J.
6 (2004)
J. Plasma
Plasma Fusion
FusionRes.
Res.SERIES,SERIES,Vol.
Vol.
6 (2004)74–79
000–000

Excitation of Zonal Flows and Fluid Closure
WEILAND Jan, DASTGEER Sheikh1, MOESTAM Robert, HOLOD Igor and GUPTA Sangeeta2
Chalmers University of Technology, EURATOM-VR Association, Gothenburg, S-41296, Sweden
1
University of California, Riverside, CA 92521, USA
2
University of Wisconsin, Madison, WI 53706, USA(Received: 9 December 2003 / Accepted: 23 February 2004)

Abstract
The excitation of zonal flows by ion temperature gradient driven modes is studied by analytical and numericalmethods using a reactive advanced fluid model. It is shown analytically that the dominant nonlinearity is the convectionin the energy equation. The excitation of zonal flows is particularly strong just above linear marginal stability leading
to a nonlinear upshift in the critical gradient for steady transport. Turbulence simulations confirm the overall
expectations and a nonlinear upshift in agreement with the Dimits nonlinear particle simulations is obtained. The
strong excitation of zonal flows just above marginal linear stability is due to the fluid resonance in the energy equation.
This is exactly where the fluid closure is made so the excitation of zonal flows depends sensitively on this closure.
Keywords:
fluid closure, zonal flows, drift waves, turbulence, transport

1. Introduction

effects of zonal flows. One is the already mentioned nonlinear
upshift.
Another is the damping of modes with long radial wave
lengths. Since the E ¥ B nonlinearity tends to isotropize
turbulence, a damping of eddies with long radial size meansthat the zonal flow acts as a general sink for long wavelengths. This sink can be caused by sheared poloidal flows in
general i.e. generated either externally or by the turbulence.
Thus, in codes without external sources for the flow, thenonlinearly generated zonal flow will be essential for damping
out long wavelengths thus generating an absorbing boundary
in this limit. Since a reflecting boundary for long wavelengths
increases the turbulence level strongly, this can introduce avery strong sensitivity of the transport level to the flow. Since,
however, long wavelengths are more easily torn apart by theflow, having an absorbing boundary for long wavelengths ismuch easier for a large radial box size. Since the effective
radial box size in the experiments is quite large, an absorbing
boundary for long wavelenths will usually be a realistic
assumption. The other aspect, i.e. the nonlinear upshift, isonly important sufficiently close to the critical gradient. In
the Cyclone case [4] the nonlinear upshift was 50 % but theexperiment from which the data were taken (D-III-D 81499)
had a gradient which was 73 % above the linear criticalgradient at half radius. Thus the experiment was well above
the nonlinear upshift and there our fluid model, without
nonlinear upshift, was in reasonably good agreement withDimits gyrokinetic simulation. This situation can be expected

One of the most important aspects of turbulent systems
is the presence of background flows [1-7]. Such flows may
be due to pressure difference as in pipe flow, rotating planets
or, in plasmas, neoclassical effects [8], pressure gradients orexternal excitation by beams. However, there can also be flow
generated by the turbulence itself [1-7]. Such flows are
generally parallel (streamers) or perpendicular (zonal flows)
to the background pressure gradient. We will here focus on
zonal flows which tend to reduce transport and thus have aregulating effect on it. The importance of flows for tokamaktransport has been recognized relatively late. The beginning
actually was in connection with the L to H mode transition
where most models involve the effects of radial electric fields
which generate rotation at the edge giving an edge transportbarrier. Later flows were also used to explain internaltransport barriers. Recently nonlinear gyrokinetic simulations
have shown strong effects of zonal flows. In particular Dimits[4] discovered a nonlinear upshift in the critical ion
temperature gradient for the onset of steady ion thermaltransport and some simulations indicate that flows have ageneral tendency to regulate transport. For very shortwavelengths ETG (electron temperature gradient) modes,
streamers were found to be very important in nonlinear
gyrokinetic simulations [9]. As will be argued in this paper,
kinetic resonances are playing an important role for the
nonlinear upshift in kinetic simulations. So how is it possible
for fluid drift wave models without zonal flows to describe
tokamak transport? First we need to distinguish two separate©2004 by The Japan Society of Plasma
Science and Nuclear Fusion Research

