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Neoclassical Transport. R. Dux. Classical Transport Pfirsch-Schlüter and Banana-Plateau Transport Ware Pinch Bootstrap Current. Why is neoclassical transport important?. Usually, neoclassical (collisional) transport is small compared to the turbulent transport. - PowerPoint PPT Presentation
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PhD Network, Garching, 28.9.2010 R. Dux
Neoclassical Transport
R. Dux
• Classical Transport
• Pfirsch-Schlüter and Banana-Plateau Transport
• Ware Pinch
• Bootstrap Current
PhD Network, Garching, 28.9.2010 R. Dux
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)
PhD Network, Garching, 28.9.2010 R. Dux
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:
Transport of particles, energy …due to collisions
un
PhD Network, Garching, 28.9.2010 R. Dux
Moments of the velocity distribution
We arrive at moments of the velocity distribution by integrating the distribution function times vk over velocity space
vdfn aa3
vdvfn
u aa
a31
aBaaaaa
a TknvduvuvfTrm
p 3
3
Ipvduvuvfm aaaaaa
� 3
0. moment: particle density
1. moment: fluid velocity
2. moment: pressure and viscosity
The formulation of the neoclassical theory is based on fluid equations, whichdescribe the time evolution of moments of the velocity distribution.
3. moment: (random) heat flux vduvuvfm
q aaaa
a32
2
PhD Network, Garching, 28.9.2010 R. Dux
Fluid equations = moments of the kinetic equation
0
aaa unt
n
Integrating the kinetic equation times vk over velocity space yields the equations of motion for the moments of the velocity distribution (MHD equations)
03vdCab
babababaab FvdCvmvdCvmF
33
ab
abaaaaaaaa
aaa
aa FpBuEneuut
unm
dt
udnm
0. moment: particle balance (conservation)
1. moment: momentum balance
In every equation of moment n appears the moment n+1and an exchange term due to collisions (here: momentum exchange, friction force)
PhD Network, Garching, 28.9.2010 R. Dux
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
• 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.
ab
abbaabb
ab
ababb
ba
ba
ab
abbaab
m
eeAn
v
vAn
vv
vv
m
ee
dt
pd20
22
3320
22
4
ln
4
ln
bbbb
vabab vdvf
vvvhvhA
dt
pd 31)()(
vdvhvfAF vaabab3)(
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
fa
fb
vx
vy
PhD Network, Garching, 28.9.2010 R. Dux
The friction force for nearly Maxwellian distributions
202
2
32/3
21)(exp)(
TvuTT v
uvvf
v
uv
v
nvf
T
baababaaab FuunmF
the average force density on the species a due to collisions with b is.
For Maxwellian velocity distributions with small mean velocity u<<vT
a
b
B
ababbaab m
n
Tk
mee2/3
22
20
ln
)4(3
24
The collision frequency is for Ta=Tb:
mTkvv
v
v
nvf BT
TT
/2exp)(2
2
32/30
For undisturbed Maxwellian velocity distributions with thermal velocity vT
The friction forces are zero.
PhD Network, Garching, 28.9.2010 R. Dux
Closure of the fluid equations
• In every equation of moment n appears the moment n+1.
