An introduction to plasma physics.

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

  • $\nabla\times B = \mu_{0}\vec{J}$
  • $\vec{J}\times\vec{B} = \nabla P$
  • $\nabla \cdot \vec{B} = 0$

Consequences of MHD Equillibrium

  • $\vec{B}\cdot \vec{\nabla p} = 0$
    • The pressure gradient is normal to the field lines (or in other words, there is no pressure gradient along the magnetic field lines)
      • A consequence of this is that you can define pressure surfaces in terms of flux instead of radial distance (ie. $p(r) = P(\Psi)$)
  • From $\nabla p = J\times B$ and $\nabla \times B = \mu J$, one can use the identity $-(\nabla \times B) \times B = \frac{\nabla B^2}{2}-B\cdot \nabla B$ and some algebra to get
    • $0 = \nabla (\frac{B^{2}}{2\mu_{0}}+p)-\frac{B\cdot \nabla B}{\mu_{0}}$
    • Let $\vec{B} = B\hat{b}$
      • $B\cdot \nabla B = B^{2}b\cdot \nabla b = B^{2}\kappa$
        • $\vec{\kappa}$ is the curvature vector, which is defined as $\frac{\vec{-R_{c}}}{R_{c}^{2}} = \vec{\kappa}$
      • $\nabla = \nabla_{\perp}+(\hat{b}\cdot \nabla)\hat{b} = \nabla_{\perp}$
    • Combining the above lines yield: $\nabla_{\perp}(p+\frac{B^{2}}{2\mu_{0}})-\frac{\vec{B^{2}}}{\mu_{0}}\vec{\kappa} = 0$
    • Don’t forget that the vectors are defined as:
      • $\vec{B} = B_{z}\hat{z}+B_{\theta}\hat{\theta}$
      • $\vec{J} = J_{z}\hat{z}+J_{\theta}\hat{\theta}$
      • and similarly for $\vec{\kappa}$

Theta Pinch

  • $B_{\theta} = J_{z} = 0$
  • Defined by the fact that current flows in the theta direction
  • $\vec{\kappa} = 0$
  • This creates a diamagnetic plasma (ie. existence of plasma creates a drop in the B field)

Z Pinch

  • $J_{\theta} = B_{z} = 0$
  • Defined by the fact that current flows in the z direction
  • $B^{2}\vec{\kappa} \rightarrow \frac{-B_{\theta}^{2}}{r} \hat{r}$ in cylindrical coordinates
  • This extra curvature term can aid in building up $\nabla p$

Screw Pinch

  • Super imposes Z and Theta Pinches on top of each other:
    • $0 = \frac{\partial}{\partial r}(P+\frac{B_{\theta}^{2}+B_{z}^{2}}{2\mu_{0}})+\frac{B_{\theta}^{2}}{r\mu_{0}}$

Toroidal Geometry

  • Very few topologies lend themselves to all the conditions of MHD. The most stringent of these is a divergence free $\vec{B}$ field
    • Can’t use a sphere due to the hairy ball theorem
    • A torus is one of the few geometries that supports two divergence free vector fields

Definitions

  • Define $\Psi$ such that $B\cdot \nabla \Psi = 0$
    • $\Psi_{tor} = \int B_{\phi}dS$
    • Can think of $\Psi_{tor}$ as the stream function of the magnetic field in the toroidal direction (ie. the long way around)
    • $\Psi_{tor} = \int B_{\phi} dS$ where $dS$ is the area of torus
      • can interpret $\sqrt{\Psi_{tor}}$ as a radius of your geometry
  • A similar definition holds for $\Psi_{pol}$ where you instead measure the flux around the short length
  • Safety factor q is related to the rotation transform
    • Can think of rotation transform as the number of poloidal transits per single toroidal transit of a field line on a toroidal flux surface
      • In math terms: $\frac{i}{2\pi} = \frac{d\psi}{d\Phi}$
    • $q = \frac{2\pi}{i}$
    • $q = \frac{rB_{\phi}}{RB_{\theta}}$
    • Having a low q factor means that you have a high plasma current, which has a higher risk of current-driven instability
  • $\beta$
    • $\beta_{Tor} = \frac{<p>}{\frac{B_{\phi}^{2}}{2\mu}}$
      • Can think of this as the magnet utilization
    • $\beta_{Pol} = \frac{<p>}{\frac{\beta_{\theta}^{2}}{2\mu}}$
      • Can think of this as the current utilization
    • $\frac{1}{\beta} = \frac{1}{\beta_{T}}+\frac{1}{\beta_{p}}$
  • Let $\epsilon = \frac{a}{R}$ where a is the minor radius and R is the major radius
    • Called the inverse aspect ratio
  • $\kappa = \frac{b}{a}$ where b is the half-height of the revolution ellipse and a is half-width
    • Called elongation
    • There is another $\kappa_{a} = \frac{Area}{\pi a^{2}}$ where area is the area of the revolving surface
  • $\Delta$ is called the triangularity

