How to calculate the Alfven number (MA or Alfven Mach number) fundamental dimensionless ratio magnetohydrodynamics MHD and plasma physics measuring flow velocity vs Alfven wave velocity ratio: MA = v / vA where vA = B / sqrt(mu_0 * rho) Alfven velocity m/s. v plasma flow velocity m/s (solar corona 100-1000 km/s; solar wind 400-800 km/s; tokamak 1-10 km/s plasma rotation; ionosphere 100-1000 m/s); B magnetic field T (Sun surface 1e-4 to 0.4; solar wind 5e-9; tokamak 5-15 T ITER; Earth 3e-5; lab plasma 0.1-10); rho mass density kg/m^3 (corona 1e-12; solar wind 1e-20; tokamak 1.6e-7; ionosphere 1e-12; lab 1e-9 to 1e-5).

Alfven velocity vA: vA = B / sqrt(mu_0 * rho) + mu_0 = 4*pi*1e-7 = 1.257e-6 T*m/A vacuum permeability + vA km/s typical corona 1000-2000 + solar wind 30-80 + tokamak 6e6 + lab plasma 1e5; vA practical units vA = 21.84 * B[T] / sqrt(rho[kg/m^3]) m/s.

Alfven waves: transverse waves B oscillates perpendicular to B_0 + propagation along B_0 at velocity vA + Hannes Alfven 1942 Nature 150:405 'Existence of electromagnetic-hydrodynamic waves' classic + Alfven waves transverse + typical scales 100-10000 km solar 1-100 m lab.

MHD regime classification: MA<0.1 sub-Alfvenic magnetic field dominant + frozen flux Alfven theorem + field lines frozen + solar corona quiescent regions; 0.1<MA<1 sub-Alfvenic magnetic dominant; 1<MA<10 trans-Alfvenic transition + shock waves possible; MA>10 super-Alfvenic kinetic dominant + magnetic field passively transported + supernova remnants astrophysical jets.

Ideal MHD equations: continuity drho/dt + div(rho*v) = 0; momentum rho*dv/dt = -grad(p) + (J x B) + rho*g; induction dB/dt = curl(v x B) - eta*nabla^2(B) where eta = 1/(mu_0*sigma) magnetic diffusivity + ideal eta=0 frozen-in flux Alfven.

Related MHD numbers: magnetic Reynolds Rm = v*L/eta magnetic diffusivity + Rm>>1 ideal frozen + Rm<<1 diffusive; magnetic Prandtl Pm = nu/eta; Hartmann Ha = B*L*sqrt(sigma/mu); plasma beta = p/(B^2/2/mu_0) thermal vs magnetic pressure; Lundquist S = vA*L/eta; Elsasser Lambda.

Frozen-in flux theorem Alfven: ideal plasma Rm>>1 magnetic field lines move with fluid + Alfven 1942 conservation + classical reference + applications solar corona loops + magnetosphere reconnection + accretion disks.

Magnetic reconnection: finite resistivity eta region + Sweet-Parker 1958 + Petschek 1964 fast reconnection + tearing instability + plasmoid + solar flares + substorms + classical reference. History: Hannes Olof Gosta Alfven (Norrkoping Sweden 1908 - Djursholm 1995) Royal Institute Technology KTH Stockholm 1940-1973 + UC San Diego 1967-1991 + Royal Swedish Academy; 1942 paper Nature 150:405 'Existence of electromagnetic-hydrodynamic waves' classic Alfven waves + 1939 cosmic radiation Phys Rev + 1950 'Cosmical Electrodynamics' Oxford Clarendon 237 pages classic + 1981 'Cosmic Plasma' Reidel 164 pages; Nobel Prize Physics 1970 'fundamental work and discoveries in magnetohydrodynamics with fruitful applications in different parts of plasma physics' shared with Louis Neel; pioneer plasma astrophysics + magnetosphere theory + double layers + Alfven number eponym + Alfven waves eponym + Royal Society fellow 1980 + Royal Astronomical Society Gold Medal 1967 + EU Plasma Physics Society Hannes Alfven Prize eponym annual since 2000.

Lyman Spitzer Jr (Toledo Ohio 1914 - Princeton 1997) Princeton 1947-1997 founder Princeton Plasma Physics Laboratory PPPL 1951; 1962 'Physics of Fully Ionized Gases' 2nd Wiley 170 pages classic + 1958 stellarator Princeton invention + Spitzer resistivity 1953 + classical plasma physics + 1946 Hubble Space Telescope first proposal inventor + interstellar medium + Bruce Medal 1973 + Crafoord Prize 1985.

