CVE method),but to distinguish between the quasi-hydrodynamic and quasi-adiabatic cases.The PDF of the Weiss field (Fig.4) reveals that it is its skewness that differentiates bestthe two fields. Indeed, it is more skewed towards negative Q for the quasi-hydrodynamiccase than for the quasi-adiabatic case: the skewness is −
11 for the former, compared to − Q is comparable for the two cases (5 and 4for the quasi-hydrodynamic case and the quasi-adiabatic case, respectively). The skewnessof the Weiss field Q appears to be a good quantitative measure to distinguish between the10 -6 -5 -4 -3 -2 -1 -120 -100 -80 -60 -40 -20 0 20 40 P ( Q ) QQuasi-hydrodynamicQuasi-adiabatic
FIG. 4: PDF of the Weiss field Q for the quasi-hydrodynamic and quasi-adiabatic velocity fields. two cases studied in the present work. In further studies it can be investigated, whether thismeasure can be used to identify coherence in different types of turbulent flows. III. CONCLUSION AND PERSPECTIVES
In conclusion, we have applied the
Coherent Vortex Extraction method to dissipativedrift-wave turbulence. The results show that we can identify the essential degrees of freedom(less than 2%) responsible for the nonlinear dynamics and transport. The coherent modescontain almost all the energy and enstrophy and contribute to more than 98% of the radialflux.Evaluating the scatter-plot of the vorticity versus the electrostatic potential, it is shownthat the coherent structures in the quasi-hydrodynamic case are close to a state of localdepletion of polarization-drift nonlinearity. In contrast, this is not the case for the quasi-adiabatic regime, where nonlinearity remains active and no sinh -functional relation betweenvorticity and electrostatic potential is observed. This depletion of nonlinearity in the quasi-hydrodynamic regime may explain the failure of the quasi-linear estimate of the radial flux .The skewness of the Weiss field yields a quantitative measure for the difference in nonlinearbehavior of the coherent structures between the quasi-hydrodynamic and quasi-adiabaticcases.The wavelet transforms, or the proper orthogonal decomposition (POD), may becomevery useful to denoise particle-in-cell simulations of plasma turbulence . A comparison ofthe performance of the POD and CVE method is currently undertaken and will be reported11n a future paper. Acknowledgments
Lionel Larchevˆeque is acknowledged for supplying and helping with a routine to computethe Weiss field. Wendel Horton is acknowledged for comments on the manuscript. Thiswork was supported by the
Agence Nationale de la Recherche under the contract
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