vm e v C e (F e1 ) = en e E d r r r d 3 vm e v δf e δẋ ˆr F S e d 3 vˆb δϕδf e, (1) where F S

Transcription

1 Turbulent Current Drive Mechanisms Christopher J. McDevitt, 1 Xianzhu Tang, 1 and Zehua Guo 1 Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM (Dated: July 11, 217) Mechanisms through which plasma microturbulence can drive a mean electron plasma current are derived. The efficiency through which these turbulent contributions can drive deviations from neoclassical predictions of the electron current profile is computed by employing a linearized Coulomb collision operator. It is found that a non-diffusive contribution to the electron momentum flux as well as an anomalous electron-ion momentum exchange term provide the most efficient means through which turbulence can modify the mean electron current for the cases considered. Such turbulent contributions appear as an effective EMF within Ohm s law, and hence provide an ideal means for driving deviations from neoclassical predictions. I. INTRODUCTION Non-inductive current drive mechanisms are essential to the realization of steady state tokamak operation and also play an important role in determining the plasma s magnetohydrodynamic (MHD) stability. The neoclassical bootstrap current is of particular interest, in part because it provides a particularly economic means of driving a substantial fraction of the plasma current, but also due to its strong impact on the MHD stability of the edge pedestal. While extensive studies have been carried out investigating how fluctuations, such as self-generated microturbulence or externally imposed resonant magnetic perturbations, can shape density and temperature profiles, and thus indirectly modify the plasma s bootstrap current, few studies have investigated the direct impact of these fluctuations on the plasma current itself. In this work we discuss the efficiency of different mechanisms through which microturbulence can modify the plasma current. The first two mechanisms that we consider arise in direct analogy with mechanisms previously discussed in the context of ion intrinsic rotation 1 3. In particular, as discussed below, an electron mean plasma current can be driven either by: (1) turbulent acceleration, (2) or an electron residual stress 4 7. The former mechanism relies on the turbulence mediated exchange of momentum between ions and electrons. Such an exchange of momentum can lead to the acceleration of electrons, and hence appears as an effective source term in Ohm s law. This mechanism corresponds to the electron analogue of the ion acceleration mechanism discussed in Ref. 8, whereby the main ions exchange energy with impurity ions. The electron residual stress, in contrast, corresponds to a term in the electron momentum flux that is independent of the parallel electron flow (electron current) and its gradient, and hence also has an appropriate form for driving an electron current. Alternatively, an additional means through which turbulence can drive a mean plasma current results from the establishment of an equilibrium between trapped and passing electrons due to resonant scattering by drift wave microturbulence 9. This mechanism is analogous to the familiar neoclassical bootstrap current except that it relies on waveparticle interactions to detrap electrons rather than Coulomb collisions. In order to determine the relative importance of these mechanisms, a mean field formulation is developed that incorporates the above turbulent mechanisms as well as a linear Coulomb collision operator. Specifically, phase space scattering by drift wave microturbulence, turbulence induced momentum exchange, the electron residual stress, and Coulomb collisions are incorporated into a mean field kinetic equation for the electron distribution function. Such a formulation allows for turbulent mechanisms to be treated on an equal footing with neoclassical effects, such that no assumption of "additivity" of turbulent and neoclassical mechanisms is made. The remainder of this paper is organized as follows. In Sec. II, a brief overview of the three turbulent current drive mechanisms addressed in this work is given. Section III describes the model of electrostatic microturbulence utilized as well as means through which the parallel symmetry of the fluctuations can be broken. Utilizing this model of microturbulence, the strength of the different turbulent current drive mechanisms is estimated, and a self-consistent calculation of the electron current including Coulomb collisions is carried out in Sec. IV. Finally, a brief summary of the main results is given in Sec. V. II. CURRENT DRIVE MECHANISMS In this section we will be interested in describing different mechanisms through which turbulence can drive plasma current. The first two mechanisms that we wish to consider do not require toroidal geometry, and hence can be understood by considering the electron parallel momentum equation in a cylinder, e.g. d 3 vm e v C e (F e1 ) = en e E + 1 r r r d 3 vm e v δf e δẋ ˆr e d 3 vˆb δϕδf e, (1) where is a flux surface average and we have assumed the fluctuations can be treated in the electrostatic limit. Aside from the collisional momentum exchange term (left hand side) and a mean parallel electric field, two contributions from turbulence can be identified [second and third terms on the right

