Come calcolare il numero di Rossby (Ro) rapporto adimensionale geophysical fluid dynamics atmosphere + ocean che misura il rapporto tra forze inerziali e forza di Coriolis (apparent due to rotation Earth + planet): Ro = u / (f * L). u velocità caratteristica m/s; L lunghezza caratteristica m (synoptic 1000 km, mesoscale 100 km, microscale 1-10 km); f = 2*Omega*sin(latitude) Coriolis parameter 1/s dove Omega = 7.2921e-5 rad/s Earth angular velocity (1 rotation/86164 s sidereal day) + latitude in degrees.

Coriolis parameter f: equatore lat=0 f=0; lat=45° f=1.03e-4 1/s; pole 90° f=1.46e-4 1/s.

Classificazione regime: Ro<0.01 strong rotation geostrophic balance hurricane + planetary scale + jet stream + Hadley cell + pressure gradient = Coriolis force balance + geostrophic wind formula u_g = -(1/(rho*f))*dp/dn; 0.01<Ro<0.1 rotation dominant synoptic large-scale weather + extratropical cyclone + anticyclone + frontal system; 0.1<Ro<1 mixed mesoscale fronts + sea breeze + lake effect snow + thunderstorm cluster; 1<Ro<10 inertia dominant small-scale convection + tornado + supercell + dust devil; Ro>10 pure inertia small-scale no Coriolis effects + bathtub vortex (urban legend) + tornado tornado cyclone direction + downburst microbursts.

Ro<<1 geostrophic balance wind blow along isobars not across (Northern Hemisphere clockwise high pressure + counter-clockwise low pressure) Buys-Ballot's law 1857.

Geostrophic wind: u_g = -(1/(rho*f)) * dp/dn pressure gradient force balance Coriolis perpendicular to wind. Storia: Carl-Gustaf Arvid Rossby (Stockholm Sweden 1898 - Stockholm 1957) Stockholm Hogskola PhD 1925 + Geophysical Institute University Bergen Norway 1919-1925 Vilhelm Bjerknes Bergen School + Stockholm 1925-1928 + USA emigrated 1926 MIT Massachusetts Institute Technology Cambridge MA 1928-1939 + University Chicago 1941-1950 + Stockholm Hogskola 1950-1957 returned + Tellus journal founder 1949; 1939 paper J Marine Research 2:38 'Relation between variations in the intensity of the zonal circulation of the atmosphere and the displacements of the semi-permanent centers of action' classic Rossby waves planetary atmospheric oscillations + 1940 paper Mon Weather Rev 68:122 Rossby phase speed; pioneer dynamic meteorology + Rossby number eponym + Rossby waves planetary + Rossby radius of deformation + Tellus journal Stockholm + 1937 paper Quart J Roy Meteorol Soc 63:339 jet stream + 1949 textbook 'On the dispersion of planetary waves'.

Gaspard-Gustave de Coriolis (Paris France 1792-1843) Ecole Polytechnique matematico fisico francese 1835 paper 'Sur les equations du mouvement relatif des systemes de corps' classic Coriolis force eponym + 1844 textbook 'Traite de la mecanique des corps solides' Coriolis force apparent inertial reference frame rotation.

Buys-Ballot Christophorus Buys-Ballot Netherlands 1857 law wind direction high-low pressure. Eredita: Holton 2004 'An introduction to dynamic meteorology' Academic Press classic textbook 4th edition + Pedlosky 1987 'Geophysical fluid dynamics' Springer classic + Vallis 2017 'Atmospheric and oceanic fluid dynamics' Cambridge classic.

