Flow Shear Suppression of Pedestal Turbulence--A First Principles Theoretical Framework
D. R. Hatch, R. D. Hazeltine, M. K. Kotschenreuther, S. M. Mahajan
aa r X i v : . [ phy s i c s . p l a s m - ph ] J un Flow Shear Suppression of Pedestal Turbulence—A FirstPrinciples Theoretical Framework
D.R. Hatch, R.D. Hazeltine, M.K. Kotschenreuther, and S.M. Mahajan Institute for Fusion Studies, University of Texas at Austin, Austin, Texas, 78712
Abstract
A combined analytic and computational gyrokinetic approach is developed to address the ques-tion of the scaling of pedestal turbulent transport with arbitrary levels of E × B shear. Due tostrong gradients and shaping in the pedestal, the instabilities of interest are not curvature-drivenlike the core instabilities. By extensive numerical (gyrokinetic) simulations, it is demonstratedthat pedestal modes respond to shear suppression very much like the predictions of a basic ana-lytic decorrelation theory. The quantitative agreement between the two provides us with a newdependable, first principles (physics based) theoretical framework to predict the efficacy of shearsuppression in burning plasmas that lie in a low-shear regime not accessed by present experiments. PACS numbers: ntroduction.– The interplay between shear flow and turbulence is a central componentin the self-organization of wide-ranging fluid and plasma systems. In neutral fluids, forexample, shear flow is a common driver of turbulence (i.e., Kelvin-Helmholtz instabilities).In contrast, the primary source of turbulence in a typical fusion plasmas is the immensefree energy contained in extreme temperature and density gradients. How does the shearflow interact with this class of drift-generated turbulence? It is found that, as opposed toits role in hydrodynamic turbulence, the shear flow, in fact, suppresses drift turbulence [1],and is thought to be the main mechanism underlying the formation and sustenance of theedge transport barrier (called the pedestal) characteristic of the tokamak H-mode [2]. Sincethe residual turbulence mediates the structure of the pedestal and, consequently, largelydetermines plasma confinement, it (and its interplay with shear flow) is of central importanceto fusion energy.The earliest theoretical investigations of shear suppression [3–5], which we will call decor-relation theories, predicted reduced fluctuation amplitudes due to the combined advectionby background shear flow and self-consistent turbulent flow. Amongst these, the analytictheory of Zhang and Mahajan [5, 6] has compared rather favorably with experimental ob-servations of shear suppression [7, 8], albeit in experimental setups, perhaps, less challengingthan a fusion-relevant H-mode pedestal.It turns out, however, that the predictions of these basic theories are in striking agree-ment with gyrokinetic simulations (using the
Gene code [9, 10]) of the pedestal, providinga sound basis for a deep fundamental understanding of the reduction of turbulence by shearflow. We can, thus, with greater confidence, apply a combination of analytical and nu-merical approaches to study the scaling of turbulence with flow shear, even in the extremeenvironment of the H-mode pedestal. In effect, we will seek a first principles (physics-based)answer to the crucial question: how does turbulence in the pedestal react to a systematicreduction of flow shear rate? Since all proposed burning plasma machines will lie in the lowshear regime, a reliable answer to this question is crucial to the future of fusion energy viaTokamaks operating in H-modes.After the decorrelation theories of shear suppression (worked out in simple geometry) wereproposed, subsequent work emphasized the importance of toroidal effects [11–15]. Toroidaleffects are indeed prominent for the conventional instabilities in the plasma core . Driven bytoroidal curvature (i.e., via resonances with the magnetic drift frequencies), such fluctuations2eak in the low magnetic field region of the torus. Interestingly, however, the dynamics ofshear suppression manifests differently under conditions characteristic of the pedestal where,due to steep gradients and geometric shaping, the relevant modes are typically not curvature-driven, and consequently are insensitive to toroidal effects [16]. It is expected, then, thatthe influence of shear flow on pedestal turbulence may be very different from what could beextrapolated from the notions pertinent to the core plasma. In fact, we reach the surprisingconclusion that, despite the substantial complexity and computational challenges involved inpedestal turbulence simulations, the early decorelation theories of shear suppression becomehighly relevant.The importance of this work should be framed as follows. First, it provides a natural ex-tension of the theory of shear suppression to a pedestal context (perhaps its most importantapplication). Second, building on recent numerical work [16, 17], it establishes the theo-retical underpinnings necessary to understand and therefore to predict/estimate pedestaltransport over the transition to lower shear burning plasma regimes.