Igor Tamm (Vladivostok 1895 - Moscow 1971) Moscow State Lebedev Institute; 1957 stellarator-tokamak Sakharov-Tamm proposal + Nobel Prize Physics 1958 Cherenkov-Frank-Tamm radiation; pioneer plasma confinement Soviet + Sakharov mentor + Cherenkov radiation theory 1937.

Andrei Sakharov (Moscow 1921 - 1989) Lebedev Institute + 1968 dissident Nobel Peace 1975; 1950 Sakharov-Tamm tokamak proposal toroidal magnetic confinement + classical reference fusion + tokamak coined Russian abbreviation 1957 + ITER international 1985 Reagan-Gorbachev + father of Soviet H-bomb 1953. Legacy: Priest 'Solar Magnetohydrodynamics' 1982 Reidel 469 pages + Goedbloed-Poedts 'Principles of MHD' 2004 Cambridge 613 pages + Biskamp 'Nonlinear MHD' 1993 Cambridge 392 pages + Wesson 'Tokamaks' 4th 2011 OUP 800 pages classic + Freidberg 'Ideal MHD' 1987 Plenum + Boyd-Sanderson 'Physics of Plasmas' 2003 Cambridge 555 pages + Kulsrud 'Plasma Physics for Astrophysics' 2005 Princeton 459 pages classic.

USA research plasma + MHD + fusion + astrophysics: Princeton Plasma Physics Laboratory PPPL Princeton 1951 600 world-leading + NSTX-U + DOE; MIT Plasma Science Fusion Center PSFC Cambridge MA 250 world-leading + Alcator C-Mod + SPARC; General Atomics San Diego 7000 + DIII-D tokamak world-leading + ITER components; Sandia National Laboratory Albuquerque NM 13000 + Z-machine pulsed power; Lawrence Livermore National Lab LLNL 7500 + NIF inertial confinement fusion world-leading 2022 ignition; Los Alamos National Lab LANL 13000; Oak Ridge National Lab ORNL 6000 + ITER U.S. office; NASA Goddard 11000 + Solar Probe + Parker; NRL Naval Research Lab 2700 + plasma physics; NSF Geosciences 4500; NCAR HAO Boulder 1500 + solar atmospheric; NOAA SWPC Boulder 100 + space weather.

USA universities plasma + MHD: Princeton AOS PPPL 5400 + Russell Kulsrud + Hantao Ji world-leading; MIT NSE PSFC 11500 + Dennis Whyte SPARC + Earl Marmar Alcator world-leading; Caltech 8800 + Paul Bellan plasma; Univ Wisconsin Madison 7000 + Cary Forest plasma + RFX MST experiment; UCLA 31000 + Walter Gekelman LAPD; UC Berkeley 8500; UCSD UC San Diego 35000 + Patrick Diamond turbulence; Univ Maryland 41000; Univ Texas Austin IFS 50000 + Wendell Horton; Univ Washington Seattle 47000; Auburn 22000; Stanford 11000.

USA industry fusion + tokamak: General Atomics 7000 + DIII-D + ITER first wall world-leading; Lockheed Martin Palmdale 122000 + compact fusion R&D; TAE Technologies Foothill Ranch CA 250 field-reversed config; Commonwealth Fusion Systems Cambridge MA 200 + SPARC ARC HTS magnets; Helion Energy Everett WA 100; Tokamak Energy UK US ops 250; Zap Energy Seattle 80 Z-pinch; Type One Energy Madison 100; Pacific Fusion 50.

USA industry astrophysics + space: NASA Goddard JPL Glenn 17000; SpaceX Hawthorne CA 13000; ULA Centennial 4500; Lockheed Martin Space 122000; Northrop Grumman Space 95000 + James Webb; Ball Aerospace Boulder 5500; Maxar 4000; Planet San Francisco 1000; Maxar Westminster 4000; L3Harris Melbourne FL 47000.

USA industry radio + microwave + plasma: Raytheon Waltham MA 195000 + radar plasma diagnostics; Northrop Grumman Falls Church 95000; L3Harris 47000; BAE Systems 32000.

Alfven formula

Definition: MA = v / vA where vA = B / sqrt(mu_0 * rho). v flow velocity m/s + B magnetic field T + rho density kg/m^3 + mu_0 = 1.257e-6 T*m/A vacuum permeability + dimensionless.