Italia ricerca atmosphere + oceanography + meteorology: ARPA Agenzia Regionale Protezione Ambiente Lombardia Milano + Emilia-Romagna Bologna + Veneto Padova + Lazio Roma + Piemonte Torino + Toscana Firenze + Sicilia Palermo + Sardegna Cagliari + 21 regional total 6000 employees + air quality + meteorology + boundary layer; ISPRA Istituto Superiore Protezione Ricerca Ambientale Roma 1500; CNR-ISMAR Istituto Scienze Marine Venezia + Pozzuolo del Friuli + Bologna + Trieste 200 ocean research + Mediterranean overturning + Adriatic Tyrrhenian + Mediterranean Sea Rossby waves; CNR-IIA Inquinamento Atmosferico Monterotondo Roma + Sesto Fiorentino 250; CMCC Centro Mediterraneo Cambiamenti Climatici Lecce + Bologna + Capua 200 climate + ocean Mediterranean Rossby waves; INGV Istituto Nazionale Geofisica Vulcanologia Roma 1999 + L'Aquila + Catania + Pisa Sicilia + Bologna 1500; CNR-INM Istituto Marina Roma + Genova 200; Aeronautica Militare Pratica di Mare Roma + Brindisi + Decimomannu Sardinia meteorological service + Italian Meteorological Society SIMM Roma; ECMWF European Centre Medium-Range Weather Forecasts Reading UK + Italy contribution + ENS ensemble forecasts + IFS Integrated Forecasting System; Italmeteo + 3B Meteo + iLMeteo private weather services.

Italia università + dynamic meteorology + Rossby waves: Sapienza Roma DICEA Civile Edile Ambientale 4000 + DIMA + DIET + dynamic meteorology + atmospheric science; Politecnico Milano DEIB + DENG 7500 + scuola Carlo Cercignani Boltzmann legacy 1939-2010 kinetic theory; Politecnico Torino DENERG + DIATI 1859 5000; Federico II Napoli DICMaPI + DII + DIETI 5000; Bologna DICAM + DEI + DIN 4000 + Hydrology Section + Bologna ARPA Emilia-Romagna; Padova DII + DEI + ICEA 1222 3000 + CISAS; Pisa DICI + DESTeC 1343 2500; Politecnico Bari DEI 1990 2500; Trento DICAM + DISI 1962; Genova DICCA + DIBRIS 1481 2000 + IIT 1700; Ca Foscari Venezia 1868 + DAIS Dipartimento Scienze Ambientali Informatica Statistica + Venezia Lagoon mixing research stratification + IUAV Venezia. Italia industria atmospheric + climate research: CMCC Lecce 200 + climate research + ocean Mediterranean Rossby waves + climate change + IPCC contributions; Fondazione Cini Venezia + ISMAR Venezia ocean Adriatic; INAF Roma 1500 + Sardinia Radio Telescope + atmospheric science + Mauna Loa Hawaii partnership; INGV 1500 + atmospheric volcanic plume Mt Etna + Stromboli + Vesuvio; ASI Agenzia Spaziale Italiana Roma 1988 + ESA contributions Sentinel + Solar Orbiter; Cosmo-SkyMed second generation + Sentinel-1 ESA + atmospheric remote sensing. Italia industria offshore + density-driven flow large-scale: Saipem Milano 1957 32000 oil&gas EPC + offshore platforms;

Eni SpA Roma 1953 32000 + Eni R&D San Donato Milanese 1500 + raffinerie 5 sites + Versalis polimeri + Plenitude rinnovabili; Tecnimont Milano Maire Tecnimont 6000 EPC; Snam Milano 1941 6000 gas grid 30000 km gasdotti; Fincantieri Trieste 1959 22000 cantieri navali + cruise liners + military vessels U212; Thales Alenia Space 7800 + Cosmo-SkyMed + Sentinel-1 satellites atmospheric. Italia wind energy + ABL Coriolis effects: ANEV Associazione Nazionale Energia Vento Roma 100 + Enercon Italia + Vestas Italia + Siemens Gamesa Italia + Nordex Italia 600 wind farms 11.6 GW installed Italia 2024 + 80% on-shore + 20% off-shore + Lazio + Abruzzo + Campania + Sicilia + Puglia + Calabria + Sardegna + 800 wind farms; ENEL Green Power Roma 1300 wind + solar + hydroelectric. Italia industria aerospatial + Coriolis: Leonardo Roma 51000 + Eurofighter Typhoon + ATR Pomigliano + Avio Aero Rivalta Torino 5000 GE Aviation + Avio Colleferro 5000 Vega Ariane + Thales Alenia Space 7800 + Airbus Italia + ESA + INAF + Solar Orbiter ESA 2020 atmosphere research.

Formula numero Rossby

Definizione: Ro = u / (f * L). u velocità m/s; L caratteristica m; f Coriolis parameter 1/s.

