Plasma start-up in tokamaks
Winter School, Marianska, January 21, 2010
Jan StockelInstitute 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!!
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/2a
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.
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 1
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
* current in primary winding of the transformer
Sequence of events before a tokamak discharge 2
Capacitor bank Toroidal Field CoilsGrid Rectifier
Capacitor bank Primary winding of transformerGrid Rectifier
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
Start-up of a tokamak discharge
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.
Toroidal electric field – how to measure?
d/dt – magnetic flux
Loop voltage Uloop = - d/dt
The toroidal electric field is measured by a single loop located along the plasma column:
E tor = Uloop/2RdIprim/dt 0
Why the E tor (loop voltage) must be as low as possible during the breakdown?
Magnetic flux through the primary windings of a tokamak transformertVloop(t) dt < max
Maximum flux max [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 max = 0.12 Vs (iron core)COMPASS max = 0.64 Vs – (air-core)
(0,4 Vs for breakdown and current ramp-up + 0,24 Vs for flat top phase)ITER max = 277 Vs – (air-core)
Iron-core transformer
Air-core transformer
H ~ Iprim
B*S
[Vs]
max
Sequence of events during a discharge
Toroidal magnetic field
Pressure of Hydrogen
Loop voltage
Uloop is high enough – Breakdown phase
50 mPa
Time
Delay
Trigger Bt
Trigger Uoh
Loop voltage [V]Uloop
Toroidal current [kA]I_plasma+ I_vessel
Plasma density ne [1018 m-3]
Start-up phase of a discharge on CASTOR
Time [ms]0 2 4 6
I_vessel = I_plasma – Uloop/Rvessel
Fully ionized plasma – "cold" (5-10 eV)
Fully ionized plasma – "hot" (~200 eV)
23 eplasmaloopplasma TIUR
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
Avalanche & Coulomb phases of breakdown
2/351051 eT
where is the degree of ionization.
Typically, the transition occurs in tokamaks at 5% ionization at Te ~5 eV
[eV]
Electron are accelerated in toroidal direction and ionize the working gas
Fully ionized plasma fills the vessel (in 0.1-10 ms – depending on the size of tokamak)
Density of charged particles increases exponentially in timeFree electron(s) appear in the vessel
EBpAp /exp( 00
Avalanche phase of breakdown – ionization length
1ionL
First Townsend coefficient [m-1]
Pressure p0 [Pa]E=Uloop/1R [V/m] For hydrogen (H2) A = 3.75, B = 99
Ionization length Lion[m]
Ionization length versus E
GOLEM
COMPASS
ITER Lion ~ 2000 m
Drift velocity & Ionization time during the avalanche
)/(109,6 4 pEvD [m/s, V/m, Pa]Approx. for 70<E/p<1500 [V/m, Pa]
Typically E/p = 80-800 Vd ~ 0.55 –2*106 m/s
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:
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:
where the ionization time ti is defined as ti ~ Lion/ vd . Typically, ti ~ 20 s 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 s !!!
HOWEVER – this appears in an ideal case, when all electrons remain inside the vessel during the avalanchel!!
ittntn exp)( 0
Connection length & Loss time during the avalanche
Example B = Bz
We can define the connection length Lcon ~ a Btor/ B
and an effective loss lime loss ~ Lcon / vd
Rate of the density increase is consequently reduced:
and eventually the breakdown may not occur when Lion ~ Lcon
In practice, the condition Lion ~ 10 x Lcon should be fulfilled
REALITYThe 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.
tvLL
tttn
tnD
conilossi
11exp11exp)(0
Stray magnetic field B from the Toroidal Field coils
View from the top
A strong vertical field Bz is created (oriented downwards) 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!!
,
Bz = 0I/2r
I = 1 kA, R = 0.4 m Bz(center) ~ 0.15 T !!
Stray magnetic field B from the vessel currentToroidal current through the tokamak vessel (without plasma) generates a vertical magnetic field inside the tokamak vessel
Rough estimate (linear approx – lower limit):
RIr
IB vesselz /10
270
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
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.
COMPASScase
Evolution of plasma current during the avalanchePlasma 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=a2 [m-2]e= 1.6*10-19 CvD(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 beIplasma (end of avalanche) ~ 8*104 x S [A].If the current flows through the whole cross section, then:
GOLEM (S~0.02 m-2) I ~ 1.6 kACOMPASS (S~0.12 m-2) I ~ 9.6 kATORE Supra (S~1.6 m-2) I ~ 130 kA
Coulomb phase of the start-up
p
eie
e nSnndtdn
0
eie nSnndtdn
00
Particle balance during the Coulomb phase is described by differential equation for electrons and neutrals
Si – rate of ionization by electrons
p – particle confinement time
– particle influx (recycling)
Solution for p infinity, 0
)/exp(/)/exp(
ie
ie tnN
tNn
enNn 0
ii NS
1
where N is initial number density of H i is the ionization time in Coulomb phase
Dynamics of atomic/molecular species
Dissociation cross section of H2 is greater than the ionization one at low Te
Result of modeling including dissociationTe = 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]
UUUvS
Tei
232,0)exp(10291,0
39,013 [m3/s]
where U = 13.6/Te [eV]
318106,5 ei TS [m3/s, eV]Approximation for Te < 10 eV:
Ionization time at the Coulomb phasefor Te = 5 eV and N = 1019 m-3
is longer than during the avalanche
sNSi
i 10010101 1519
Power losses due to collisions with atomic hydrogen
[J/eV] 10602,1 19
exi SS eV 10,eV 6,13 exi
][W/m 10776,3 30
18ieloss SnnP
2/0 Nnne
][W/m 1044,9 3219i
MAXloss SNP
][W/m )()( 300 exexeiieloss SnnSnnP
ionization losses excitation losses
Ionization and excitation rates are comparable
Energy loss per a single ionization/excitation of H0
Energy losses are maximum when
eV],s[m 106,5 13318 ei TS
][W/m 520 33219 e
MAXloss TNP
N=1*1019 and Te = 6 eV Ploss ~ 200 kW/m3
1919 10NN
Power balance at start-up
][W/m 2
322
RaIV
P pres
loopOH
Power losses due to the collisions have to be compensated by ohmic heating
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
CASTOR N= 1019 m-3 and POH~200 kW/m-3 Te ~ 7.2 eVThis number is an upper limit for Te – some fraction of POH is consumed to heat electrons
][W/m 520 33219 e
MAXlossOH TNPP
] W/m[eV, 520
33
1
219
NPT OH
e
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!!
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