2 hand side of Eq. (1)]. The first turbulent contribution can be readily identified as an electron momentum flux, whereas the second turbulent contribution cannot be expressed in a conservative form, and thus appears as an effective source term. Due to quasi-neutrality, this latter contribution can be verified to vanish when summed with the ion momentum equation, and can thus be recognized as a means of electron-ion momentum exchange. The specific mechanism through which these two turbulent contributions can drive deviations from neoclassical predictions will be described in detail in Secs. II A and II B below. The third mechanism we are interested in considering results from the establishment of an equilibrium between trapped and passing electrons due to resonant scattering by drift wave microturbulence 9. This mechanism, which requires toroidal geometry, is closely analogous to the familiar neoclassical bootstrap mechanism except that it relies on waveparticle interactions to detrap electrons rather than Coulomb collisions. This mechanism thus requires an accurate description of turbulent and collisional phase space fluxes, and is somewhat more involved. A detailed description of this mechanism is outlined in Sec. II C. A. Electron Momentum Flux The turbulent transport of electron parallel momentum in the electrostatic limit can be written as Π e m e R d 3 vv δf e δv EB ˆr pass = m e R d 3 vv δg e δv EB ˆr pass. (2) where ( ) is a temporal average. Here, we have decomposed the perturbed electron distribution function into its adiabatic and non-adiabatic components δf e = (e/t e ) Fe M δϕ + δg e, where Fe M is taken to be a centered Maxwellian, and noted that the adiabatic component vanishes after integration by parts. As noted in Refs. 5,6, the electron momentum flux can generally be expressed as Π e = χ ϕ u e r + V u e + π e, (3) where u e is the mean parallel electron flow. The first term in Eq. (3) can be recognized as an electron viscosity, the second term is a pinch of electron momentum, whereas the last term will be referred to as an electron residual stress. Since the electron residual stress is independent of the electron flow and its gradient, it has an ideal form for driving an electron parallel flow, and hence a mean electron current. We will thus be interested in considering this latter contribution in more detail below. A brief motivation of the physics underlying the electron residual stress can be obtained by considering the linearized response to an electrostatic perturbation, e.g. ( e ) δg e,m,n, = ( ) F M v /qr (m nq) + i e where e ct e eb 1 L n [1 + η e ( v 2 2v 2 T e 3 2 )] m r. eδϕ m,n, T e. Here, η e L n /L T e, L 1 n d ln n/dr and L 1 T e d ln T e /dr, v T e T e /m e, and m and n are the poloidal and toroidal mode numbers, respectively. Since we are at present only interested in computing the electron residual stress, we will take Fe M to be a centered Maxwellian. Substituting Eq. (4) into the electron momentum flux, yields π e = im e ρ s B m,n, m r eδϕ m,n, T e 2 (4) d 3 vv ( e ) ( v /qr ) (m nq) + i F M e. (5) Taking the limit, then allows ( ) π e = 2π 2 m qr eδϕ m,n, m e ρ s B r m nq k m,n, T e [ ] dµb ( e ) Fe M v, (6) =/k where k = (m nq (x)) /qr. Note that Eq. (6) has odd parity about the rational surface (e.g. odd in k, which changes sign as one traverses a rational surface). Thus, if one were to perform a spatial average across the mode rational surface, this term would yield zero net momentum transport if δϕ m,n, 2 is symmetric with respect to the surface r m,n, where r m,n is the radial location of a rational surface defined by q (r m,n ) = m/n. However, for the case where δϕ m,n, 2 is not even in r m,n a finite contribution to the electron momentum flux, and hence a finite parallel electron flow can be anticipated. While this local picture requires significant modification when considering extensions to global geometry, the simple k symmetry breaking paradigm discussed here will provide a useful means for identifying scenarios in which the electron residual stress is capable of driving a significant electron current. Indeed, nonlinear global simulations carried out in Ref. 1, show that the ion residual stress correlates closely with the spectral averaged k. It s also useful to note that while the electron residual stress has a factor of 1/k 2 that diverges as one approaches the rational surface, the magnitude of the residual stress is regularized by the exponential convergence of the Maxwellian, e.g. ( ) Fe M v exp 2 /k 2 =/k v2 T e as k. A particular feature of the electron residual stress, distinct from the ion residual stress, is that it can be shown to be tightly localized about the rational surface. This property can be understood by considering the form of the passing electron resonance in cylindrical geometry, e.g. k v =. (7) 2 2

3 Assuming v v T e and e, and noting that in a sheared magnetic geometry k = k θ x/l s, where L s = qr /ŝ, and ŝ = rd ln q/dr, allows Eq. (7) to be approximated by e k y x L s v T e = x L s L n me m i ρ s. (8) Here x describes the width of the electron Landau layer. Thus, for a normal magnetihear regime where m i /m e > L s /L n > 1, the passing electron resonance, and hence the resulting electron residual stress (in the limit ) will also be localized to a layer larger than the electron gyroradius, but smaller than an ion gyroradius. As discussed in more detail in Sec. IV A, the presence of such a localized electron momentum flux will tend to drive fine scale corrugations in the electron current about rational surfaces, with low order rational surfaces likely to see particularly strong responses. Note that while in toroidal geometry a magnetic drift term will also be present, such a term is generally subdominant, and hence will only quantitatively modify the above estimate. B. Turbulent Electron-Ion Momentum Exchange Aside from the electron momentum flux, an additional means through which turbulence can modify the electron current, corresponds to the exchange of momentum with ions via the last term in Eq. (1). It will thus be useful to sketch a brief derivation of the mechanism through which microturbulence can mediate the exchange of momentum between electrons and ions. Following an analogous calculation