introduction to plasma physics fulvio zonca · a.a.2019-20 technologiesfornuclearfusion...

A.A. 2019-20 TECHNOLOGIES FOR NUCLEAR FUSION PART I – Physics Lecture 2 – 1

Lecture 2

INTRODUCTION TO PLASMA PHYSICS

Fulvio Zonca

http://www.afs.enea.it/zonca

ENEA C.R. Frascati, C.P. 65 - 00044 - Frascati, Italy.

Institute for Fusion Theory and Simulation, Zhejiang University, Hangzhou 310027, P.R.C.

March 11.th, 2020

TECHNOLOGIES FOR NUCLEAR FUSIONPART I – Physics

INTRODUCTION TO PLASMA PHYSICS

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SyllabusMain areas that will be explored are:

• Lecture 1

i) Classification of plasmas, Debye length

ii) Collisions between charged particles

iii) Collisional slowing down, plasma resistivity

iv) Fusion reactor scheme

v) Power balance, Lawson Criterion

vi) Ideal ignition temperature

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• Lecture 2

i) Solution of the power balance equation

ii) Main heating and loss terms

iii) Scaling laws for the energy confinement time

iv) Tokamak and stellarator

v) Brief introduction to inertial confinement fusion

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Solution of the power balance equation

It is possible to reconsider the power balance of a fusion reactor (lecture 1)and describe the evolution of the thermal energy content as

d

dt3nT =

n2

4〈σv〉Eα + Padd −

3nT

τE(n, T )

If additional heating is slowly increased, the solution of the power balanceequation can be understood as a sequence of quasi steady-states, for whichthe r.h.s. nearly balances to zero.

Largest uncertainty is coming from the (n, T ) dependence of τE. Assuminggiven τE , one can estimate the power needed to approach ignition.

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General behavior expected and addi-tional power needed for ignition tem-perature at 10 keV.

Contours of additional power re-quired to maintain (n, T ) constant inthe (n, T ) plane. Trajectory to igni-tion requiring minimum power is onewhich goes over the saddle point: theCordey pass.

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Stability of ignition condition: consider the ignition condition

3T

τE(n, T )=

n

4〈σv〉Eα

and consider a small change ∆T of the operation temperature.

Expanding the power balance about the operation temperature

3nd

dt∆T =

[

n2

4

d 〈σv〉

dTEα −

3n

τE(n, T )+

3nT

τ 2E(n, T )

dτEdT

]

∆T

Substituting the power balance at ignition

3nd

dt∆T =

n2

4〈σv〉

Eα

T

[

T

〈σv〉

d 〈σv〉

dT− 1 +

T

τE

dτEdT

]

∆T

If the quantity in square parentheses is positive, the operation temperaturegrows exponentially.

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The condition for stability is

T

τE

dτEdT

< 1−T

〈σv〉

d 〈σv〉

dT

If the system is not stable at igni-tion temperature, it will be neces-sary to provide a feedback controlfor adjusting one of the knobs inthe power balance equation.

Possibilities are density, D-T mix-ture and confinement time (e.g.through Padd). Another impor-tant factor is the accumulation ofhelium ash.

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Main heating and loss terms

Schematic representation of plasma heating processes (in addition to nuclearfusion self-heating; lecture 1).

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Ohmic heating: we know from lecture 1 that plasmas have finite resistivity,defined as

η =me

nee2τe−i90

=8πm

1/2e Ze2 ln Λ

(3kT )3/2

The power density dissipated by plasma current density due to finiteresistivity is (V is the plasma volume, κ is the plasma elongation andZeff =

∑

i niZ2i /

∑

i niZi)

PΩ

V= JE = ηJ2 =

1

ηE2 ≃

2.8× 10−15ZeffI2[A]

κ2a4[m]T3/2e [keV ]

[

MW/m3]

Note that PΩ ∝ T−3/2e . Thus, it is questionable whether a reactor can be

heated to the temperatures needed for fusion reactions to occur using onlythe Ohmic heating since its efficiency decreases with increasing temperature.

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Additional heating and current-drive systems: from radio-frequency wavesand neutral beam injection (last module of Part I).

Neutral beam injection as high energy particles into thermonuclear plasmasis the most efficient additional heating to date.

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Loss terms generally consist of losses connected to radiative processes andof losses due to plasma transport.

In lecture 1, we have described the combined effect of power loss due toplasma radiation, PR and, of power loss due to plasma transport, PL, intro-ducing the concept of energy confinement time.

It is typical to use this approach (of introducing an appropriate confine-ment time) for transport losses (see later), which are largely dominated byfluctuation induced processes on micro- to macro-scales.

