Turbulent Convection and Anomalous
Cross-Field Transport in Mirror
Plasmas
V.P. Pastukhov and N.V. Chudin
Outline
1. Introduction.
2. Theoretical model.
3. Results of simulations for GAMMA 10
and GDT conditions.
4. Discussion and comments.
Introduction.
• anomalous particle and energy transport is one of the
crucial problems for magnetic plasma confinement;
• low-frequency (LF) fluctuations and the associated
transport processes in a wide variety of magnetic plasma
confinement systems exhibit rather common features:
- frequency and wave-number spectra are typical for a
strong turbulence;
- intermittence;
- non-diffusive cross-field particle and energy fluxes;
- presence of long-living nonlinear structures
(filaments, blobs, streamers, etc.);
- self-organization of transport processes (“profile
consistency”, LH-transitions, transport barriers, etc.)
 LF convection in magnetized plasmas is quasi-2D;
 inverse cascade plays an important role in the nonlinear
evolution and leads to formation of large-scale dominant
vortex-like structures;
 direct dynamic simulations of the structured turbulent
plasma convection and the associated cross-field plasma
transport appear to be a promising and informative method;
 relatively simple adiabatically reduced one-fluid MHD
model demonstrate a rather good qualitative and
quantitative agreement with many experiments;
 mirror-based systems are very convenient both for
experimental and theoretical study of the structured LF
turbulent plasma convection. Application to tandem mirror
and GDT plasmas is reasonable;
Theoretical model
• plasma convection in axisymmetric or effectively
symmetrized shearless magnetic systems;
• magnetic field can be presented as:
• stability of flute-like mode :
• convection near the MS-state for the flute-like mode:
S = const ;
• ASM-method and adeabatic velocity field;
• small parameter
additional small parameter   rU  / 2U (   1) in
paraxial systems admits considerable deviation from the MS
state S = const
• characteristic frequencies of the
adiabatic convective motion
are much less than the characteristic frequencies of stable
magnetosonic
incompressible Alfven
longitudinal acoustic waves
• small parameter
additional small parameter   rU  / 2U (   1) in
paraxial systems admits considerable deviation from the MS
state S = const
• characteristic frequencies of the
adiabatic convective motion
are much less than the characteristic frequencies of stable
magnetosonic
incompressible Alfven
longitudinal acoustic waves
Set of reduced equations
• adiabatic velocity field has the form:
~
where:   0 (t , )  (t , , ) and
are plasma potential and frequency of sheared rotation;
• generalized dynamic vorticity is the canonical momentum:
• magnetic configuration is characterized by form-factors:
and U
Simulations for symmetrized mirrors
Applicability reasons
• all equations are obtained by flux-tube averaging; as a
result, effectively symmetrized sections (like in GAMMA 10)
gives symmetrized contributions to linear terms in the
reduced equations;
• axisymmetric central and plug-barrier cells gives a
dominant contribution to the flux-tube-averaged nonlinear
inertial term (Reynolds stress);
• non-axisymmetric anchor cells with anisotropic plasma
pressure contribute mainly to linear instability drive and can
be effectively accounted in a flux-tube-averaged form;
• in addition to a standard MHD drive we can model a
“trapped particle” drive assuming that only harmonics with
sufficiently high azimuthal n-numbers are linearly unstable
due to a pressure-gradient.In other words we can
assume   0 for small n and   0 for higher n;
• as a first example we present simulations for GAMMA 10
conditions with a weak MHD drive and without FLR and
line-tying effects.
