PHYSICS OF PLASMAS 16, 062306 共2009兲
Structures generated in a temperature filament due to drift-wave
convection
M. Shi, D. C. Pace, G. J. Morales, J. E. Maggs, and T. A. Carter
Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA
共Received 3 March 2009; accepted 17 April 2009; published online 9 June 2009兲
A simplified numerical study is made of the structures that are formed in a magnetized temperature
filament due to oscillatory convection from large amplitude drift waves. This study is motivated by
a recent experiment 关D. C. Pace, M. Shi, J. E. Maggs et al., Phys. Plasmas 15, 122304 共2008兲兴 in
which Lorentzian-shaped temporal pulses are observed. These pulses produce a broadband,
exponential frequency power spectrum. The model consists of an electron heat transport equation in
which plasma convection arising from pressure-gradient driven drift-waves is included. It is found
that above a critical wave amplitude, spatially complex structures are formed, which give rise to
temporal pulses having positive and negative polarities at different radial positions. The temporal
shape of the pulses can be fit by a Lorentzian function. The associated spatial structures exhibit
temporally oscillatory heat plumes 共positive polarity兲 and cold channels 共negative polarity兲. The
idealized effect of a static flow on these structures is explored. Depending on the flow direction
共relative to the azimuthal propagation of the drift waves兲, the temporal Lorentzian pulses can be
suppressed. © 2009 American Institute of Physics. 关DOI: 10.1063/1.3147863兴
I. INTRODUCTION
A recent experimental study1,2 documented the longterm evolution of a controlled, electron temperature filament3
embedded in the center of a large magnetized plasma generated in the Large Plasma Device 共LAPD-U兲 共Ref. 4兲 operated
by the Basic Plasma Science Facility 共BaPSF兲 at the University of California, Los Angeles. The term “filament” refers to
a structure whose length scale along the confinement magnetic field is much larger than in the transverse direction. For
example, in the LAPD-U experiment, the axial length of a
typical temperature filament is about 8 m while the transverse dimension is about 5 mm. The surrounding plasma is
18 m long and 75 cm in diameter, thus it can be considered
as an infinite medium when describing the behavior of the
filament.
It has been found in the temperature filament
experiments1–3 that after an early stage, accurately described
by the classical theory of transport,5–7 the system develops
coherent drift-Alfvén modes8–11 driven by the cross-field
pressure gradient.12–15 After the coherent modes reach substantial amplitude, a complex regime appears characterized
by fluctuations with a broadband frequency spectrum. It has
been determined experimentally that the broadband frequency spectrum is exponential in nature for frequencies below the ion cyclotron frequency. Furthermore, the origin of
the exponential spectrum has been traced to temporal pulses
having a Lorentzian functional shape. The width of the
pulses is found to be a fraction of one period of the driftAlfvén oscillations. The pulses can display negative or positive polarity; negative polarity is predominantly observed
near the center of the temperature filament and positive polarity in the outer regions. Similar temporal signatures and
exponential frequency spectra have also been found in the
LAPD-U 共Ref. 2兲 using a totally different plasma configura1070-664X/2009/16共6兲/062306/12/$25.00
tion, namely, a limiter-edge density gradient.16 Exponential
spectra have been observed in a wide range of devices with
various plasma parameters,17–20 thus suggesting that the underlying character of the phenomena is universal.
Although the experimental study of Pace et al.2 clearly
identified the temporal signature of the pulses, the existing
diagnostics are not capable of imaging the spatial structures
associated with Lorentzian pulses of positive and negative
polarities in different regions of the temperature filament.
Thus, it is of interest to undertake an exploratory numerical
study that sheds light on the possible morphology of structures that arise in such a system when drift-Alfvén waves are
excited. The present investigation uses a simple model to
assess the essential elements that must be included in a more
comprehensive description of the phenomena. The study also
aims to provide insight in the planning of future experiments
into the type of spatial structures21 that need to be resolved
by diagnostic tools.
The model studied considers an electron heat transport
equation in which a convection pattern due to idealized drift
waves is included. In the absence of waves, the transport
equation describes quantitatively the classical behavior of the
temperature filament. When the waves are activated, it is
found that, above a critical wave amplitude, oscillatory structures having positive and negative polarities are generated.
The temporal shape of the structures can be fit by Lorentzian
functions. The associated spatial structures are a complex
arrangement of hot and cold temperature plumes, or channels, which rotate and oscillate at the wave frequency. Hot
temperature channels correspond to positive polarity Lorentzian pulses and cold channels to negative polarity. The idealized effect of a static flow on the structures is explored.
Plasma flow in the azimuthal direction counter to the propagation of the drift waves can suppress the Lorentzian pulses.
The manuscript is organized as follows. Section II pre-
16, 062306-1
© 2009 American Institute of Physics
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062306-2
Phys. Plasmas 16, 062306 共2009兲
Shi et al.
1.4
2.5
Experiment
1.2
2.0
0.8
Te (eV)
r (cm)
1
0.5
0.6
0.5
0.2
200
300
0.0
1
3.
4
100
1.0
0.5
2.5
3
4.5
5
1.5
1.5
2
2.5
42 3 3.51.51
5
0.4
0.
1
Theory
5
400 500
z (cm)
600
700
800
FIG. 1. Two-dimensional electron temperature contours predicted by the
transport code at a time t = 3.6 ms after the heating source is turned on. Note
the disparity in the scales used to display the axial 共z兲 and radial dependencies 共r兲. The contour numbers refer to electron temperature values in eV.
The contours for the cold ambient plasma surrounding the filament are not
shown.
sents the transport model used in the convection studies. The
numerical scheme used in the survey studies is described in
Sec. III. Section IV explores the properties of the nonlinear
structures generated by the transport model. The sensitivity
of the structures to the drift-wave radial profiles is presented
in Sec. V. Conclusions are given in Sec. VI.