as in the previous subsection, this contribution can be approximated by: M e d 3 vδf eˆb δϕ = e d 3 vδg eˆb δϕ v 2 2 T e = m e (m nq) eδϕ m,n, qr T m,n, e d 3 i ( e ) v ( ) F M v /qr (m nq) + i e, (9) where we have used Eq. (4) to approximate the electron perturbation and again noted that the adiabatic contribution vanishes. Taking the limit, and carrying out the parallel velocity integration yields M = 2π 2 m e v 2 T e m,n, dµb [ ( e ) F M e (m nq) m nq eδϕ m,n, T e ] v =/k. (1) Analogous to the residual stress, Eq. (1) has odd parity about the rational surface, and will thus not give a net contribution if δϕ m,n, 2 is an even function of r m,n. Thus, similar to the case of the electron residual stress above, symmetry breaking will be required in order to drive a transfer of momentum between electrons and ions. 2 C. Turbulence Driven Bootstrap Current While the two mechanisms discussed above can be qualitatively described via a simple model in cylindrical geometry, a more subtle means through which turbulence can modify the mean plasma current can be understood by considering a kinetic formulation in toroidal geometry. It will be convenient to begin by considering the gyrokinetic Vlasov equation 11 F e t + Ẋ F e + v F e v = C e (F e ), (11) where the characteristic equations can be expressed as B Ẋ = v B v = 1 B m e B c eb ˆb (me µ B e J ϕ), (12a) (m e µ B e J ϕ), (12b) and B B v (m e c/e) ˆb. It will be useful to express the distribution function in terms of a mean and a fluctuating piece, e.g. F e (Z, t) = F e (Z) + δf e (Z, t). (13) Here, the gyrokinetic coordinates are given by Z ( X, v, µ ), and the mean component of the distribution function is assumed to be independent of time and the toroidal angle variable. Making this decomposition, and expanding the mean distribution function in the ratio of the Larmor radius over an equilibrium scale length, e.g. F e = F e + F e1 +, allows (see Ref. 12 for details) [ ] I (ψ) F e B F e1 v = B [ ( )] C turb e (δf e ) + C e Fe. ce ψ v (14) Here we have defined the turbulent collision operator as Ce turb (δf e ) = v B ( Γ x + with the phase space fluxes given by ) v 2 Γ v 2, (15) µ Γ x Ẋ B v δf e, Γ v 2 2 e m e ( v ˆb + vde ) δϕ B v δf e. where v de [ ) (16) µb (ˆb ln B + v 2 ˆb ] / cs, and we have noted the relation 13 ( I (ψ) v de F e = v ˆb v ce ) F e. ψ We have also introduced the coordinate transformation ( X, v, µ ) ( X, v 2, µ ) such that v = σ v 2 2µB with σ = v / v. This expression can be further reduced by assuming the collisional and fluctuation induced scattering rates 3

4 are small compared to the bounce/transit motion of an electron, allowing an auxiliary expansion F e1 = F () (1) e1 +F e1 +. At lowest order Eq. (14) can be written as [ ] B F () e1 v I (ψ) F e =, (17) ce ψ which implies F () e1 = v I (ψ) F e ce ψ + H e, (18) where H e = H e ( ψ, v 2, µ, σ ) is a flux surface function. At next order, Eq. (14) is given by B F (1) e1 = B v [ C turb e (δf e ) + C e ( F () e1 )]. (19) After flux surface averaging, the solubility constraint can be shown to be given by B [ ( )] Ce turb (δf e ) + C e F () e1 =. (2) v In the absence of turbulence, Eq. (2) provides the familiar collisional solubility constraint used to compute the neoclassical bootstrap current in the banana regime 13,14. Physically such a constraint implies that Coulomb collisions provide a means of establishing an equilibrium between trapped and passing electrons, and hence drive a passing electron current. In contrast, if one considers the collisionless limit, it is apparent that the turbulent collision operator will provide a means of establishing such an equilibrium between trapped and passing electrons, and thus provide an alternate means of driving a passing electron current. In order to provide a more explicit motivation of the turbulent bootstrap mechanism it will be useful to approximate the turbulent collision operator via quasilinear theory 12,15, yielding B Ce turb (δf e ) = ( v Z D QL F ) e1, (21) Z where D QL is a diffusion tensor arising from wave-particle interactions, whose detailed form will be discussed in more detail in Sec. IV D, and we have dropped the superscript on the mean distribution function for convenience. Substituting Eq. (21) into the solubility constraint [Eq. (2)] in the collisionless limit, yields an equation for the mean distribution function, e.g. ( = Z D QL F ) e1. (22) Z Thus, the velocity space component of the quasilinear diffusivity will provide a means of diffusing electrons across the trapped-passing boundary and hence establishing a smooth equilibrium between trapped and passing electrons. An example case is shown in Fig. 1. Here the top figure shows the electron current whereas the bottom figure displays the electron <j e B>/eB n e v /v the v /v the Figure 1. (top) The solid red curve indicates the electron current computed in the collisionless limit whereas the dashed black curve is the neoclassical bootstrap current shown for comparison. The vertical blue lines indicate the location of rational surfaces. (bottom) Velocity space distribution of the non-maxwellian portion of the electron distribution at.55. The parameters for this example case were chosen to be a/l max n = a/l max T e = a/l max T i = 1, n /a = T i /a = T e /a =.1, q =, q 1 = 2.5, the toroidal mode number was taken to be n = 15, ρ ρ s /a = 1/25, and a/r =.32. velocity space distribution at.55. From the velocity space distribution, a smooth transition between the trapped and passing regime is evident, suggesting that electron detrapping by microturbulence provides a means through which a collisionless bootstrap current can by established. As can be observed from the top plot in Fig. 1, the resulting collisionless current is comparable to, but not equal to, the neoclassical bootstrap current. In this case the neoclassical bootstrap current is computed utilizing a linearized Fokker-Planck collision operator 16 whose implementation and comparison with analytic formulas of the bootstrap current is described in Ref. 12. The calculation of the collisionless bootstrap mechanism discussed here is more comprehensive than that reported in Ref. 12. Specifically, while within the present calculation a finite parallel wavenumber was retained, Ref. 12 considered the limit of strongly localized eigenmodes such that k. 4

