Plasma Physics Kinetic plasma description A. Flacco Structure from - - PowerPoint PPT Presentation

Plasma Physics

Kinetic plasma description

• A. Flacco

Structure

• from particle description to PDF

3

• Kinetic Description

5

• Vlasov Equation

9

• nth-order momentum equation

10

• A. Flacco/ENSTA - PA201: Introduction

Page 2 of 21

Lagrangian description

• f a set of interacting particles

The complete description of a set on N particles must take in account:

• 1. the equations of movement

    

dxi dt = vi dvi dt = qi mi [Em (xi, t) + vi × Bm (xi, t)]

• 2. the Maxwell equations (with sources)

  

∇ · Em (xi, t) = ρ/ε0 ∇ × Bm (xi, t) = µ0Jm (xi, t) + µ0ε0 ∂Em ∂t (xi, t)

• A. Flacco/ENSTA - PA201: Introduction

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Lagrangian description

Sources

  

∇ · Em (xi, t) = ρ/ε0 ∇ × Bm (xi, t) = µ0Jm (xi, t) + µ0ε0 ∂Em ∂t (xi, t)

    

ρm (xi, t) = ρext +

• i

qiδ (x − xi (t)) Jm (xi, t) = Jext +

i qiviδ (x − xi (t))

• A. Flacco/ENSTA - PA201: Introduction

Page 4 of 21

Eulerian description

Fluid equations describe a plasma at the thermal equilibrium (independent variables are space and time, being the temperature fixed.) Such a model works extremely well, unless deformations of the velocity distribution can live sufficiently long time. Eg. low collision rate (low density, high

• temperature. . . )

The velocity distribution is a function of seven independent variables: fα,micr (r, v, t) =

• i

δ [r − ri (t)] δ [v − vi (t)] This is a single particle distribution function: correlations among particles are not taken in account.

• A. Flacco/ENSTA - PA201: Introduction

Page 5 of 21

One Particle Distribution Function

The density of particles with velocity between v and v + d3v is written as: fα,micr (r, v, t) d3v which means than particle density can be calculated by integrating the distribution function over velocities: nα (r, t) =

• fα,micr (r, v, t) d3v

The normalized distribution function is defined as: f ′ = 1 n (r, t) f

• A. Flacco/ENSTA - PA201: Introduction

Page 6 of 21

Distribution function and Evolution

from Liouville to Vlasov Following the usual (in statistical mechanics) path, the evolution in time of the fα () function shall be ruled by the Liouville theorem: dfα dt = ∂fα ∂t +

• i

∂fα

∂xi ∂xi ∂t + ∂fα ∂pi ∂pi ∂t

• = 0

hence ∂fα ∂t + (v · ∇) fα + Fα mα ∂fα ∂v = 0 However, two points should be addressed before naming this equation:

• Fα = qα (E + v × B), hence the dependence on the subscript α;
• fα,micr () −

→ fα () ?

• A. Flacco/ENSTA - PA201: Introduction

Page 7 of 21

Distribution function and Evolution

fmicr → f By definition fα,micr will have strongly

• scillating terms.

Let’s put fα,micr = fα,micr + ˜ fα,micr where by construction ˜ fα,micr = 0 In a similar manner are defined:

• Emicr

= Emicr + ˜ Emicr Bmicr = Bmicr + ˜ Bmicr Renaming the averaged quantities to fα, E and B, the equation is rewritten as:

• ∂fα

∂t + ∂˜ fm ∂t

• + (v · ∇)

fα + ˜ fm

• + qα

mα [E + ~ Em + v × (B + ~ Bm)]

• ∂fα

∂v + ∂˜ fm ∂v

• = 0
• A. Flacco/ENSTA - PA201: Introduction

Page 8 of 21

Distribution function and Evolution

Vlasov and collisions Taking the average of the entire equation we obtain ∂fα ∂t + (v · ∇) fα + qα mα [E + v × B] ∂fα ∂v = qα mα

˜

Em + v × ~ Bm

∂˜

fm ∂v

• r

∂fα ∂t + (v · ∇) fα + qα mα [E + v × B] ∂fα ∂v =

∂fα

∂t

• c

In conditions where collisions can be neglected the last term is dropped, resulting in the well known Vlasov Equation: ∂fα ∂t + (v · ∇) fα + qα mα [E + v × B] ∂fα ∂v = 0

• A. Flacco/ENSTA - PA201: Introduction

Page 9 of 21

fα distribution function

Let ψ (v) a function of the velocity v. The averaged value of ψ (v) on fα is then: ψ (v)α =

• ψ (v) fαd3v
• fαd3v

We will than call

• vm

x vn y vr z fαd3v

the m − n − r order momentum of fα. In particular:

• ψ (v) = 0: zeroth order momentum
• ψ (v) = mαv: first order momentum
• etc. . .
• A. Flacco/ENSTA - PA201: Introduction

Page 10 of 21

Distribution function

• 10
• 5

5 10

• 10
• 5

5 10 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 f(vx,vy) vx vy f(vx,vy)

