An introduction to plasma physics.
Logistics Grading Scheme MHD Equillibrium Consequences of MHD Equillibrium Theta Pinch Z Pinch Screw Pinch Toroidal Geometry Definitions Forces Grad Shafranov Equation Stream Functions Grad Shafranov Derivation Sketch Aspect Ratio Expansion Logistics Location: Mudd 825 Time: 1:10-2:40 Book: MHD Stability of Tokamaks by Zohm Grading Scheme Projects: 60% Homework: 40% MHD Equillibrium Suprisingly useful since particles in plasma undergo gyration, which enhances number of collisions and causes discrete particle to behave more fluid-like Setting time derivatives in MHD equations to zero This is a good assumption, since deviations from this result in motion with timescales on the order of microseconds (called the Alfven time $\tau_{A}$) Hence, if you are physically observe the plasma, good chance it is in EQ $\vec{v} = 0$ is the magneto-static limit (or that $v« v_{a}$) where $v_{a}$ is the Alfven speed (which in turn is defined as size of detector divided by $\tau_{a}$) In space, the magneto-static approximation is normally not valid Define the plasma beta as $\beta = \frac{p}{\frac{B^{2}}{2\mu}}$ or the thermal energy divided by the magnetic field energy Equation of state $P = n_{e}T_{e}+n_{i}T_{i}$ Momentum equation: $\rho(\frac{\partial}{\partial t}\vec{v}+\vec{v}\nabla\cdot \vec{v}) = \vec{J}\times B =\nabla P$ Maxwell’s equations $\nabla \times \vec{B} = \mu_{0}\vec{J}$ $\nabla \cdot B = 0$ $v \times B +E =0 \rightarrow E = 0$ $\frac{\partial }{\partial t} B = -\nabla \times E \rightarrow \frac{\partial }{\partial t} B =0$ Continuity $\frac{\partial}{\partial t}\rho + \nabla \cdot v = 0$ 0=0 Energy $\frac{\partial}{\partial t}P+\vec{v}\cdot\nabla p -c_{s}^{2}(\frac{\partial}{\partial t}\rho + \vec{v}\cdot \nabla \rho) = 0$ To Summarize, for MHD equilibrium, we have that...