Radiation losses are consequence of the power emitted by a charged particlesubject to acceleration. In the non-relativistic limit the Larmor equationapplies

Prad =2

3

q2|v|2

c3

Due to the dependence on |v|2, radiation losses are mostly due to electrons.Relativistic generalization is given by Lienard-Wiechert potentials.

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Radiation losses: are essentially given by the following three processes

• Bremsstrahlung: the radiation lost by a particle due to the accelera-tion by Coulomb collisions

• Cyclotron emission: the radiation lost by a particle due to the accel-eration produced by the helical motion in the ambient B field

• Impurity radiation: the radiation lost by partially ionized impurityatoms

– line radiation: due to radiative decay of an electron from oneexcited level to lower excited or ground state, after collisionalexcitation

– recombination radiation: due to the recombination of a freeplasma electron with an ionized atom, to a lower ionization stateof the same atom

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Bremsstrahlung: quantum mechanics is needed for accurate calculation.A good estimate can be obtained using the Rutherford cross section andnoting that, in Coulomb collisions,

|v| ≃Ze2

meb2; ⇒ Pbrems = nine|ve|

∫ bmax

bmin

Prad2πbdb .

Pbrems ≃ 4.8× 10−37Z2ni[m−3]ne[m

−3]T 1/2e [keV ]

[

W/m3]

.

Cyclotron emission: accurate calculation implies taking into account therelativistic electron distribution function, the absorption and re-emission ofcyclotron radiation by the plasma and the reflection of the radiated powerfrom surrounding wall. An approximated expression is given by

Pcycl ≃ 6.2× 10−17B2[T ]ne[m−3]Te[keV ] (1. + Te[keV ]/204 + ...)

[

W/m3]

.

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Impurity radiation: in the rangeof temperatures of fusion interest,the line radiation and recombina-tion radiation can be fitted as

Pline ≃ 1.8×10−38Z4ni[m

−3]ne[m−3]

T1/2e [keV ]

[

W

m3

]

.

Prec ≃ 4.1×10−40Z6ni[m

−3]ne[m−3]

T3/2e [keV ]

[

W

m3

]

.

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Scaling laws for the energy confinement time

Power losses due to transport processes can be due to Coulomb collisionsas well as fluctuations.

Collisional transport can be understood as resultant of a diffusion processby which particles jump from a magnetic field line to nearby ones, sinceconstants of motion are broken by Coulomb collisions.

It is possible to accurately calculate collisional transport (not subject of thislecture) and to experimentally verify theoretical predictions.

As anticipated above, transport losses are largely dominated by fluctuationinduced processes on micro- to macro-scales.

To date, a first principle calculation of fluctuation induced transport lossesis not yet feasible, although important progress has been made. Addressingthese issues is outside the present scope.

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It is necessary to resort to empirical representations of the confinementtime, which take the form of the product of powers of various parametersinvolved.

The lack of understanding of the underlying physics is such that it is notpossible to find one single scaling law of general validity. One rather findsa patchwork of scaling expressions, which apply in different regimes of op-eration

• Ohmically heated plasmas

• Additionally heated plasmas: L-mode (low)

• Additionally heated plasmas: H-mode (high)

• Enhanced of improved confinement plasmas

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Ohmically heated plasmas At low density, τE is found to scale as (a/Rand minor/major radii and q is the safety fac-tor; MKS units)

τE = 0.07(n/1020)aR2q

As density is increased, the linear improve-ment with n is lost and τE saturates at (Ais the atomic mass, I the plasma current andκ = b/a is the plasma elongation)

nsat/1020 = 0.06(I/106)RA0.5κ−1a−2.5

By controlling the density and maintaininga peaked density profile it has been possibleto extend the linear density scaling to the socalled Improved Ohmic Confinement (IOC)regime.

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L-mode confinement: when plasmas are heated by either neutral beaminjection or radio-frequency waves, τE is found to degrade with the injectedpower.

This tendency is captured by the Goldston scaling

τG = 0.037(I/106)R1.75κ0.5a−0.37(P/106)−0.5

This scaling was obtained before large tokamak were operational. Althoughit works well also on big machines, an extended database was proposed toimprove the predictive capability for ITER: the ITER89-P scaling

τ ITER89−PE = 0.048(I/106)0.85R1.2a0.3κ0.5(n/1020)0.1B0.2A0.5(P/106)−0.5

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H-mode confinement: in tokamaks with a divertor, an abrupt transition andimprovement in the confinement time is found for sufficient applied power,leading to edge pedestals in temperature and density.

H-mode was originally discovered on the ASDEX tokamak and then repro-duced in all other devices with same features.