5 Such an asymptotic limit allows for the turbulent bootstrap mechanism to be formally decoupled from the electron residual stress and electron-ion momentum exchange terms (both of which require k ), such that the turbulent bootstrap mechanism could be analyzed independently. In contrast, for the plasma current shown in Fig. 1, we have artificially turned off the electron residual stress and electron-ion momentum exchange terms, by setting contributions to the turbulent collision operator proportional to the Maxwellian distribution function to zero [i.e. the terms given by Eqs. (51e) and (51f) below] in order to isolate the turbulent bootstrap mechanism. The retention of a finite parallel wavenumber requires assuming an explicit form for the drift wave eigenmode. While the results discussed in Sec. IV below utilize eigenmodes computed from a numerical eigenmode solver (see Sec. III), here for convenience we have used a simple slab like model of the drift eigenmodes whose form is given by: δϕ n, = m Φ m,n e imθ exp [ ( xm,n w ) 2 ], (23) where Φ m,n is the amplitude of the fluctuations (taken to be constant), x m,n r r m,n is the distance to a rational surface, and w is the radial width of the eigenmode which is taken to be equal to the distance between rational surfaces, e.g. w = 1/ (nq ). The retention of a finite parallel wavenumber modifies the form of the quasilinear diffusion coefficients in two important ways. First, the inclusion of a finite k allows for the full electron resonance condition to be retained, which can be written schematically as De v k =, (24) where De is the electron magnetic drift. The inclusion of the passing electron Landau resonance [third term in Eq. (24)] can be shown to lead to a strongly spatially varying resonance condition. Specifically, for a sheared slab geometry with k = k θ x m,n /L s, with 1/L s = ŝ/ (qr ), ŝ = rq /q, and normalizing the resonance condition by e = k θ ρ s ( /L n ), yields e De e ( mi Ln m e L s ) ( ) ( ) v xm,n =, (25) v T e where for thermal electrons > De. Hence, for a physical mass ratio of m i /m e = 4, only low energy electrons will be able to match the resonance condition away from a rational surface, whereas very close to a rational surface the resonance condition will be dominated by suprathermal electrons. The length scale delineating these two regimes is approximately given by x defined in Eq. (8) above. An additional difference in comparison to the quasilinear transport coefficient derived in Ref. 12 is that the first term in the velocity flux given by Eq. (16) was also neglected in Ref. 12, due to the assumption k. The inclusion of a more accurate resonance condition as well as the complete velocity space flux can be seen to lead to a somewhat more complex current profile, where the detailed structure of the current is sensitive to the locations of rational surfaces as indicated by Fig. 1. ρ s III. MODEL OF ELECTROSTATIC MICROTURBULENCE In order to obtain a more accurate description of the efficiency of the different electron current drive mechanisms it will be useful to consider a more comprehensive model of the drift wave eigenmode. In the long wavelength limit, an eigenvalue equation for electrostatic drift waves and ion temperature gradient (ITG) modes is given by 17 : where ) = ρ 2 s 2 φ n, c2 2 s (ˆb φn, 2 2τ Di φ n, ( ) ( ) e τ δne,n, + φ n,, (26a) pi pi n i iv T i ρ i (ˆb ψ ) ln δϕ n, ln n ψ, (26b) Di iv T i ρ i (ˆb ln B ) ln δϕ n,, (26c) Here τ = T e /T i, pi = i (1 + η i ), η i L n /L T i, i = e /τ and φ n, e δϕ n, /T i. When deriving Eq. (26a) we have assumed the ordering ( ti, bi, Di ), where ti and bi are the ion transit and bounce frequencies respectively, as well as ρ 2 i k2 1. Here, the first term in Eq. (26a) is a finite Larmor radius term, the second term represents coupling to an ion sound wave, and the third term introduces toroidal coupling between different poloidal harmonics. The fourth and fifth terms in Eq. (26a) correspond to the lowest order portion of the drift wave dispersion relation, where for adiabatic electrons (δn e,n, φ n, /τ), balancing these two terms yields e. In order to retain the trapped electron mode, it will be necessary to incorporate a model for the non-adiabatic electron response. This is accomplished by utilizing a trapped gyro- Landau-fluid model 18 to approximate the electron density perturbation. A simple two moment model is given by: 19 = δnt e,n, n e f (ε, θ) φ n, τ Dif (ε, θ) φ n, 3 2 τ δp t e,n, Di, (27a) p e = δpt e,n, e (1 + η e ) f (ε, θ) φ n, + 5 p e τ 2 Dif (ε, θ) φ n, 5 2 τ δp t ( e,n, δp t ) e,n, Di + 2τ Di ζ D δnt e,n,, p e p e n (27b) where quantities marked with the superscript "t" indicate trapped particle quantities, f (ε, θ) is the trapped electron population at a given poloidal angle, and ζ D =.7 + i.8σ t with σ t Di / Di. The passing electron distribution will be approximated as adiabatic, such that the total electron perturbation can be written as δn e,n, = [1 f (ε, θ)] n φ n, /τ + δn t e,n,. (28) 5

