Unlike the troposphere, the stratosphere does not feel the thermal inertia of the oceans and without dynamics would be close to radiative equilibrium at all latitudes with a thermal structure and underlying annual cycle determined by a balance between heating due to absorption of incoming radiation (mostly solar UV) and cooling due to infrared emissions . Maximum radiative heating occurs at the summer pole, while the maximum cooling is located at the winter pole. At the equinoxes the maximum heating shifts to the Equator with cooling at both poles. Consequently, the winter pole is relatively cold, while the summer pole is relatively warm. This leads to a meridional temperature gradient and, by gradient wind balance, an eastward

Throughout this review, eastward and westward will be used when referring to wind direction rather than westerly and easterly, respectively, as traditionally chosen by practical meteorologists.

Concurrent with 's calculations, deduced from theory that the upward propagation of Rossby waves into the stratosphere can only occur through zonal mean winds that are eastward relative to the wave phase speed, and then only if those winds are not too strong. Even when these conditions are met, all but the largest-scale waves are inhibited from propagating. The planetary-scale waves that can propagate are mainly forced by surface topography and land–sea contrasts , with some additional forcing resulting from baroclinic processes . Based on the observed climatological zonal mean zonal winds (Fig. a and b) this then explains why the circulation in the summer stratosphere is relatively zonal symmetric and why the deviations from zonal symmetry are larger in the NH winter than the SH winter. In general, the zonal asymmetries are dominated by the first and second zonal wave harmonics as illustrated by the example daily maps of geopotential height at 10 hPa shown in Fig. . Also noticeable in the figure is the absence of zonal asymmetries in summer (Fig. b and c) and the weaker wave amplitudes in the SH compared to the NH (cf. Fig. a and d), which is due to the difference in the surface topography and the land–sea contrast between the hemispheres.

also showed that the steady, non-dissipated Rossby waves they considered had no effect on the mean flow. Similar theoretical relationships between mountain (gravity) waves and the mean flow were derived by and form the basis for a non-acceleration theorem, or Eliassen–Palm theorem, which states that, to second order in wave amplitude, waves that are steady and conservative (i.e. waves that are not time varying and not subjected to any forcing or dissipation) do not change the mean flow. Following further theoretical developments in the 1960s and 1970s e.g., introduced a generalized form of the Eliassen–Palm and Charney–Drazin theorems expressed in terms of the transformed Eulerian mean (TEM) equations, which are now widely used for analysing wave–mean-flow interactions (Sect. 8 in ). The generalization of is ubiquitous as it is relevant to both gravity and Rossby waves and is applicable in the tropics as well as the extratropics.

derived the TEM equations by defining a residual mean meridional circulation, 1v‾*,w‾*=v‾-ρ0-1∂∂zρ0v′θ′‾θ‾z,w‾+1acosϕ∂∂ϕcosϕv′θ′‾θ‾z, and transforming the zonal mean thermodynamic and zonal momentum equations in latitude, log-pressure coordinates (ϕ,z) to 2∂θ‾∂t+v‾*a∂θ‾∂ϕ+w‾*∂θ‾∂z=Q‾-ρ0-1∂∂zρ0v′θ′‾θ‾ϕaθ‾z+w′θ′‾ and 3∂u‾∂t+v‾*1acos⁡ϕ∂∂ϕu‾cos⁡ϕ-2Ωsin⁡ϕ+w‾*∂u‾∂z=∇⋅Fρ0acos⁡ϕ+X‾, respectively. Here an overbar refers to a zonal average and a prime the deviation therefrom or eddy. Zonal, meridional and vertical velocities are denoted by u, v and w, respectively; θ is potential temperature; Q‾ is the zonal mean diabatic (radiative) heating; ρ0=ρsexp⁡(-z/H) is the basic state density with H defined as scale height for log-pressure coordinates, generally taken to be ∼7 km, and ρs is a sea level reference density. The rotation rate and radius of the Earth are Ω and a, respectively. On the right-hand side of Eq. () F is the Eliassen–Palm flux vector defined as 4F(ϕ)=ρ0acos⁡ϕu‾zv′θ′‾θ‾z-u′v′‾ and 5F(z)=ρ0acos⁡ϕ2Ωsin⁡ϕ-u‾cos⁡ϕϕacos⁡ϕv′θ′‾θ‾z-u′w′‾, with 6∇⋅F=1acos⁡ϕ∂∂ϕF(ϕ)cos⁡ϕ+∂∂zF(z).

In the thermodynamic Eq. () the second term on the right-hand side can generally be neglected such that, for steady-state conditions, the equation is isomorphic to that used by to calculate the diabatic circulation. used this TEM equation to estimate the residual mean meridional circulation (v‾*,w‾*) for steady conditions, and, as might be expected, his results agree remarkably well with those of . From this deduced that, rather than the Eulerian mean meridional circulation (v‾,w‾), it is this residual mean circulation (v‾*,w‾*) incorporating dynamical contributions from the eddy heat and momentum fluxes that is responsible for driving the stratosphere away from radiative equilibrium. Full theoretical details of the actual dynamical mechanisms involved are presented in the classic paper on downward control of the diabatic circulation by and are not repeated here. One aspect that is worth noting is that the residual mean meridional circulation approximates Lagrangian transport and, on an hemispheric scale, forms the advective component of the Brewer–Dobson circulation, which also includes two-way mixing .