• 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)
ab
ababaaaaaaa
aa uunmpBuEnedt
udnm
momentum balance
ababaaab
a
uunmF
0
PhD Network, Garching, 28.9.2010 R. Dux
The ordering
)()(
)( 23
O
abababaa
O
aaaa
O
aaa uunmpBuEnedt
udnm
momentum balance
1Be
m
a
aa
ca
a
We assume :
• the strong magnetic field limit (magnetized plasma)
• to be close to thermal equilibrium
• temporal equilibrium
1,,
Tnca
Ta
Tn
a
L
v
L
Lowest order of : no friction
aaaa pBuEne
)0(0
Next order of : include friction with lowest order fluid velocities
ab
ababaaaaa uunmBune )0()0()1(0
)1()0(aaa uuu
Expand fluid velocity
PhD Network, Garching, 28.9.2010 R. Dux
The lowest order perpendicular fluid velocities
Cross product with B-field yields perpendicular velocities:
aaaa pBuEne
)0(0 ab
ababaaaaa uunmBune )0()0()1(0
22)0(
, Bne
Bp
B
BEu
aa
aa
ababab
a
aa BuBu
Be
mu
)0()0(2
)1(,
aa
aa ne
pEBu
)0( ab
ababa
aa uu
e
mBu )0()0()1(
2BuBBu
ab aa
a
bb
bab
a
aa ne
p
ne
p
Be
mu
2)1(,
ExB drift + diamagnetic drift
Velocity in direction of pressure gradients(ExB drops out, friction only due to diamagnetic drift)
zero order first order
PhD Network, Garching, 28.9.2010 R. Dux
The lowest order perpendicular current density
22)0(
, Bne
Bp
B
BEu
aa
aa
The zero order perpendicular current is consistent with the MHD equilibrium condition.
22
alityQuasineutr 0
2)0(
,)0(
B
Bp
B
Bpne
B
BEunej
a
a
aaaa
aaa
Bjp
)0(
BFB
uneab
abaaa
2
)1(,
1
ab
abab
ababaaaaa FuunmBune )0()0()1(
01
onconservati momentum 0
2)1(,
)1(
BF
Bunej
a abab
aaaa
The first order perpendicular current is zero. The fluxes are ambipolar.
zero order
first order
PhD Network, Garching, 28.9.2010 R. Dux
Particle picture for the ambipolarity of the radial flux
2qB
Bprrgc
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.
The position of the gyro centre and the gyrating particle are related by.
gcb
a
b
a
b
a
agca r
e
e
Be
Bp
Be
Bpr
2onconservati momentum
2
0 gcbb
gcaa rere
0 gcb
gca rr
No net transport for collisions within one species (Faa=0 in the fluid equation).
The same argument does not hold for the energy transport (exchange of fast and slow particle within one species).
PhD Network, Garching, 28.9.2010 R. Dux
The classical radial particle flux (structure)
ab aa
a
bb
bab
a
aa ne
p
ne
p
Be
mu
2)1(,
ab a
b
b
b
b
aaaab
a
Ba
ab aa
aa
bb
bbab
a
Baaa
e
e
T
T
n
n
e
enn
Be
Tkm
ne
nTTn
ne
nTTn
Be
kmn
122
2
ab
aba
abab
ca
Ta
abab
aa
aB
abab
a
BaaCL
v
mBe
mTk
Be
TkmD
22
2
2
2
22222 aaa
CLD 2
2
The radial particle flux density is thus (for equal temperatures):
aaCLa
aCLa nvnD
It has a diffusive part and a convective part like in Fick’s 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.
TTTTknp baaBaa
PhD Network, Garching, 28.9.2010 R. Dux
A cartoon of classical flux
2, Bne
Bpu
aa
aadia
Diamagnetic velocity depends onthe charge and causes frictionbetween different species, thatdrive radial fluxes.
Be
mTk
mBe
v
a
aaB
aa
Taa
2
PhD Network, Garching, 28.9.2010 R. Dux
The classical diffusion coefficient
ab
aba
BaaCL Be
TkmD
22 a
b
B
ababbaab m
n
Tk
mee2/3
22
20
ln
)4(3
24
ab
bbabab
B
aCL nem
TkBD 2
2/1220
ln1
)4(3
24
• 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 maeme 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 frictionforce from the shifted Maxwellian is too crude...
