Lundquist magnetic Reynolds number calculator plasma MHD

The Lundquist number S calculator determines the fundamental dimensionless parameter of magnetohydrodynamics (MHD) measuring the ratio of resistive timescale τ_η to Alfvén timescale τ_A in a magnetized plasma, per the formulation by Swedish physicist Stig Lundquist 1949 Physical Review 76, 1805 "Experimental Investigations of Magneto-Hydrodynamic Waves" Uppsala University. The Lundquist number S is a variant of the magnetic Reynolds number Rm built with Alfvén velocity v_A instead of flow velocity v. The classic formula: S = μ₀·L·v_A / η = τ_η / τ_A, with L = characteristic length scale (m), v_A = B / √(μ₀·ρ) Alfvén velocity (m/s), η = magnetic resistivity (Ω·m), μ₀ = 4π×10⁻⁷ T·m/A vacuum permeability. Equivalently: S = L·v_A / D_m with D_m = η/μ₀ magnetic diffusivity (m²/s). The general magnetic Reynolds number: Rm = μ₀·L·v / η with v flow velocity (Lundquist uses v=v_A specifically). MHD physics: for S >> 1 → ideal MHD (frozen-flux Alfvén theorem 1942, magnetic field lines frozen in plasma move with it); S << 1 → resistive diffusion dominates (field lines relax via ohmic dissipation);

S ~ 1 → magnetic reconnection (topology change). Typical S values:

• Solar corona L=10⁸ m, v_A=10⁶ m/s, η=10⁻⁶ Ω·m → S ≈ 10¹³ (ideal MHD)
• Tokamak ITER L=2 m, v_A=10⁷ m/s, η=10⁻¹⁰ Ω·m → S ≈ 10⁹
• Solar wind L=10⁸ m, v_A=5×10⁴ m/s, η=10⁻⁹ Ω·m → S ≈ 10¹²
• Lab plasma L=0.1 m, v_A=10⁵ m/s → S ≈ 10⁵
• Galactic dynamo L=10²⁰ m → S ≈ 10²⁰. Magnetic reconnection: Sweet-Parker 1957-58 τ_rec = τ_A · √S (slow for large S, collisional scaling); Petschek 1964 τ_rec ~ τ_A · ln(S) (fast, X-point with shocks); Plasmoid instability Loureiro 2007 for S > 10⁴ turbulent reconnection regime. Physical applications:

• MHD plasma simulations (GIZMO, FLASH, PLUTO INAF code)
• Solar flares CME magnetic reconnection
• Tokamak instability ELMs Edge Localized Modes
• Magnetar magnetosphere SGR
• Accretion disks AGN/X-ray binaries
• Galactic dynamo amplification μG fields
• Magnetic reconnection diagnostics solar flares. Italian applications:

• ENEA Frascati DTT Divertor Tokamak Test 2028 (Italian Energy Agency Centro Ricerche Frascati Rome)
• FTU Frascati Tokamak Upgrade 1989-2020 (8 T field deuterium, highest density tokamak ENEA)
• ITER France 2025 Cadarache + Ansaldo Nucleare Genoa (Italy 25% ENEA + INFN + CNR)
• INFN Frascati 1957 LNF National Labs Bruno Touschek ADA 1961 storage rings
• INAF IAPS Rome 2010 Astrophysics Planetology Space plasma MHD physics
• Solar Orbiter ESA 2020 ASI 25% corona-chromosphere reconnection Metis EUI
• BepiColombo ESA-JAXA 2018 ASI 35% Mercury magnetosphere MHD plasma
• AGILE satellite ASI 2007 AGN gamma-ray flares blazars. Italian space plasma research: INAF Trieste 1898 solar physics IBIS Interferometric BIdimensional Spectropolarimeter; Athena ESA 2035 X-IFU INAF Palermo 25% AGN cluster ICM reconnection; LiteBIRD ESA 2032 INAF + ASI CMB B-modes inflation; Cineca Marconi100 30 PFlops + Leonardo 250 PFlops 2023 N-body MHD plasma simulations 5th TOP500 worldwide; ICTP Trieste 1964 + SISSA 1978 MHD astrophysical plasma. Italian Physics Nobel: Carlo Rubbia 1984 CERN W and Z bosons UA1 + ENEA president 1999-2005 nuclear fusion. PNRR 2021-2026: Mission 2 green hydrogen nuclear fusion; Mission 4 Component 2 research €6.9B

Italian Lundquist number plasma table

(Source: Lundquist 1949 Phys. Rev. 76, 1805 + Sweet 1958 Parker 1957 + Petschek 1964 + Loureiro 2007 plasmoid + ENEA Frascati 2025 + INAF IAPS Rome).

Italian Lundquist calculation examples

• Solar corona X-class flare reconnection Solar Orbiter ESA 2020 ASI 25% Metis EUI: L = 10⁸ m (1 solar radius), v_A = 10⁶ m/s (B = 10 G ρ = 10⁻¹² kg/m³), η = 10⁻⁶ Ω·m (Spitzer T=10⁶ K). S = μ₀·L·v_A/η = (4π×10⁻⁷)·10⁸·10⁶/10⁻⁶ = 1.26×10¹³. τ_A = L/v_A = 100 s, τ_η = L²/(η/μ₀) = 1.3×10¹⁹ s (>> universe age). Sweet-Parker τ_rec = τ_A·√S ≈ 3.5×10⁷ s = 1 year (TOO SLOW for observed CME 1 hour). Petschek τ_rec = τ_A·ln(S) ≈ 3,000 s = 50 min (matches CME observed)
• Tokamak ITER France 2025 Cadarache ENEA 25% + Ansaldo Nucleare Genoa: L = 2 m, v_A = 10⁷ m/s (B = 5.3 T D-T plasma), η = 10⁻¹⁰ Ω·m (T = 150 mln K). S = (4π×10⁻⁷)·2·10⁷/10⁻¹⁰ = 2.5×10⁹. Plasmoid instability for S > 10⁴ → turbulent reconnection ELMs Edge Localized Modes (controlled via RMP Resonant Magnetic Perturbations)
• FTU Frascati Tokamak Upgrade 1989-2020 ENEA 8 Tesla: L = 0.9 m, v_A = 5×10⁶ m/s, η = 10⁻⁹. S = 5.7×10⁶
• DTT Divertor Tokamak Test ENEA Frascati 2028: L = 1.2 m, v_A = 8×10⁶ m/s, η = 10⁻¹⁰. S = 1.2×10⁸
• Mercury magnetosphere BepiColombo MPO ESA-JAXA 2018 ASI 35%: L = 2×10⁶ m, v_A = 2×10⁵ m/s, η = 10⁻⁸. S = 5×10¹⁰. Substorm reconnection magnetotail Mercury (1 g vs Earth 30%)
• AGN accretion disk Athena ESA 2035 X-IFU INAF Palermo 25%: L = 10¹³ m, v_A = 10⁵ m/s, η = 10⁻⁸. S = 1.3×10¹⁵. MRI Magnetorotational Instability Balbus-Hawley 1991 angular momentum transport
• Lab plasma INFN Frascati LNF 1957: L = 0.1 m, v_A = 10⁵ m/s, η = 10⁻⁵. S = 1.3×10⁵. Z-pinch magnetic confinement experiments

Italian plasma fusion MHD research

Italian MHD plasma research:

• ENEA Centro Ricerche Frascati Rome 1960 Italian Agency Energy New Technologies Sustainable Economic Development: FTU Frascati Tokamak Upgrade 1989-2020 8 Tesla highest field record, deuterium plasma record density; DTT Divertor Tokamak Test 2028 first-of-its-kind divertor exhaust heat resolution (€500 mln PNRR + Eurofusion); RFX-mod2 Padua 2024 Reversed Field Pinch Consorzio RFX CNR-INFN-University 2 MA plasma current
• INFN National Institute Nuclear Physics Frascati 1957 LNF Laboratori Nazionali Bruno Touschek ADA 1961 + ADONE 1969 e+e- storage rings (DAΦNE 2002); 23 national sections (Rome Sapienza/Tor Vergata, Milan Bicocca, Padua, Pisa, Bologna, Turin, Naples, Pavia, Genoa, Bari, Trieste, Catania, Lecce, Cagliari, Perugia); €50M annual budget
• INAF National Institute Astrophysics Rome 2002 Monte Porzio Catone (16 observatories): IAPS Rome 2010 Astrophysics Planetology Space plasma MHD + space missions; Trieste 1898 solar physics IBIS Sunspot 2003 + Planck CMB 2009-2018; Capodimonte Naples 1812 VST KIDS dark matter; Catania-Etna 1880 solar; Padua 1761 Galileo Asiago 1979 Copernico 182 cm; Brera Milan 1762; Arcetri Florence 1872; Bologna Loiano 1936; Palermo 1790; Cagliari 1899 SRT 64 m radio
• Italian solar telescopes: IBIS Interferometric BIdimensional Spectropolarimeter Sacramento Peak NM AURA INAF Arcetri-Catania magnetic photosphere-chromosphere spectroscopy; EST European Solar Telescope 4 m Canary Islands 2030 Italy 15% INAF + ENEA; SST Swedish Solar Telescope INAF collaboration
• ASI Rome 1988 plasma missions: Solar Orbiter ESA 2020 ASI 25% 0.28 AU minimum solar distance, Italian instruments Metis coronagraph INAF Padua-Turin + EUI INAF + STIX X-ray; Parker Solar Probe NASA 2018 9.86 R_sun 2024; BepiColombo ESA-JAXA 2018 ASI 35% MPO Mercury orbiter 2026 magnetosphere MHD plasma SERENA INAF + ISA accelerometer; AGILE ASI 2007-2024 gamma-ray observatory INAF + INFN AGN flares blazar; Athena ESA 2035 X-IFU calorimeter INAF Palermo 25% AGN cluster ICM accretion disk magnetic reconnection; LiteBIRD JAXA-ESA 2032 CMB B-modes inflation INAF Trieste + ASI
• ITER International Thermonuclear Experimental Reactor France Cadarache 2025 first plasma 2025 D-T fusion 500 MW (Italy 25% via EU EUROfusion + ENEA + INFN + CNR + Consorzio RFX Padua; Ansaldo Nucleare Genoa vacuum vessel + TF coils; Walter Tosto Chieti mechanical components; SIMIC Camerano; Criotec Turin; OCEM Energy Bologna)
• Cineca Bologna 1969 Marconi100 30 PFlops + Leonardo HPC pre-exascale 250 PFlops 2023 5th TOP500 worldwide MHD plasma magnetar AGN GRMHD simulations
• ICTP Trieste 1964 International Centre Theoretical Physics Abdus Salam Nobel 1979 + SISSA Trieste 1978 MHD astrophysics. Italian Physics Nobel: Carlo Rubbia 1984 CERN Pavia University W and Z bosons UA1 + ENEA president 1999-2005 nuclear fusion project. Italian R&D tax bonuses 2025-2026: R&D Tax Credit L. 160/2019 art. 1 c. 200 up to 25% research innovation plasma fusion expenses; Patent Box L. 178/2020 50% income from MHD software algorithm patents; Industry/Transition 5.0 Tax Credit DL 39/2024 art. 38 up to 35% energy efficiency fusion green hydrogen investments; PNRR Mission 2 Component 2 hydrogen nuclear fusion; Mission 4 Component 2 research €6.9B fundamental and applied

Updated July 2026