Coriolis parameter f: f = 2 * Omega * sin(latitude) 1/s. Omega = 2*pi/86164 = 7.2921e-5 rad/s Earth sidereal day. Equator f=0; latitude 30° f=7.29e-5; latitude 45° f=1.03e-4; pole 90° f=1.46e-4 1/s. Mediterranean ~40° f=9.4e-5; Northern Italy ~45° f=1.03e-4; Sicily ~37° f=8.78e-5.

Geostrophic wind: u_g = -(1/(rho*f)) * dp/dn Buys-Ballot's law 1857. Wind blows along isobars with low pressure on left (Northern Hemisphere). Pressure gradient force balanced by Coriolis force perpendicular to wind. Geostrophic balance Ro<<0.1 valid synoptic + planetary.

Significato fisico: Ro<>1 inertia dominant + Coriolis negligible + small-scale tornado.

Numeri correlati: Ekman number Ek = nu/(f*L^2) viscous boundary layer rotating; Burger number Bu = (R_d/L)^2 stratification rotation; Richardson Ri = N^2/(du/dz)^2; Atwood At density contrast; Reynolds Re inertia/viscosity. Rossby radius of deformation R_d = sqrt(g*H)/f = N*H/f shallow water + stratified.

Storia e contesto

Carl-Gustaf Arvid Rossby (Stockholm Sweden 1898 - Stockholm 1957) Stockholm Hogskola PhD 1925 + Geophysical Institute University Bergen Norway 1919-1925 Vilhelm Bjerknes Bergen School polar front theory + Stockholm 1925-1928 + USA emigrated 1926 MIT Massachusetts Institute Technology Cambridge MA 1928-1939 + University Chicago 1941-1950 + Stockholm Hogskola 1950-1957 returned director International Meteorological Institute Stockholm + Tellus journal founder 1949 atmospheric oceanic dynamics journal; 1939 paper J Marine Research 2:38 'Relation between variations in the intensity of the zonal circulation of the atmosphere and the displacements of the semi-permanent centers of action' classic Rossby waves planetary atmospheric oscillations Rossby phase speed c = u - beta*L^2/(4*pi^2) westward planetary wave + 1940 Mon Weather Rev 68:122 + 1937 Quart J Roy Meteorol Soc 63:339 jet stream + 1949 textbook 'On the dispersion of planetary waves'; pioneer dynamic meteorology + Rossby number eponym + Rossby waves planetary + Rossby radius of deformation + jet stream prediction + Bergen School polar front theory legacy.

Gaspard-Gustave de Coriolis (Paris France 1792-1843) Ecole Polytechnique professor 1816-1838 + Conservatoire Arts Metiers Paris 1838-1843 director matematico fisico francese; 1835 paper 'Sur les equations du mouvement relatif des systemes de corps' classic Coriolis force eponym apparent inertial force in rotating reference frame F_C = -2*Omega x v + 1844 textbook 'Traite de la mecanique des corps solides' classic mechanics rotation.

Buys-Ballot Christophorus Buys-Ballot (Netherlands 1817-1890) Utrecht University 1857 paper Buys-Ballot's law wind direction high-low pressure Northern Hemisphere clockwise high counter-clockwise low + KNMI Royal Netherlands Meteorological Institute De Bilt founder 1854.

Vilhelm Bjerknes Norway 1862-1951 Bergen Geophysical Institute 1917 + Bergen School polar front theory 1919-1922 + dynamic meteorology Charney 1947 quasi-geostrophic. Eredita: Charney 1947 + Phillips 1956 + Rossby waves dispersion relation + Charney-Drazin 1961 stratospheric Rossby wave propagation; Holton 2004 'An introduction to dynamic meteorology' Academic Press 535 pages classic textbook 4th edition graduate; Pedlosky 1987 'Geophysical fluid dynamics' Springer 710 pages classic textbook GFD; Vallis 2017 'Atmospheric and oceanic fluid dynamics' Cambridge 924 pages classic textbook AOFD modern.