Decorrelation Theories—
The decorrelation theories of shear suppression begin with ageneric fluid equation of the form ∂ t ξ + ¯ v ( x ) ∂ y ξ + ˜ v ( x, y, t ) ∂ x ξ = q ( x, y, t ) , (1)where x is a radial coordinate, y is the corresponding binormal coordinate, ˜ v is the fluctuating E × B velocity, ¯ v ( x ) is the macroscopic steady state shear flow, q is a gradient-driven sourceterm, and ξ is a fluid quantity like density or temperature. Here we analyze the perpendiculartemperature fluctuations (i.e. ξ = ˜ T ⊥ —hereafter denoted by ˜ T ) so that Eq. 1 may be viewedas a simple analog to Eq. 11 from Ref. [18], which is derived from a moment expansion of thegyrokinetic equations. The decorrelation theories are based on the so-called clump theory described in Ref. [19], and apply basic turbulence closures to solve for properties of thetwo point correlation function (note that Ref. [6] derives similar results via an alternativeapproach to clump theory).For the purpose of this study, we have reproduced a calculation very similar to theoriginal ZM theory [5] (due to the close connection, we will refer to our model simply as theZM theory—details can be found in Appendix A). The calculation arrives at the followingrelation describing the reduction of turbulence by shear flow, P ( P −
13 )( P −
1) = 23 W P α , (2)3here P − represents the reduction in fluctuation amplitudes, P − ≡ ∆ x ∆ x h ˜ T ih ˜ T i , (3)and W is the normalized shear rate W = γ E × B τ c Θ . (4)In these expressions, 0 subscripts denote shear free quantities, γ E × B = dv/dx is the shearrate, the brackets represent ensemble averages (in practice, averages over space and time), τ c is the shear-free correlation time, ∆ x,y is the correlation length in the x, y direction(respectively), Θ = ∆ x / ∆ y accounts for anisotropy, and α is a near-unity scaling parameter,which will be described below.The ZM theory has two features that distinguish it from other decorrelation theories.Both are indispensable for the quantitative comparisons that will be described below. First,the theory is non-asymptotic in shear rate, describing shear suppression seamlessly acrossthe weak and strong shear limits. Second, and most importantly, it accounts for the fact thatfluctuation levels and nonlinear diffusivity are intimately connected and are both sensitivelydependent on shear rate. This is accomplished via an ad hoc expression relating the two: D = D ∗ h ˜ T i α , (5)where D is the nonlinear diffusivity, D ∗ is a constant proportionality factor, and α is the rele-vant scaling parameter related to the strength of the turbulence ( α ∼ . / . α , Eq. 2 predicts the relative suppression of turbulent fluctuation amplitudes.In essence, the theory captures the nonlinear decorrelation of turbulence when subject si-multaneously to a background shear flow and a self-consistent turbulent flow. By balancingthis decorrelation with a generic gradient drive, an expression is derived for the reductionof turbulence by shear flow. Notably, the theory neglects parallel dynamics (e.g. Landaudamping), zonal flows [20], toroidal effects, non-local (i.e., global) effects [21–24], details ofthe driving instability, coupling with damped eigenmodes [25–28], and non-monotonic flowprofile variation, all of which are included in our simulations. Thus, to the extent thatsimulation and theory agree, it can be concluded that the underlying mechanism of shearsuppression in the pedestal is described by a few relatively simple ingredients. Presently, wemake such comparisons. 4 .92 0.94 0.96 0.98 ρ tor T ( k e V ) T(keV) n ( m − ) n(10 m −3 ) 0.92 0.94 0.96 0.98 1.00 ρ tor | γ E × B ( c s / a ) | FIG. 1: Representative profiles of density and temperature (a) and E × B shear rate (absolutevalue) (b). The dashed vertical lines denote the simulation domain for gyrokinetic simulations. Comparisons between Theory and Simulation—
The gyrokinetic
