Page 1. On the nonlinear turbulent dynamics of shear-flow decorrelation and zonal flow generation. G. R. Tynan, R. A. Moyer, and M. J. Burin. Department of ...
PHYSICS OF PLASMAS
VOLUME 8, NUMBER 6
JUNE 2001
On the nonlinear turbulent dynamics of shear-flow decorrelation and zonal flow generation G. R. Tynan, R. A. Moyer, and M. J. Burin Department of Mechanical and Aerospace Engineering, Jacobs School of Engineering, University of California, San Diego, La Jolla, California 92093-0417
C. Holland Department of Physics, University of California, San Diego, La Jolla, California 92093
共Received 27 January 2000; accepted 24 January 2001兲 Sheared flows, thought to be generated by turbulence in magnetized fusion plasmas, are predicted to mediate the transport of mass, momentum, and heat across the shear flow region. In this paper we show that an examination of three-wave coupling processes using the bispectrum and bicoherence of turbulent fields provides an experimentally accessible test of the turbulence-generated shear flow hypothesis. Results from the Continuous Current Tokamak 共CCT兲, Princeton Beta Experiment– Modified 共PBX–M兲, and DIII–D tokamaks indicate that the relative strength of three-wave coupling increases during low-mode to high-mode (L – H) transitions and that this increase is localized to the region of strong flow and strong flow shear. These results appear to be qualitatively consistent with the turbulence-generated shear flow hypothesis. © 2001 American Institute of Physics. 关DOI: 10.1063/1.1357220兴
I. INTRODUCTION
have focused upon the effect of sheared flows on turbulent statistical properties such as amplitude, correlation length, and phase angles. Since these are the only currently available predictions, experimental studies have necessarily been focused on studying the response of these turbulent statistical properties to flow shear. However, these statistical properties merely reflect the final result of shear flow-turbulence interactions; they do not provide any insight into the underlying interaction between the turbulence and the mean sheared flow. Thus the available experiment–theory comparisons, as detailed and exhaustive as they are, have not examined the critical underlying operative effect of shear flow on the nonlinear dynamics of plasma turbulence 共although we note that recent work published during the review of this paper begins to redress this issue11兲. There are two broad goals of this paper. First, in a straightforward way we show how turbulence-generated sheared flows, generated via the Reynolds stress term in the momentum equation, can be viewed as a mode-coupling problem mediated by three-wave interactions. As a result, shear flow generation from turbulence can be observed in the bispectrum and bicoherence computed between the shear flow and the turbulent fields, providing an experimentally accessible test that shear flows can be generated from turbulence in plasmas. Second, we use existing Langmuir probe measurements of turbulent density and potential fields, obtained from L – H transitions in the Continuous Current Tokamak 共CCT兲, Princeton Beta Experiment-Modified 共PBXM兲, and DIII–D tokamaks, to search for the expected changes in the nonlinear coupling between the shear flow and turbulence during and after the formation of an H-mode in these devices. The results indicate that the degree of nonlinear coupling between the shear flow and the turbulence increases during the formation of a shear flow region in a
The formation of regions of reduced cross-field turbulent particle, momentum, and ion heat transport in confinement devices has been attributed to E⫻B shear flow decorrelation of turbulence and turbulent transport.1,2 The origin of the shear flow is not yet clear, but it is generally thought to be driven by turbulent Reynolds stress in a type of nonlinear self-organization phenomena.3 Turbulence-generated flows are also thought to be related to the interaction between Rossby waves and zonal flows in planetary atmospheres.4 Numerical simulations of plasma turbulence in tokamak devices suggest that turbulence can also generate sheared flows in these systems5 and thus the phenomena may also represent a form of self-organization in two-dimensional fluids which has wider scientific interest. For a recent summary of the subject in the context of magnetized fusion plasmas see, e.g., the recent review.6 A recent summary of the subject in the context of planetary atmospheric dynamics is also available.4 In magnetized fusion plasmas sheared flows have also been predicted to occur via neoclassical ion-orbit loss mechanisms.7,8 However, experiments in the DIII–D tokamak showed that the rotation direction of majority ion species was inconsistent with the orbit-loss mechanism,9 casting doubt upon this mechanism for shear flow generation and leading to the general acceptance of the view that shear flows are generated from turbulence in tokamak devices. Detailed experimental studies of the interaction between turbulence and large-scaled shear flows are in the initial stages and no broadly accepted experimental results have yet emerged.10 Thus theoretical approaches and experiments to address the key underlying interactions between turbulence and shear flows are of interest. Published predictions of shear-flow/turbulence interactions using either analytic theory or numerical simulations 1070-664X/2001/8(6)/2691/9/$18.00
