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MHD Concepts and Equations Handout – Walk-through

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 variations in time. Change due to motions of fluid through the element. 2) The Pressure Tensor.

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MHD Concepts and Equations Handout – Walk-through

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  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 variations in time Change due to motions of fluid through the element

  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: ‘Shear’ Stresses Perpendicular and Parallel Pressures 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: • ρq = qn; ρqv = 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; • ρqE 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 Resistive term (cf V=IR; η = resistivity Hall term Electron inertia Electric field in a moving plasma Electron anisotropy 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 = nkBT • Adiabatic (no heat exchange) ρ = nm (mass density) P = Cργ (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:

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