Kein Folientitel

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Neoclassical Transport
R. Dux

• Classical Transport
• Pfirsch-Schlüter and Banana-Plateau Transport
• Ware Pinch
• Bootstrap Current

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Why is neoclassical transport important?

• Usually, neoclassical (collisional) transport is
  small compared to the turbulent transport.
• Neoclassical transport is important
• when turbulent transport becomes small -
  transport barriers (internal, edge barrier in
  H-modes) - central part of the plasma, where
  gradients are small
• to understand the bootstrap current and the
  plasma conductivity
• transport in a stellarator (we do not cover
  this)

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Transport of particles, energy due to collisions

• The gradients of density, temperature and
  electric potential in the plasma disturb the
  Maxwellian velocity distribution of the
  particles, which would prevail in thermodynamic
  equilibrium.
• The disturbance shall be small.
• Coulomb collisions cause friction forces between
  the different species and drive fluxes of
  particles and energy in the direction of the
  gradients.
• Coulomb collisions drive the velocity
  distribution towards the local thermodynamic
  equilibrium and the fluxes try to diminish the
  gradients.
• We seek for linear relations between the fluxes
  and the thermodynamic forces (gradients).
• We concentrate on the particle flux

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Moments of the velocity distribution
The formulation of the neoclassical theory is
based on fluid equations, whichdescribe the time
evolution of moments of the velocity distribution.
We arrive at moments of the velocity distribution
by integrating the distribution function times vk
over velocity space
0. moment particle density
1. moment fluid velocity
2. moment pressure and viscosity
3. moment (random) heat flux
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Fluid equations moments of the kinetic equation
Integrating the kinetic equation times vk over
velocity space yields the equations of motion for
the moments of the velocity distribution (MHD
equations)
0. moment particle balance (conservation)
1. moment momentum balance
In every equation of moment n appears the moment
n1and an exchange term due to collisions (here
momentum exchange, friction force)
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The friction force due to collisions
The force on particle a with velocity va due to
collisions with particles b with velocity vb
averaged over all impact parameters
vy

• formally equal to the attractive gravitational
  force (in velocity space)
• This result for point like velocity
  distributions can be extended to an arbitrary
  velocity distribution fb of particles b using
  a potential function h.

fa
vx
fb
The average force density on all the particles a
with velocity distribution fa is obtained by
integrating the force per particle over the
velocity distribution
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The friction force for nearly Maxwellian
distributions
For undisturbed Maxwellian velocity distributions
with thermal velocity vT
The friction forces are zero.
For Maxwellian velocity distributions with small
mean velocity ultltvT
the average force density on the species a due to
collisions with b is.
The collision frequency is for Ta?Tb
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Closure of the fluid equations

• In every equation of moment n appears the moment
  n1.
• At one point one has to close the fluid
  equations by expressing the higher order
  moments in the lower ones
• In neoclassical theory one considers the first
  four moments density, velocity, heat flux,
  ???-flux
• To estimate classical particle transport we use
  a simple approximation and just care about
  density and velocity (first two moments)

momentum balance
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The ordering
momentum balance

• We assume
• the strong magnetic field limit (magnetized
  plasma)
• to be close to thermal equilibrium
• temporal equilibrium

Expand fluid velocity
Lowest order of ?? no friction
Next order of ?? include friction with lowest
order fluid velocities
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The lowest order perpendicular fluid velocities
first order
zero order
Cross product with B-field yields perpendicular
velocities
ExB drift diamagnetic drift
Velocity in direction of pressure gradients (ExB
drops out, friction only due to diamagnetic
drift)
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The lowest order perpendicular current density
zero order
The zero order perpendicular current is
consistent with the MHD equilibrium condition.
first order
The first order perpendicular current is zero.
The fluxes are ambipolar.
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Particle picture for the ambipolarity of the
radial flux
The position of the gyro centre and the gyrating
particle are related by.
A collision between a and b changes the
directionof the momentum vector and the position
of the gyro centre changes by.
The displacement is ambipolar.
No net transport for collisions within one
species (Faa0 in the fluid equation).
The same argument does not hold for the energy
transport (exchange of fast and slow particle
within one species).
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The classical radial particle flux (structure)
The radial particle flux density is thus (for
equal temperatures)
It has a diffusive part and a convective part
like in Ficks 1st law
The classical diffusion coefficient is identical
to the diffusion coefficient of a randomwalk
with Larmor radius as characteristic radial step
length and the collision frequency as stepping
frequency.
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A cartoon of classical flux
Diamagnetic velocity depends on the charge and
causes friction between different species,
that drive radial fluxes.
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The classical diffusion coefficient

• The classical diffusion coefficient is (nearly)
  independent of the charge of the
• species.
• In a pure hydrogen plasma DCL is the same for
  electrons and ions.
• For impurities, collisions with electrons mae?me
  can be neglected compared to
• collisions with ions.
• The diffusion coefficient decreases with 1/B2
  due to the quadratic dependence on the Larmor
  radius
• Our expression for the drift is still not the
  final result, since the friction
• force from the shifted Maxwellian is too crude...

