PHYSICS OF PLASMAS 18, 055709 (2011)
Numerical simulation and analysis of plasma turbulence the Large Plasma
Devicea)
M. V. Umansky,1,b) P. Popovich,2 T. A. Carter,2 B. Friedman,2 and W. M. Nevins1
1
Lawrence Livermore National Laboratory, Livermore, California 94550, USA
Department of Physics and Astronomy and Center for Multiscale Plasma Dynamics, University of California,
Los Angeles, California 90095-1547, USA
2
(Received 28 November 2010; accepted 26 January 2011; published online 4 May 2011)
Turbulence calculations with a 3D collisional fluid plasma model demonstrate qualitative and
semi-quantitative similarity to experimental data in the Large Plasma Device [W. Gekelman et al.,
Rev. Sci. Inst. 62, 2875 (1991)], in particular for the temporal spectra, fluctuations amplitude,
spatial correlation length, and radial particle flux. Several experimentally observed features
of plasma turbulence are qualitatively reproduced, and quantitative agreement is achieved at
the order-of-magnitude level. The calculated turbulence fluctuations have non-Gaussian
characteristics, however the radial flux of plasma density is consistent with Bohm diffusion.
Electric polarization of density blobs does not appear to play a significant role in the studied case.
Turbulence spectra exhibit direct and inverse cascades in both azimuthal and axial wavenumbers
C 2011 American
and indicate coupling between the drift instability and Kelvin-Helmholtz mode. V
Institute of Physics. [doi:10.1063/1.3567033]
I. INTRODUCTION
Plasma turbulence is an important phenomenon for magnetic fusion energy applications and beyond. In particular,
designing magnetic fusion reactors could be done more efficiently if computational models were available with the
capability to predict characteristics of plasma turbulence for
future devices. Certainly, such models, to be trusted to do
any predictions, first of all should be able to reproduce observations in existing experiments. This makes validation of
turbulence models an important area in present-day fusion
energy science.
Linear plasma devices such as the Large Plasma Device
(LAPD),1 Controlled Shear Decorrelation Experiment
(CSDX),2 Versatile Instrument for studies on Nonlinearity,
Electromagnetism,
Turbulence
and
Applications
(VINETA),3 Large Mirror Device (LMD),4 Helicon-Cathode
Dual-Source Basic Plasma Physics Device (HelCat),5 and
Mirabelle6 offer an opportunity to validate turbulence and
transport simulations in simple geometry and with boundary
conditions and plasma parameters with reasonable relevance
to tokamak edge and scrape-off-layer plasmas. Thanks to
their low temperature, these devices are highly diagnosable,
providing for detailed comparison against code predictions.
A relatively new aspect of linear devices is using them
for validation of fusion plasma modeling codes. This in part
motivated several recent studies where plasma turbulence
codes using fluid (collisional or gyro-fluid) models were
applied to linear devices.7–11 These previous studies were
largely successful and revealed important results. For example, the importance of ion-neutral collisions for turbulence in
LMD was demonstrated through simulation studies;7 and
simulations were able to reproduce some of the main experia)
Paper GI3 4, Bull. Am. Phys. Soc. 55, 109 (2010).
Invited speaker. Electronic mail: umansky1@llnl.gov
mental features of formation and propagation of turbulent
structures in the VINETA device.9–11
LAPD is a linear device in which a hot cathode discharge is used to produce a plasma on a solenoidal magnetic
field (B 0:1 T). The LAPD plasma is 17 m long and has
a radius of r 0:3 m (vacuum chamber radius is 0.5 m).
Characteristic density is 2:5 1012 cm3 , electron temperature 5 eV, and ion temperature under 1 eV.
LAPD plasma lies well within the domain of validity of
the collisional fluid plasma model, at least for spatial and
temporal scales associated with drift turbulence studied here.
Here the figures of merit are k? qci , kei =Ljj , x=mei , where
k? ¼ 2p=k? and x ¼ 1=s are the characteristic perpendicular wavenumber and frequency of the turbulence. For validity of the collisional fluid model all these dimensionless
parameters should be substantially smaller than one. It is
instructive to compare these characteristic parameters for
LAPD and edge plasmas in some major tokamak experiments, which are summarized in Table I. The parameters
used in Table I for LAPD are k? m=a 0:3 cm1 for
characteristic azimuthal number m 10; and the reciprocal
time scale, x x 4 104 rad=s.