0.0
The equation describing the spatiotemporal evolution of
the electron temperature Te due to classical transport resulting from Coulomb collisions in the presence of a magnetic
field is6
3
2 n共⳵tTe
ជ D · ⵜTe兲 + 共nTe兲 ⵜ · Vជ D = − ⵜ · qជ + Q,
+V
共1兲
ជ D is the electron flow velocwhere n is the plasma density, V
ity, qជ is the electron heat flux, and Q = Qei + Qext is the total
local heat input consisting of: the loss of electron heat to
ions,
Qei = −
3me n
共Te − Ti兲,
M ␶e
共2兲
and the external heat source Qext. In Eq. 共2兲 me and M refer
to the electron and ion mass, respectively, Ti is the ion temperature, and ␶e is the electron collision time given by
␶e =
3冑meT3/2
e
4冑2␲␭e4n
,
共3兲
with e the quantum of charge and ␭ the Coulomb logarithm.
In the absence of a relative velocity between the ions and the
electrons, as is the case in the situation considered, the electron heat flux is given by
1.0
r (cm)
1.5
2.0
2.5
FIG. 2. Radial dependence of electron temperature at an axial position
z = 384 cm and at a time t = 1.3 ms after heating source is turned on. The
solid curve is the experimental measurement and the dashed curve is the
prediction of the transport code.
qជ = − ␬储ⵜ储Te − ␬⬜ⵜ⬜Te −
5 cnTe
ẑ ⫻ ⵜTe ,
2 eB
共4兲
where B is the strength of the magnetic field pointing along
the unit vector ẑ in the z-direction and c is the speed of light.
In Eq. 共4兲 the parallel and transverse thermal conductivities are given by
␬储 = 3.16
nTe␶e
,
me
␬⬜ = 4.66
II. FORMULATION OF TRANSPORT MODEL
0.5
nTe
,
me␶e⍀2e
共5兲
共6兲
where ⍀e is the electron cyclotron frequency.
In the study of the electron temperature filament one
considers a pure temperature gradient situation in which the
plasma density is spatially uniform. Thus, the rightmost term
in Eq. 共4兲 is divergence-free and does not contribute to the
time evolution described by Eq. 共1兲.
Equation 共1兲 has been previously solved numerically2,3,22
ជ D = 0, and for an exfor the case of no electron flow, i.e., V
ternal heat source corresponding to the injection of a small
electron beam into the afterglow plasma generated in the
LAPD-U. The numerical solutions described the time evolution of the temperature filament in two spatial dimensions
共r , z兲, as is appropriate to an azimuthally symmetric heat
source 共the injected beam兲. Here z refers to the direction
along the confinement magnetic field and r to the transverse
radial coordinate. During the early stage of evolution, when
fluctuations are not present, the numerical solutions have
been found to quantitatively reproduce the experimental observations and parameter scaling within experimental uncertainties. For completeness and to provide a better perspective
for the present study, Figs. 1 and 2 display typical results
obtained with the code. Detailed comparisons between experimental results and transport modeling are found in the
references.1,3
Figure 1 displays two-dimensional contours of the electron temperature, Te, in the 共r , z兲 plane at a time, t = 3.6 ms
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Phys. Plasmas 16, 062306 共2009兲
Structures generated in a temperature filament…
after beam injection, when a steady-state temperature filament has been formed. Note the disparity in the axial 共z兲 and
radial 共r兲 scales used in the display; the actual filament is an
extremely narrow and long tube of elevated temperature. The
contours for the cold ambient plasma surrounding the filament are not shown.
Figure 2 shows the radial temperature profile at axial
position z = 384 cm and at time t = 1.3 ms after beam injection. The solid curve is the experimental result and the
dashed curve is the code result.
The inclusion of a convective flow due to drift waves
destroys the azimuthal symmetry of the filament, thus making the numerical study a formidable challenge involving
temporal evolution at a transport time scale in three spatial
dimensions. In order to make progress toward the assessment
of the convective effects, a reduced model is useful. The
logic of the model is to resolve the important spatial dimensions. In this case the spatial structures of interest are in the
共x , y兲 plane across the confinement magnetic field. Indeed,
axial variations exist in the filament temperature profile and
in the drift waves, but the axial scale length is several thousands times larger than the radial scale length. Thus, a twodimensional treatment is a reasonable approximation in the
study of cross-field structures. However, since axial thermal
conduction is large, care must be exercised in retaining this
feature within the desired two-dimensional description. To
incorporate the large axial heat conduction, the axial heat
diffusion term in Eq. 共1兲 is replaced by an average local
relaxation term, i.e.,
⳵ z共 ␬ 储 ⳵ zT e兲 → − 具 ␬ 储 典
Te
,
L2
共7兲
where L is the characteristic length of the filament and 具␬储典 is
the average axial thermal conductivity of the filament.
Another issue that must be addressed in constructing a
realistic, axially reduced model pertains to the heating
source. In the experiment the heating source is an electron
beam that thermalizes as it propagates along the z-direction,
away from the beam injector. In practice this creates a heat
source region, about 1 m in length, that causes temperature
increases downstream due to simultaneous axial and radial
classical transport. Obviously, removing the axial dependence does away with this essential element. The difficulty is
resolved by introducing a radially dependent, effective heat
source Qeff. This source is extracted numerically by running
the transport code without convection in the 共r , z兲 configuration to obtain a steady-state temperature filament of the type
illustrated in Fig. 1. From the steady-state profile, the heat
flux qជ ss and the electron-ion loss 共Qei兲ss can be obtained.