Italia ricerca atmospheric + ocean + dynamic meteorology: ARPA Agenzia Regionale Protezione Ambiente 21 regional 6000 + air quality + meteorology + boundary layer + Pasquill stability classes A-F + ARPA Lombardia + Emilia-Romagna + Veneto + Lazio + Piemonte + Toscana + Sicilia + Sardegna; ISPRA Roma 1500 + atmospheric dispersion + groundwater + boundary layer Pasquill classes; CNR-ISMAR Istituto Scienze Marine Venezia + Pozzuolo del Friuli + Bologna + Trieste 200 ocean research + thermohaline mixing + stratification + Mediterranean + Adriatic + Tyrrhenian + Ionian + estuary salinity gradient + Mediterranean Sea overturning circulation MOC + Rossby waves Mediterranean; CNR-IIA Inquinamento Atmosferico Monterotondo Roma + Sesto Fiorentino 250 atmospheric pollutants + dispersion; CMCC Centro Mediterraneo Cambiamenti Climatici Lecce + Bologna + Capua 200 climate + ocean Mediterranean Rossby waves + climate change + IPCC contributions Italian; INGV Istituto Nazionale Geofisica Vulcanologia Roma 1999 + L'Aquila + Catania + Pisa Sicilia + Bologna 1500 + atmospheric volcanic plume + Mt Etna + Stromboli + Vulcano + Vesuvio + Phlegraean Fields; CNR-INM Istituto Marina Roma + Genova 200 idrodinamica navale + density-driven flow + INSEAN bacino 470 m; INFN Roma 1951 5000 + Frascati LNF 800; Aeronautica Militare Pratica di Mare Roma + Brindisi + Decimomannu Sardinia + meteorological service + Italian Meteorological Society SIMM Roma;

ECMWF European Centre Medium-Range Weather Forecasts Reading UK + Italy contribution + ENS ensemble forecasts + IFS Integrated Forecasting System; Italmeteo + 3B Meteo + iLMeteo private weather; ENAV Roma + Direzione Generale Aviazione Civile DGAC + Aeronautica Militare aviation meteorology.

Italia università atmosfera + ocean + dynamic meteorology: Sapienza Roma DICEA Civile Edile Ambientale 4000 + DIMA + DIET + dynamic meteorology + atmospheric science; Politecnico Milano DEIB + DENG + DCMC + LCM 7500 + scuola Carlo Cercignani Boltzmann legacy 1939-2010 kinetic theory mathematical Politecnico Milano DAER; Politecnico Torino DENERG + DIATI 1859 5000 + DIATI Ambiente Territorio Infrastrutture; Federico II Napoli DICMaPI + DII + DIETI 1224 5000; Bologna DICAM + DEI + DIN 1088 4000 + Hydrology Section + Bologna ARPA Emilia-Romagna; Padova DII + DEI + ICEA 1222 3000 + CISAS Centro Studi Attività Spaziali; Pisa DICI + DESTeC + DII Bioingegneria 1343 2500; Politecnico Bari DIChim + DEI 1990 2500; Trento DICAM + DISI 1962; Genova DICCA + DIBRIS 1481 2000 + IIT Italian Institute Technology 1700; Ca Foscari Venezia 1868 + DAIS Dipartimento Scienze Ambientali Informatica Statistica + Venezia Lagoon mixing research stratification Rossby Mediterranean + IUAV Venezia; Salerno DIIN 1968; Firenze DIEF 1321; Cagliari DIMCM 1606. Italia hurricane Mediterranean medicane + tropical-like cyclones: ARPA Sicilia + Calabria + Puglia + Sardegna + Mediterranean medicane (Mediterranean Hurricane) Apollo September 2021 + Numa November 2017 + Trudy + Daniel September 2023 Greece Libia 4500 deaths + medicanes Rossby Ro~0.01-0.05 mesoscale tropical-like cyclone Mediterranean basin warm sea surface temperature SST + Coriolis low-mid latitude Mediterranean.

Italia oceanography Adriatic + Tyrrhenian Mediterranean overturning circulation MOC: CNR-ISMAR Venezia + Trieste + Bologna + Pozzuolo + Adriatic deep water formation + Levantine Intermediate Water LIW + Mediterranean overturning circulation EMOC eastern Mediterranean overturning circulation + Rossby waves propagation Mediterranean basin + Eastern Mediterranean Transient EMT 1990s climate shift; CMCC Lecce 200 + Mediterranean climate + ocean Rossby waves + GCM Global Climate Models contribution + IPCC AR6 chapter 8.

