Atmospheric Dynamics (Deep)

Atmospheric dynamics is the geophysical-fluid-dynamics subfield concerned with the large-scale motions of the atmosphere — the planetary-scale waves, the jet streams and storm tracks, the Hadley and Walker overturning, the monsoons, the tropical waves and cyclones, and the stratospheric circulations — and with how those motions transport heat, momentum, water vapour, and tracers. The discipline rests on the primitive equations of rotating stratified fluid, supplemented by the quasi-geostrophic, balanced, and shallow-water approximations that yielded the canonical theories of midlatitude weather. This note compiles the equations, the foundational theorems (Charney, Eady, Rossby, Hoskins-McIntyre-Robertson, Andrews-Holton-Leovy), the standard diagnostics (potential vorticity, Eliassen-Palm flux, wave-activity), the climatologies of jets, cells, monsoons, tropical waves, blocking, and tropical cyclones, and the climate-change signatures emerging across CMIP6.

See also

• physical-climate-system — atmosphere-ocean-ice general circulation foundations.
• atmospheric-chemistry-and-aerosols — chemistry and aerosols transported by the dynamics here.
• atmospheric-chemistry-and-radiative-transfer — radiative drivers of the circulation.
• extreme-event-attribution — heatwaves, atmospheric rivers, blocking attribution.
• hydrology-and-water-cycle — monsoons, atmospheric rivers, water-vapour transport.
• regional-climate-and-downscaling — RCM dynamics inherits these large-scale features.
• climate-sensitivity-and-feedbacks — dynamical feedbacks in jet shifts and Hadley expansion.
• paleoclimate-and-deep-time — paleo circulation reconstructions and proxy signals.

1. The governing equations

The atmosphere is a stratified, rotating, compressible fluid. Conservation of mass, momentum, energy, and a thermodynamic equation of state govern its evolution.

1.1 Primitive equations

In pressure coordinates and on a rotating sphere with latitude φ and longitude λ:

$\frac{Du}{Dt}-fv=-\frac{1}{\rho a\cos \phi }\frac{\partial p}{\partial \lambda }+F_u$

$\frac{Dv}{Dt}+fu=-\frac{1}{\rho a}\frac{\partial p}{\partial \phi }+F_v$

$\frac{\partial p}{\partial z}=-\rho g\text{(hydrostatic balance)}$

$\nabla \cdot \mathbf{v}+\frac{\partial \omega }{\partial p}=0\text{(continuity in pressure coordinates)}$

$\frac{D\theta }{Dt}=\frac{\theta }{T}\frac{Q}{c_p}$

with $\theta =T(p_0/p{)}^{R/c_p}$ the potential temperature, $f=2\Omega \sin \phi$ the Coriolis parameter, $\omega =Dp/Dt$ the vertical pressure velocity, and $F$ the friction. The hydrostatic primitive equations are the dynamical core of every GCM (CESM, E3SM, GFDL AM4, IPSL-CM6A, HadGEM3, MPI-ESM, MIROC, NorESM, ICON, EC-Earth, CMCC) although newer dynamical cores (FV3 in GFDL and E3SM, ICON-NWP in DWD-MPI, MPAS in NCAR) are non-hydrostatic to permit km-scale resolution.

1.2 Geostrophic and thermal-wind balance

For Rossby number $Ro=U/(fL)\ll 1$, the horizontal momentum equations collapse to geostrophic balance:

$f{u}_g=-\frac{1}{\rho }\frac{\partial p}{\partial y},f{v}_g=\frac{1}{\rho }\frac{\partial p}{\partial x}$

Combining with hydrostatic balance yields the thermal-wind equation in pressure coordinates:

$\frac{\partial u_g}{\partial \ln p}=\frac{R}{f}\frac{\partial T}{\partial y},\frac{\partial v_g}{\partial \ln p}=-\frac{R}{f}\frac{\partial T}{\partial x}$

Pole-to-equator temperature gradients in the troposphere drive westerlies that strengthen with altitude — the polar-front jet and subtropical jet. Reversal in the stratosphere (summer hemisphere cooler aloft) produces easterly summer-hemisphere stratospheric winds; winter hemisphere stratosphere retains strong westerlies.

1.3 Potential vorticity

Ertel 1942 PV:

$Q=\frac{(\zeta +f)\cdot \nabla \theta }{\rho }$

is conserved on isentropic surfaces under adiabatic, frictionless flow. The Hoskins-McIntyre-Robertson 1985 Quarterly Journal of the Royal Meteorological Society 111 review established PV thinking as the dominant theoretical framework for synoptic-scale dynamics: invertibility (given PV plus boundary conditions, the balanced wind and mass fields are recoverable), conservation under adiabatic flow, and ability to track stratosphere-troposphere exchange via tropopause folds (the 2-PVU surface as dynamical tropopause).

