Plasma Physics Kinetic plasma description A. Flacco
Structure • from particle description to PDF 3 • Kinetic Description 5 • Vlasov Equation 9 • n th -order momentum equation 10 A. Flacco/ENSTA - PA201: Introduction Page 2 of 21
Lagrangian description of a set of interacting particles The complete description of a set on N particles must take in account: 1. the equations of movement d x i = v i d t d v i q i = [ E m ( x i , t ) + v i × B m ( x i , t )] d t m i 2. the Maxwell equations (with sources) ∇ · E m ( x i , t ) = ρ/ε 0 ∂ E m ∇ × B m ( x i , t ) = µ 0 J m ( x i , t ) + µ 0 ε 0 ( x i , t ) ∂ t A. Flacco/ENSTA - PA201: Introduction Page 3 of 21
Lagrangian description Sources ∇ · E m ( x i , t ) = ρ/ε 0 ∂ E m ∇ × B m ( x i , t ) = µ 0 J m ( x i , t ) + µ 0 ε 0 ( x i , t ) ∂ t � ρ m ( x i , t ) = ρ ext + q i δ ( x − x i ( t )) i J ext + � J m ( x i , t ) = i q i v i δ ( x − x i ( 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: f α, micr ( r , v , t ) = � δ [ r − r i ( t )] δ [ v − v i ( t )] i 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 + d 3 v � is written as: f α, micr ( r , v , t ) d 3 v which means than particle density can be calculated by integrating the distribution function over velocities: � f α, micr ( r , v , t ) d 3 v n α ( r , t ) = The normalized distribution function is defined as: 1 f ′ = 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 f α () function shall be ruled by the Liouville theorem: � ∂ f α � � d f α d t = ∂ f α ∂ x i ∂ t + ∂ f α ∂ p i ∂ t + = 0 ∂ x i ∂ p i ∂ t i hence ∂ f α ∂ t + ( v · ∇ ) f α + F α ∂ f α ∂ v = 0 m α However, two points should be addressed before naming this equation: • F α = q α ( E + v × B ), hence the dependence on the subscript α ; • f α, micr () − → f α () ? A. Flacco/ENSTA - PA201: Introduction Page 7 of 21
Distribution function and Evolution f micr → f By definition f α, micr will have strongly In a similar manner are defined: oscillating terms. Let’s put � � E micr � + ˜ E micr = E micr � B micr � + ˜ f α, micr = � f α, micr � + ˜ = f α, micr B micr B micr where by construction Renaming the averaged quantities to f α , E and B , the equation is rewritten as: � ˜ f α, micr � = 0 � � ∂ t + ∂ ˜ ∂ f α f m + ( v · ∇ ) � � f α + ˜ f m ∂ t � � ∂ v + ∂ ˜ ∂ f α + q α f m [ E + ~ E m + v × ( B + ~ B m )] = 0 m α ∂ v 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 �� ˜ � � ∂ ˜ ∂ f α [ E + v × B ] ∂ f α ∂ t + ( v · ∇ ) f α + q α ∂ v = q α f m E m + v × ~ B m m α m α ∂ v or � ∂ f α � ∂ f α ∂ t + ( v · ∇ ) f α + q α [ E + v × B ] ∂ f α ∂ v = m α ∂ t c In conditions where collisions can be neglected the last term is dropped, resulting in the well known Vlasov Equation: ∂ f α [ E + v × B ] ∂ f α ∂ t + ( v · ∇ ) f α + q α ∂ v = 0 m α A. Flacco/ENSTA - PA201: Introduction Page 9 of 21
f α distribution function Let ψ ( v ) a function of the velocity v . The averaged value of ψ ( v ) on f α is then: � ψ ( v ) f α d 3 v � ψ ( v ) � α = � f α d 3 v We will than call � z f α d 3 v v m x v n y v r the m − n − r order momentum of f α . In particular: • ψ ( v ) = 0: zeroth order momentum • ψ ( v ) = m α v : first order momentum • etc. . . A. Flacco/ENSTA - PA201: Introduction Page 10 of 21