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The perturbed Maxwellian (more than just a shift)
The B-field shows in the y-direction. All gyro
centres on a Larmor radius around the point of
origin contribute to the velocity distribution.
There shall be a gradient of n and T in the
x-direction.
We calculate the perturbation by an expansion of
the Maxwellian in the x-direction
old
The perturbation of the Maxwellian has an extra
term besides the diamagnetic velocity, which we
have neglected so far. It leads to the
diamagnetic heat fluxand an extra term in the
friction force, the thermal force.
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The thermal force
There is a diamagnetic heat flux connected with
the temperature gradient
This leads to new terms in the friction force
which are proportional tothe temperature
gradient and are called the thermo-force.
Ion-ion collisions, equal directions of ??p and
?T For magtmb ? the thermo-force is in the
opposite direction than the ?p-term
Also for a simple hydrogen plasma the two forces
are opposite.
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The thermal force
The reason for thermal force is the inverse
velocity dependence of the friction
force Collisions with higher velocity
difference are less effective than collisions
with lower velocity difference. This lowers
the friction force due to the differences in the
diamagnetic velocity.
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The classical radial particle flux (final result)
For a heavy impurity in a hydrogen plasma
(collisions with electrons can be neglected)
inward
outward (temperature screening)
In equilibrium the impurity profile is much more
peaked than the hydrogen profile (radial flux0)
For a pure hydrogen plasma
ion and electron flux into the same direction and
of equal size!
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End of classical transport
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Look ahead to neoclassical transport

• Similar
• Classical and neo-classical particle fluxes have
  the same structure - diffusive term drift
  term - larger drift for high-Z elements (going
  inward with the density gradient) - temperature
  screening
• The neo-classical diffusion coefficients are
  just enhancing the classical value by a
  geometrical factor.
• Different
• A coupling of parallel and perpendicular
  velocity occurs due to the curved geometry.
• The neo-classical transport is due to friction
  parallel to the field (not perpendicular).
• additional effects due to trapped particles
  bootstrap current and Ware pinch

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The Tokamak geometry
Helical field lines trace out magnetic surfaces.
the poloidal flux is 2??? and ??RBp

• The safety factor q gives the number of toroidal
  turns of a field line during one poloidal
  turn.
• The length of a field line from inboard to
  outboard is ?qR
• The transport across a flux surface is much
  slower than parallel to B.
• We assume constant density and temperature on
  the flux surface.

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Flux surface average
Density and temperature are (nearly) constanton
a magnetic flux surface due to the much faster
parallel transport and the transport problemis
one dimensional.
We calculate the flux surface average of a
quantity G
Tokamak
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Flux surface average of the transport equation
The average of the divergence of the flux is
calculatedusing Gauss theorem
We have to determine the surface averages
The one dimensional equation is then
which are linear in the thermodynamic forces
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Two contributions to the radial flux
Take the toroidal component of the momentum
equation, multiply with R and form a flux surface
average. This leads to an expression for the
radial flux due to toroidal friction forces
We can calculate this term by just forming the
flux surface average from the old result
We need to know the differences ofthe parallel
flowvelocities to get the friction forces.
The classical diffusion flux with correct flux
surface average
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Divergence of the lowest order drift
One contribution two the parallel flows arises
from the divergence of the diamagnetic and ExB
drift
pressure and particle density andelectric
potential are in lowest order constant on flux
surface
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The lowest order drifts are not divergence free
From the continuity equation
and the divergence of the diamagn. drift
udi
ude
we find, that ions pile up on the top and
electrons on the bottom of the flux surface
(reverses with reversed B-field). In the
particle picture this is found from the torus
drifts (curvature, grad-B drift). This leads
to a charge separation.
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Coupling of parallel and perpendicular dynamics
The separation of charge leads to electric
fields along the field lines and a current is
driven which preventsfurther charge separation.
Parallel electron and ion flows build upto
cancel the up/down asymmetry.The parallel and
perpendicular dynamics are coupled. The
remaining charge separation leadsin next order
to a small ExB motion andcauses radial
transport.
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Coupling of parallel and perpendicular heat flows
A similar effect appears for the diamagnetic
heat flow, which causestemperature perturbations
inside theflux surface which is counteracted
byparallel heat flows leading in higherorder to
a radial energy flux.
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The Pfirsch-Schlüter flow
diamagnetic velocity
Pfirsch-Schlüter velocity
form of total velocity (divergence free)