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© 2001 American Institute of Physics
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confined plasmas, and that the increases are spatially localized to the shear region. Results from the DIII–D device also suggest that the increase is temporally localized to the L – H transition. These initial results appear to be qualitatively consistent with the generation of shear flows from turbulence. However, as pointed out in the discussion section of this paper, additional work is needed in order to definitively demonstrate that shear flow is generated from turbulence. II. FLOW GENERATION FROM TURBULENCE AS A MODE-COUPLING PROBLEM
The generation of a sheared flow from turbulence as an explanation of the formation of transport barriers in magnetic fusion devices was first proposed by Diamond and Kim.3 Following their work, in the absence of magnetic fluctuations the azimuthal momentum balance for an incompressible twodimensional plasma fluid with azimuthal flow u in a cylindrical plasma is
u u ru ⫹ ⫽⫺ u , t r
共1兲
where u r u is the Reynolds stress and is the damping rate of the mean flow due to dissipative processes 共note that this is not a viscous damping effect, and that we have not yet averaged over any relevant time scales兲. In this highly simplified model, density fluctuations and axial 共i.e., toroidal兲 flows are neglected. These points must be addressed in a complete model of the tokamak low-mode to high-mode (L – H) transition; however we neglect them here for the purposes of clarity and simplicity. Mean shear flow generation can then occur when the Reynolds stress term leads to a local concentration of azimuthal momentum, in a similar manner to the formation of a strong density and pressure gradient due to the reduction of turbulent particle and heat transport. This mechanism can be set into the framework of mode coupling by considering the spatial Fourier transform of the incompressible azimuthal momentum balance equation, which is given as
u 共 k兲 ⫹ik r t
兺
k1 ,k2 k⫽k1 ⫹k2
u 共 k1 兲 u r 共 k2 兲 ⫽⫺ u 共 k兲 ,
共2兲
where u i (k) is the ith component of the transformed velocity u at wave number k . Multiplying this expression by the complex conjugate u * (k) results in one new equation, taking the complex conjugate of this new result, then adding these two equations together gives the evolution of the kinetic energy of the flow u 2 (k) as
兩 u 共 k兲 兩 2 ⫺2k r t
兺
k1 ,k2 k⫽k1 ⫹k2
⫽⫺ 兩 u 共 k兲 兩 2 .
Im关 u * 共 k 兲 u 共 k1 兲 u r 共 k2 兲兴 共3兲
An expression governing the evolution of the magnitude squared of the shear of the velocity u (k) is obtained by multiplying by k r2 to obtain
兩 U 共 k兲 兩 2 ⫺2k r3 t
兺
k1 ,k2 k⫽k1 ⫹k2
Im关 u * 共 k 兲 u 共 k1 兲 u r 共 k2 兲兴
⫽⫺ 兩 U 共 k兲 兩 2 ,
共4兲
where we have defined U⫽ u / r⫽ik r u (k). Next, consider a zonal flow having kZ ⫽k rZ rˆ and u Zˆ ⫽u with other components nearly vanishing, and with spatial scales which are larger than those of turbulence, i.e., with 兩 krZ 兩 ⬍ 兩 k1 兩 , 兩 k2 兩 , and with a duration Z ⬃1/k rZ u Z which exceeds the turbulent fluctuation cross-correlation time scale, i.e., with Z ⬎ corr . A much more detailed discussion of the characteristics of zonal flows has been given elsewhere.12 This separation of scales is highly idealized, and zonal flows in fact span from these large range spatio-temporal scales down to the spatio-temporal scales approaching the turbulence scales. Also note that this separation of scales implies that we have assumed that an inertial range exists in wave number space lying between the shear-flow scale and the turbulence scales. The primary result of this model—that the generation of shear flow via the turbulent Reynolds stress is manifested as an increase in the rate of energy transfer from turbulence into the shear flow—is unchanged by this simplification. Averaging Eq. 共4兲 over the zonal flow time scale then gives the evolution of the shear flow amplitude in terms of the interaction between the shear flow and the turbulence:
具 U 共 kZ 兲 典 2 ⫺2k r3 t
兺
k1 ,k2 kZ ⫽k1 ⫹k2
Im关 具 u * 共 kZ 兲 u 共 k1 兲 u r 共 k2 兲 典 兴
⫽⫺ 2 具 U 共 kZ 兲 典 2 ,
共5兲
where the second term on the left-hand side is the power transferred between the 共large-scale兲 shear flow and smallscale turbulent fluctuations, and where the brackets signify the time average 共or, equivalently, an ensemble average兲 over the shear flow time scale. For low beta plasmas with significant fluctuation amplitudes for k s Ⰶ1, the fluid velocities can be taken to be given by the electrostatic E⫻B drift speed, i.e., u(k)⫽⫺(ik (k)⫻B)/B 2 where 共k兲 is the Fourier transformed electrostatic potential. Taking B along the z axis and defining the normalized potential as (k) ⫽e (k)/k B T e we can then write Eq. 共5兲 as 1 具 2 共 kZ 兲 典 ⫺ P kturb⫽⫺ 具 2 共 kZ 兲 典 , Z 2 t
共6兲
where P kturb⫽ Z
k BT e eB
兺
k1 ,k2 kZ ⫽k1 ⫹k2
k 1 r k 2 Re具 * 共 kZ 兲 共 k1 兲 共 k2 兲 典
is the power transferred from the small-scale turbulent fluctuations into the larger-scaled flow 共P kturb⬎0 denotes a Z driven zonal flow; P kturb⬍0 denotes a zonal flow damped by Z turbulence兲. We also emphasize again here that kZ denotes the zonal flow wave number which is oriented primarily in the radial direction.