Alfven velocity: vA = B/sqrt(mu_0*rho) = 21.84*B[T]/sqrt(rho[kg/m^3]) m/s + typical corona 1000-2000 km/s + solar wind 30-80 + tokamak 6e6 + lab plasma 1e5.

Related MHD numbers: magnetic Reynolds Rm = v*L/eta; magnetic Prandtl Pm = nu/eta; Lundquist S = vA*L/eta; plasma beta = 2*mu_0*p/B^2.

Typical values: solar corona quiescent v=100 km/s vA=1000 km/s MA=0.1 sub-Alfvenic; solar wind v=450 vA=80 MA=5.6 super-Alfvenic; tokamak v=10 km/s vA=6000 MA=1.7e-3; ionosphere ITER MA<<1.

Frozen-in flux theorem: Rm>>1 + magnetic field lines frozen to fluid + Alfven 1942 + corona loops + accretion disks.

History and context

Hannes Olof Gosta Alfven (Norrkoping Sweden 1908 - Djursholm 1995) Royal Institute Technology KTH Stockholm 1940-1973 + UC San Diego 1967-1991 + Royal Swedish Academy member; 1942 paper Nature 150:405 'Existence of electromagnetic-hydrodynamic waves' classic Alfven transverse waves + 1939 cosmic radiation Phys Rev + 1950 'Cosmical Electrodynamics' Oxford Clarendon 237 pages classical reference + 1950 'On the origin of cosmic rays' Tellus 2:84 + 1981 'Cosmic Plasma' Reidel 164 pages + 1986 'Plasma Universe' classical; Nobel Prize Physics 1970 'fundamental work and discoveries in magnetohydrodynamics with fruitful applications in different parts of plasma physics' shared with Louis Neel; pioneer plasma astrophysics + magnetosphere theory + double layers + Alfven number eponym + Alfven waves eponym + Alfven layer ionospheric + Member US National Academy + Royal Society fellow 1980 + Royal Astronomical Society Gold Medal 1967 + EU Plasma Physics Society Hannes Alfven Prize annual since 2000.

Lyman Spitzer Jr (1914 - 1997) Princeton 1947-1997 founder Princeton Plasma Physics Laboratory PPPL 1951; 1962 'Physics of Fully Ionized Gases' 2nd Interscience-Wiley 170 pages classic + 1958 stellarator Princeton invention + Spitzer resistivity 1953 + classical reference + 1946 Hubble Space Telescope first proposal inventor + interstellar medium classical + Bruce Medal 1973 + Crafoord Prize 1985.

Igor Tamm (1895 - 1971) Moscow Lebedev; 1957 Sakharov-Tamm tokamak proposal + Nobel Prize Physics 1958 Cherenkov-Frank-Tamm radiation; pioneer Soviet plasma + Cherenkov radiation theory 1937.

Andrei Sakharov (Moscow 1921 - 1989) Lebedev + 1968 dissident Nobel Peace 1975; 1950 Sakharov-Tamm tokamak toroidal magnetic confinement + classical reference fusion + tokamak Russian abbreviation 1957 + ITER international 1985 Reagan-Gorbachev + father of Soviet H-bomb 1953. Legacy: Priest 'Solar MHD' 1982 Reidel 469 pages + Goedbloed-Poedts 'Principles of MHD' 2004 Cambridge 613 pages + Biskamp 'Nonlinear MHD' 1993 Cambridge 392 pages + Wesson 'Tokamaks' 4th 2011 OUP 800 pages classical + Freidberg 'Ideal MHD' 1987 Plenum + Boyd-Sanderson 'Physics of Plasmas' 2003 Cambridge 555 pages + Kulsrud 'Plasma Physics for Astrophysics' 2005 Princeton 459 pages classic.

USA research: PPPL Princeton 600 world-leading + NSTX-U + DOE; MIT PSFC 250 world-leading + Alcator C-Mod + SPARC; General Atomics 7000 + DIII-D world-leading; Sandia 13000 Z-machine; LLNL 7500 NIF 2022 ignition; LANL 13000; ORNL 6000 ITER US; NASA Goddard 11000 Solar Probe; NRL 2700.

Universities: Princeton AOS PPPL 5400 + Russell Kulsrud + Hantao Ji world-leading; MIT NSE 11500 + Dennis Whyte SPARC + Earl Marmar; Caltech 8800 + Paul Bellan; Wisconsin 7000 + Cary Forest MST; UCLA 31000 + Walter Gekelman LAPD; UCSD 35000 + Patrick Diamond.

