1. Introduction
The sudden stratospheric warming (SSW), or simply sudden warming, refers to the phenomenon of an abrupt temperature rise by several tens of kelvins in a short period in the high latitudes of the stratosphere, and, in most extreme cases, a reversal of zonal-mean westerly winds associated with the stratospheric polar night jet [see Butler et al. (2015) and references therein]. Discovered in 1952 by the German scientist Richard Scherhag when analyzing radiosonde data; it has been linked to other atmospheric phenomena such as the quasi-biennial oscillation (QBO) (Gray et al. 2004; Charyulu et al. 2007), North Atlantic Oscillation (NAO) [particularly the negative NAO phase (Baldwin and Dunkerton 2001)], and ENSO teleconnection (Ineson and Scaife 2009; Butler et al. 2014), and has been evidenced to impact the Atlantic storm track (Thompson et al. 2002), equatorial tropospheric convective activity (Kodera 2006), Arctic and Antarctic ozone variability (Schoeberl and Hartmann 1991), transport of tropospheric CO2 and pollutants (Jiang et al. 2013), North Atlantic Ocean circulation (Reichler et al. 2012), and tropospheric planetary and synoptic-scale eddies (Hitchcock and Simpson 2016), to name a few. It may exert an effect on our daily lives by warming Greenland, eastern Canada, and southern Eurasia and bringing about extreme cold air outbreaks in parts of North America and Eurasia (e.g., Thompson et al. 2002; Nath et al. 2016).
An attempt to give the SSW an unambiguous definition turns out to be challenging. For over 60 years, scientists have been endeavoring to define and characterize it but still have not reached an agreement. In their comprehensive review, Butler et al. (2015) remarked that a well-accepted definition, and hence classification, is “at best ambiguous and at worst nonexistent.” Usually referred to in the literature are major warming and minor warming, plus additional ones, such as Canadian warming. [Note, here we should distinguish SSW and stratospheric final warming (SFW); the latter is characteristic of the breakdown of the polar vortices (Black et al. 2006; Black and McDaniel 2007; Sheshadri et al. 2014)]. In such characterizing, a significant rise in temperature is generally required, but in recent decades zonal wind reversal has been the dominant basis for the definition of major warmings (e.g., McInturff 1978). These discrepancies have led to different SSW identifications. For example, application of the seven different definitions described in Butler et al. (2015) to the 1958–2014 NCEP Reanalysis data results in 26–46 major SSWs, or 0.46–0.8 per winter. In this study, it is not our intention to run into the debate of the definition; we will focus on one major warming event, namely, the December 2012–January 2013 SSW, which has been unanimously identified by all the seven definitions in Butler et al. (2015). We want particularly to investigate how the temperature is abruptly increased.
Since the discovery of the SSW event, much effort has been invested in explaining its generating mechanism. Classically it is believed that SSWs are due to the interaction of the upward-propagating planetary waves (Charney and Drazin 1961; Dickinson 1968) with the zonal winds. Specifically, the waves from the troposphere act to decelerate the polar night jet, giving rise to the distortion/breakdown of the polar vortex (Matsuno 1970, 1971). This wave–mean flow interaction is illustrated in the semispectral model of Holton (1976, 1980); it has been further studied by Robinson (1985, 1988) and evidenced in other studies, such as that of Harada et al. (2010).
On the other hand, Trenberth (1973) found that the nonzonal heating may also account for a weaker westerly jet and a considerable warming in the polar night stratosphere, while Sjoberg and Birner (2012) emphasized the role of transient forcing and, particularly, the scale of transient forcing. They found that the frequency of SSW occurrences drops as the temporal forcing scale is reduced.
Apart from the external formation mechanisms, in another line of work it is suggested that SSWs may have intrinsic origins; they can occur without precursor tropospheric pulse of planetary wave energy. The self-tuned resonance mechanism (Plumb 1981; McIntyre 1982; Dritschel and McIntyre 2008; Esler and Matthewman 2011; Matthewman and Esler 2011; Albers and Birner 2014) and catastrophe theory (Chao 1985) are such examples. Plumb (1981) pointed out that the temporal growth could result from the resonance between the stationary waves and the slowing-down progressive waves and that the SSWs with the polar vortex displaced and those with the polar vortex split must have quite different generating mechanisms. To understand how nature may select the mechanism(s) for the SSW formation, dynamical diagnostics make an important methodology. Particularly, the Lorenz energy cycle diagnostics prove to be a powerful approach (Lorenz 1955). In this regard, the energetics were first studied by Reed et al. (1963), and this was followed by Julian and Labitzke (1965), Perry (1967), and Trenberth (1973), among others. These multiscale energetics studies, however, are all global in that the resulting energetics are averaged over the domain of concern, without distinguishing between spatial locations. The limitation of global energetics has long been recognized, and there has been a long-lasting effort to overcome this by introducing local energetics studies, such as those of Holopainen (1978), Plumb (1983), and, recently, Liang and Robinson (2005), Murakami (2011), and Liang (2016). [Particularly, the empirical method of Murakami (2011) was applied by Zuo et al. (2012) to study the January 2009 sudden warming event.]
The difficulty of energetics analysis lies in the following two aspects. First, atmospheric processes tend to occur locally in time; an SSW event spans a short period in the year and is not stationary even during that period. The time-mean decomposition essentially cannot have nonstationary processes appropriately separated. An alternative is to take zonal mean, but that invokes another issue: that is, a loss of spatial localization in longitude. Besides, atmospheric processes usually involve more than just two ranges of scales. Now a common practice is to rely on filters to achieve the decomposition. However, how filtered fields can be used to express multiscale energy and energetic terms (any quadratic properties) is by no means trivial; actually, it is a profound problem in functional analysis. This is because filtered fields are reconstructions in physical space, while multiscale energy is a concept in phase space that is related to physical energy through the renowned Parseval relation (cf. Liang and Anderson 2007). The second difficulty is the relaxation of the global integration/average from the Lorenz-type energetic terms. It has long been argued that the thus-obtained energetics, particularly the energy transfers between the mean and eddy fields that are most important in the energy cycle diagnostics, are ambiguous (Holopainen 1978, Plumb 1983). Furthermore, the widely used transfer (i.e., the energy extraction via Reynolds stress against basic profile) does not yield the expected diagnosis with benchmark problems (Liang and Robinson 2007); see the following section for more details.
Recently it has been found that the above two difficulties actually can be overcome in a unified approach [see Liang (2016) for a review] within the framework of a newly developed functional analysis apparatus called multiscale window transform (MWT; Liang and Anderson 2007). Liang and Anderson realized that, for some specially devised orthogonal filters, a filtered field has correspondingly a transform coefficient, just like that in Fourier transform and inverse Fourier transform. It has been proved that the transform coefficient then can be combined to represent the multiscale energetics. Using this, Liang (2016) rigorously proved, through a reconstruction of some atomlike quantities, that the transfer processes can be unambiguously separated from the intertwined nonlinear transports. The resulting transfers bear a Lie bracket form, just like the Poisson bracket in Hamiltonian mechanics. This formalism, which was first proposed in Liang and Robinson (2005) in a half-empirical way (rigorously proved later on), has led to a new diagnostic methodology, namely, the localized multiscale energy and vorticity analysis (MS-EVA). MS-EVA has been applied with success in many real oceanic and atmospheric diagnoses and engineering problems, such as wake control (e.g., Liang and Wang 2004). See section 2 for a brief introduction.
We will apply the new multiscale energetics formalism to diagnose the SSW processes. We choose a particular case (i.e., the December 2012–January 2013 SSW case) for this purpose. This SSW has been identified by all the seven definitions listed in Butler et al. (2015); it is very special in that, in the course of warming, the polar vortex is not only displaced but also split. Since 1980, there are only three cases (1985/86, 1987/88, and 2012/13) that are like this one (Liu and Zhang 2014; Nath et al. 2016). Besides, this event has an extraordinarily long duration. According to Nath et al. (2016), it lasts for more than 38 days, greatly exceeding the climatological mean. For this reason, there have been many studies with this case (e.g., Liu and Zhang 2014; Tripathi et al. 2016; Coy et al. 2015; De Wit et al. 2015; Taguchi 2016; Nath et al. 2016; Attard et al. 2016). A faithful energetics diagnosis is expected to help us gain more understanding of this unusual event.