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Phys. Plasmas, Vol. 8, No. 6, June 2001
On the nonlinear turbulent dynamics of shear-flow . . .
The interpretation of this equation is straightforward and leads to important physical insight into the dynamics of flow generation. In steady state the flow amplitude is then determined by balancing flow generation with flow damping. In two-dimensional fluids kinetic energy is preferentially transferred to larger scale lengths and, eventually, it condenses into the largest scale available to the system 共see, e.g., Hasegawa and Wakatani13兲. Thus in the case where flow damping dominates the mode-coupling effect, then the system will have little or no mean shear flow. However, if the inverse energy transfer rate due to three-wave coupling can be increased relative to the flow damping rate 共e.g., by increasing the turbulence amplitudes driven by increased free-energy sources and/or by increasing the strength of the three-wave coupling between turbulent fluctuations and larger-scaled shear flow兲 then a transition to a state of large flow/small amplitude turbulence can ensue. From the perspective of mode-coupling, larger-scaled slowly varying zonal flows can be generated by nonlinear interactions of turbulent fluid velocity fluctuations with 兩 k1 兩 ⬃ 兩 k2 兩 ⬎ 兩 kZ 兩 and with correlation times which are short compared to the zonal flow time scale mediated via three-wave coupling satisfying the constraint kZ ⫽k1 ⫹k2 . Such nonlinear interactions should be reflected in an increase in the real part of the bispectrum B k Z 共 k 1 ,k 2 兲 ⬅
兺
k1 k2 kZ ⫽k1 ⫹k2
具 u * 共 kZ 兲 u 共 k1 兲 u r 共 k2 兲 典
computed between the turbulent scales and the zonal flow scale. We emphasize that this conclusion is simply the representation in k space of the turbulent Reynolds-stress interacting with the mean flow, and we also note that similar conclusions have been recently made11 using a drift wave action formalism. As mentioned earlier, the separation of scales implies the existence of an inertial range in k space. Thus the energy transferred into the large-scale mean flow can be related to the energy which is pumped into the inertial scale as shown schematically in Fig. 1. Inclusion of the nonlinear convective derivative in the fluid momentum equation leads to the nonlinear coupling of modes which, in turn, lead to the eventual saturation of the turbulent fluctuation spectrum. When combined with the ion continuity equation and the electron parallel dynamics this coupling of modes can be expressed as14 1 k T 共 k ,k 兲 . ⫽ 共 ␥ k ⫹i k 兲 k ⫹ t 2 k 1 ,k 2 k 1 2 k 1 k 2
兺
共7兲
Without flow shear this three-wave coupling equation describes the rate of change of the normalized fluctuation amplitude at ( k ,k) due to the linear growth rate, ␥ k , the linear dispersion relation, k , and the quadratically nonlinear wave–wave coupling coefficient T k (k 1 ,k 2 ) which transfers power from fluctuations at ( k 1 ,k1 ) and ( k 2 ,k2 ) which satisfy the Manley–Rowe criteria. In the presence of a mean sheared flow the definitions of ␥ k and k can be expanded to include a Doppler shift of the real frequency and a reduction in the linear growth rate due to velocity shear.1 In addition, changes in the three-wave coupling coefficient T k (k 1 ,k 2 )
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FIG. 1. Schematic representation of 共a兲 growth of modes and subsequent transfer of turbulent energy to dissipation scale via cascade process, 共b兲 corresponding idealized model of unstable region, separated from dissipation regions by an inertial range, with corresponding growth or damping rates and energy transfer coupling coefficients.
due to velocity shear have also been predicted.15 Thus theoretical estimates of ␥ k and T k (k 1 ,k 2 ) can be made; alternatively they can be estimated from experimental data using established techniques.14,16 If one multiplies this three-wave coupling equation by the complex conjugate of k and then performs a timeaverage over the fast-scaled fluctuations an energy dynamics equation is obtained which describes the dynamics of turbulent energy transfer in 共,k兲 space due to three-wave interactions:
兩 k兩 2 T k 共 k 1 ,k 2 兲 具 * ⫽2 ␥ k 兩 k 兩 2 ⫹Re k k 1 k 2 典 . 共8兲 t k 1 ,k 2
兺
This model simply says that the fluctuation energy contained in the kth mode is formed by a balance of the linear growth 共damping兲 rate of this mode balanced by the energy transfer away from 共into兲 this mode. The energy transferred out of the turbulent fluctuations can be related to the energy transferred into the larger scaled shear flow. In toroidal devices poloidal plasma rotation 共analagous to azimuthal cylindrical rotation兲 with a spatial scale of the order the plasma minor radius, a 共e.g., zonal flows with k⬜ i Ⰶk rZ a⬍1兲 can be damped by ion-neutral interactions, by transit-time magnetic pumping effects,17 or by collisions between trapped and passing particles;2 this effect is represented in Eq. 共1兲 by the damping term . In the region 1/L n ⬍k⬍1/ i free energy is released at the linear growth rate. In a two-dimensional Navier–Stokes fluid system which lacks vortex stretching fluctuation energy is transferred preferentially to small k 共i.e., to large spatial scales兲.
