Significance Weather extremes are becoming more frequent and severe in many regions of the world. The physical mechanisms have not been fully identified yet, but there is growing evidence that there are connections to planetary wave dynamics. Our study shows that, in boreal spring-to-autumn 2012 and 2013, a majority of the weather extremes in the Northern Hemisphere midlatitudes were accompanied by highly magnified planetary waves with zonal wave numbers m = 6, 7, and 8. A substantial part of those waves was probably forced by subseasonal variability in the extratropical midtroposphere circulation via the mechanism of quasiresonant amplification (QRA). The results presented here support the overall hypothesis that QRA is an important mechanism driving many of the recent exceptional extreme weather events.

Abstract In boreal spring-to-autumn (May-to-September) 2012 and 2013, the Northern Hemisphere (NH) has experienced a large number of severe midlatitude regional weather extremes. Here we show that a considerable part of these extremes were accompanied by highly magnified quasistationary midlatitude planetary waves with zonal wave numbers m = 6, 7, and 8. We further show that resonance conditions for these planetary waves were, in many cases, present before the onset of high-amplitude wave events, with a lead time up to 2 wk, suggesting that quasiresonant amplification (QRA) of these waves had occurred. Our results support earlier findings of an important role of the QRA mechanism in amplifying planetary waves, favoring recent NH weather extremes.

Recent years have seen an increasing number, severity, and spatial scale (covered area) of summer extremes in the Northern and Southern Hemispheres (NH and SH), such as the European heat wave in 2003, the Russian heat wave and the Indus river flood in Pakistan in 2010, and the heat waves in the United States in 2011 (1⇓–3). Model projections suggest that weather extremes may become even more intense, more frequent, and longer in the climate of the future (4). Several conceptual mechanisms have been proposed to explain the physical basis of the extremes and the reason for the increase in their frequency of occurrence.

As shown by Coumou et al. (5) and Comou and Robinson (6), the observed long-term increase in frequency of extreme heat events can, on a global scale, be explained purely thermodynamically as a response to a shift in the mean surface temperatures to warmer values. Likewise, general trends toward higher annual maximum daily rainfall are consistent with an overall rise in atmospheric moisture associated with warmer air (7⇓–9).

Recent global climate change is also likely to affect large-scale atmospheric circulation patterns, with strong nonlinear feedbacks between thermodynamic and dynamic components of the climate system (10, 11). This could potentially alter the frequency of extremes on seasonal to subseasonal timescales. In one of the first studies on this issue, Schär et al. (12) developed a stochastic concept of regional blocking events for the explanation of the 2003 European heat wave as a result of the observed climatic warming trend, which shifts and widens the probability distribution of summer temperatures. Luterbacher et al. (13) estimated a return period for this type of extreme event as being about 100 y in the European region, taking climatic warming into account. However, a number of severe summer extremes have already occurred since then in the NH, particularly in Europe. The anomalous atmospheric circulation patterns accompanying these extremes were of hemispheric scale (14⇓⇓⇓⇓⇓⇓⇓–22). They encircled the entire NH and persisted over nearly the whole summer, which is fundamentally different from conventional regional blocking events with about a 10-d life span (23). This shows that not only purely stochastic regional mechanisms of extremes are at work (3). Based on the NH annular mode (NAM) (see ref. 24), Tachibana et al. (14) showed that an anomalously strong positive summer NAM (as occurred specifically during the 2003 European heat wave) accounted well for hemispheric weather regimes with anomalously high midlatitude blocking activity between strongly marked polar and subtropical jets, over the period 1958–2005. Black et al. (15) analyzed basic factors that likely contributed to the summer 2003 European heat wave, examining large-scale atmospheric flow, regional heat budget at the top of the atmosphere, and sea surface temperature. As a key factor, the authors of ref. 15 identified a Rossby wave train of alternating-sign stream function anomalies, spreading north-northeast from the source region in tropical America across the Atlantic and farther into Eurasia. This pattern resembles those discussed conceptually in ref. 25. Similar to ref. 15, Cassou et al. (16) argue that the anomalously warm June 2003 in western Europe could be related to wetter-than-average conditions in the Caribbean that triggered the occurrence of a Rossby wave train pattern stretching from the Caribbean across the Atlantic, whereas the anomalous August 2003 could be associated with a summer NAO-like pattern and enhanced monsoon over the Sahel, which might have been compensated dynamically by anomalously strong downdrafts over Europe. Hong et al. (17) and Lau and Kim (18) described a Rossby wave train spanning Eurasia during the catastrophic 2010 Pakistan flood and Russian heat wave, with the southern branch spreading through northern Pakistan and being accompanied by heavy monsoon surges there.

