Contrib. Plasma Phys. 34 (1994) 2/3, 232-246

Scrape-Off Layer Turbulence Theory Ronald H. Cohen, Nathan Mattor and Xueqiao Xu Lawrence Livermore National Laboratory, Livermore, CA, USA.

Abstract A review is presented of recent developments in the theory of fluctuations

in tokamak scrape-off layers. A brief summary of typical experimentd features is presented, followed by derivation of a flute-mode dispersion relation including conducting-wall, curvature, axial E x B shear and radial compression drives. Non- linear results are reviewed, including arguments for and evidence of inverse cascading, penetration of SOL-generated turbulence into the closed-flw-surface edge plasma, and large-amplitude fluid excursions, comparable to observed SOL widths. Effects of divertor and limiter geometry, including x-point effects, are discussed. The po- tential importance of atomic physics drives for radiative and gas-target divertors is noted, as is the possibility that SOL turbulence originates in the core plasma. Conse- quences of SOL turbulence, including broadening of the SOL and dissipating parallel momentum (of potential importance for gas-target divertors) are outlinedl, and the possibility of actively exciting turbulence to achieve these objectives is discussed.

1. Introduction It is by now well-established that tokamak scrape-off layers (SOL’S) exhibit

large-amplitude fluctuations in potential, density, and, where measured, electron temperature. The study of these fluctuations is important, as they play a central role in determining the scrape-off layer width, and thus the magnitude of the peak heat flux on the divertor and the effectiveness of the SOL as a shield for neutral atoms. As will be noted below, SOL turbulence may also play a central role in enabling novel solutions to the divertor heat flux problem, such as gas-target and radiative divertors. There has been a number of theoretical studies in recent years [l-111, in part reviewed last year by Nedospasov [12]. In this review we shall survey the principal theoretical models for SOL fluctuations, with particular emphasis on developments published since or not emphasized in the presentation of Ref. 12.

The organization of this review is as follows. In Sec. 2 we briefly summarize trends in the experimental data. Sec. 3 surveys the major instability mechanisms. A set of nonlinear 2-dimensional fluid equations, which contains most of the mecha- nisms, is written down, and a unified dispersion relation (for flute modes) obtained. We also note instabilities which do not fit into this framework and indicate the role of non-flute effects. In Sec. 4, we summarize nonlinear results. In Sec. 5, we discuss consequences of SOL turbulence, including determination of the SOL width and its potential importance in enabling gas-target divertors. We also discuss the possibility of actively inducing SOL turbulpce, and summarize the paper.

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DIII-D

TEXT

2. Experimental Data

We briefly review here some important experimental results. In this discussion and in the following sections, we distinguish between the SOL and the edge; the latter term will refer to the outermost region of closed flux surfaces. As a setting for turbulence, the SOL differs markedly from both the core and the edge. Equilibrium densities and temperatures are of course lower than in the other two regions, and tend to have shorter radial scale lengths than in the core (on the order of 1 - 5 cm). The SOL has other obvious unique features, including field lines ending on solid materials, electrical potentials dominated by this contact (and hence, usually, positive radial electric fields E d , and a relatively high concentration of neutral atoms and impurities. A correlary to open field lines, relevant to linear mode structure and probably nonlinear effects as well, is the absence of rational surfaces (and flux surfaces in general). Finally, at least in diverted plasmas, the SOL is characterized by substantial variations of equilibrium quantities along magnetic field lines [13] .

Fluctuations which have been explicitly identified as occurring in the SOL have been reported on several tokamaks, including ASDEX [14-171, TEXT [18], TFTR [19], Phaedrus-T [20], Tokamak de Varennes [21] , Tore Supra [22], TEXTOR [23], DIII-D [24], and TJ-1 [25]. The measurements of Refs. 18-21 were taken by or in collaboration with the TEXT group, who report [18] a general tendency for e$/T > ;i/n > Pe/Te. Other groups have found e$/T x ii/n [14,25,24] and Pe/Te x ;i/n [25], while the ASDEX group reports [15] that, depending on the measurement technique, they can obtain Pe/Te greater or less than 6/n. On TEXT and Phaedrus- T [26], ii and Pe are found to be in phase in the SOL and out of phase in the edge, while the reverse is true on TJ-1 [25] . It seems to be generally the case [17, 24, 181 that the electrostatically driven particle flux (ij,+) can account for most if not all of the total flux inferred from equilibrium profiles.

L

0 9.