6 Taken together with Eq. (26a), Eqs. (27a)-(28) provide a closed system for computing the drift wave/itg eigenmode structure. As noted in Refs. 2 22, equilibrium E B shear provides a robust means of driving k symmetry breaking, and hence it will be necessary to incorporate this physics into the above eigenmode solver. This can be accomplished via introducing the Doppler shifted frequency such that EB, where EB iv () EB ln δϕ n,, (29) where v () EB is the equilibrium E B flow, which is generally a spatially varying quantity. When solving the above system it will be useful to express the magnetic field as B = α ψ, (3) where ψ is the poloidal flux and the coordinate α can be written in terms of the physical toroidal angle ϕ (not to be confused with the electrostatic potential) as α = ϕ θ dθˆq, ˆq ˆb ϕ ˆb θ. (31) Here θ corresponds to the poloidal angle at fixed ϕ. In the non-orthogonal coordinates (ψ, θ, α) it is easy to see Z δ φ m x R R x ˆb α =, ˆb ψ =. (32) In this coordinate system, the various gradients and frequencies can be approximated by 2 2 r 2 + 2inq ŝθ r r n2 q 2 ( 1 + ŝ 2 r 2 θ 2), (33a) 1 ˆb qr θ, i v T i ρ i nq r EB c B nq r 1 L n, ϕ () r, (33b) (33c) (33d) Di v T i R nq r ρ i (cos θ + ŝθ sin θ) i v T i R ρ i sin θ r. (33e) This system can be solved by using Eqs. (27a), (27b) and (28) to express Eq. (26a) as a polynomial eigenvalue problem. An example solution to this system is shown in Fig. 2. Here we have assumed density and temperature profiles of the form: A(r) = A exp [ A L max tanh A ( r r L max A )], (34) where A represents a profile quantity, i.e. (n e, T e, T i ), L max A is the maximum gradient, A characterizes the width of the Figure 2. (top) Real part of trapped electron eigenmode structure in the poloidal plane. (bottom) Magnitude of poloidal harmonics of the trapped electron eigenmode. The parameters for this example case were chosen to be a/l max n = a/l max T e = 2, a/l max T i = 1/5, n/a = T i/a = T e/a =.2, q =, q 1 = 2.5, the toroidal mode number was taken to be n = 25, ρ ρ s/a = 1/25, and a/r =.32. gradient region, and r is the radial midpoint of the simulation domain. The radial profile of the safety factor is assumed to be q = q + q 1 r, and for simplicity, the radial electric field is approximated by the diamagnetic term in the radial force balance equation, e.g. E r 1 Zen i p i r. (35) The parameters for this example case were chosen to be a/l max n = a/l max T e = 2, a/l max T i = 1/5, n /a = T i /a = T e /a =.2, q =, q 1 = 2.5, the toroidal mode number was taken to be n = 25, ρ ρ s /a = 1/25, and a/r =.32 where R is the major radius. Due to the fairly flat ion temperature gradient, the most unstable mode for these parameters can be identified as a trapped electron mode. Here a solution that is strongly ballooning on the weak field side is clearly evident. More insight into the solution can be gained by extracting the poloidal harmonics of the 2D eigenmodes. Such a plot is shown in Fig. 2, where each poloidal harmonic 6

7 is approximately localized about its respective rational surface, with small offsets evident for modes at the radial extremes of the fluctuation envelope. These small shifts result both from the presence of E B shear, as well as the variation of the mean profiles. As discussed in Sec. IV, such asymmetries about the rational surface provide a useful proxy for identifying broken symmetry in the background fluctuation spectrum. π e /n e m e c 2 s.5 1 x 1 5 IV. RESULTS.5 Within this section we will describe how the above turbulent current drive mechanisms can act to drive deviations from the neoclassical bootstrap current. We will begin by providing a more detailed description of the electron residual stress as well as the turbulent electron-ion momentum exchange terms using solutions from the eigenmode solver discussed in Sec. III. Subsequently, an order of magnitude estimate of the strength of various turbulent mechanisms is given with respect to the collisional resistivity. Finally, the explicit form of the quasilinear transport coefficients used to estimate the turbulent collision operator are given, and the plasma current is computed including both turbulence as well as Coulomb collisions. A. Structure of the Electron Residual Stress It will be useful to first consider the radial structure of the electron residual stress. Here the electron residual stress is computed by numerically inverting the linearized drift kinetic equation ( de + i v ) qr θ + i δg e,n, = e T e F M e ( e ) δϕ n,, (36) and substituting the result into the electron momentum flux given by Π e = m e R d 3 vv δg e δv EB ψ, (37) pass where the form of the electrostatic potential is taken from the numerical solution of the eigenmode equation defined in Sec. III, and is a broadening term. A plot of the radial profile of the electron residual stress is shown by the red curve in Fig. 3. The parameters used are the same as for the example eigenmode calculation shown in Fig. 2, except with five n-mode numbers given by n = (21, 23, 25, 27, 29), all of which are assumed to have the same amplitude. In agreement with the simple analytic estimate given by Eq. (6) the electron residual stress rapidly changes sign as a rational surface is traversed, where from the analytic expression of the residual stress derived above, the spatial length scale associated with these variations can be estimated by x m e /m i (L s /L n ) ρ s, Figure 3. The red curve plots the spatial structure of the residual stress, whereas the black curve corresponds to the window average of the electron residual stress, with the width of the window is taken to be the distance between rational surfaces. where L s = qr /ŝ. If one further notes that the electron residual stress appears inside a spatial divergence, such a rapid variation of the electron residual stress about rational surfaces will result in a large, localized EMF. Such an EMF is plotted in Fig. 4, where large spikes in the effective EMF centered around rational surfaces are evident. One thus anticipates that the electron residual stress will provide a robust