Branstator (26) proposed a mechanism of generation of predominantly zonally oriented Rossby wave trains. He showed that a sufficiently intense quasizonal subtropical jet could act as a waveguide for a quasistationary zonal wave number 5. The mechanism was subsequently applied to explain several important features of some of the recent summer extremes (18⇓⇓–21, 27).

Francis and Vavrus (28) have suggested a conceptual model of deceleration and increase in the north−south meridional extent of eastward-propagating midlatitude planetary Rossby waves over the North America/North Atlantic sector caused by the recently observed decrease in the midlatitudinal westerly winds. In their study, these authors rely on observational evidence of recent Arctic amplification (AA), i.e., a strong increase in lower-tropospheric temperatures in the Arctic compared with that over the total NH (see, e.g., ref. 29). The authors of ref. 28 hypothesized that AA, consistent with polar sea ice loss, might favor a decrease in midlatitude westerlies and therefore lead to increased probabilities of persistent extreme regional weather events in the NH midlatitudes. At least in summer, the NH westerlies and storm tracks have really weakened over 1979–2015 (30, 31).

Screen (32) explored the influence of Arctic sea ice on European summer climate using a state-of-the-art atmospheric model in view of the historically unprecedented sequence of six consecutive wet summers from 2007 to 2012 in northern Europe, which featured a marked southward shift of the polar jet stream. He found that prescribed sea ice loss in the model caused a southward shift of the summer jet stream and increased northern European precipitation. An anomalous Rossby wave-4 train is reported by Hanna et al. (33), when studying the exceptional Greenland ice sheet melt in summer 2012.

Petoukhov et al. (34) proposed a common mechanism for generating persistent high-amplitude quasibarotropic planetary-scale wave patterns of the NH midlatitude atmospheric circulation with zonal wave numbers m = 6, 7, and 8 that can explain a number of the major NH summer extremes over the 1980–2011 period (34, 35). Petoukhov et al. (34) showed that these patterns could result from the trapping of free quasistationary barotropic Rossby waves with zonal wave numbers k equal or close to the three integer values indicated above, within the predominantly zonally oriented midlatitude waveguides. Unlike the one considered in Branstator (26) for zonal wave number 5, the formation of these waveguides is based on a specific change in the latitudinal shape (and not the magnitude) of the quasizonal extratropical winds at the equivalent barotropic level (EBL). The midlatitude waveguides considered in ref. 34 can favor an onset of midlatitude quasiresonant amplification (QRA) of these waves, causing a strong increase in the atmosphere’s dynamical response to quasistationary midtroposphere external forcing with zonal wave numbers m = 6, 7, and 8 (Calculation of the Meridional Wave Number). The reason for the change in the shape and positions of atmospheric jet streams is, for now, an issue of debate, with AA as one of the potential candidates (see, e.g., refs. 36 and 37).

Recently, Screen and Simmonds (38) showed that, during 1979–2012, months with extreme weather (in terms of high anomalies of land surface temperature and land precipitation) in the NH midlatitudes were commonly accompanied by zonally elongated midlatitude trains of quasistationary midtropospheric planetary waves, predominantly with m = 5−7. Their findings also suggest that amplified quasistationary waves with these wave numbers increased the probabilities of extreme weather events over the NH midlatitudes.

QRA Mechanism of Planetary Wave Reinforcement presents a brief description of the QRA mechanism proposed in ref. 34. In Spring to Autumn Weather Extremes, we investigate, within the framework of QRA, severe regional weather extremes that occurred in boreal spring-to-autumn 2012 and 2013 in the NH. We show that a considerable portion of these extremes could be favored by QRA events for the midlatitude waves with zonal wave numbers 6, 7, and 8. In Discussion, we discuss the results presented in Spring to Autumn Weather Extremes and briefly touch on the issue of high-amplitude quasistationary midlatitude waves with zonal wave numbers 4 and 5.