Gene simulations de-scribed here are designed to include a wide range of relevant pedestal effects while alsofacilitating clear comparisons with theory. To this end, we employ an adiabatic electronapproximation in order to reduce computational demands and limit the dynamics to the iontemperature gradient (ITG) driven turbulence of interest. We note that this ITG turbulenceis not the dominant pedestal transport mechanism in most present day experiments preciselydue to its suppression by shear flow. The most important fluctuations are likely electrontemperature gradient turbulence [29–33], microtearing modes [17, 33], and low-n (toroidalmode number) magnetic fluctuations [34–38]—all of which are expected to be much lesssensitive to shear flow than ITG.Recent related work (Refs. [16, 17]) explores the implications of the expected ρ ∗ scalingof pedestal flow shear ( ρ ∗ is the ratio of the sound gyroradius to the minor radius ρ s /a ).The pedestal shear rate is effectively determined by the self-organization of the pedestalby means of force balance between the radial electric field and the pressure gradient. Thisforce balance, which is well-described by neoclassical theory and well-founded experimen-tally [39], results in shear rates that scale linearly with ρ ∗ : γ E × B av Ti ∝ ρ ∗ ( v T i is the ionthermal velocity). Refs. [16, 17] identify two classes of pedestal transport from gyrokineticsimulations—one that scales close to the expected gyroBohm ρ ∗ scaling , and a second (ITGturbulence) that is small throughout most of the experimentally accessible parameter spacebut has an unfavorable ρ ∗ scaling due to its sensitivity to shear flow. These studies predictthe latter mechanism—shear-sensitive ITG turbulence—to be relevant on JET (which has5ccess to the lowest values of ρ ∗ among active experiments) and to become increasingly im-portant in the transition to low- ρ ∗ regimes. To summarize, ITG turbulence in the pedestalis suppressed by the strong shear rates characteristic of present-day experiments. We areaddressing the question of the manner in which it re-emerges as shear rates decrease.In order to make comparisons with the ZM theory (Eq. 2), the time- and box-averagedsquared temperature fluctuation amplitude, h ˜ T i , and the radial and binormal correlationlengths (∆ x , and ∆ y ) are calculated from simulation data. The correlation lengths arecalculated for temperature fluctuations at the top of the torus where the fluctuation levelspeak. The heat flux, which is embedded in the corresponding temperature moment of thegyrokinetic nonlinearity, provides an appropriate proxy for the nonlinear diffusivity D . Thescaling factor α (recall Eq. 5) is extracted from a comparison of Q i and h ˜ T i and lies in therange α = [0 . , .
97] for the cases studied here. In order to connect the shear rate W (Eq. 4)with its corresponding quantity from the simulations, we use the inverse linear growth rateat k y ρ s = 0 . τ c , the standard definition of the E × B shear rate used in the Gene code [32], and an anisotropy factor Θ = ∆ x / ∆ y defined bythe correlation lengths. One free parameter is used to scale the shear rate and is selected tominimize the discrepancy between simulation and theory. Encouragingly, this free parameterremains of order unity in all cases studied (varying from 0 .
58 to 2 . ρ ∗ pedestal setup described in Ref. [16], which uses JET profile shapes [40]in conjunction with ITER geometry and projected ITER pedestal parameters. Profiles forthis base case are shown in Fig. 1 (a).The extensive simulation campaign described below entails scans of E × B shear rate forfour different scenarios, which are designed to isolate various effects and gauge variation inshear suppression dynamics. The first case, the local constant shear (LCS) case, is designedto match the assumptions of the ZM theory as closely as possible by employing a localapproximation (i.e., taking plasma parameters, gradients, and shear rate at a single radiallocation and neglecting effects from radial profile variation). A comparison between theoryand simulation is shown in Fig. 2 (a); it exhibits a remarkable quantitative prediction ofthe simulations by the theory. As demonstrated with the global constant shear (GCS) case,which includes self consistent global profile variation (but retains a radially constant shearrate), the theory is robust to the addition of global effects (see Fig. 2 (b)). The global fullshear (GFS) case additionally includes non-monotonic flow profiles whose shapes are set6y the standard neoclassical expression for the radial electric field (we define the pedestalshear rate to be the radially averaged quantity). This is particularly significant since itintroduces a region of zero shear in the simulation domain (see Fig. 1 (b)), raising thepossibility of non-trivial interactions between the turbulence and the flow profile. We notein this case anomalous behavior in the low shear limit, as shown in Fig. 2 (c), where adiscontinuity in P − between the low shear and the zero shear cases is observed. Extendedsimulations targeted at reducing statistical uncertainty produce very minor differences influctuation