Forces

  • Define little r as the minor radius, $R_{0}$ as the major radius, $\theta$ as the angle from x on the revolving circle, and Z as the torus height
    • $R = R_{0}+r\cos\theta$
    • $Z = Z_{0}+r\sin\theta$
  • From aspect ratio expansions, can derive the following type of innate forces
  • Tire tube force: higher pressure in the interior of the creates an expansive force in the plane of the revolving ring
  • Hoop force: The different magnetic fields along the R direction causes a force to expand radially outwards
  • In order to counteract these forces, need an external B field to restore equillibrium to system (which is what Grad Shafranov equation tells you)

Grad Shafranov Equation

Stream Functions

  • Any divergence free field (like B) has a stream function
    • $\frac{-\partial \Psi}{\partial x} = B_{y}$
    • $\frac{\partial \Psi}{\partial y} = B_{x}$

Grad Shafranov Derivation Sketch

  1. Start from the simplified MHD equillibrium equations
  2. Use $\nabla \cdot B = 0$ to define stream function like above
  3. Express B with the stream function $\Psi$
  4. Plug into Ampere’s law for the J equation
  5. Use J equation to define F and $\Delta^{*} \Psi$
  6. Plug B&J into force balance to get G-S equation

While all of that is just vector algebra and kind of tedious, Step 5 is kind of opaque.

  • F is the flux function. It’s defined as $F = RB_{\phi}$
  • $\Delta^{*} \Psi$ is the Stokes Operator. It’s definedas $\Delta^{*} \Psi= -\frac{1}{R}\frac{\partial^{2}\Psi}{\partial z^2}-\frac{\partial}{\partial R}\frac{1}{R}\frac{\partial \Psi}{\partial R}$

The above definitions make sense within the context of the derivation. Trust me.

Anyhow, once you get to the end, you find the final form of the Grad Shafranov equation: $\Delta^{*} \Psi= R^{2}\mu_{0}\frac{\partial P}{\partial \Psi}+ F\frac{\partial F}{\partial \Psi}$

  • Stokes operator defines the surfaces of constant flux
  • 1st term on RHS defines the pressure term
  • 2nd term on RHS defines the current term

Aspect Ratio Expansion

  • To gain intuition of equation, suppose that the aspect ratio $\epsilon = \frac{a}{R_{0}}$ is very small (ie. the major radius of the tokamak is much larger than the minor radius)
  • Suppose that you can expand the G-S solution as a power series w.r.t $\epsilon$: $\phi(r,\theta) = \phi_{0}(r) = \sigma_{n=1}^{\infty} \epsilon^{n} \phi_{n}(r,\theta)$
    • From the above, make the ansatz: $\phi(r,\theta) \approx \phi_{0}(r) + \phi_{1}(r)\cos\theta = \phi_{0}(r) + \frac{\phi_{0}}{\partial r}\Delta(r)\cos\theta$
  • Can taylor expand F around $\phi_{0}$ yielding: $F = F(\phi_{0})+\epsilon \phi_{1}\frac{\partial F}{\phi} |_\phi$
  • Can expand $\frac{1}{R_{0}+r\cos\theta}\approx \frac{1}{R_{0}}(1-\epsilon \frac{r}{a} \cos \theta)$
  • Similar expansions happen for other terms
  • Doing a lot of math, you get the following findings:
    • You recover the screw pinch equation from the 0th order part
    • The 1st order part solution implies circular flux surfaces which are displaced from the center (Shafranov shift)
  • Normally, you just use a computer to solve the G-S equation as needed
    • Useful when placing wire to provide external B field