Discussion Overall, the results of our calculations of the amplitudes of the waves with m = 6, 7, and 8 in the field of the midlatitude meridional velocity at 300 hPa suggest that 12 of the 17 HPA events during May−September of 2012 and 2013 substantially coincided or were preceded by QRA events up to 2 wk earlier (see Fig. 1). The QRA amplitudes calculated by Eqs. 1a and 1b match—with an accuracy of the order of 1 m⋅s−1—the amplitudes of the observed HPA events. These results strongly suggest that, in spring-to-autumn 2012 and 2013, the QRA mechanism played an important role in generating HPA events for wave numbers m = 6, 7, and 8 that were accompanied by regional weather extremes, causing serious damage for society. However, four occurrences of HPA events marked by open circles in Fig. 1 cannot be explained by the QRA mechanism. Also, in one case denoted by the open square in Fig. 1, QRA predicted an HPA event that did not happen. This attests that the QRA, as described here in a quasilinear approximation, is not, of course, the only mechanism for generating regimes of high-amplitude midlatitude waves with m = 6, 7, and 8. Other important competing mechanisms exist that can drive high-amplitude midlatitude extratropical planetary waves, like the Branstator mechanism (26), El Niño−Southern Oscillation (51), and North Atlantic Oscillation (NAO) (37). The choice of the mean flow and the scale separation between the mean flow and the stationary waves is critical. In our paper, we have dealt with the zonally averaged zonal winds as the mean flow and excluded from our consideration packets of quasistationary planetary waves trapped in predominantly meridional elongated waveguides, specifically those originating in the tropics. On the other hand, a large number of recent NH summer extremes occurred as circumglobal chains of alternating-in-longitude regional droughts and floods, embracing a large part of the midlatitude belt (see Figs. 2 and 3). A substantial part of these extremes could be favored by zonally elongated trains of the midtroposphere planetary waves (see, e.g., refs. 21, 34, and 38). In this paper, we investigated such wave trains with zonal wave numbers m = 6, 7, and 8, triggered by the QRA events. Our results showed that the time shift between the QRA event and respective HPA event could be up to about half a month. This is close to the “integral e-fold time scale” of excitation and decay of the persistent teleconnection anomalies observed over the North Pacific, North Atlantic, and Siberia sectors of the NH (52). The QRA mechanism is considered in the present paper at the conceptual level: In the working equation for the azonal stream function, zonally averaged zonal flows and azonal forcing at the EBL are prescribed using observational data (refs. 41 and 46). For that reason, the QRA mechanism as discussed here is only a diagnostic, and not a predictive, theory of the zonally elongated planetary wave amplification. As to the quasistationary planetary waves with m = 4 and m = 5 mentioned in the Introduction, a wave action of these waves can propagate far to the extratropics even under normal conditions (25, 42). In the present paper, we analyzed the May−September time series of the amplitudes of the observed waves with m = 4 and m = 5 over the 1980–2013 time range (www.pik–potsdam.de/∼petri/extr_2012_2013.html#movies). The numbers of the HPA events for m = 4 and m = 5 were 16 and 20 in the first 11 y, 17 and 15 in the second 11-y period, and 20 and 25 in the last 12 y. The obtained results might indicate an increase in the number of such events for the waves with m = 4 and m = 5 in the last decade or so, compared with the previous ones, but this conclusion needs further verification.

Conclusions We show that, in May−September 2012 and 2013, the majority (12 of 17) of HPA events, for midlatitude wave numbers m = 6, 7, and 8, with the observed amplitudes exceeding 1.5 SDs from the 1980–2013 May−September climatology, occurred when the resonance conditions for these wave numbers were fulfilled, within up to 2 wk preceding an HPA event. In all 12 cases, the wave amplitudes predicted by the QRA theory (Eqs. 1a and 1b) were in a good agreement with the observed amplitudes of the related HPA events, which favored strong regional weather extremes in the NH midlatitudes.