GAMMA 10 experiments
(c)
GAMMA 10 experiments
Simulations with low
sheared rotation
(c)
Vortex-flow contours
Pressure fluctuations contours
GAMMA 10 experiments
Simulations with low
sheared rotation
(c)
Vortex-flow contours
Pressure fluctuations contours
Turbulence suppression by high on-axis sheared-flow vorticity
Transport barrier is formed in experiments by
generation of sheared flow layer with high vorticity
X-Ray Tomography
Te Increase
Ti Increase
Turbulence
(Note; No Central ECH)
4 keV
Suppress
Potential
Vorticity
5 keV
Cylindrical Laminar ExB Flow
due to Off-Axis ECH Confines
Core Plasma Energies
ExB flow; Barrier Formation
Common Physics Importance for ITB and
H-mode Mechanism Investigations
Comparison of simulations with experiments
Soft X-ray tomography
(experimint)
Without shear flow layer
With shear flow layer
Comparison of simulations with experiments
Simulations with
low shear W = -1
Soft X-ray tomography
(experimint)
Without shear flow layer
With shear flow layer
Comparison of simulations with experiments
Simulations with
low shear W = -1
Soft X-ray tomography
(experimint)
Without shear flow layer
With shear flow layer
Simulations with
high shear W = - 6
Results of simulations for regime with a peak of
dynamic vorticity maintained near x=0.4 (r =7cm)
w0 0.5
S0
0.0
-0.5
-1.0
-1.5
-2.0
-2.5
0.0

0.5
x
1.2
1.0
0.8
0.6
0.4
0.2
0.0
1.0
0.5
x
0.5
0
0.0
0.5
x 1.0
-0.5
-0.1
 -0.2
0.4
-0.3
-0.4
-0.5
-0.6
-0.7
-0.8
-0.9
0.3
0.2
0.1
0.0
0.0
0.5
x 1.0
25 30 35 40 45 50 55 60 65 70 75 80 85
t
0.0
0.5
x 1.0
Chord-integrated pressure
 p dy
0
Profiles of dynamic vorticity w0 , entropy
function S 0 , plasma potential  0 ,
and plasma rotation frequency 
(corresponds to soft X-ray
tomography in GAMMA 10
experiments)
Evolution of well-developed convective flows and
fluctuations in the regime with peak of w0 .
Results of simulations for regime with a potential
biasing near x=0.7 (near r =10cm for GDT)
0.9
S0 0.8
w0 2
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
1
0
-1
-2
-3
0.0
0.5
x 1.0
0.02
 0.00
x
0.5
0.0
0.5
x 1.0
-0.5
 0.4
0.3
0.2
0.1
0.0
-0.1
-0.02
-0.04
-0.06
-0.08
-0.10
-0.12
0.0
0.5
x 1.0
0
25 30 35 40 45 50 55 60 65 70 75 80 85
t
0.0
0.5
x 1.0
Chord-integrated pressure
 p dy
0
Profiles of dynamic vorticity w0 , entropy
function S 0 , plasma potential  0 ,
and plasma rotation frequency 
(corresponds to soft X-ray
tomography in GAMMA 10
experiments)
Evolution of well-developed convective flows and
fluctuations in the regime with potential biasing
Discussion and comments (1)
• sheared plasma rotation in axisymmetric or effectively symmetrized
paraxial mirror systems can strongly modify nonlinear vortex-like
convective structures;
• this result was demonstrated by simulations for a weak MHD drive,
but the similar and even stronger effect was obtained for the “trapped
particle” drive as well;
• as a rule, the rotation does not stabilize plasma completely, however,
the cross-field convective transport reduces significantly and the plasma
confinement becomes more quiet
• the most quiet regimes were obtained in regimes where a peak of
vorticity was localised at the axis;
• the above favorable results were obtained even without FLR and linetying effects, which can additionally improve the plasma confinement;
Discussion and comments (2)
• in additional simulations with   0 for all harmonics
(i.e. without any MHD or “trapped particale” drives) low nnumber fluctuations in the core disappear, while fluctuations
with higher n-numbers still exist in both examples;
• accounting the above we can conclude that the core vortex
structures were mainly driven by pressure gradient, while the
edge vortex structures were maintained by Kelvin-Helmholtz
instability generated by sheared plasma rotation;
• we can also conclude that the main effect of the sheared
plasma rotation results from a competition between pressure
driven and Kelvin-Helmholtz driven vortex structures.