Distribution function 0.09 0.08 0.07 0.06 f(v x ,v y ) f(v x ,v y ) 0.05 0.04 0.03 0.02 0.01 0 -10 -5 10 0 5 v y 0 5 -5 v x 10 -10 A. Flacco/ENSTA - PA201: Introduction Page 11 of 21
Distribution function 10 0.09 0.08 0.07 0.06 f(v x ,v y ) f(v x ,v y ) 0.05 0.04 0.03 5 0.02 0.01 0 -10 -5 10 0 5 v y 0 0 5 v y -5 v x 10 -10 -5 -10 -10 -5 0 5 10 v x A. Flacco/ENSTA - PA201: Introduction Page 12 of 21
f α 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 ) d f α = 0 d t or, explicitely � ψ ( v ) ∂ f α [ E + v × B ] ∂ f α ∂ t + ψ ( v ) ( v · ∇ ) f α + ψ ( v ) q α ∂ v d 3 v = 0 m α Termwise averaging of ψ () function on the Vlasov equation gives: � � ∂ − n q ∂ψ + ∂ψ ∂ t [ n � ψ � ] + ∇· [ n � ψ v � ] ∂ v · E ∂ v · ( v × B ) = 0 m � �� � � �� � � �� � � �� � 2 1 3 4 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 + ∇· ( n u ) = 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 t G ( x , t ) = ∂ G d ∂ t + ∂ G d x d t = ∂ G ∂ G ∂ t + u x ∂ x ∂ x d G d x = ∂ G ∂ t + ( u · ∇ ) G ≡ D G D t Equation of Continuity: ∂ n α + ∇ · ( n α u α ) = 0 ∂ t and, in terms of convective derivative D D t n α + n α ( ∇ · u α ) = 0 A. Flacco/ENSTA - PA201: Introduction Page 15 of 21
First order equation: ψ = m v 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 + m u [ ∇· ( n u )] + ∇· ( mn � ww � ) Finally using the convective derivative definition and the continuity equation, the usual form is obtained: mn D u D t = nq [ E + u × B ] − ∇· P A. Flacco/ENSTA - PA201: Introduction Page 16 of 21
First order equation: ψ = m v 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 = ∇· 1 p = ∇ p Being p = mn � w 2 � it holds, for maxwellian velocity distribution p = 3 nk B T A. Flacco/ENSTA - PA201: Introduction Page 17 of 21
Second order equation: ψ = 1 2 mv 2 Heat equation The energy evolution is calculated by averaging the kinetic energy. Following the terms numbering we get: ✿ 0 1 2 m ∂ � n � v 2 �� + 1 2 m ∇· � n � v v 2 �� ✘ + ✘✘✘✘ − nq u · E v · ( v × B ) = 0 ∂ t � �� � � �� � � �� � 3: Acceleration 1: Energy variation 2: Heat flux 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 2 mv 2 Terms decomposition From the previous calculation: • Being � v 2 � = � ( u + w ) · ( u + w ) � = u 2 + � w 2 � we get: � 1 | v − u | 2 �� = ∂ 2 mnu 2 + 1 2 mn � 1 ∂ t • By decomposing the third order term � v v 2 � = � ( u + v ) | u + v | 2 � = u u 2 + u � w 2 � + � w w 2 � + 2 u · � ww � we get: Q P · u � �� � � �� � 1 1 1 2 m � w 2 � 2 m � w w 2 � 2 mu 2 u 2 = + u + + mn � ww � · u � �� � � �� � � �� � � �� � System Macroscopic Heat convection Heat conduction work flux energy flux for compres- sion/expansion A. Flacco/ENSTA - PA201: Introduction Page 19 of 21
Second order equation: ψ = 1 2 mv 2 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 � w 2 � + ∇ · Q + P : ∇ u = 0 D t where the two definitions for pressure, P ≡ mn � ww � , and for heat flux, 2 m � w w 2 � Q ≡ 1 . 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
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