Calculation of the Meridional Wave Number l and the Dimensionless Stationary Wave Number K s 2 a 2 for the Quasistationary Barotropic Free Midlatitude k Wave Trapped Within the Quasiresonant Waveguide The dimensional meridional wave number l of the quasistationary barotropic midlatitude free waves with nondimensional zonal wave number k ≈ 6−8 (the k wave) is calculated from the equation (see equation S5 in ref. 34) l 2 = 2 Ω ⁡ cos 3 ⁡ ϕ a u _ − cos 2 ⁡ ϕ a 2 u ¯ d 2 u ¯ d ϕ 2 + sin ⁡ ϕ ⁡ cos ⁡ ϕ a 2 u _ d u ¯ d ϕ + 1 a 2 − ( k a ) 2 . [S1]In Eq. S1, Ω is Earth’s rotation angular velocity, a is Earth’s radius, ϕ is the latitude, and u ¯ is the 15-d running mean of zonally averaged zonal wind at the EBL. For any given k , Eq. S1 determines l 2 as a function of u ¯ . A square of the nondimensional stationary wave number, K s 2 a 2 , shown in the left y axis in Figs. 2B and 3B, is given by the following equation (see equations S1b and S2−S4, with the accompanying text, in ref. 34) K s 2 a 2 ≡ β ˜ a 2 ⁡ cos 2 ⁡ ϕ / u ¯ = 2 Ω a ⁡ cos 3 ⁡ ϕ u ¯ − cos 2 ⁡ ϕ u ¯ d 2 u ¯ d ϕ 2 + sin ⁡ ϕ ⁡ cos ⁡ ϕ u ¯ d u ¯ d ϕ + 1 , [S2]where β ˜ is the meridional gradient of the absolute vorticity multiplied by cos ⁡ ϕ , so that the difference between K s 2 a 2 and k 2 just gives the value of l 2 a 2 (see Eq. S1). If K s 2 a 2 = k 2 at any midlatitude, under positive value of zonally averaged zonal wind at that latitude, the result is a TP at which the corresponding l 2 passes zero. In the case of two TPs in the midlatitudes for the same k, this k wave becomes trapped within the waveguide between the two TPs, provided K s 2 − k 2 ( = l 2 a 2 ) > 0 in between. In so doing, the (maximum) value of l reached in the waveguide’s interior is used for comparison with the meridional wave numbers l m of the partial waves dominantly contributing to the external forcing with a given zonal wave number m (see Basic Necessary Conditions and Assumptions). The examples of the waveguides are shown in Figs. 2B and 3B, for the cases of the QRA events considered in Spring-to-Autumn Weather Extremes 2012 and 2013 in the Context of QRA. In so doing, the right axis for k in Figs. 2B and 3B scales quadratically with k. On the other hand, as discussed in ref. 40, the barotropic orographic and thermal forcing with zonal wave numbers m is the major extratropical source for forced quasistationary barotropic dynamical waves with these zonal wave numbers. The persistent barotropic vertical structure of these forced waves with m = 6, 7, and 8 on a monthly time scale in the extreme years is clearly documented in corresponding maps (see, e.g., figures S1 and S2 in ref. 34). The corresponding working quasilinear wave equation for the barotropic azonal stream function Ψ m ′ of the forced waves with m = 6, 7, and 8 (m waves) with nonzero right-hand side (forcing + eddy friction) yields (34) u ˜ ∂ ∂ x ( ∂ 2 Ψ m ′ ∂ x 2 + ∂ 2 Ψ m ′ ∂ y 2 ) + β ˜ ∂ Ψ m ′ ∂ x = 2 Ω ⁡ sin ⁡ ϕ ⁡ cos 2 ⁡ ϕ T ˜ u ˜ ∂ T m ′ ∂ x − 2 Ω ⁡ sin ⁡ ϕ cos 2 ⁡ ϕ H κ u ˜ ∂ h o r , m ∂ x − ( k h a 2 + k z H 2 ) ( ∂ 2 Ψ m ′ ∂ x 2 + ∂ 2 Ψ m ′ ∂ y 2 ) , [S3]where x = a λ and y = a ⁡ ln [ ( 1 + sin ⁡ ϕ ) / cos ⁡ ϕ ] are the coordinates of the Mercator projection of Earth’s sphere, with λ as the longitude, H is the characteristic value of the atmospheric density vertical scale, T ˜ is a constant reference temperature at the EBL, T m ′ is the m component of azonal temperature at this level, u ˜ = u ¯ / cos ⁡ ϕ , κ is the ratio of the zonally averaged module of the geostrophic wind at the top of the PBL to that at the EBL (53), h o r , m is the m component of the large-scale orography height, and k h and k z are the horizontal and vertical eddy diffusion coefficients. Using a scale−magnitude analysis method, these are calculated as k h ≈ U ˜ L and k z ≈ W ˜ H ˜ ≈ ( U ˜ / L ) ( H R o ) 2 , in terms of the characteristic values of the horizontal, U ˜ , and vertical, W ˜ , velocities and horizontal, L , and vertical, H R o , spatial scales of the baroclinic eddies efficiently contributing to the atmospheric near-surface and internal “eddy friction” (see SI text section A.3 in ref. 34). Following ref. 53, we assign U ˜ = κ u ¯ . The first two terms on the right side of Eq. S3 describe, respectively, the external thermal and orographic forcing at the EBL (see SI text section A.3 in ref. 34). The wave number m is an integer here so that this wave is a solution to the full wave equation with periodic boundary conditions, i.e., fitting around Earth in a latitude band without discontinuity.