From these, a radially dependent, effective heat source can
be constructed, i.e.,
Qeff = 具ⵜ · qជ ss − 共Qei兲ss典,
共8兲
where the brackets refer to an axial average.
The solid curve in Fig. 3 displays the radial dependence
of the effective heat source used in the numerical study. The
source has contributions from the axial conduction 共dotted
curve兲, the radial heat transport 共dashed curve兲, and heat loss
5
3.5
x 10
radial
axial
Ion
Total
3
Power Density (erg/(cm2⋅ s))
062306-3
2.5
2
1.5
1
0.5
0
−0.5
0
0.2
0.4
0.6
r (cm)
0.8
1
1.2
FIG. 3. Radial dependence of the effective heat source used in the reduced
model. The solid curve is total heat source, dashed curve is the contribution
from radial transport, dotted curve is due to axial transport, and dash-dotted
is due to heat transfer to ions.
to the ions 共dash-dotted curve兲. It should be noted that the
use of Eq. 共8兲 ensures that when the convection terms are
turned off, or their effect is not important, the temperature
profile automatically relaxes to the steady-state value. Such a
relaxation is illustrated later in Fig. 11. With the approximations previously described, the relevant transport equation
takes the form
3
n共⳵tTe + vx⳵xTe + vy⳵yTe兲
2
= ⳵ x共 ␬ ⬜⳵ xT e兲 + ⳵ y 共 ␬ ⬜⳵ y T e兲 − ␬ 储
Te
+ Qei + Qeff .
L2
共9兲
ជ D = vxx̂ + vy ŷ, must
Next, a model for the flow velocity, V
be provided in order to solve Eq. 共9兲. Ideally, a calculation of
the nonlinear evolution of drift-Alfvén waves driven by the
temperature gradient should be used to evaluate this quantity.
Since that is a formidable task, in this exploratory study, the
waves are represented by simplified oscillatory potentials
⌽ whose form is chosen to incorporate the important
experimentally observed features. The required drift is related to the potential waveforms through the expression
ជ D = −共c / B兲ⵜ⬜⌽ ⫻ ẑ. A sensitivity survey is made later to
V
assess how the structures generated by the flows depend on
the form chosen for the potentials.
A feature incorporated in the choice of the potential
waveforms is motivated by an interesting experimental observation depicted in Fig. 11 of Ref. 8. It is observed in that
earlier study that an initially unstable drift-Alfvén mode having azimuthal mode number m = 1 evolves, slowly and progressively, into a mode having larger m-number; eventually
modes with m = 6 and 7 are seen. The drift-Alfvén modes are
driven by the radial electron temperature gradient associated
with the filament and have the property that their frequency
is insensitive to the value of the mode number m, as expected
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062306-4
Phys. Plasmas 16, 062306 共2009兲
Shi et al.
5
III. NUMERICAL SCHEME
Te
4
Φ (m=6)
3
Te (eV), Φ
The numerical technique used to solve Eq. 共9兲 consists
of a splitting scheme that separates the heat flux terms into
three individual parts:
Φ1(m=1)
6
⳵tT = D1共T兲 + D2共T兲 + D3共T兲,
2
共11兲
where
1
0
D1共T兲 =
2
⳵x共␬⬜⳵xT兲,
3n
共12兲
D2共T兲 =
2
⳵y共␬⬜⳵yT兲,
3n
共13兲
−1
(a)
−2
0
0.2
0.4
0.6
0.8
1
r (cm)
FIG. 4. Radial dependence of the two azimuthal modes used in the numerical study and described by Eq. 共10兲. The radial wave functions are superimposed on the equilibrium temperature profile of the filament 共solid curve兲.
theoretically12 and documented experimentally.8 Since the
radial wave function of modes having higher m-values peaks
at a larger radius than that for the m = 1 mode, it is suggestive
that when a low and a high-m number mode are present,
radially extended structures can be generated. Eventually the
radial structures could result in negative and positive polarity
features of the type exhibited by the Lorentzian pulses observed in the laboratory. It is the purpose of this modeling
study to explore this conjecture.
It should be noted further that since the different azimuthal modes have equal frequency, the structures generated
by them can lead to pulses having a unique temporal width.
Such a feature is evident in the laboratory studies.1
To explore the effects of convective flows driven by
modes of the type previously described, a model potential
waveform,
⌽共r, ␪,t兲 = Re兵关A1J1共k1r兲ei␪ + A6J6共k6r兲ei6␪兴e−i␻te−␣r其,
共10兲
is considered. In Eq. 共10兲 Am represents the constant amplitude of mode m, Jm is the Bessel function of order “m”
having wave number km, and the exponentially decaying
term, exp共−␣r兲, accounts for the radial localization of the
mode to the temperature filament.
The dashed and dotted curves in Fig. 4 illustrate the
radial dependence of the two azimuthal modes used in the
numerical study and described by Eq. 共10兲. The radial wave
functions are superimposed on the equilibrium temperature
profile of the filament, represented by the solid curve. The
corresponding two-dimensional, color contour of the total
potential associated with these modes is shown in Fig. 5.
Superimposed on the potential contour are black arrows indicating the local direction of the convective flow velocity
that is used in the numerical study of Eq. 共9兲.
D3共T兲 = − 共vx⳵xT + vy⳵yT兲 +
冉
冊
2
T
− ␬储 2 + Qei + Qeff .
3n
L
共14兲
For simplicity, the subscript “e” is dropped in representing
the electron temperature T.
For each time step, the contribution from each part is
evaluated separately to obtain an intermediate value for T.
This procedure is implemented consecutively, using the intermediate value obtained from the previous piece as the initial value for the next.
