Transcript Plasma start-up in tokamaks Winter School - Golem

Plasma start-up in tokamaks
Jan Stockel
Institute of Plasma Physics, Prague, [email protected]
Tokamak plasma has to be "hot". We will talk today on physics
of the transition from an empty tokamak vessel to fully ionized,
but "cold" plasma. In general, underlying physics is quite
complex. We focuse here on selected issues:
•Basic hardware/diagnostics to get plasma in a tokamak
•Basic physiscs of the start-up phase of a tokamak dischage
Any questions during my talk are welcome!!
Winter School, Marianska, January 21, 2010
Tokamak basics
Tokamak is composed of
three basic components
•
•
•
Large transformer with primary winding
Plasma ring as secondary winding
Coils for confinement of plasma ring by
magnetic field (toroidal solenoid)
Electric current I generated in the plasma ring by the transformer
•
delivers the ohmic power Pohmic = I2Rplasma to plasma (heating)
•
generates the poloidal magnetic field in the plasma ring Bpoloidal ~I/2pa
However, before ohmic heating, we have to fill the tokamak vessel by fully
ionized plasma. This period is called as start-up or breakdown phase of the
tokamak discharge. It is a complex process, which begins from well
evacuated toroidal vessel. This talk just tackle about basic physics.
Sequence of events before a tokamak discharge 1
1. Tokamak is pumped down to to the pressure < 10-4 - 10-3 Pa.
2. Baking of the vessel to 150-2500 and glow discharge cleaning is required!
3. Then, the tokamak vessel is filled by a working gas – but what pressure?
4. At normal conditions (105 Pa, T = 273.15 K) we have n0 = 2,7×1025 m−3 H2
molecules (Loschmit number), i.e. 5.4 ×1025 m−3 atoms is available
5. We require the plasma density ~ 1018 - 1019 m-3 at the breakdown phase.
Density of neutral atomic hydrogen should be comparable.
6. Therefore, the initial pressure of H2 should be in the range ~ 0.02 - 0. 2 Pa
Sequence of events before a tokamak discharge 2
1. Some free electrons have to be generated inside the vessel by some
extermal source (pre-ionization)
•
electron gun
•
VUV lamp
•
cosmic radiation background
•
eventually RF assisted pre-ionization
2. At least, two power supplies (capacitor banks) to be activated (charged)
to drive:
* current in the toroidal magnetic field coils
Grid
Rectifier Capacitor bank
Toroidal Field Coils
* current in primary winding of the transformer
Grid
Rectifier Capacitor bank
Primary winding of transformer
Start-up of a tokamak discharge
1. A trigger pulse is applied to start the data acquisition system
 Experimental data are collected
2. A trigger pulse is applied to discharge the capacitor bank UBt
to toroidal field coils
 Toroidal magnetic field is generated inside the vessel
3. Wait until a reasonable level of the toroidal magnetic field is
reached. (GOLEM – a typical time delay is 1 - 4 ms)
4. A trigger pulse is applied to discharge the capacitor bank Uoh to
primary winding of the transformer
 Time-dependent current in the primary winding generates
the toroidal electric field inside the vessel
Toroidal electric field – how to measure?
Toroidal electric field E tor is required plasma breakdown in tokamaks
and for inductive current drive.
E tor is generated by transformer (iron- or air-core) by primary current
I(t), which has to vary in time.
dy/dt – magnetic flux
The toroidal electric field is
measured by a single loop
located along the plasma
column:
dIprim/dt  0
E tor = Uloop/2pR
Loop voltage Uloop = - dy/dt
Why the E tor (loop voltage) must be as low as possible
during the breakdown?
Magnetic flux through the primary windings of a tokamak transformer
F(t) = Vloop(t) dt < Fmax
Maximum flux Fmax [Weber = Voltseconds] is limited either by quality of the iron
core transformer or by mechanical properties of the central solenoid (air-core
transformer)
CASTOR/GOLEM
COMPASS
F ~ B*S [Vs]
ITER
Fmax
Fmax = 0.12 Vs (iron core)
Fmax = 0.64 Vs – (air-core)
(0,4 Vs for breakdown and current ramp-up + 0,24 Vs for flat top phase)
Fmax = 277 Vs – (air-core)
Iron-core
transformer
Air-core
transformer
H ~ Iprim
Sequence of events during a discharge
Pressure of Hydrogen
50 mPa
Toroidal magnetic field
Trigger Bt
Loop voltage
Trigger Uoh
Delay
Uloop is high enough – Breakdown phase
Time
Start-up phase of a discharge on CASTOR
R plasma = U loop I plasma  Te
3 2
Loop voltage [V]
Uloop
Fully ionized plasma – "hot" (~200 eV)
Toroidal current [kA]
I_plasma+ I_vessel
Fully ionized plasma – "cold" (5-10 eV)
Plasma density
ne [1018 m-3]
0
I_vessel = I_plasma – Uloop/Rvessel
2
4
Time [ms]
6
Avalanche & Coulomb phases of breakdown
Plasma start-up can be divided into two phases with different
underlying physics. Therefore, they have to be treated separately.
