Role of quasiresonant planetary wave dynamics in recent boreal spring-to-autumn extreme events

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.

QRA Mechanism of Planetary Wave Reinforcement

Rossby waves (also known as planetary waves) are ubiquitous in ocean and atmosphere. In the atmosphere, we can distinguish forced and free Rossby waves. Forced Rossby waves occur as a response of the midtroposhere and high-troposphere atmospheric circulation to the external diabatic and orographic forcing (25, 39, 40), which arises, e.g., from the thermal contrast between land and oceans as well as from mountain ranges. They occur at various wave numbers and are quasistationary and principally barotropic. Free extratropical Rossby waves with zonal wave numbers about 6 to 8 mostly occur as high-amplitude, fast traveling waves (the so-called synoptic transients responsible for much of the weather variability in the extratropics); once established, they can freely propagate predominantly to the east with a phase speed c ≈ 6−12 m⋅s⁻¹ without maintenance from external forcing. In contrast to these fast traveling waves, the quasistationary extratropical barotropic free waves with zonal wave numbers 6 to 8 are normally weak, with a magnitude of the meridional wind velocity less than 1.5–2.5 m⋅s⁻¹ (25, 34, 35, 39, 40).

However, using National Centers for Environmental Prediction−National Center for Atmospheric Research (NCEP-NCAR) reanalysis data (41), Petoukhov et al. (34) showed that, during a number of recent NH extremes in July and August, certain persistent high-amplitude atmospheric wave patterns with barotropic vertical structure evolved, to which the quasistationary component of midlatitude barotropic free waves with zonal wave numbers k ≈ 6−8 made an exceptionally large contribution. These authors also presented an explanation for the emergence of these unusual high-amplitude wave patterns, namely that they were due to the mechanism of QRA of planetary waves.

In ref. 34, the QRA mechanism is applied to quasistationary and circumglobal planetary waves, i.e., those that wrap around the planet in the midlatitudes. These waves are solutions to the full wave equation, including forcing, for integer values of the zonal wave number m (we will refer to them as m waves). This corresponds to the solutions of the forced oscillator, with m as the analog of the frequency of the external force. In addition, there are also solutions to the homogeneous wave equation (without forcing terms), i.e., free waves. These waves with zonal wave numbers k ≈ 6−8 (they can be noninteger; we will call these waves k waves) usually experience strong meridional dispersion, and, for that reason, their energy disperses rapidly. However, under specific conditions, two strongly reflecting points [the so-called turning points (TPs)] can emerge for these waves in the midlatitudes, at which the square of their meridional number, $l^2$, changes sign, with $l^2>0$ between the TPs. This leads to the development of a waveguide for these free waves, which traps them in the midlatitudes and prevents their strong dispersion, with the location of the southern and northern boundaries of the waveguide closely related to the position of the TPs. Planetary waves with these wave numbers thus become the favored free waves of the system, analogous to the natural frequency of an oscillator. Then, in cases where these favored free k-wave numbers are close to those of the m waves pushed by the forcing, i.e., if k is close to m, a resonance can arise resulting in large m-wave amplitudes, analogous to the forced oscillator when the frequency of the external force is close to the natural frequency (34).

Whether a waveguide develops for a particular wave number k depends on the latitudinal shape of the zonal mean midtroposphere zonal winds $\overline{u}$ in the extratropical atmosphere. The equations for Rossby waves (Calculation of the Meridional Wave Number, Physics of the Parameter, and Calculation of the Amplitudes) show that this can occur if a set of necessary conditions are met: $\overline{u}>0$ in the midlatitude region; the highest value of $l$ within the waveguide is in the range of the meridional wave numbers $l_m$ dominantly contributing to the external forcing with a given m, which provides closeness of the k waves to respective m waves not only in terms of the zonal but also the meridional wave numbers, favoring the QRA of the m waves; the total latitudinal width of the waveguide is no less than the characteristic spatial scale of the relevant Airy function (25), which is used as the boundary condition at its southern and northern boundaries; and latitudinal distribution of $l$ is sufficiently smooth in the waveguide, and both TPs lie within a midlatitude region of ∼ 25°N–30°N and ∼65°N−70°N, as the necessary condition for the application of quasilinear Wentzel−Kramers−Brillouin (WKB) method (25) when solving the equations for Rossby waves. The above necessary conditions for QRA are combined in ref. 34 in a set of four necessary conditions i−iv that are discussed in the main text and supporting information of that paper (see also Basic Necessary Conditions and Assumptions for details).

In this paper, we deal with the 15-d running means over May−September of the studied waves. In that case, the amplitude, ${\tilde{A}}_m^{Ort}$, of the m component of the midlatitude external forcing experiences generally noticeably higher variations in time, compared with those of monthly means investigated in ref. 34. In view of this, we pose here an additional necessary condition that the minimum value of ${\tilde{A}}_m^{Ort}$ should exceed some threshold number, ${\tilde{A}}_{m,\min }^{Ort}$, allowing us to use the quasilinear WKB method when solving the equations for the studied Rossby waves (see Additional Necessary Condition of the Minimum Amplitude).

