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MHD Concepts and Equations Handout – Walk-through

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

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………..

………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

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

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.

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

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

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).

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

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