10 100 frequency (kHz)

Fig. 1. Typical power spectra of potential fluctuations for DIII-D, TEXT and ASDEX. (DIII-D and ASDEX spectra unpublished and provided by R. Moyer and H. Niedermeyer, respectively; TEXT data [27] was re-formatted by Tsui.)

The frequency spectra tend to be broad-band and monotonically decreasing, from around the lowest frequencies sampled (a few kHz) to well above 100 kHz. The L-mode spectra tend to exhibit a flattening below about 10 kHz and a steepening

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above 100 (see Fig. 1 for typical examples), but are otherwise devoid of significant structure. These features persist independent of SOL configuration (note the simi- larity of spectra from diverted DIII-D and limited TEXT in Fig. 1). Spectra for H mode have been reported on DIII-D, and tend to have more, though not necessarily reproducible, features, and also exhibit a faster fall-off above 100 kHz [24] .

The ASDEX team reports [15] a high degree of correlation (- 85%, with no phase lag) between fluctuations in the mid-region of an SOL field line and near the divertor floor, indicating very long parallel wavelengths and high coherence along field lines. The poloidal correlation length is on the order of 2-3 cm. In double null operation, the turbulence level is much higher on the outboard SOL than on the inboard one, suggesting a major role for curvature. The turbulence level on the outboard side is of comparable level for single and double-null operation.

Limited and diverted tokamaks differ substantially in the length of SOL field lines. They also tend to differ in the relative and absolute thickness of their scrape- off layers, with limited tokamaks being at least a factor of two wider (see, e.g., the summary in Ref. 18, and Refs. 13, 27, and 28). The combination of these effects implies that the particle and heat fluxes required to account for the observed widths in limited plasmas is substantially larger in the limited machines than the diverted machines.

In light of the current interest in gas-target divertors, it is of interest to know whether high neutral densities can cause enhanced turbulent transport. This ques- tion has been explored recently in the ATF torsatron [29] and the PISCES linear reflex-arc experiment [30]. In ATF, an increase in fluctuation level W M I observed as the temperature drops to below 10 eV (where ionization becomes negligible and the neutral density at the position of observation increases). This is suggestive of a turbulence process that is sensitive to the neutral density (but probably not the ionization rate.) On PISCES, a transport level several times that attributable to ion-neutral collisions and increasing with neutral density has been reported. The same experiment also shows, with increasing neutral density, a tendency to shift from high plasma density all the way to the end wall to a state where the plasma density drops to a small value at the ends, leading to the question of whether neutral- related transport (turbulent or otherwise) could trigger a bifurcation in the divertor equilibrium. Note that turbulent transport related to neutral density was observed on similar devices many years ago [3i].

A natural question to ask theoretically is whether there is any causal connection between turbulence in the SOL, edge and core regions. The experimental evidence is not conclusive. Spectra in the SOL and edge seem to be significantly different [18] , but this could be attributable to the significantly different E x B rotation velocities. Beam emission spectroscopy on TFTR [32] suggests that there is essentially no corre- lation between edge and core fluctuations, but perhaps a causal connection between edge and SOL fluctuations, This study found a radius, about 5 cm inside the last closed flux surface, across which fluctuations have near zero correlation. Between two points not spanning this radius, there is large correlation.

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In this section we review the major instability mechanisms which have been considered for SOL plasmas. We begin by deriving a dispersion relation for flute modes which includes several of the mechanisms. While the fluctuations are not expected to be strictly flute, particularly in diverted configurations (where the field line length is especially long), the result illustratively displays the interplay of the various drives, and the nonlinear fluid equations with which we begin will be useful in the discussion of nonlinear effects in the next section. We will make note of non-flute and other neglected effects as is appropriate. The drive mechanisms to be included are curvature [1,2], rotation of the plasma relative to conducting endwalls [4-51, axial [6] and radial [7] variation of the E x B rotation rate, and radial variation of the field line length (more precisely, a tilt of the end plates relative to field lines as observed in a poloidal cross section) [S] , Curvature is a familiar source of instability; the issue for the SOL is the interplay between curvature drive and incomplete line tying through the Debye sheath, as discussed long ago by Kunkel and Guillory [33]. The conducting-wall drive is less familiar, although the idea is also old [34]. One way of viewing it is that the rotation of the plasma results in a negative sheath resistance, as will be discussed below. Axial shear was also noted to be an instability drive in Ref. 34. Radial variation of the field line length, coupled with partial reflection at the (tilted) sheath, results in a parallel compression in response to a fluctuating diamagnetic drift, possibly amplifying the existing pressure perturbation.