In the 15-d running means of the m waves, we have situations (very rare, however) of coexistent, closely adjacent, or even overlapping in latitude, midlatitude waveguides, each satisfying all of the above-mentioned necessary conditions of QRA, with, however, low amplitudes of the observed m waves triggered within up to 2 wk. The analysis of the results shows that, in all those cases, the distance between the adjacent boundaries of the waveguides is lower than the characteristic spatial scale of the relevant Airy functions. This presumably violates a high reflection of the wave energy at these boundaries, causing, instead, a noticeable interaction, including the nonlinear interference, between the m waves trapped in the waveguides. We exclude all those cases from our consideration, posing an additional necessary condition on the minimum allowable distance between multiple coexisting waveguides (see Additional Necessary Condition Posed for details).

When all of the above necessary conditions for quasiresonance are met, then the QRA amplitude, ${\tilde{A}}_m$, for the meridional velocity in the midlatitude m wave at the EBL is described in dependence on the thermal and orographic forcing amplitude ${\tilde{A}}_m^{Ort}$ at this level (Calculation of the Amplitudes) by the following equation (see equation S14 in ref. 34):

$${\tilde{A}}_m=\frac{{\tilde{A}}_m^{Ort}}{{\left\{{\left[{(k/a)}^2-{(m/a)}^2\right]}^2+C_{1,m}^2{\left(L/a^2+R{o}^2/L\right)}^2{(m/a)}^2\right\}}^{1/2}}.$$ [1a]

In Eq. 1a, a is the Earth's radius, and $L$ and $Ro$ are, respectively, the characteristic Rossby radius and Rossby number for the eddies contributing effectively to atmospheric near-surface and internal “eddy friction” described by the second term in curly braces in the denominator of the right-hand side of Eq. 1a (Calculation of the Meridional Wave Number). A nondimensional parameter $C_{1,m}\ge 0$ in Eq. 1a represents the ratio of the zonally averaged module of the geostrophic wind at the top of the planetary boundary layer to that at the EBL, $\overline{u}$ (34) (see Physics of the Parameter for more detail).

As follows from Eq. 1a, in the absence of atmospheric friction, a classic resonance takes place, with ${\tilde{A}}_m\to \infty$ if $k\to m$, whereas, for realistic values of L, Ro, and $C_{1,m}$, ${\tilde{A}}_m$ reaches a high but finite quasiresonant value if $k=m$. This is the essence of the QRA mechanism of the planetary wave reinforcement proposed in ref. 34. QRA can be regarded as an extension of the Haurwitz-type mechanism (42) of a strong increase in the amplitude of the midlatitude atmospheric barotropic wave system response to stationary external barotropic thermal forcing, with a spatial frequency m approaching the natural stationary spatial frequency k of the wave system, to the case of external barotropic thermal and orographic forcing under a latitude-dependent $\overline{u}$ and an integer m over the midlatitude belt on the spherical Earth.

In the present paper, unlike ref. 34, in some rare cases, the QRA amplitudes ${\tilde{A}}_m$ of the 15-d running means calculated by Eq. 1a reached very high values about 10–15 m⋅s⁻¹. On the other hand, the theory of the wave breaking process in the middle and high atmosphere, which is the basic one limiting ${\tilde{A}}_m$ from above in the extratropics (see, e.g., ref. 43), requires substantially lower values of the amplitudes of the considered m waves. Following this requirement, we assign the maximum allowable value for ${\tilde{A}}_m$, $Ag\left(m\right)$, the latter being the amplitude of the m wave in the process of wave breaking (43) (see Amplitude of Wave Breaking), so that, if ${\tilde{A}}_m$ calculated by Eq. 1a exceeds $Ag\left(m\right)$, we put

$${\tilde{A}}_m=Ag\left(m\right).$$ [1b]

We note that this condition does not present an additional necessary condition of QRA but rather just limits the predicted wave amplitude in cases of resonance.

The theory of QRA, based on the equations for Rossby waves, thus allows us to specify the necessary conditions under which a QRA of planetary waves can occur. The explanatory power of QRA theory can only be established empirically, by comparing the times of resonance conditions and the estimated amplitudes to the high-amplitude waves actually observed. This will be done in this paper.

