Tokamak Energy Confinement

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Tokamak Energy Confinement
Tokamaks do not behave as predicted by
neoclassical theory. The energy confinement, as
measured by the confinement time, is found to be
much shorter than the neoclassical value. As a
result, an empirical representation of the
confinement time has been widely used.

• Global energy confinement time
• Definition and significance
• Various operation modes
• Confinement scaling laws
• Transport and Energy loss mechanisms
• Fluctuations and turbulence
• Radiation losses

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Global Energy Confinement Time

• Definition

• To predict the performance of future devices, the
  energy confinement time is one of the most
  important parameter
• Since tokamak transport is anomalous, empirical
  scaling laws for energy confinement are necessary
• Empirical scaling laws regression analysis from
  available experimental database.

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Tokamak Operation Modes

• Ohmically heated plasmas
• Ohmic energy confinement scaling
• Auxilliary heated plasma operation modes
• L-mode
• H-, VH-, CH-, CDH-modes
• Super-shot, high-li mode, hot ion mode, PEP
• ERS, NCS

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Ohmic Energy Confinement Scaling

• Linear dependency on density
• Alcator (or INTOR) scaling
• ?Esec0.5 a2 (n/1020 )
• neo-Alcator (or Goldston)
• ?Esec0.071 a1.04 R2.04 (n/1020 )q0.5
• very promising, but conflict with neoclassical
  theory
• Saturation at higher density range
• nsat0.06x 1020 IpR A0.5 a-2.5 ?-1
• partly due to increased radiative losses.
• Improved confinement w/o density saturation
  observed at ASDEX (peaked density profile)

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Ohmic Energy Confinement Scaling

• Improved confinement w/o density saturation
  observed at ASDEX (peaked density profile)
• New scaling law from C-Mod
• Similar scaling to L-mode
• ?E? M0.3Ip Ptot-0.5
• neo-Alcator scaling at low density
• ( nlt 1.5x 1020 , klt 1.35)
• ?E 0.07 aqR2 (n/1020 )k0.5

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Auxilliary-heated Plasmas L-mode

• During the early phase of tokamak heating
  experiments, the plasma confinement was found to
  degrade very rapidly from the ohmic value with
  application of auxiliary heating. The typical
  observation is that the ?E increases with the
  plasma current(note that it is opposite to the
  ohmic), decreases with the applied power,
  increases with R and a little dependence with
  density and minor radius.
• Goldston scaling ?Gsec0.037 I R1.75 a-0.37
  P-0.5
• ITER89-P scaling
• ?Esec0.048 I0.85 R1.2 a0.3 ?0.5 (n/1020 )0.1
  B0.2 A0.5 /P0.5
• Confinement scaling valid from ohmic to strongly
  additionally heated plasmas
• Goldston ?E,G -2 ?OH -2 ?AUX,G -2
• linear offset EEOH EAUX or ?E, EOH / Ptot
  ?inc
• transport model for anomalous electron (Rebut,
  Lallia and Watkins)

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L-mode Confinement
Using the power balance relation, Goldston
confinement time takes the approximate form
ITER89-P scaling ?Esec0.048 I0.85 R1.2 a0.3
?0.5 (n/1020 )0.1 B0.2 A0.5 /P0.5
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Auxilliary-heated Plasmas H-mode

• In 1982 IAEA meeting,a new improved confinement
  regime in diverted ASDEX plasmas was reported
  (F. Wagner et al.) which was termed H-mode(high
  mode). Compared to the L-mode, the energy
  confinement similar to that of ohmic plasma was
  recovered. The reported H-mode confinement time
  typically had the twice that of the L-mode.

The improvement in the energy confinement comes
about mainly due to the increased density while
keeping the central temperature relatively
unchanged (note that since the L-mode has little
density dependence, this is considered to be a
breakthrough). The density and electron
temperature profile during H-mode are broader and
are characterized of having pedestal at the
plasma edge.
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H-mode Characteristics

• H-mode has a similar scaling law, but with 1-2
  times improvement over L-mode ?H H ?E ITER
  89-P or
• ?Th ITER H93-Psec0.053 I1.06 R1.9 a-0.11 ?0.66
  (n/1020 )0.17 B0.32 A0.41 /P0.67
• Reduced edge recycling and often with ELM (edge
  localized mode) activities.
• Broader density and temperature profiles give a
  larger stored energy and high beta discharges.
• However, not much central temperature and
  density increase shows less favorable for
  producing neutrons.
• H-mode transition seems to be strongly related
  to the formation of radial electric field (or
  poloidal rotation).
• Power threshold for H-mode increases with the
  product of density(0.5-1.0) , magnetic field, and
  surface area(0.5-1.0) .
• Various H-modes VH (DIIID), CH(PBX-M),
  CDH(ASDEX-U)

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Other Improved Confinement Modes

• Super-Shot(TFTR) neutral beam injection into a
  low density plasma, two oppositely injected
  balanced beams, low edge recycling, peaked
  density profile, H3
• Hot Ion Mode(JET) similar mode to TFTR super
  shot, high power NBI in low density target
  plasmas, beryllium-conditioned wall, centrally
  peaked density and temperature profiles, highest
  fusion triple product in JET, QDT1
• High-li Mode peaked current density profile
• PEP(Pellet Enhanced Performance) or High ?p
  H-mode H3.8, peaked pressure profiles by the
  injection of hydrogen pellets(JET, JT60-U)
• VH Mode H3.6, boronized wall(DIII-D),
  beryllium surfaces(JET), edge temperature
  pedestal and a high edge bootstrap current
• Enhanced Reversed Shear or Negative Central
  Shear
• TFTR (PRL 75(1995)4417) and DIII-D (PRL
  75(1995)4421)

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Enhanced Reversed Shear(ERSTFTR)Negative
Central Shear(NCSDIII-D)

• Current density profile can be optimized to be
  desirable for confinement, stability, and
  bootstrap current.
• --gt reversed magnetic shear i.e. hollow current
  density profile

reversed shear
current density profile hollow profile
stabilize
micro-instabilities trapped electron modes MHD
instabilities short-wavelength ballooning
modes and resistive tearing modes
high pressure gradient
strong off-axis bootstrp current
core confinement
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TFTR
DIII-D
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Scaling Laws

• Theory
• scale invariance technique
• (Connor-Taylor or Kadomtsev constraints)
• exploits the invariance of the governing
  equations under scale transformations
• no information on geometrical ratios and the
  safety factor
• Experimental scaling relations
• Bohm diffusion
• gyro-Bohm scaling
• ITER89-P scaling law gives Bohm scaling with
• Simple physics model
• beam fueling profile

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Scaling Laws Based on Theory
Vlasov (or F-P) equation with quasi-neutrality
condition (or Poissons eqn)
Three scale transformations
Confinement time
three constraints
Scaling relation
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Experimental Scaling Relations
Bohm diffusion time
More generally

• Gyro-Bohm scaling confinement governed by
  small scale turbulence on the sclae of Larmor
  radius

• Bohm scaling turbulence on the scale of a

• ITER89-P scaling law gives Bohm scaling with

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Scaling Law Based on Physics Model