3 Plasma Physics
Vector algebra for plasma fields, fluid operators, the Lorentz force, two-fluid and single-fluid MHD, resistive MHD, equilibrium theory, and tokamak/FRC configurations.
3.1 Vector Algebra
Cross product of two vectors: wraps Mathlib’s crossProduct.
Dot product of two vectors: wraps Mathlib’s dotProduct.
Pointwise cross product of two vector fields: \((\mathbf{F}\times \mathbf{G})(x) = \mathbf{F}(x)\times \mathbf{G}(x)\).
Pointwise dot product of two vector fields.
Scalar–vector field multiplication: \((c\mathbf{F})(x) = c(x)\, \mathbf{F}(x)\).
Pointwise addition of vector fields.
Pointwise subtraction of vector fields.
Pointwise negation of a vector field.
Time-dependent pointwise cross product.
Time-dependent scalar–vector multiplication.
Time-dependent pointwise addition.
\(\mathbf{a}\times \mathbf{b} = -(\mathbf{b}\times \mathbf{a})\).
Anti-commutativity at the field level.
\(\mathbf{a}\cdot (\mathbf{a}\times \mathbf{b}) = 0\).
\((\mathbf{a}\times \mathbf{b})\cdot \mathbf{b} = 0\).
\(\mathbf{F}(x)\cdot (\mathbf{F}(x)\times \mathbf{G}(x)) = 0\).
\((\mathbf{F}(x)\times \mathbf{G}(x))\cdot \mathbf{G}(x) = 0\).
3.2 Fluid Operators
Advective derivative of a scalar field: \((\mathbf{v}\cdot \nabla )f = \sum _i v_i\, \frac{\partial f}{\partial x_i}\).
Advective derivative of a vector field: \(((\mathbf{v}\cdot \nabla )\mathbf{F})_j = \sum _i v_i\, \frac{\partial F_j}{\partial x_i}\).
Material derivative of a scalar field: \(\frac{D f}{D t} = \frac{\partial f}{\partial t} + (\mathbf{v}\cdot \nabla )f\).
Material derivative of a vector field: \(\left(\frac{D \mathbf{F}}{D t}\right)_j = \frac{\partial F_j}{\partial t} + ((\mathbf{v}\cdot \nabla )\mathbf{F})_j\).
3.3 Lorentz Force
Lorentz force per unit volume: \(\mathbf{f} = \rho _c\, \mathbf{E} + \mathbf{J}\times \mathbf{B}\).
\(\mathbf{f}_L\cdot \mathbf{B} = \rho _c(\mathbf{E}\cdot \mathbf{B})\) because \((\mathbf{J}\times \mathbf{B})\cdot \mathbf{B} = 0\).
3.4 Two-Fluid Model
A plasma species with charge \(q\) and mass \(m {\gt} 0\).
Two-fluid equations for a single species: continuity and momentum.
3.5 Ideal MHD
Physical constants: vacuum permeability \(\mu _0 {\gt} 0\) and adiabatic index \(\gamma {\gt} 1\).
\(\mu _0 \ne 0\).
\(\mu _0 {\gt} 0\).
\(\gamma {\gt} 0\).
The ideal MHD system: mass conservation, momentum (\(\rho \frac{D \mathbf{v}}{D t} = \mathbf{J}\times \mathbf{B} - \nabla p\)), adiabatic energy, induction (\(\frac{\partial \mathbf{B}}{\partial t} = \nabla \! \times \! (\mathbf{v}\times \mathbf{B})\)), \(\nabla \! \cdot \! \mathbf{B} = 0\), and Ampère’s law (\(\nabla \! \times \! \mathbf{B} = \mu _0\mathbf{J}\)).
If \(\frac{\partial \mathbf{B}}{\partial t} = -\nabla \! \times \! \mathbf{E}\) and \(\mathbf{E} = -\mathbf{v}\times \mathbf{B}\) (ideal Ohm’s law), then \(\frac{\partial \mathbf{B}}{\partial t} = \nabla \! \times \! (\mathbf{v}\times \mathbf{B})\).
\(\mathbf{J} = (1/\mu _0)\, \nabla \! \times \! \mathbf{B}\).
\(\nabla \! \cdot \! \mathbf{B} = 0\) at all times (from the solenoidal axiom).