Spring-to-Autumn Weather Extremes 2012 and 2013 in the Context of QRA

In this paper, we extend the investigation of recent NH regional extremes in the framework of QRA to the years 2012 and 2013. These years are marked by a large number of strong regional extremes in the NH during May−September (see, e.g., refs. 44 and 45). We show that a considerable number of these extremes could have been favored by the QRA mechanism proposed in ref. 34, as was shown there for many July−August extremes observed during the last three decades. Fig. 1 displays the time series of a standard planetary wave diagnostic, namely the observed wave amplitudes ${\tilde{A}}_{m,obs}$ over May−September 2012 and 2013 of the 15-d running means for zonal wave numbers m = 6, m = 7, and m = 8 of the meridional wind velocity at 300 hPa averaged over the 37.5°N−57.5°N latitude range, diagnosed from daily NCEP-NCAR reanalysis data (41). As shown in ref. 34, 15-d running means of the studied waves meet the condition of quasistationarity, with the characteristic values of longitudinal phase velocity about 1–2 m⋅s⁻¹. As seen from Fig. 1, there occurred 17 high-peak-amplitude (HPA) events (see the set of filled and open circles in Fig. 1), with amplitudes exceeding 1.5 SDs above the 1980–2013 climatology. Of these events, nine occurred in 2012 and eight occurred in 2013; 13 of the 17 HPA events marked by filled circles in Fig. 1 occurred within up to 2 wk of QRA events, i.e., times during which all of the necessary conditions for QRA were satisfied and the amplitudes ${\tilde{A}}_m$, calculated with the use of Eqs. 1a and 1b (predicted QRA amplitudes), exceeded 1.5 SDs above the corresponding 1980−2013 climatology (see Fig. 1). These QRA amplitudes are marked by filled squares in Fig. 1. As seen from Fig. 1, ${\tilde{A}}_m$ are in a good agreement with the observed amplitudes ${\tilde{A}}_{m,obs}$ of respective HPA events.

Fig. 1.

Time series of the observed amplitudes (in meters per second) of zonal wave numbers m = 6 (black), m = 7 (red), and m = 8 (blue) for the 15-d running means of the meridional wind velocity at 300 hPa averaged over 37.5°N−57.5°N for May−September 2012 and 2013, based on daily reanalysis data (41). The filled circles designate the observed HPA events exceeding the 1.5 SD (dashed horizontal lines) above the 1980–2013 climatology (solid horizontal lines), when the QRA mechanism was at work within up to 1.5–2 wk time offset relative to respective HPA event. The amplitudes of these QRA events are marked by colored filled squares. The open circles denote the observed HPA events when the QRA mechanism was not in action. The open square marks the high amplitude of the QRA event for m = 7, when the corresponding QRA-predicted HPA event did not happen.

In 12 of these 13 cases, the quasiresonance conditions (QRA event) preceded or occurred substantially simultaneously with the respective HPA event (we call it a combined QRA−HPA event), supporting a causal link. According to refs. 44 and 45, severe NH midlatitude regional extremes occurred in connection with these QRA−HPA events.

A large number of HPA events over May−September (nine in 2012 and eight in 2013) for the waves with m = 6, 7, and 8 is typical for the years with the most severe extremes in the NH midlatitudes, e.g., 2003, 2006, and 2010, with 7, 10 and 11 HPA events, respectively (see www.pik–potsdam.de/∼petri/extr_2012_2013.html#movies, which presents time series of ${\tilde{A}}_{m,obs}$ for m = 4, 5, 6, 7, and 8 for May−September 1980–2013).

It is instructive to compare these numbers with those characteristic of a set of the years during 1979–2012 with no or only one major regional extreme event (in terms of land surface temperature and land precipitation anomalies) in the NH midlatitudes, from late April/early May to late September/early October, as reported yearly since 1993 in the World Meteorological Organization statements on the status of the global climate (see also ref. 38). According to www.pik–potsdam.de/∼petri/extr_2012_2013.html#movies, out of this set of 10 y, the year 1981 saw the largest number, four, of HPA events for the waves with m = 6, 7, and 8 in total, and only one of those four events was preceded by a QRA event. These years are thus clearly linked to a near-absence of quasiresonant wave amplification.