1. Avalanche phase – degree of ionization is low. Collisions between
electrons and hydrogen molecules dominate. Electrons obey a drift
velocity vD II Etor, which is higher than their thermal velocity. Plasma
current is still low, and the rotational transform is negligible.
2. Coulomb phase – collisions between charged particles dominate.
Plasma current is sufficiently high and magnetic surfaces and the
confinement is expected to increase significantly.
Transition between these two phase occurs when

1 
 5 10 5 Te
3/ 2
[eV]
where  is the degree of ionization.
Typically, the transition occurs in tokamaks at 5% ionization at Te ~5 eV
Electron
Fully ionized
are plasma fills
Density of charged
accelerated
the
vessel
(in
in 0.1-10
toroidal
ms –
Free
electron(s)
particles increases
direction
depending
ionize
the
size
theof
appear and
inon
the
vessel
exponentially in time
working
tokamak)gas
Avalanche phase of breakdown – ionization length
First Townsend coefficient  [m-1]
 = Ap0 exp(  Bp 0 / E
ITER Lion ~ 2000 m
Ionization length Lion [m]
Lion =  1
Pressure p0 [Pa]
E=Uloop/1pR [V/m]
COMPASS
For hydrogen (H2)  A = 3.75, B = 99
GOLEM
Ionization length versus E
Drift velocity & Ionization time during the avalanche
Electrons obtain a drift velocity vd between ionization collisions, which
depends on the ratio of the toroidal electric field and pressure of molecular
hydrogen E/p . Only approximation of vd is available for H2:
Approx. for 70<E/p<1500 [V/m, Pa]
Typically
E/p = 80-800
vD = 6,9 104 ( E / p) [m/s, V/m, Pa]
Vd ~ 0.55 –2*106 m/s
Note: For E/p > 500 , the electron distribution function becomes strongly non-Maxwellian and a
significant fraction of electron can run-away!
Temporal evolution of plasma density is:
n(t ) = n0 exp
t
ti
where the ionization time ti is defined as ti ~ Lion/ vd .
Typically, ti ~ 20 ms at p0 ~30 mPa
Example: Our final goal is to reach degree of ionization 5%, i.e. the plasma density
5x1017 m-3 with just a single electron inside the tokamak vessel (n0=1 m-3).
This occurs during the time interval t = 17 x ln5 x 20 x 10-6 ~ 550 ms !!!
HOWEVER – this appears in an ideal case, when all electrons remain
inside the vessel during the avalanchel!!
Connection length & Loss time during the avalanche
REALITY
The magnetic field is not strictly toroidal during the start-up. It always has a
perpendicular component B, which significantly impacts trajectories of
charged particles during the avalanche phase of the discharge.
Example
B = Bz
We can define the connection length
Lcon ~ a Btor/ B 
and an effective loss lime
tloss ~ Lcon / vd
Rate of the density increase is consequently reduced:
1 1 
1
1 
t = exp  
vDt
= exp  
n0
 ti tloss 
 Li Lcon 
and eventually the breakdown may not occur when Lion ~ Lcon
n(t )
In practice, the condition Lion ~ 10 x Lcon should be fulfilled
Stray magnetic field B from the Toroidal Field coils
View from the top
,
A strong vertical field Bz is created
(oriented downwards)
Bz = m0I/2pr
I = 1 kA, R = 0.4 m
Bz(center) ~ 0.15 T !!
Installation of Return Current Conductor
significantly reduces the Bz field
Nevertheless, a small fraction of Bz (<1 mT)
could still exists inside the tokamak vessel
because of imperfect alignment of TF coils
and the return conductor!!
Stray magnetic field B from the vessel current
Toroidal current through the tokamak vessel (without plasma) generates
a vertical magnetic field inside the tokamak vessel
Rough estimate (linear approx – lower limit):
Bz =
m0 I vessel
= 10 7 I / R
2pr
For 2r = R = 0.8 m and I = 2 kA  Bz ~ 0.25 mT
For GOLEM a~0.08 m, Btor ~ 0.25 T, B  ~ 0.25 mT  Lcon ~ 8 m only !!
