9.3 Intermediate shocks and rotational discontinuities

Intermediate shocks and rotational discontinuities are crucial MHD structures in space plasmas. They differ in how they change plasma properties and magnetic fields, playing key roles in energy transfer and magnetic topology changes.

Intermediate shocks transition flow from super- to sub-Alfvénic, changing field magnitude and plasma density. Rotational discontinuities only rotate the magnetic field, preserving its magnitude and plasma properties. Both are important in magnetic reconnection and space plasma dynamics.

Intermediate Shocks vs Rotational Discontinuities

Definitions and Key Characteristics

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• Intermediate shocks involve changes in tangential magnetic field component and transition from super-Alfvénic to sub-Alfvénic flow
• Rotational discontinuities rotate magnetic field vector without changing magnitude or plasma density
• Both structures manifest as non-linear wave structures in magnetized plasmas
• Intermediate shocks classified based on normal flow velocity changes relative to characteristic MHD speeds (fast, intermediate, slow)
• Rotational discontinuities propagate at Alfvén speed maintaining constant normal components of magnetic field and velocity
• These structures play crucial roles in energy transfer and magnetic field topology changes in space plasmas (solar wind, magnetosphere)

Classification and Propagation

• Intermediate shocks types determined by relationship between upstream and downstream flow speeds and characteristic MHD wave speeds
  • Slow-intermediate shocks: upstream flow super-slow and sub-intermediate, downstream flow sub-slow
  • Fast-intermediate shocks: upstream flow super-fast, downstream flow sub-intermediate and super-slow
• Rotational discontinuities always propagate at local Alfvén speed $v_A = B / \sqrt{\mu_0 \rho}$ where B is magnetic field strength and ρ is plasma density
• Intermediate shock propagation speed varies depending on type and plasma parameters
  • Generally between slow and fast magnetosonic speeds
• Rotational discontinuities maintain constant propagation speed in homogeneous plasma

Properties of Shocks vs Discontinuities

Plasma Parameter Changes

• Intermediate shocks change plasma density and pressure
  • Density increases across shock front
  • Pressure rises due to compression and heating
• Rotational discontinuities maintain constant density and pressure
• Both rotate magnetic field, but intermediate shocks also change field magnitude
• Intermediate shocks compress plasma and produce entropy
• Rotational discontinuities remain non-compressive and isentropic
• Intermediate shocks can convert between MHD wave modes (Alfvén to magnetosonic)
• Rotational discontinuities primarily affect Alfvén wave polarization

Energy and Momentum Transfer

• Intermediate shocks dissipate significant energy through irreversible processes
  • Particle heating
  • Magnetic field annihilation
• Rotational discontinuities involve minimal energy dissipation
  • Energy transfer primarily through field rotation
• Both structures important in magnetic reconnection
  • Intermediate shocks often associated with reconnection outflows
  • Rotational discontinuities can separate reconnection inflow and outflow regions
• Momentum transfer across intermediate shocks follows modified Rankine-Hugoniot relations for MHD
• Rotational discontinuities conserve momentum flux with only directional changes

Plasma Parameter Changes Across Shocks and Discontinuities

Velocity and Magnetic Field Changes

• Intermediate shocks decrease normal velocity component
  • Tangential velocity components may change direction and magnitude
• Magnetic field undergoes rotation and magnitude change across intermediate shocks
  • Normal component remains constant due to divergence-free condition $\nabla \cdot \mathbf{B} = 0$
• Rotational discontinuities maintain constant normal velocity and magnetic field components
  • Tangential components rotate while preserving magnitude
• Both structures conserve tangential electric field components
  • Ensures consistency with Faraday's law $\nabla \times \mathbf{E} = -\partial \mathbf{B}/\partial t$

Thermodynamic Parameter Changes

• Plasma density increases across intermediate shocks
  • Follows from mass conservation $\rho_1 v_{n1} = \rho_2 v_{n2}$
• Pressure rises in intermediate shocks due to compression and heating
  • Satisfies momentum conservation $p_1 + \rho_1 v_{n1}^2 + B_{t1}^2/(2\mu_0) = p_2 + \rho_2 v_{n2}^2 + B_{t2}^2/(2\mu_0)$
• Rotational discontinuities preserve density, pressure, and temperature
  • No compression or expansion occurs
• Entropy production occurs in intermediate shocks
  • Irreversible processes lead to increase in specific entropy
• Rotational discontinuities maintain constant entropy
  • Isentropic nature preserves adiabatic invariants

Conditions for Shock and Discontinuity Existence

Plasma and Field Parameters

• Intermediate shocks require sufficiently high plasma beta $\beta = \frac{2\mu_0 p}{B^2}$
  • Allows necessary compression and magnetic field changes
  • Typically $\beta > 1$ for intermediate shock formation
• Incident flow must be super-Alfvénic for at least one characteristic MHD wave speed
  • Ensures shock formation through nonlinear steepening
• Rotational discontinuities exist in both high and low plasma beta regimes
  • No density or pressure changes required
• Both structures need guide magnetic field component parallel to discontinuity surface
  • Enables rotation of field vector

Stability and Formation Mechanisms

• Intermediate shock stability depends on balance between nonlinear steepening and dispersive effects
  • Steepening tends to sharpen shock front
  • Dispersion acts to spread out disturbance
• Rotational discontinuities more readily formed in collisionless plasmas
  • Kinetic effects support field rotation without dissipation
  • Ion and electron dynamics crucial for maintaining structure
• Intermediate shocks can form through nonlinear wave steepening or collision of simpler MHD discontinuities
  • Slow shocks merging with rotational discontinuities
• Rotational discontinuities often result from large-amplitude Alfvén waves or magnetic field line draping
  • Solar wind interactions with planetary magnetospheres (Earth's magnetopause)