Since the D1 and D2 terms contain second-order spatial
derivatives, an implicit scheme is used to evaluate the corresponding intermediate state T̃. Applying the center-in-space
differentiation yields the difference equation associated with
D1共T兲,
冋
T̃i,j − Ti,j 2 1 1
=
共␬i+1,j + ␬i,j兲共Ti+1,j − Ti,j兲
⌬t
3n ⌬x2 2
册
1
− 共␬i,j + ␬i−1,j兲共Ti,j − Ti−1,j兲 .
2
共15兲
The subscript “i” indexes the x-direction and the subscript
“j” the y-direction. This set of difference equations is then
solved for every value of j, using the tridiagonal inversion
algorithm. The boundary condition used is that the temperature gradient is zero at the edges of the calculation grid. A
similar procedure is used to evaluate the intermediate state
generated by D2, using the values of temperature generated
from the D1 calculation.
The contribution from D3 requires a different treatment
because it contains first-order spatial derivatives. A simple
method to treat this type of equation is the “upwind
differencing”23 scheme. For example, in the x-direction the
derivative is given by the expression
Ux共T̃兲i,j =
冦
T̃i,j − T̃i−1,j
, vx,i ⬎ 0
⌬x
T̃i+1,j − T̃i,j
, vx,i ⬍ 0
⌬x
冧
,
共16兲
where T̃ is the intermediate state obtained previously using
D1 and D2.
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062306-5
Phys. Plasmas 16, 062306 共2009兲
Structures generated in a temperature filament…
0
"
0.4
0.6
4
0.3
0.4
0.2
0.2
0.5
(a)
0.1
0
0
0
-0.1
-0.2
-0.4
x
−0.5
-0.3
−0.5
-0.4
-0.6
"
x
-0.2
-0.6
-0.4
-0.2
0
x (cm)
0.2
0.4
0.6
0
FIG. 5. 共Color兲 Two-dimensional 共x , y兲, color contour 共the color bar is in
units of volts兲 of the total potential associated with the modes shown in Fig.
4. Superimposed on the potential contour are black arrows indicating the
local value of the convection flow velocity that is used in the numerical
study of Eq. 共9兲.
0.5
(b)
0.5
y (cm)
y (cm)
2
0
−0.5
In the upwind method the direction of differencing is
always against the direction of the flow. Furthermore, in
evaluating the contribution from D3 the two convection
terms can be treated simultaneously because the “upwind
differencing” is an explicit scheme, which does not involve
matrix solving. Thus, the resulting difference equation has
the form
T̂i,j − T̃i,j
= − 关vx,iUx共T̃兲i,j + vy,jUy共T̃兲i,j兴
⌬t
冉
冊
2
T̃
+
− ␬储 2 + Qei + Qeff
3n
L
−0.5
0
0.5
(c)
0.5
0
−0.5
−0.5
,
共17兲
i,j
where Ux and Uy represent the upwind differencing in the xand y-directions as given by Eq. 共16兲. T̂ is then the initial
value used to advance the next time step. The next section
uses numerical studies based on this scheme to explore the
properties of the structures generated by convection due to
drift waves.
IV. STRUCTURE GENERATION
The two-dimensional 共x , y兲 spatial pattern of the temperature filament in the presence of small amplitude
drift waves is shown in Fig. 6 for three different times in
the evolution: 共a兲 initial configuration 共t = 0兲, 共b兲 after half
a wave period 共t = 0.01 ms兲, and 共c兲 after five wave
periods 共t = 0.1 ms兲. The amplitude of the waveforms is
A1 = A2 = 0.03 V. This corresponds to a maximum drift velocity VD = 0.3V␪ , where V␪ represents the phase velocity of
the modes in the ␪-direction of a cylindrical coordinate system in which the confinement magnetic field is aligned with
the z-axis. In terms of a scaled potential fluctuation level this
amplitude corresponds to e˜␸ / Te ⬇ 1%.
0
x (cm)
0.5
FIG. 6. 共Color兲 Two-dimensional 共x , y兲 contours of electron temperature for
different times. Top color scale is in eV. Drift waves are turned on at t = 0
with small amplitude A1 = A6 = 0.03 V. 共a兲 t = 0, 共b兲 t = 0.01 ms 共half wave
period兲, 共c兲 t = 0.1 ms 共five wave periods兲. The white X symbols in panel 共a兲
mark the locations of the temporal signals shown in panels 共a兲 and 共b兲 of
Fig. 7.
The drift waves are turned on at time t = 0 when the
temperature filament has reached a steady-state profile 共as in
Fig. 1兲. The wave frequency is 50 kHz, consistent with experimental observations and linear stability analysis.12 The
color display corresponds to constant electron temperature
contours; the color scale at the top of the figure is in eV. It is
seen that at this small amplitude level no new spatial structures are formed; the filament essentially displays linear temperature oscillations. Near the center of the filament, one
observes an off-center distortion associated with the rotation
of the m = 1 mode, while at large radial position, one notices
ripples due to the m = 6 mode.
The local temporal behavior associated with the spatial
pattern of Fig. 6 is shown in Fig. 7 for three different radial
positions: 共a兲 r = 1.85 mm, 共b兲 r = 3.85 mm, and 共c兲
r = 9.85 mm. The location of the first two positions is
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062306-6
Phys. Plasmas 16, 062306 共2009兲
Shi et al.
4.2
(a)
4
0
3.8
Te (eV)
0.05
(a)
0.1
(b)
0
1.8
x
x
1.7
−0.5
1.6
1.5
0
4
0.5
3.6
3.4
0
1.9
2
0.05
−0.5
0.1
0
(c)
0.45
0.5
(b)
0.5
0.43
0
0.05
time (ms)
0.1
FIG. 7. Local temporal behavior associated with the spatial pattern of Fig. 6
for three different radial positions: 共a兲 r = 1.85 mm, 共b兲 r = 3.85 mm, and 共c兲
r = 9.85 mm. Sinusoidal temperature oscillations are induced by the small
amplitude drift waves.
y (cm)
0.44
0
−0.5
−0.5
marked with a white “X” in panel 共a兲 of Fig. 6. The third
position is outside the range shown in panel 共a兲. It is seen
that, for small amplitude drift waves, the electron temperature exhibits sinusoidal oscillations without distortion. The
induced temperature oscillations have a maximum peak-topeak amplitude of the order of 10% of the local temperature
of the filament.