In the following, we first briefly introduce the MWT-based multiscale energetics analysis (i.e., MS-EVA), and then the data that will be used (section 3). The MS-EVA is set up in section 4, and sections 5 and 6 provide the analysis results. This study is summarized in section 7.
2. A brief introduction of multiscale window transform and localized multiscale energetics
As we see, it is a rather profound problem to have the local energy of a time-dependent filtered field faithfully represented. In fact, this issue has just been addressed in the development of MWT (Liang and Anderson 2007), with the aid of the established connection between filter banks and wavelets (Strang and Nguyen 1997).
MWT is a functional analysis tool that decomposes a function space into a direct sum of orthogonal subspaces, each with an exclusive range of scales (in time or in space, depending on the problem in question), while preserving its local properties. Such a subspace is termed a scale window, or simply a window. MWT is developed for a faithful representation of the multiscale energies on the resulting scale windows and hence make multiscale energetics analysis possible. This is a feature lacked in the traditional filters, the outputs of which are fields in physical space, while multiscale energy is a concept in phase space that is related to its physical space counterpart through the Parseval equality. Liang and Anderson (2007) realized that, just as in the Fourier transform and inverse Fourier transform, there exists a transfer-reconstruction pair for a class of specially devised orthogonal filters. This pair is the very MWT and its peer [i.e., multiscale window reconstruction (MWR)]. Loosely speaking, the MWR of a series S(t) results in a filtered series, while the corresponding MWT coefficients can give the energy of that filtered series. For a brief introduction, see the section 2 of Liang (2016).
In MWT, a scale window is bounded below and above by two scale levels. For a series with a time duration of τ, a scale level j corresponds to a period 2−jτ. The time steps of the series hence need to total to a number of the power of 2. In this study, as discussed in section 4, we impose three scale windows that characterize, respectively, the background fields, the fields on the scales of SSW events, and the field on smaller scales. They are between scale levels 0–j0, j0–j1, and j1–j2. For the sake of easy reference, we will denote them with 
Given a time series [S(t)], application of MWT yields the MWT coefficient, which we will write as 
Multiscale energetic terms (m2 s−3) in Eqs. (6) and (7). If total energetics (W) are to be computed, the resulting integrals with respect to (x, y, and p) should be divided by g. Besides, all terms are to be multiplied by 
The MS-EVA Eqs. (6) and (7) are thus fundamentally different from the classical ones. By collecting the MS-EVA terms, the energetic processes can be classified into four categories: energy transport (flux divergence), canonical transfer, buoyancy conversion, and dissipation/diffusion. Interestingly, the first three are all in some conservative form: a transport vanishes if integrated over a closed domain, a canonical transfer vanishes if summarized over windows and locations, while a buoyancy conversion mediates between KE and APE within each individual window. Figure 1 schematizes these processes with a three-window decomposition.
Energy flowchart for a three-window decomposition. The superscripts 0, 1, and 2 stand for the mean, SSW, and synoptic-scale windows, respectively. The values 
Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-17-0002.1
Energy flowchart for a three-window decomposition. The superscripts 0, 1, and 2 stand for the mean, SSW, and synoptic-scale windows, respectively. The values
Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-17-0002.1
Energy flowchart for a three-window decomposition. The superscripts 0, 1, and 2 stand for the mean, SSW, and synoptic-scale windows, respectively. The values
Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-17-0002.1
3. Data
The data we will be using are the reanalysis product ERA-Interim provided by the European Centre for Medium-Range Weather Forecasts (ECMWF), including temperature, geopotential, and wind (u, υ, ω). The time resolution is 6 h. Vertically there are 37 levels, from 1000 to 1 hPa. In the horizontal direction, we will use a resolution of 2° × 2° for the sake of computational economy. We have also tried the 1° × 1° resolution, and the results are similar.