Fig. 2 gives an example of a QRA−HPA event that began as a QRA event with 10 August 2012 as the central date, followed, with a lag of 5 d, by an HPA event during the strong heat waves in the western and eastern United States, accompanied by severe flooding in the central United States that occurred simultaneously with destructive flooding in central China and severe droughts in eastern and western China (44). The Fig. 2A shows a map of this HPA event. Fig. 2B plots the curve of the nondimensional stationary wave number $K_s^2{a}^2=k^2+l^2{a}^2$ (left y axis; see Calculation of the Meridional Wave Number) for the QRA event. Any points of intersection of this curve with the horizontal line $k=const$ in Fig. 2 mark the latitudinal position of the TP for the corresponding k wave, so that, in the case of two TPs, their abscissa specify the latitudinal positions of the waveguide’s boundaries for the trapped k wave (Calculation of the Meridional Wave Number), provided the difference $K_s^2{a}^2-k^2$ is positive between the TPs and all of the above-mentioned necessary conditions for QRA are met within the waveguide. In that case, $K_s^2{a}^2-k^2$ yields the value of $l^2{a}^2>0$ within the waveguide (see Calculation of the Meridional Wave Number). Eq. 1a then gives the value of the QRA amplitude ${\tilde{A}}_m$ of the forced m wave with m close to k. This situation is indeed found for the central date shown in Fig. 2 within the waveguide for the free wave with $k\approx 7.05$. The QRA amplitude ${\tilde{A}}_7\approx (5.6\pm 0.8)$ m⋅s⁻¹ calculated by Eq. 1a with the use of ${\tilde{A}}_7^{Ort}$ derived from the temperature (41) and orography (46) data sets is close to the observed amplitude ${\tilde{A}}_{7,obs}\approx (5.4\pm 1.3)$ m⋅s⁻¹ reached in the following HPA event lagged by 5 d (see Fig. 1). Refer to Estimation of the Error Bars for a discussion of the error bars on these amplitudes. Notice that the observed amplitude of m = 7 for the central date of the QRA event (10 August 2012) was markedly lower than ${\tilde{A}}_7$ (see Fig. 1). This is characteristic of a number of the QRA−HPA events in May−September 2012 and 2013, where HPA event occurs about 5–15 d after the QRA event (in the 15-d running mean diagnostics used here).

Fig. 2.

The QRA event with 10 August 2012 as the central date, followed, with a lag of 5 d, by the observed HPA event during heat waves in the western and eastern United States and severe flooding in the central United States, accompanied by flooding in central China and droughts in eastern and western China (44). (A) Map (in meters per second) of the HPA event. (B) Nondimensional stationary wave number squared (the curve, left y axis) and resonance zonal wave number k (the straight line, right y axis) at the QRA event. All of the necessary conditions for the QRA event were met for the free wave with zonal wave number $k\approx 7.05$ within the midlatitude waveguide whose boundaries are marked by the vertical solid lines in B. The QRA amplitude ${\tilde{A}}_7\approx (5.6\pm 0.8)$ m⋅s⁻¹ of the forced m = 7 component matches well the observed ${\tilde{A}}_{7,obs}\approx (5.4\pm 1.3)$ m⋅s⁻¹ for the HPA event lagged by 5 d (see Fig. 1).

Our second example is the central European flooding in May−June 2013, which was one of the strongest NH regional extremes (47). Its causal chain can be traced from autumn 2012, which featured record-low sea ice cover in the Arctic. This situation persisted into the winter 2012/2013, with high temperatures in the Arctic and very cold northern continents (48). Late snowmelt over the region in late April/early May, followed by heavy rains in late May/early June, resulted in extremely high water levels in the Danube, Elbe, and Rhine, with coastal flooding in early/middle June there (49). A strong snowmelt in late April/early May and torrential rains in late May/early June could have been caused by the occurrence of persistent quasibarotropic high-amplitude QRA structures with zonal wave numbers m = 6 and m = 7 in the field of the NH midlatitude meridional velocity. The phase of these waves led to a northward flow over central Europe.

Fig. 3 illustrates one of these structures with m = 7 for 2 May 2013 as the central date of the QRA event. The HPA event peaked 1 d later. The necessary conditions for QRA were met for the quasistationary free wave with zonal wave number $k\approx 6.8$ within the midlatitude waveguide (see Fig. 3B). The QRA amplitude ${\tilde{A}}_7\approx (4.4\pm 0.6)$ m⋅s⁻¹, calculated by Eq. 1a with the use of the forcing amplitude ${\tilde{A}}_7^{Ort}$ derived from refs. 41 and 46 data sets, matches well the observed amplitude ${\tilde{A}}_{7,obs}\approx (4.5\pm 1.3)$ m⋅s⁻¹ for the following HPA event (see Fig. 1). We note that a similar situation has been observed again during the devastating flood in southeastern Europe in May 2014 (50).

Fig. 3.

The QRA event with 2 May 2013 as the central date during the catastrophic flood in central Europe. (A) Map (in meters per second) of the following observed HPA event for m = 7 with the central date shifted 1 d later. (B) Nondimensional stationary wave number squared (the curve, left y axis) and resonance zonal wave number k (the straight line, right y axis) at the QRA event. All of the necessary conditions for the QRA event were met for the free wave with zonal wave number $k\approx 6.8$ within the midlatitude waveguide whose boundaries are shown by the vertical solid lines in B. The QRA amplitude ${\tilde{A}}_7\approx (4.4\pm 0.6)$ m s⁻¹ matches well the observed ${\tilde{A}}_{7,obs}\approx (4.5\pm 1.3)$ m⋅s⁻¹ of the following HPA event (see Fig. 1).