For drift-frequency-range modes in the SOL, we expect vte/L > ( w - W E ~ B ) N

w. >> v,i/L, where L is the field line length. Hence an appropriate set of fluid equations for the electron density n, displacement 4 [defined by (at + V E . V)< = GE, where e, is a unit vector in the radial direction and VE is the E x B drift velocity], and electron and ion temperatures Te and T, are the following:

111. Survey of Instability Mechanisms

These equations are, respectively, (electron) particle conservation, current con- servation, and the heat equations for each species. Here, ds denotes an incre- ment of length along a field line, an overline denotes a field-line average, vpi = -VPi x b/nrniw,i is the ion diamagnetic drift velocity, P denotes total pressure (summed over species), n is the curvature vector, ’ denotes differentiation with re- spect to the radial coordinate 2, a denotes the angle of inclination of the end plates

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relative to field lioes in a poloidal cross section (assumed symmetric in s, and so, for strongly tilted plates, approximately equal to L'/L, where L is field-line length), vi is the ionization rate, and ST is the heating rate. In writing these equations we have as- sumed, for simplicity, that the density is constant along field lines. The subscripts 0 refer to unperturbed quantities. Quantities without subscripts or tildes are fully non- linear ones. We have assumed collisionally dominated (filled loss-cone) end loss, so that the ion and electron end loss currents and the temperature end loss currents are, respectively, j n i = nc,, j n e = nute exp(-e4/Te), j,e = nvte(9 + 2Te)eq(-e4/Te), and j ~ i = criTc,, where c, is the sound speed, which we take as [(Te + Ti)/m;]'/'], 4 is the plasma potential, and a; M 2.5.

The displacement equation (2) is just the field-line integral of 0.j = 0. In writing it for the case of inclined end plates, we must account [8] for the fact that it is the plasma current nonnal to the plate that must be equated to the normal projection of the sheath current j n i - j n e . Hence, there are end-wall contributions from the ion diamagnetic current, and also, because of the frequency ordering, from the ion E x B current. (By keeping only these two terms, we are assuming L' /L << T' /T , n'/n, etc.)

Because of the ordering of the frequency relative to the ion transit time (valid, in particular, at the frequencies characteristic of peak linear growth), the source and end-loss terms are unimportant except in the vorticity equation, and hence will be dropped elsewhere for the following discussion. However, the neglected terms are important for some purposes, as will be discussed later.

Linearizing this system and utilizing the relation V * ( = 0 yields the following dispersion relation (in the radially local approximation):

where P s d is the gyroradius defined using the sound speed at the divertor or limiter plate, where WE 2 W E - WE and similarly for w*i, and where we haxe written R 3 w - WE. The first term in this equation is ion inertia; the second is the drive from axial shear in the E x B rotation and ion diamagnetism; the third is the drive from radial shear in these frequencies; the fourth is the curvature drive; the fifth is the drive from radial tilt of the divertor plates, and the sixth is the sheath impedance term, which is an instability drive as long as WE # 0. It is readily verified that any of these terms alone can drive instability. As noted above, the conducting- wall drive can be interpreted in terms of a negative sheath resistance; we note that d/jii a w / ( w - W E ~ B ) , which can be negative when W E ~ B # 0.

The radial shear drive is that of the Kelvin-Helmholtz instability. Nonlinearly, this drive must compete with the tendency of velocity shear to tear apart eddies, reducing turbulence [35]. For a gas-target divertor, the magnetic field may be parallel to close-fitting walls of the divertor slot chamber. Under such circumstances, one

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expects a radial gyrosheath, and strong Kelvin-Helmholtz activity could be expected within a few gyroradii of the wall. (See also the discussion in Sec. IV).

It is noteworthy that the various drives included in Eq. (4) are not the source of distinct modes of the SOL; rather, they are all contributors to the same pair of modes, and, except for the radial shear drive (which for large negative k, can weakly destabilize both branches) all contribute to a single unstable mode. Thus, for example, while the curvature drive alone would create a wave in the ion diamagnetic direction (in the plasma frame), and the conducting-wall drive favors the electron diamagnetic direction, the real frequency for a given kl varies smoothly through zero as the balance between drives is shifted. It is also interesting to note the relative phases of these drives: the conducting-wall and radial shear drives are in phase and hence strictly additive (or subtractive), while the remaining drives appear with an additional i phase shift and thus, for example, could never completely cancel the destabilization by one of the first group of terms.