PhD Network, Garching, 28.9.2010 R. Dux
The perturbed Maxwellian (more than just a shift)
We calculate the perturbation by an expansion of the Maxwellian in the x-direction:
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.
mTkvv
v
v
nvf BT
TT
/2exp)(2
2
32/30
22
2
02
2
01
2'
2
5''
2
3'
T
z
a
B
Tdia
c
z
T v
v
Be
Tk
v
vuf
v
T
T
v
v
T
T
n
nff
czgc vx
cL vr B
x
z
Tn
201
2
T
diaz
v
uvff
100
00
0
00 ff
v
x
ffx
x
fff
c
z
xgc
x
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.
old
PhD Network, Garching, 28.9.2010 R. Dux
The thermal force
There is a diamagnetic heat flux connected with the temperature gradient
2)0(
, 2
5
Be
BTkpq
a
aBaa
2
flux)heat (diamagn. force-thermo velocity)(diamagn. force slipping
, 2
3
B
B
me
m
me
mTk
ne
p
ne
pnmF
aa
ab
bb
abB
aa
a
bb
babaaab
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 ma>mb the thermo-force is in the opposite direction than the p-term
2, 2
3
B
BTk
n
p
e
nmF B
eieei
Also for a simple hydrogen plasma the two forces are opposite.
PhD Network, Garching, 28.9.2010 R. Dux
The thermal force
The reason for thermal force isthe 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 diamagneticvelocity.
PhD Network, Garching, 28.9.2010 R. Dux
The classical radial particle flux (final result)
abaab
CLab a
ab
a
b
b
ab
b
b
b
aabCL
aba
abCLa
aCL D
m
m
e
e
m
m
T
T
n
n
e
eDnDn
21
2
31
2
3 2
T
T
n
nZnn
H
HZZ
ZHZZCL 2
1
2
2
iCL
eieeieeCL T
Tnn
T
T
n
nnn
22
22
1
2
22
For a heavy impurity in a hydrogen plasma (collisions with electrons can be neglected):
inward outward (temperature screening)
For a pure hydrogen plasma:
ieieie 22 ion and electron flux into the same direction and of equal size!
T
T
n
nZ
n
n
H
H
Z
Z
2
1In equilibrium the impurity profile is much more peaked than thehydrogen profile (radial flux=0)
PhD Network, Garching, 28.9.2010 R. Dux
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
PhD Network, Garching, 28.9.2010 R. Dux
The Tokamak geometry
ttt eZReRBB
),(
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.
PhD Network, Garching, 28.9.2010 R. Dux
Flux surface average
dSG
VdS
GVG
'
11
Density and temperature are (nearly) constanton a magnetic flux surface due to the much faster parallel transport and the transport problemis one dimensional.
V VV
dSG
VdSrG
VGdV
VG
11
pp
dlB
G
VG
'
2pp dlRddSRB
t
n
We calculate the flux surface average of a quantity G
Tokamak:
PhD Network, Garching, 28.9.2010 R. Dux
Flux surface average of the transport equation
dSG
VG
'
1
''
1V
Vt
n
t
n
'VdSSddVV
'
''
''
V
VV
V
VV
The average of the divergence of the flux is calculatedusing Gauss theorem:
The one dimensional equation is then:
We have to determine the surface averages:
q
which are linear in the thermodynamic forces
a
aa T
np
PhD Network, Garching, 28.9.2010 R. Dux
Two contributions to the radial flux
Take the toroidal component of the momentum equation, multiply with R and forma flux surface average. This leads to an expression for the radial flux due to toroidal friction forces:
ab
abp
abt
aabtab
aa B
FRB
B
FRB
eRF
e
transportclassical
,
transportalneoclassic
||,,
11
We can calculatethis 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.
ab
aba
a
Bapa
n
e
Tkm
B
BR
22
22
Diffusion CL
The classical diffusion flux with correct flux surface average:
eB
Be
B
Be ptt
||
PhD Network, Garching, 28.9.2010 R. Dux
Divergence of the lowest order drift
22222
)0(, B
B
Bne
Bp
B
B
Bne
Bp
B
BEu a
aa
a
aa
aa
a
aaa
p
ne
1
B
Bu
BB
B
BB
BB
BB
u
a
a
aa
aa
)0(,
3
0
22
0
2)0(
,
2
2
1
0
00
j
BBB
pressure and particle density andelectric potential are in lowest order constant on flux surface
One contribution two the parallel flows arises from the divergenceof the diamagnetic and ExB drift:
PhD Network, Garching, 28.9.2010 R. Dux
The lowest order drifts are not divergence free
)0(,
)0(, aaaa
a ununt
n
B
Buu aa
)0(,
)0(, 2
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.
