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Massachusetts Institute of Technology Subject 6.651J/8.613J/22.611J 1 December 2005 R. Parker Due: 8 December 2005 Problem Set 9 Problem 1. In this problem we’ll develop a general (if a bit formal) derivation of the Vlasov equation. As we discussed in class, conservation of particles requires r r D( fΔr Δv ) = 0, Dt where the symbol D/Dt means that the time derivative is to be taken along a particle orbit in phase space. Specifically, r r r r r r r r D( f ) f (r + v Δt , v + a (r , v , t )Δt , t + Δt ) − f (r , v , t ) = lim(Δt → 0) . Dt Δt a) Show that the RHS of this expression reduces to r ∂f r + v ⋅ ∇f + a ⋅ ∇ v f ∂t b) Now consider the term r r r r r r Δr Δvrr + vrΔt ,vr + arΔt ,t + Δt − Δr Δvrr ,vr ,t D (Δr Δv ) = lim(Δt → 0) Δt Dt As indicated, the rate of change of the phase space volume is to be calculated along a particle trajectory in phase space. For small Δt along this trajectory, the particle orbit is simply given by r r r r ' = r + v Δt r r r r r v ' = v + a (r , v , t )Δt r r r r These equations define a simple transformation of variables between the r , v and r ' , v ' coordinates. Accordingly, a well-known mathematical result is that the volume elements are related by the Jacobian of the transformation defined as the determinant of the matrix ∂x′j ⎤ ⎡ ⎢cij = ⎥. ∂xi ⎦ ⎣ r r D(Δr Δv ) and complete the derivation of Vlasov’s equation. Use this result to calculate Dt Problem 2. In deriving the fluid energy equation in class, we got to the following point: r r m r 2 r t r⎞ r 3 5 ∂ ⎛1 ⎛1 ⎞ 2 2r ⎜ mnu + nkT ⎟ + ∇ ⋅ ⎜ mnu u + nkTu + Π ⋅ u ⎟ + ∇ ⋅ q − qE ⋅ nu = ∫ dv v ∑ C ( f , f β ) . 2 2 2 ∂t ⎝ 2 ⎠ ⎝2 ⎠ β It was then stated that this equation could be transformed into the simpler form: r r ∂u 3 d kT n + p∇ ⋅ u = −Π ij i − ∇ ⋅ q + Q , 2 dt ∂x j where the summation convention applies and Q= m r 2 dww ∑ C ( f , f β ) . 2∫ β Show that the second form follows from the first by using the continuity equation, the result of dotting r the momentum equation with u , and the identity t t r r ∂u − u ⋅ (∇ ⋅ Π ) + ∇ ⋅ (Π ⋅ u ) = Π ij i . ∂x j Problem 3. Use the 1-D model of the SOL developed in class to calculate the temperature at the divertor plates for Alcator C-Mod using the following assumptions and parameters: Conduction limited regime ( nu is neglible) Temperature equilibrated ( Te = Ti ≡ T ) Length of divertor channel l = 10 m Temperature channel input = 50 eV Power density at channel input = 500 MW/m2. Sketch T (x) where x is distance along the divertor channel from the input.