4. MS-EVA setup
To set up the MS-EVA, we first need to determine the scale window bounds. This is achieved through wavelet spectral analysis. Choose a series of temperature at the North Pole spanning April 2010–November 2015 with a time step of 6 h. We remove the mean over the duration and consider only the temperature anomaly. This totals to 213 = 8192 data points. (MWT requires that the number of time steps be a power of 2.) This series has the period of concern (January 2013) lying in the middle, with two ends far away enough to avoid the possible boundary effect.
The wavelet power spectrum [with respect to the orthonormal basis built by Liang and Anderson (2007)] is shown in Fig. 2. Also shown is the time series. From the spectrum, clearly there are two peaks. One is at j = 2, corresponding to a period of 365.2 days, which is the very annual signal. Another corresponds to the SSW event, j = 5–8 (11.4–91.3 days). We have also tried Fourier spectral analysis, but only the annual signal stands out; the SSW signal is mostly disguised. This is a very good example against using Fourier analysis for nonstationary signals.
(top) Wavelet spectrum for the (bottom) 10-hPa polar temperature time series. The mean has been removed prior to the spectral analysis.
Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-17-0002.1
(top) Wavelet spectrum for the (bottom) 10-hPa polar temperature time series. The mean has been removed prior to the spectral analysis.
Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-17-0002.1
(top) Wavelet spectrum for the (bottom) 10-hPa polar temperature time series. The mean has been removed prior to the spectral analysis.
Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-17-0002.1
Based on the above results, we identify three bounds j0 = 4, j1 = 8, and j2 = jmax = 13. They divide the spectrum into three scale windows, which we will refer to as, respectively, mean window, sudden-warming-scale window (or SSW window), and synoptic-scale window. From this, the SSW window contains signals with periods from 16 to 256 days. The mean-scale, SSW-scale, and synoptic-scale reconstructions of the series are shown in Figs. 3a–c. Clearly we can see the annual cycle in Fig. 3a and the sudden warming events in Fig. 3b. Figure 3c shows the temperature variabilities associated with the synoptic eddies.
(a) The temperature anomaly T (in blue; with the mean over 23 Apr 2010– 30 Nov 2015 removed) at the North Pole and 10 hPa from 23 Apr 2010 to 30 Nov 2015, and its mean-scale reconstruction (in red), (b) SSW-scale reconstruction, and (c) synoptic-scale reconstruction. The scale window bounds are referred to in the text. (d) The zoom of the above in the period 1 Dec 2012–9 Feb 2013 with the blue solid line being the T in (a), the blue dashed line the T in (b), and the red line the u wind component at 10 hPa.
Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-17-0002.1
(a) The temperature anomaly T (in blue; with the mean over 23 Apr 2010– 30 Nov 2015 removed) at the North Pole and 10 hPa from 23 Apr 2010 to 30 Nov 2015, and its mean-scale reconstruction (in red), (b) SSW-scale reconstruction, and (c) synoptic-scale reconstruction. The scale window bounds are referred to in the text. (d) The zoom of the above in the period 1 Dec 2012–9 Feb 2013 with the blue solid line being the T in (a), the blue dashed line the T in (b), and the red line the u wind component at 10 hPa.
Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-17-0002.1
(a) The temperature anomaly T (in blue; with the mean over 23 Apr 2010– 30 Nov 2015 removed) at the North Pole and 10 hPa from 23 Apr 2010 to 30 Nov 2015, and its mean-scale reconstruction (in red), (b) SSW-scale reconstruction, and (c) synoptic-scale reconstruction. The scale window bounds are referred to in the text. (d) The zoom of the above in the period 1 Dec 2012–9 Feb 2013 with the blue solid line being the T in (a), the blue dashed line the T in (b), and the red line the u wind component at 10 hPa.
Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-17-0002.1
5. The 2013 SSW fields and their reconstructions
a. A brief description of the original fields
In Fig. 3, we show the temperature evolutions at the North Pole. Figure 3d is the zoom of the period 1 December 2012–9 February 2013. Clearly, in early January 2013, the expected low temperature is replaced by a high temperature (the blue solid line in the figure). The warming is so rapid that the temperature anomaly is increased from −25 K on 1 January to 25 K (comparable to the highest of the year) on 6 January. During 7–12 January, it oscillates within a small range. After that, it begins to decline. Besides this major warming, from the figure we see there exist actually two minor warmings early in December and around 24 December. The zonal winds change correspondingly. We look at the 10-hPa wind at 84°N, 180°. In Fig. 3d, a positive value indicates a westerly wind. One can see that, during the period of the sudden warming, the zonal wind is weakened greatly. By 7 January, the westerly wind at 84°N, 180° has essentially changed to an easterly wind.
The evolution of the spatial distribution is shown in Fig. 4. Early in December 2012, the polar region at 10 hPa is occupied by a large-scale cold center, which lies more in the Atlantic Ocean sector. A warm center first appears over the Eurasian continent. As time goes by, it moves poleward; in the meantime, the cold center moves toward the Western Hemisphere. As of 29 December, the warm center reaches 60°N. It develops rapidly in the following days, covering Siberia and the region to the north. The cold center appears in the form of a comma, pushed by the developing warming center to North America, making a dipolar structure in the polar region. As the warm center is pushed northward, the cold center in the Western Hemisphere is split into two halves. By 9 January, the temperature field in the polar region is characterized by two pairs of cold–warm dipoles. The cold centers lie over Alaska and Atlantic–western Europe, while the warm centers are over the European continent and Canada. Afterward, the Eurasian warm center gets weakened and moves toward Canada, which is eventually merged with the warm center on 21 January. Clearly, this process is very special in that the polar vortex is not just displaced but also split into two and then four vortices.
Temperature anomaly (K) at 10 hPa in the Northern Hemisphere; (a),(b),(c): 10, 21, and 29 Dec 2012; (d),(e): 9 and 13 Jan 2013; and (f) 14 Feb 2013.
Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-17-0002.1
Temperature anomaly (K) at 10 hPa in the Northern Hemisphere; (a),(b),(c): 10, 21, and 29 Dec 2012; (d),(e): 9 and 13 Jan 2013; and (f) 14 Feb 2013.
Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-17-0002.1
Temperature anomaly (K) at 10 hPa in the Northern Hemisphere; (a),(b),(c): 10, 21, and 29 Dec 2012; (d),(e): 9 and 13 Jan 2013; and (f) 14 Feb 2013.
Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-17-0002.1
b. Multiscale reconstructions
The multiscale reconstructions of the temperature, wind, and geopotential fields on the three scale windows, especially on the sudden warming window, allow us to visualize better the warming event. To get a feeling of this, Fig. 5 shows a snapshot of the decomposition of the 10-hPa temperature in the Northern Hemisphere on 4 January 2013. Obviously, the mean or background field T~0 (Fig. 5b) is characterized by a single cold center, and the SSW part T~1 is characterized by a single warm center (Fig. 5c). Note on the T~0 field the polar vortex is not centered at the pole; in boreal winter, it sits more over the Atlantic side because of the Aleutian high (e.g., O’Neill et al. 2015).
(a) The 10-hPa temperature anomaly (K) on 4 Jan 2013, and (b) its reconstructions in the mean window, (c) SSW-scale window, and (d) synoptic-scale window.
Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-17-0002.1
(a) The 10-hPa temperature anomaly (K) on 4 Jan 2013, and (b) its reconstructions in the mean window, (c) SSW-scale window, and (d) synoptic-scale window.
Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-17-0002.1
(a) The 10-hPa temperature anomaly (K) on 4 Jan 2013, and (b) its reconstructions in the mean window, (c) SSW-scale window, and (d) synoptic-scale window.
Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-17-0002.1
The decomposed temperature fields at 10 hPa evolve differently in different windows. In the mean-scale window there is only a slowly varying cold polar vortex, which lies over Greenland. This feature is just like that in Fig. 5b and hence is not shown here. The varying background implies that the warming process is nonstationary.