In Table I for tokamak edge turbulence characteristic values k? ¼ 1cm, x ¼ 1MHz are assumed; and the tokamak
edge plasma parameters used are, for C-Mod:
ne ¼ 5 1013 cm3 , Te; i ¼ 40 eV, Ljj ¼ 300 cm; for DIII-D:
TABLE I. Figures of merit for validity of collisional plasma fluid model.
DIII-D
C-Mod
NSTX
LAPD
k? qci
kei =Ljj
x=mei
1.0
0.1
1.0
0.1
1.0
0.1
0.1
0.01
1.0
0.1
0.1
0.01
b)
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18, 055709-1
C 2011 American Institute of Physics
V
055709-2
Umansky et al.
Phys. Plasmas 18, 055709 (2011)
ne ¼ 1 1013 cm3 , Te; i ¼ 100 eV, Ljj ¼ 800 cm; for NSTX:
ne ¼ 5 1012 cm3 , Te; i ¼ 20 eV, Ljj ¼ 500 cm.
As can be seen in Table I, for some tokamak edge
plasmas applicability of the collisional fluid model may be
questionable; however for drift turbulence in LAPD, the
application of a fluid model seems reasonable. LAPD therefore provides a unique test bed for the validation of simulation based on fluid models.
The BOUT code was developed in the late 1990s for
modeling edge turbulence in tokamaks.12 BOUT solves for
the time evolution of a set of plasma fluid variables in realistic 3D divertor tokamak geometry. The version of the code
used for the present simulations was developed from the
original BOUT including algorithmic improvements and verification testing.13 For the purposes of LAPD modeling the
code had to be slightly modified for cylindrical geometry.14
II. PHYSICS MODEL
The BOUT physics model is based on a set of fluid
equations that represent conservation of density, electron and
ion momentum and electric charge14
ð@t þ ve rÞ n ¼ 0;
1
nme ð@t þ ve rÞ ve ¼ rpe ne E þ ve B
c
(1)
nme mei ðve vi Þ nme men ve ;
(2)
1
nmi ð@t þ vi rÞ vi ¼ ne E þ vi B nmi min vi ;
c
r J ¼ 0;
J ¼ enðvik vek Þ þ enðvi? ve? Þ;
(3)
(4)
where pe ¼ nkB Te . A friction term due to ion-neutral collisions (elastic and charge-exchange) is included in the ion
momentum equation. All terms involving finite ion temperature effects are neglected. The friction forces in the electron
momentum equation are due to electron-ion (mei ) and electron-neutral collisions (men ). However, as Coulomb collisions
are dominant for the electrons (mei men ), electron-neutral
collisions are ignored.
The following simplifying assumptions are made, which
are relevant for LAPD plasma parameters: constant magnetic
field B ¼ B0 z, cold ions tke tki , Te Ti , and no background parallel flows; electron temperature fluctuations are
neglected.
Following the assumptions and procedure standard for
derivations of low-frequency plasma turbulence models15–17
one can come up with the following set of equations equivalent to Eqs. (1)–(4); these are the model equations that are
actually solved in BOUT:14
@t N ¼ vE rN rk ðvke NÞ;
@t vke ¼ vE rvke l
Te0
rk N þ lrk / me vke ;
N0
(5)
(6)
@t - ¼ vE r- rk ðNvke Þ þ b rN rv2E =2 min -;
(7)
where the potential vorticity is defined as
def
- ¼ r? ðNr? /Þ:
(8)
A viscosity term is added to introduce dissipation for nonlinear calculations.18 All the quantities here are normalized
using the Bohm convention, i.e., density, temperature and
magnetic field are normalized to reference values nx , Tex (the
maximum of the corresponding equilibrium profiles), and
B0 , the axial magnetic field. Frequencies and time derivatives are normalized to Xcix ¼ eB0 =mi c: @^t ¼ @t =Xcix ,
^ ¼ x=Xcix ; pvelocities
x
ffiffiffiffiffiffiffiffiffiffiffiffiffi are normalized to the ion sound
speed Csx ¼ Tex =mi ; lengths – to the ion sound gyroradius
qsx ¼ Csx =Xcix ; electrostatic potential to the reference electron temperature: /^ ¼ e/=Tex .