From the continuity equation:
and the divergence of the diamagn. drift
B
Bun
t
naa
a
)0(
,2
ude
udi
PhD Network, Garching, 28.9.2010 R. Dux
Coupling of parallel and perpendicular dynamics
The separation of charge leads to electric fields along the field linesand 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.
PhD Network, Garching, 28.9.2010 R. Dux
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.
PhD Network, Garching, 28.9.2010 R. Dux
The Pfirsch-Schlüter flow
diamagnetic velocity
Pfirsch-Schlüter velocity
eB
RBp
neu pa
aaa
1)0(,
||2
)0(||,
11e
B
B
BRB
p
neu t
a
aaa
form of total velocity (divergence free)
BKu ta
eR)0(
0)0(||, Bu a
• 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)
2ˆ BBu
PhD Network, Garching, 28.9.2010 R. Dux
The Pfirsch-Schlüter transport
Pfirsch-Schlüter velocity
||2
)0(||,
11e
B
B
BRB
p
neu t
a
aaa
We use the shifted Maxwellian friction force and calculate the radial flux
ab
a
aa
b
bbab
a
aatPSa
p
ne
p
nee
nm
BBRB
1111
22
2
ab
abt
aa B
FRB
e||,1
22222
2 11
BBBBR
RBg
p
tPS
...8
29
16
3912
231
11
1 4222
2
2
22
qqgPS
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.
PhD Network, Garching, 28.9.2010 R. Dux
The Pfirsch-Schlüter flux pattern
The Pfirsch-Schlüter velocity
||2
)0(||,
11e
B
B
BRB
p
neu t
a
aaa
changes its direction at the top/bottomof the flux surface.
Also the radial fluxes change direction.
The flux surface average is theintegral over opposite radial fluxesat the inboard/outboard side.
the flux vectorscan also showinward/outward at the outboard/inboardside
PhD Network, Garching, 28.9.2010 R. Dux
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 < 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
PhD Network, Garching, 28.9.2010 R. Dux
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 (a<<ca)
The mean free path increases with T2 and can rise to a few kilometerin the centre. Thus, we arrive at a situation, where the mean free pathis long against the length of a complete particle orbit on the flux surfaceonce around the torus.
The trapped particle orbits become very important in that regime, sincethey introduce a disturbance in the parallel velocity distribution for a givenradial pressure gradient. This extra parallel velocity ‘shift’ will lead to a new contribution in the parallel friction forces and to another contributionto the radial transport, the so called Banana-Plateau term.
PhD Network, Garching, 28.9.2010 R. Dux
Particle Trapping
22|| 2
1
2
1 mvmvE
B
mv
2
2
leads to particle trapping. At the low field side, v||has a maximum.
LFSLFSLFS B
Bvvv 122
||2||
21
111||
LFS
HFS
LFSB
B
v
vv|| becomes zero on the orbit for all particles with
For a magnetic field of the form
cos1/cos10
0
0
B
Rr
BB
Conservation of particle energy and magn. moment
PhD Network, Garching, 28.9.2010 R. Dux
Fraction of trapped particles
In all these estimates the inverse aspect ratio=r/R is considered to be a small quantity.
The fraction of trapped particles is obtainedby calculating the part of the spherical velocitydistribution, which is inside the trapping cone.
46.046.114
31
max/1
0
2
B
tB
dBf
tf 46.046.1
ft only depends on the aspect ratio.
PhD Network, Garching, 28.9.2010 R. Dux
Trapped particle orbits
The bounce movement together with thevertical torus drifts leads to orbits with a banana shape in the poloidal cross section.
The trapped particles show larger excursionsfrom 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 inthe direction of the plasma current (co).
