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The large-scale movement of the atmosphere, including the tropical Hadley cell and the midlatitude Ferrel cell.

Named after William Ferrel, the thermally indirect middle-latitude mean atmospheric overturning: near-surface flow is poleward and the upper return equatorward. It is maintained largely by eddy forcing and surface stress. Its Eulerian mean thermal transport can oppose the poleward eddy contribution; it is not a second thermally direct Hadley cell.

The low-latitude regions of Earth absorb more incoming solar radiation than they emit as outgoing thermal radiation, whereas high latitudes have a net annual radiation deficit. The obliquity of solar incidence and the distribution of daylight create a strong latitude dependence of absorbed solar energy; emitted thermal radiation varies less strongly because terrestrial temperature contrasts are smaller. The terrestrial atmosphere and ocean therefore carry energy from the tropical surplus toward the polar deficits. Otherwise the low latitudes would keep warming and the high latitudes cooling until a different local radiative equilibrium was reached.

For a steady, zonally averaged energy balance, let $R(\varphi )$ be net downward radiation at the top of the terrestrial atmosphere and $H(\varphi )$ the total northward energy transport across latitude $\varphi$. A narrow spherical band and conservation of energy giveThe poleward direction is northward in the Northern Hemisphere and southward in the Southern Hemisphere. Seasonal internal energy storage modifies the instantaneous balance, but not the annual need for poleward redistribution.

Several circulation structures contribute in addition to the transient eddies produced by baroclinic instability of the middle latitudes:

The Brewer-Dobson circulation additionally redistributes energy in the stratosphere, with tropical ascent and extratropical descent, although most total atmospheric heat transport occurs lower down. An Eulerian mean middle-latitude Ferrel cell is thermally indirect and can oppose the poleward transport by eddies; listing every meridional circulation as a direct poleward heat carrier would be incorrect.

Use eastward $x$, northward $y$, pressure coordinate $p$ increasing downward, and geopotential $\Phi =gz$. Horizontal derivatives are taken on an isobaric surface. The horizontal geostrophic balance and vertical hydrostatic balance arewhere $R$ is the specific gas constant of dry air and the final equality uses the ideal gas law. At fixed latitude $f$ is independent of $p$. Cross-differentiating givesConsequently the pressure-coordinate thermal wind equations areEquivalently, $f{\mathbf{e}}_z\times {\partial }_{\ln p}{\mathbf{u}}_g=R{\nabla }_{p}T$. Notice the sign: increasing eastward velocity upward means decreasing eastward velocity with $\ln p$, and hence poleward-decreasing temperature in the Northern Hemisphere.

For the stratospheric polar vortex estimate, the original diagram must be read on its logarithmic pressure axis. Near $60^{\circ }$ in the winter hemisphere, adjacent $10\mathrm{m}{\mathrm{s}}^{-1}$ wind contours suggest an increase of roughly $20\mathrm{m}{\mathrm{s}}^{-1}$ through a pressure decade surrounding $10\mathrm{mb}$: for example, about $10\mathrm{m}{\mathrm{s}}^{-1}$ near $30\mathrm{mb}$ and $30\mathrm{m}{\mathrm{s}}^{-1}$ near $3\mathrm{mb}$. These are coarse graphical readings, not a sounding. ThusWith $|f|=2\Omega \sin 60^{\circ }\simeq 1.26\times 10^{-4}{\mathrm{s}}^{-1}$ and $R\simeq 287\mathrm{J}\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{K}}^{-1}$, the poleward temperature gradient is approximatelyIf the vortex edge is taken to span roughly $50^{\circ }$ to $70^{\circ }$, its width is $a(20\pi /180)\simeq 2.2\times 10^6\mathrm{m}$, givingThe estimate depends on the width assigned to the edge and the graphical vertical shear; a range of several to roughly fifteen kelvin is more justified than a precise value. The sign remains a cold winter pole in either hemisphere. The diagram does not by itself prescribe a unique pair of edge latitudes.

At low latitudes, horizontal free-atmospheric temperature gradients are usually much smaller than at the winter vortex edge. For a comparable finite wind vertical shear, the rearranged thermal-wind balance, $T_y=(f/R)u_{\ln p}$, is consistent with weaker temperature gradients as $|f|$ decreases. However, geostrophic balance loses validity near the equator because the Coriolis parameter vanishes; division by $f$ cannot be used there to derive a prediction. The weak temperature gradient approximation is better motivated by rapid horizontal internal gravity wave adjustment, together with tropical convection and the Hadley cell. Thermal-wind balance describes the balanced extratropical relation and its tendency approaching the tropics; it is not a complete causal explanation of equatorial thermal uniformity.

The annual, zonal-mean absorbed sunlight falls from the tropics towards the poles much more strongly than outgoing thermal radiation. Thus Earth's top-of-atmosphere radiation budget has a tropical surplus and a polar deficit, with changes of sign around $35^{\circ }$ latitude in a basic sketch. The curves below are illustrative, rather than measured data; their area-weighted global means agree.

In a statistically stationary climate, local conservation of energy requires the surplus to be exported. If $R(\varphi )$ is the net downward radiative heat flux and $\mathcal{H}(\varphi )$ the northward total power carried across the latitude circle, thenHere $a$ is Earth's radius. The terrestrial atmosphere and ocean circulation carry this energy poleward by moving warm and cold fluid, including atmospheric transport of latent heat. Without this transport the assumed steady radiation budget could not persist: the tropics would warm and polar regions cool until the budget changed. The tropical atmospheric contribution includes the Hadley cell: rising warm air near the equator, poleward flow aloft, subtropical descent, and an equatorward lower return.

For a parcel moving without an axial torque, conservation of angular momentum applies in an inertial frame. The specific axial angular momentum at latitude $\varphi$ is“Initially at rest” means $u=0$ relative to the rotating surface at the equator, so $M=\Omega a^2$. Neglecting atmospheric height compared with $a$ givesWith $\Omega =7.292\times 10^{-5}{\mathrm{s}}^{-1}$ and $a=6.371\times 10^6\mathrm{m}$, the eastward velocity at $30^{\circ }\mathrm{N}$ isThis is the ideal angular-momentum-conserving upper branch of the Hadley cell. Real jet streams exchange angular momentum through eddies and friction, so this estimate is not an observational claim about the actual wind speed.

For a zonal background $U(y)$, its relative vorticity is $\overline{\xi }=-U_y$. The meridional absolute vorticity gradient is thereforeAt $30^{\circ }\mathrm{N}$,A localized jet stream of velocity scale $U=134\mathrm{m}{\mathrm{s}}^{-1}$ and width $L$ hasFor $L=1000$--$2000\mathrm{km}$ this is about $1.7$--$6.8$ times the planetary contribution, so wind curvature cannot generally be neglected. The coefficient in this scaling depends on the chosen jet profile and width convention.

Linearizing the vorticity equation now gives the Rossby-wave equation for a sheared zonal currentFor a qualitative estimate at the jet core, freeze the local coefficients. At a smooth maximum $U_{yy}<0$, so $Q_y>\beta$: the restoring absolute vorticity gradient is stronger. The local Rossby-wave dispersion relation in a zonal jet gives, for the same local $U$ and zonal wavenumber $k$ with $l=0$,soThe envelope in the diagram would advance eastward faster, while individual phase contours would advance more slowly eastward, become stationary if $Q_y=U{k}^2$, or move westward if $Q_y>U{k}^2$. The separation between envelope and phase speeds increases. On a plot with time downward, compare horizontal displacement at fixed time: the envelope moves farther right, while a given phase moves less far right, or left after reversal. If the frequency changes sign for a fixed signed $k$, conjugate the mode to restore the convention $\omega \ge 0$; the physical propagation is unchanged.

This conclusion concerns the concave-down jet core. On a flank with $U_{yy}>0$ the correction has the opposite sign; the value at the edge of the Hadley cell alone does not specify the curvature everywhere. Indeed the angular-momentum-conserving profile in 1(a) is a rising inner branch, not a complete smooth jet maximum. A global sheared-current normal mode requires its meridional structure and boundary conditions; the local calculation is the qualitative estimate requested.

Persistent low-latitude surface winds with a westward component, associated with the lower branch of the Hadley cell. Their weakening or reversal in the equatorial Pacific can excite equatorial Kelvin waves that contribute to the development of El Niño.