As the amplitude of the drift waves is increased, the
temperature filament develops new spatiotemporal structures, which are not simply the linear superposition of oscillations driven by the m = 1 and m = 6 modes. The new structures become clearly identifiable when the ratio VD / V␪
becomes larger than unity. Figure 8 shows the spatial patterns exhibited by the temperature filament when the driftwave amplitude is increased to a value of VD = 0.3 V, corresponding now to VD = 3V␪ and e˜␸ / Te ⬇ 10%. The format in
the display is similar to that of Fig. 6. It is seen that structures having protruding arms at large radii and penetrating
channels near the center are formed. The protruding arms
exhibit the underlying m = 6 symmetry of the outermost drift
wave shown in Fig. 4. It should be noted that the development of the nonlinear structures causes an enhanced energy
transport that lowers the central temperature of the filament.
This effect is clearly evident when comparing panel 共c兲 in
Figs. 6 and 8 since they are both displayed with the same
color scale.
0
0.5
(c)
0.5
0
−0.5
−0.5
0
x (cm)
0.5
FIG. 8. 共Color兲 Two-dimensional 共x , y兲 contours of electron temperature for
different times. Top color scale is in eV. Drift waves are turned on at t = 0
with large amplitude A1 = A6 = 0.3 V. 共a兲 t = 0, 共b兲 t = 0.01 ms, and 共c兲
t = 0.1 ms. The white X symbols in panel 共a兲 mark the locations of the
temporal signals shown in panels 共a兲 and 共b兲 in Fig. 9, to be compared with
Fig. 6.
Figure 9 displays interesting temporal features associated with the large amplitude structures; they should be contrasted with the small amplitude behavior shown in Fig. 7.
Panel 共a兲 shows that near the center of the filament
共r = 1.85 mm兲, the sinusoidal signals develop narrow, nega-
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062306-7
Phys. Plasmas 16, 062306 共2009兲
Structures generated in a temperature filament…
5
5
Final
Initial
(a)
4
4
Te (eV)
3
2
1
0
0.05
0.1
1.8
(b)
3
2
1
Te (eV)
1.6
0
1.4
−1
−0.5
1.2
1
0
0.05
0.55
0.1
(c)
0.5
0.45
0.4
0
0.05
time (ms)
0.1
FIG. 9. Local temporal behavior associated with the spatial pattern of Fig. 8
generated by large amplitude drift waves. Three different radial positions are
sampled: 共a兲 r = 1.85 mm, 共b兲 r = 3.85 mm, and 共c兲 r = 9.85 mm, to be compared with Fig. 7.
tive pulses having a temporal width on the order of one-fifth
of the period of the drift-wave oscillations. As seen in panel
共b兲, similar narrow pulses appear at larger radii
共r = 3.85 mm兲, but these now have positive polarity. Far out
at the edge of the filament 共r = 9.85 mm兲, panel 共c兲 shows
sinusoidal oscillations superimposed on a rising temperature,
i.e., the structures are transferring energy to the surrounding
cold plasma.
The global rearrangement of the temperature profile induced by the large amplitude drift waves is illustrated in Fig.
10. The dashed curve corresponds to the steady-state, electron temperature profile before the large amplitude drift
waves are turned on. The solid curve is the profile, averaged
over one wave period, after five oscillation cycles. It is seen
that at the center of the filament the temperature experiences
a large decrease and the profile flattens, while at large radii
the temperature increases.
0
x (cm)
0.5
1
FIG. 10. The change in the radial structure of the temperature. The dashed
line is the initial radial profile of the temperature. The solid line is the radial
structure averaged over the fifth period of the oscillation.
It should be emphasized that the transport behavior being explored is not an initial value relaxation problem. A
steady heat source is being applied; a structured radial temperature profile develops as a consequence of the combined
effects of Coulomb collisions and convection due to drift
waves. When the drift waves are turned off, only classical,
collisional transport is present. Since the heat source continues to be applied, the temperature profile returns to the initial
steady state. The recovery toward the steady state is shown
in Fig. 11. Panel 共a兲 is the two-dimensional temperature contour at t = 10 ␮s 共a half wave period兲 after the drift waves are
shut off. It shows that the fine-scale spatial structures have
almost disappeared within one-half period of oscillation.
Panel 共b兲 illustrates the recovery at three different radial positions, r = 1.85 mm 共blue curve兲, r = 3.85 mm 共green curve兲,
and r = 9.85 mm 共red curve兲. They indicate that full recovery
to the steady state requires more than five wave periods, i.e.,
100 ␮s.
An enlarged display of the convectively generated structures is shown in Fig. 12. The left panel displays the entire
cross section of the temperature filament, while the right
panel is a blow-up highlighting the shape of a cold temperature channel that develops in the region near the center. It is
these channels that display the negative-polarity pulses.
Next, a test is made of the temporal shape associated
with the convectively generated pulses. As discussed in the
introduction, laboratory experiments conclusively identified
that an ensemble of pulses exhibiting a Lorentzian functional
form,
L共t兲 =
A␶2
,
共t − t0兲2 + ␶2
共18兲
produces a broadband spectrum exhibiting an exponential
frequency dependence. In Eq. 共18兲 ␶ represents the pulse
width 共half width at half maximum兲 and t0 is the time at
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062306-8
Phys. Plasmas 16, 062306 共2009兲
Shi et al.