5 4.57~10

Fig. 2. Plot of growth rates for individual F-

drives for TEXT parameters. C.W. is conducting wall drive; cum. is curvature; KH is Kelvin-Helmholtz. *

‘8 v) v

0 k (cm-’) 49.6 Y

In Fig. 2 we plot the growth rate vs. wavenumber for several of the drives. The parameters chosen are representative of TEXT. It is seen that the conducting-wall drive is the largest, but the other drives are not insignificant. For a diverted SOL such as DIII-D, when account is taken of the average magnetic shear (but not the x- point effects discussed below), the growth rates for the conducting-wall and curvature drives are nearly equal [9], and the radial shear drive is also competitive. The k l ’ s at which the mixing-length estimate ~/k: maximize are also comparable, suggesting comparable diffusivities (x N 1 m2 for DIII-D) characteristic wavelengths ( k l p N

O.l), and fluctuation levels eJ/T, cv 0.2 - 0.5). The axial shear drive is relatively weak in the flute limit; however, as is shown in Ref. 6, finite lcll increases growth. For the case of a “single-ended field line” (where correlations with the opposite end are lost), the growth rate is maximum when the parallel wavelength is of order of the length of the potential-variation region, and the resulting mixing-length diffusivity is comparable to the others.

It is instructive to consider the dispersion relation (4) in the context of the double-null divertor asymmetry data on ASDEX. The inner leg has favorable curva- ture and lower temperatures. The former effect contributes a direct stabilizing effect on the inner leg; the latter reduces all the drives on the inner leg compared to the outer.

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When electron energy end loss is included in Eq. (3), we find tha.t at long wavelengths, electron energy end loss stabilizes the conducting-wall mode, while short wavelengths are also stable as a result of finite-L&mor-radius (FLR) effects coupling with the energy end loss. Similar effects should hold for the other drives.

All of the modes have phase velocities typically smaller than the equilibrium E x B velocity, and will thus be observed in the lab frame to rotate in the direction of V E , i . e . in the ion diamagnetic direction. This is opposite from the situation in the closed-flux-surface edge, because of the change in sign of E,. The fact that the conducting-wall drive tends to push the mode in the electron direction in the plasma frame may be relevant to observations that the direction observed in the lab frame sometimes reverses [17].

The flute analysis above provides a reasonable description for conducting-wall and curvature-driven modes in moderate-size limited tokamaks, but finite kil effects become important for larger and, especially, diverted tokamaks. At the simplest level [5], finite Ic l l effects increase with increased field-line length and beta; the relevant pa- rameter for the conducting-wall mode is KT = P' / ' (L /LT) ' /~ . The flute dispersion relation is accurate for KT 6 1. For larger values (as is typical of machines in the DIII-D class and larger), the growth rate for the fundamental (in parallel structure) mode decreases, but higher harmonics start to become important.

At a next level of complication, there are two significant features attached to diverted plasmas. One is that field lines are strongly inclined relative to the divertor plates, intersecting at an angle whose tangent is about BPI&. This results in no change to the dispersion relation in the flute limit, but we find a significant increase in growth rate (and a modest decrease in the wavenumber at the peak growth rate) for KT ;S 1.

The other significant feature is the presence of the x point, resulting in strong shear of SOL field lines as they pass near it. Farina, Pozzoli and Ryutov [lo] have pointed out that, for modes which are flute-like through the x-point region, the strong twisting of field lines transforms a moderate poloidal wavenumber on one side of the x point to a veiy large radial wavenumber on the other (as one follows a field line). Hence, flute-like modes would experience significantly reduced growth due to the large polarization term on the high-l, side of the x point. They concluded that modes on one side of the x point would be isolated from those on the other. This result appears to be at odds with the observation on ASDEX [15] of a high degree of correlation of fluctuations at the midplane of the SOL with those at the divertor floor. One possible resolution is that ASDEX, with its highly localized x- point region, does not have sufficient fanning. We have been examining this issue, and our preliminary findings indicate another likely resolution: in the neighborhood of an x point the solution is non-flute-like and in fact non-eikonal; but the solution matches on to low-kll, moderate-kl eikonal solutions far from the x point on either side.

There are other potentially significant turbulence mechanisms in the SOL which are not described by the equation set (1)-(3). These include atomic-physics-driven (ionization, radiation-condensation) turbulence, parallel velocity-shear-driven turbu-

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lence, ion-temperature-gradient-driven (7;) modes, and turbulence propagating into the SOL from the core.