Calculation of the Amplitudes A ˜ m O r t of the External Forcing Variable A ˜ m O r t in the right-hand side of Eq. 1a designates the amplitude of the 15-d running means of the external thermal+orographic barotropic forcing with zonal wave number m at the EBL, averaged over the midlatitude belt Δ = 37.5°−57.5°N. The values for A ˜ m O r t are derived from the daily data on temperature at 300 hPa from ref. 41 and the geographic distribution of the orography from ref. 46, the latter data coarsened to 10° × 15° resolution as in ref. 53. The use of current observed forcing precludes the use of the QRA theory for weather forecasting purposes; it is a diagnostic tool used to elucidate the mechanism that has led to large planetary wave amplitudes.

Basic Necessary Conditions and Assumptions for the Midlatitude QRA Mechanism In the present paper, several basic necessary conditions and requirements are formulated for the QRA mechanism to arise in the midlatitudes. Existence of a Waveguide. Condition 1: two TPs should occur in the midlatitudes for the considered free waves (see Calculation of the Meridional Wave Number), with l 2 > 0 in the latitudinal range (waveguide) between the TPs and l 2 ≤ 0 outside but in its close vicinities, with u ¯ > 0 within the waveguide and in its close vicinities (necessary condition i in ref. 34). When the zonal wavenumber of the trapped free k wave supported by the waveguide coincides with or is close to that of the m wave driven by the forcing, i.e., when k ≈ m, this latter wave can grow to large amplitude in a resonance process, as it is forced but not dispersed. For resonance to exist, we thus require that k is close to m: Condition 2: | k − m | < C m , with a parameter C m estimated using the equation (see Eq. 1a) A ˜ m , e = A ˜ m , S D O r t { [ C m ( 2 m + C m ) ] 2 + C 1 , m 2 ( L / a + R o 2 a / L ) 2 m 2 } 1 / 2 , [S4]where A ˜ m , e and A ˜ m , S D O r t are, respectively, the mean +1 SD value of A ˜ m and the maximum value of A ˜ m O r t , over the 1980–2013 time range. In that case, the value of C m , which satisfies Eq. S4, gives a good estimation of the maximum possible deviation of k from m over 1980–2013 for which the amplitude A ˜ m of the considered m wave is still higher than A ˜ m , e . Substitution of the relevant values of A ˜ m , e , A ˜ m , S D O r t , C 1 , m , L , and R o into [S4] results in the values of C m ≈ 0.2−0.25, with the change in the value of C m being caused by substitution in Eq. S4 of different values of m = 6, 7, and 8 and uncertainty in the value of C 1 , m (see Physics of the Parameter). (Necessary condition 2 is formulated in ref. 34 when deriving the basic equation S14 for the resonance amplitudes of the planetary waves in that paper. In the present paper, we formalize this condition 2 using Eq. S4. Applicability of the Quasilinear WKB Method. Eq. 1a is derived applying a quasilinear WKB method (25), which is valid if the change in the meridional wavelength Λ ϕ = 2 π / l of the free wave trapped in the waveguide over a distance Λ ϕ /(4−8)π is less than Λ ϕ (25, 34). This implies the following additional necessary conditions: Condition 3: | d l − 1 / a d ϕ | < 1−2 within the quasiresonant waveguide’s interior (necessary condition ii in ref. 34). Condition 4: The total latitudinal width of the quasiresonant waveguide is no less than the characteristic scale Δ A of respective Airy functions (25, 34) at the southern and northern lateral boundaries of the waveguide, with the position of the southern and northern boundaries north of ∼25°N−30°N and south of ∼65°N−70°N, respectively, and with Δ A ≈ 2.25−3.75° calculated using the corresponding equations S17 and S18 in ref. 34, at realistic values of the atmospheric parameters (necessary condition iii in ref. 34). Permissible Range ( l m i n , l max ) of the Values of l Within the Waveguide's Interior Ensuring Proximity of l to the Meridional Wave Numbers l m Dominantly Contributing to the External Forcing. Resonance requires not only the proximity of zonal wave numbers k and m but, likewise, a closeness of the maximum value of the meridional wave number l of k wave, calculated with the use of Eq. S1 for the waveguide’s interior, to the meridional wave numbers l m of the partial waves dominantly contributing to the external forcing. Condition 5: l ∼ l m In that case, l m entering the expression for the external forcing (see the right side of equation S6 in ref. 34) can be replaced by the maximum (highest) value of l , which is calculated with the use of Eq. S1 for the waveguide’s interior. This makes Eq. 1a a resonance equation with [ ( k / a ) 2 − ( m / a ) 2 ] 2 in its denominator. The sequential steps in the transition to the resonance equation with [ ( k / a ) 2 − ( m / a ) 2 ] 2 in its denominator are described in ref. 34 by equations S8−S12. On the other hand, by virtue of the dominant contribution from the partial waves with l m to the total amplitude A ˜ m O r t of the external forcing from a full set of the partial forcing waves with a given m, just A ˜ m O r t can be used, to a first approximation, as the forcing in the right side of Eq. 1a for A ˜ m . In so doing, A ˜ m O r t is calculated within the indicated Calculation of the Amplitudes A ˜ m O r t of the External Forcing section midlatitude belt Δ applying a simplified “strip-by-strip” algorithm of computation of the external forcing, presented in SI sections A.1, A.3, and A.5 of ref. 34. When using this algorithm, A ˜ m O r t is described by the area-weighted sum of the contributions from all of the latitudinal strips entering Δ . As regards the meridional wave numbers of the above-mentioned dominant partial waves of the external forcing with a given m = 6, 7, and 8, the observational and model results of the 1D Fourier transform of the midtroposphere extratropical atmospheric fields attest that the characteristic values of l m for these waves (see, e.g., refs. 41 and 46) can change within the l min ≈ 0.25 ⋅ 10 − 6 m−1 < l m < l max ≈ 2.7 ⋅ 10 − 6 m−1 range. Just this ( l min , l max ) interval specifies the possible range for the highest values of l within the waveguide’s interior between the TPs, in order that the condition l ∼ l m be obeyed (necessary condition iv in ref. 34).

