9.1 Rankine-Hugoniot relations and shock jump conditions

MHD shocks are like traffic jams for space plasma. They happen when super-fast stuff slams into slower stuff, causing a sudden change in speed, density, and magnetic fields.

Rankine-Hugoniot_relations_0### are the math that describes these cosmic pile-ups. They help us figure out how plasma properties change across the shock, connecting the calm before with the chaos after.

Rankine-Hugoniot Relations for MHD Shocks

Derivation and Fundamental Concepts

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• Rankine-Hugoniot relations describe physical property relationships across MHD shock waves
• Derived from conservation laws (mass, momentum, energy) and Maxwell's equations
• MHD equations written in conservative form express conserved quantity changes across shock discontinuity
• Derivation applies integral form of conservation laws to control volume enclosing shock front
• Taking limit as control volume thickness approaches zero yields algebraic relations between pre-shock and post-shock quantities
• Magnetic fields introduce additional terms compared to hydrodynamic shocks
• Final set includes equations for conservation of mass flux, momentum flux (with magnetic pressure), energy flux, and magnetic flux

Mathematical Framework and Applications

• Integral form of conservation laws applied to shock-enclosing control volume
  • Mass conservation: $\int_S \rho \mathbf{v} \cdot d\mathbf{S} = 0$
  • Momentum conservation: $\int_S (\rho \mathbf{v}\mathbf{v} + p\mathbf{I} - \frac{1}{\mu_0}\mathbf{B}\mathbf{B}) \cdot d\mathbf{S} = 0$
  • Energy conservation: $\int_S \left(\rho\mathbf{v}\left(\frac{1}{2}v^2 + \frac{\gamma}{\gamma-1}\frac{p}{\rho}\right) + \frac{1}{\mu_0}\mathbf{E}\times\mathbf{B}\right) \cdot d\mathbf{S} = 0$
• Limit process yields jump conditions across infinitesimally thin shock
• Resulting equations relate upstream (subscript 1) to downstream (subscript 2) quantities
  • Mass flux: $\rho_1 v_{1n} = \rho_2 v_{2n}$
  • Momentum flux: $\rho_1 v_{1n}^2 + p_1 + \frac{B_{1t}^2}{2\mu_0} = \rho_2 v_{2n}^2 + p_2 + \frac{B_{2t}^2}{2\mu_0}$
  • Energy flux: $\rho_1 v_{1n}\left(\frac{1}{2}v_1^2 + \frac{\gamma}{\gamma-1}\frac{p_1}{\rho_1}\right) + \frac{1}{\mu_0}(\mathbf{E}_1\times\mathbf{B}_1)_n = \rho_2 v_{2n}\left(\frac{1}{2}v_2^2 + \frac{\gamma}{\gamma-1}\frac{p_2}{\rho_2}\right) + \frac{1}{\mu_0}(\mathbf{E}_2\times\mathbf{B}_2)_n$
• Applications include analyzing solar wind interactions with planetary magnetospheres and astrophysical jet propagation

MHD Shock Jump Conditions

Problem-Solving Techniques

• Shock jump conditions relate upstream (pre-shock) and downstream (post-shock) plasma properties
• Form nonlinear algebraic equation system solved simultaneously for post-shock conditions
• Key variables include density, velocity, pressure, temperature, and magnetic field components
• Shock strength characterized by upstream Mach number (defined using fast, Alfvén, or slow MHD wave speeds)
• Solution methods involve iterative techniques or specialized nonlinear equation system solvers
• Graphical techniques (Hugoniot curve) visualize possible shock solutions and identify physically realizable states
• Careful consideration of shock geometry required, particularly angle between magnetic field and shock normal

Practical Applications and Examples

• Solar wind interaction with Earth's bow shock
  • Upstream conditions: $n_1 = 5 \text{ cm}^{-3}, \mathbf{v}_1 = 400 \text{ km/s}, \mathbf{B}_1 = 5 \text{ nT}$
  • Apply jump conditions to determine downstream plasma density, velocity, and magnetic field strength
• Interstellar medium shock waves
  • Analyze density compression and temperature increase across supernova remnant shock front
• Magnetic reconnection in solar flares
  • Use jump conditions to estimate energy release and particle acceleration in reconnection outflow regions
• Laboratory plasma experiments
  • Predict plasma conditions in Z-pinch devices or tokamak edge localized modes (ELMs)