The equations are solved in a cylindrical annular domain,
with the inner and outer radii 15 cm and 45 cm. The axial
domain size is taken 17 m, matching the length of LAPD. The
azimuthal size is taken either full 2p, or a fraction of it, i.e., p
or p=4, neglecting low azimuthal modes for a faster calculation. Boundary conditions on the azimuthal and axial boundaries are taken periodic; on the radial boundaries the boundary
conditions enforce zero radial flux of plasma density. The radial density profile ni0 ðrÞ is fitted to experimental LAPD profile. The electron temperature is taken constant, 5 eV, across
the domain; the ion temperature is taken zero.
III. LINEAR ANALYSIS
The linear dispersion relation is obtained by linearizing
Eqs. (5)–(7) (see details in Ref. 14), and assuming solutions
of the linearized equations sought in the form f ðxÞ ¼ f ðrÞ
expðimh þ ikk z ixtÞ, where ðr; h; zÞ are the cylindrical
coordinates; kz ¼ 2pnz =Lk is the parallel (axial) wavenumber, nz is the axial mode number. Denoting f 0 ¼ @r f and
~¼x
introducing the Doppler-shifted frequency x
ðmh =rÞ/00 , the 1D equation for radial eigenfunctions of the
perturbed potential uðrÞ is derived as
C2 ðrÞ/00 þ C1 ðrÞ/0 þ C0 ðrÞ/ ¼ 0;
(9)
~ are functions of equilibrium
where the coefficients Ci ðr; xÞ
~ (full expressions for Ci are presented in
quantities and of x
Ref. 14).
Equation (9) is a 2nd order ordinary differential equation
(ODE) in r. Supplemented with homogeneous boundary conditions on the radial boundaries it forms an eigenvalue prob~ that enters the coefficients
lem, where the eigenvalue is x
~
Ci ðr; xÞ.
From the linear dispersion relation it is found that in this
system there are two classes of modes: those with kz 6¼ 0 and
kz ¼ 0. The former are driven by the radial pressure gradient
and they can be recognized as the resistive drift wave instability (DW). The flute-like modes ðkz ¼ 0Þ are driven by the
/0 terms, i.e., the plasma azimuthal rotation. They are either
the rotation-driven interchange modes (IC) which are destabilized by uniform rotation due to the centrifugal force; or
the Kelvin-Helmholtz instability (KH) destabilized by nonuniform rotation. Of course, these individual instability
branches can be separated only in some limiting cases, in
055709-3
Numerical simulation and analysis of plasma turbulence in LAPD
general the solutions of the linear dispersion relation are
hybrids of all three modes.
A reduced model containing only the IC mode is
obtained by simplifying Eqs. (5)–(7) to the small subset
where kz is set 0 and the neutral damping and E B terms
are dropped from the vorticity equation
@t N ¼ vE rN;
(10)
@t - ¼ b rN rv2E =2:
(11)
The destabilizing term is the right-hand side of the vorticity
equation (11), which is formally equivalent to the magnetic
field curvature term in the full two-fluid model.12 For uniform rotation with angular velocity Xrot it can be written as
b rN Fc , where Fc ¼ X2rot r is the centrifugal force per
unit mass. A general volumetric radial force Fr can also
drive the interchange instability; in this case the mechanism
is charge separation by the F B drift, and the driving term
appears in the vorticity equation as b rN ðFr =mNÞ.
The KH mode is obtained from the general model, Eq.
(7) by setting kz ¼ 0 and eliminating the centrifugal force
term v2E driving the IC mode. The resulting linear dispersion
relation is compared with BOUT solution of the corresponding initial value problem
@t - ¼ vE r-;
(12)
where vE and - contain the stationary background and timedependent perturbation parts, vE ¼ vE0 ðr; h; zÞ þ vE1
ðr; h; z; tÞ, - ¼ -0 ðr; h; zÞ þ -1 ðr; h; z; tÞ.
The linear dispersion relation has been used for detailed
BOUT verification.14 An example of verification calculation
for the KH mode in sinusoidal /0 field is shown in Fig. 1
where analytic solution from solving the eigenvalue problem, Eq. (9), is shown by lines and BOUT data points are
shown by circles.
An important conclusion from the linear analysis is that
for LAPD conditions for the azimuthal mode numbers 10
where turbulence appears to be strongest all three modes
Phys. Plasmas 18, 055709 (2011)
(DW, IC, and KH) can have comparable growth rates and
thus all three physical mechanisms are potentially important
for turbulence in LAPD.14
IV. NONLINEAR CALCULATIONS
For the nonlinear calculations Eqs. (5)–(8) are used.