Corresponding author’s e-mail: elfjw@elmagn.chalmers.se
174

Weiland J. et al., Excitation of Zonal Flows and Fluid Closure

3. The fluid closure

to occur generally for collisionless plasmas with strong
heating. Thus a fluid model which assumes absorbing
boundary for long wavelengths should be applicable for mostpresent day tokamak experiments in steady state.
We will in the present paper analyze the effects of zonal
flows in kinetic and fluid simulations. In particular the effectsof the fluid and kinetic resonances turn out to be important.
This will naturally connect to the questions of the fluid
closure which will be discussed in detail.

For comparison with nonlinear kinetic theory in generaland, in particular, for the nonlinear upshift, it is important toanalyze the relations between the fluid and kinetic resonances.
When used in combination with the ion continuity equationand the usual low frequency fluid drifts, the temperature
perturbation eq. (3) leads to a three pole density response in
the presence of parallel ion motion. If we include higher ordermoments, we get a higher order density response in terms ofpoles (fluid resonances). In general we have a density
response of the type:

2. Fluid model
The main part of this work will be based on the reactive
fluid model first applied to ITG (ion temperature gradient)transport in [10]. It uses the collisionless parts of theBraghinskii fluid model where the argument for the closure
is nonlinear effects in velocity space rather than collisions.
This closure has been discussed in several papers and in thebook [11]. Also a collisionless derivation has been made foranisotropic temperatures [12]. For isotropic temperatures wehave the collisionless energy equation:dnw - w * + ...
ef= w *a nn(w - a1w D )(w - a 2w D )....(w - a nw D )(w - k|| vth ) Te

(4a)
for a fluid response of order n + 1. Such a response can also
be written in the form:dn=nw - b jw * efÂw -a w T
j

D

(4b)

e

(1a)
by means of splitting into partial fractions. Now eq. (4b) isof the form of the response due to a sum of cold beams. It
was shown by Dawson [16] that the usual Landau damping
is recovered when such a sum is extended to infinity when
the cold beams correspond to a discretization of a Maxwellian
velocity distribution. Thus we conclude that the kinetic
resonance will emerge if we go to infinite order n in eq. (4a).
The kinetic resonance will give an imaginary part corresponding to linear drift resonances and Landau-damping. The
imaginary part is, thus, expected to be the main addition
obtained by going to infinite order. Gyro-Landau models[17,18] thus include a complex heat flow. The simplest generalization of eq. (1b) was obtained in [18]. There the closure
was made by adding a complex heat flux to eq. (1b) andmatching this to linear kinetic theory. Basically a gyrofluid
model is obtained by the replacement

where

(1b)
We note that thermal force effects are collisional although
they appear as reactive in the Braghinskii formulas. Thus they
have been excluded. It is the heat flow given by eq. (1b) thatdefines the closure. It means that we can express the heatflow in the lower moments n and T which means that wehave a closed system. Another type of closure is developedin [13] (see also [14, Sec. 11]) on the basis of the stochastic
treatment in [15]. Using the relation:(2)
where V* and VD are the diamagnetic and magnetic drifts, we
obtain the temperature perturbation:

È 2 dni w *edTi
w
2 ef 
=+(hi - )  ,ÍTi
w - 5w Di / 3 Î 3 niw
3 Te 

q Æ q* + iqgL ,(5)where qgL is the Gyro-Landau resonance representing theinfluence of infinitely many higher order moments. There arealso higher order gyro-Landau models that add the gyroLandau resonance in the equation for a higher moment. It is
obvious that this gyro-Landau resonance will differ from that
added directly to the energy equation. This means that The
fluid resonances contribute a part of the full linear kinetic
resonance. Of course, once the system has been closed wecan, by eliminating higher order moments algebraically,
calculate a new qgl to be used in eq. (5). Thus the formulationin eq. (5) is quite general.
We will now briefly recall the collisionless fluid derivation [12] of eq. (1). The derivation actually leads to anisotropic
temperatures with one energy equation for the parallel temperature perturbation and one for the perpendicular. The(3)

where w* and wD are diamagnetic and magnetic drift frequencies. Equation (3) is comparatively simple but implicit due tothe presence of the density perturbation. For Boltzmann
electrons, however, eq. (3) directly gives a simple relation
between temperature and potential perturbations. We will here
only consider this case.
Equation (3) contains the fluid resonance at w = (5/3)wDi.
Since we keep it we can describe both the adiabatic regimew >> wDi and the isothermal regime w << wDi. However wepostulate that due to nonlinear effects in velocity space we
can use eq. (3) also when w @ (5/3)wDi. This is actually the”advanced” feature of the present fluid model.
275