0.6 (a)
2.5
y (cm)
0.4
2
0.2
0
1.5
0.2
x
0.4
x
1
0.5
0.6
0.5
0
x (cm)
0.5
4
(b)
Te (eV)
3
r=1.85mm
r=3.85mm
r=9.85mm
2
1
0
0
0.05
time (ms)
0.1
FIG. 11. 共Color兲 Recovery toward steady state after drift waves are shut off.
共a兲 Two-dimensional 共x , y兲 temperature contours 10 ␮s 共one half wave period兲 after shut off. The color bar is in units of eV. 共b兲 Temporal recovery at
different radial positions r = 1.85 mm 共blue兲, r = 3.85 mm 共green兲, and
r = 9.85 mm 共red兲. The white X symbols in panel 共a兲 mark the locations of
the temporal signals 共blue and green兲 shown in panel 共b兲.
convection. The solid curves correspond to Eq. 共18兲 and the
dots to the numerical results. The four signals in the left
panels are negative polarity pulses sampled near the center of
the filament. They are associated with the penetrating cold
channels of the type illustrated on the right panel of Fig. 12.
The four signals in the right panels are positive polarity
pulses sampled in the outer region of the filament. They are
associated with the protruding arms reflecting the m = 6 symmetry of the underlying drift mode. From the best fits to the
pulses the average pulse width is found to be 具␶典 = 1.9 ␮s,
which corresponds to a scaling frequency f s = 83 kHz. The
full temporal pulse width 共full width at half maximum兲 found
in the model corresponds to approximately one-fifth of the
drift-wave period while in the experiment it is about onefourth. The spread in pulse widths found in the model is
about 0.3 ␮s, which is relatively narrow. This behavior is
consistent with the experimental observation of ensembleaveraged exponential spectra with a single scaling frequency.
That is, the linear slope of the power spectrum in a semilog
display is uniquely related to the average pulse width. It is
seen from Fig. 13 that the Lorentzian fits work better for the
negative polarity pulses than for the positive. The positive
pulses exhibit a slightly faster rise than predicted by the
Lorentzian shape. Such an asymmetry in the pulse shape has
been identified in some of the experimental data.
For completeness, it should be mentioned that, in fluid
mechanics, the dimensionless number characterizing the solutions of advective-diffusive heat equations 关such as Eq.
共9兲兴 is the Péclet number, the ratio of the advective to diffusive terms.24 The larger the Péclet number 共⬎1兲 the more
convection dominates and temperature gradients are steeper.
For Eq. 共9兲 the Péclet number can be written
which peak amplitude A is attained. Mathematically, the amplitude of the frequency spectrum of a signal of the type
given by Eq. 共18兲 is
兩L̃共f兲兩 = ␲A␶e
−f/f s
共19兲
,
where the scaling frequency is related to the pulse width by
f s = 1 / 2␲␶.
Figure 13 shows Lorentzian functional fits to temporal
pulses associated with the structures generated by drift-wave
2.5
1
0.5
共20兲
−0.05
−0.1
0.4
−0.15
0.2
y (cm)
y (cm)
e⌽
.
Te
For the two representative values of the peak amplitude of
the potential, 0.03 and 0.3 V considered in this section, the
Péclet numbers correspond to 3 and 30, respectively. The
interesting structures emerge25 at the larger Péclet number.
0.6
2
1.5
Pe = .32 ⍀e␶e
0
−0.2
−0.2
−0.25
−0.4
−0.3
−0.6
−0.5
0
x (cm)
0.5
−0.2
−0.1
0
x (cm)
0.1
0.2
FIG. 12. 共Color兲 Enlarged display of the convectively generated structures. Shown are two-dimensional 共x , y兲 temperature contours. The color bar is in units
of eV. Left panel displays the entire cross section of the temperature filament. The right panel is a blow-up of a cold temperature channel in the region near
the center.
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062306-9
Phys. Plasmas 16, 062306 共2009兲
Structures generated in a temperature filament…
1.5
0.6
0.4
1
y (cm)
0.2
0
0.5
−0.2
0
−0.4
−0.6
−0.5
0
x (cm)
0.5
FIG. 14. 共Color兲 Two-dimensional 共x , y兲 potential contours 共the color bar is
in units of volts兲 with the total drift-velocity represented by the black arrows. Drift-wave amplitude of A1 = A2 = 0.3 V. Now an additional rigid azimuthal flow with angular speed of 105 rad/ s is added in the direction opposite to the azimuthal phase propagation of the drift waves.
described Lorentzian structures. The additional rigid azimuthal flow has an angular speed of 105 rad/ s in the direction opposite to the azimuthal phase propagation of the drift
waves. This angular speed is about one-half of the phase
speed of the m = 1 mode and three times that of the m = 6
mode. The rotation imposed is equivalent to adding a static
potential proportional to r2.
Figure 15 illustrates the effect produced by the additional counterflow. The top panel corresponds to the temporal evolution in a region near the center of the filament where
negative polarity pulses are generated. The bottom panel corresponds to positive polarity pulses at larger radii. To make
the effect more visible the extra rotation is turned on and off.
FIG. 13. Lorentzian functional fits to temporal pulses generated by driftwave convection. Solid curves correspond to Eq. 共18兲 and the dots to the
numerical results. The left panels are negative polarity pulses sampled near
the center of the filament. The right panels are positive polarity pulses
sampled in the outer region of the filament. The vertical temperature scale is
in eV and the horizontal time scale is milliseconds 共ms兲.