Tabella valori Rossby

• Ro = u / (f * L): definizione
• f = 2*Omega*sin(latitude): Coriolis parameter
• Omega = 7.2921e-5 rad/s: Earth angular
• Equator f=0: lat 0°
• lat 30°: f=7.29e-5
• lat 45°: f=1.03e-4
• Pole 90°: f=1.46e-4
• Mediterranean ~40°: f=9.4e-5
• Sicily ~37°: f=8.78e-5
• Northern Italy 45°: f=1.03e-4
• Ro<0.01 strong rotation: geostrophic hurricane planetary
• 0.01-0.1 rotation dominant: synoptic large-scale
• 0.1-1 mixed mesoscale: fronts sea breeze
• 1-10 inertia: small-scale tornado
• Ro>10 pure inertia: micro-scale
• Hurricane Ro ~0.01-0.05: tropical cyclone
• Mid-latitude cyclone Ro ~0.1
• Tornado Ro ~10-100: small-scale
• Geostrophic wind u_g = -(1/(rho*f))*dp/dn
• Buys-Ballot 1857: wind direction law
• Rossby Sweden USA 1898-1957: dynamic meteorology
• Rossby 1939 J Marine Res 2:38: Rossby waves
• Coriolis France 1792-1843: 1835 paper Coriolis force
• Bjerknes Norway 1862-1951: Bergen polar front
• Holton 2004 + Pedlosky 1987 + Vallis 2017: textbooks
• Italia ARPA 6000: 21 regional + Pasquill
• Italia ISPRA 1500: atmospheric dispersion
• Italia CMCC Lecce 200: climate Mediterranean Rossby
• Italia CNR-ISMAR 200: Mediterranean overturning EMOC
• Italia ECMWF Reading UK: ENS IFS
• Italia Aeronautica Militare: meteorology Pratica di Mare
• Italia ANEV 100: 11.6 GW wind

Esempi numerici dettagliati

Esempio 1 - Synoptic weather mid-latitude. u=10 m/s, L=1e6 m (1000 km synoptic), latitude=45° f=1.03e-4. Ro = 10/(1.03e-4 * 1e6) = 0.097. Rotation dominant synoptic + Northern Italy mid-latitude cyclone + Atlantic frontal systems. Italia: ARPA Lombardia + Emilia-Romagna + Veneto + Lazio + Aeronautica Militare + ECMWF Reading.

Esempio 2 - Hurricane tropical cyclone. u=50 m/s (winds), L=1e5 m (100 km radius), latitude=25° f=6.16e-5. Ro = 50/(6.16e-5*1e5) = 8.1. Inertia dominant + tropical cyclone winds + Coriolis effect curvature. Italia: medicane Mediterranean Apollo September 2021 + Numa November 2017 + Daniel September 2023 Greece Libia.

Esempio 3 - Tornado Italy. u=100 m/s + 360 km/h, L=100 m diameter funnel, latitude=40° f=9.37e-5. Ro = 100/(9.37e-5*100) = 10672. Pure inertia small-scale + Coriolis negligible. Italia: tornado Veneto + Emilia-Romagna + Lombardia + Trentino + 50 tornadoes/anno + Treviso 2007 EF3 4 deaths + Apulia 2014 EF4 trade fair Trani.

Esempio 4 - Ocean current Mediterranean. u=1 m/s Mediterranean Liguro-Provencal current, L=1e5 m 100 km mesoscale, latitude=30° f=7.29e-5. Ro = 1/(7.29e-5*1e5) = 0.137. Mixed mesoscale Mediterranean ocean current + thermohaline. Italia: CNR-ISMAR Venezia + CMCC Lecce 200 + Mediterranean overturning EMOC + Adriatic deep water formation.

Esempio 5 - Lab demo small-scale. u=0.01 m/s, L=0.1 m, latitude=45°. Ro = 0.01/(1.03e-4*0.1) = 970. Pure inertia Coriolis negligible lab scale. Italia: Politecnico Milano DEIB + DENG + Sapienza DICEA dimostrazione laboratory experiments rotating tank vs lab table.