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)

$$l^2=\frac{2\mathrm{\Omega }{\cos }^3\varphi }{a\stackrel{\_}{u}}-\frac{{\cos }^2\varphi }{a^2\overline{u}}\frac{d^2\overline{u}}{d{\varphi }^2}+\frac{\sin \varphi \cos \varphi }{a^2\stackrel{\_}{u}}\frac{d\overline{u}}{d\varphi }+\frac{1}{a^2}-{\left(\frac{k}{a}\right)}^2.$$ [S1]

In Eq. S1, $\mathrm{\Omega }$ is Earth’s rotation angular velocity, $a$ is Earth’s radius, $\varphi$ is the latitude, and $\overline{u}$ 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 $\overline{u}$.

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)

$$K_s^2{a}^2\equiv \tilde{\beta }\mathrm{ }{a}^2{\cos }^2\varphi /\overline{u}=\frac{2\mathrm{\Omega }a{\cos }^3\varphi }{\overline{u}}-\frac{{\cos }^2\varphi }{\overline{u}}\frac{d^2\overline{u}}{d{\varphi }^2}+\frac{\sin \varphi \cos \varphi }{\overline{u}}\frac{d\overline{u}}{d\varphi }+1,$$ [S2]

where $\tilde{\beta }$ is the meridional gradient of the absolute vorticity multiplied by $\cos \varphi$, 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 ${\mathrm{\Psi }}_m^{\prime }$ of the forced waves with m = 6, 7, and 8 (m waves) with nonzero right-hand side (forcing + eddy friction) yields (34)

$$\begin{array}{l}\tilde{u}\frac{\partial }{\partial x}\left(\frac{{\partial }^2{\mathrm{\Psi }}_m^{\prime }}{\partial x^2}+\frac{{\partial }^2{\mathrm{\Psi }}_m^{\prime }}{\partial y^2}\right)+\tilde{\beta }\frac{\partial {\mathrm{\Psi }}_m^{\prime }}{\partial x}=\frac{2\mathrm{\Omega }\sin \varphi {\cos }^2\varphi }{\tilde{T}}\tilde{u}\frac{\partial T_m^{\prime }}{\partial x}\\ -\frac{2\mathrm{\Omega }\sin \varphi {\cos }^2\varphi }{H}\kappa \tilde{u}\frac{\partial h_{or,m}}{\partial x}-\left(\frac{k_h}{a^2}+\frac{k_z}{H^2}\right)\left(\frac{{\partial }^2{\mathrm{\Psi }}_m^{\prime }}{\partial x^2}+\frac{{\partial }^2{\mathrm{\Psi }}_m^{\prime }}{\partial y^2}\right),\end{array}$$ [S3]

where $x=a\lambda$ and $y=a\ln [(1+\sin \varphi )/\cos \varphi ]$ are the coordinates of the Mercator projection of Earth’s sphere, with $\lambda$ as the longitude, $H$ is the characteristic value of the atmospheric density vertical scale, $\tilde{T}$ is a constant reference temperature at the EBL, $T_m^{\prime }$ is the m component of azonal temperature at this level, $\tilde{u}=\overline{u}/\cos \varphi$, $\kappa$ is the ratio of the zonally averaged module of the geostrophic wind at the top of the PBL to that at the EBL (53), $h_{or,m}$ 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 $k_h\approx \tilde{U}L$ and $k_z\approx \tilde{W}\tilde{H}\approx (\tilde{U}/L){\left(HRo\right)}^2$, in terms of the characteristic values of the horizontal, $\tilde{U}$, and vertical, $\tilde{W}$, velocities and horizontal, $L$, and vertical, $HRo$, 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 $\tilde{U}=\kappa \mathrm{ }\overline{u}$. 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.

Physics of the Parameter $C_{1,m}$ in the Denominator of Eq. 1a

Following ref. 53, we assign a nondimensional parameter $C_{1,m}\ge 0$ in Eq. 1a a value of κ (see Calculation of the Meridional Wave Number). Based on NCEP-NCAR reanalysis data (41), we prescribe κ the value of 0.31 ± 0.05 (see Estimation of the Error Bars for more details regarding estimation of the error bars for κ). We notice that the above value of this parameter is rather close to the value 0.4 ascribed in ref. 53 (see ref. 40 for some discussion on the issue). Let us note also that C_1,m is assigned a unit value in ref. 34, assuming that the atmospheric friction effectively acts throughout the whole troposphere and, specifically, just below the midlatitude tropopause. However, our results attest that this difference in the description of the atmospheric friction in ref. 34 and the present paper has but a minor influence on the values of the amplitudes ${\tilde{A}}_m$ of the resonance waves. This is due to a rather small contribution of the friction term to ${\tilde{A}}_m$ in the denominator of Eq. 1a, under the condition that ${\tilde{A}}_m$ should be less than the amplitude of the m wave, $Ag\left(m\right)$, in the wave breaking process assigned by Eq. 1b (see Amplitude of Wave Breaking).