3.6 MHD Equilibrium
Static MHD equilibrium: \(\nabla p = \mathbf{J}\times \mathbf{B}\) with \(\nabla \! \times \! \mathbf{B} = \mu _0\mathbf{J}\) and \(\nabla \! \cdot \! \mathbf{B} = 0\).
\(\mathbf{B}\cdot \nabla p = 0\): magnetic field lines lie on pressure surfaces.
\(\mathbf{J}\cdot \nabla p = 0\): current lines lie on pressure surfaces.
\(\nabla p = (1/\mu _0)(\nabla \! \times \! \mathbf{B})\times \mathbf{B}\).
\(\nabla p = (1/\mu _0)((\mathbf{B}\cdot \nabla )\mathbf{B} - \nabla (|\mathbf{B}|^2/2))\): separates magnetic tension and magnetic pressure.
3.7 Cylindrical Coordinates
Major radius \(R = x_0\) from a point \((R, \varphi , Z)\).
Vertical coordinate \(Z = x_2\).
A scalar field is axisymmetric: \(\frac{\partial f}{\partial \varphi } = 0\).
Poloidal flux function \(\psi (R, Z)\) with \(C^2\) smoothness and axisymmetry.
The Grad-Shafranov operator: \(\Delta ^*\psi = \frac{\partial ^2 \psi }{\partial R^2} - \frac{1}{R}\frac{\partial \psi }{\partial R} + \frac{\partial ^2 \psi }{\partial Z^2}\).
3.8 Grad-Shafranov Equation
The Grad-Shafranov equation for tokamak equilibrium: \(\Delta ^*\psi = -\mu _0 R^2 p'(\psi ) - g(\psi )g'(\psi )\).
Pressure profile \(p(x) = p(\psi (x))\).
If \(\psi (x) = \psi (y)\), then \(p(x) = p(y)\): pressure is constant on flux surfaces.
\(g\) is constant on flux surfaces.
Magnetic field from flux: \(\mathbf{B} = (-\frac{1}{R}\frac{\partial \psi }{\partial Z},\; \frac{g(\psi )}{R},\; \frac{1}{R}\frac{\partial \psi }{\partial R})\).
The Grad-Shafranov equation implies \(\nabla p = \mathbf{J}\times \mathbf{B}\) in cylindrical geometry (\(R \ne 0\)).
When \(p(\psi ) = p_0\psi \) and \(g^2(\psi ) = g_0^2 + 2\alpha \psi \) are linear, the GS equation becomes \(\Delta ^*\psi = -\mu _0 R^2 p_0 - \alpha \).
3.9 Resistive MHD
Ohm’s law parameters: resistivity \(\eta \ge 0\).
Resistive MHD: same as ideal but with \(\mathbf{E} + \mathbf{v}\times \mathbf{B} = \eta \mathbf{J}\). Induction becomes \(\frac{\partial \mathbf{B}}{\partial t} = \nabla \! \times \! (\mathbf{v}\times \mathbf{B} - \eta \mathbf{J})\).
When \(\eta = 0\), the resistive induction equation reduces to the ideal one.
\(\frac{\partial \mathbf{B}}{\partial t} = \nabla \! \times \! (\mathbf{v}\times \mathbf{B}) - \eta \, \nabla \! \times \! \mathbf{J}\): splits convection and diffusion terms.
3.10 FRC Equilibrium
FRC equilibrium: radial profiles \(p(R)\), \(B_z(R)\), \(B_\theta (R)\) satisfying radial pressure balance with field reversal at the separatrix.
With \(B_\theta = 0\): \(\frac{dp}{dR} + \frac{1}{\mu _0}B_z\frac{dB_z}{dR} = 0\).
\(p(R) = (B_{\mathrm{ext}}^2 - B_z(R)^2)/(2\mu _0)\) when total pressure is conserved.
\(\beta = 1\) at the separatrix: \(2\mu _0 p(R_s) = B_{\mathrm{ext}}^2\).
RMF drive parameters: amplitude \(B_\omega {\gt} 0\), frequency \(\omega {\gt} 0\).
A Rotamak: FRC sustained by a rotating magnetic field with 1D Ampère’s law \(dB_z/dR = -\mu _0 J_\theta \).
Inside the separatrix where \(B_z\) decreases (\(dB_z/dR {\lt} 0\)), the toroidal current \(J_\theta {\gt} 0\).