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In a magnetized plasma similar dynamics will occur provided that the E⫻B convection velocity dominates over the polarization drift velocity corrections. With this assumption in mind, consider an idealized model of the plasma turbulence-mean flow interaction which has the turbulence spatial scales separated from the mean flow scale by an inertial range as shown schematically in Fig. 1共a兲. The growth rate and energy transfer rate which would correspond to this model is shown schematically in Fig. 1共b兲. For a stationary turbulent spectrum, Eq. 共8兲 gives the fluctuation power transferred from the turbulent wave number range 1/L n ⬍k⬍1/ i into the inertial scale range, P k 0 , as P k 0 ⫽Re
兺
k 1 ,k 2 k 0 ⫽k 1 ⫹k 2
T k 0 共 k 1 ,k 2 兲 B k 0 共 k 1 ,k 2 兲 ,
共9兲
where the bispectrum computed on turbulent space-time scales is defined as B k 0 (k 1 ,k 2 )⬅ 具 k* k 1 k 2 典 and we have 0 explicitly noted that the summation occurs for fluctuations above the cutoff wave number k 0 which defines the lower bound of the unstable wave number range and the upper bound of the inertial wave number range. For this idealized model in which the E⫻B nonlinearity dominates and in which there is a separation of scales between the fluctuations and the zonal flow as shown schematically in Fig. 1, the power transferred out of the turbulent fluctuation range into the inertial range must be equal to the power transferred from the inertial range into the large-scale mean flow range, i.e., P kturb⫽ P k 0 . Solving Eq. 共6兲 for P kturb in Z Z the equilibrium case and equating the result to P k 0 then gives ⫺ 兩 2 共 kZ兲 兩 2 ⫽Re
兺
k 1 ,k 2 k 0 ⫽k 1 ⫹k 2
T k 0 共 k 1 ,k 2 兲 B k 0 共 k 1 ,k 2 兲 , 共10兲
where the terms on the right-hand side are evaluated over the turbulent spatial scales with k⬎k 0 . Expressing the largescale flow in terms of the E⫻B drift velocity and the bispectrum in terms of the bicoherence, bˆ k2 共 k 1 ,k 2 兲 ⬅ 0
兩 B k 0 共 k 1 ,k 2 兲 兩 2
flow by using the shear-decorrelation model1 along with a simple mixing length estimate of the turbulence amplitude in the presence of a density gradient. This model is obtained when the turbulence is homogenous 共i.e., 兩 k 兩 Ⰶ1/L n 兲, the turbulence amplitude saturates at the mixing length amplitude ˜n /n⬇1/k⬜ L n , the turbulent particle flux is taken as a diffusive process with 具˜n˜v r 典 /n 0 ⫽⫺D(1/n 0 )“n 0 ⫽⫺D/L n ˜ /n), and the transport diffusion coefficient is ⬇⫺Dk⬜ (n equivalent to the turbulent eddy diffusivity D⫽ 41 ⌬ t ⌬ r2 where ⌬ t is the turbulent decorrelation rate and ⌬ r is the radial turbulent decorrelation length 共⌬ t and ⌬ r are evaluated in the absence of shear flow兲. The result of single-field two-point models of plasma turbulence in a sheared flow is a predicted reduction in squared normalized fluctuation ampli˜ 2 of the form18 tude ˜n 2k /n 20 ⬇e ˜ 2k /kT e ⬅ k ˜ 2兩 1 k V⬘ ⬇ s, 2 ˜ k 兩 V ⬘ ⫽0 1⫹ 共 sh /⌬ t 兲
where sh⫽⌬ r k V ⬘ is the rate at which a shear rate V ⬘ decorrelates a perturbation of radial extent ⌬ r and poloidal wave number k . 1 In the limit of strong shear ( shⰇ⌬ t ) theory predicts s⫽2/3 while s⫽2 in the limit of weak shear ( sh ⬃⌬ t ). 18 Thus shear decorrelation theory indicates that, assuming that the underlying free-energy source 共e.g., the cross-field gradient scale length兲 remains unchanged and thus the inherent turbulent drive is unchanged, strong shearing rates 共as compared to the inherent decorrelation rate of unsheared turbulence兲 tend to reduce the saturated turbulence levels. We now proceed to close our model. Using this result along with our previous assumption regarding the separation of turbulence scales, we approximate the turbulence amplitudes as 兩 k 0 兩兩 k 1 兩兩 k 2 兩 ⬇ 兩 k 兩 3 . In the presence of strong flow shear the turbulence amplitude reduction is approxi3 3 mated as 兩 k 兩 V ⬘ / 兩 k 兩 V ⬘ ⫽0 ⬇(⌬ t / sh) 2 where 兩 k 兩 V ⬘ is the fluctuation amplitude with flow shear and 兩 k 兩 V ⬘ ⫽0 is the fluctuation amplitude in the absence of flow shear. Using this approximation we then write ⫺ 关 共 kz兲兴 2
兩 k 0兩 2兩 k 1 k 2兩 2
and the sign of the real part of the bispectrum, this expression becomes ⫺ 兩 共 kz 兲 兩 2 ⫽
兺