Calculation of the Amplitudes ${\tilde{A}}_m^{Ort}$ of the External Forcing

Variable ${\tilde{A}}_m^{Ort}$ 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 $\mathrm{\Delta }$ = 37.5°−57.5°N. The values for ${\tilde{A}}_m^{Ort}$ 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 $l^2\le 0$ outside but in its close vicinities, with $\overline{u}>0$ 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:$\left|k-m\right|<C_m$, with a parameter $C_m$ estimated using the equation (see Eq. 1a)

$${\tilde{A}}_{m,e}=\frac{{\tilde{A}}_{m,SD}^{Ort}}{{\left\{{\left[C_m(2m+C_m)\right]}^2+C_{1,m}^2{\left(L/a+R{o}^{2}a/L\right)}^2{m}^2\right\}}^{1/2}},$$ [S4]

where ${\tilde{A}}_{m,e}$ and ${\tilde{A}}_{m,SD}^{Ort}$ are, respectively, the mean +1 SD value of ${\tilde{A}}_m$ and the maximum value of ${\tilde{A}}_m^{Ort}$, 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 ${\tilde{A}}_m$ of the considered m wave is still higher than ${\tilde{A}}_{m,e}$. Substitution of the relevant values of ${\tilde{A}}_{m,e}$, ${\tilde{A}}_{m,SD}^{Ort}$, $C_{1,m}\mathrm{ }$, $L$, and $Ro$ 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 ${\mathrm{\Lambda }}_{\varphi }=2\pi /l$ of the free wave trapped in the waveguide over a distance Λ_ϕ/(4−8)π is less than ${\mathrm{\Lambda }}_{\varphi }$ (25, 34). This implies the following additional necessary conditions:

Condition 3: $\left|d{l}^{-1}/ad\varphi \right|$ < 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 ${\mathrm{\Delta }}_A$ 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 $\left({\mathit{l}}_{\mathit{m}\mathit{i}\mathit{n}}{,\mathit{l}}_{\mathit{max}}\right)$ of the Values of $\mathit{l}$Within the Waveguide's Interior Ensuring Proximity of $\mathit{l}$to the Meridional Wave Numbers ${\mathit{l}}_{\mathit{m}}$ 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, $\mathrm{ }{l}_m$ 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 $\left[{(k/a)}^2-{(m/a)}^2\right]{\mathrm{ }}^2$ in its denominator. The sequential steps in the transition to the resonance equation with $\left[{(k/a)}^2-{(m/a)}^2\right]{\mathrm{ }}^2$ 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 $\mathrm{ }{l}_m$ to the total amplitude ${\tilde{A}}_m^{Ort}$ of the external forcing from a full set of the partial forcing waves with a given m, just ${\tilde{A}}_m^{Ort}$ can be used, to a first approximation, as the forcing in the right side of Eq. 1a for ${\tilde{A}}_m$. In so doing, ${\tilde{A}}_m^{Ort}$ is calculated within the indicated Calculation of the Amplitudes ${\tilde{A}}_m^{Ort}$ of the External Forcing section midlatitude belt $\mathrm{\Delta }$ 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, ${\tilde{A}}_m^{Ort}$ is described by the area-weighted sum of the contributions from all of the latitudinal strips entering $\mathrm{\Delta }$. 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 $l_{\min }\approx 0.25\cdot {10}^{-6}$ m⁻¹ < $l_m$ <$l_{\max }\approx 2.7\cdot {10}^{-6}$ m⁻¹ range. Just this ($l_{\min },l_{\max }$) interval specifies the possible range for the highest values of $\mathrm{ }l$ 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 ${\tilde{A}}_m^{Ort}$

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: ${\tilde{A}}_m^{Ort}$ is not lower than a certain threshold minimum value, ${\tilde{A}}_{m,\min }^{Ort}$.

This condition is a direct consequence of applying a quasilinear WKB method for the description of the QRA. For that, we require that ${\tilde{A}}_m^{Ort}$ 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 ${\tilde{A}}_{m,\min }^{Ort}$ 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

${\tilde{A}}_m\le {\tilde{A}}_{m,\max }={\tilde{A}}_{m,b}$, where ${\tilde{A}}_{m,b}$ is the amplitude reached by the planetary Rossby waves in the wave breaking process within the midlatitude belt $\mathrm{\Delta }$. The above process is the basic one limiting ${\tilde{A}}_m$ in the midlatitudes (43). To obtain the equation for ${\tilde{A}}_{m,b}$, 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, ${\mathrm{\Delta }}_{QRA}^{\mathrm{int}}$ (34), with $\varphi =\varphi {\mathrm{ }}_0$ as the central latitude (43)

$$C_{gm}^{y}A=\frac{\rho (m/a)l_0}{K^2}{\overline{u}}_0\frac{\overline{q^{\prime 2}}\mathrm{ }}{\overline{q_y}}.$$ [S5]