Additional Necessary Condition of the Minimum Amplitude of the External Forcing A ˜ m O r t In the absence of significant external forcing at wave number m, no high-amplitude waves will result. In the present paper, we thus apply Condition 6: A ˜ m O r t is not lower than a certain threshold minimum value, A ˜ m , min O r t . This condition is a direct consequence of applying a quasilinear WKB method for the description of the QRA. For that, we require that A ˜ m O r t should be higher than the sum of the nonlinear terms in the original nonlinear barotropic vorticity equation on a sphere (see, e.g., ref. 54) for a given 15-d interval. This allows one to neglect these latter terms in the working equation while retaining the forcing term. For realistic values of the parameters of the waves with m = 6, 7, and 8, the respective values of A ˜ m , min O r t are about (1.1−2.5) × 10−13 m−1 s−1. The indicated variation in the values of this parameter are determined basically by changes in the values of zonal wind within the waveguide and the meridional and zonal wave numbers of the resonance waves.

Additional Necessary Condition Posed on Closely Positioned (Double) and Overlapping Waveguides In rare cases, closely positioned (double) waveguide or even overlapping waveguides resulted from our calculations. We detected five such situations: the first three for waves m = 6 and m = 7 over the periods from 30 May 2012 to 8 June 2012, from 10 August 2013 to 12 August 2013, and from 15 August 2013 to 22 August 2013; the fourth one for waves m = 7 and m = 8 from 6 August 2013 to 9 August 2013; and the fifth for waves 6, 7, and 8 on 13 August 2013 and 14 August 2013, as the central dates. The necessary condition 1 was violated, as there existed propagation instead of reflection of the wave activity at the common border of the adjacent single waveguides. Also, condition 5 on the “smoothness” of the waveguide’s latitudinal shape was not satisfied. In the cases of the overlapping waveguides for different m, the nonlinear interference of the waves could occur. As a result, in all these cases, the full set of resonance conditions was not satisfied, and the amplitudes of the observed waves were markedly lower than 1.5 SD above the corresponding 1980–2013 climatology. In view of that, we posed the necessary. Condition 7: In case of two closely adjacent midlatitude waveguides, their angular distance with latitude has to exceed at least a value of 5°, and the overlapping waveguides for different m must be completely ruled out of consideration.