Conservation Laws Across MHD Shocks

Mass, Momentum, and Energy Conservation

• Mass conservation requires constant mass flux: $\rho_1v_{1n} = \rho_2v_{2n}$
• Momentum conservation includes thermal and magnetic pressure terms
  • Tensor equation accounts for anisotropic pressure in magnetic fields
  • $\rho_1\mathbf{v}_1\mathbf{v}_1 + p_1\mathbf{I} + \frac{1}{2\mu_0}B_1^2\mathbf{I} - \frac{1}{\mu_0}\mathbf{B}_1\mathbf{B}_1 = \rho_2\mathbf{v}_2\mathbf{v}_2 + p_2\mathbf{I} + \frac{1}{2\mu_0}B_2^2\mathbf{I} - \frac{1}{\mu_0}\mathbf{B}_2\mathbf{B}_2$
• Energy conservation accounts for kinetic, thermal, and magnetic energy fluxes
  • Includes work done by electromagnetic forces
  • $\rho_1\mathbf{v}_1\left(\frac{1}{2}v_1^2 + \frac{\gamma}{\gamma-1}\frac{p_1}{\rho_1}\right) + \frac{1}{\mu_0}(\mathbf{E}_1\times\mathbf{B}_1) = \rho_2\mathbf{v}_2\left(\frac{1}{2}v_2^2 + \frac{\gamma}{\gamma-1}\frac{p_2}{\rho_2}\right) + \frac{1}{\mu_0}(\mathbf{E}_2\times\mathbf{B}_2)$
• Magnetic field parallel component continuous across shock
• Normal component may change to satisfy divergence-free condition
• Entropy increases across shock, consistent with second law of thermodynamics
• Conservation laws couple plasma dynamics and electromagnetic fields

MHD Shock Types and Properties

• Fast shocks
  • Compress both plasma and magnetic field
  • Increase in magnetic field strength across shock
  • Example: Earth's bow shock in solar wind
• Intermediate shocks
  • Rotate magnetic field direction
  • No change in field strength
  • Example: Rotational discontinuities in solar wind
• Slow shocks
  • Compress plasma but expand magnetic field
  • Decrease in magnetic field strength across shock
  • Example: Slow-mode shocks in solar flare reconnection outflows
• Switch-on and switch-off shocks
  • Magnetic field component appears or disappears across shock
  • Example: Switch-on shocks in strongly magnetized accretion disks

MHD vs Hydrodynamic Shock Jump Conditions

Key Differences and Additional Complexities

• MHD shock jump conditions include magnetic field terms, absent in hydrodynamic shocks
• Anisotropic pressure effects in MHD due to magnetic fields, unlike isotropic pressure in hydrodynamic shocks
• Multiple propagation modes in MHD (fast, intermediate, slow) due to plasma-magnetic field interaction
  • Hydrodynamic shocks have single mode
• Magnetic field provides additional pressure support in MHD
  • Potentially alters compression ratio across shock compared to hydrodynamic cases
• MHD shock structure depends on angle between magnetic field and shock normal
  • Leads to parallel, perpendicular, and oblique shock configurations
• Energy partitioning in MHD shocks includes magnetic energy conversion to thermal and kinetic energy
  • Process absent in hydrodynamic shocks
• Switch-on and switch-off shocks unique to MHD
  • Magnetic field components can appear or disappear across shock
  • No analogue in hydrodynamics

Comparative Analysis and Examples

• Compression ratio limitations
  • Hydrodynamic shocks: Maximum compression ratio of 4 for strong shocks in ideal gas
  • MHD shocks: Can exceed factor of 4 due to magnetic pressure effects
• Shock heating mechanisms
  • Hydrodynamic: Solely through compression and viscous dissipation
  • MHD: Additional heating through magnetic reconnection and wave-particle interactions
• Shock propagation speeds
  • Hydrodynamic: Single characteristic speed (sound speed)
  • MHD: Multiple characteristic speeds (fast, Alfvén, and slow magnetosonic speeds)
• Examples illustrating differences
  • Solar coronal mass ejection (CME) propagation
    • MHD treatment necessary to capture magnetic field evolution and shock formation
  • Supernova remnant expansion
    • Early stages require MHD approach, later stages may be approximated hydrodynamically
  • Interstellar shock waves
    • MHD effects crucial in understanding cosmic ray acceleration and magnetic field amplification