Resistive and Viscous Magnetohydrodynamics: Dimensionless Numbers, Heating, and Key Properties

Summary

This document covers the formulation of viscous and resistive magnetohydrodynamics (MHD) in dimensionless form. It introduces key dimensionless parameters such as the Reynolds number, magnetic Reynolds number, Lundquist number, and magnetic Prandtl number as ratios of physical effects and timescales. It also addresses non-ideal effects including flux freezing violation, Ohmic and viscous heating, and core properties of resistive MHD systems.

Viscosity… it’s such a drag!

• Viscosity transports momentum between parts of the fluid that are in relative motion • Viscosity results from particle collisions • The momentum equation with viscosity is given by ρ(∂/∂t + V · ∇)V = (J × B)/c − ∇p + ρν∇²V (13) Here, ν is the kinematic viscosity

Putting the momentum equation in dimensionless form

• Define V ≡ V₀Ṽ… where V₀ is a characteristic value and Ṽ is dimensionless, and use V₀ ≡ L₀/t₀ • Use these definitions in the momentum equation while including only the viscosity term on the RHS ρ ∂V/∂t + V · ∇V = ρν∇²V (14) (V₀/t₀) ∂Ṽ/∂t̃ + (V₀²/L₀) Ṽ · ∇̃Ṽ = (νV₀/L₀²) ∇̃²Ṽ ∂Ṽ/∂t̃ + Ṽ · ∇̃Ṽ = (ν/(V₀L₀)) ∇̃²Ṽ (15) where the Reynolds number is Re ≡ V₀L₀/ν (16)

The Reynolds number gauges the importance of the advective term compared to the viscous term

• We can rewrite the momentum equation with only viscosity as ∂Ṽ/∂t̃ + Ṽ · ∇̃Ṽ (advection) = (1/Re) ∇̃²Ṽ (viscous diffusion) (17) • A necessary condition for turbulence to occur is if Re ≫ 1 (18) so that the viscous term is negligible on scales ~ L₀. • In astrophysics, usually Re ≡ L₀V₀/ν ≫≫ 1 • The only scale in this equation is the viscous scale

Physics of cooking, part 2!

• The Reynolds number is Re ≡ L₀V₀/ν (19) • If you want to make peanut butter turbulent, you can either

Get a humongous vat of it (increase L₀), or
Stir it up really quickly (increase V₀) • Alternatively, since viscosity is a function of temperature (as in a plasma), you could also heat it up (decrease ν)

Putting the induction equation in dimensionless form

• The induction equation with uniform resistivity is ∂B/∂t = ∇ × (V × B) + D_η∇²B (20) • Again, define B ≡ B₀B̃… with V₀ ≡ L₀/t₀ (B₀/t₀) ∂B̃/∂t̃ = (V₀B₀/L₀) ∇̃ × (Ṽ × B̃) + (D_ηB₀/L₀²) ∇̃²B̃ (21) ∂B̃/∂t̃ = ∇̃ × (Ṽ × B̃) + (D_η/(L₀V₀)) ∇̃²B̃ (22)

Defining the magnetic Reynolds number and Lundquist number

• We define the magnetic Reynolds number as Rm ≡ L₀V₀/D_η (23) • The Alfvén speed is V_A ≡ B/√(4πρ) (24) • We define the Lundquist number as S ≡ L₀V_A/D_η (25) where we use that the characteristic speed for MHD is V₀ = V_A

Rm and S gauge the relative importance between the advection term and the resistive diffusion term

• We can write the induction equation as ∂B̃/∂t̃ = ∇̃ × (Ṽ × B̃) + (1/Rm) ∇̃²B̃ (26) • If Rm ≫ 1 then advection is more important (ideal MHD limit) • If Rm ≪ 1 then diffusion is more important • Usually in astrophysics, Rm ≫≫ 1 • Example: T ~ 10⁴ K, L₀ ~ 1 pc, V ~ 1 km/s. Then Rm ~ 10¹⁶ Interstellar plasmas are extremely highly conducting!

Re, Rm, and S can also be expressed as the ratio of timescales

• The Reynolds number is Re ≡ L₀V₀/ν = (viscous timescale) / (advection timescale) (27) • The magnetic Reynolds number is Rm ≡ L₀V₀/D_η = (resistive diffusion timescale) / (advection timescale) (28) • The Lundquist number is S ≡ L₀V_A/D_η = (resistive diffusion timescale) / (Alfvén wave crossing time) (29)

The relative importance of viscosity vs. resistivity is given by the magnetic Prandtl number

• The magnetic Prandtl number is delivered by: Pm ≡ ν/D_η = (resistive diffusion timescale) / (viscous timescale) (30) where ν and D_η are both in units of a diffusivity: length²/time • Usually Pm ≉ 1, but people doing simulations (like me) often set Pm = 1 for simplicity (sigh) • In plasma turbulence, the Prandtl number determines which scale is larger: the viscous dissipation scale or the resistive dissipation scale • Helps determine the physics behind dissipation of energy in turbulence!

Is there flux freezing in resistive MHD?

• Short answer: no! • Long answer: let’s modify the frozen-flux derivation! • The change of flux through a co-moving surface bounded by a contour C is dΨ/dt = −c ∮_C (E + (V × B)/c) · dl (31) • In ideal MHD, the integrand is identically zero. • In resistive MHD, the integrand is ηJ! • The change in flux becomes dΨ/dt = −c ∮_C ηJ · dl (32) This is generally ≠ 0, so flux is not frozen-in.

Viscous and resistive heating

• Ohmic (resistive) heating is given by Q_η = E · J = ηJ² (33) This shows up as a source term in the energy equation • Heating preferentially occurs in regions of very strong current, but those regions typically have small volumes • Viscous heating is of the form Q_ν = ρν ∇V^T : ∇V (34)

Key Properties of Resistive MHD

• Magnetic topology is not preserved • Mass, momentum, and energy are conserved • Helicity and cross-helicity are approximately conserved • There is Ohmic (resistive) and viscous heating • There’s dissipation! • The scales in the problem are set by viscosity and resistivity