means of driving corrugations in the plasma current on a length scale comparable to x. Such fine scale corrugations were originally predicted by Ref. 23 in the context of an electromagnetic formulation and numerically verified in Ref. 4. An important subtlety, is that the strength of the fine scale EMF structures will depend sensitively on the number of energy carrying modes present, and thus the turbulent spectrum of the microturbulence. Specifically, for a fixed turbulent energy, as the number of modes involved is increased, the amount of energy carried by each mode will be reduced. Thus, the magnitude of each local EMF corrugation will generally decrease, and the number of such structures will increase. This is true everywhere except near low order rational surfaces, where m/n harmonics tend to pile up with a subsequent rarefaction of rational surfaces in the adjacent regions. As a result, local corrugations in the EMF profile will be most prominent at such surfaces, as also evidenced in Ref. 23. This physics is illustrated in Fig. 5, where nine n-modes have been included. In comparison to the five mode case shown in the top panel of Fig. 4, the magnitude of the local EMF structures have been significantly reduced, with a proportional increase in the number of such structures. The exception to this trend is at the m/n = 4/3 surface (.533), where the magnitude of the EMF is comparable to the five mode case. Note that while the m/n = 3/2 surface is located at =.6, no turbulent EMF is present at this surface, since in this example case we have chosen to consider odd n modes. While the rapid spatial variation of the electron residual stress can be anticipated to drive localized corrugations about rational surfaces, since the sign of the residual stress flips as 7

8 a.π e /n e m e c 2 s.2 a.π e /n e m e c 2 s a.π e /n e m e c 2 s x Figure 4. (top) The divergence of the electron residual stress term. (bottom) Window average of divergence of electron residual stress term. one traverses a rational surface, the net impact of the residual stress on the current profile can be expected to approximately vanish after averaging across a rational surface. In particular, as indicated by the simple analytic estimate given by Eq. (6), a form of symmetry breaking in the turbulence spectrum will be required to drive a net deviation from neoclassical predictions of the current profile. As noted in Sec. III, the presence of E B shear, as well as profile shearing provide a robust means of symmetry breaking and thus driving a net current 1,2 22, This can be more clearly illustrated by considering a spatial window average of the electron residual stress term as indicated by the black curve in Fig. 3. Here, the spatially averaged residual stress can be observed to be predominantly negative for <.55 and predominantly positive for >.55. Such a non-vanishing average is also evident when considering the divergence of the residual stress (see the bottom plot of Fig. 4), which will result in a finite average effective EMF in Ohm s law. Thus, in addition to driving corrugations in the current profile near rational surfaces, an average deviation in the current profile can also be anticipated Figure 5. Same as the top plot in Fig. 4, but with nine n-modes, i.e. n = (17, 19, 21, 23, 25, 27, 29, 31, 33). The m/n = 4/3 surface is located at.533. B. Structure of the Electron-Ion Momentum Exchange Term A more detailed estimate of the form of the electron-ion momentum exchange term can be derived by solving Eq. (36) and substituting the result into the expression M = e d 3 vδg eˆb δϕ. (38) A plot of the radial form of the electron-ion momentum exchange term is given in Fig. 6. Here, while the magnitude of the electron-ion momentum exchange term is much smaller than the divergence of the electron residual stress (as also observed in Ref. 23 ), the window averaged value of this term is smaller, but still roughly comparable to the window averaged electron residual stress for these parameters. Thus, this turbulent contribution will likely be able to drive a net deviation of the electron current comparable to that driven by the electron residual stress discussed above. C. Order of Magnitude Estimate of Turbulent Impact on Plasma Current Before carrying out an explicit calculation of the electron current, it will be useful to consider an order of magnitude estimate of the strength of the turbulent contributions in comparison to the collisional term. Considering the electron momentum equation (neglecting the inductive electric field for simplicity) ( ) ne m e u e + Π e + M =.51 n em e u e, (39) t τ e where the second and third term on the left hand side correspond to the momentum flux and electron-ion momentum exchange terms due to turbulence, respectively, whereas the right hand side describes collisional friction. For simplicity 8

9 am e /n e m e c 2 s x Figure 6. The red curve plots the spatial structure of the electron-ion momentum exchange term. The black curve is computed by taking a window average of the red curve, with the width of the window is taken to be the distance between rational surfaces. we have assumed cylindrical geometry, and used Braginskii s expression for the collision friction between electrons and a single ion species, where we have assumed the ions to be stationary. While the inclusion of toroidal geometry would enhance the collisional drag by a factor related to the trapped particle fraction in the banana regime, the simple form used in Eq. (39) will provide an order of magnitude comparison between the turbulent and collisional terms. A more complete calculation will be carried out below using a linearized Fokker-Planck collision operator in toroidal geometry in Sec. IV E. Normalizing the collisional drag by n e m e c 2 s/a allows the