levels, but a persistent ( ∼ Scan of ρ ∗ — While valuable for the purpose of theoretical verification, the three casesexamined thus far may be characterized as idealized (and somewhat artificial) setups thatexploit the flexibility of our simulation capabilities to independently scan E × B shear rates.In an experimental context there is little external control over the shear rate. As describedabove, the shear rate is set by the self-organization of the pedestal by means of force balancebetween the radial electric field and the pressure gradient, resulting in direct proportionalitybetween ρ ∗ and pdestal shear rates. Consequently, the most experimentally relevant simula-tion scenario (called the global rho star [GRS] case) involves a fully self-consistent scan of ρ ∗ ,which holds the pedestal width (in magnetic flux coordinates) fixed along with all other di-mensionless parameters (i.e., safety factor q, ν ∗ , β ). In this scenario, the E × B shear profileis determined self-consistently from the density and temperature profiles using the standardneoclassical expression [41, 42]. This scenario involves an additional level of complexity sinceit conflates the effects of shear suppression with intrinsic ρ ∗ effects, which are well-known toindependently affect turbulence levels (i.e., produce deviations from gyroBohm scaling) as ρ ∗ is raised above a certain threshold [21–24]. We address this additional complexity with astraightforward modification to the ZM theory. We assume that finite ρ ∗ effects are limitedto two mechanisms—1) E × B flow shear, and 2) ρ ∗ effects manifest in the linear instability7rive. The latter enters the theory while balancing the decorrelation and gradient drive h ˜ T /T i τ c = γ lin ( ρ ∗ )( v T i /a ) DL , (6)where γ lin is the ρ ∗ -dependent linear growth rate, τ c is the decorrelation time, and L is amacrosopic gradient scale length. Note that the decorrelation theories (e.g., [4, 5]) use Eq. 6without the inclusion of the linear growth rate (see Appendix A for additional details). Withthis generalization, we find excellent agreement with simulation results, as shown in Fig. 2(d).Clearly, the agreement between simulation and theory is substantial in all four casesstudied. This agreement strongly supports a quantitative connection between the simula-tions and the ZM theory, particularly in light of the following observations: 1) There isgood agreement in all four cases studied, which represent substantial variation in physicalcomprehensiveness. 2) The agreement has been achieved using only a single free parame-ter, which remains order unity in all cases. 3) As an additional test, we make comparisonswhile neglecting various aspects of the theory. The discrepancy increases significantly whenthe anisotropy facter Θ and/or the the radial correlation lengths are left out of the theory.The results described here suggest that the scaling of pedestal shear suppression of ITGturbulence is determined by the basic ingredients in the decorrelation theory, namely theinterplay between turbulent and background advection—both balanced by a gradient drivemechanism. By extension, the complex physics included in the simulations (zonal flows,Landau damping, damped eigenmodes, global profiles effects, non-monotonic shear profiles,etc.)—which are important for determining absolute fluctuation levels—have little influenceon the scaling of turbulence with shear flow. High Shear Limit—
We now examine the scaling of turbulence in the high shear regime.The appropriate scaling can be readily derived by taking the P ≫ P − = (2 / / (2 α − W / (2 α − . (7)Fig. 3 shows a comparison between this high shear scaling and fits to the asymptotic simu-lation data points. Note that the asymptotic scaling is strongly dependent on the α factor.The self-consistent values of α produce a much better match than either the weak turbu-lence ( α = 1 translating to W − ) or strong turbulence ( α = 0 . W − )limits. This high shear scaling has implications for the ρ ∗ dependence of pedestal transport.8 W (d). c =0.62α =0.81GRS DataTheory P − c =2.06α =0.88(a). LCS DataTheory c =1.2α =0.97(b). GCS DataTheory W P − c =0.58α =0.86(c). GFS DataTheory
FIG. 2: Comparisons between Eq. 2 and simulations. The factor α (Eq. 5) and the free parameter c (used to scale the shear rate W ) are denoted for each case. W -1 -1.32-1.45 GRS
Eq. 2Eq. 6Fit -2 -1 P − -1.84-1.62 LCS
Eq. 2Eq. 6Fit -2 -1 -1.6 -1.92 GCS
Eq. 2Eq. 6Fit W -1 P − -1.38 -1.58 GFS
Eq. 2Eq. 6Fit
FIG. 3: Comparisons between Eq. 7 and simulations.
