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alfvén mach number wikipedia jump to content main menu main menu move to sidebar hide navigation main page contents current events random article about wikipedia contact us contribute help learn to edit community portal recent changes upload file special pages search search appearance donate create account log in personal tools donate create account log in contents move to sidebar hide top 1 derivation from the magnetohydrodynamics momentum equation 2 see also 3 references toggle the table of contents alfvén mach number 3 languages català español français edit links article talk english read edit view history tools tools move to sidebar hide actions read edit view history general what links here related changes upload file permanent link page information cite this page get shortened url switch to legacy parser print export download as pdf printable version in other projects wikidata item appearance move to sidebar hide from wikipedia the free encyclopedia dimensionless quantity in plasma physics the alfvén mach number also known as the alfvén number alfvénic mach number and magnetic mach number m a or m m is a dimensionless quantity representing the ratio of the relative velocity of a fluid to the local alfvén speed it is used in plasma physics where it is analogous to the mach number but based on alfvén waves rather than sound waves alfvén and mach were physicists who studied shock waves along with the sonic mach number the alfvén mach number is frequently used to characterize shock fronts 1 2 3 and turbulence 4 5 in magnetized plasmas 6 7 m a u u a displaystyle mathrm m_ a frac u u_ mathrm a where m a is the alfvén mach number u is the flow velocity and u a is the alfvén speed 8 when u m a the flow is referred to as sub alfvénic and when u m a the flow is referred to as super alfvénic 2 derivation from the magnetohydrodynamics momentum equation edit ignoring viscosity and external body forces the momentum equation for magnetohydrodynamics mhd is ρ v t v v p j b displaystyle rho left frac partial mathbf v partial t mathbf v cdot nabla mathbf v right nabla p mathbf j times mathbf b where j b displaystyle mathbf j times mathbf b is the lorentz force using the low frequency form of ampère s law μ 0 j b displaystyle mu _ 0 mathbf j nabla times mathbf b which neglects the displacement current the momentum equation becomes ρ v t v v p 1 μ 0 b b displaystyle rho left frac partial mathbf v partial t mathbf v cdot nabla mathbf v right nabla p frac 1 mu _ 0 left nabla times mathbf b right times mathbf b we may nondimensionalize the momentum equation by introducing the nondimensional variables 9 page needed v u v x l x b b b ρ ρ 0 ρ t l u τ displaystyle mathbf v u widehat mathbf v quad mathbf x l widehat mathbf x quad mathbf b b widehat mathbf b quad rho rho _ 0 widehat rho quad t frac l u tau where u displaystyle u is the characteristic velocity l displaystyle l is the characteristic length b displaystyle b is the characteristic magnetic field strength and ρ 0 displaystyle rho _ 0 is the characteristic mass density therefore the inertial and lorentz terms in the momentum equation respectively scale as ρ v v ρ 0 u 2 l 1 μ 0 b b b 2 μ 0 l ρ v v 1 μ 0 b b u 2 b 2 μ 0 ρ 0 displaystyle rho mathbf v cdot nabla mathbf v sim frac rho _ 0 u 2 l quad frac 1 mu _ 0 left nabla times mathbf b right times mathbf b sim frac b 2 mu _ 0 l implies frac rho mathbf v cdot nabla mathbf v frac 1 mu _ 0 left nabla times mathbf b right times mathbf b sim frac u 2 b 2 mu _ 0 rho _ 0 recognizing that the alfvén speed u a b μ 0 ρ 0 displaystyle u_ mathrm a b sqrt mu _ 0 rho _ 0 we may see that this quantity is the square of the alfvén mach number m a u u a displaystyle mathrm m_ a u u_ mathrm a hence the alfvén mach number naturally arises from comparing the inertial and magnetic terms in the mhd momentum equation and expresses the ratio of inertial to magnetic forces in a plasma see also edit plasma parameters magnetic reynolds number lundquist number references edit droege w schlickeiser r 1986 particle acceleration in solar flares the astrophysical journal 305 909 bibcode 1986apj 305 909d doi 10 1086 164305 1 2 treumann r a 2009 fundamentals of collisionless shocks for astrophysical application 1 non relativistic shocks the astronomy and astrophysics review 17 4 409 535 bibcode 2009a arv 17 409t doi 10 1007 s00159 009 0024 2 kang hyesung ryu dongsu 2013 diffusive shock acceleration at cosmological shock waves the astrophysical journal 764 1 95 arxiv 1212 3246 bibcode 2013apj 764 95k doi 10 1088 0004 637x 764 1 95 cho jungyeon lazarian a october 2003 compressible magnetohydrodynamic turbulence mode coupling scaling relations anisotropy viscosity damped regime and astrophysical implications monthly notices of the royal astronomical society 345 1 325 339 arxiv astro ph 0301062 bibcode 2003mnras 345 325c doi 10 1046 j 1365 8711 2003 06941 x padoan paolo jimenez raul juvela mika nordlund åke 2004 the average magnetic field strength in molecular clouds new evidence of super alfvénic turbulence the astrophysical journal 604 1 l49 l52 arxiv astro ph 0311349 bibcode 2004apj 604l 49p doi 10 1086 383308 de keyser johan roth michel de sterck hans poedts stefaan 2001 a survey of field aligned mach number and plasma beta in the solar wind the 3 d heliosphere at solar maximum vol 97 pp 201 204 bibcode 2001ssrv 97 201d doi 10 1007 978 94 017 3230 7_31 isbn 978 90 481 5723 5 cite book journal ignored help tofflemire benjamin m burkhart blakesley lazarian a 2011 interstellar sonic and alfvénic mach numbers and the tsallis distribution the astrophysical journal 736 1 60 arxiv 1103 3299 bibcode 2011apj 736 60t doi 10 1088 0004 637x 736 1 60 beresnyak a october 2023 2023 nrl plasma formulary washington dc naval research laboratory davidson peter a 2016 introduction to magnetohydrodynamics cambridge texts in applied mathematics 2nd ed cambridge university press pp 112 120 v t e fluid mechanics fluid statics hydraulics archimedes principle fluid dynamics computational fluid dynamics aerodynamics navier stokes equations boundary layer entrance length dimensionless numbers alfvén mach archimedes atwood bagnold bejan biot bodenstein bond brinkman capillary cauchy chandrasekhar damköhler darcy dean deborah dukhin eckert ekman eötvös euler froude galilei graetz grashof görtler hagen iribarren kapitza keulegan carpenter knudsen laplace lewis mach marangoni morton nusselt ohnesorge péclet prandtl magnetic turbulent rayleigh reynolds magnetic richardson roshko rossby rouse schmidt scruton sherwood shields stanton stokes strouhal stuart suratman taylor ursell weber weissenberg womersley this plasma physics related article is a stub you can help wikipedia by adding missing information v t e retrieved from https en wikipedia org w index php title alfvén_mach_number oldid 1320015605 categories dimensionless numbers of fluid mechanics plasma parameters plasma physics stubs hidden categories articles with short description short description is different from wikidata cs1 errors periodical ignored wikipedia articles needing page number citations from october 2025 all stub articles this page was last edited on 2 november 2025 at 06 13 utc page was rendered with parsoid text is available under the creative commons attribution sharealike 4 0 license additional terms may apply by using this site you agree to the terms of use and privacy policy wikipedia is a registered trademark of the wikimedia foundation inc a non profit organization privacy policy about wikipedia disclaimers contact wikipedia legal safety contacts code of conduct developers statistics cookie statement mobile view search search toggle the table of contents alfvén mach number 3 languages add topic