Te (eV)
0
3.5
0.05
0.1
time (ms)
0.15
0.2
0.25
0.3
3
2.5
2
1.5
No Flow
Flow Turned-on
No Flow
1.6
1.4
Te (eV)
V. SENSITIVITY STUDIES
1.2
It is of interest to explore what is the effect produced on
the structures generated by drift-wave convection by an additional rotation of the entire filament. This is a situation
reminiscent of what might be encountered in H-mode26–30
studies in which an explicit rotation is induced.31–35 Figure
14 displays the resulting two-dimensional potential contours
with the total drift-velocity represented by the superimposed
black arrows. In this case a drift-wave amplitude of A1 = A2
= 0.3 V is used, the same as that leading to the previously
1
0.8
0
0.05
0.1
0.15
0.2
time (ms)
0.25
0.3
FIG. 15. Suppression of the Lorentzian pulses by the rigid counterflow. Top
panel corresponds to the temporal evolution in a region near the center of
the filament where negative polarity pulses are generated. The bottom panel
corresponds to positive polarity pulses at larger radii. To make the effect
more visible the extra rotation is turned on and off as indicated.
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062306-10
Phys. Plasmas 16, 062306 共2009兲
Shi et al.
1
5
Φ (m=1)
1
3
y (cm)
Te
1
0
x
x
e
Φ,T (eV)
0.5
Φ6 (m=6)
4
1.5
(a)
0.5
2
0.5
1
1
1
3
0.2
0.4
0.6
r (cm)
0.8
1
1.2
FIG. 16. Radial dependence of the two azimuthal modes used to test sensitivity to nonoscillatory behavior. The radial wave functions are superimposed on the equilibrium temperature profile of the filament 共solid curve兲.
To be contrasted with Fig. 4.
0
0.5
x (cm)
1
1.85 mm
4.35 mm
9.85 mm
(b)
2.5
Te (eV)
0
0
0.5
2
1.5
1
0.5
It is evident by examining the middle curves in the top and
bottom panels that when the external rotation is active the
pulses disappear. The system returns to a near sinusoidal
behavior, similar to that of the linear oscillations obtained for
small amplitude drift waves. The right curves in the top and
bottom panels illustrate that when the additional rotation is
removed, the pulses return. In terms of an experimentally
relevant power spectrum, the sequence of events depicted by
Fig. 15 imply that the system experiences a transition from
broadband to coherent behavior and back to broadband, as
the rotation is turned on and off. It should be emphasized that
in this study the drift waves continue to be driven at the same
amplitude while the rotation is present, thus the effect that
destroys the structures is not that the drift-wave amplitude is
reduced by the rotation.
It is found that adding a rigid rotation in the same direction as the drift-wave propagation does not significantly alter
the Lorentzian pulses. Thus the details of that case are not
shown.
Next, the sensitivity of the Lorentzian pulses to the radial shape of the potential structure is explored. As before,
two azimuthal mode numbers, m = 1 and m = 6, are considered, but now the potential does not oscillate in the radial
direction. The radial behavior is shown in Fig. 16, superimposed on the electron temperature profile. The corresponding
nonlinear structure generated is illustrated in Fig. 17. It is
seen in panel 共a兲 that a starlike shape is formed instead of the
complex, azimuthally folded structure shown in Fig. 12. This
is a consequence of the azimuthal convection being in the
same direction due to the lack of radial oscillations in the
potential structure. Panel 共b兲 shows the temporal behavior at
different radial positions. While nonlinear distortions are
formed by these nonoscillatory potential structures, no evidence is found for the generation of narrow Lorentzian
pulses of either polarity.
0
0
0.05
time (ms)
0.1
FIG. 17. 共Color兲 Structure formed by nonoscillatory radial eigenfunctions
shown in Fig. 16. 共a兲 Two-dimensional 共x , y兲 radial contour at t = 0.1 ms.
The color bar is in units of eV. 共b兲 Temporal behavior at different radial
locations r = 1.85 mm 共blue兲, r = 4.35 mm 共green兲, and r = 9.85 mm 共red兲.
The white X symbols in panel 共a兲 mark the locations of the temporal signals
共blue and green兲 shown in panel 共b兲.
The next issue of interest is the role of the two modes,
m = 1 and m = 6, in creating the Lorentzian pulses. This feature is tested by exciting each mode separately. Figure 18
illustrates the behavior induced by the m = 1 mode. Panel 共a兲
shows that the two-dimensional temperature contour
develops a channel of low temperature near the center and
remains symmetric at larger radii; the behavior is to be compared to the left panel of Fig. 12. Panel 共b兲 displays the time
evolution of the electron temperature at three different radial
positions. The thicker color curves are with the m = 1 mode
alone while the thin black curves show the behavior with
both modes present. It is seen that the narrow, negative
pulses are still generated at r = 1.85 mm, albeit with smaller
amplitude. The positive, narrow pulses at r = 3.85 mm are
not formed in the absence of the m = 6 mode, instead a sinusoidal oscillation takes place.
The analogous effect of the m = 6 mode is demonstrated
in Fig. 19. Panel 共a兲 shows that in this case a starlike pattern
is formed with protruding temperature plumes at large radii.
Near the center, the temperature filament remains symmetric.