Time integration of the dynamic equations with BOUT
results in exponential growth of linear unstable modes and
then comes to nonlinear saturation and steady-state turbulence. For plasma density perturbation n~i the axisymmetric
component m ¼ 0 is fully or partially suppressed to maintain
the axisymmetric density profile ni0 close to the time-average
profile in the experiment. This can be justified a posteriori
by noticing that the variation of the m ¼ 0 component of density is slow compared to the characteristic turbulence timescale.
For the electric potential perturbations /^ the turbulence
develops a substantial m ¼ 0 component which results in
non-uniform poloidal rotation; this is further referred to as
the zonal flow.
The equation describing dynamics of this zonal flow
component can be derived from the vorticity equation by
averaging it over the azimuthal angle18
1
r? / 2
min ðNVh Þ l@r ð-Þ:
N@h
@t ðNVh Þ ¼ ð-Vr Þ þ
2
r
(13)
In the right-hand side of Eq. (13) the first two terms correspond to the Reynolds stress, the other two describe the neutral drag and viscous damping.
One can estimate the ratio of the neutral drag term to the
viscous damping term in Eq. (13) as
min ðNVh Þ
min =mii
:
l@r ð-Þ
ðqci =aÞ2
(14)
For LAPD, using min ¼ 1e3 1=s, mii ¼ 5e5 1=s, qci ¼ 0:2 cm,
a ¼ 30 cm, we find the ratio is 45, which points to the
importance of neutral drag for LAPD turbulence.
V. COMPARISON WITH LAPD DATA
Detailed results of the initial comparison of BOUT simulations with LAPD data are presented by Popovich et al.18
In summary, the model agrees with experimental data qualitatively and semi-quantitatively. Several qualitative features
are in agreement:
(i)
FIG. 1. (Color online) Eigenvalue solution and BOUT results for KH instability in sinusoidal potential.
In both the experiment and simulations the power spectrum of fluctuations is exponential,19 SðfÞ / expðaf Þ,
for frequencies higher than 5 kHz. However the
exponent a is smaller in the simulation than in LAPD
data by a factor 2. The neutral drag term min appears
to be very important for matching the temporal spectra.
If the neutral density nN is taken much smaller than the
experimental value than the power spectrum indicates
the presence of coherent features not observed in the
experimental data. If nN is taken much higher than the
experimental value than turbulence is strongly damped.
055709-4
(ii)
(iii)
(iv)
(v)
Umansky et al.
Overall, using experimental range of nN appears to
work best for matching the power spectrum.
The spatial correlation length in the plane perpendicular to the magnetic field is on the order of magnitude
but 2–3 times larger than in the experiment.
The calculated fluctuations amplitude n~rms ðrÞ as a
function of radial coordinate has form similar to the
experimental data - peaking at r 25 cm and similarly decaying inward and outward. However in the
simulations the fluctuations amplitude is a factor of
2 smaller than in the experimental data.
The skewness of density perturbations grows radially
in both LAPD data and BOUT results, with rather
similar values, ranging from 0.5 at r ¼ 15 cm to 1.0
at r ¼ 45 cm.
The effective plasma source inferred from the radial
particle flux in the simulations, SðrÞ ¼ r C, is negative inside of r 28 cm and positive outside. This radius, r ¼ 28 cm, matches the radius of plasma central
column in LAPD where plasma is formed by ionization of neutrals. The magnitude of the source is
within factor of 2–3 from the estimated ionization
source (inside of the central column), and volumetric
sink due to the parallel plasma streaming (outside of
the central column). The level of transport is consistent with Bohm diffusion, which has been previously
found to match LAPD data for non-biased plasmas.20
Matching the average radial electric field between the
simulation and LAPD is a separate issue. In the simulation
the azimuthal average electric field hEr ih varies slower than
turbulence eddies, although its magnitude and sign do
change with time; and the magnitude of the average potential
drop across the simulation domain is on the order of 2–3 V
(Te 5 eV).
In the experiment the time-average electric plasma
potential is measured, as shown in Fig. 2 where plasma profiles from LAPD are plotted versus the radial coordinate.