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H-mode scalings similar to ITER89-P have been proposed.

τ ITERH93−PE = 0.053(I/106)1.06R1.9a−0.11κ0.66(n/1020)0.17B0.32A0.41(P/106)−0.67

when the plasma edge is affected by the so-called Edge Localized Modes(ELMs), another expression is often used

τIPB98(y,2)E = 0.145(I/106)0.93R1.39a0.58κ0.78(n/1020)0.41B0.15A0.19(P/106)−0.69

Scalings are similar to those of the L-mode. Thus, it is useful to introducethe H factor, relating L-mode and H.mode confinement

H =τEτLE

The H factor is typically 2, but it is generally reduced in the presence ofELMs.

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Enhanced of improved confinement plasmas: further improvement of τEwith respect to H-mode have been obtained by suitably tailoring plasmaprofiles

• controlling current density profile

• controlling plasma rotation

• controlling plasma impurities

• by choice of proper plasma facing components (e.g. wall material,including liquid metal)

A detailed analysis of improved confinement regimes is beyond the presentscope.

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Tokamak and stellarator

Toroidal curvature (lecture 1) is not simply good.

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In a tokamak this is corrected by the presence of a plasma current.

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Schematic view of a tokamak.

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The International Thermonuclear Experimental Reactor in construction inCadarache, France.

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In a stellarator the beneficial effect of twisted field lines is obtained by theexternal coils at the cost of loosing the toroidal symmetry.

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The Wendelstein 7-X stellarator in Greifswald, Germany.

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Advantages of stellarators w.r.t. tokamaks

• Stationary plasma operations (no transformer/no induced current)

• No net plasma current and, therefore, essentially free of current driveninstabilities

Disadvantages of stellarators w.r.t. tokamaks

• Complex magnetic field coils

• Curved coils lead to large forces and require strong supporting struc-tures

• Difficult to make compact devices

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Brief introduction to inertial confinement fusion

Inertial confinement relies on the dynamic compression of a D−T fuel pelletobtained by high energy laser or heavy ion beams.

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Extreme conditions for inertial fusion are found only in stellar interiors andnuclear weapon tests

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There are two principal approaches to compression in inertial confinementfusion

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Radiation (x-rays or laser) or particle (ion) energy deposition rapidly heatsthe surface of the pellet, forming a surrounding plasma envelope

Fuel is compressed by the rocket-like blow-off of the hot surface material

During the final part of the capsule implosion, the fuel core reaches densityand temperature satisfying the Lawson criterion and ignites

Thermonuclear burn spreads rapidly through the compressed fuel, yieldingmany times the input energy

Indirectly-driven central hot spot ignition is the scenario currently fol-lowed for demonstration on the National Ignition Facility (NIF) in the US(Livermore, CA) and on the Laser Megajoule (LMJ) in France (Le Barp,Aquitaine).

Scientific break-even reached on NIF in 2014 [Nature 506, 343].

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X-rays enhance implosion symmetry and reduce hydrodynamic instabilityat a cost in efficiency

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Fast ignition is an approach to ICF which decouples compression from ig-nition

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Fast ignition relies on the possibility of overheating a pre-compressed targetwith help of an external match. This was proposed in the 60’2, but is madepossible by the recent development of picosecond petawatt-class laser.

Physics basis is not clear yet. Key issue is the energy transport of relativisticelectron beams at ultra-high current (GA) from the laser interaction regionat critical density ncrit to the compressed fuel core at n/ncrit ≈ 105.

Two options for bringing the igniter beam closer to the fuel core:

• laser beam bores a hole into the over-dense plasma

• use a metallic cone to guide the igniter beam

European efforts on fast ignition have been recently merged in the proposalof a European High Power laser Energy Research facility (HiPER).

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Shock ignition is an approach to ICF which aims at reducing the high im-plosion velocities (350-400 km/s) of central hot spot ignition, and thereforethe role of hydrodynamic instabilities

Ignition at lower implosion velocities (240-300 km/s) is achieved by a strongconvergent shock (produced by a ∼ 106W/cm2 laser pulse) precisely timedto reach fuel at stagnation. Experimental and numerical studies are under-way.

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Exercises

Justify the Bremsstrahlung expression

Pbrems = nine|ve|

∫ bmax

bmin

Prad2πbdb

Show how you can use the Larmor equation to calculate this expression.

Show that in a non-uniform B field the mirror force acting on a particle isF = −µ∇B, with µ the magnetic moment. Show that, in the presence ofan electric field, particles are subject to the drift motion vE = cE×B/B2.

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introduction to plasma physics fulvio zonca · a.a.2019-20 technologiesfornuclearfusion...

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