2. Rossby waves

Rossby 1939 Journal of Marine Research 2 derived the dispersion relation for waves of large-scale flow in a homogeneous rotating fluid on a beta plane (f = f0 + βy):

$\omega =Uk-\frac{\beta k}{k^2+l^2}$

with k and l zonal and meridional wavenumbers, U the mean zonal flow, β the planetary vorticity gradient. The eastward phase speed is U − β/(k^2 + l^2) — Rossby waves propagate westward relative to the mean flow. Group velocity is eastward for typical mid-latitude scales; wave activity propagates downstream.

2.1 Stationary Rossby waves

The atmosphere supports stationary Rossby waves with phase speed equal and opposite to the mean flow:

$U=\frac{\beta }{k^2+l^2}$

For typical mid-latitude U ≈ 20 m s⁻¹ and β ≈ 1.6 × 10⁻¹¹ m⁻¹ s⁻¹, the stationary wavenumber is k_s ~ wavenumber 4–6. Topographic forcing (Tibetan Plateau, Rockies) and thermal forcing (land-sea contrast, North Atlantic warm pool) excite stationary waves that organize the climatological jet meanders. Charney-Eliassen 1949 and subsequent work showed that the observed Northern Hemisphere stationary-wave amplitude is consistent with forced quasi-linear theory.

2.2 Sphere and beta-plane formulations

On the full sphere, the dispersion relation generalizes to spherical-harmonic eigenvalue problems. Held-Hou 1980 and Held 1983 Annual Review of Fluid Mechanics 15 elaborated. Rossby wave propagation follows ray-theory equations:

$\frac{dx}{dt}=c_{gx},\frac{dk}{dt}=-\frac{\partial \omega }{\partial x}$

Waves refract toward higher PV gradients and turn back at critical latitudes where U = c (the wave phase speed).

2.3 Vertical propagation: Charney-Drazin

Charney-Drazin 1961 Journal of Geophysical Research 66 demonstrated that planetary Rossby waves propagate upward into the stratosphere only when:

• The mean zonal wind is westerly (eastward), AND
• The mean wind is below a critical value depending on wavenumber.

In summer, easterly stratospheric winds block upward planetary-wave propagation; the stratosphere is quiescent. In winter, westerly winds allow vertical propagation; the planetary waves deposit easterly momentum upon dissipation, decelerating the polar vortex. Strong wave bursts can produce Sudden Stratospheric Warmings.

3. Baroclinic instability

3.1 Charney 1947 and Eady 1949

The midlatitude storm track is born of baroclinic instability — the conversion of mean available potential energy (stored in horizontal temperature gradients) to eddy kinetic energy. Charney 1947 Journal of Meteorology 4 and Eady 1949 Tellus 1 derived the instability conditions for the quasi-geostrophic equations.

The Eady model assumes a uniformly stratified Boussinesq fluid with constant vertical shear $U(z)=\Lambda z$ between rigid lids; growth rate:

$\sigma =0.31\frac{f\Lambda }{N}$

with N the Brunt-Väisälä frequency. For typical mid-latitude U_top − U_bottom ~ 40 m s⁻¹ and N ≈ 0.01 s⁻¹, σ ~ 1/(2 days), with most unstable wavelength ~3 900 km — matching observed cyclone spacing.

The Charney model removes the upper lid and adds the β effect, producing a more complex spectrum and a meridional propagation mode. Both models predict the observed scale, propagation speed, and growth rate of mid-latitude cyclones; the full nonlinear evolution requires numerical simulation.

3.2 Baroclinic life cycles

Simmons-Hoskins 1978 Journal of the Atmospheric Sciences 35 identified two canonical baroclinic life cycles in non-divergent barotropic simulations:

• LC1. Cyclonic wave breaking; equatorward wave propagation; mean jet shift poleward; sub-tropical jet stronger.
• LC2. Anticyclonic wave breaking; poleward wave propagation; mean jet shift equatorward; sub-tropical jet weaker.

Thorncroft-Hoskins-McIntyre 1993 Quarterly Journal of the Royal Meteorological Society 119 extended to the full lifecycle. Climate change appears to shift the global balance toward LC1 over the SH and toward LC2 in some NH sectors (Riviere 2009 Journal of the Atmospheric Sciences 66).

3.3 Storm-track theory

The storm-track maxima emerge from the interaction of baroclinic forcing, β-deflection, and surface friction. Hoskins-Valdes 1990 Journal of the Atmospheric Sciences 47 showed that the Atlantic and Pacific storm tracks are sustained against radiative damping by the convergence of moist static energy at the western boundary currents. Chang-Lee-Swanson 2002 Journal of Climate 15 reviewed storm-track climatologies.

4. Quasi-geostrophic theory

Charney 1948 Geophysical Public Papers Norway 17 derived the quasi-geostrophic (QG) equations as an asymptotic expansion in Rossby number. The QG equations are the workhorse of synoptic-scale theory.

4.1 The QG omega equation

Combining QG vorticity and thermodynamic equations yields the omega equation:

$\left({\nabla }^2+\frac{f^2}{\sigma }\frac{{\partial }^2}{\partial p^2}\right)\omega =\frac{f}{\sigma }\frac{\partial }{\partial p}[{\mathbf{v}}_g\cdot \nabla ({\zeta }_g+f)]+\frac{R}{\sigma p}{\nabla }^2[{\mathbf{v}}_g\cdot \nabla T]$

This diagnoses vertical motion ω from the differential vorticity advection and Laplacian of temperature advection — the classical synoptic ingredients of weather forecasting.