One can see in Fig. 2 that in the experiment the main
drop of the electric plasma potential occurs in the outer part of
the domain where there is little plasma density. This suggests
that in the experiment Er is set by a combination of turbulence
effects and boundary conditions, e.g., the sheath on the endplates. These boundary conditions are not included in the present model; on the other hand, the variation of the electric
plasma potential in the central part of the plasma is on the
scale of 2–3 V, which is consistent with the simulation.
The discrepancy between the model and experiment
may be due to missing certain physics in the model, in particular, temperature fluctuations and proper boundary conditions at end-plates and side-walls. Still, the overall level of
qualitative and semi-quantitative agreement lends a strong
evidence of the model’s relevance to LAPD.
VI. ANALYSIS OF COMPUTED TURBULENCE
A. Intermittency and density blobs
In LAPD data the PDFs of turbulent fluctuations in general are non-Gaussian.26 Consistent with that, in the BOUT
Phys. Plasmas 18, 055709 (2011)
FIG. 2. (Color online) Experimental LAPD profiles of average plasma density, electron temperature, and electric potential.
simulations, the PDFs of fluctuating quantities of interest plasma density, electric potential, and radial particle flux exhibit non-Gaussian features. The commonly used statistics,
such as the skewness and kurtosis of the PDF, show this
quantitatively, see Fig. 3.
Such non-Gaussian features of turbulence fluctuations
are often associated with formation and coherent radial
motion of density “blobs” (see the review paper by Krasheninnikov21 and references therein), in tokamak edge plasma
and other systems, including linear devices.
Turbulent fluctuations of density lead to formations of
localized density peaks. A distinct feature of density blobs is
that they are electrically polarized, and thus are capable of
moving through the magnetic field due to the E B
advection.
In toroidal devices, the radial motion of density blobs is
believed to be caused by polarization due to currents associated with particle drifts, in particular the curvature drift. For
linear devices with straight axial field where observations of
“blobby” transport were reported it has been proposed that
centrifugal force or “neutral wind”21,22 may lead to drift currents and cause polarization.
Indeed, consider the term b rN rv2E =2, corresponding to the centrifugal force, in the vorticity equation, Eq. (4).
For a density blob shown schematically in the plane perpendicular to the magnetic field in Fig. 4 it can be easily seen
that the centrifugal force causes electric polarization in the h
direction. This, in turn, leads to outward radial motion of the
blob due to the E B velocity. On the other hand, a radial
force of some other nature, e.g., the neutral wind, can similarly cause blob polarization and radial transport. Our model,
Eqs. (1)–(4), does not have neutral wind or any other
055709-5
Numerical simulation and analysis of plasma turbulence in LAPD
Phys. Plasmas 18, 055709 (2011)
tures. Examining BOUT results one can find both examples
of what looks like polarized density blobs, see Fig. 5(a); and
density peaks without electric polarization, see Fig. 5(b).
Blob polarization is accompanied by localized azimuthal
electric field and radial convection velocity VEr . Thus coincidence of a density peak and VEr peak is a measure of blob
polarization.
For quantitative measure of polarization we use the correlation function which is defined as
corrðX; YÞ ¼
E½ðX lX ÞðY lY Þ
;
rX rY
(15)
where the numerator is the covariance of X and Y and the denominator is the product of their standard deviations.
For each time-slice we calculate the correlation function
n; V~Er Þ is clearly
of n~ and V~Er . The correlation function corrð~
related to the radial particle flux, and therefore its time-average value is expected to be positive.
To investigate the effects of a general volumetric radial
force, such as the neutral wind, we also conduct turbulence
calculations adding a radial force term to the vorticity equation. We observe that an outward radial force increases the
correlation of n~ and V~Er (and the radial flux), compared to
the base case. A radially inward force decreases the value of
correlation function, see Fig. 6.
Removing the centrifugal term v2E from the model does
not significantly affect the value of the correlation function,
which indicates that plasma rotation, i.e., the average electric
field hEr i, in the studied case is too small to make time-average transport by polarized density blobs a significant player.
B. Spectrum in azimuthal wavenumber
FIG. 3. (Color online) PDFs of turbulent fluctuations at a particular radial
location. Also shown are values of the skewness c1 and kurtosis c2 .
volumetric radial force, however the centrifugal force is
included by the v2E term.