PhD Network, Garching, 28.9.2010 R. Dux
Trapped particle orbits
Conservation of canonical toroidal momentum
consteRvmp aa
yields for low aspect ratio an estimate for the radial width of the banana on the low-field side
papa
Tb
T
bpaaaa
mBe
vw
vvv
wRBeRvmeRvm
||
022
The width scales with the poloidalgyro radius (= Larmor radius evaluatedwith the poloidal field).
bw
PhD Network, Garching, 28.9.2010 R. Dux
The banana current
apa
TbtTt mBe
vwnnvv ||,
dr
dp
eB
dr
dn
meB
vv
dr
dnwvun
p
p
TT
tbtt
12/3
||,||
The banana current is in the co-direction for negative radial pressure gradient: dp/dr < 0.
Collisions try to cancel the anisotropy in the velocity distribution.
Consider a radial density gradient.• On the low-field side, there are more co-movingtrapped particles than counter moving particles,leading to a co-current density • effect is similar to the diamagnetic current
PhD Network, Garching, 28.9.2010 R. Dux
Time scales
The velocity vector is turned by pitch angle scattering.The collision frequency is the characteristic value for an angle turn of1:
b
aba
To scatter a particle out of the trappedregion it needs on average only an angle. Due to the diffusive nature of theangle change by collisions the effectivecollision frequency is:
aa
effa
2,
The distance from LFS to HFS along the field line is: LqR
A passing particle with thermal velocity vTa needs a transit time:Ta
T v
qR
A trapped particle has lower parallel velocity and needs the longer bounce time: Ta
TB
v
qR
PhD Network, Garching, 28.9.2010 R. Dux
Collisionality
The collisionality is the ratio of the effectivecollision frequency to the bounce frequency
2/3
,
02/300,*
amfpTa
a
Ta
a
b
effaa
qR
v
qR
v
qR
• The summation includes a. • higher collisionality for high-Z. • strong T-dependence
322
2
2/3024
22
22/3
020
*
mlnkeV
m109.4
ln1
)4(3
4
bbbabab
a
a
bbbabab
aB
aa
nZmm
Z
T
qR
nemmTk
eqR
Schlüter-Pfirsch:high
plateau :medium1
banana:low 1
2/3*
2/3*
*
a
a
a
PSedge @ Imp
plateaucenter @ Imp
plateauedge @ He,
bananacenter @ He,
PhD Network, Garching, 28.9.2010 R. Dux
Random walk estimate
If the collisionality is in the banana regime, wecan estimate the diffusion coefficient. The diffusion is due to the trapped particles.
qwr
nn
pb
eff
t
2rn
nD eff
tBP
CLBP Dqq
D2/3
22
In the banana regime the transportof trapped particles dominates by a large factor. This banana-plateau contribution becomes small at highcollisionalities.
This estimate works only if the step length (banana width) is small againstthe gradient length.
PhD Network, Garching, 28.9.2010 R. Dux
Exchange of momentum: trapped passing
The loss of trapped particles into the passingdomain creates a force density onto the passing particles
dr
dp
eBmunmF a
p
aatataeffpt
12/3,||
The passing particles loose momentum to the trapped particles in a fraction nt/n of all collisions
aaaatp unmF ||
aa
paa
aa
paaaatppt
udr
dp
Ben
udr
dp
BennmFFF
||
coeff.viscosity
||
1
1
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.
Simple model for banana regime:
PhD Network, Garching, 28.9.2010 R. Dux
The 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):
||||ˆ aa BFB
ab
ababaaa
aataa uunm
p
neRBu ˆˆ
1ˆ
||2||2
)0(||, ˆ
11e
B
Bue
B
B
BRB
p
neu at
a
aaa
This integration constant û of the parallel fluid velocity is calculated from the flux surface averaged parallel momentum balance.
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.
ab
abt
aa B
FRB
e||,1
ab
ababaat
aa uunm
B
RB
eˆˆ
12
PhD Network, Garching, 28.9.2010 R. Dux
Radial banana plateau flux (Hydrogen)
We calculate the û for a Hydrogen plasma using the simple viscosity estimate inthe case of low collisionality.
ieieit
iieii
eieiet
eeiee
uup
en
RBu
uup
en
RBu
ˆˆˆ
ˆˆˆ
i
eeiie
i
eeeiieeei m
m
m
m
iet
ei
pp
en
RBuu
1ˆˆ
0ˆ
ˆˆˆ
iti
eieiet
eei
p
en
RBu
uup
en
RBu
p
e
m
B
RBuunm
B
RB
eeiet
eieiet
e 22
2
2BP 1ˆˆ
1
2/3
2
2
2
2222
2
BBRB
RBg
p
tBP
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 walkarguments.