4.2 QG potential vorticity

$q={\nabla }^2\psi +f+\frac{\partial }{\partial p}\left(\frac{f^2}{\sigma }\frac{\partial \psi }{\partial p}\right)$

conserved in adiabatic flow. The Rossby-Charney-Stern theorem (Stern 1961) states that necessary conditions for baroclinic instability include the meridional QGPV gradient changing sign in the interior, or the QGPV gradient and surface temperature gradient having opposite signs at the boundary.

5. Wave-mean flow interaction

5.1 Eliassen-Palm flux

Eliassen-Palm 1961 Geofysiske Publikasjoner 22 derived the wave-momentum flux F that characterizes Rossby-wave forcing of the mean flow. In QG coordinates:

$\mathbf{F}=(F^{(\phi )},F^{(z)})=\left(-\overline{u^{\prime }{v}^{\prime }},\frac{f}{\sigma }\overline{v^{\prime }{\theta }^{\prime }}\right)$

Its divergence ∇·F is proportional to the rate of change of zonal-mean angular momentum by waves. The Andrews-Holton-Leovy 1987 Middle Atmosphere Dynamics textbook provides the canonical formulation in transformed Eulerian mean (TEM) coordinates.

5.2 The transformed Eulerian mean (TEM)

In TEM, the zonal-mean residual circulation $({\bar{v}}^{\ast },{\bar{w}}^{\ast })$ approximates the Lagrangian-mean motion. The TEM momentum equation:

$\frac{\partial \bar{u}}{\partial t}=f{\bar{v}}^{\ast }+\frac{1}{{\rho }_{0}a\cos \phi }\nabla \cdot \mathbf{F}+\bar{X}$

separates the wave-driving (∇·F) from the residual-circulation Coriolis term. In steady state, the residual circulation is driven by wave-momentum fluxes (“downward control,” Haynes-Marks-McIntyre-Shepherd-Shine 1991 Journal of the Atmospheric Sciences 48).

6. The stratosphere

6.1 The Brewer-Dobson circulation

Brewer 1949 Quarterly Journal of the Royal Meteorological Society 75 and Dobson 1956 Proceedings of the Royal Society A 236 inferred from water-vapour and ozone tracers a slow meridional overturning in the stratosphere: ascent in the tropics through the tropical tropopause layer (~14–18.5 km, Fueglistaler et al. 2009 Reviews of Geophysics 47), poleward transport in both hemispheres, and descent at high latitudes (especially the winter polar regions). The Brewer-Dobson circulation is driven primarily by Rossby-wave drag from the troposphere (the “downward control” mechanism, Haynes et al. 1991). CMIP6 models project a robust strengthening of the Brewer-Dobson circulation under warming (Butchart 2014 Reviews of Geophysics 52).

6.2 Quasi-Biennial Oscillation (QBO)

The equatorial stratospheric zonal wind oscillates between easterly and westerly phases with a period of 28 ± 2 months — first identified by Reed-Campbell-Rasmussen-Rogers 1961 Journal of Geophysical Research 66 and Veryard-Ebdon 1961 Meteorological Magazine 90. Mechanism: upward-propagating equatorial Kelvin and Rossby-gravity waves, with critical-layer absorption (Lindzen-Holton 1968; Holton-Lindzen 1972 Journal of the Atmospheric Sciences 29). The QBO modulates extratropical climate via the Holton-Tan mechanism (Holton-Tan 1980 Journal of the Atmospheric Sciences 37): easterly QBO phase weakens the NH polar vortex via altered Rossby-wave refraction. The QBO was unprecedently disrupted in 2015–2016 (Newman-Coy-Pawson-Lait 2016 Geophysical Research Letters 43; Osprey-Butchart-Knight-Scaife-Hamilton-Anstey-Schenzinger-Zhang 2016 Science 353), surprising the community.

6.3 Sudden Stratospheric Warmings (SSW)

Major SSW: zonal-mean zonal wind at 60°N 10 hPa reverses from westerly to easterly. Occur on average every other NH winter (Charlton-Polvani 2007 Journal of Climate 20). Triggered by amplified upward planetary-wave flux from the troposphere, often associated with tropical (MJO, ENSO) or stratospheric (QBO) preconditioning. Downward coupling: after major SSW, the AO/NAO shifts toward negative phase for 1–2 months (Baldwin-Dunkerton 2001 Science 294), increasing the probability of cold-air outbreaks over Eurasia and the eastern US. Notable recent SSWs: January 2009, January 2013, February 2018 (“Beast from the East”), January 2021, and the unusually weak February 2024 vortex disruption that contributed to the strong NH cold snap that month.