Amplitude of Wave Breaking After reaching a certain size, waves will not grow further but will break. We thus implement condition A ˜ m ≤ A ˜ m , max = A ˜ m , b , where A ˜ m , b is the amplitude reached by the planetary Rossby waves in the wave breaking process within the midlatitude belt Δ . The above process is the basic one limiting A ˜ m in the midlatitudes (43). To obtain the equation for A ˜ m , b , we use the parameterization of the wave activity A during the wave breaking proposed in ref. 43. In the considered case of quasiresonant m waves, this parameterization reads, within the waveguide’s interior, Δ Q R A int (34), with ϕ = ϕ 0 as the central latitude (43) C g m y A = ρ ( m / a ) l 0 K 2 u ¯ 0 q ′ 2 ¯ q y ¯ . [S5]In [S5], ρ and C g m y are, respectively, the air density and the m-wave group velocity in the meridional direction (25, 43), q ′ is the quasigeostrophic wave potential vorticity, and q y ¯ is the zonal mean gradient of potential vorticity, all four variables being calculated for ϕ = ϕ 0 at the EBL; u ¯ 0 and l 0 in [S5] are, respectively, u ¯ and l at ϕ = ϕ 0 , and K 2 reads (43) K 2 = ( m / a ) 2 + l 0 2 + ( f 0 / N 0 ) 2 ( 1 / 4 H 2 ) , [S6]where H is the atmospheric density scale height, and f 0 and N 0 are, respectively, the Coriolis parameter and the Brunt−Väisäla frequency at the EBL for ϕ = ϕ 0 . According to ref. 43, the following relations result for saturated (breaking) planetary waves: | q ′ y | ¯ = | l 0 q ′ | ¯ = q ¯ y . [S7]Substitution of [S7] into [S5] yields C g m y A = ρ ( m / a ) K 2 ( q ′ 2 ¯ ) 1 / 2 . [S8]On the other hand, from the dispersion relation for the m waves in the WKB approximation (25, 43), we can write, within Δ Q R A int , q y ¯ = K 2 u ¯ 0 , [S9]so that [S5] can be rewritten as follows: C g m y A = ρ ( m / a ) l 0 u ¯ 0 2 . [S10]Equating [S8] and [S10], we have ( q ′ 2 ¯ ) 1 / 2 = u ¯ 0 K 2 l 0 . [S11]For the considered quasibarotropic m waves, q ′ 2 ¯ in [S11] reads (cf. ref. 43) q ′ 2 ¯ = ( ∂ 2 Ψ ′ / ∂ x 2 + ∂ 2 Ψ ′ / ∂ y 2 ) 2 ¯ | ϕ = ϕ 0 [S12]within Δ Q R A int . In [S12], Ψ ′ is the corresponding perturbation stream function over the total waveguide’s width, Δ Q R A (34). Following ref. 34, consider Ψ ′ given by Ψ ′ = Ψ 0 ⁡ sin m a x ⁡ cos l m ( y − y 0 ) . [S13]In [S13], l m ≈ l 0 , and y 0 corresponds to the latitude ϕ 0 (34). Applying respective equations for the meridional, v ′ m , and zonal, u ′ m , velocities v ′ m = ∂ Ψ ′ / ∂ x u ′ m = − ∂ Ψ ′ / ∂ y [S14]in the m wave and using the identity sin 2 m a x ¯ = cos 2 m a x ¯ = 1 2 [S15]that is valid for any integer m and x ranging over the [ 0 , 2 π a ] interval in a Mercator projection of the sphere, the equation for v ′ m 2 ¯ yields, in the case of the quasiresonant m wave breaking within Δ Q R A , v ′ m 2 ¯ ( y ) = v ′ m , b 2 ¯ ( y ) = v ′ m , b 2 ¯ ( y 0 ) cos 2 l 0 ( y − y 0 ) , [S16]where v ′ m , b 2 ¯ ( y 0 ) is given by v ′ m , b 2 ¯ ( y 0 ) = K 4 [ K 2 + l 0 2 + l 0 4 / ( m / a ) 2 ] 2 K 2 l 0 2 u ¯ 0 2 . [S17]Then, on the strength of A ˜ m , b 2 ( y 0 ) = 2 v ′ m , b 2 ¯ ( y 0 ) , the latitudinal averaging of [S17] over the Δ Q R A range results in the following estimate of the maximum allowable value, A ˜ m , b 2 ( Δ Q R A ), for the amplitude of the 15-d-mean m component of the meridional velocity at the EBL over Δ Q R A , A ˜ m , b 2 ( Δ QRA ) = 2 K 4 [ ( K 2 + l 0 2 + l 0 4 ) / ( m / a ) 2 ] 2 K 2 l 0 2 u ¯ 0 2 〈 cos 2 l 0 ( y − y 0 ) 〉 Q R A , [S18]where 〈 X 〉 Q R A stands for the latitudinal averaging of X over Δ Q R A . Accounting for the characteristic spatial scale Δ A of the relevant Airy function (see Basic Necessary Conditions and Assumptions) at the waveguide’s boundaries, the averaging of [S18] over Δ yields the following estimate for A ˜ m , max , at typical values of l 0 ≈ (0.3−0.5) × 10−6 m−2, Δ Q R A / Δ ≈ 0.3−0.5, and Δ A / Δ ≈ 0.2−0.25: A ˜ m , max ≈ K 2 [ ( K 2 + l 0 2 + l 0 4 ) / ( m / a ) 2 ] K l 0 u ¯ 0 Δ Q R A 2 Δ . [S19]Finally, substituting relevant values of f 0 , N 0 , and H into [S6] and using [S19] results in A ˜ m , max ≈ (0.3−0.7) u ¯ 0 for the considered m waves with m = 6, 7, and 8. The indicated variation in the values of A ˜ m , max are determined basically by changes in the values of zonal wind within the waveguide, the meridional and zonal wave numbers of the resonance waves, and the width of the waveguide’s interior, Δ Q R A int . We note that this section does not present an additional necessary condition of QRA but rather gives a receipt of transition from Eq. 1a to Eq. 1b for A ˜ m given in QRA Mechanism of Planetary Wave Reinforcement, in the case when A ˜ m calculated by Eq. 1a exceeds A ˜ m , max .