The parallel velocity shear instability in a limited SOL plasma is studied in Ref. 36. This instability [3?] is driven by the free energy of the radial shear in the equilibrium parallel velocity q , and, in a fluid model, is damped by sound waves and by a cross-field density gradient. The instability criterion is L, < 21;,[crvll/c, - 1]1/2/a, where a = pdqR/L,a and L, is the radial scale length for 2111. Kinetic analysis shows an additional stabilization from ion Landau damping [38]. Though for most tokamak plasma parameters, estimates with L, z L, imply stability (except for CCT), nonlinear fluid simulations indicate that the self-consistently evolved 2111

profile is narrow enough to permit a strong instability. Atomic physics effects are strong in the SOL, and may play a role in driving

fluctuations. Theories of atomic instabilities have been devised primarily for closed flux regions, but similar mechanisms may persist in the SOL. Here, we mention two of the better known such drives; there are others. First, impurity radiation [39] enters the equations in the form: me]% = -Irad + . . .. Perturbing this equation in ?1 and f', and assuming Te06 = -noT shows that the combination of radiation and condensation drives fe if dlnIrad/dInTe < 1, which is often the case for impurities of various ionization states. Since the strength of the drive is strongly dependent on particular edge conditions, whereas the observed SOL fluctuations are not, radiative drive is probably not a significant contributor to turbulence in conventional SOL'S. On the other hand, it might play a significant role in a radiative divertor.

A second atomic drive that has received attention is ionization [40]. This enters the equations as a density source: an/% = uin + . . . Perturbing this equation and neglecting temperature dependence of vi suggests that i? grows at rate vi. This simple analysis neglects sinks and temperature perturbations. In the closed flux surface region, the sink is provided primarily by turbulent transport. Whether or not ionization adds to the turbulent flux present without an ionization term depends on the nature of the existing turbulence. The issue is further complicated in the SOL by the addition of end loss as a sink. Whether ionization adds to the level of turbulence in this circumstance is not yet resolved. A first attempt at a calculation for the SOL suggests a diffusion coefficient of order viAtz [41], where Ax is the radial mode width.

[9], ionization in the magnetized sheath reduces the growth rates (and increases the most important wavenumbers) for the conducting-wall mode, because of reduction in ErO and because ionization appears as a sink term in the Te equation.

Another neutral-density-related turbulence drive, which may be quite relevant to PISCES but (at least as formulated) largely irrelevant for tokamak divertors, is the so-called E x B instability [31], described some twenty years ago. This mode arises from the difference in electron and ion E x B drifts resulting from the difference in frequency of collisions with neutrals. When the electric field is in the direction of the density gradient, a test density perturbation is amplified, implying instability.

An alternative scenario for SOL fluctuations is that they are generated in the

As noted in Ref.

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core (by one of the many possible instabilities there) and propagate out to the edge and SOL [42]. This is supported by the observation that the absolute level of ii tends to be roughly constant over regions extending from the core to the far reaches of the SOL, suggesting some sort of causal link, like propagation. While experiments have generally reported zero propagation [43], it is argued [42] that the twisted field structure of a tokamak tends to produce modes with large components of opposite propagation, and net propagation results from a small difference between largely canceling components, which may be beyond the resolution of present day diagnostics.

IV. Nonlinear Theory and Simulation Nonlinear studies of SOL turbulence are in their infancy, with little analytic

work beyond the mixing-length level and only a few nonlinear simulation studies. But there has been considerable work on nonlinear aspects of modes in the closed- flux-surface region; some of this, particularly work on resistive pressure-gradient- driven modes [44] and atomic-physics-driven turbulence [40,39], may be applicable with some modification to the SOL.

In nonlinear analyses of core turbulence, one considers a particular instability, uses ingredients from statistical renormalization theory to estimate the correlation width and time of the dominant fluctuations, and uses these to determine turbu- lent fluxes. The result is usually a local transport coefficient (such as a turbulent convective velocity or diffusivity), which can then be placed in a transport code to determine the resuiting profiles. However, some evidence suggests that trans- port from SOL turbulence may be fundamentally nonlocal, making many standard theoretical approaches inapplicable. In particular, experimental observations of cor- relation lengths (e.g., Ref. 15), computational results [46] and theoretical estimates of fluid displacement magnitudes (see Sec. V) suggest that there is no separation of equilibrium and fluctuation scales in the SOL. Furthermore, several studies, in- cluding experiments [45, 241, theories [ll] and simulations [46] , suggest that SOL turbulence undergoes an inverse cascade, wherein fluctuation energy transfers con- tinuously to ever larger scale lengths (terminating at equilibrium scale lengths, or, for a simulation, perhaps at the box size). For an inverse cascade, one would expect a power spectrum which extends broadly (but falling) from wavenumbers correspond- ing to equilibrium scales to wavenumbers at which the fluctuation energy is injected, and a sharper fall-off above, consistent with the experimental spectra of Fig. 1 and the computed spectrum of Fig. 3.