6.4 11-year solar cycle and the QBO modulation

Solar-cycle UV variations of ~6% in the 200-nm band drive ~2% variations in stratospheric ozone heating, modulating the polar vortex via the Kodera-Yamazaki 1992 mechanism. Solar-cycle composites (Camp-Tung 2007 Geophysical Research Letters 34) show ~1°C polar stratospheric temperature variations at solar max-min difference.

6.5 The Southern Hemisphere annular mode and ozone

Antarctic ozone depletion (1980s–1990s) drove cooling of the Antarctic stratosphere and a poleward shift of the SH tropospheric jet (Polvani-Waugh-Correa-Son 2011 Journal of Climate 24). Ozone recovery under the Montreal Protocol is now opposing the greenhouse-gas-forced jet shift, producing an intriguing 21st-century SAM future evolution (Eyring-Arblaster-Cionni-Sedlacek-Perlwitz-Young-Bekki-Bergmann-Cameron-Smith-Collins-Faluvegi-Gottschaldt-Horowitz-Kinnison-Lamarque-Marsh-Saint-Martin-Shindell-Sudo-Szopa-Watanabe 2013 Journal of Geophysical Research 118).

7. Tropospheric jets

7.1 Eddy-driven and subtropical jets

Two distinct jets in each hemisphere:

• Subtropical jet (STJ). Located ~30° latitude near the tropopause; angular-momentum-conserving boundary of the Hadley cell. Strongest in winter.
• Eddy-driven jet (EDJ) / polar-front jet. Located ~45–60° latitude; sustained by convergence of eddy momentum fluxes from baroclinic systems. Lower-tropospheric maximum vs the upper-tropospheric STJ. Distinguishable via PV thinking and surface winds.

In the SH the two often merge into a single jet; in the NH (Atlantic and Pacific) they are often distinct. Lee-Kim 2003 Journal of the Atmospheric Sciences 60 and Schneider 2006 Annual Review of Earth and Planetary Sciences 34 reviewed the dynamics.

7.2 Annular modes

The leading mode of variability of the extratropical zonal-mean zonal flow is the annular mode (Northern: NAM/AO; Southern: SAM). Defined as the first EOF of zonal-mean zonal wind anomalies, or geopotential at 500 hPa. North Atlantic Oscillation (NAO) is the Atlantic-sector projection; Pacific-North American (PNA) pattern is the Pacific-sector teleconnection. SAM is the Southern Annular Mode (Marshall 2003 Journal of Climate 16 station-based index). Climate change projections: persistent positive SAM (driven by GHG forcing partially offset by ozone recovery in 21st century); ambiguous AO/NAO with weak NH jet poleward shift in CMIP6.

7.3 Jet variability and stationary patterns

Beyond annular modes:

• NAO. Hurrell 1995 Science 269 station-based index; first EOF of N Atlantic SLP. Positive NAO: stronger westerlies, wet NW Europe, cool SE Canada; negative NAO: blocking, cold Europe.
• PNA. Wallace-Gutzler 1981 Monthly Weather Review 109 4-point teleconnection. ENSO-modulated.
• AO. Thompson-Wallace 1998 Geophysical Research Letters 25; NAM at 1000 hPa.
• SAM. Thompson-Wallace 2000 Journal of Climate 13.

7.4 Climate change response

CMIP6 multi-model mean projects:

• NH eddy-driven jet shift poleward under SSP5-8.5, particularly in summer; winter Atlantic shows weak shift due to opposing forcings (Arctic amplification weakens the gradient).
• SH eddy-driven jet shift poleward in summer; trend slows or reverses with ozone recovery.
• STJ intensification and poleward Hadley-cell expansion (Hu-Fu 2007 Atmospheric Chemistry and Physics 7; Lu-Vecchi-Reichler 2007 Geophysical Research Letters 34) of ~1° per °C global warming.
• Storm-track shifts. Atlantic storm track shifts equatorward and intensifies on its southern flank; Pacific storm track extends eastward (Yin 2005 Geophysical Research Letters 32; Chang-Yau 2016 Climate Dynamics 47).

8. The Hadley, Walker, and Ferrel cells

8.1 Hadley cell

The mean meridional overturning circulation extends from the ITCZ to ~30° in each hemisphere. Held-Hou 1980 Journal of the Atmospheric Sciences 37 derived the angular-momentum-conserving Hadley cell theory: the poleward edge sits where the angular-momentum-conserving zonal wind equals the geostrophic wind, giving the latitude scaling $\sin {\phi }_H\propto (gH{\Delta }_h{)}^{1/2}/(\Omega a)$ where Δ_h is the equator-to-pole potential-temperature difference and H scale height. Modern observations (Stachnik-Schumacher 2011 Journal of Geophysical Research 116) show Hadley cell width 25–35° per hemisphere.

Hadley-cell expansion under warming (~0.3–0.5° per decade, Hu-Fu 2007) shifts subtropical aridity poleward into the Mediterranean, southwest US, southern Africa, and Australia. Detection-attribution studies (Davis-Birner 2017 Geophysical Research Letters 44) reconcile across reanalyses.