Weiland J. et al., Excitation of Zonal Flows and Fluid Closure

nonlinear parts of these equations will contain curvature terms
that tend to isotropize the temperature perturbations. Becauseof this we will assume isotropic temperature perturbation.
However, the main ingredient in this derivation is to ignore
the irreducible part of the four velocity correlation. We will
here formally include it in the form:

source filling in a Maxwellian distribution at the typical phasevelocity of our turbulence. We note that e.g. ion cyclotron
heating will have a frequency at least two orders of magnitude
above the typical frequency of drift wave turbulence and will
not be resonant even with the drift waves with the shortest
frequencies. These will furthermore be strongly damped by
viscosity. Now deriving fluid equations from eq. (8) we willobtain a transport equation of the type eq. (7) where the external sources will not contribute to SG. We have one morecase where SG will not contribute. This is if SG averages out
on the transport time scale. This will be the case in a rather
coherent situation with particle trapping. The energy exchange
between wave and particle will then clearly average to zeroin a bounce period. As we know, the nonlinear effects will all
the time work in the direction of taking particles out of
resonance and as we can see here, there are several possible
ways in which this can happen. This leads us to the following
closure strategy: We include all moments that have sourcesÆin the experiment [11]. This means that we include also q*
since it depends only on n and T. Now the fluid resonances
associated with the moments that have sources in the experiment will be maintained due to these sources. Since theseresonances are a part of the full kinetic resonance, this meansthat the velocity distribution does not have to become completely flat even in the quasilinear limit when G relaxes tozero. Another relevant aspect is that lower order moments
sometimes can give sources for higher order moments. An
example is Ohmic heating where a current gives a heatsource. We note, however, that this is a completely random
process. Such a process would not give a source for G. A
further support for this point is also that the closure G = 0
was obtained for a semiconductor plasma by using theentropy principle for an isolated system [19]. Furthermore
collisions gave a source only for the reducible part of G. Thisagain focuses interest on the external sources. The generalclosure problem for driven systems is, of course, quite complicated. It is, however, possible to study idealized models ofsuch systems. The most simple case is actually to replace thesources by fixed gradients. In steady state the sources ofdensity and temperature will exactly balance the transport insuch a way that the gradients became stationary. The realsources, may in general, also have other effects. Thus fixing
the gradients is equivalent to applying ideal sources whichhave no other influence than to maintain the gradients. An
example of such a study was the Cyclone project [4].

< vi v j vk vl > = < vi v j >< vk vl >+ < vi vk >< v j vl > +..... + G ,where

G = < vi v j vk vl > irr .

(6)This means that G is the part of the four velocity correlationthat can not be reduced to products of two velocity correlations. G is, in fact, the first higher order fluid moment thathas been omitted from our model. Just as for any other fluid
moment there must be a transport equation for it. We write itin the form:G 
= ( c G )G + SG ,
t r
r

(7)

where cG is the effective transport coefficient for G and SG isthe source.
Our argument for closure is that there will be no source(SG = 0) in eq. (7) if the heat source is not resonant with the
turbulent fluctuations we consider. The velocity space pictureis that nonlinear effects will continuously take particles outof resonance. For comparison we note that for modes driven
externally by velocity space resonances, the source will beresonant with the fluctuation (e.g. for Fishbone or TAE
modes). In this case there will be a source in velocity space
which will continuously fill in resonant particles.
We can write the drift kinetic equation in the form:(8)
Here Sv is the source in velocity space and qc is the charge.
Such a source will always be present in connection withheating and/or current drive. For perturbations, we are usuallyconsidering eq. (8) in (w, k) space. The velocity distribution
would then relax towards a state where the source exactly
balances the nonlinear velocity space terms for a resonant
external source. If, however, S v does not have Fouriercomponents in the (w, k) regime of the considered turbulence,
it will not contribute. Thus we can introduce the term
“correlated source” for the case when fluctuations are driven
by the source in velocity space and “uncorrelated source” forthe case when Sv does not have Fourier components in theregime of our turbulence. For the background distributionfunction we furthermore assume that there will not be a