In contrast to the more familiar forward cascade, where turbulent transport acquires a diffusive nature, transport from the inverse cascade is not even approxi- mately diffusive, since step sizes from turbulent convection are not separable from equilibrium scales. There is no standard approach to assess the transport from an inverse cascade. A recent study Ill] postulated that the inverse energy cascade is driven by the forward cascade of the second invariant [enstrophy, for the E x B convective nonlinearities which dominate Eqs. (1-3)]; hence the inverse cascade of a given parcel of energy comes to a halt when the accompanying second invariant

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is depleted. Thus, the turbulence acts to supply the equilibrium with energy in a form where the accompanying enstrophy is at a minimum, which can be calculated by variational means. For the model considered- in Ref. 11, the result is that tur- bulence drives a minimum enstrophy state, in which the electron temperature and electrostatic potential acquire an exponential profile.

Shear flow generation by edge turbulence has received much attention, and it is important to note that this acquires an entirely different character in the SOL. By virtue of the fact that field lines are connected to endplates, the equilibrium potential is determined by the sheath constraint eq5o = AT,, with A x const.; thus, the E x B flow profile in the SOL is closely tied to the T, profile and transport. It is difficult to say whether momentum transport or thermal transport is a stronger factor in determining these profiles; probably both of these play a role. Clearly thermal flux from the core ultimately plays an important role in maintaining the T, profile, and supplying fluctuation energy. On the other hand, the inverse cascade scenario, which is consistent with the Reynolds stress description of shear flow generation in Ref. 47 implies that the Reynolds stress drives both the Te and #o profiles. SOL equilibrium flows can have a strong impact on edge rotation by enabling parallel flows there; this in turn would impact edge turbulence [48].

Nonlinear simulation results for SOL turbulence have been reported by Xu [46] for the conducting-wall drive. For curvature drive, a brief study was reported by Gerhauser and Claassen [49] ; first results from a new simulation by Benkadda, Garbet and Verga [50] are reported at this meeting. As noted in Sec. 111, McCarthy e t al. have performed simulations for the parallel velocity shear instability [36]. These are all 2-D fluid simulations. In addition, particle simulations have been performed [51] for Kelvin-Helmholtz instabilities with a wall parallel to field lines (as in the gas-target divertor chamber example discussed in Sec. 111); saturated vortices and a particle flux comparable to mixing-length levels are observed.

The simulation model of Ref. 46 consists of a set of electrostatic equations for 4 and T, in a shearless magnetized slab. The model has an edge region, with periodic boundary conditions along the magnetic field, and an SOL region in which the magnetic field is taken to be of finite length with model (logical sheath) boundary conditions at divertor (or limiter) plates. The simulation results show that the observed linear instability agrees well with theory, and that a saturated state of turbulence is reached. In saturated turbulence, clear evidence of the expected long- wavelength mode penetration into 'the edge (with skin depth of the order of the poloidal wavelength, as calculated in Ref. 52) is seen, and an inverse cascade of wave energy (toward both long wavelengths and low frequencies) is observed.

For typical DIII-D SOL parameters, Ted = 25eV, W*i/wExB = 0.6, e+/T,d = 3.5, midplane radial temperature scale length LTe = 1.5cm, and L = 20 m, the simulation saturates at eJ/T, N 39.1% and ?,/Te - 14.5%, and an electron thermal diffusivity xsim N 0.3 m2 sec-' is obtained. The fluctuation levels are comparable to, but x is an order of magnitude smaller than, mixing-length estimates for the same parameters. Thus there appears to be a tendency.for large fluctuations but small turbulent transport. Yet the i j ~ - f , correlation is large (- 0.7). We will return

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to these observations in Sec. V. The DIII-D simulation yields the laboratory-frame power spectrum shown in

Fig. 3. Because of the finite simulation volume, the minimum resolvable E x B frequency, and hence the minimum lab-frame frequency, is around 10 kHz. The spectrum is qualitatively similar to the experimental spectrum of Fig. 1.

1000 -

Fig. 3. Potential power spectrum in laboratory frame from 2-D conducting- wall simulation, for DIII-D parameters.

