6.1 MHD equations and approximations

Magnetohydrodynamics (MHD) is the study of electrically conducting fluids interacting with magnetic fields. It combines fluid dynamics and electromagnetism to describe the behavior of plasmas in various settings, from lab experiments to astrophysical phenomena.

MHD equations and approximations form the foundation for understanding plasma behavior. Key concepts include the Reynolds number, magnetic Reynolds number, and ideal MHD assumptions, which help simplify complex plasma dynamics for analysis and modeling.

Magnetohydrodynamics Fundamentals

MHD Concepts and Equations

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• Magnetohydrodynamics (MHD) studies the dynamics of electrically conducting fluids interacting with magnetic fields
• Ideal MHD equations describe the behavior of perfectly conducting fluids in magnetic fields
• Reynolds number measures the ratio of inertial forces to viscous forces in fluid flow
• Magnetic Reynolds number quantifies the relative importance of magnetic field advection to magnetic diffusion

Fluid Dynamics in MHD

• MHD combines principles of fluid dynamics and electromagnetism
• Ideal MHD assumes infinite electrical conductivity and negligible viscosity
• Reynolds number helps determine flow regime (laminar or turbulent)
• Magnetic Reynolds number indicates whether magnetic fields are frozen into the fluid motion

Applications and Significance

• MHD applies to astrophysical plasmas (solar wind, accretion disks)
• Ideal MHD equations form the basis for many plasma physics models
• Reynolds number aids in scaling laboratory experiments to real-world phenomena
• Magnetic Reynolds number determines the effectiveness of magnetic field generation in dynamo processes

MHD Conservation Laws

Continuity and Mass Conservation

• Continuity equation expresses conservation of mass in fluid flow
• Describes how fluid density changes with time and space
• Accounts for compressibility effects in MHD fluids
• Relates fluid velocity to density variations

Momentum Conservation and Forces

• Momentum equation balances forces acting on the fluid
• Incorporates pressure gradients, magnetic forces, and gravitational effects
• Describes fluid acceleration due to various forces
• Couples fluid motion with electromagnetic fields

Magnetic Flux Conservation

• Frozen-in flux theorem states magnetic field lines move with the fluid
• Describes conservation of magnetic flux in ideal MHD
• Explains why magnetic field lines appear to be "frozen" into highly conducting plasmas
• Leads to important consequences for plasma confinement and astrophysical phenomena

Electromagnetic Principles in MHD

Faraday's Law and Magnetic Induction

• Faraday's law describes how changing magnetic fields induce electric fields
• Governs the generation of electromotive forces in moving conductors
• Explains the induction of currents in MHD fluids
• Forms the basis for many MHD phenomena (dynamo effect, magnetic reconnection)

Ohm's Law and Conductivity

• Ohm's law relates electric current density to electric and magnetic fields
• Describes the response of a conducting fluid to electromagnetic forces
• Incorporates effects of fluid motion on current generation
• Simplifies to ideal MHD limit for perfectly conducting fluids

Alfvén's Theorem and Wave Propagation

• Alfvén's theorem states magnetic field lines behave like elastic strings in ideal MHD
• Describes the propagation of Alfvén waves in magnetized plasmas
• Explains how magnetic tension forces contribute to plasma dynamics
• Provides insights into energy transport in astrophysical plasmas