k 1 ,k 2 k 0 ⫽k 1 ⫹k 2
T k 0 共 k 1 ,k 2 兲 兩 k 0 兩兩 k 1 兩兩 k 2 兩
⫻bˆ k 0 共 k 1 ,k 2 兲 sgn共 Re共 B k 0 共 k 1 ,k 2 兲兲兲 . 共11兲 This result simply expresses a balance between mean-flow generation, driven by an inverse cascade from small-scale homogenous turbulence, and mean-flow damping due to large-scale nonturbulent processes. The right-hand side 共RHS兲 of Eq. 共11兲 contains turbulence amplitudes which are evaluated in the presence of the sheared flow. We can relate this sheared flow turbulence amplitude to the turbulence amplitudes in the absence of shear
共12兲
⬇
兺 k ,k 1
冉 冊 sh ⌬t
2
T k 0 共 k 1 ,k 2 兲 兩 k 兩 V ⬘ ⫽0 bˆ k 0 共 k 1 ,k 2 兲 3
2
k 0 ⫽k 1 ⫹k 2
⫻sgn共 Re共 B k 0 共 k 1 ,k 2 兲兲兲 .
共13兲
III. SUMMARY OF MODEL RESULTS
The results contained in either Eq. 共6兲 共if data from zonal flow and turbulence scales is available兲 or Eq. 共13兲 共if data from only the turbulence scales is available兲 constitute the main results of our ad hoc model, and they provide an experimentally accessible test of shear flow generation from turbulence. We summarize again the key conclusion of this model. If shear flow is driven by three-wave turbulent-scale/ shear-flow interactions, then an increase in the sheared zonal flow amplitude must be accompanied by an increase in either
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Phys. Plasmas, Vol. 8, No. 6, June 2001
On the nonlinear turbulent dynamics of shear-flow . . .
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turbulence amplitudes, by an increase in the turbulent-scale bicoherence bˆ k2 (k 1 ,k 2 ) and/or by an increase in the 0 turbulent-scale coupling coefficient T k 0 (k 1 ,k 2 ) or by a decrease in the effective flow damping rate 共due, for example, to the nonlinear response of neoclassical damping to radial electric fields2兲. If data across the range of shear-flow/ turbulence scales are available, then one could alternatively search for a increase in the bispectral quantity B k2 (k 1 ,k 2 ). Z Turbulence amplitudes are usually observed to decrease momentarily after the formation of mean sheared flows in the H-mode transition19 but are not usually observed to increase immediately prior to the transition. Thus an increase in the degree of three-wave coupling between turbulent scales and shear-flow scales is required if the shear flow is generated by the turbulent Reynolds stress. Observation of the spontaneous generation of a shear-flow region without an increase in the three-wave coupling would be inconsistent with the Reynolds-stress hypothesis of shear-flow generation. IV. COMPARISON WITH EXPERIMENTAL RESULTS
These wave-number domain results can be transformed into the frequency domain using a frozen-flow hypothesis. This hypothesis is valid when the propagation time ␦ t of the fluctuations through the sampling volume d is short relative to the turbulence autocorrelation time ⬃10 sⰇ ␦ t ⫽d/ v . Here d⬃0.5 cm and v is the fluctuation poloidal phase velocity which is approximately 1 – 3⫻105 cm/s (⬃1 ⫻106 cm/s) during L mode 共H mode兲, respectively. We thus estimate ␦ t⬇2 – 5 s in L mode and ␦ t⭐1 s in H mode. The E⫻B fluctuation propagation velocity VE is such that kZ •VE ⬇0; as a result the shear-flow fluctuations are not Doppler shifted and will retain the low-frequency variation f Z ⬃1/ Z of the shear flow. However, the turbulent fluctuations have a strong Doppler shift and thus have a lab-frame frequency f 1,2⬇k1,2•VEⰇ f Z . As a result the anticipated increase in bˆ k2 (k 1 ,k 2 ) should be manifested as an increase in Z bˆ 2 ( f , f ) with the frequencies given above satisfying the fZ
1
FIG. 2. 共a兲 Time evolution of raw floating potential during biased H-mode experiment in CCT. In this experiment the bias voltage exceeds the transition threshold immediately, resulting in a change in turbulence amplitude observed at t⬃13 ms. In this initial stage the Langmuir probe is located just outside of the last-closed flux surface on open field lines. A few ms later the LCFS moves out past the probe, resulting in a large reduction of the DC floating potential. Strong Doppler shifts are observed in the frequency spectra of the floating potential fluctuations in the region of strong dc electric field.
