Fundamentals of Plasma Physics
290 Chapter 9. MHD equilibria Thus, the system of equations can be summarized as (Bellan 1992) ∂ ∂t (χ r^2 ) +U·∇ (χ r^2 ) = υ∇· ...
9.9 Dynamic equilibria:flows 291 Onceχ,ψandIhave been determined, it is possible to determine the pressure and electrostatic pot ...
292 Chapter 9. MHD equilibria stagnation region Ir,zconst. r,zconst. z z Ir,zconst. r,zconst. high pressure low pr ...
9.9 Dynamic equilibria:flows 293 hand ifJpolwere not parallel toBpol,these surfaces would not be coincident and there would be a ...
294 Chapter 9. MHD equilibria Integration and using the boundary condition thatP=0atr=agives P(r)= λ^2 ψ^20 4 π^2 μ 0 a^2 ( 1 − ...
9.10 Assignments 295 can be imagined as a sort of “zipper” which collimates theflux surfaces. Collimation is often seen in curre ...
296 Chapter 9. MHD equilibria Shafranov equation describes axisymmetric equilibria and so can be used to charac- terize an axisy ...
9.10 Assignments 297 ne/ 2 .(Hint: the mean energy per degree of freedom isκT/ 2 so the mean energy of a particle moving in thre ...
10 Stability of static MHD equilibria Solutions to Eq.(9.49), the Grad-Shafranov equation, (or to some more complicated coun- te ...
10.1 The Rayleigh-Taylor instability of hydrodynamics 299 The equation of motion for this system is m d^2 x dt^2 =±κx (10.1) whe ...
300 Chapter 10. Stability of static MHD equilibria adequate to support the inverted water. From a mathematical point of view, th ...
10.1 The Rayleigh-Taylor instability of hydrodynamics 301 The perturbation is assumed to have the form v 1 =v 1 (y)eγt+ik·x (10. ...
302 Chapter 10. Stability of static MHD equilibria Substitution of Eq.(10.17) into Eq.(10.19) gives the dispersion relation γ^2 ...
10.2 MHD Rayleigh-Taylor instability 303 onyand thus rotates as a function ofy.In this latter case, the magnetic field lines are ...
304 Chapter 10. Stability of static MHD equilibria Theycomponent is found by dotting withˆyand then using the vector identity∇·( ...
10.2 MHD Rayleigh-Taylor instability 305 Finitek·B 0 opposes the effect of the destabilizing positive density gradient, reducing ...
306 Chapter 10. Stability of static MHD equilibria Jzand∂Bz/∂y∼Jx;this shear inhibits interchange instabilities. However, as wil ...
10.3 The MHD energy principle 307 perfectly conducting w all Gaussian pillbox equilibrium surface x x perturbed surface vacuu ...
308 Chapter 10. Stability of static MHD equilibria ∂ρ ∂t +∇·(ρU)=0 (10.40) P∼ργ. (10.41) Ohm’s law, Eq.(10.37), and Faraday’s la ...
10.3 The MHD energy principle 309 Eq.(10.49) can be written as ∂ ∂t ( ρU^2 2 + P γ− 1 + B^2 2 μ 0 ) +∇· ( ρ U^2 2 U+ E×B μ 0 + γ ...
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