To investigate blob polarization in the simulated plasma
turbulence it is convenient to overlay contours of plasma
density and electric potential, to see if monopole density
structures are accompanied by dipole electric potential struc-
FIG. 4. (Color online) Schematic of blob polarization due to the centrifugal
force or a general volumetric radial force.
Density perturbations can be represented in the spectral
form
X
(16)
n~ðx; tÞ ¼
n^ expðimh þ ikz z þ ikr r ixtÞ:
Here we consider jnk j2 for steady-state turbulence, averaged
over time for the saturated turbulence state.
The first thing of interest is the dependence of the spectral amplitude on the azimuthal wavenumber kh ¼ m. For the
r and z dimensions one can use either averaging or slicing to
reduce the number of independent variables to the single
variable kh .
The plot of jnk j2 versus kh is shown in Fig. 7. Calculations results are shown for three azimuthal domain sizes, 2p,
p, p=4, the number of azimuthal grid points is 256, 128, 32,
respectively. For all three cases the results overlap well for
> 10.
kh One observation that one can draw from Fig. 7 is that
the high-frequency part of the spectrum follows a power-law
close to jnk j2 kh6 . This power-law form is an evidence of
a direct cascade to higher azimuthal wavenumbers. The
power-law has a breakpoint at kh 20, where the function
weakly grows monotonically to smaller wavenumbers. Note
that m ¼ 0 component of n~ is suppressed in this simulation
for maintaining the average density profile n0 ðrÞ, so the spectral amplitude is identically zero for m ¼ 0.
055709-6
Umansky et al.
Phys. Plasmas 18, 055709 (2011)
FIG. 5. (Color online) (a) Polarized density blob. Electric potential exhibits
dipolar structures on top of monopolar
density peak structures. (b) Unpolarized
density blob. There is no visible dipolar
electric potential structures on top of
density peaks.
The breakpoint in the spectrum in Fig. 7 is consistent
with the linear spectrum of the global drift instability for this
case, see Fig. 8. As shown in Fig. 8, the fastest-growing linear drift mode for this system has nz ¼ 1 (i.e. the fundamental axial mode), and kh 20. This fastest-growing drift
mode is the primary linear instability, driven by the gradient
of the underlying density profile ni0 ðrÞ, that pumps energy
into the system. Apparently, the energy further spreads to
higher azimuthal wavenumbers due to the direct cascade;
and to lower azimuthal wavenumbers, due to an inverse cascade. The inverse cascade is seen in the part of the curve to
the left of the breakpoint in Fig. 7.
C. Spectrum in axial wavenumber
Another characteristic of saturated turbulence is the dependence of the spectral amplitude on the axial wavenumber
kz . The r and h dimensions are eliminated by either averaging or slicing.
Figure 9 shows the plot of jnk j2 versus kz . Same data are
shown in the semi-log and log-log scale. One can observe
that jnk j2 follows a kjj4 power-law. The nz ¼ 0 mode cannot
FIG. 6. (Color online) Correlation function of fluctuating n~ and V~Er calculated as a function of time for the reference case and for cases with inward
and outward radial force.
be shown in the log-log plot; same data in the right of Fig. 9
plotted in the semi-log scale show that in the saturated turbulence state the nz ¼ 0 mode has much stronger amplitude
than the nz ¼ 1 corresponding to the fastest linearly unstable
mode in the system.
D. Time-evolution of the wavenumber spectrum
In Fig. 10 several consecutive time slices are shown
describing the time-evolution of jnk j2 in perpendicular wavenumber space. On the left side the nz ¼ 0 mode is shown and
on the right the nz ¼ 1 mode is shown. The time-evolution
starts from a small perturbation seed and in the first frame
there is a finite amplitude perturbation corresponding to the
fastest growing drift-wave mode in this system, with nz ¼ 1
and kh 25, consistent with the linear spectrum of the DW
mode in Fig. 8. In the next frame one can observe that
FIG. 7. (Color online) Spectrum in the azimuthal wave number kh . The
power-law indicating cascade to higher wavenumbers is well reproduced for
different azimuthal domain sizes.