2
sum condition f Z ⫽ f 1 ⫾ f 2 . Based upon the predictions of this model, we have reexamined turbulence data taken in three different tokamak H-mode experiments. Results from all three experiments provide support for the prediction that the bispectrum or bicoherence is an important quantity to examine. The discussion below summarizes these results. H-mode transitions were triggered by external electrode biasing in the CCT tokamak and Langmuir probes have been used to study the evolution of the turbulence and turbulent particle transport. For detailed discussions of the biasing techniques, the effect of biasing upon edge turbulence and transport, and upon poloidal asymmetries the reader should consult the literature.20–22 In the experiments discussed here, a Langmuir probe with tips measuring floating potential and ion saturation current was located at the outside midplane of the CCT device. Discharges were created with a toroidal magnetic field of 3 kG, a toroidal plasma current of 36–38 kA, a line-averaged density of 2–4⫻1012 cm⫺3, and a loop voltage of ⬃1.2 volts. Assuming Spitzer resistivity for the
parallel current the average electron temperature of the plasma was estimated to be ⬃100 eV. The plasma position and shape were chosen such that prior to the H mode the probe tips were located just outside of the last closed flux surface 共LCFS兲 关raw floating potential signal is shown in Fig. 2共a兲兴. After the H mode is triggered via electrode biasing, the plasma pressure increases due to increased confinement. This effect, combined with an intentional outward movement of the plasma, causes the probe to scan across the LCFS, onto closed field lines, and into the strong Er region. The Doppler shift caused by the E⫻B drift of the fluctuations is easily observed 关see Fig. 2共b兲兴. The net effect is to provide a radial scan of the time-stationary turbulence properties during the H-mode phase. A comparison of the bicoherence computed from the Ohmic phase and from the region of strong Er during the H-mode phase of the discharge is shown in Figs. 3共a兲 and 3共b兲, respectively 共for a detailed discussion of the bispectrum, bicoherence, and the interpretation and symmetry properties of these statistical measures of turbulence see,
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FIG. 4. Summed bicoherence in biased CCT H mode. The summed bicoherence increases substantially above the Ohmic levels in region localized to the strong Er/strong Er shear. Particularly strong increases occur for lower frequencies, which correspond to fluctuations with larger scale lengths.
FIG. 3. 共a兲 Bicoherence during Ohmic CCT discharge 共maximum value of 0.25 shown in black/dark gray兲, 共b兲 bicoherence of CCT biased H-mode discharge 共maximum value of 0.77 shown in black/dark gray兲. A marked increase in bicoherence is observed over the 1–200 kHz range of frequencies during electrode biasing. This increase is localized to the region of strong electric field.
total bicoherence increases in a narrow region during a biased CCT H mode. These observations suggest that coupling between turbulence and mean flows, mediated via the Reynolds stress, does increase during biased H-mode experiments in CCT, and that the changes are localized to the region of strong Er and strong Er shear. Measurements of turbulence from an L mode and a spontaneous H mode with P beam⬃ P thres in the PBX–M tokamak have been previously reported25 共here P beam refers to the
e.g., Ref. 23 and references therein兲. An increase in the bicoherence is observed over a ⬃200 kHz region of frequency space. The propagation speed of the fluctuations is marginally high enough in the Ohmic phase, and is clearly high enough in the H-mode phase, such that the turbulent fluctuations do not significantly evolve during their passage past the probe 共i.e., the ‘‘frozen flow hypothesis’’ discussed above is roughly valid兲. Thus frequency and aziumuthal wave number can be related to each other simply. This result provides a qualitative indication of the increase in three-wave coupling of a range of wave numbers during a time-stationary biased H mode in CCT. The summed bicoherence b 2 ( f 3 )⫽ 兺f 1 , f 2 b 2f ( f 1 , f 2 ) f 3⫽ f 1⫾ f 2
3
provides a measure of the importance of three-wave coupling at the frequency f 3 . The total bicoherence b 2 ⫽ 兺 f 3 b 2 ( f 3 ) then provides a measure of the total bicoherence—i.e., a measure of the relative strength of nonlinear coupling— across the entire spectrum. The summed bicoherence increases significantly over the Ohmic value as seen in Fig. 4. The increased summed bicoherence occurs over a broad range of frequencies which correspond to a broad spatial scale range. Previously published results of the total bicoherence24 from externally induced H modes in the CCT tokamak device, reproduced in Fig. 5 below, indicate that the
FIG. 5. Total bicoherence data from CCT tokamak device. 共a兲 Time evolution of bicoherence at outside midplane during biased H-mode transition, 共b兲 deconvolved radial profile of bicoherence during the H mode. The bicoherence appears to increase during the H-mode phase, consistent with the ad hoc model presented in this paper 共the total bicoherence has a value of ⬃0.02 during the Ohmic phase兲. The increase appears to be spatially localized to the region of strong dc radial electric field and/or electric field shear. Figures reproduced from Van Milligen et al. 共Ref. 24兲 with permission.