8.2 Walker circulation

The Walker circulation is the equatorial zonal overturning across the Pacific (rising over the Maritime Continent, sinking over the eastern Pacific). Quantified by the Southern Oscillation Index (SOI) — pressure difference Tahiti minus Darwin. ENSO is the dominant interannual variability of the Walker cell: El Niño weakens it (anomalous rising over the central Pacific), La Niña strengthens it. CMIP6 multi-model projection of long-term Walker weakening (Vecchi-Soden-Wittenberg-Held-Leetmaa-Harrison 2006 Nature 441) is partly contradicted by observed Walker strengthening 1980–2014 (England-McGregor-Spence-Meehl-Timmermann-Cai-Sen Gupta-McPhaden-Purich-Santoso 2014 Nature Climate Change 4) — the “Pacific dipole bias” challenge.

8.3 Ferrel cell

The mid-latitude Ferrel cell is thermally indirect (rising in the cold, sinking in the warm — the opposite of a heat engine) and is dynamically driven by eddy momentum fluxes. The TEM residual circulation is small in midlatitudes precisely because eddies set the cell.

9. Monsoons

A monsoon is a large-scale seasonally reversing circulation driven by land-sea thermal contrast modulated by orography and ITCZ migration. Held-Hou theory extended to off-equatorial heating (Lindzen-Hou 1988 Journal of the Atmospheric Sciences 45) and Chou-Neelin (2003) “moist static energy” framework explains the monsoon onset latitude.

9.1 Asian summer monsoon

The largest monsoon system; encompasses the Indian summer monsoon (June-September) and East Asian summer monsoon (June-August). Indian monsoon rainfall ~80% of annual total. Mechanisms: differential heating between elevated Tibetan Plateau and Indian Ocean; cross-equatorial flow turning northward as the Somali Jet (Findlater 1969); orographic forcing on the Western Ghats and Himalayan foothills. Monsoon depressions and low-pressure systems drive most rainfall (Hunt-Fletcher 2019 Quarterly Journal of the Royal Meteorological Society 145).

9.2 West African monsoon

WAM brings ~80% of Sahel annual rainfall in June-September. Driven by ITCZ migration and West African jet structures: African Easterly Jet (~600 hPa, around 15°N) and Tropical Easterly Jet (~150 hPa, around 5–10°N). African Easterly Waves are Rossby-wave disturbances along the AEJ; serve as seeds for Atlantic tropical cyclogenesis.

9.3 North American monsoon

NAM brings summer rainfall to southwestern US and northwestern Mexico (July-September). Driven by the moisture flux from the Gulf of California and Gulf of Mexico under thermal-low forcing over the Sonoran Desert. NAM rainfall accounts for ~50% of annual rainfall in Arizona and New Mexico. CMIP6 projects NAM weakening under warming due to enhanced subtropical subsidence (Pascale-Boos-Bordoni-Delworth-Kapnick-Murakami-Vecchi-Zhang 2017 PNAS 114).

9.4 South American monsoon

SAMS is centered over the Amazon and South Atlantic Convergence Zone. Onset October-November, peak December-February. Driven by the South American Low-Level Jet east of the Andes.

9.5 Australian monsoon

Northern Australia: December-March; reverses to easterlies in the dry season.

9.6 ITCZ shifts

The annual-mean ITCZ sits ~5°N reflecting NH-SH energy-flux asymmetry (Frierson-Hwang-Fuckar-Seager-Kang-Donohoe-Maroon-Liu-Battisti 2013 Nature Geoscience 6). Anthropogenic forcing (NH aerosol cooling) shifted ITCZ southward through 20th century; declining aerosols may reverse this. Paleoclimate (8.2 ka, LIA, Heinrich events) record southward ITCZ shifts during NH cooling — consistent with energy-flux framework.

10. Tropical waves and the MJO

10.1 Equatorially trapped waves

Matsuno 1966 Journal of the Meteorological Society of Japan 44 derived the equatorially trapped wave solutions of the shallow-water equations on an equatorial β-plane. The wave families:

• Kelvin waves. Eastward-propagating, non-dispersive, eastward zonal wind anomalies coupled to temperature.
• Equatorial Rossby waves. Westward-propagating, dispersive.
• Mixed Rossby-gravity waves (Yanai waves). Westward (low k) or eastward (high k) propagation.
• Inertio-gravity waves. Eastward and westward propagating.

Wheeler-Kiladis 1999 Journal of the Atmospheric Sciences 56 — observational identification of Convectively Coupled Equatorial Waves (CCEWs) in the OLR (Outgoing Longwave Radiation) spectrum. The “Wheeler-Kiladis diagram” — wavenumber-frequency plot — is the canonical diagnostic.