For the TEM equations the non-acceleration relations derived by reduce to the simple expression ∇⋅F=0; i.e. the forcing term on the right-hand side of the zonal momentum Eq. () is zero. F indicates the direction and magnitude of wave propagation in the meridional plane and, for small-amplitude waves, represents the flux of wave activity . The observed climatological winter mean ∇⋅F/(ρ0acos⁡ϕ) is negative throughout the stratosphere (Fig. ) and therefore has a westward acceleration (eastward deceleration) effect on the circumpolar vortex. This explains why the polar night jets (Fig. a and b) are weaker than would be expected from purely radiative considerations (Sect. ). The arrows in Fig.  indicate that Rossby waves are propagating upward from the troposphere, though on reaching the stratosphere they turn equatorward. Essentially this is because of the variation in latitude of the refractive index (due mainly to its dependence on the Coriolis parameter 2Ωsin⁡ϕ) in the wave equation , which determines the direction of propagation of Rossby waves in the (ϕ,z) plane . As expected, the wave fluxes are stronger in the northern winter than in the southern winter due to the relatively weak surface forcing in the SH.

Rossby waves in the stratosphere are often transient and almost certainly thermally damped , but the main cause of negative ∇⋅F and, therefore, an eastward deceleration (westward acceleration) of the mean flow is wave breaking, at least in the extratropics. Waves propagating vertically, which do not encounter a critical layer (i.e. where the phase speed approaches that of the mean flow and propagation is inhibited), attain large amplitudes due to the exponentially decreasing density with height, and the waves eventually break. Waves also break on encountering a critical layer either horizontally or vertically. Both gravity and Rossby waves experience amplitude growth with height. In the extratropical (and tropical) stratosphere, most gravity waves predominantly propagate upward

Vertically propagating gravity waves in the extratropics typically have wave lengths that are too small to be resolved by the reanalysis used for Fig.  or, indeed, nearly all current climate models, and, therefore, their effect on the mean flow is included in the term X‾ in Eq. () and not ∇⋅F.

Like potential temperature (θ), potential vorticity (PV) is a quasi-conservative quantity in the stratosphere. With θ as the vertical coordinate, 7PV=-gζθ+2Ωsin⁡ϕ∂θ∂p, where ζθ is the relative vorticity evaluated along isentropic surfaces (surfaces of constant θ) and g gravitational acceleration. In the stratosphere isentropic surfaces are quasi-horizontal. PV generally increases poleward due to the Coriolis parameter 2Ωsin⁡ϕ in Eq. (), and it is these gradients that support Rossby wave propagation. A key advance in our understanding came when presented coarse-grained maps of the isentropic distribution of PV derived from newly available satellite observations which showed that PV gradients in the middle stratosphere were not uniform at all latitudes. Instead they identified a “main vortex” with steep PV gradients at its edge surrounded by a “surf zone” exhibiting relatively weak large-scale gradients (Fig. ). Today the main vortex is known as the “polar vortex”. argued that the tongues of PV seen drawn out from the main vortex in their maps (cf. Fig. b) gave the first “reasonably convincing direct view of the breaking of planetary scale Rossby waves”, and went on to claim these were, in fact, the “world’s largest breaking waves”.

PV maps provide a useful diagnostic for separating the reversible effects of Rossby wave propagation, such as the distortion and displacement of the polar vortex, from the irreversible effects of the waves breaking. The latter is characterized by tongues of PV drawn from the polar vortex (e.g. Fig. ) and the eventual non-linear mixing of PV into the surf zone. A corollary of this is an erosion of the vortex with a sharpening of the PV gradients at its edge (Fig. ). Therefore, both reversible and irreversible processes can contribute to the variability of the polar vortex.

conjectured that wave breaking and the concomitant vortex erosion occur almost continuously throughout the winter. By simply measuring the size of the vortex in terms of the area enclosed within contours of constant PV on the isentropic maps (Fig. a), confirmed this for the winter of 1978/79, with the confirmation subsequently extended to all northern winters from 1964–1982 by . Figure a shows that, with wave breaking, the vortex weakens throughout winter compared to what would be expected if there were no waves and the stratosphere evolved radiatively (Fig. b). If the amplitudes of the waves become sufficiently large, the vortex is displaced off the pole or even split in two, and usually this corresponds with the occurrence of an SSW (Sect. ).

In terms of the interannual standard deviation of the monthly and zonal mean zonal winds, the maximum variability of the polar night jet occurs in high latitudes in the NH winter but in the SH winter is displaced toward mid-latitudes and does not extend down into the lower stratosphere (Fig. a and c, respectively). Instead the variability in the SH spring (Fig. d) is more similar to that of the NH winter, albeit slightly weaker, though stronger than the corresponding variability seen in the NH spring (Fig. b). This is what is expected for variability resulting from Rossby wave forcing from the troposphere. The mid-winter SH jet is generally strong enough (Fig. ) to limit Rossby wave propagation (Sect. ) and hence the variability, and the jet only becomes comparable in strength to the NH mid-winter jet as it weakens in spring (Fig. c). In contrast, by April in the NH the switch to the summer westward circulation has often already commenced (Fig. c), and, again, this inhibits Rossby wave propagation.