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Phys. Plasmas, Vol. 8, No. 6, June 2001
FIG. 6. Comparison of summed bicoherence from PBX–M L-mode 共shot 294 588兲 and H-mode 共shot 294 593兲 discharges with P beam ⬃ P thres⬃2 MW. Data are obtained from just inside the LCFS in both cases.
neutral beam heating power and P thres refers to the heating power threshold needed to trigger the L – H transition兲. In that earlier work it was concluded that the flow shear changed by a modest amount 共2–3 times increase兲 moving across the L – H transition, suggesting that flow shear might be a critical parameter in a phase-transition phenomena. Analysis of this data using the bispectral technique 共see Fig. 6兲 indicates that an increase in the bicoherence does occur just inside the separatrix during a time-stationary spontaneous H mode in PBX–M operating near the L – H transition power threshold. No such changes are observed on the openfield line region during the L – H transition. Unfortunately there are no probe data available from just inside the separatrix during an L – H transition on the PBX–M device. Fortunately such data are available from the DIII–D device. An examination of the bicoherence of potential and density fluctuations from the open field line region do not show significant changes in the bicoherence as the plasma moves from L mode to H mode state. Furthermore, an examination of time-stationary H-mode data obtained just inside the separatrix with heating power well above the L – H transition threshold heating power does not show a significant increase in bicoherence. However, an examination of data taken just inside the separatrix just before and during an L – H transition show interesting signatures. Figure 7 shows the time evolution of the raw turbulent floating potential and ion saturation current fluctuations measured with a stationary Langmuir probe just inside the LCFS during the L – H transition. The evolution of the bicoherence computed from the Isat fluctuations, shown in Figs. 8共a兲–8共c兲, indicates that there is a marked increase in bicoherence immediately prior to and just after the L – H transition in DIII–D. Similar observations occur in the bicoherence computed from floating potential fluctuations. Furthermore, the increases die away later in time as the pressure gradient builds up in the H mode. Thus there appears to be an increase in the degree of nonlinear turbulence interactions during the L – H transition. Results taken just outside the LCFS on open field lines show
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FIG. 7. 共a兲 Probe position history, 共b兲 raw floating potential evolution, and 共c兲 raw ion saturation current evolution during an L – H transition in DIII–D. An L – H transition occurs at 1522.6 ms.
no such signature during the transition. Thus the increased coupling appears to be localized to the strong Er and strong shear region. Comparing these DIII–D results, taken with the heating power P heat well above the L – H transition power threshold P threhold , with PBX–M results obtained at P threhold suggests that the strength of this interaction may depend upon the quantity ( P heat⫺ P threhold)/ P threshold . A more detailed discussion of these new DIII–D results, along with a comparison with more global L – H transition diagnostics, will be published elsewhere.26 We note that our estimated auto-bicoherence is well above the statistical significance level, but that, as suggested by published work,27 the absolute value of our auto-bicoherence estimator has probably not converged due to the short record lengths available. We have taken care to use identical record lengths and number of realizations for all of the work discussed in this paper to ensure that we can reliably estimate relative changes in the auto-bicoherence. Note also that the density fluctuation signals appear to provide a clear indication of changes in threewave coupling, suggesting that other spatially localized diagnostics which measure density fluctuations might be capable of searching for similar nonlinear dynamics in the interior of hot magnetized plasmas as a possible zonal flow diagnostic. V. DISCUSSION AND CONCLUSIONS
An examination of published theory of mean-flow/ turbulence interactions indicates that sheared flow can be driven by the turbulent Reynolds stress. Unfortunately a direct study of the Reynolds stress during an L – H transition is difficult. By recasting this mechanism in terms of three-wave coupling, we have shown that the bispectrum or bicoherence can effectively permit one to study the same mechanism in a more experimentally accessible way. This conclusion is similar to those recently published by other workers.11 The
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FIG. 8. 共Color兲 Bicoherence computed from Isat data obtained from DIII–D discharges just inside the LCFS. 共a兲 Early L-mode phase 共12–15 ms before L – H transition兲, 共b兲 late L-mode phase 共0–3 ms before L – H transition兲, 共c兲 early H-mode phase 共1.8–4.8 ms after L – H transition兲, and 共d兲 late H-mode phase. The bicoherence increases substantially just before the L – H transition. The increase is sustained for a period after the L – H transition, and then dies away.