right hand side of Eq. (39) to be written as am Brag n e m e c 2 s ( a =.51 τ e ) ( u e ). (4) The electron collision frequency and the dimensionless collisionality νe are related by ( ) ( ) a mi a ε 3/2 νe. (41) τ e m e qr Furthermore, assuming the mean electron flow to be comparable to the bootstrap current, allows the second term in Eq. (4) to be estimated u e ubs e f t B B θ ρ s L n. (42) Considering the parameters m i /m e = 4, νe = 1 2, a/l n = 2, q = 2, ρ ρ s /a = 1/25, =.5, and a/r =.32, allows the above two ratios to be estimated as ( ) a.5, τ e u e.5, (43) which suggests the collisional friction is of the order am Brag n e m e c 2 s 1 4. (44) Comparing this estimate with the effective EMF resulting from the residual stress shown in the bottom plot of Fig. 4, suggests that for this low collisionality regime, turbulence is capable of driving significant deviations from the neoclassical bootstrap current. D. Calculation of Quasilinear Transport Coefficients The mean plasma current will be computed by solving the solubility constraint given by Eq. (2). As discussed in Sec. II C, the turbulent collision operator will be estimated by quasilinear theory. This can be accomplished by expressing the electron perturbation by R δg = L (Max) F e il (x) F e1 il (v2 ) F e1 v 2, (45) µ where we have defined the quantities [ ) ] R = + i (v ˆb + vde + i, (46a) L (Max) L (x) = ( e ) eδϕ T e, (46b) = c B (ˆb δϕ ), (46c) L (v2 ) = 2 e m e ( v ˆb + vde ) δϕ. (46d) Here F e is the lowest order mean distribution function, taken to be a Maxwellian, and F e1 corresponds to the non- Maxwellian component of the mean distribution function, whose form is given by Eq. (18), and is Doppler shifted by the E B drift given by Eq. (29). Retention of terms related to F e1 is necessary in order to incorporate the turbulent bootstrap mechanism discussed above. The phase space quasilinear transport coefficients can then be computed by numerically inverting Eq. (45), and substituting the result into the spatial and velocity space fluxes given by Eq. (16). Explicitly, after numerical inversion, Eq. (45) can be written as [ ] δg = R 1 L (Max) F e il (x) F e1 il (v2 ) F e1 v 2. µ (47) Substituting this expression into the spatial flux given by Eq. (16), yields B Γ x ψ = L (x) ψr 1 L (Max) F e v B i i L (x) v ψr 1 L (x) B L (x) ψr 1 L (v2 ) v F e1 µ F e1 v 2. (48) 9

10 Noting that the mean distribution function can be decomposed as I (ψ) F e F e1 = v ce ψ + H e = f e + H e, allows Eq. (48) to be written in the form Γ x ψ = D xx H e ψ D xv H e v 2 + Sx M µ and similarly for the velocity flux H e Γ v 2 = D vv H e v 2 D vx µ ψ + SM v + S NM x, + S NM v, (49) (5) where the mean field transport coefficients are given by: [ ] B D xx R i L (x) ψr 1 L (x) ψ, (51a) v D xv = D vx R S NM x S M x [ i [ B D vv R i L (v v R i S NM v S M v R [ i ] B L (x) ψr 1 L (v2 ), (51b) v B L (x) v 2 ) B L (x) ψr 1 L (v2 ) v [ B R i L (v v B i L (v v [ B L (x) v 2 ) [ B R L (v v R 1 L (v2 ) ] ψr 1 L (x) f e 2 ) R 1 R 1 L (v2 ) f e v 2 µ L (x) f e f e v 2 µ ], (51c) ]. (51d). (51e) ] ψr 1 L (Max) F e, (51f) 2 ) R 1 ] L (Max) F e. (51g) Here, the lowest order contribution to the electron residual stress is given by Eq. (51f), whereas the electron-ion momentum exchange term is contained in Eq. (51g). The remaining terms describe real space and velocity space diffusion of the non-maxwellian distribution F e1, as well as their cross terms. These latter terms provide a means of establishing a collisionless equilibrium between trapped and passing electrons, and hence drive the turbulent bootstrap current discussed in Sec. II C. From the solubility constraint given by Eq. (2), an explicit expression for the mean electron distribution can be written as 1 V ψ V Γ x ψ v 2 Γ v 2 µ B ( ) + C e Fe1 =, (52) v where the phase space fluxes are defined above, and the linearized Coulomb collision operator employed is described in detail in Ref. 12. The boundary condition on the unknown portion of the distribution function H e (ψ) can be derived by first noting that the odd portion of H e (ψ) vanishes in the trapped region due to symmetry arguments 14. The even portion of H e (ψ) has a mathematically analogous form as the Maxwellian distribution (e.g. it is a flux surface function that is even in v ), hence it will provide a small correction to the electron residual stress and the electron-ion momentum exchange terms [Eqs. (51f) and (51g)]. For simplicity we will neglect this correction in the present work and leave its possible role in turbulent current drive to future work. Thus, we will only solve for the odd portion of H e (ψ), where we will enforce He odd (ψ) = at the trapped-passing boundary. When solving for the mean electron distribution function it will ( be convenient to perform the coordinate transformation X, v 2, µ ) (X, v, κ), where κ is a normalized pitch-angle defined by κ [1 λ (1 ε)] / (2ελ) and λ 2µB /v 2. The normalized variable κ has a range given by (, ), where the trapped-passing boundary is located at κ = 1. The details of how the fluxes given by Eqs. (49) and (5) transform into the new coordinate system are given in Ref. 12. In addition, we have assumed the magnetic geometry to be described by unshifted circular flux surfaces. E. Calculation of Electron Current In this section the mean electron current in the presence of both turbulence and Coulomb collisions is computed by solving Eq. (52). An example case is shown in Fig. 7. Here the red curve corresponds to the electron current computed including all turbulent terms for ν = 1 2 and a fluctuation level of eδϕ/t e 1 2. The remaining parameters are given by a/l max n = a/l max T e = 2, a/l max T i = 1/5, n /a = T i /a = T e /a =.1, q = 2.5r, ρ ρ s /a = 1/25, and a/r =.32. Five toroidal mode numbers were included whose values are given by n = (21, 23, 25, 27, 29). Due to the relatively flat ion temperature gradient, the most unstable mode in this case is given by a trapped electron mode. Strong corrugations of the current profile are evident around rational surfaces, which as discussed above, are driven primarily by the electron residual stress. Note that the turbulent spatial diffusive flux of parallel momentum has played an important 1