Industry fusion: General Atomics 7000; Commonwealth Fusion Systems 200 SPARC ARC HTS; TAE Technologies 250 FRC; Helion Energy 100; Lockheed Martin 122000; Tokamak Energy 250; Zap Energy 80.

Astrophysics space: NASA 17000; SpaceX 13000; ULA 4500; Lockheed Martin Space 122000; Northrop Grumman 95000 JWST; Ball Aerospace 5500; Maxar 4000.

Radar plasma: Raytheon 195000; Northrop Grumman 95000; L3Harris 47000; BAE 32000.

Alfven values table

• MA = v/vA: definition
• vA = B/sqrt(mu_0*rho): Alfven velocity
• vA = 21.84*B/sqrt(rho): practical
• MA<0.1: ultra sub-Alfvenic frozen-in
• 0.1<MA<1: sub-Alfvenic magnetic dominant
• 1<MA<10: trans-Alfvenic shock
• MA>10: super-Alfvenic kinetic
• plasma beta = 2*mu_0*p/B^2
• Rm = v*L/eta: magnetic Reynolds
• Lundquist S = vA*L/eta

5 numerical examples

1. Solar corona quiescent loops: v=100 km/s + B=1e-3 T + rho=1e-12 kg/m^3 + vA = 890 km/s + MA = 0.11 sub-Alfvenic frozen-in.
2. Solar wind 1 AU: v=450 km/s + B=5e-9 T + rho=8.4e-21 kg/m^3 + vA = 49 km/s + MA = 9.2 super-Alfvenic.
3. Tokamak ITER plasma: v=10 km/s + B=5 T + rho=1.7e-7 kg/m^3 + vA = 11000 km/s + MA = 9.1e-4 ultra sub-Alfvenic.
4. Earth ionosphere: v=500 m/s + B=3e-5 T + rho=1e-12 kg/m^3 + vA = 27000 km/s + MA = 1.9e-5 ultra sub-Alfvenic.
5. Lab plasma RFX: v=10 km/s + B=0.1 T + rho=1e-8 kg/m^3 + vA = 890 km/s + MA = 0.011 sub-Alfvenic.

Limitations

Ideal MHD: Rm>>1 + frozen-in + collisional plasma + scales > Larmor radius + > Debye length + classical Alfven 1950.

Plasma beta: beta = 2*mu_0*p/B^2 distinct + low-beta corona ITER + high-beta solar wind ICME.

Compressibility: fast Alfven + slow Alfven + magnetosonic waves + sound vs Alfven distinct.

Anisotropy: B preferred direction + Goldreich-Sridhar 1995 turbulence + parallel vs perpendicular cascade.

Common mistakes

• Confuse MA and beta: MA kinetic/Alfven + beta thermal/magnetic distinct + Goedbloed-Poedts 2004.
• Mu_0 units: mu_0 = 4*pi*1e-7 = 1.257e-6 T*m/A SI + Gaussian cgs differs sqrt(4*pi) factor + Jackson 1999.
• Hall MHD: ion Larmor scale + frequency > Omega_ci + Hall extension + classical scale dependence.
• Ion-electron mass: vA electronic vs ionic + electron Alfven ~ 43*ion sqrt(m_p/m_e) + plasma physics distinct.
• Magnetosonic: fast slow magnetosonic distinct + cusp + perpendicular propagation + Goedbloed-Poedts 2004.

Applications

1. Solar corona heating: Alfven dissipation + Parker nanoflares + NASA + Solar Orbiter ESA.
2. Solar wind: Solar Probe Parker + NASA + space weather NOAA SWPC.
3. Tokamak fusion: ITER + DIII-D + General Atomics + Princeton PPPL world-leading.
4. Stellarator W7-X SPARC: Commonwealth Fusion Systems + MIT PSFC.
5. Earth magnetosphere: Van Allen + NASA Goddard + space weather.
6. Astrophysical jets: AGN + microquasars + NASA Hubble JWST.
7. Accretion disks: black holes + neutron stars + magnetorotational MRI.
8. Interstellar medium: galactic magnetic fields + supernova remnants + Cassini.
9. Space weather: solar flares + ICME + NOAA SWPC + Boulder.
10. Lab plasma: PPPL NSTX-U + DIII-D + LAPD UCLA world-leading.

Updated July 2026