For large collisionalities the viscosity decreases with collision frequency and the banana-plateau flux becomes small.
PhD Network, Garching, 28.9.2010 R. Dux
The bootstrap current
The difference of the û for a Hydrogen plasma from the simple viscosity estimateyields a parallel current.
r
p
Br
p
RBB
BRB
B
Bjj
pp
t
1
1
ˆ22||
2|| ˆˆ
B
Buuenj ei
iet
ei
pp
en
RBuu
1ˆˆ
dr
dp
Buenj
ptt
12/3||,banana||,
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 currentof the passing particles.
r
Tnk
r
Tnk
r
nTTk
Bj i
Be
BieBp
bs 42.069.044.2
A better expression correct to order
Finally, a dependence on the collisionality has to enter .
PhD Network, Garching, 28.9.2010 R. Dux
Effects on the conductivity
Trapped particles do not carry any current.
Only the force on the passing particlesgenerates a current.
The corrections disappear for high collisionalityof the electrons.
2
*11
CSP
1nnp
These two effects lead to a neo-classical correction on the Spitzer conductivity due to the trappedparticles.
Momentum is lost by collisions with ionsor by collisions with trapped particles.
|||||| uu
dt
dueeei
PhD Network, Garching, 28.9.2010 R. Dux
Ware Pinch
Conservation of canonical toroidal momentum
constemRvp a
At the banana tips, the toroidal velocityis zero. All turning points of the bananaare on a surface with
The movement of this surface of const.flux yields a radial movement of the banana orbit.
const
pt RBvREvtdt
d
field electric induced
p
ttware B
Enf)( *
2B
BEv ptE
The Ware pinch is much larger than the classical pinch:
• co: acceleration• counter: de-acceleration• no equal stay above/below equator• radial drifts do not cancel
PhD Network, Garching, 28.9.2010 R. Dux
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.
BPPS,CL,
DriftDiffusion
11
x aba
abx
b
bb
aaabx n
T
TH
n
ne
enD
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).
PhD Network, Garching, 28.9.2010 R. Dux
The text book picture of neo-classical diffusion
aaa
CLD 2
2
CLBP Dq
D2/3
2
CLPS DqD 2
BPPSCLcoll DDDD
BP contribution decreases inplateau regime and is zero inPS regime: DBP at *=1 is roughly equal to DPS at *=-3/2
Thus, there is a collisionality region with roughly constant D (plateau regime).
PhD Network, Garching, 28.9.2010 R. Dux
Transport coefficients due to collisions (example 1)
62.0
T5.2
240
52
keV5005.0
m101 320
,
trap
t
De
eSiHe
f
B
.ε
.q
T
nn
nn
change of collisionality bychange of T at a fixed position
PhD Network, Garching, 28.9.2010 R. Dux
Transport coefficients due to collisions (example 2)
T7.5
3
keV32)0(
m101
m45.8
95
320
t
e
B
q
T
n
R
T5.2
3.3
keV5.2)0(
m101
m65.1
95
320
t
e
B
q
T
n
R
ITER-FDR (old)
ASDEX Upgrade
PhD Network, Garching, 28.9.2010 R. Dux
Standard neoclassical theory does not work, ...
• near the axisthe 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 >> 1 - leads to asymmetries of the density on the flux surface• ...
PhD Network, Garching, 28.9.2010 R. Dux
(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
The kinetic equation
vdxdtvxfNd a336 ,,
bbaab
a
a
aaa ffCv
fBvE
m
e
x
fv
t
f,
the time derivative alongthe particle orbit
prescribed macroscopicfields
Collision operator
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.