1. MHD Concepts and Equations Handout – Walk-through

2. 1) The Convective Derivative (hopefully recap) Scalars, A: Vectors, A: Total change of a parameter within a fluid element Change due to motions of fluid through the element Change due to variations in time

3. 2) The Pressure Tensor We know that particle behaviour is different parallel to B than perpendicular to B. This leads to pressure anisotropy: (necessarily) More generally, plasmas may experience shear stresses (e.g. viscosity) which transfer, for example, x-directed momentum flux in the y- or z-directions –Leads to a pressure tensor………..

4. ………for example in a field aligned coordinate system (z || B): Assuming an ideal gas: Perpendicular and Parallel Pressures ‘Shear’ Stresses defines 2 temperatures for a plasma

5. 3) Mass Conservation Mass is a conserved quantity in a non- relativistic plasma, so we have the MHD mass conservation or continuity equation for e.g. number density n: This implies that mass in a given volume of space changes only if there is a net mass flux into or out of that volume (cf. Gauss Theorem) RHS represents possible sources or sinks of the quantity

6. 4) Charge Conservation Charge is also a conserved quantity, the MHD charge continuity equation: ρ q = qn; ρ q v = j For (quasi-)time stationary plasma ∂/∂t = 0: → current divergence = 0; current flowing into a volume is balanced by current flowing out; J || is field-aligned current, which are important in the magnetosphere.

7. 5) Momentum Equation Momentum is a conserved quantity This is the MHD Eqn of motion (cf. F = ma) – represents plasma pressure gradients; –ρ q E is electric field force (usually neglected as no net charge density in plasma) –j x B is the magnetic field or Lorentz force; –Other forces appear on RHS if applicable. + any other forces acting on the plasma (e.g. gravity) The convective derivative of the momentum nmv Forces, which are sources and sinks of momentum

8. 6) A Generalized Ohms Law ‘Ideal MHD’ η = 0 This is a very good approximation for many space plasma applications → ‘frozen-in flux’ approximation Electric field in a moving plasma Hall term Electron anisotropy Electron inertia Resistive term (cf V=IR; η = resistivity These terms are small on long length and time scales and/or in regions of weak currents → often ignored → E + v  B = 0

9. 7) Equations of State An equation representing conservation of energy; this can take different forms, depending on assumptions: –Simplest - Ideal gas: P = nk B T –Adiabatic (no heat exchange) ρ = nm (mass density) P = Cρ γ (γ is ratio of specific heats) –(more rigorous versions have sources and sinks of energy on the RHS – heat flux, radiative terms, ohmic heating, etc).

10. 8) Maxwells Equations These relate the electromagnetic parameters and can be used to close the system of equations: