How to calculate Atwood number At Rayleigh-Taylor density

Calculate Atwood number At = (rho_h-rho_l)/(rho_h+rho_l) Rayleigh-Taylor instability density contrast 2 fluids. At=0 Boussinesq; At<0.1 weak; 0.1-0.5 standard; At=1 vacuum. ICF fusion + supernova + ocean mixing. NIF LLNL + LANL + Caltech + MIT + NASA.

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How to calculate Atwood number (At) dimensionless Rayleigh-Taylor instability + density-driven flows 2 stacked fluids ratio between heavy + light density: At = (rho_h - rho_l) / (rho_h + rho_l). Range 0<=At<=1.

Regime classification: At0.9 extreme vacuum/dense ICF supernova.

George Atwood (UK 1746-1807) Trinity College Cambridge + Atwood machine 1784.

Sir Geoffrey Ingram Taylor 1950 Proc Roy Soc A 201:192 RT eponym + Manhattan Project Los Alamos. Lord Rayleigh 1883 first instability.

USA universities + national labs ICF + RT research: NIF National Ignition Facility LLNL Livermore CA 2009 + 192 lasers 1.8 MJ + 2022 first ignition Q>1 inertial fusion confinement; OMEGA Laser Facility University Rochester NY 1995 60 lasers 30 kJ ICF research; NIST Boulder CO + Gaithersburg MD 1901 3000; MIT Plasma Science Fusion Center Cambridge MA 1976 200 + SPARC Commonwealth Fusion + Alcator C-Mod retired; Stanford 1885 17000 + Stanford SLAC; Caltech Pasadena 1891 2300 + GALCIT; Princeton Plasma Physics Laboratory PPPL 1951 600 + NSTX-U + Lyman Spitzer Stellarator legacy; General Atomics San Diego CA 1955 5500 + DIII-D National Fusion Facility; UC Berkeley LBNL 1868 45000; Harvard SEAS 1847 + David Weitz soft matter; Cornell Ithaca NY 1865; UPenn 1740; UMN 1851; Texas A&M 1876; Michigan 1817; Georgia Tech 1885; Wisconsin Madison 1848.

USA national labs RT + ICF: LLNL Lawrence Livermore CA 1952 + NIF + Magnetic Mirror; LANL Los Alamos NM 1943 12000 + Trinity supercomputer + nuclear weapons + plasma; SNL Sandia Albuquerque NM + Livermore CA 14000 + Z Machine 26 MA pulsed power Z-pinch fusion; ORNL Oak Ridge TN 5000 + Spallation Neutron Source SNS + ITER Test Blanket Module DCLL; ANL Argonne IL 5000 + Advanced Photon Source APS; PNNL Richland WA 5000; INL Idaho Falls 5000 nuclear; PPPL Princeton 600 + NSTX-U; NRL Naval Research Laboratory Washington DC 1923 2500 + Plasma Physics Division.

USA companies fusion + ICF: Commonwealth Fusion Systems Cambridge MA 2018 800 SPARC tokamak compact high-field + ARC commercial + REBCO superconductor; TAE Technologies Foothill Ranch CA 2017 200 Norman field-reversed configuration FRC; Helion Energy Everett WA 2013 200 Polaris pulsed magnetic; General Fusion Burnaby BC Canada 2002 200 acoustic; Tokamak Energy UK 2009 ST40 spherical tokamak; Lockheed Martin Skunk Works Compact Fusion Reactor CFR; First Light Fusion UK + LASNUS USA inertial fusion; Avalanche Energy Seattle WA 2019 200; Realta Fusion Madison WI 2022; Marvel Fusion Munich Germany USA office + laser inertial.

USA NewSpace + atmospheric entry: SpaceX Hawthorne CA 2002 13000 + Falcon 9 Falcon Heavy Starship + Raptor methalox + Dragon + Starlink 5000+ LEO; Boeing 156000 + 702 satellite bus + Starliner crewed; Lockheed Martin Sunnyvale CA Space Systems + Orion crew capsule Mars + GPS Block IIIF + AEHF; Northrop Grumman 95000 + Cygnus + James Webb Space Telescope JWST 2021; Blue Origin Kent WA 2000 Bezos + New Shepard + New Glenn + BE-4 methalox; Virgin Galactic Mojave CA 2004 SpaceShipTwo; Sierra Space Lone Tree CO 2021 Dream Chaser + LIFE habitat 2026; Rocket Lab Long Beach CA 2006 Electron + Neutron 2025; Astra Space Alameda CA 2016; ULA United Launch Alliance Centennial CO 2006 Boeing-Lockheed Atlas V + Delta IV + Vulcan Centaur.

USA hypersonic + atmospheric entry RT testing: NASA Langley Research Center Hampton VA 1917 3500 + wind tunnels + thermal protection systems TPS Apollo Mars rovers; NASA Ames Research Center Mountain View CA 1939 2300 + arc jet 60 MW + Pleiades supercomputer; AEDC Arnold Tullahoma TN Air Force wind tunnels Mach 14 + 16 + arc heated; CUBRC Buffalo NY hypersonic shock tunnel LENS-X 7-30 km/s entry flow.

USA astrophysics supernova RT: NASA JPL Caltech Pasadena CA 1936 6000 + Voyager + Cassini + DSN; NASA Goddard Greenbelt MD 1959 10000 + SOHO + STEREO + Parker Solar Probe + Solar Orbiter ESA + Chandra X-ray Observatory; NRAO Charlottesville VA + VLA Socorro NM + ALMA Chile + GBT Green Bank WV; NSO National Solar Observatory Boulder CO + Sunspot NM + DKIST Maui HI 4 m largest solar telescope; HAO High Altitude Observatory NCAR Boulder CO 1500; Caltech Owens Valley OVRO 40 m.

USA petrochemical density-driven oil-water-gas: ExxonMobil Spring TX 62000 + Chemical 22000 + Baytown 590k + Beaumont + Baton Rouge + Joliet refineries + cryogenic LNG; Chevron 45000 + Phillips Chemical 8000 + Pascagoula MS + Richmond CA refineries; Phillips 66 14000; Marathon Petroleum 17000; Valero 10000; Halliburton 40000 oilfield + density flow; Schlumberger 99000 + reservoir; Baker Hughes 55000; Dow Chemical 36000; DuPont 23000; LyondellBasell 19000; Chemours 6000.

USA atmospheric + ocean mixing climate: NASA Goddard 10000 + climate; NOAA Silver Spring MD 1970 12000 + Space Weather Prediction Center Boulder CO + Climate Prediction Center; USGS Reston VA 9000 + USGS Climate + hydrology; Lamont-Doherty Earth Observatory Columbia University NY 1949; WHOI Woods Hole Oceanographic Institution MA 1930 1000 + ocean + atmosphere; Scripps Institution Oceanography UC San Diego La Jolla CA 1903 1000.

Atwood formula

At = (rho_h - rho_l) / (rho_h + rho_l). Range 0-1. RT growth gamma = sqrt(At*g*k).

History and context

George Atwood UK 1746-1807 Trinity College Cambridge + Atwood machine 1784 + 1784 textbook. Lord Rayleigh 1883 Proc London Math Soc 14:170 + Theory of Sound 1877. G.I. Taylor 1950 Proc Roy Soc A 201:192 RT eponym + Manhattan Project. Sharp 1984 Physica D 12:3 RT review + Lindl 1998 Springer ICF + Cabot-Cook 2006 Nature Physics + Layzer 1955 ApJ classic. USA universities + national labs: NIF LLNL Livermore CA 2009 + 192 lasers 1.8 MJ + 2022 first ignition Q>1; OMEGA University Rochester 1995 60 lasers 30 kJ; NIST Boulder + Gaithersburg 1901 3000; MIT Plasma Science Fusion Center 1976 200 + SPARC; Stanford SLAC; Caltech 1891 2300 GALCIT; Princeton PPPL 1951 600 NSTX-U; General Atomics 1955 5500 DIII-D; UC Berkeley LBNL 1868 45000; Harvard SEAS Weitz; Cornell + UPenn + UMN + Texas A&M + Michigan + Georgia Tech + Wisconsin BSL textbook. National labs: LLNL 1952 + NIF + Magnetic Mirror; LANL 12000 Trinity supercomputer; SNL 14000 Z Machine 26 MA pulsed; ORNL 5000 SNS ITER TBM DCLL; ANL Argonne; PNNL 5000; INL 5000 nuclear; PPPL 600; NRL 2500 Plasma Physics. USA fusion companies: Commonwealth Fusion Cambridge 2018 800 SPARC ARC REBCO; TAE Technologies 200 Norman FRC; Helion Energy 200 Polaris pulsed; General Fusion Canada 200 acoustic; Tokamak Energy UK ST40; Lockheed Skunk Works CFR; First Light Fusion + LASNUS + Avalanche Energy 200 + Realta Fusion + Marvel Fusion Munich. USA NewSpace: SpaceX 13000 Starlink Raptor; Boeing 156000 + Starliner;

Lockheed 116000 Orion; Northrop 95000 Cygnus + JWST 2021; Blue Origin BE-4; Virgin Galactic; Sierra Space Dream Chaser + LIFE 2026; Rocket Lab Electron + Neutron 2025; Astra; ULA Vulcan Centaur. USA hypersonic: NASA Langley 3500 + Ames 60 MW arc jet + AEDC Mach 14-16 + CUBRC LENS-X 7-30 km/s. USA astrofisica: NASA JPL 6000 + Goddard 10000 + Parker Solar Probe + Solar Orbiter + Chandra X-ray; NRAO + ALMA + GBT + DKIST Maui 4 m; NSO Boulder; HAO NCAR; Caltech OVRO. USA petrochimica density: ExxonMobil 62000 + Chemical 22000 + Baytown 590k bpd + Halliburton 40000 + Schlumberger 99000 + Baker Hughes 55000 + Dow 36000 + DuPont 23000 + LyondellBasell 19000 + Chemours. USA atmospheric ocean: NASA Goddard 10000 climate + NOAA 12000 SWPC Boulder + USGS 9000 + Lamont-Doherty Columbia 1949 + WHOI Woods Hole 1930 1000 + Scripps La Jolla 1903 1000.

Atwood table

  • At = (rho_h - rho_l) / (rho_h + rho_l)
  • Range 0-1
  • At<0.01 Boussinesq
  • 0.01-0.1 low contrast
  • 0.1-0.5 moderate RT
  • 0.5-0.9 high asymmetric
  • At>0.9 extreme ICF supernova
  • RT growth gamma=sqrt(At*g*k)
  • Water-air At=0.997
  • Oil-water At=0.111
  • Salt-fresh At=0.012
  • Supernova SN1987A At~0.95-0.99
  • ICF DT capsule At~0.99
  • Atwood Cambridge 1746-1807
  • Lord Rayleigh 1883
  • G.I. Taylor 1950 Proc Roy Soc
  • Sharp 1984 Physica D 12:3
  • Lindl 1998 Springer ICF
  • USA NIF LLNL 2009: 192 lasers 1.8 MJ Q>1 2022
  • USA OMEGA Rochester 1995: 60 lasers 30 kJ
  • USA SNL Z Machine: 26 MA pulsed Z-pinch
  • USA Princeton PPPL 1951: 600 NSTX-U Spitzer
  • USA MIT 1976: 200 SPARC Plasma Center
  • USA General Atomics 1955: 5500 DIII-D
  • USA Commonwealth Fusion 2018: 800 SPARC ARC REBCO
  • USA TAE 2017: 200 Norman FRC
  • USA NASA Langley 1917: 3500 wind tunnels TPS
  • USA AEDC Tullahoma: Mach 14-16
  • USA CUBRC LENS-X: 7-30 km/s
  • USA SpaceX 2002: 13000 Starlink Raptor

Numerical examples

Example 1 - Water-air interface. At=0.997. Extreme. USA: shipping + climate research NOAA + WHOI.

Example 2 - Oil-water. At=0.111. Moderate. USA: ExxonMobil 62000 + Halliburton 40000 + Schlumberger oil-water separation.

Example 3 - Supernova. At=0.980. Extreme. USA: NASA JPL + Goddard + Chandra X-ray + NRAO + ALMA + DKIST.

Example 4 - ICF DT capsule. At=0.990. Extreme. USA: NIF LLNL Livermore 192 lasers 1.8 MJ 2022 ignition + OMEGA Rochester + SNL Z Machine.

Example 5 - Brackish-freshwater. At=0.012. Boussinesq. USA: WHOI Woods Hole + Scripps + USGS + NOAA estuary thermohaline.

Limits

  1. Two-fluid assumption multi-fluid stratified.
  2. Linear vs nonlinear bubble-spike.
  3. Surface tension Etvos competition.
  4. Viscosity Reynolds RT cutoff.
  5. Compressibility supersonic + MHD.
  6. Magnetic field MHD-RT mode.
  7. Stratification continuous Brunt-Vaisala.
  8. Time-dependent acceleration g(t).
  9. Mode coupling KH RM.
  10. Numerical CFD DNS LES WENO.

Common errors

At range: 0-1 strict.

Boussinesq vs full: At<0.1.

Linear vs nonlinear: small vs late time.

RT vs Richtmyer-Meshkov: gravity vs shock.

Surface tension viscosity: Etvos Reynolds.

Practical applications

  1. ICF inertial fusion: USA NIF LLNL 2009 192 lasers 1.8 MJ + 2022 ignition Q>1 + OMEGA Rochester 60 lasers + SNL Z Machine 26 MA + LANL Trinity supercomputer.
  2. Magnetic confinement fusion: PPPL Princeton 600 NSTX-U + MIT Plasma Center 200 SPARC + General Atomics 5500 DIII-D + Commonwealth Fusion Cambridge 800 + TAE 200 Norman + Helion 200 + Tokamak Energy UK + Lockheed Skunk Works CFR.
  3. Atmospheric entry capsule + spacecraft: NASA Langley 3500 + Ames 60 MW arc jet + AEDC Mach 14-16 + CUBRC LENS-X 7-30 km/s + SpaceX Starship + Blue Origin BE-4.
  4. Astrofisica supernova RT: NASA JPL 6000 + Goddard 10000 + Chandra X-ray + NRAO + ALMA + GBT + DKIST 4 m + NSO Boulder + HAO NCAR.
  5. Oil-water-gas density flow: ExxonMobil 62000 Baytown 590k + Halliburton 40000 + Schlumberger 99000 + Baker Hughes 55000 + Dow 36000 + DuPont 23000.
  6. Atmospheric ocean mixing climate: NASA Goddard 10000 + NOAA 12000 SWPC + Climate Prediction Center + WHOI 1930 1000 + Scripps La Jolla 1903 1000 + Lamont-Doherty Columbia.
  7. Volcanic magma chambers: USGS Reston 9000 + USGS Volcano Observatories + Hawaii + Yellowstone + Cascades + Alaska Volcano Observatory.
  8. Plasma fusion + Z-pinch: SNL Z Machine + LANL plasma + LLNL NIF + ORNL ITER + ANL APS + PNNL.
  9. Marine engineering ship hydrodynamics: General Dynamics Bath Iron Works + HII Newport News VA + Ingalls Pascagoula + ExxonMobil offshore Hibernia + Chevron Big Foot Gulf Mexico.
  10. University research RT instability: USA NIST Boulder + MIT 11000 + Stanford 17000 + Caltech 2300 GALCIT + Berkeley LBNL 45000 + Princeton 1746 + Wisconsin BSL + Cornell + Texas A&M + Michigan + Georgia Tech.

Updated July 2026

Frequently Asked Questions

What is the Atwood number?

At = (rho_h - rho_l) / (rho_h + rho_l) dimensionless Rayleigh-Taylor instability + density-driven flows ratio between heavy + light density 2 stacked fluids. Range 0-1. At=0 Boussinesq; At<0.1 small contrast atmosphere ocean; 0.1-0.5 standard RT; At>0.9 extreme ICF supernova vacuum-dense. George Atwood Cambridge UK 1746-1807 + G.I. Taylor 1950 Proc Roy Soc A 201:192. USA: NIF LLNL Livermore 2009 192 lasers 1.8 MJ + 2022 ignition Q>1 + OMEGA Rochester + SNL Z Machine 26 MA + Princeton PPPL 600 NSTX-U + MIT 200 SPARC.

How to calculate Atwood?

Procedure: (1) At = (rho_h - rho_l) / (rho_h + rho_l); (2) rho_h heavy phase kg/m^3; (3) rho_l light phase kg/m^3; (4) range 0-1. Examples: water-air At=0.997 extreme; oil-water At=0.111 moderate; saltwater-freshwater At=0.012 Boussinesq; ICF DT capsule At~0.99. USA: NIF LLNL Livermore + OMEGA Rochester + SNL Z Machine + General Atomics DIII-D + Commonwealth Fusion 800 SPARC.

What flow regimes does At classify?

At=0 Boussinesq equal density; At<0.01 small contrast atmosphere ocean Rayleigh-Benard; 0.01-0.1 low weakly nonlinear; 0.1-0.5 moderate standard RT instability; 0.5-0.9 high asymmetric supernova; At>0.9 extreme vacuum/dense ICF capsule + atmospheric entry. USA: research NASA JPL + Chandra X-ray + NIF LLNL 2009 ICF + Princeton PPPL + MIT Plasma Center + General Atomics + Caltech GALCIT.

What is Rayleigh-Taylor instability?

Gravitational instability 2 stacked fluids where heavy fluid (top) accelerates down through light fluid (below) generating bubble-spike pattern. Linear growth gamma = sqrt(At*g*k) wavenumber k surface waves. Lord Rayleigh 1883 + G.I. Taylor 1950 RT eponym. Examples: cherry on top of cream + ICF DT capsule + supernova SN1987A + atmospheric entry capsule. USA: research NIF LLNL ICF 192 lasers 1.8 MJ 2022 ignition + NASA + Princeton PPPL + Caltech GALCIT.

Who was George Atwood?

George Atwood (Westminster London UK 1746-1807) Trinity College Cambridge 1764-1784 fellow + Mathematics tutor English mathematician Cambridge fluid dynamics + mechanics; 1784 textbook 'A Treatise on the Rectilinear Motion and Rotation of Bodies' classic mechanics + Atwood machine 1784 famous experiment demonstration Newton's 2nd law F=ma gravity acceleration measurement; Atwood number eponym fluid mechanics density ratio later 1950s established R.D. Richtmyer Los Alamos + David Sharp 1968 paper Rocky Mountain Mathematics Journal. USA: research NIST + MIT + Princeton + Wisconsin BSL textbook.

What is ICF inertial confinement fusion?

Inertial Confinement Fusion ICF intense laser/X-ray compression DT capsule cryogenic deuterium-tritium 5 kg/m^3 + ablator polystyrene 1000 kg/m^3 At~0.99 + RT instability mitigation. Goal Q=net energy gain ignition. NIF National Ignition Facility LLNL Livermore CA 2009 + 192 lasers 1.8 MJ + 2022 first ignition Q>1; OMEGA University Rochester 1995 60 lasers 30 kJ; SNL Z Machine 26 MA pulsed Z-pinch fusion. USA: National Academy ICF research + LANL Trinity supercomputer.

Applications of At?

Applications: ICF inertial fusion USA NIF LLNL 2009 192 lasers 1.8 MJ + 2022 ignition Q>1 + OMEGA Rochester + SNL Z Machine 26 MA + LANL Trinity; magnetic confinement fusion Princeton PPPL 600 + MIT 200 SPARC + General Atomics 5500 DIII-D + Commonwealth Fusion Cambridge 800; atmospheric entry capsule NASA Langley + Ames 60 MW arc jet + AEDC Mach 14-16 + CUBRC LENS-X 7-30 km/s + SpaceX Starship + Blue Origin BE-4; astrofisica supernova RT NASA JPL + Goddard + Chandra X-ray + NRAO + ALMA + DKIST 4 m + HAO NCAR; oil-water flow ExxonMobil 62000 + Halliburton 40000 + Schlumberger 99000.

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