055709-7
Numerical simulation and analysis of plasma turbulence in LAPD
Phys. Plasmas 18, 055709 (2011)
where Sðx1 ; x2 Þ is the bispectrum,
^ 1 ; k1 ÞYðx
^ 2 ; k2 Þi;
Sðx1 ; x2 Þ ¼ hY^ ðx3 ; k3 ÞYðx
FIG. 8. (Color online) Linear growth rate of DW instability as a function of
the azimuthal wavenumber for different axial wavenumbers.
subsequently a mode with nz ¼ 0 develops. The nz ¼ 0
mode appears at m 50 and m 0, suggesting that a quadratic nonlinearity links this mode to the primary drift-wave
instability with nz ¼ 1. The last frame corresponds to the saturated steady-state of turbulence, where both nz ¼ 0 and
nz ¼ 1 modes end up approximately isotropic in the kr and
kh plane.
The quadratic nonlinearity involved here apparently
points to the Reynolds stress. The physical picture emerging
here is as follows: (i) initially, the fastest DW grows with
nz ¼ 1 and m ¼ 25; (ii) next, Reynolds stress appears; (iii) this
drives modes with nz ¼ 0 and m ¼ 50 and m ¼ 0 components.
(18)
and h i stands for the ensemble-average.
The utility of bicoherence comes from the fact that it is
a measure of the strength of nonlinear coupling between considered modes.
Observations in Fig. 10 suggest that during the turbulence ramp-up a strong coupling should exist between drift
modes with nz ¼ 1 and mh 25; and flute-like mode nz ¼ 0
and mh 0. The conjecture is that this interaction should
remain significant in fully saturated turbulence. Thus for the
bicoherence calculation we consider the following three
modes in saturated turbulence spectrum: W1 ¼ ð1; 25Þ,
W2 ¼ ð1; 24Þ, and W3 ¼ ð0; 1Þ; where the first number
nz is the axial mode number, and the second one mh is the
azimuthal mode number. For the mode W3 we use m ¼ 1
rather than m ¼ 0 since the azimuthal-average component of
density perturbation is subtracted out in the BOUT calculations used here, and thus the m ¼ 0 mode is just not present.
For modes W1 and W2 the numbers nz and mh are chosen
such that these are close to the peak of the drift mode spectra
and they satisfy the wavenumber matching rules:
m1 þ m2 ¼ m3 , and nz1 þ nz2 ¼ nz3 .
The calculated bicoherence is shown in Fig. 11 as a contour plot versus the frequencies of modes x1 and x2 ; the frequency of the third mode satisfies the frequency matching
condition x1 þ x2 ¼ x3 . The strong peak of bicoherence in
the upper-left corner indicates strong coupling of the two
drift modes W1 and W2 , and a flute-like mode W3 with a
much lower frequency. This is consistent with the arguments
based on observations of turbulence ramp-up in Fig. 10.
E. Bicoherence analysis
Bicoherence analysis is a promising techniques used for
analysis of nonlinear interactions in plasma turbulence.23–25
^ 1 ; k1 Þ, Yðx
^ 2 ; k2 Þ, Yðx
^ 3 ; k3 Þ,
For three Fourier modes Yðx
satisfying the sum rules x1 þ x2 ¼ x3 , k1 þ k2 ¼ k3 , the
bicoherence is defined as
jSðx1 ; x2 Þj
bðx1 ; x2 Þ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; (17)
^ 1 ; k1 ÞYðx
^ 2 ; k2 Þj2
^ 3 ; k3 Þj2 jYðx
jYðx
VII. DISCUSSION
Turbulence simulations of LAPD are conducted with a
3D reduced collisional fluid plasma model. Our physics
model is similar to the classical Hasegawa-Wakatani (HW)
model.15,27,28 However, our model is more general: it is 3D,
i.e., retains the full parallel dynamics; and it allows for a
general ni0 profile, with large variations across the domain
(five-fold in our case).
FIG. 9. (Color online) Spectrum in the
axial wavenumber kjj . Same data are
shown in log-log scale (left) and semilog scale (right).
055709-8
Umansky et al.
Phys. Plasmas 18, 055709 (2011)
FIG. 11. (Color online) Bicoherence is plotted vs. frequencies of modes
W1 ¼ ð1; 25Þ, W2 ¼ ð1; 24Þ, where the first number ny is the axial mode
number, and the second one mh is the azimuthal mode number. The strong
peak of bicoherence in the upper-left corner indicates strong coupling of two
drift modes W1 and W2 , and a flute-like mode W3 (0,1) with a much lower
frequency.
FIG. 10. (Color online) Several consecutive time slices showing evolution
of jnk j2 in the kr and kh coordinates. On the left side the nz ¼ 0 flute-like
mode is shown, on the right the fundamental nz ¼ 1.