In [S5], $\rho$ and $C_{gm}^y$ are, respectively, the air density and the m-wave group velocity in the meridional direction (25, 43), $q\prime$ is the quasigeostrophic wave potential vorticity, and $\overline{q_y}$ is the zonal mean gradient of potential vorticity, all four variables being calculated for $\varphi =\varphi {\mathrm{ }}_0$ at the EBL; ${\overline{u}}_0$ and $l_0$ in [S5] are, respectively, $\overline{u}$ and $l$ at $\varphi =\varphi {\mathrm{ }}_0$, and $K^2$ reads (43)

$$K^2={(m/a)}^2+l_0^2+{(f_0/N_0)}^2(1/4H^2),$$ [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 $\varphi =\varphi {\mathrm{ }}_0$. According to ref. 43, the following relations result for saturated (breaking) planetary waves:

$$\overline{\left|{q\prime }_y\right|}=\overline{\left|l_{0}q\prime \right|}={\overline{q}}_y.$$ [S7]

Substitution of [S7] into [S5] yields

$$C_{gm}^{y}A=\frac{\rho (m/a)}{K^2}{\left(\overline{q^{\prime 2}}\right)}^{1/2}.$$ [S8]

On the other hand, from the dispersion relation for the m waves in the WKB approximation (25, 43), we can write, within ${\mathrm{\Delta }}_{QRA}^{\mathrm{int}}$,

$$\overline{q_y}=K^2{\overline{u}}_0,$$ [S9]

so that [S5] can be rewritten as follows:

$$C_{gm}^{y}A=\frac{\rho (m/a)\mathrm{ }\mathrm{ }}{l_0}{\overline{u}}_0^2.$$ [S10]

Equating [S8] and [S10], we have

$${\left(\overline{q^{\prime 2}}\right)}^{1/2}=\frac{\overline{u}{\mathrm{ }}_0{K}^2}{l{\mathrm{ }}_0}.$$ [S11]

For the considered quasibarotropic m waves, $\overline{q^{\prime 2}}$ in [S11] reads (cf. ref. 43)

$$\overline{q^{\prime 2}}={\overline{{(\partial {\mathrm{ }}^2\mathrm{\Psi }\prime /\partial \mathrm{ }{x}^2+\partial {\mathrm{ }}^2\mathrm{\Psi }\prime /\partial \mathrm{ }{y}^2)}^2}|}_{\varphi ={\varphi }_0}$$ [S12]

within ${\mathrm{\Delta }}_{QRA}^{\mathrm{int}}$. In [S12], $\mathrm{\Psi }\prime$ is the corresponding perturbation stream function over the total waveguide’s width, ${\mathrm{\Delta }}_{QRA}$ (34). Following ref. 34, consider $\mathrm{\Psi }\prime$ given by

$$\mathrm{\Psi }\prime ={\mathrm{\Psi }}_0\sin \frac{m}{a}x\cos \mathrm{ }{l}_m(y-y_0).$$ [S13]

In [S13], $\mathrm{ }{l}_m\approx l_0$, and $y_0$ corresponds to the latitude $\varphi \mathrm{ }{\mathrm{ }}_0$ (34). Applying respective equations for the meridional, ${v\prime }_m$, and zonal, ${u\prime }_m$, velocities

$$\begin{array}{l}{v\prime }_m=\partial \mathrm{\Psi }\prime /\partial \mathrm{ }x\\ {u\prime }_m=-\partial \mathrm{\Psi }\prime /\partial \mathrm{ }y\end{array}$$ [S14]

in the m wave and using the identity

$$\overline{{\sin }^2\frac{m}{a}x}=\overline{{\cos }^2\frac{m}{a}x}=\frac{1}{2}$$ [S15]

that is valid for any integer m and x ranging over the $\left[0,\mathrm{ }2\pi a\right]$ interval in a Mercator projection of the sphere, the equation for $\overline{{v\prime }_m^2}$ yields, in the case of the quasiresonant m wave breaking within ${\mathrm{\Delta }}_{QRA}$,

$$\overline{{v\prime }_m^2}\left(y\right)=\overline{{v\prime }_{m,b}^2}\left(y\right)=\overline{{v\prime }_{m,b}^2}\left(y_0\right)\cos \mathrm{ }{\mathrm{ }}^2\mathrm{ }l{\mathrm{ }}_0(y-y_0),$$ [S16]

where $\overline{{v\prime }_{m,b}^2}\left(y_0\right)$ is given by

$$\overline{{v\prime }_{m,b}^2}\left(y_0\right)=\frac{K^4}{{\left[K^2+l_0^2+l_0^4/{(m/a)}^2\right]}^2}\frac{K^2}{l_0^2}{\overline{u}}_0^2.$$ [S17]