Estimation of the Error Bars for the Values of A ˜ m , o b s and A ˜ m in the HPA Events The peak values of the amplitudes A ˜ m , o b s for zonal wave numbers m = 6, m = 7, and m = 8 shown by filled and open circles in Fig. 1 represent corresponding coefficients of the 1D Fourier decomposition of the 15-d running means of the circumglobal meridional velocity field at 300 hPa averaged over the 37.5°N−57.5°N range during the HPA events, computed on the basis of daily NCEP-NCAR reanalysis data (41). These data include some errors that appear while processing the random raw atmospheric data with the use of the applied model and assimilation method (see, e.g., ref. 55). We take 1 SD of the values of A ˜ m , o b s over 1980–2013 as a representative measure of the error bars for this quantity. The QRA amplitudes A ˜ m (see Eq. 1a) are calculated here with the use of the amplitudes A ˜ m O r t of the external thermal and orographic forcing whose values are derived from the daily data on temperature at 300 hPa from ref. 41 and the orography data from ref. 46, the latter data being coarsened to 10° × 15° resolution as in ref. 53. Also, the NCEP-NCAR data for the module of the ratio of the wind speed u ¯ P B L at the top of the planetary boundary layer to that at the EBL u ¯ enters the denominator of Eq. 1, via the parameter κ. All these input data are subject to some uncertainties. Assuming that | u ¯ P B L | and | u ¯ | are normally distributed with nearly vanishing densities at zero, the distribution density for their ratio s is given by the following equation (see, e.g., ref. 55): δ ( s ) = σ A 2 μ B + σ B 2 μ A s 2 π ( σ A 2 + σ B 2 s 2 ) 3 / 2 exp [ − ( μ A − μ B s ) 2 2 ( σ A 2 + σ B 2 s 2 ) ] , [S20]where σ A and σ B are, respectively, 1 SD for | u ¯ P B L | and | u ¯ | , and μ A and μ B are their means over the 1980–2013 time interval. Substitution of corresponding values σ A , σ B , μ A , and μ B in [S19] allows one to estimate δ ( s ) for different values of s ( κ , in our case), yielding, finally, κ = 0.31 ± 0.05 . At the second step, we calculate, in the same manner, the value of A ˜ m , substituting, in the numerator of Eq. 1, A ˜ m O r t with the above-mentioned estimation of the error bars for this quantity, and κ = 0.31 ± 0.05 in the denominator, which results in the values about 0.6–1.0 m⋅s−1 for the error bars of the predicted QRA amplitudes of the considered m waves.

Footnotes Author contributions: V.P. and H.J.S. designed research; S.R., D.C., and K.K. performed research; S.P. and K.K. analyzed data; and V.P. and S.R. wrote the paper.

Reviewers: R.E.B., The Norwegian Meteorological Institute; and D.K., University of Melbourne.

The authors declare no conflict of interest.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1606300113/-/DCSupplemental.