An alternative approach for analysing the interannual variability of the polar night jets is to use empirical orthogonal functions (EOFs; ). For reanalysis data from 1980 to 1999 the first EOF explains 87 % of the variance in the zonal mean zonal wind at 50 hPa in the NH, while at the same level in the SH the first two EOFs explain 59 % and 35 % of the variance, respectively . Variations in the jet strength are represented by the first mode of variability or EOF, while the second mode corresponds to a meridional shift of the jet. The two local maximums in the latitudinal structure of the interannual standard deviation for July (Fig. c) are then simply a manifestation of the larger contribution from the second EOF in the SH, as compared to the NH, which results from year-to-year fluctuations in the latitude of the jet maximum.

Because the strength of the polar night jet modulates Rossby wave propagation from the troposphere (Sect. , ) there is potential for the jet strength to vacillate. A weakening followed by reversal of the eastward jet due to wave–mean-flow interactions diminishes wave propagation from the troposphere. This then reduces the deceleration effect of the waves, and radiative effects restore the jet's strength. More wave propagation is then allowed, and wave–mean-flow interactions begin to weaken the jet again; thus a vacillation cycle is created. Stratospheric vacillations of this form were first identified in idealized numerical experiments and subsequently in a general circulation model (GCM) where they had a timescale of approximately 100 d . Establishing the role of vacillations in the observed variability is more challenging. found that for the 39 years of ERA-Interim a single fixed-amplitude sine wave with a 120 d period provided a reasonable fit in the middle stratosphere (10 hPa, 55–65∘ N) to the sub-seasonal zonal wind variations from the mean seasonal cycle over the period November through March if a best-fit phase and constant offset were calculated for each of the 39 years. An important outstanding question is how much of the variability represented by the single sine wave is purely internal (i.e. would occur if the wave flux from the troposphere remained constant) and how much is the result of variations in the upward wave flux itself, at the tropopause ?

In contrast, concluded that the interannual variability of Arctic lower-stratospheric temperatures during spring is almost completely determined by the variability of the (meridional) eddy heat flux near the tropopause slightly earlier in the year. For large-scale Rossby waves the heat flux (v′T′‾, where T is temperature) is approximately proportional to the vertical component of the EP flux (F(z), Eq. ) and, hence, the vertical flux of wave activity. Over the 22 years of 1979–2001 observed that mean temperatures poleward of 60∘ N at 50 hPa and averaged from 1 to 16 March were significantly correlated with the 45–75∘ N mean eddy heat flux at 100 hPa averaged from 15 January to 28 February. In particular, they found that a strong planetary wave eddy heat flux in the 100–400 hPa and 45–75∘ N region during January-February results in a warm March polar lower stratosphere, while a weak planetary wave eddy heat flux results in a cold March polar lower stratosphere. noted that their result is consistent with a theoretical framework based on linear wave propagation and also robust to the choice averaging periods and latitude ranges. A similar relationship exists between Antarctic lower-stratospheric temperatures in spring and the eddy heat flux emerging from the troposphere slightly earlier in the year, and for both hemispheres this relationship appears robust across a range of model simulations .

During the extended winter season, temperatures and zonal winds at high latitudes in the middle stratosphere can vary quite significantly from day to day in both hemispheres, as well as from year to year. For example, when the standard deviations of the daily variations in mean temperature poleward of 60∘ N and zonal mean zonal wind at 60∘ N are calculated at 10 hPa for October to March for each winter and then averaged for the 41 winters from 1979/80 to 2019/20, the values are ∼7 K and ∼13 m s-1, respectively, and comparable to the inter-annual spread in daily values shown by the light grey shading in Fig. a and b.

As noted above, the strongest inter-annual (and also daily) variability in the SH occurs roughly 2 months later in the annual cycle than in the NH. Therefore, the corresponding 6-month period in the SH is June to November, and for the 41 winters of 1980 to 2020 the averages of the standard deviations of the daily 10 hPa time series of the mean temperature poleward of 60∘ S and zonal mean zonal wind at 60∘ S are ∼4 K and ∼9 m s-1, respectively, which is slightly weaker than their NH counterparts. In both hemispheres daily variability is not uniform over the 6 months but includes periods when temperatures and winds evolve more steadily, similar to the summer months (Fig. ) when the dynamical forcing is weak or nonexistent. At other times significant and rapid rises in temperature and deceleration of eastward polar night jet (e.g. purple curves in Fig. ) are observed, especially in the NH. Corresponding rapid falls in temperature and/or a sudden acceleration of the jet are not observed as the upward-propagating Rossby waves can only decelerate the polar night jet (Sect. )

Even though there are no rapid falls in zonal mean temperatures, locally rapid temperature rises can be followed by rapid cooling (e.g. ) if, for instance, the polar vortex is displaced from and then back toward the pole.