11 <j e B>/eB n e <j e B>/eB n e Figure 7. Plot of the electron current computed with and without turbulent contributions. The solid red curve is the unaveraged electron current including turbulent terms, the solid black curve is the window averaged value of this current, and the dashed black curve is the reference neoclassical bootstrap current. role in broadening and reducing the amplitude of such corrugations. Had it been neglected, the amplitude of these corrugations would be significantly larger than that shown in Fig. 7. In addition to these large localized deviations, the spatial window averaged electron current deviates modestly from the neoclassical bootstrap current. Such an average deviation from the neoclassical current results primarily from the residual stress as well as the electron-ion momentum exchange term. A plot of the averaged electron current as a function of fluctuation intensity is shown in Fig. 8. Here, while for a moderate level of fluctuations ( eδϕ/t e ), only a modest deviation from neoclassical predictions is evident. However, for the case of a larger fluctuation amplitude ( eδϕ/t e ), a much larger deviation from neoclassical predictions is evident. It will also be useful to consider a case with a steeper ion temperature gradient such that the ITG mode is destabilized. Specifically, the ion temperature gradient is taken to be a/l T i = 2 whereas the density gradient is given by a/l n = 1. The remaining parameters are the same as those used in Fig. 7. Such a case is shown in Fig. 9. Here, while the magnitude of the deviation from the nominal neoclassical curve is similar to the previous case, the fractional deviation is significantly larger. This is due to the electron neoclassical bootstrap current being significantly smaller, both as a result of the weaker density gradient and the stronger ion temperature gradient 14. An additional feature which is evident for the case of ITG is that three extrema are present, in contrast to the two extrema present in the case of drift waves. Such a pattern is similar to the effective EMF resulting from the divergence of the electron residual stress (see Fig. 4), suggesting that the electron residual stress provides the dominant turbulent current drive mechanism in this case. A plot of the electron current for different fluctuation amplitudes is given in Fig. 1. Here, the deviation of the electron current from the Figure 8. Comparison of the averaged electron current for different fluctuation amplitudes. The dashed black curve is the nominal neoclassical electron current, the solid green curve is for eδϕ/t e 5 1 3, the solid black curve is for eδϕ/t e 1 2, and the solid red curve is for eδϕ/t e The remaining parameters are the same as those used in Fig. 7 <j e B>/eB n e Figure 9. Same as Fig. 7, except with a steeper ion temperature gradient and shallower density gradient such that an ITG mode is the most unstable mode. neoclassical current can be observed to increase rapidly as the fluctuation amplitude is increased. V. CONCLUSIONS Several mechanisms through which microturbulence can directly impact the electron current have been derived. The specific mechanisms treated correspond to an electron residual stress, anomalous electron-ion momentum exchange, as well as a turbulent analogue of the bootstrap current. The efficiency through which these turbulent mechanisms can modify the electron current is determined via utilizing a mean field formulation that incorporates the above mechanisms as well 11

12 <j e B>/eB n e Figure 1. Comparison of the averaged electron current for different fluctuation amplitudes. The dashed black curve is the nominal neoclassical electron current, the solid green curve is for eδϕ/t e 5 1 3, the solid black curve is for eδϕ/t e 1 2, and the solid red curve is for eδϕ/t e The remaining parameters are the same as those used in Fig. 9. as a linearized Coulomb collision operator. This formulation allows for the turbulent mechanisms to be treated on an equal footing with neoclassical effects, such that no assumption of "additivity" of turbulent and neoclassical mechanisms is made. It is found that even very modest levels of microturbulence are capable of driving localized deviations from neoclassical predictions in the vicinity of rational surfaces. Such fine scale corrugations, in the electron current are shown to result from the rapid variation of the electron residual stress about a rational surface. As discussed in Sec. IV A above, due to the large number of modes resonant at low order rational surfaces, current corrugations will be robustly driven at these surfaces. The impact on MHD mode stability is thus likely to be most prominent for modes localized about such low order rational surfaces. Aside from the generation of fine scale corrugations, for larger amplitude fluctuations, or lower collisionalities, it is shown that broken symmetries in the turbulence spectrum can drive significant average deviations from neoclassical predictions. This work has identified both E B shear as well as profile shearing as effective means of symmetry breaking. While the magnitude of the electron-ion momentum exchange term is found to be small compared to the contribution to Ohm s law resulting from the residual stress, the averaged value of this contribution is of comparable magnitude. 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