As shown in Fig. 3, turbulence reduction scales roughly as shear rate to the − / P − and W ) to more-intuitive quantities (gyroBohm-normalized heat flux Q/Q GB and ρ ∗ ) produces (empirically)the rough scaling Q/Q GB ∝ ρ − ∗ , which indicates a transport mechanism that is independentof ρ ∗ . This scaling is roughly observed for the ρ ∗ scan described above as well as for similar9cans described in Refs. [16, 17], suggesting its robustness. This fundamental theoreticalprediction must be considered when extrapolating to low ρ ∗ regimes. Discussion—
The present study demonstrates that the underlying mechanism of pedestalshear suppression involves a relatively small set of transparent physical ingredients, whichare insensitive to parameter variations and difficult to modify. We have, thus, the makingsof a first-principles, physics-based theory of turbulence suppression (via shear flow) thatcan be exploited to estimate/predict turbulent transport in regimes that have not beenexperimentally probed yet. In fact, ITER and all the future burning plasma experimentsfall in the low shear regime not accessed in most current experiments.We end the paper by pointing out that the simulations/theory suggest several possibleroutes for controlling turbulent transport in pedestals. Since the relevant suppression pa-rameter W is the shear rate normalized to the linear growth rate, the most promising routeto optimized pedestal performance in low ρ ∗ regimes is through the minimization of ITGgrowth rates. As demonstrated in Refs. [16, 17], this can be accomplished by at least twomechanisms—1) ion dilution via impurity seeding, and 2) reduction of η i (the ratio of thedensity and temperature gradient scale lengths) through the separatrix boundary condition.The latter route will rely heavily on optimization of divertor performance. APPENDIX A—THEORY SUMMARY
Here we succinctly outline a derivation of the Zhang-Mahajan (ZM) shear suppressiontheory [5, 6] used in this study. This derivation is intended to be more accessible thanthe rigorous but very involved calculation described in [6]. Our simplified derivation firstreproduces the orbit equations from Biglari-Diamond-Terry [4] using the standard clumptheory [19]. Thereafter, the distinctive elements of the ZM theory are applied, namely anon-asymptotic (in shear rate) treatment of the problem and the use of an ansatz relating thenonlinear diffusivity and fluctuation amplitude (described below). The original ZM theorybuilds on an alternative set of orbit equations, which stem from an independent approachthat does not rely on the standard clump theory. The model described below exhibits onlyminor qualitative differences from the original ZM theory.We use coordinates ( x, y ), where x represents the radial direction and y varies in somedirection (other than that of B ) on the magnetic surface. Parallel gradients, along with10uch geometrical details as magnetic curvature, are neglected.A perturbed fluid quantity, such as temperature or density, is denoted by ξ ( x, y, t ) andassumed to satisfy the equation ∂ t ξ + ¯ v ( x ) ∂ y ξ + ˜ v ( x, y, t ) ∂ x ξ = q ( x, y, t ) (8)Here ¯ v ( x ) is the shear flow, a slowly varying equilibrium flow in the y -direction, while˜ v ( x, y, t ) ≪ ¯ v is a turbulent flow. We assume that the turbulent flow is incompressible,and that the variation of ξ on flux surfaces is smaller than its radial variation, so the x -component of the turbulent velocity dominates. Rather than trying to solve (8), we derivefrom it an approximate equation for the correlation function C ≡ h ξ ( x , y , t ) ξ ( x , y , t ) i ≡ h ξ ξ i where the angle brackets denote a statistical average. Standard renormalization methods,based primarily on the random phase approximation, yield the evolution equation( ∂ t + ω s x − ∂ y − − ∂ x − ( k i x i − ) D∂ x − ) C = Q (9)where x − = x − x is the relative coordinate, Q is the source, ω s ≡ ¯ v ′ c is the shearing rate, D is a turbulent diffusion coefficient and k is a spectral-averaged wave number, related tothe width ∆ of an eddy: k x,y = 1 / ∆ x,y . The corresponding Green’s function G ( x − , t ; x ′− , t ′ )satisfies the homogeneous version of (9) with initial data G ( x − , t ; x ′− , t ) = δ ( x − − x ′− ). Weare content to study the moments M ij ( t ) ≡ Z d x G ( x , t ; x , x i x j where ( x , x ) = ( x − , y − ) and x is some initial value. Integration by parts yields thedynamical moment equations ∂ t M = 2 Dk ⊥ (cid:0) M + sin θM (cid:1) (10) ∂ t M = ω s M + 2 Dk ⊥ M (11) ∂ t M = 2 ω s M (12)Here k ⊥ ≡ k x and sin θ ≡ k y /k x . Denoting the characteristic time for change in M ij by τ c and introducing the nominal diffusion time τ D ≡ ( k ⊥ D ) − = ∆ D we obtain the characteristicequation z ( z − z −
6) = 4( ω s τ D ) sin θ (13)11here z ≡ τ D /τ c . Since the gradients relax through turbulent diffusion, the source forturbulence is measured by D/L . This observation leads to the estimate1 τ c * ˜ T T + = D/L (14)We concentrate on the fastest relaxation rate, z = 6, where the subscript refers to zeroshear. The effects of shear are displayed through the ratio zz = ∆ ∆ h ˜ T i h ˜ T i ≡ P (15)Following [5], we adopt the ansatz D = D ∗ h ˜ T i γ , where D ∗ is independent of the turbulencelevel and scale. Then (13) becomes P (cid:18) P − (cid:19) ( P −
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