4. Zonal flows
As it turns out, the description of the nonlinear (Dimits)upshift, which is an effect of zonal flows, puts exceptional
requirements on the fluid closure. The reason for this is thatthe Dimits upshift is caused by the convective nonlinearity inthe energy equation which is sensitive to the fluid resonance.
Actually the resonance in the energy equation is exactly at
marginal linear stability in the strong ballooning limit. It is
given by [20]:376Weiland J. et al., Excitation of Zonal Flows and Fluid Closure

5w = w Di ,3hith =(9a)2 10+ e n ; e n = 2 Ln / R.
3 3

(9b)

Since there is a pole in the temperature perturbation at marginal stability, it will be large just above marginal stability.
ÆThen also the convective nonlinearity vE · —T will be large
and there will be a strong nonlinear excitation of zonal flow.
We have calculated the nonlinear coupling factor for thezonal flow, using two different orderings in the Taniuti
reductive perturbation method [21]. We consider only selfinteraction of a mode at the correlation length. Nonlinear
simulations have shown that this is also the linearly fastest
growing mode [22]. With a standing wave structure in theradial (x) direction with mode number km and propagating
wave in the poloidal (y) direction we can write the(2)dimensionless (normalized by T/e) potential F0 of the flowas [6,7]:

F 0( 2 ) = km Lk y2 r s 2 T f1(1)2

sin 2 km x .

Fig. 1dotted line is the quasilinear result [4] and the full line is thenonlinear simulation from [7]. Our simulations also confirmed
the detuning of zonal flows by magnetic shear predicted by
our analytic calculations [6]. This trend, i.e. an increase oftransport with magnetic shear, is also in agreement withsimulations of ETG turbulence [9] as it seems for similarreasons.
It is very interesting to compare the above results for
excitation of zonal flows in a fluid model with the corresponding kinetic results. The main paper to compare with ishere the Cyclone paper [4] where particle collisions were notincluded. As pointed out above, there is complete agreementwith the nonlinear upshift between our fluid simulation [7](Fig. 1) and Dimits simulation. The upshift in the kinetic
simulation was, however, sensitive to the convergence withregard to number of particles. Thus only half the upshift was
obtained with 4 million particles while all simulations with 8
million particles or more got the published upshift. The
maximum number of particles was 134 million particles. It
has recently been pointed out [24,25] that it is the resonant
regimes in phase space that are most sensitive to the convergence with regard to number of particles and that a particle
loading with more particles in the regimes of wave-particle
resonances reduces the total number of particles needed. Since
convergence with regard to number of particles was obtainedin [4] above R/Lt = 7 (experimental gradient) also with 4
million particles we conclude that wave particle resonances
are important in the nonlinear upshift regime and more
specifically: the wave particle resonance is important for
obtaining the correct nonlinear upshift. Reducing the numberof particles in the resonant regime decreases the nonlinear
upshift. Thus we may conclude that the nonlinear upshift isdue to the kinetic resonance in kinetic simulations. As wehave already seen, the nonlinear upshift is due to the fluid
resonance in an advanced fluid simulation. We then conclude
that obtaining the correct nonlinear upshift in a fluid
simulation is a sensitive test on the closure.

(10)

Here f1(1) is the dimensionless drift wave potential, here assumed to be of order e, L is the system size in x and T is thenonlinear coupling factor. It contains the denominator of eq.
(3) in the denominator but also a factor w (1 + ky2 rs2) – w L
where w L = w r – igd, w r is the real part of the local eigenfre5 en s
) w * is the magnetic shear damping
quency and gd = (1 + —3t 4 qs(s is the usual shear parameter, qs the safety factor and t =Te/Ti). Thus we have a resonance which can be detuned bothby linear instability and by magnetic shear damping. Boththese features show up in nonlinear turbulence simulations[7].
The resonant ordering gives a Zakharov like system
where, however, the low frequency equation has only first
derivatives. It is [23]: (compare also the resonant ordering in
[21])
(1) 2

 f1
F 0( 2 )
F 0( 2 )
- cf
= cn
t
x
x

.