10.2 Madden-Julian Oscillation (MJO)

Madden-Julian 1971 Journal of the Atmospheric Sciences 28; Madden-Julian 1972 Journal of the Atmospheric Sciences 29 identified a 40–50-day eastward-propagating intraseasonal oscillation in tropical deep convection. The MJO is the dominant intraseasonal mode of tropical variability:

• Eastward propagation at ~5 m s⁻¹ across the Maritime Continent and western Pacific.
• Lifecycle 30–60 days; Wheeler-Hendon 2004 Monthly Weather Review 132 RMM (Real-time Multivariate MJO) index based on OLR and zonal wind at 850 and 200 hPa. Phase 1–8 sectors trace MJO across the tropics.
• Modulates tropical-cyclone genesis, monsoon active-break cycles, atmospheric-river landfalls, extratropical-stratosphere coupling.
• Difficult to simulate in GCMs (Hung-Lin-Tsai-Wang-Yi-Cheng-Sui 2013 Journal of Climate 26 CMIP5 evaluation); CMIP6 modest improvement (Ahn-Kim-Ham-Stan 2020 Journal of Climate 33).

Theories of MJO: moisture mode (Adames-Kim 2016 Journal of the Atmospheric Sciences 73); skeleton model (Majda-Stechmann 2009 PNAS 106); WISHE-related (Emanuel 1987 Journal of the Atmospheric Sciences 44); multi-scale aggregation (Mapes 2000 JAS 57). No single accepted theory.

10.3 ENSO and tropical Pacific variability

El Niño / Southern Oscillation cycles between warm (El Niño) and cool (La Niña) Pacific SST states on 2–7 yr timescales. Bjerknes 1969 Monthly Weather Review 97 identified the positive feedback (warm east Pacific → weaker trades → less upwelling → warmer east Pacific). Modern theory: delayed-oscillator (Suarez-Schopf 1988 Journal of the Atmospheric Sciences 45), recharge-discharge (Jin 1997 Journal of the Atmospheric Sciences 54). Recent strong El Niños: 1982–83, 1997–98, 2015–16, 2023–24. Recent triple-dip La Niña 2020–22. ENSO diversity: Eastern Pacific (canonical, EP) vs Central Pacific (CP / “Modoki”) El Niños.

11. Tropical cyclones

11.1 Climatology

Annual global ~80–90 named tropical cyclones across seven basins: North Atlantic, Eastern Pacific, Western Pacific (most active), North Indian Ocean, South Indian Ocean, Australian, South Pacific. Atlantic 2020 season (30 named storms, record); Pacific 2024 (11 named-storm Atlantic, Hurricane Helene + Milton causing >$120 B losses combined per NOAA NCEI).

11.2 Genesis indices

Necessary conditions (Gray 1968 Monthly Weather Review 96): warm SST (>26.5°C in mixed layer ≥50 m), low vertical wind shear, mid-tropospheric moisture, low-level vorticity, Coriolis parameter |f| sufficient (not too close to equator). Composite indices:

• Genesis Potential Index (GPI, Emanuel-Nolan 2004 26th AMS Conference on Hurricanes). $\text{GPI}=|10^5\eta {|}^{3/2}(H/50{)}^3(V_{pot}/70{)}^3(1+0.1V_s{)}^{-2}$ combining low-level absolute vorticity, mid-level RH, potential intensity, and 850-to-200-hPa shear.
• Tropical Cyclone Potential Intensity (Emanuel 1986, 1988 Journal of the Atmospheric Sciences 43, 45). $V_{max}^2=(C_k/C_D)(T_s-T_o)/T_o\cdot (k_s^{\ast }-k)$ — Carnot-cycle theory.

11.3 Rapid intensification

Rapid intensification (RI): max wind speed increase ≥30 kt in 24 hr. Hurricane Wilma (October 2005), Hurricane Patricia (October 2015, fastest RI on record, 100 kt in 24 hr), Hurricane Otis (October 2023, intensified from tropical storm to Cat 5 in 12 hr, devastating Acapulco), Hurricane Milton (October 2024, intensified from Cat 1 to Cat 5 in 22 hr over Gulf of Mexico). Climate-change attribution: warmer SST and higher PI → higher conditional probability of RI (Kossin-Knapp-Olander-Velden 2020 PNAS 117; Bhatia-Vecchi-Knutson-Murakami-Kossin-Dixon-Whitlock 2019 Nature Communications 10).

11.4 Climate scaling

Knutson-McBride-Chan-Emanuel-Holland-Landsea-Held-Kossin-Srivastava-Sugi 2010 Nature Geoscience 3 and Knutson-Camargo-Chan-Emanuel-Ho-Kossin-Mohapatra-Satoh-Sugi-Walsh-Wu 2020 BAMS 101 update — expected ~5–10% increase in TC mean intensity per °C warming, ~10–15% increase in TC precipitation rate (Clausius-Clapeyron + dynamics), small change in TC frequency (with substantial uncertainty). Observed Atlantic intensity increases consistent with this scaling (Kossin-Olander-Knapp 2013 Journal of Climate 26; Emanuel 2020 PNAS 117).

11.5 Observation

Reconnaissance aircraft: NOAA Aircraft Operations Center (AOC) WP-3D Orion and Gulfstream IV; USAF 53rd WRS “Hurricane Hunters” with C-130J. NOAA HRD (Hurricane Research Division, AOML Miami) operates the field program. Satellite: GOES-16/18/19 with ABI imager; SSMI/S microwave; CYGNSS (Cyclone Global Navigation Satellite System) for ocean-surface winds. Recent surface-based: SAILDRONE Saildrone Explorer drones in hurricane Sam (2021), Fiona (2022), Idalia (2023). HAFS (Hurricane Analysis and Forecast System, NOAA, operational 2023) is the new operational hurricane model.

