Graduate
  1. Classical mechanics  1.1. Advanced Kinematics  1.1.1. Generalized coordinate systems  1.1.2. Parametric equations of motion  1.1.3. Non-inertial frames and fictitious forces  1.1.4. Covariant formulation of momentum  1.1.5. Action-angle variable  1.2. Lagrangian and Hamiltonian mechanics  1.2.1. Principle of minimum action  1.2.2. Euler–Lagrange equations  1.2.3. Noether's theorem and conservation laws  1.2.4. Normalized Speed  1.2.5. Canonical conversion  1.2.6. Poisson bracket and Hamilton–Jacobi theory  1.2.7. Hamiltonian chaos and integrability  1.3. Rigid body mobility  1.3.1. Euler's equations of motion  1.3.2. Principal moments of inertia  1.3.3. Gyroscopic motion and precession  1.3.4. Inertia tensor  1.3.5. Stability of rotational speed  1.4. Nonlinear dynamics and chaos  1.4.1. Phase space and stability analysis  1.4.2. Poincaré sections and bifurcation theory  1.4.3. Strange Attractors and Fractals  1.4.4. Lyapunov exponent  1.4.5. KAM theorem and quasi-periodic motion
  2. Electromagnetism  2.1. Advanced Electrodynamics  2.1.1. Green's function and potential theory  2.1.2. Multipole expansion  2.1.3. Image charging method  2.1.4. Boundary conditions and uniqueness theorem  2.2. Electromagnetic wave propagation  2.2.1. Plane waves in dielectrics and conductors  2.2.2. Waveguides and cavity resonators  2.2.3. Radiation pressure and optical tweezers  2.2.4. Polarization and Jones calculus  2.3. Relativistic electrodynamics  2.3.1. Covariant formulation of electrodynamics  2.3.2. Lienard–Wiechert potentials  2.3.3. Synchrotron radiation and bremsstrahlung  2.3.4. Electromagnetic field tensor and Lorentz transformations  2.4. Plasma physics  2.4.1. Magnetohydrodynamics  2.4.2. Debye shielding and plasma oscillations  2.4.3. Alfvén waves and tokamak confinement  2.4.4. Plasma instabilities and turbulence
  3. Statistical mechanics and thermodynamics  3.1. Advanced Thermodynamics  3.1.1. Legendre transforms and thermodynamic potentials  3.1.2. Fluctuation–dissipation theorem  3.1.3. Onsager interpersonal relations  3.1.4. Critical events and phase transitions  3.2. Quantum statistical mechanics  3.2.1. Density Matrix and Ensemble Theory  3.2.2. Bose–Einstein condensates  3.2.3. Quantum phase transition  3.2.4. Fermi–Dirac and Bose–Einstein statistics  3.2.5. Partition functions and grand canonical ensembles
  4. Quantum mechanics  4.1. Advanced wave mechanics  4.1.1. WKB approximation  4.1.2. Path integral formulation  4.1.3. Quantum harmonic oscillator and coherent states  4.2. Angular momentum and spin  4.2.1. Spherical Harmonics  4.2.2. Clebsch–Gordan coefficient  4.2.3. Wigner–Eckart theorem  4.2.4. Spin–orbit coupling  4.3. Quantum scattering theory  4.3.1. Born approximation  4.3.2. Partial wave analysis  4.3.3. Optical theorem  4.3.4. S-matrix theory  4.4. Quantum field theory  4.4.1. Second Quantization  4.4.2. Feynman diagrams and propagators  4.4.3. Renormalization theory  4.4.4. Gauge theory and Yang–Mills fields  4.4.5. Spontaneous symmetry breaking and the Higgs mechanism
  5. General relativity and cosmology  5.1. Tensor calculus and differential geometry  5.1.1. Einstein field equations  5.1.2. Schwarzschild and Kerr metrics  5.1.3. Geodesics and Christoffel symbols  5.2. Cosmology and the Universe  5.2.1. The FLRW metric and cosmic inflation  5.2.2. Dark energy and structure formation  5.2.3. Cosmic microwave background radiation
  6. Condensed matter physics  6.1. Band structure and transport theory  6.1.1. Block theorem and the Kronig–Penney model  6.1.2. Fermi surface topology  6.1.3. Quantum Hall Effect  6.2. Superconductivity  6.2.1. BCS theory and Cooper pairs  6.2.2. Meissner effect and flux quantization  6.2.3. The Josephson Effect and SQUIDs  6.3. Topological phases of matter  6.3.1. Topological Insulators  6.3.2. Majorana fermions in topological phases of matter
  7. Nuclear and Particle Physics  7.1. Quantum Chromodynamics (QCD)  7.1.1. Quark–gluon plasma  7.1.2. Asymptotic freedom  7.1.3. Confinement and hadronization  7.2. Beyond the Standard Model  7.2.1. Grand Unified Theories  7.2.2. Supersymmetry and extra dimensions  7.2.3. Neutrino oscillations