To maintain the axisymmetric density profile ni0 close
to the experimental LAPD profile, for plasma density perturbation n~i the axisymmetric component m¼0 is fully or partially suppressed. Estimating the time-scale of m ¼ 0
component variation as ni0 =r C where C is the simulated
radial turbulent flux, one can find that s is on the order of ms,
which is an order of magnitude larger than the characteristic
turbulence time, sturb 2p=x . Thus the variation of the
m ¼ 0 component of density is slow compared to the characteristic turbulence time-scale.
Overall, the present numerical modeling results demonstrate qualitative and semi-quantitative similarity to LAPD
data. The simulations reproduce qualitative trends in temporal spectral, spatial correlation length, fluctuation amplitude,
and turbulent radial particle flux. Quantitatively, the simula-
tion reproduces experimental data by the order-of-magnitude. It is quite possible that extending the model by adding
equations for Te and Vjji , electromagnetic terms and sheath
boundary conditions will improve even further the agreement between simulation model and experimental data.
In the simulations, the radial particle flux appears to be
close to Bohm diffusion, consistent with experimental data.
However, the PDFs of turbulent fluctuations indicate nonGaussian features and intermittency, which are often associated with the presence of coherent structures which are convectively transported.
Non-Gaussian statistics of turbulent fluctuations in
boundary plasmas is often associated with transport by radially moving density blobs that are electrically polarized and
move by E B advection.21 However, in the studied case
the blob transport apparently is not playing a significant role
since the only blob polarization mechanism that is possible
in our model is the centrifugal force due to the self-generated
radial electric field ðEr Þ which is rather small; and the centrifugal force is shown to have little effect on the simulation
results.
On the other hand, polarized blobs have been observed
in LAPD.26 However, in the experiment the radial electric
field ðEr Þ can be large in the outer part of the plasma, see
Fig. 2; and as mentioned in Sec. V this may be due to effects
not included in the present model. Therefore the centrifugal
force can be large in the experiment, and, besides, in the
experiment other mechanisms such as the neutral wind22
may be playing a role. Therefore the issue of blob
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Numerical simulation and analysis of plasma turbulence in LAPD
polarization and transport in LAPD requires further investigation which will be reported in the future.
The present analysis of turbulence spectra indicates
direct and inverse cascades in the azimuthal and axial wavenumbers. Such cascades in the azimuthal wavenumber have
been long observed in quasi-2D models such as the Hasegawa-Wakatani model,27,28 where the parallel dynamics is
described by a single parallel wavenumber. The observed
here cascade in the axial wavenumber is a remarkable feature of a 3D model where full parallel dynamics is retained.
Observation of the spectra time-evolution and bicoherence analysis of saturated turbulence point to the coupling
between the drift-waves instability and flute-like modes. As
pointed out in Sec. III, there are two types of linear flute-like
modes with nz ¼ 0 supported by Eqs. (5)–(7): KH and IC
modes. However, one can rule out the IC modes as not playing a significant role here since the results are little affected
by removing from the equations the v2E term responsible for
the IC mode. Thus it is apparently the coupling of drift-wave
instability and Kelvin-Helmholtz modes which is behind the
observed time-evolution of wavenumber spectrum.
VIII. CONCLUSIONS
Turbulence calculations with a 3D collisional fluid
plasma model demonstrate qualitative and semi-quantitative
similarity to experimental data in the LAPD device, in
particular for the temporal spectra, fluctuations amplitude,
spatial correlation length, and radial particle flux. Several
experimentally observed features of plasma turbulence are
qualitatively reproduced, and quantitative agreement is
achieved at the order-of-magnitude level. The calculated turbulence fluctuations have non-Gaussian characteristics, however the radial flux of plasma density is consistent with
Bohm diffusion. Electric polarization of density blobs does
not appear to play a significant role in the studied case.
Turbulence spectra exhibit direct and inverse cascades in
both azimuthal and axial wavenumbers and indicate coupling
between the drift instability and Kelvin-Helmholtz mode.
ACKNOWLEDGMENTS
This work was supported by DOE Fusion Science
Center Cooperative Agreement DE-FC02-04ER54785, NSF
Phys. Plasmas 18, 055709 (2011)
Grant PHY-0903913, and by LLNL under DOE Contract
DE-AC52-07NA27344. BF acknowledges support through
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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