model is experimentally testable and thus provides a means to examine the underlying physics of shear-flow/turbulence interactions. This simple model captures the essential feature of the zonal flow-turbulence interactions as embodied in the previous Reynolds-stress model.3 If some small ‘‘seed’’ sheared flow is momentarily generated, an increase in the energy transfer rate from small scale to large-scale fluctuations should occur 共assuming that large-scale damping remains unchanged兲. As a result, the large-scale mode amplitude 共i.e., the sheared flow兲 would increase, thereby leading to a runaway condition. The key experimental observable of the mechanism which is thought to lead to zonal flow generation thus emerges: one should observe an increase in the inverse transfer rate of fluctuation energy with increasing flow shear. Such an increase in the inverse cascade rate should be manifested in an increase in the bispectrum 共or bicoherence兲 of the turbulence-zonal flow coupling as described in Eq. 共5兲. In a turbulent plasma which is dominated by the E⫻B convective nonlinearity and which exhibits a well-defined inertial wave number range, such an increase in the turbulence-flow coupling is also manifested by an increase in the bispectrum 共or bicoherence兲 coupling the linearly unstable regime to the beginning of the inertial range. Thus it seems reasonable that the most direct test of the zonal flow-turbulence interaction
model is to examine the effect of flow shear on turbulent bispectra and on energy cascade rates. We have performed such an initial study using archived edge turbulence data from H-mode transitions in the CCT, PBX–M, and DIII–D tokamak devices. The results appear to be qualitatively consistent with the expectations of the Reynolds-stress driven shear-flow hypothesis. In particular in all three devices an increase in the bicoherence, corresponding to increased nonlinear interactions across the spectrum, is observed to occur in the H mode. The results indicate that these changes are localized to the region of strong Er and strong shear flow, and are also temporally localized to the L – H transition region. We emphasize that these results do not prove the hypothesis of turbulence-generated shear flow—they are merely consistent with it. A clear proof of the hypothesis requires additional work. First, a measurement of the direction of energy transfer 共i.e., towards small scales or towards large scales兲 via three-wave coupling, possibly using existing techniques,14,16 is needed. Second, during the formation of the shear flow the rate of inverse energy transfer should increase, and this increase should be of sufficient magnitude to overcome flow damping. Third, these changes should be spatially localized to the region of shear-flow generation. The resolution of the energy transfer rate and direction using es-
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Phys. Plasmas, Vol. 8, No. 6, June 2001
tablished techniques may also permit one to distinguish between predicted changes in phase-coherent mode coupling 共as evidenced in the bispectrum and bicoherence兲 and predicted changes in the coupling coefficient. Other workers have reported increases in the bispectrum and/or bicoherence after the L – H transition.28 However, these papers have only reported an increase in nonlinear coupling after the transition is completed—to our knowledge our results are the first to report changes in coupling during the transition itself. In addition our results provide additional details on the frequency scales involved in the increase in three-wave interactions, and they provide additional insight into the spatial localization of this effect. Clearly this single field model must be extended to the nonadiabatic case 共i.e., with separate fluctuating density and potential fields兲, and to plasmas with finite electron temperature fluctuations in order to be compared with real experimental data. A theory is needed in which other nonlinearities 共i.e., the polarization drift兲 are included and in which the shear flow and turbulence scales are not widely separated is also needed. The admittedly crude ad hoc closure scheme used here also should be improved. In addition, the more complicated evolution of plasma rotation in toroidal devices during the L – H transition should be incorporated into a more complete model of evolution of the bispectrum and bicoherence due to three-wave turbulence–zonal flow interactions. It would also appear that an examination of these nonlinear turbulent dynamics would be ideally suited for an experiment in which the turbulence drive, the flow shear, and the magnetic shear can be independently controlled, and in which the effect of shear flow on nonlinear turbulence dynamics can then be studied would permit a more fundamental test of the shear decorrelation hypothesis. Such smallscale experiments should be coupled with similar studies of the turbulent energy transfer dynamics in fusion devices which exhibit spontaneous transport barrier formation, and with identical studies of mode coupling in direct numerical simulations of self-generated shear flows. Furthermore, since the underlying dynamics are thought to be identical, similar studies could also be performed on rotating fluid experiments which undergo mean-flow generation from turbulence. An observation of similar mode-coupling mechanisms in plasma and rotating fluid systems would then unite the nonlinear dynamics and provide useful scientific insights into the process of mean flow generation in turbulent 2D fluid systems. ACKNOWLEDGMENTS
The authors wish to acknowledge valuable discussions with P. H. Diamond and S. Krasheninnikov. C.H. performed
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this research under an appointment to the Fusion Energy Sciences Fellowship Program, administered by Oak Ridge Institute for Science and Education under a contract between the U.S. Department of Energy and the Oak Ridge Associated Universities.
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