12. Atmospheric rivers

12.1 Definition and detection

Zhu-Newell 1998 Monthly Weather Review 126 identified Atmospheric Rivers (ARs) as long, narrow corridors of strong horizontal water-vapour transport (IVT integrated vapour transport >250 kg m⁻¹ s⁻¹). Typical AR: 2000+ km long, 400–500 km wide, contains ~20% of meridional water-vapour transport globally in any given snapshot. Detection algorithms: Guan-Waliser 2015 Journal of Geophysical Research 120 (tIVT-based object detection); Rutz-Steenburgh-Ralph 2014 Monthly Weather Review 142.

12.2 AR scale

Ralph-Rutz-Cordeira-Dettinger-Anderson-Reynolds-Schick-Smallcomb 2019 BAMS 100 introduced an AR scale (AR1 “weak” to AR5 “exceptional”) based on IVT magnitude and duration. ARkStorm scenario (USGS, 2010; updated ARkStorm 2.0, 2022) describes a 1-in-200-year megastorm hitting California with sustained AR3-AR5 conditions for weeks, producing trillion-dollar losses — a real risk underscored by the 2022–23 California winter that delivered 12 ARs in 3 weeks (Corringham-Cayan-Tardy-Pierce-Schick 2024 Science Advances).

12.3 West coast impacts

Pacific Northwest, California, and Southeast Alaska: ARs deliver >50% of annual precipitation in just ~10 events per year. CW3E (Center for Western Weather and Water Extremes, Scripps) operates the AR observatory and forecast experiments. The Pineapple Express is an iconic Hawaii-to-West-Coast AR.

12.4 Other regions

UK and Northern Europe: AR landfalls drive Storm Desmond December 2015 and Storm Eunice February 2022. Iberian Peninsula. Eastern Asia. South America (Pacific side). Antarctica: ARs deliver episodic surface mass balance to West Antarctica (Wille-Favier-Dufour-Gorodetskaya-Turner-Agosta-Codron 2019 Nature Geoscience 12).

13. Atmospheric blocking

13.1 Definition and typology

Blocking = quasi-stationary high-pressure system that disrupts the westerly flow. Classical typology (Rex 1950 Tellus 2):

• Omega block. High flanked by two cyclonic lobes; shape of Greek letter Ω.
• Dipole / Rex block. Anticyclone-cyclone dipole with the high to the north.

Indices: Tibaldi-Molteni 1990 Tellus A 42 1-D geopotential index (500 hPa). 2-D indices: Davini-Cagnazzo-Gualdi-Navarra 2012 Climate Dynamics 39; Schalge-Schiemann-Wernli 2011 Climate Dynamics 36 based on PV reversal.

13.2 Mechanisms

• Wave-breaking origin. Anticyclonic Rossby-wave breaking (LC2) deposits low-PV air in the high latitudes; cyclonic wave breaking deposits high-PV equatorward. Wave breaking is a key blocking mechanism (Pelly-Hoskins 2003 Journal of the Atmospheric Sciences 60).
• Eddy-mean flow feedback. Synoptic eddies reinforce blocking ridges through fluxes of low-PV air poleward.
• Latent-heating amplification. Pfahl-Schwierz-Croci-Maspoli-Grams-Wernli 2015 Nature Geoscience 8 — diabatic heating within warm conveyor belts upstream of blocking is essential for block establishment.

13.3 Impacts

Blocking drives multi-week extreme weather:

• Russia 2010 heatwave / wildfire. Blocking high stalled over European Russia ~6 weeks; ~55 000 excess deaths in Moscow region (Schär 2016 Nature Geoscience 9).
• Western Europe 2003 heatwave. ~70 000 excess deaths (Robine-Cheung-LeRoy-vanOyen-Griffiths-Michel-Herrmann 2008 Comptes Rendus Biologies 331).
• PNW 2021 heat dome. Lytton British Columbia 49.6°C (Philip-Kew-vanOldenborgh-Anslow-Seneviratne-Vautard-Coumou-Ebi-Arrighi-Singh-vanAalst-Pereira Marghidan-Wehner-Yang-Li-Schumacher-Hauser-Bonnet-Luu-Lehner-Gillett-Tradowsky-Vecchi-Rodell-Stull-Howard-Otto 2022 Earth System Dynamics 13).
• Western Europe summer 2022. Multiple heat domes; UK 40°C exceedance July 2022 (Christidis-Stott 2022 attribution).
• 2024 cold-snap January. SSW-triggered AO collapse in the US.

13.4 Climate change response

CMIP6 multi-model: weak overall change in blocking frequency with persistent under-simulation bias (Davini-D’Andrea 2020 Journal of Climate 33). Increased persistence and intensification of summer blocking via amplified Arctic warming reducing the meridional temperature gradient (Francis-Vavrus 2012 Geophysical Research Letters 39 — contested; Blackport-Screen 2020 Science Advances 6).