Stray magnetic field from the air-core transformer
COMPASS
case
Strong vertical field is generated, when
the primary current flows only through
the central solenoid
The vertical field is significantly reduced,
when the primary current flows also
through properly distributed poloidal
coils.
Evolution of plasma current during the avalanche
Plasma current grows exponentially with approx. the same rate as the
plasma density during the avalanche phase
Iplasma = S*e*ne(t)*vD(t)
where
S is the cross section of the current channel S=pa2 [m-2]
e= 1.6*10-19 C
vD(t) =const ~ 106 m/s is the average drift velocity
At the end of the avalanche phase, the plasma density is * nmax
with the degree of ionization  = 0.05 and nmax = 1019 m-3
So, the plasma current at the end of the avalanche phase should be
Iplasma (end of avalanche) ~ 8*104 x S [A].
If the current flows through the whole cross section, then:
GOLEM
COMPASS
(S~0.02 m-2)
(S~0.12 m-2)
TORE Supra (S~1.6 m-2)
I ~ 1.6 kA
I ~ 9.6 kA
I ~ 130 kA
Coulomb phase of the start-up
Particle balance during the Coulomb phase is described by differential equation
for electrons and neutrals
dne
n
= ne n0 Si  e
dt
tp
dn0
= ne n0 Si  ne 
dt
Si – rate of ionization by electrons
tp – particle confinement time
 – particle influx (recycling)
Solution for tp  infinity,   0
exp( t / t i )
ne = N
N / ne  exp( t / t i )
n0 = N  ne
where
N is initial number density of H
ti is the ionization time in Coulomb phase
ti =
1
NSi
Dynamics of atomic/molecular species
Dissociation cross section of H2 is greater than the ionization one at low Te
Result of modeling including
dissociation
Te = 6 eV, NH2 ~ 7x 1018 m-3
Ionization rate for atomic hydrogen
The ionization rate is a steep function of
guessed electron temperatures (3 – 10 eV]
Si = v
Te
= 0,29110
13
U 0,39 exp( U )
[m3/s]
0,232  U
where U = 13.6/Te [eV]
Approximation for Te < 10 eV:
18
Si = 5,6 10 Te
3
[m3/s, eV]
Ionization time at the Coulomb phase
for Te = 5 eV and N = 1019 m-3
1
ti =
 10 19 1015 = 100 ms
NSi
is longer than during the avalanche
Power losses due to collisions with atomic hydrogen
Ploss =  (ne n0 Si  i )   (ne n0 S ex  ex ) [W/m 3 ]
ionization losses
Si  Sex
 = 1,602 1019 [J/eV]
excitation losses
i = 13,6 eV,  ex  10 eV
Energy loss per a single ionization/excitation of H0
Ionization and excitation rates are comparable
Ploss = 3,776 10 18 ne n0 Si
[W/m 3 ]
Energy losses are maximum when ne = n0 = N / 2
Ploss
MAX
Ploss
= 9,44 1019 N 2 Si [W/m 3 ]
MAX
2
 520 N19 Te
3
Si = 5,6 1018Te
3
[m3s 1 , eV]
N19 = N 10 19
[W/m 3 ]
N=1*1019 and Te = 6 eV Ploss ~ 200 kW/m3
Power balance at start-up
Power losses due to the collisions have to be compensated by ohmic heating
res
POH =
Vloop I p
[W/m 3 ]
2p 2 Ra 2
CAST0R- Ohmic power during start up:
Volume 0.1 m3, Vloop~ 10 V,
Ip ~ 2 kA

Poh ~ 200 kW/m-3
The electron temperature can be roughly estimated from the power balance
POH  Ploss
MAX
2
 520 N19 Te
 POH 

Te  
2 
 520 N19 
1
3
[W/m 3 ]
3
[eV, W/m 3 ]
CASTOR N= 1019 m-3 and POH~200 kW/m-3  Te ~ 7.2 eV
This number is an upper limit for Te – some fraction of POH is consumed to heat electrons
Conclusions
I tried to explain some underlying physics of the plasma start-up in
tokamaks.
• Two phases were defined
* Avalanche phase
* Coulomb phase
• We focus on the avalanche phase
• importance of ionization and connection lengths (stray magnetic fields)
Many relevant features were not discussed at all (role of impurities, RF
assisted pre-ionization, runaway electrons, plasma current ramp-up, …..)
More information is available in many publications upon request at
[email protected]
Thanks for attention those who did not sleep, but also to sleepers, who
did not snore!!