1000 Frequency(kH2)

V. Discussion In this section we discuss applications of turbulence theory results for the SOL,

and summarize the paper. One important application of any calculation of turbulent fluxes in the SOL

is to the determination of the value and scaling of the SOL width (in temperature and density). This is usually done by applying a multi-scale approximation, distin- guishing between fluctuation and equilibrium (transport) time scales. The electron temperature-gradient width is then established from the condition that heat can diffuse from its source (the separatrix) only as far radially as it can in an axial con- finement time, or &(LTe)/nTeLTe = 2qc,/L, where qT, is the energy per electron removed by end loss. For the conducting-wall drive with a mixing-length diffusive approximation for the radial heat flux [5] , this prescription yields the scaling L T ~ oc L 2 / 5 p ~ ~ 5 X ~ ' 0 , where AT is the ratio of the midplane and divertor electron temper- atures, while for the curvature drive, one obtains [17] LTe cc (Te /R) ' /3 (L /B)2 /3 . One can proceed one step further and eliminate the temperature in favor of the ap- plied power, using energy conservation; then, for example, for the conducting-wall drive, one obtains [5] L T ~ cc (B/Bp)s/ l2L;{; L ' 1 l 2 (P /nR) ' /6B- ' /2 , where Lpol is the poloidal length of a field line. When the constants are re-introduced, these formu- lae do a reasonable job of reproducing the SOL widths in diverted tokamaks. But, as noted above, the SOL widths tend to be somewhat larger in limiter machines, apparently at odds with these scalings.

There is a further complication. In the conventional picture described above, fluxes comparable to the mixing-length estimates are required to account for observed SOL widths in typical diverted tokamaks. Yet, as noted in Sec. IV, the turbulent heat flux found in simulations of the conducting-wall mode for DIII-D is well below the mixing-length level (as estimated from the ICi at peak linear growth). This set

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of observations could be a result of an inverse cascade, which would render f i E / i smaller than the mixing-length estimate. Then similar results would be expected regardless of which of the drives in Eqs. (1-3) dominates.

A possible resolution of the apparent discrepancies between SOL widths and mixing-length scalings and computed fluxes lies in the non-diffusive nature of in- versely cascading turbulence. In fact, in the aforementioned simulation code, we have implemented tracer particles which indicate large oscillations in the particle position, comparable to the SOL width (see Fig. 4). This is consistent with the mixing-length estimate of the saturation level, ~ / L T ~ N l / k , L ~ ~ , which should be O(1) for an inverse cascade. This convective motion, coupled with end loss at a random phase of the oscillations, could then account for much of the observed SOL width. Note that this picture would be consistent with Endler e t d s interpretation [16] of ASDEX fluctuations as curvature-driven convective cells with a width of order of the SOL width.

Fig. 4. Tracer-panicle orbits for fluid simulation with DIII-D parameters given in text. x and z denote radial and parallel (along B) coordinates.

20

- 8 10

0 Wcm> 4

SOL turbulence may play a crucial role in establishing the feasibility of gas- target divertors. First, the scaling of the thermal diffusivity with neutral density, electron and ion temperatures and their scale lengths may play a critical role in determining conditions for cooling the plasma to a few eV or less at the divertor floor. Second, moth nonlinear turbulence theories tend to imply anomalous viscosi- ties comparable to thermal diffusivities. In the context of a tight-fitting divertor, such a level would be sufficient to dissipate much of the ion momentum to the walls. This is an important consideration, since, from integrating the sum of the electron and ion parallel momentum equations, the upstream sum of plasma pressure and di- rected kinetic energy must be balanced by neutral pressure and cross-field transport, which is difficult to arrange with strictly classical processes and a reasonable level of neutrals .

The strong parametric dependence of the atomic-physics-based turbulent trans- port processes discussed above opens the question of transport-driven bifurcation phenomena in the SOL. Since such processes tend to scale with atomic physics rates (e.g. ionization or radiation rates), turbulence associated with any classical process which by itself can lead to bifurcations, is likely to lead to bifurcations over an expanded parameter range. Thus, for example, a strongly radiating plasma with non- monotonic radiation rate is a strong candidate for a turbulence-driven bifurcation.

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Diamond and co-workers [41] are exploring a preditor-pray model in which ionization- enhanced turbulence might lead to a bifurcation between a state where hot plasma extends to the divertor floor and a state where the end is neutral-dominated and cold; this would be analogous to their recent L-H transition model [53] . They are also examining the possibility that at least one of these states is not steady-state.