Panel 共b兲 shows that no narrow, temperature pulses are
formed at r = 1.85 mm. Positive pulses exhibiting a nonlinear distortion are formed at r = 3.85 mm, corresponding to
the arms of the star, but they do not display the narrow time
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0.6
2
0
-0.2
x
-0.4
x
1.5
1
-0.6
0.5
-0.6 -0.4 -0.2
0
x (cm)
0.2
5
0.4
2.5
2
0
-0.2
x
1.5
-0.4
x
1
-0.6
0.6
0.5
-0.6 -0.4 -0.2
5
3
0
0.2
x (cm)
(b)
0.4
0.6
1.85 mm
3.85 mm
9.85 mm
4
T (eV)
e
4
3
0.2
1.85 mm
3.85 mm
9.85 mm
(b)
3.5
(a)
0.4
2.5
0.2
Te (eV)
0.6
3
(a)
0.4
y (cm)
Phys. Plasmas 16, 062306 共2009兲
Structures generated in a temperature filament…
y (cm)
062306-11
3
2
2
1
1
0
0
0
0
0.02
0.04
0.06
time (ms)
0.08
0.1
FIG. 18. 共Color兲 Effect of mode m = 1. 共a兲 Two-dimensional 共x , y兲 temperature contour develops a lower temperature channel near the center and remains symmetric at larger radii. The color bar is in units of eV. To be
compared with the left panel of Fig. 12. 共b兲 Time evolution of the electron
temperature at three different radial positions. The thicker color curves are
with the m = 1 mode alone, and thin black curves are with both modes
present. The white X symbols in panel 共a兲 mark the locations of the temporal
signals 共blue and green兲 shown in panel 共b兲.
signature on the time scale of one-fifth of the wave period, as
shown in the bottom panel of Fig. 13.
VI. CONCLUSIONS
A simplified model that combines classical heat transport
with idealized convection due to drift waves has been used to
explore the spatiotemporal structures formed in an electron
temperature gradient. The specific situation considered consists of an ideal hot-electron filament embedded in an infinite, magnetized cold plasma. Such an entity has been generated and studied in the laboratory under controlled
conditions in the LAPD-U environment. A highlight of the
previously published experimental results is the development
of fluctuations exhibiting a broadband frequency spectrum
with a characteristic exponential frequency dependence for
frequencies below the ion-cyclotron frequency. Such a spectrum has been determined in the laboratory to arise from
temporal pulses having a unique Lorentzian shape. In the
present study it is found, from a numerical survey of the
simplified transport model, that drift-wave convection, above
a critical value, can indeed generate such temporal features.
The critical value corresponds to the convection speed be-
0.02
0.04
0.06
Time (ms)
0.08
0.1
FIG. 19. 共Color兲 Effect of mode m = 6. Format is the same as in Fig. 18. To
be compared with the left panel of Fig. 12.
coming comparable to the phase velocity of the drift waves
across the magnetic field.
The present study identified that the temporal Lorentzian
pulses are associated with complex spatial structures consisting of an arrangement of hot and cold temperature plumes, or
channels, which rotate and oscillate at the wave frequency.
Hot temperature channels correspond to positive polarity
Lorentzian pulses and cold channels to negative polarity. The
development of the structures causes an enhanced energy
transport that lowers the central temperature of the filament.
The properties of the Lorentzian pulses described by the
simplified convection model are found to be consistent with
the major experimental observations previously published.
Both positive and negative polarity pulses having a Lorentzian shape are generated; negative polarity pulses appear
near the filament center and positive polarity at the outer
edge. The narrow temporal pulses are embedded in the sinusoidal oscillations of the underlying coherent drift waves. In
the laboratory, such temporal mixing is manifested by the
clear observation of fluctuation spectra showing narrow
“eigenmode peaks” on top of a “broadband noise” baseline
having an exponential frequency dependence, as documented
in Fig. 6 of Ref. 1. The temporal width of the Lorentzian
pulses obtained in the model corresponds to approximately
one-fifth of the drift-wave period while in the experiment it
is about one-fourth. The spread in pulse widths found in the
model is relatively narrow. This behavior is consistent with
the experimental observation of ensemble-averaged expo-
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062306-12
Phys. Plasmas 16, 062306 共2009兲
Shi et al.
nential spectra with a single scaling frequency, i.e., displaying a unique linear slope in semilog displays.
From a sensitivity survey it is concluded that the simultaneous observation of negative and positive polarity pulses,
at well-separated radial locations, is likely associated with a
mixture of low- and high-m eigenmodes. The peak convection arising from such modes is radially well separated and
causes a different effect in the cold and hot regions of the
filament. The resulting structure is globally connected and
should not be confused with individual, radially traveling
solitary pulses. The survey also indicates that radially standing eigenmodes do not generate clear Lorentzian pulses. This
feature requires a radial phase variation in the convection
pattern.
This study considered two drift modes with different azimuthal and radial structure, but with the same frequency. A
richer dynamics might be expected in the case of the same
modes with incommensurate frequencies. This situation requires longer computational run times and thus was not explored in this preliminary survey, but should be considered in
future, more comprehensive studies.
The addition of global rotation superimposed on the
drift-wave convection pattern is found to quench the generation of the Lorentzian pulses when the rotation is opposite to
the direction of propagation of the drift wave across the magnetic field. Rotation in the same direction of phase advance
does not alter the Lorentzian pulses. Since this observation
provides an example of quenching of broadband noise without switching off the underlying coherent mode, it deserves
to be explored in more detail using a self-consistent calculation of the drift waves. An understanding of the quenching
phenomena in this relatively simple and fundamental system
may be helpful to studies of H-modes in fusion devices.
Since exponential frequency spectra have been observed
in widely different experimental arrangements involving
cross-field pressure gradients, the results of the present exploratory survey suggest that more detailed studies should be
undertaken, experimentally and theoretically, that link the
formation of spatiotemporal structures driven within a fraction of the drift-wave period to the phenomena of anomalous
transport.
ACKNOWLEDGMENTS
D. C. Pace and T. A. Carter acknowledge support from
NSF CAREER Grant No. PHY-0547572 and DOE Fusion
Science Center Cooperative Agreement No. DE-FC0204ER54785. J. E. Maggs and G. J. Morales’s work was performed under the auspices of the BaPSF, which is jointly
supported by a DOE-NSF cooperative agreement.
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