In the decade following 's first observation of a mid-winter stratospheric warming (see Introduction), many more reports of anomalous warming events appeared in the literature (e.g. ), and by the early 1960s such events had acquired the name “stratospheric sudden warmings” , though today it is considered more appropriate to refer to these events as “sudden stratospheric warmings” . Initially SSWs were simply categorized as either mid-winter warmings or final warmings which are a manifestation of the year-to-year variability in the timing and vertical structure of the spring transition from an eastward to westward circulation in the middle- and high-latitude stratosphere . In the 1960s warmings were further separated into “minor” and “major” . “Minor” warmings occur when a significant temperature increase is observed (i.e. at least 25 K in a period of a week or less) at any stratospheric level in any area of the wintertime hemisphere but the criteria for a major warming are not met. A warming is “major” if at 10 hPa or below the latitudinal mean temperature increases poleward from 60∘ latitude and an associated circulation reversal is observed (i.e. mean eastward winds poleward of 60∘ latitude are succeeded by mean westward winds in the same area). These criteria for distinguishing major from minor warmings are still used today, and the occurrences of major warmings can be identified in Fig. b and d as the times when the zonal winds fall below the zero wind line in the figures. This occurs only once in the SH (in 2002); i.e. only one major warming has been observed in the SH since 1979 and indeed since records began

In mid-September 2019 a significant weakening of the SH polar vortex occurred, but as no actual wind reversal was observed poleward of 60∘, this event was not classified as a major SSW.

and are independently credited with the discovery of quasi-periodic reversals of the prevailing winds at the Equator from eastward to westward and back again roughly every 28 months in the lower and middle stratosphere. Corresponding oscillations were also identified in temperature and ozone, and soon after its discovery this irregular oscillation became known as the “quasi-biennial oscillation” or QBO . The original discovery was based on single station data near the Equator, but similar behaviour is seen for the zonal means. For the zonal mean zonal wind, the signal is centred at the Equator with a latitudinal half width of about 12∘ . The most iconic image of the QBO is that of a Hovmöller diagram showing alternate descending layers of eastward and westward winds such as in Fig. . In this figure repeated transitions in monthly averaged zonal mean zonal winds at the Equator appear to originate in the upper stratosphere and then propagate downward, at least until 2016. These repeating irregular cycles have, in fact, been continuously observed since the 1950s before being unexpectedly interrupted in 2016 . A second interruption was observed in 2019/20 in the zonal wind above Singapore (, Fig. 1a), though the occurrence of this interruption is not so apparent in the zonal mean zonal wind shown in Fig. .

In the years following its discovery, various dynamical mechanisms were proposed for the QBO, but it was difficult to explain key aspects such as (i) the QBO's downward propagation from 10 to 40 hPa without loss of amplitude and (ii) the super rotation of the atmosphere at the Equator during the eastward phase without vertical transport of momentum by equatorial waves. were the first to consider such processes. They argued that with a broad spectrum of gravity waves propagating up from the troposphere there would be (i) eastward acceleration of the mean flow in regions of eastward vertical shear from the waves with eastward phase speeds and (ii) westward acceleration in regions of westward vertical shear from the waves with westward phase speeds. Moreover, since selective filtering by the mean flow meant that only waves with westward phase speeds could propagate through strong eastward winds, and vice versa for waves with eastward phase speeds, the shear zones would descend over time. This proposed mechanism was corroborated with results from idealized numerical experiments .

Instead of gravity waves, considered vertical transport of momentum by thermally and mechanically damped upward-propagating planetary-scale Kelvin and Rossby-gravity waves and also obtained a realistic QBO in an idealized numerical model. However, subsequent studies suggest that the observed large-scale waves are unable to fully explain the observed QBO accelerations . Most likely a combination of equatorial Kelvin and Rossby-gravity waves and small-scale gravity waves cause the alternating eastward and westward accelerations, though for the shear zones to descend, the wave forcings (accelerations) have to exceed the effects of mean upward advection by the residual mean meridional or diabatic circulation (Sect. ), especially in the lower stratosphere .

The two-way interaction between the mean flow and vertical fluxes of wave momentum is now well established as the fundamental process leading to the QBO, but major challenges remain in determining the details of the waves involved. Small-scale gravity waves are, in particular, difficult to observe , and hence the precise partitioning of the various wave types in the forcing of the QBO is uncertain. Estimating the wave momentum flux from the available observations is also problematic as it relies on numerous assumptions . In addition, the relative importance of the different wave dissipation mechanisms (e.g. damping or critical layer wave breaking that occurs when the phase speed is close to the wind speed) is poorly understood .

Strictly, the QBO is not a wave but a sequence of alternating eastward and westward wind regimes propagating downward. Therefore, the cycles are not pure sinusoids and display variability from cycle to cycle . Many cycles show the descent of the region of westward vertical shear apparently stalling around 30 hPa followed by the eastward winds persisting longer in the lower stratosphere (Fig. ). Some of the variability most likely results from variations in the sources (mostly convection) of the tropical waves , which are not well understood. Another possibility suggested by was that laterally propagating Rossby waves from the NH winter could modulate the strength of the QBO during its eastward phase, though evidence from idealized simulations did not support this. Hence, prior to 2016, the consensus was the laterally propagating Rossby waves from the extratropics had only a small effect (e.g. ) on the QBO. Since 2016, horizontal wave momentum transport from the winter hemisphere has been implicated with initiating interruptions to the QBO cycling in 2016 and 2019, and this has prompted renewed interest in the role of the extratropics in modulating tropical stratospheric variability (see Sect. ). A tendency for the phases of the QBO and annual cycle to align is thought to arise from the annual cycle in the tropical upwelling in the equatorial lower stratosphere that opposes the QBO's downward progression .

Due to thermal wind balance, a corresponding QBO signal is also observed in the temperatures at the Equator, with warm and cool anomalies where there is eastward and westward vertical wind shear, respectively . Maintaining the balance requires a secondary residual meridional circulation with ascent at the Equator in the regions of westward vertical shear and descent in regions of eastward vertical shear (Fig. ). noted that vertical advection associated with this secondary circulation could account for some of the observed phase asymmetries by, for instance, retarding the downward propagation of the westward shear zone (cf. the stalling noted above) and speeding up the descent of the eastward shear zone. The secondary circulation also acts to concentrate the strongest eastward accelerations close to Equator , which again is consistent with the observed meridionally narrower eastward phase compared to westward phase .

Vertical advection by the secondary circulation along with the QBO temperature anomalies is the primary reason for a similar period oscillation in ozone, which was first seen in subtropical data by . At the Equator, transport (advection) is the dominant process in the lower stratosphere, and the ozone and wind QBO anomalies are generally in phase. Above about 15 hPa the effects of the temperature-dependent ozone chemistry becomes more relevant and the ozone anomalies change sign . Feedbacks from these ozone anomalies, in turn, modulate the dynamical variability. For example, in model simulations ozone-related diabatic feedbacks prolong the QBO period and strengthen the QBO signal in temperature and zonal wind in the lower and middle stratosphere. However, without a better understanding of the precise role of chemical processes in the ozone QBO, there is uncertainty in the strength of the feedbacks, and this remains an area of active research .

provided the first evidence of the QBO possibly influencing the high-latitude stratosphere when they reported a QBO signal in polar temperatures (and ozone) in both hemispheres. With more years of observations (1962–1977) found that the strength of the NH winter polar vortex correlated well with the phase of the equatorial QBO: a warmer, weaker vortex coincided with westward equatorial winds at 50 hPa, whereas when the winds were eastward the vortex was colder and stronger. A corollary of this so-called “Holton–Tan (HT) effect” first reported by is that the occurrence of an SSW is more likely when the QBO winds are westward at 50 hPa than when the winds are eastward . also noted a QBO signal in the SH zonal wind, this time in spring. An illustration of the HT effect in both hemispheres is presented in Fig. . Comparing this figure with the inter-annual standard deviation of the monthly mean zonal winds (Fig. ) highlights the importance of QBO teleconnections in modulating the variability of the polar vortex.

A full understanding of the underlying mechanisms for the QBO polar vortex teleconnection remains elusive, despite numerous investigations into the HT effect following its discovery . noted that the zero zonal wind line at the edge of the tropical QBO was typically further poleward during the westward phase than in the eastward phase and argued that this contributed to confining the extratropical planetary waves more toward high latitudes, thereby leading to the warmer, weaker vortex. This is often referred to as the “Holton–Tan mechanism”. In contrast, suggested that enhanced polar refraction of planetary waves results from the QBO's secondary meridional circulation modifying the Rossby wave refractive index in the subtropics. As the extratropical planetary wave activity is further modulated by the strength of the vortex (Sect. ), this has, so far, hampered attempts to isolate which of these two proposed mechanisms for the HT effect is more relevant . More recently, have suggested an alternative tropospheric pathway for the boreal winter in which a convection anomaly resulting from the QBO effect on the tropical troposphere (see Sect. ) generates a Rossby wave train into the mid-latitude troposphere, which, in turn, influences the upward propagation of planetary waves into the polar vortex.

Model studies suggest that, in general, the HT effect is sensitive to the state of the polar vortex , though in some models QBO teleconnections are also effected by circulation biases . A dependency on the vortex strength is one possible reason for the observed intra-seasonal variations and inter-hemisphere differences in extratropical QBO response: in the NH the HT effect is most robust in December and January , while in the SH the winter polar vortex is much stronger (Sect. ) and the response is more distinctive around the time of the final warming . Interactions found between the QBO stratospheric teleconnections and the solar cycle and El Niño–Southern Oscillation (ENSO; ) make the causes of these differences uncertain as both the QBO and polar vortex variability are, separately, impacted by these longer timescale modes of variability. The QBO altitudes which exert the strongest influence on the extratropics is another uncertain aspect .

In February 2016 a shallow layer of westward winds began to form at the Equator within the decaying eastward phase of the QBO . At the time this was a complete surprise, and over the next few months the usual oscillation in the equatorial winds appeared to cease (Fig. ), raising concerns among leading researchers of the QBO entering a “death spiral” and disappearing altogether. We now know that this did not happen, and by late 2016 the regular cycling of the equatorial winds had recommenced (Fig. ), albeit with the phase progression shifted from what would be expected if the interruption had not occurred .

Compared to previously observed QBO cycles, the anomalous behaviour in 2016 was unprecedented . It could not be accounted for by the established canonical model of the QBO, which relies only on the vertical transport of momentum (Sect. ). Instead, and attributed the anomalous westward acceleration near 40 hPa to enhanced horizontal EP flux, F(ϕ) (Eq. ), entering the tropics from the NH. Arguably this was the first observational evidence of the extratropical stratosphere directly influencing the variability of the tropical stratosphere, as opposed to indirectly though the meridional circulation. Further evidence for the direct influence was obtained when the QBO was surprisingly interrupted again in December 2019, though this time the horizontally propagating Rossby waves came from the SH .

A dynamical explanation for the interruptions is still in its infancy. While several studies have examined the role of different wave types and found that tropical mixed Rossby-gravity waves have an important role in the anomalous westward accelerations , others have stressed the importance of feedback mechanisms. Nonetheless, all conclude that during the disrupted cycles there was a significant extratropical influence on the variability in the tropical stratosphere. One consequence of this direct influence by the extratropical waves is that the paradigm in which the tropical zonal winds retain memory from year to year (the low-latitude “fly-wheel” effect; ), and thereby provide a mechanism for inter-annual variability, is possibly less appropriate than previously thought and may need to be reconsidered once a full understanding of the QBO interruptions is obtained.

First attempts to investigate stratospheric dynamics in a general circulation model (GCM) were by , though the model they used was only hemispheric with a simulation of less than 1 year. In general, early GCMs typically neglected the stratosphere based simply on its small fraction (∼17 %) of the atmospheric mass and need to optimize use of available computing resources. When the stratosphere did feature in GCMs (e.g. ), long simulations of the variability were not feasible due, again, to computational constraints. Computers used for modelling the atmosphere evolved into supercomputers around 1980

In the 1960s the computing community was already using the term “supercomputer”, but it only came into common usage in atmospheric science with the introduction of vector machines capable of around 100 MFLOPS.

Predicting stratospheric variations can be done either empirically using statistical methods or, as considered here, dynamically using numerical models. pioneered attempts to develop numerical predictions of the stratosphere, but their 14 d GCM simulation for March 1965 was not successful at capturing the sudden warming that occurred. The February 1979 SSW was the first to be successfully simulated and also the first observed from space using newly introduced operational temperature soundings of the stratosphere . The successful simulation of the 1979 SSW by was initialized 5 d prior to the peak of the warming but relied on prescribed lower boundary conditions at the tropopause. In subsequent numerical forecasts and found this SSW was predictable up to 10 and 5 d ahead, respectively.

With more SSWs and concomitant case studies the current consensus is that SSWs can, in general, be deterministically predicted 1 to 2 weeks in advance , with higher skill more likely from models with enhanced representation of the stratosphere . For a single model predictive skill varies between SSWs due to differences in the phenomenology and generation mechanisms for vortex-splitting and vortex-displacing events (Sect. ). These differences contribute to determining if the predictability arises mostly from the stratosphere or from the troposphere . Predictability arising from the troposphere is sensitive to model biases affecting planetary waves and depends on their propagation timescales , whereas predictability arising from the stratosphere is more likely to depend on the background stratospheric state prior to an SSW . In general, vortex splitting appears more difficult to forecast than a vortex displacement .

Compared to the troposphere the extratropical stratosphere exhibits extended predictability , though for extreme vortex events such as an SSW or, alternatively a polar vortex intensification, the predictability is still limited by initial condition uncertainty and model errors . Among forecast systems there is, nonetheless, an affinity between higher skill in the stratosphere and higher skill in the troposphere , and most systems consistently predict polar vortex intensification and final warmings better than SSWs . For the NH winter/spring stratosphere deterministic forecast skill is generally limited to a range of 1 to 2 weeks . In the SH perturbations to the polar vortex in winter persist into spring due to the reduced intraseasonal variability (Fig. ) resulting from the weaker planetary wave forcing . Hence, the SH stratosphere is potentially predictable on longer timescales, as is evident in the significant skill found by in the prediction of the strength of the Antarctic stratospheric polar vortex in spring at 1-month-average lead times (Fig. ).

During the boreal winter the extratropical stratosphere has also been found to be predictable in a probabilistic sense on seasonal timescales well beyond the deterministic range of forecast skill . noted that in seasonal forecasts capable of reproducing the overall climatological frequencies of SSWs and strong polar vortex (SPV) events well, the proportion of ensemble members predicting the occurrence of an SSW or SPV event varied from year to year (Fig. ). For winters in which an SSW (SPV) was “observed” (based on proxy observations derived from the forecast ensemble – see , for details), the proportion of ensemble members forecasting a SSW (SPV) increased, on average, by 12 %, indicating potential probabilistic forecast skill on a seasonal timescale with significance beyond the 95 % level . Most likely contributors to this probabilistic skill are the ENSO and the QBO, though, on average, the latest state-of-the-art seasonal forecast systems overestimate the anomalous wave forcing of the polar vortex due to ENSO while underestimating the modulation of the strength of the vortex by the QBO, at least for a hindcast period when the observed interannual variability of the polar vortex was significantly affected by the QBO .

Prediction skill of the QBO itself is very high in the lower to middle stratosphere on sub-seasonal to seasonal timescales . Out to 1 year, high skill is still obtained for predictions of the phase progression , though initialized forecasts are generally less successful at maintaining adequate QBO amplitudes . This is largely attributed to weaknesses in all models in predicting the development of a sufficiently strong westward phase which, in turn, can be linked to inadequacies in overall representation of the QBO in the models . Beyond 12 months, decadal forecast systems have been found to exhibit predictive skill up to 4 years for the QBO phase , with potential for extending this further by improving the representation and prediction of ozone variability in the models . Enhanced ENSO prediction skill also has potential for extending lead times for skillful QBO predictions as strong ENSO events have been found to lock the QBO phase across ensemble members in uninitialized simulations .

Exploiting the skillful multi-year QBO predictions is currently frustrated by deficiencies in the amplitudes and vertical structure of the predicted QBOs . Multi-model results show that similar deficiencies are already present on the seasonal-to-annual timescale and almost certainly contribute to a lack of skill in predicting the QBO's extratropical winter teleconnections beyond the first month . The two recent interruptions to the QBO (Sect. ) add further uncertainty to the robustness of the QBO as a source of predictability as the predictable signal was lost during each interruption before re-emerging a few months later but with a significant shift in phase from what would have been expected . Neither interruption was predicted by operational seasonal forecasts , though for the 2016 event found skill at lead times of around 1 month. Predictability on this timescale is consistent with the events initiated from the extratropics (Sect. ) but much reduced from the timescales associated with the QBO's uninterrupted quasi-regular cycling.

When first discovered SSWs he also presented tentative evidence that their effects reach the surface. Similarly, over a decade after co-discovering the QBO found evidence suggesting that it features in determining the character of the NH tropospheric circulation. The earliest model results linking tropospheric circulation changes to variations in the stratospheric polar vortex are those published by , with concurrent theoretical advances further supporting the possibility of a downward influence. Nonetheless, up until the end of the twentieth century, the prevalent view was that the dynamical coupling between the troposphere and stratosphere was mostly one-way with the stratosphere responding to forcing from below.

The paradigm shift to a two-way coupling arose from analysis of the Northern Annular Mode and Southern Annular Mode (NAM and SAM, respectively; ). These are the leading patterns of the large-scale variability of the extratropical circulation and dominate variability on intraseasonal to interdecadal timescales. A manifestation of the annular modes in the troposphere is a fluctuation in the latitudinal position of the jet stream, while in the stratosphere the characteristic feature is a fluctuation in the strength of the polar vortex . Annular modes in the stratosphere and troposphere appear to be coupled with occurrences of a weaker polar vortex correlated with equatorward shifts of the jet stream and vice versa for occurrences of a stronger polar vortex . The discovery that NAM anomalies progress downward from the stratosphere all the way to the Earth's surface is most often credited to , though other studies had previously noted stratospheric circulation anomalies extending down into the upper troposphere . Downward progression of SAM anomalies from the stratosphere to the surface was first reported by in association with SH climate change.

Persistence times for annular mode anomalies are longest in the stratosphere, but in months when the polar vortex is most active (January–February in the NH and November–December in the SH – see Sect. ) and strongly coupled to the troposphere, timescales in the troposphere lengthen . In these months anomalous values of the annular modes in the stratosphere are often followed by anomalies of the same sign in the troposphere, which sometimes persist for up to 2 months in the NH (e.g. Fig. ) and even longer in the SH . A slight delay before the occurrence of the concomitant tropospheric anomalies (Fig. ) suggests the possibility that the anomalies are propagating downward , and, indeed, model studies have demonstrated that there is a genuine physical downward influence with the stratosphere affecting behaviour in the troposphere and at the surface as reviewed by .

On the other hand, annular modes derived from EOFs are not necessarily physical modes of variability , and formulating a dynamical theory for the downward propagation of anomalies is still a work in progress. Given the comparatively light weight of the stratosphere, its downward influence almost certainly results from a sensitivity of tropospheric dynamics to changes in the stratosphere with eddy feedbacks fundamental for enhancing the tropospheric response and contributing to its persistence. An alternative perspective for the stratosphere's downward effect on the Atlantic jet is the strong response results from a regime shift between three preferred jet positions rather than a simple latitudinal movement in jet position . Proposed mechanisms for communicating the dynamical effects from the stratosphere to the troposphere include non-local PV inversion , the downward effect of an induced secondary meridional circulation , and the effects resulting from upward-propagating waves either being reflected back downward or changing the refractive properties of the lower stratosphere through wave–mean-flow interactions . However, none of these proposed dynamical mechanism can account for all aspects of the stratosphere's downward influence.

One complicating factor is that the anomalous stratospheric events are quite diverse and do not always extend downward into the troposphere or, indeed, to the surface . In the NH the diversity is largely associated with the different types of SSWs (Sect. ), with a surface impact only observed for about two-thirds of the events . Characteristics which determine whether an SSW has a downward influence remain elusive, and it is still not possible to foresee at the onset of an SSW if it is likely to be followed by a discernible surface impact . When there is a downward influence the surface impacts of split events are seen, on average, about 1 week sooner than the impacts of displacement events, at least for reanalysis data . On the other hand, model results from indicate no significant dependency on SSW type and show that there is probably insufficient evidence to establish a difference between impacts by split and displacement events.

Following from studies of descending annular mode anomalies, it is now apparent that there is an affinity between occurrences of extreme surface weather and the strength of the polar vortex , and also the phase of the QBO due to its teleconnections to the polar vortex (Sect. ). Extreme weather events involving a significant contribution from the stratosphere have been catalogued by , and they found them to be wide ranging with the stratospheric connection more firmly established for some than others. The connection in the NH is most prominent and robust in the Atlantic basin , with a weakening of the polar vortex, or SSW, often associated with cold weather over Europe and the eastern USA and cold air outbreaks over the northeastern Atlantic . In contrast, an unusually strong Arctic polar vortex, as often occurs during the QBO eastward phase (Sect. ), is associated with stronger surface winds and more intense cyclones with increased the risk of storms and extreme rainfall over northwest Europe .

A weakening of the SH polar vortex and negative SAM index in the lower stratosphere generally leads to hot dry extremes over Australia , cooler wetter conditions over southern South America and higher Antarctic surface temperatures, except for the Palmer Peninsula, which is more likely to experience cold dry conditions . A positive SAM index often brings significant cooling to Antarctica and much of Australia, warming to the Palmer Peninsula, and warm dry conditions to southern South America, New Zealand and Tasmania, while Australia and South Africa are more likely to experience anomalously wet conditions .

The influence of the stratosphere on the tropical troposphere was independently reviewed recently by , and, hence, only a selection of the more significant dynamical developments are covered here. First indications of a possible coupling between the stratosphere and tropical troposphere came when noted a correlation between the frequency of Atlantic hurricanes and the phase of the QBO, though somewhat earlier had tentatively suggested that the QBO had a role in determining the frequency of Pacific typhoons. Later studies , however, questioned the robustness of such relationships, and a convincing dynamical explanation is still wanting .

Tropical dynamics differ from extratropical dynamics due to the smallness of the Coriolis parameter, which implies that, unlike in the extratropics (Sect. ), the vertical scales inferred from balance dynamics are generally too shallow to potentially explain a coupling between the stratosphere and troposphere. Instead, proposed mechanisms for a tropical pathway between the stratosphere and troposphere are commonly based on the paradigm that variability at the tropopause and in the lower stratosphere modulates behaviour throughout the depth of the tropical troposphere . While the variability in the tropical stratosphere is dominated by the QBO (Sect. ), 3 km, or so, above the tropopause the descending alternating eastward and westward wind regimes begin to weaken significantly (Fig. ) and do not penetrate into the troposphere. Driven by the induced secondary circulation, the concomitant temperature QBO (Sect. ) extends slightly lower with an identifiable signal observed up to ∼15∘ on either side the Equator near the tropopause, albeit small (∼0.5 K) compared to the dominant annual cycle (Sect. ). The resultant temperature and circulation anomalies at the tropopause and in the lower stratosphere exert a downward influence on tropical convection, which in turn plays a major role in coupling the QBO to the tropical troposphere . Spatial variations in convection partially explain why the QBO signal in the troposphere is no longer predominately zonally symmetric as it was in the stratosphere but, so far, none of the physical mechanisms proposed can account for all aspects of the large geographical variations in the QBO's downward influence on the tropical troposphere .

The QBO's influence on tropical convection is the main reason for its involvement with the Madden–Julian oscillation (MJO), which dominates intraseasonal variability in the tropical troposphere . During the NH winter up to 40 % of the inter-annual variations in the MJO amplitude are explained by a coupling to the QBO , with the MJO stronger and more persistent, by up to 10 d, when the QBO is westward in the lower stratosphere . In contrast, an eastward QBO phase in the lower stratosphere favours a weaker-than-normal MJO during NH winter .

Knowledge of the QBO–MJO relationship is only just emerging , and there is still plenty to do in developing a working mechanism . The relationship is, nonetheless, particularly relevant as it implies a potential pathway coupling the stratosphere to the extratropical troposphere (cf. Sect. ) via the MJO's teleconnections to high latitudes . The corresponding ramifications for the extratropics of this modulation of the MJO's global teleconnections by the QBO are reviewed in , and, therefore, this aspect of the QBO–MJO relationship is not considered here.

The tropical troposphere and MJO are also affected by SSWs . One proposed mechanism is that the increased wave driving during an SSW strengthens the mean meridional (Brewer–Dobson) circulation and cools the tropical lower stratosphere and tropopause. The anomalies created then impact tropical convection in a similar way to the anomalies resulting from the QBO's secondary circulation. Tropopause level PV intrusions from the subtropics during an SSW also affect the upper tropical troposphere and represent an alternative subtropical pathway from the stratosphere to tropical troposphere .

Ramifications of the stratosphere's influence on the tropical troposphere include an enhancement of convective activity and precipitation over the tropical west Pacific and a suppression over the equatorial central and east Pacific when the QBO is westward in the lower stratosphere, at least for the annual mean . During SSWs equatorial convection is enhanced south of the Equator and suppressed to the north, especially for split vortex events , while tropical cyclones are generally more intense . Monsoon intensity and rainfall appear to be sensitive to the QBO phase with, for example, the East Asian winter monsoon weaker, on average, when the QBO is westward compared to when it is eastward at 70 hPa . Also, over East Asia MJO-related precipitation anomalies are between 40 % and 70 % larger during the QBO westward phase than during the eastward phase, depending on the phase of the MJO .

The stratosphere's powerful downward influences and longer prediction timescales (Sect. ) are reasons why the stratosphere matters for tropospheric predictions . Models with better-resolved stratospheres generally produce superior simulations of the surface climate and more skillful forecasts on sub-seasonal to seasonal timescales (e.g. Fig. ; ), with more accurate stratospheric initial conditions and variability further enhancing forecast skill. In the extratropics, predictive skill is enhanced following SSWs, and, in April, skill can be gained from a knowledge of the vertical structure of the final warming . In the tropics, MJO predictability is influenced by the QBO .

Another reason why the stratosphere matters for extratropical predictions is the pathways it provides to potential sources of extended range skill lying outside the stratosphere such as ENSO , Eurasian snow cover and Arctic sea ice . Multi-year skill arising from the solar cycle and volcanic eruptions also involves the stratosphere, though the possible mechanisms are not considered here. A concurrent independent perspective of these aspects of the stratosphere's role in long-range prediction can, instead, be found in .