Graduate → Electromagnetism → Plasma physics ↓


Magnetohydrodynamics


Magnetohydrodynamics (MHD) is the study of the dynamics of electrically conducting fluids. Examples of such fluids include plasmas, liquid metals, salt water, and electrolytes. The field of MHD attempts to understand the behavior of these fluids in magnetic and electric fields. At its core, MHD combines the principles of hydrodynamics (fluid dynamics) and electromagnetism.

Introduction to MHD

Fluids that conduct electricity enable fascinating interactions between fluid motion and electromagnetic fields. MHD plays an important role in many disciplines such as astrophysics, geophysics, and engineering, including the design of fusion reactors and everyday applications such as electromagnetic pumps.

Theory and equations

Magnetohydrodynamic equations

The basic set of equations governing the behavior of magnetohydrodynamic fluids is a combination of the Navier–Stokes equations from fluid dynamics and Maxwell's equations from electromagnetism.

Continuity equation

The continuity equation ensures mass conservation in a fluid:

∂ρ/∂t + ∇ • (ρv) = 0

where ρ is the density of the fluid and v is the velocity of the fluid.

Momentum equation

The momentum equation presents a force balance, involving pressure, viscous, gravitational, and electromagnetic forces:

ρ(∂v/∂t + v • ∇v) = -∇p + j × B + ρg + ∇ • τ

where p is the pressure, j is the current density, B is the magnetic field, g is the gravitational acceleration, and τ is the stress tensor.

Induction equation

The induction equation is derived from Faraday's law of induction and describes how the magnetic field evolves with time within a fluid:

∂B/∂t = ∇ × (v × B) - ∇ × (η∇ × B)

where η is the magnetic diffusivity.

Ohm's law for MHD

Ohm's law in the context of MHD is given as:

j = σ(E + v × B)

Here, σ is the electrical conductivity, E is the electric field, and we account for fluid flow by including v × B term.

Practical applications of MHD

Astrophysical applications

In astrophysics, MHD helps explain phenomena such as solar flares, the behavior of stars, the interstellar medium, and even entire galaxies. For example, the Sun's corona exhibits behavior that can be understood through MHD theories.

Geophysical applications

The Earth's liquid outer core conducts electricity, and MHD is important in understanding the origin and changes of the geomagnetic field. Such study is vital for navigation that relies on magnetic fields.

Engineering and technology

MHD is used in the development of magnetic confinement fusion reactors such as tokamaks, which aim to control plasma behaviour to reach the conditions necessary to sustain nuclear fusion.

Visualize the concept

MHD with simple fluid particles

Particle 1 Particle 2 B Field

The visualization shows two electrically charged particles within a fluid under a magnetic field, represented by the red curve. As these particles move, they generate currents that can interact with the magnetic field and create complex behaviors studied under MHD.

Challenges and opportunities in MHD

Mathematical complexity

MHD involves complex partial differential equations that are often very challenging for analytical solutions. Numerical simulations become essential in addressing these complex dynamics.

Fluid instability

MHD fluids can exhibit instabilities that can lead to phenomena such as solar explosions or disruptions in fusion reactors. Understanding and controlling these instabilities is critical to advancing applications.

Conclusion

Magnetohydrodynamics provides a fundamental understanding of the behavior of conducting fluids in magnetic and electric fields. Its principles are widely applied and are critical to innovations in technology, astronomy, and Earth science. Ongoing research and development in MHD continues to push the boundaries of knowledge and applications in physics and engineering.


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Magnetohydrodynamics

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