Limiti e considerazioni

1. Beta plane vs f-plane: f-plane Ro = u/(f*L) constant Coriolis; beta plane f varia con latitude beta = df/dy = (2*Omega/R_Earth) * cos(lat). Rossby waves require beta plane.
2. Equatorial limit: f→0 latitude=0°. Ro→infinity equatorial regions Coriolis negligible. Equatorial waves Kelvin Rossby Yanai mixed.
3. Stratification: Burger number Bu = (R_d/L)^2 stratification rotation. Quasi-geostrophic theory + Rossby radius of deformation R_d = N*H/f.
4. Time-dependent: variable rotation Omega(t) Earth precession + nutation + length-of-day variation Coriolis varies.
5. Friction Ekman layer: viscous boundary layer Ekman number Ek = nu/(f*L^2) bottom friction modify geostrophic.
6. Topography: mountains + bathymetry + coast generate vorticity stretching + topographic Rossby waves.
7. Compressibility: high Mach + supersonic + atmospheric high altitude compressible.
8. Magnetic field: MHD coupled + magnetorotational instability.
9. Numerical CFD: WRF Weather Research Forecasting + ROMS Regional Ocean Modeling System + CESM Community Earth System Model + ECMWF IFS.
10. Mode coupling: Rossby + Kelvin + Yanai + gravity waves coupling + double diffusion.

Errori comuni

Latitude 0 equatore: f=0 Ro infinite. Equatorial regions special equatorial wave dynamics not Rossby.

Beta plane vs f-plane: Rossby waves require beta plane variable f. f-plane constant doesn't capture wave propagation.

Geostrophic balance Ro<<0.1: violation requires non-geostrophic effects ageostrophic + cyclostrophic + gradient wind balance.

Northern vs Southern Hemisphere: Coriolis sign opposite. Cyclone Northern counter-clockwise; Southern clockwise. Buys-Ballot's law sign changes.

Inertial frequency 2f Coriolis: inertial oscillation frequency f Hz; Coriolis parameter f 1/s. Confondere.

Applicazioni pratiche

1. Synoptic weather forecasting: Italia ARPA 21 regional 6000 + Aeronautica Militare Pratica di Mare Roma + ECMWF Reading UK + ENS ensemble forecasts + IFS Integrated Forecasting System + Italmeteo + 3B Meteo private.
2. Mediterranean medicane tropical-like cyclones: ARPA Sicilia + Puglia + Sardegna + Calabria + Mediterranean Apollo September 2021 + Numa November 2017 + Daniel September 2023 Greece Libia 4500 deaths.
3. Mediterranean overturning circulation MOC: CNR-ISMAR Venezia + CMCC Lecce 200 + Adriatic deep water formation + Levantine Intermediate Water LIW + Eastern Mediterranean Transient EMT.
4. Wind energy ABL Coriolis effects: ANEV 100 + Enercon Italia + Vestas + Siemens Gamesa + Nordex 600 + 11.6 GW Italia + ENEL Green Power 1300 + ERG 4000 + offshore Sardegna proposal.
5. Tornado Veneto + Emilia-Romagna + Lombardia: ARPA monitoring + Aeronautica Militare radar + 50 tornadoes/anno Italia + Treviso 2007 EF3 + Trani 2014 EF4 + INGV seismic.
6. Aviation jet stream: ENAV Roma + Aeronautica Militare + Direzione Generale Aviazione Civile DGAC + clear-air turbulence CAT + jet stream forecasting + Eurocontrol Brussels.
7. Climate change Mediterranean: CMCC Lecce 200 + Fondazione Cini Venezia + ASI Sentinel + ESA Solar Orbiter + IPCC contributions AR6 + COP climate negotiations.
8. Volcanic plume atmospheric: INGV Roma 1500 + L'Aquila + Catania Mt Etna + Stromboli + Vesuvio + Phlegraean Fields + atmospheric volcanic ash dispersion.
9. Marine engineering Mediterranean: Saipem 32000 EPC + Eni 5 raffinerie + Snam 30000 km gasdotti + Fincantieri 22000 cantieri navali + Mediterranean shipping routes Coriolis.
10. Universita ricerca atmospheric + ocean: Politecnico Milano DEIB + DENG + LCM 7500 + Sapienza DICEA + Bologna DICAM Hydrology + Padova ICEA CISAS + Pisa DICI + Ca Foscari Venezia DAIS + IIT Genova 1700.

Aggiornato a luglio 2026