• A. Flacco/ENSTA - PA201: Introduction

Page 11 of 21

Distribution function

• 10
• 5

5 10

• 10
• 5

5 10 vy vx

• 10
• 5

5 10

• 10
• 5

5 10 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 f(vx,vy) vx vy f(vx,vy)

• A. Flacco/ENSTA - PA201: Introduction

Page 12 of 21

fα distribution function

General momenta equation Let ψ ≡ ψ (v). In order to calculate its evolution on the distribution function, we take its average on the Vlasov equation:

• ψ (v) dfα

dt

• = 0
• r, explicitely
• ψ (v) ∂fα

∂t + ψ (v) (v · ∇) fα + ψ (v) qα mα [E + v × B] ∂fα ∂v d3v = 0 Termwise averaging of ψ () function on the Vlasov equation gives: ∂ ∂t [n ψ]

• 1

+ ∇· [n ψv]

• 2

−n q m

• ∂ψ

∂v · E 3 + ∂ψ ∂v · (v × B)

• 4
• = 0
• A. Flacco/ENSTA - PA201: Introduction

Page 13 of 21

Zeroth order equation: ψ = 1

Continuity equation From trivial calculations, and having set u = v we have ∂n ∂t + ∇· (nu) = 0 This is the continuity equation, already derived elsewhere.

• A. Flacco/ENSTA - PA201: Introduction

Page 14 of 21

Continuity Equation

Convective Derivative Convective derivative: d dt G (x, t) = ∂G ∂t + ∂G ∂x dx dt = ∂G ∂t + ux ∂G ∂x dG dx = ∂G ∂t + (u · ∇) G ≡ DG Dt Equation of Continuity: ∂nα ∂t + ∇ · (nαuα) = 0 and, in terms of convective derivative D Dt nα + nα (∇ · uα) = 0

• A. Flacco/ENSTA - PA201: Introduction

Page 15 of 21

First order equation: ψ = mv

Momentum equation By substitution it is obtained ∂ ∂t mn v + ∇· [nm v v] − nq E + v × B = 0 The term in vv is expanded through the usual substitution of v = v + w = u + w, giving m∇· [n vv] = m∇· [n (u + w) (u + w)] = m∇· [n uu + n ww + n uw + n uu + n wu] = mn(u · ∇) u + mu [∇· (nu)] + ∇· (mn ww) Finally using the convective derivative definition and the continuity equation, the usual form is obtained: mn Du Dt = nq [E + u × B] − ∇·P

• A. Flacco/ENSTA - PA201: Introduction

Page 16 of 21

First order equation: ψ = mv

Pressure term In previous equation it has been introduced the pressure tensor P = mn ww Diagonal terms represent pressure in the common sense, while off-diagonal, cross-relating different components of the thermal velocity represent viscosity. Considering an isotropic, non viscous plasma, the term can be simplified to ∇·P = ∇·1p = ∇p Being p = mn w2 it holds, for maxwellian velocity distribution p = 3nkBT

• A. Flacco/ENSTA - PA201: Introduction

Page 17 of 21

Second order equation: ψ = 1

2mv2 Heat equation The energy evolution is calculated by averaging the kinetic energy. Following the terms numbering we get: 1 2 m ∂ ∂t

• n

v2

• 1: Energy variation

+ 1 2 m∇· n vv2

• 2: Heat flux

− nqu · E

3: Acceleration

+✘✘✘✘

✘ ✿ 0

v · (v × B) = 0 A more precise insight in the terms’ meaning is obtained from the usual velocity decomposition, defining the thermal velocity: w = v + v

• A. Flacco/ENSTA - PA201: Introduction

Page 18 of 21

Second order equation: ψ = 1 2mv2

Terms decomposition From the previous calculation:

• Being

v2 = (u + w) · (u + w) = u2 + w2 we get: 1 = ∂ ∂t

1

2 mnu2 + 1 2 mn |v − u|2

• By decomposing the third order term
• vv2

= (u + v) |u + v|2 = uu2 + u w2 + ww2 + 2u · ww we get: 2 = 1 2 mu2u

Macroscopic energy flux

+ 1 2 m w2 u

• Heat convection

+

Q

• 1

2 m ww2

• Heat conduction

+

P·u

• mn ww · u
• System

work flux for compres- sion/expansion

• A. Flacco/ENSTA - PA201: Introduction

Page 19 of 21

Second order equation: ψ = 1 2mv2

Terms reduction The final (and more compact) form of the third order equation is obtained taking in account the first two and reducing. It holds 1 2 mn D Dt

• w2

+ ∇ · Q + P : ∇u = 0 where the two definitions for pressure, P ≡ mn ww, and for heat flux, Q ≡ 1

2 m

ww2 . The equation can be read as “convective derivative of the thermal energy equals the thermal flux plus the work flux of the system, for expansion or compression”

• A. Flacco/ENSTA - PA201: Introduction

Page 20 of 21

Conclusions

• Probability density function to describe the plasma
• Vlasov equation (collisions yet to be defined)
• Different order momenta are calculated on f ()
• Fluid equations hierarchy
• A. Flacco/ENSTA - PA201: Introduction

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