14. Climate change diagnostics

14.1 Pole-to-equator temperature gradient

Warming amplification in the Arctic (Arctic Amplification, AA = ratio of Arctic to global warming) ~3.8 in 1979–2021 (Rantanen et al. 2022 Communications Earth and Environment 3). AA reduces the lower-tropospheric meridional temperature gradient; aloft (upper troposphere) tropical warming exceeds polar warming, increasing the upper-tropospheric gradient. Net effect on jets and storm tracks remains an active debate (Cohen-Screen-Furtado-Barlow-Whittleston-Coumou-Francis-Dethloff-Entekhabi-Overland-Jones 2014 Nature Geoscience 7; Blackport-Screen 2020).

14.2 Stratospheric circulation changes

CMIP6 robust signals:

• Brewer-Dobson circulation strengthening. Tropical upward mass flux at 70 hPa increases ~2% per decade.
• Tropical tropopause cooling and rising. Tropopause altitude rising ~50–80 m per decade.
• Polar-vortex changes. Weakening NH and persistent SH strengthening (with ozone recovery moderating).

14.3 Dynamical signals in extremes

The “thermodynamic vs dynamic” partitioning of extreme-precipitation increases (Pfahl-O’Gorman-Fischer 2017 Nature Climate Change 7) finds that most CMIP6 global mean extreme-precipitation increase is thermodynamic (~7%/°C Clausius-Clapeyron, with dynamical contributions adding regional spread). Heatwave intensification is thermodynamic + soil-moisture feedback amplification (Seneviratne-Lüthi-Litschi-Schär 2006 Nature 443).

15. Open problems

• Pacific dipole bias. CMIP6 models simulate too cold east Pacific and too warm western Pacific, biasing Walker circulation, ENSO statistics, monsoon teleconnections.
• MJO simulation. Most CMIP6 models still under-simulate MJO amplitude and propagation.
• Storm-track sensitivity to Arctic amplification. Whether and how AA modulates mid-latitude weather extremes is contested (Cohen et al. 2014 vs Blackport-Screen 2020).
• Hadley-cell expansion attribution. Observed expansion exceeds CMIP6 multi-model mean; tropical SST patterns, ozone, and aerosols all contribute uncertainly.
• Tropical-cyclone genesis frequency. CMIP6 global mean projects neutral-to-slight decrease in named storms but increase in major hurricanes; observed Atlantic increase exceeds projections.
• Stratospheric water vapour trend. Tropical TTL cold-point temperature trend and its feedback on stratospheric H2O.
• Atmospheric blocking and persistence. No consensus on whether blocks will become more or less frequent and persistent under warming.
• QBO disruption mechanism and future. 2015–16 disruption surprised the community; recurrence under warming unclear.

Further reading

• Pierrehumbert, R. T. 2010. Principles of Planetary Climate.
• Holton, J. R. and G. Hakim 2013. An Introduction to Dynamic Meteorology (5th ed.).
• Wallace, J. M. and P. V. Hobbs 2006. Atmospheric Science: An Introductory Survey (2nd ed.).
• Trenberth, K. E. (ed.) 2009. Climate System Modeling.
• Held, I. M. 2005. “The Gap between Simulation and Understanding in Climate Modeling.” BAMS 86.
• Stocker, T. F. 2011. Introduction to Climate Modelling.
• Andrews, D. G., J. R. Holton, and C. B. Leovy 1987. Middle Atmosphere Dynamics.
• Vallis, G. K. 2017. Atmospheric and Oceanic Fluid Dynamics (2nd ed.).
• Schneider, T. 2006. “The General Circulation of the Atmosphere.” Annual Review of Earth and Planetary Sciences 34.
• Held, I. M. and B. J. Hoskins 1985. “Large-scale eddies and the general circulation of the troposphere.” Advances in Geophysics 28.
• Hoskins, B. J., M. E. McIntyre, and A. W. Robertson 1985. “On the use and significance of isentropic potential vorticity maps.” Quarterly Journal of the Royal Meteorological Society 111.
• Charlton, A. J. and L. M. Polvani 2007. “A new look at stratospheric sudden warmings.” Journal of Climate 20.
• Knutson, T. et al. 2020. “Tropical cyclones and climate change assessment.” BAMS 101.
• Ralph, F. M. et al. 2019. “A scale to characterize the strength and impacts of atmospheric rivers.” BAMS 100.
• IPCC AR6 WG1 Ch 8 (Douville et al. 2021) — “Water Cycle Changes.”
• IPCC AR6 WG1 Ch 11 (Seneviratne et al. 2021) — “Weather and Climate Extreme Events in a Changing Climate.”

Adjacent

• physical-climate-system
• atmospheric-chemistry-and-aerosols
• atmospheric-chemistry-and-radiative-transfer
• hydrology-and-water-cycle
• regional-climate-and-downscaling
• extreme-event-attribution
• climate-sensitivity-and-feedbacks