Then, on the strength of ${\tilde{A}}_{m,b}^2\left(y_0\right)=2\overline{{v\prime }_{m,b}^2}\left(y_0\right)$, the latitudinal averaging of [S17] over the ${\mathrm{\Delta }}_{QRA}$ range results in the following estimate of the maximum allowable value, ${\tilde{A}}_{m,b}^2$(${\mathrm{\Delta }}_{QRA}$), for the amplitude of the 15-d-mean m component of the meridional velocity at the EBL over ${\mathrm{\Delta }}_{QRA}$,

$${\tilde{A}}_{m,b}^2\left({\mathrm{\Delta }}_{\text{QRA}}\right)=\frac{2K^4}{{[(K^2+l_0^2+l_0^4)/{(m/a)}^2]}^2}\frac{K^2}{l_0^2}{\overline{u}}_0^2{\langle {\cos }^2{l}_0(y-y_0)\rangle }_{QRA},$$ [S18]

where ${\langle X\rangle }_{QRA}$ stands for the latitudinal averaging of X over ${\mathrm{\Delta }}_{QRA}$. Accounting for the characteristic spatial scale ${\mathrm{\Delta }}_A$ of the relevant Airy function (see Basic Necessary Conditions and Assumptions) at the waveguide’s boundaries, the averaging of [S18] over $\mathrm{\Delta }$ yields the following estimate for ${\tilde{A}}_{m,\max }$, at typical values of l₀ ≈ (0.3−0.5) × 10⁻⁶ m⁻², ${\mathrm{\Delta }}_{QRA}/\mathrm{\Delta }$ ≈ 0.3−0.5, and ${\mathrm{\Delta }}_A/\mathrm{\Delta }$ ≈ 0.2−0.25:

$${\tilde{A}}_{m,\max }\approx \frac{K^2}{[(K^2+l_0^2+l_0^4)/{(m/a)}^2]}\frac{K}{l_0}{\overline{u}}_0\frac{{\mathrm{\Delta }}_{QRA}}{2\mathrm{ }\mathrm{\Delta }}.$$ [S19]

Finally, substituting relevant values of $f_0$, $N_0$, and $H$ into [S6] and using [S19] results in ${\tilde{A}}_{m,\max }$ ≈ (0.3−0.7)${\overline{u}}_0$ for the considered m waves with m = 6, 7, and 8. The indicated variation in the values of ${\tilde{A}}_{m,\max }$ 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, ${\mathrm{\Delta }}_{QRA}^{\mathrm{int}}$.

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 ${\tilde{A}}_m$ given in QRA Mechanism of Planetary Wave Reinforcement, in the case when ${\tilde{A}}_m$ calculated by Eq. 1a exceeds ${\tilde{A}}_{m,\max }$.

Estimation of the Error Bars for the Values of ${\tilde{A}}_{m,obs}$ and ${\tilde{A}}_m$ in the HPA Events

The peak values of the amplitudes ${\tilde{A}}_{m,obs}$ 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 ${\tilde{A}}_{m,obs}$ over 1980–2013 as a representative measure of the error bars for this quantity.

The QRA amplitudes ${\tilde{A}}_m$ (see Eq. 1a) are calculated here with the use of the amplitudes ${\tilde{A}}_m^{Ort}$ 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 ${\overline{u}}_{PBL}$ at the top of the planetary boundary layer to that at the EBL $\overline{u}$ enters the denominator of Eq. 1, via the parameter κ. All these input data are subject to some uncertainties. Assuming that $\left|{\overline{u}}_{PBL}\right|$ and $\left|\overline{u}\right|$ 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):

$$\delta \left(s\right)=\frac{{\sigma }_A^2{\mu }_B+{\sigma }_B^2{\mu }_{A}s}{\sqrt{2\pi }{({\sigma }_A^2+{\sigma }_B^2{s}^2)}^{3/2}}\exp \left[-\frac{{({\mu }_A-{\mu }_{B}s)}^2}{2({\sigma }_A^2+{\sigma }_B^2{s}^2)}\right],$$ [S20]

where ${\sigma }_A$ and ${\sigma }_B$ are, respectively, 1 SD for $\left|{\overline{u}}_{PBL}\right|$ and $\left|\overline{u}\right|$, and ${\mu }_A$ and ${\mu }_B$ are their means over the 1980–2013 time interval. Substitution of corresponding values ${\sigma }_A$, ${\sigma }_B$, ${\mu }_A$, and ${\mu }_B$ in [S19] allows one to estimate $\delta \left(s\right)$ for different values of $s$ ($\kappa$, in our case), yielding, finally, $\kappa =0.31\pm 0.05$. At the second step, we calculate, in the same manner, the value of ${\tilde{A}}_m$, substituting, in the numerator of Eq. 1, ${\tilde{A}}_m^{Ort}$ with the above-mentioned estimation of the error bars for this quantity, and $\kappa =0.31\pm 0.05$ 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.