• not completely determined
• another velocity will be added
  later this is also divergence free since
  div(B)0
• it is caused by trapped particles
  (?Banana-Plateau transport)

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The Pfirsch-Schlüter transport
Pfirsch-Schlüter velocity
We use the shifted Maxwellian friction force and
calculate the radial flux
The result has the same structure as in the
classical case. The fluxes are enhanced by a
geometrical factor.
For concentric circular flux surfaces with
inverse aspect ratio ?r/R
The Pfirsch-Schlüter flux is a factor ?2q2 larger
than the classical flux.
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The Pfirsch-Schlüter flux pattern
The Pfirsch-Schlüter velocity
changes its direction at the top/bottomof the
flux surface. Also the radial fluxes change
direction. The flux surface average is
the integral over opposite radial fluxes at the
inboard/outboard side.
the flux vectors can also show inward/outward at
the outboard/inboardside
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Strong Collisional Coupling

• Temperature screening in the Pfirsch-Schlüter
  regime similar to classical case
• consequence of the parallel heat flux, which
  develops due to the non-divergence free
  diamagnetic heat flux.
• temperature screening is reduced for strong
  collisional coupling of temperatures of
  different fluids (that happens typ. for T lt
  100eV) - energy exchange time comparable to
  transit time on flux surface - up/down
  asymmetry of temperatures reduced due to
  collisions - weaker parallel heat flows -
  reduced or even reversed radial drift with
  temperature gradient

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Regime with low collision frequencies
For the CL and PS transport, we were just using
the fact, that the mean free path is large
against the Larmor radius (??altlt?ca) The mean
free path increases with T2 and can rise to a few
kilometer in the centre. Thus, we arrive at a
situation, where the mean free path is long
against the length of a complete particle orbit
on the flux surface once around the torus. The
trapped particle orbits become very important in
that regime, since they introduce a disturbance
in the parallel velocity distribution for a
given radial pressure gradient. This extra
parallel velocity shift will lead to a new
contribution in the parallel friction forces and
to another contribution to the radial transport,
the so called Banana-Plateau term.
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Particle Trapping
Conservation of particle energy and magn. moment
leads to particle trapping. At the low field
side, v?has a maximum.
For a magnetic field of the form
v? becomes zero on the orbit for all particles
with
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Fraction of trapped particles
The fraction of trapped particles is obtained by
calculating the part of the spherical
velocity distribution, which is inside the
trapping cone.
In all these estimates the inverse aspect
ratio?r/R is considered to be a small quantity.
ft only depends on the aspect ratio.
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Trapped particle orbits
The bounce movement together with the vertical
torus drifts leads to orbits with a banana shape
in the poloidal cross section. The trapped
particles show larger excursions from the
magnetic surface, since the verticaldrifts act
very long at the banana-tips. On the outer
branch of the banana the current carried by the
particle is always in the direction of the plasma
current (co).
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Trapped particle orbits
Conservation of canonical toroidal momentum
yields for low aspect ratio an estimate for the
radial width of the banana on the low-field side
The width scales with the poloidal gyro radius (
Larmor radius evaluatedwith the poloidal field).
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The banana current

• Consider a radial density gradient.
• On the low-field side, there are more co-moving
• trapped particles than counter moving particles,
• leading to a co-current density
• effect is similar to the diamagnetic current

The banana current is in the co-direction for
negative radial pressure gradient dp/dr lt
0. Collisions try to cancel the anisotropy in
the velocity distribution.
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Time scales
The velocity vector is turned by pitch angle
scattering. The collision frequency is the
characteristic value for an angle turn of????1
To scatter a particle out of the trapped region
it needs on average only an angle ?????. Due to
the diffusive nature of the angle change by
collisions the effective collision frequency is
The distance from LFS to HFS along the field line
is L?qR
A passing particle with thermal velocity vTa
needs a transit time
A trapped particle has lower parallel velocity
and needs the longer bounce time
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Collisionality
The collisionality is the ratio of the
effective collision frequency to the bounce
frequency