Since turbulence in the SOL has a generally beneficial effect, it is interesting to consider the possibility of actively driving turbulence. Because of the accessibility of the field-line ends, a logical candidate for active excitation is the conducting-wall mode. We envision applying r.f. bias to a segmented set of plates at the divertor floor, with a frequency and toroidal wavenumber chosen to be nearly in resonance. The linear theory for this is straightforward: the plasma potential responds to a driven fluctuation at a level proportional to the inverse of the plasma dielectric at the driven frequency. But the real question is the nonlinear response. We have addressed this by applying end-wall r.f. bias in our 2-D electrostatic simulation code. By applying e#bjas/Te = 0.5, at, a frequency approximately 0.12 times that at the maximum growth rate ( i e . , an applied frequency of about 10 kHz for DIII-D) and a kl related to w by approximately satisfying the linear dispersion relation, we find that the fluctuation level of potential e$/Te is tripled (to about 0.8), and the temperature fluctuation Te and the radial heat flux qz are doubled. We also observe that, at long wavelengths, the fluctuation level inside the separatrix is significantly increased, consistent with the theoretical [52] and numerical [46] experience for unbiased SOL'S. For off-resonant biasing, we have not found any significant change in the turbulence.

Another possibility for active control of SOL turbulence is DC biasing of the end plates. Antisymmetric biasing (drawing a current from one plate to another) has only a weak (stabilizing) effect on the conducting-wall mode [54], anti probably on the others as well. Nedospasov [55] suggests that symmetric biasing of both plates so as to pull a net ion current to the plates, would enhance turbulence. Such biasing multiplies the sheath impedance by the factor (1 - jo/nec,)-' , which, for 0 < jo/nec, < 1, increases the growth rate for the pure curvature-driven mode (WE = 0). This same factor, as well as the accompanying decrease in W E , reduces the growth rate for the pure conducting-wall mode, while radial variation of the wall potential 1561 has the opposite effect. Furthermore, perturbation of the process leading to equilibrium radial current can be stabilizing or destabilizing. Finally, the change in the we profile also alters the Kelvin-Helmholtz drive and the velocity- shear suppression of turbulence. Hence even the sign of the change in the growth rate depends on the relative importance of these effects for the specific SOL parameters. The experiments cited in Refs. 55 and 57 show an increase (decrease) in fluctuation level for small positive (negative) bias current, but the data for the full range of bias considered in Ref. 57 suggests that shear suppression dominates.

In summary, there are many comparable drives for turbulence in typical SOLS, but not that many different modes. Mixing-length estimates of the resultant turbu- lence levels and transport coefficients are comparable with those observed on exper- iments. Because of the large E x B drift in the SOL, one would expect to observe (in the lab frame) modes propagating in the ion diamagnetic direction, whereas the

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reverse is expected in the closed-flux-surface edge. This is also in accord with exper- imental observations. For any of the drives included in Eqs. (1-3), the linear terms in these equations suggest that i i /n and Pe IT, should be in phase (as observed in TEXT and Phaedrus-T [26], but not TJ-1 [25]) and in a ratio of L n / L ~ , , which again is roughly in accord with observations. The relative fluctuation amplitudes of 6 and ii would be in the ratio w+,, /n, which depends on the combination of drives (and is, for example, about 1 for the conducting-wall drive at frequencies below that of peak linear growth). Nonlinearly, the resultant turbulent energy is expected to in- versely cascade from wavelengths corresponding to peak linear growth to equilibrium scales; this has been demonstrated for nonlinear simulations of the conducting-wall mode and appears to be consistent with experimental spectra. Active control of the SOL width by r.f. biasing of endplates appears to be possible. Atomic physics-driven turbulence may play a significant role in gas-target and radiative divertors, by in- fluencing mean plasma profiles, and possibly inducing bifurcations and non-steady stationary states.

Acknowledgments We thank L.D. Pearlstein, T.D. Rognlien, D.D. Ryutov, G.D. Porter, P.H. Di-

amond, G. Staebler, and H. Tsui for useful conversations, and H. Niedermeyer, M. Endler, X. Garbet and R.A. Moyer for valuable electronic mail exchanges. We espe- cially thank H. Niedermeyer, M. Endler, R.A. Moyer and H. Tsui for making available the plots in Fig. 1 in advance of publication or in modified form from a publication. This work was performed at Lawrence Livermore National Laboratory under the auspices of the U.S. Department of Energy under Contract No. W-7405-ENG-48.

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