The nonlinear upshift shown as the difference between
threshold for transport in a quasilinear model and anonlinear simulation including zonal flow. Parametersfrom ref. [7] corresponding to data from ref. [4].

(11)

We note that eq. (11) becomes of the same form as eq. (10)in the quasistationary case when the time derivative can beignored. We can then integrate eq. (11) with respect to x. A
resonance of the same type as in eq. (10) is thus also present
in eq. (11). The behaviour of the couplingfactor cn at marginal
linear stability was studied in [23] where also a gyrofluid
resonance was included in one case for comparison. As
expected, the gyrofluid term had a significant detuning effecton the resonance giving a much smaller |cn| than with ourreactive model. This seems to be the reason why the IFSPPPL model got only half the kinetic nonlinear upshift in the
Cyclone simulations [4] while recent simulations [7] with ourmodel gave the same nonlinear upshift as Dimits nonlinear
kinetic simulations [4]. This is shown in Fig. 1 where the477Weiland J. et al., Excitation of Zonal Flows and Fluid Closure

5. Collisions

the closure can be studied and should be understood first. The
nonlinear upshift due to zonal flows was found to be due tothe kinetic resonance in nonlinear kinetic simulations and dueto the fluid resonance in a fluid simulation. The comparisonof these simulations is thus a sensitive test of the fluid closure.
Concerning general influence of zonal flows in experiments, it seems that a situation with absorbing boundaries in
k-space and gradients above the nonlinear upshift region ismost likely. In transient experiments [32] with time varying
heating, however, the nonlinear upshift may become more
important, in particular for stiffness, since the profiles may
transiently enter the nonlinear upshift regime.

As pointed out above, the Cyclone simulations had ideal
sources and no collisions. This means ideal conditions toapply our fluid closure (the present closure would work also
in a collision dominated case but this is not realistic for
application to tokamak cores). A remaining question is that
turbulence which cascades to smaller and smaller scales cancause an apparent effective dissipation. We can describe thisas an additional diffusion acting in a similar way as viscosity.
Such an effect was recently added to the Mattor Parker
system. Mattor and Parker [26] introduced a nonlinear closure
into a system of three interacting slab ITG modes. The nonlinear closure essentially meant including a nonlinear frequency shift into the plasma dispersion function. The solution
was very similar to the usual three wave interaction but theclosure gave a slight reduction in the nonlinear oscillations.
The addition of diffusion to this system gave a slow decreasein time of the oscillations indicating relaxation to a nonlinear
equilibrium [27]. Qualitatively the time development became
similar to that in the Cyclone simulations. The final transportlevel can be considerably lower than for the gyrofluid,
Hammet-Perkins closure [18] just as the Cyclone simulations
gave an asymptotic transport level that was considerably
lower than that obtained in the gyrofluid simulations.
In real systems we will also have effects of electron-ion
collisions. Here an important aspect is the result in [19] thatcollisions only give a source for the reducible part of G. It isalso very instructive to read ref. [28] where the beam-plasma
instability was simulated using a particle code. There the only
observable effect of collisions was to damp the wave. Since
drift waves are maintained by gradients in real space thiswould be of no consequence as long as the turbulence levelis not strongly modified. We recall that ITG modes are fairly
insensitive to collisions. The weak nature of the turbulencein [28] is not an important limitation here. In fact nonlinear
detuning of wave particle resonances are likely to be stronger
in strong turbulence [29,30]. A direct comparison between
collision frequency and trapping frequency or inverse time ofquasilinear flattening also shows that effects of collisions
would be subdominant for typical drift wave saturation amplitudes in core tokamak plasmas [31]. Another aspect ofcollisions is that they can damp zonal flows in the nonlinear
upshift regime [5].

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6. Discussion
We have here discussed recent developments in fluid
modelling and compared to nonlinear kinetic results. In
particular the fact that fluid resonances form part of the
kinetic resonance is very important in understanding how areactive fluid model can be in agreement with a nonlinear
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578

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