• The summation includes a.
• higher collisionality for high-Z.
• strong T-dependence

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Random walk estimate
If the collisionality is in the banana regime,
we can estimate the diffusion coefficient. The
diffusion is due to the trapped particles.
In the banana regime the transportof trapped
particles dominates by a large factor. This
banana-plateau contribution becomes small at
high collisionalities.
This estimate works only if the step length
(banana width) is small againstthe gradient
length.
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Exchange of momentum trapped?? passing
Simple model for banana regime
The loss of trapped particles into the
passingdomain creates a force density onto the
passing particles
The passing particles loose momentum to the
trapped particles in a fraction nt/n of all
collisions
In the fluid equations, this is the contribution
of the viscous forces to the parallel momentum
balance. The contribution increases with the
collision frequency in the banana regime and
decreases with 1/? in the PS regime.
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The parallel momentum balance
This integration constant û of the parallel fluid
velocity is calculated from the flux surface
averaged parallel momentum balance.
The PS flow drops out and one gets a system of
equations for the û (here it is written for the
shifted Maxwellian approach)
Thu û are functions of the viscosity
coefficients, collision frequencies and the
pressure gradients. Once a solution has been
obtained, one can calculatethe banana-plateau
contribution to the radial transport.
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Radial banana plateau flux (Hydrogen)
We calculate the û for a Hydrogen plasma using
the simple viscosity estimate in the case of low
collisionality.
This flux has the same form as the classical
hydrogen flux enhanced by a rather large
geometrical factor
This is just the same estimate, we got with our
random walk arguments.
For large collisionalities the viscosity
decreases with collision frequency and the
banana-plateau flux becomes small.
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The bootstrap current
The difference of the û for a Hydrogen plasma
from the simple viscosity estimate yields a
parallel current.
This is the a very rough estimate for the
bootstrap current density. It is a factorof 1/?
larger, than the banana current which is
initiating the bootstrap current of the passing
particles.
A better expression correct to order ??
Finally, a dependence on the collisionality has
to enter .
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Effects on the conductivity
Trapped particles do not carry any current. Only
the force on the passing particles generates a
current.
Momentum is lost by collisions with ionsor by
collisions with trapped particles.
These two effects lead to a neo-classical
correction on the Spitzer conductivity due to
the trapped particles.
The corrections disappear for high
collisionalityof the electrons.
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Ware Pinch
Conservation of canonical toroidal momentum
At the banana tips, the toroidal velocityis
zero. All turning points of the banana are on a
surface with
The movement of this surface of const. flux
yields a radial movement of the banana orbit.

• co acceleration
• counter de-acceleration
• no equal stay above/below equator
• radial drifts do not cancel

The Ware pinch is much larger than the classical
pinch
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The total radial flux due to collisions
The total radial flux induced by collisions is a
sum of three contributions classical(CL) ,
Pfirsch-Schlüter(PS) and banana-plateau (BP) flux.
temperature screening
For low collisionalities the BP-term dominates at
high collisionalities the PS-term. For each term
the drifts increase with the charge ratio times
the diffusioncoefficient. There are numerical
codes available to calculate the different
contributions(NCLASS by W. Houlberg, NEOART by
A. Peeters).
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The text book picture of neo-classical diffusion
BP contribution decreases in plateau regime and
is zero in PS regime DBP at ??1 is roughly
equal to DPS at ????-3/2 Thus, there is a
collisionality region with roughly constant D
(plateau regime).
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Transport coefficients due to collisions (example
1)
change of collisionality by change of T at a
fixed position
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Transport coefficients due to collisions (example
2)
ITER-FDR (old)
ASDEX Upgrade
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Standard neoclassical theory does not work, ...

• near the axis
• the banana width is assumed to be smallagainst
  the radial distance to the axis
• for very strong gradients, with gradientlength
  smaller than the banana width
• for high-Z impurities in strongly
  rotatingplasmas, with toroidal Mach numbers gtgt
  1 - leads to asymmetries of the density on the
  flux surface
• ...

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The End
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The kinetic equation

• (Neo-)classical transport can be explained by the
  combination of particle orbits and
• Coulomb collisions
• The particle density in phase space is given by a
  velocity distribution
• The kinetic equation is the Fokker-Planck
  equation

Collision operator
prescribed macroscopicfields
the time derivative alongthe particle orbit
The electric and magnetic fields are static and
only fluctuations with a length scale smaller
than the Debye length are considered. These
fluctuations are considered within the collision
operator.