In the sudden-warming-scale window, as shown in Fig. 6, a warm center first appears on 21 December, which moves poleward as time goes by, and then extends throughout the whole polar region. This process culminates on 12–14 January, after which the warm center moves to the Western Hemisphere. Prior to the arrival of the warming early in December, on the SSW scale there appears an obvious cold center over the Eurasian continent (Fig. 6a). If we examine the temperature reconstruction in this scale window for a year without SSW, generally there are no such strong centers. This shows that, during a warming period, some places may actually experience a cooling. This cooling phenomenon has also been identified prior to many other warming events over the past 35 years from the 10-hPa temperature time series at locations (say, 60°N, 180°). A careful discussion, however, is left to future studies.
As in Fig. 4, but for the 10-hPa SSW-scale temperature (K).
Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-17-0002.1
As in Fig. 4, but for the 10-hPa SSW-scale temperature (K).
Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-17-0002.1
As in Fig. 4, but for the 10-hPa SSW-scale temperature (K).
Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-17-0002.1
The evolution of the 10-hPa SSW-scale zonal wind is shown in Fig. 7. In late December on the SSW window, the midstratosphere is still controlled by a westerly wind anomaly. But an easterly wind anomaly, though weak, begins to develop over South Asia and Canada. The easterly wind anomaly is strengthened rapidly in early January and soon takes over the region north of 60°N. It completely replaces the polar westerly wind anomaly on 7 January.
Time evolution of the 10-hPa SSW-scale zonal wind (m s−1): (a) 20 and (b) 30 Dec 2012, and (c) 7 and (d) 14 Jan 2013.
Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-17-0002.1
Time evolution of the 10-hPa SSW-scale zonal wind (m s−1): (a) 20 and (b) 30 Dec 2012, and (c) 7 and (d) 14 Jan 2013.
Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-17-0002.1
Time evolution of the 10-hPa SSW-scale zonal wind (m s−1): (a) 20 and (b) 30 Dec 2012, and (c) 7 and (d) 14 Jan 2013.
Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-17-0002.1
Shown in Fig. 8 is the 10-hPa geopotential field on the SSW-scale window. Its evolution is consistent with the SSW-scale temperature. Prior to the sudden warming, there is a negative center over the polar region. On 14 December, a positive center first appears over Canada. This steers the polar vortex toward Siberia. Afterward, the polar vortex grows and migrates toward Greenland (8–18 January) and then becomes gradually weakened. The whole process comes to an end in late February and then completely disappears on the SSW window (until the next warming event).
Time evolution of the 10-hPa SSW-scale geopotential (J kg−1): (a) 14 Dec 2012, (b) 7 and (c) 14 Jan 2013, and (d) 17 Feb 2013.
Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-17-0002.1
Time evolution of the 10-hPa SSW-scale geopotential (J kg−1): (a) 14 Dec 2012, (b) 7 and (c) 14 Jan 2013, and (d) 17 Feb 2013.
Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-17-0002.1
Time evolution of the 10-hPa SSW-scale geopotential (J kg−1): (a) 14 Dec 2012, (b) 7 and (c) 14 Jan 2013, and (d) 17 Feb 2013.
Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-17-0002.1
6. Energetics
a. Sudden-warming-scale energy
Generally, on the SSW-scale window, KE and APE have distributions similar to zonal wind and temperature, respectively (not shown). To show their vertical distribution evolutions, we integrate them on the pressure levels over the whole Northern Hemisphere and plot the results in Fig. 9 (the integration is over the spherical surface with the meridional weight taken into account). As shown in Fig. 9a, the SSW-scale APE is limited in December–January above 100 hPa, and mostly above 30 hPa. In contrast, besides the maxima in the vertical, the SSW-scale KE has, at 50–5 hPa, three peaks in its evolution: one in early December, one in late December, and the strongest in January.
Time–pressure distributions of the horizontally integrated (over the whole Northern Hemisphere) SSW-scale (a) APE and (b) KE.
Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-17-0002.1
Time–pressure distributions of the horizontally integrated (over the whole Northern Hemisphere) SSW-scale (a) APE and (b) KE.
Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-17-0002.1
Time–pressure distributions of the horizontally integrated (over the whole Northern Hemisphere) SSW-scale (a) APE and (b) KE.
Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-17-0002.1
b. Multiscale energy cycles
1) SSW-scale energetic balance
To better understand the processes underlying the warming event, we integrate the energetic terms over the Northern Hemisphere from 60° to 84°N, and from 100 to 10 hPa. (We do not choose 90°N since the pole is a singular point.) The vertical integration is with respect to pressure, and hence the final result should be divided by g to ensure that the energetics have the units of energy change rate [see Liang (2016) for details]. We have also tried the integration from 30° to 84°N; the resulting bulk energetics are similar.
Notice that the sudden warming is essentially about the APE burst on the SSW window; we hence pay special attention to the SSW window energetics. Figure 10 displays the time evolution of the bulk SSW-scale energetic terms during the period 1 December 2012–9 February 2013. For comparison, superimposed is a time series of the SSW-scale temperature at the pole averaged over 60°–84°N, 100–10 hPa (blue dashed line). From Fig. 10 (top panel), the APE balance on the SSW window is mainly among buoyancy conversion, transport, and baroclinic canonical transfer. For the SSW-scale KE (middle panel), the balance is among pressure work, buoyancy conversion, and barotropic canonical transfer. The residuals account for the dissipation and other unresolved processes and are generally small. Note that throughout the duration the time rates of change of APE and KE are well correlated (bottom panel of Fig. 10). Besides, the former is in phase with b1, indicating the importance of buoyancy conversion. From the figure, the averaged SSW-scale polar temperature generally follows the variation of 
The balance among the (top) APE and (middle) KE energetics (W) on the SSW-scale window integrated from 60° to 84°N and from 100 to 10 hPa: (top) 
Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-17-0002.1
The balance among the (top) APE and (middle) KE energetics (W) on the SSW-scale window integrated from 60° to 84°N and from 100 to 10 hPa: (top)
Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-17-0002.1
The balance among the (top) APE and (middle) KE energetics (W) on the SSW-scale window integrated from 60° to 84°N and from 100 to 10 hPa: (top)
Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-17-0002.1
2) Energy flow paths in different stages
We use the above polar temperature time series to divide the duration 1 December 2012–9 February 2013 into four subperiods: 1–27 December, 28 December–10 January, 11–25 January, and the period after 25 January. Accordingly, the sudden warming period is partitioned into three stages, namely, precursor stage (1–27 December), rapid warming stage (28 December–10 January), maintaining stage (11–25 January), and decaying stage (after 25 January). In these stages the underlying dynamics are quite different. In the precursor stage, there are two minor short-period warmings (which are more evident at 30 hPa): one between 1 and 7 December and another between 20 and 27 December. This has also been documented in the literature (e.g., Attard et al. 2016; Nath et al. 2016). The SSW-scale APE balances for both warmings are among buoyancy conversion, canonical transfer, and transport. But the energy flow paths are different; hence, two substages may be further distinguished. We choose the following days: 3 December (precursor stage, first warming), 22 December (precursor stage, second warming), 4 January (rapid warming stage), 17 January (maintenance stage), and 8 February, which marks the end of the event, to draw the energy flow charts. The results are shown in Fig. 11.
The typical multiscale energy cycles (109 W) for (a),(b) the precursor stage (3 and 22 Dec, respectively); (c) rapid warming stage (4 Jan); (d) maintaining stage (17 Jan); and (e) decaying stage (8 Feb).
Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-17-0002.1
The typical multiscale energy cycles (109 W) for (a),(b) the precursor stage (3 and 22 Dec, respectively); (c) rapid warming stage (4 Jan); (d) maintaining stage (17 Jan); and (e) decaying stage (8 Feb).
Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-17-0002.1
The typical multiscale energy cycles (109 W) for (a),(b) the precursor stage (3 and 22 Dec, respectively); (c) rapid warming stage (4 Jan); (d) maintaining stage (17 Jan); and (e) decaying stage (8 Feb).
Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-17-0002.1
We first examine the precursor stage. The first warming (Fig. 11a) results from the collaboration of the APE transport 
The rapid warming stage is between late December and early January. Figure 11c depicts the energy flow on a typical day, 4 January. Here the warming is mainly the resultant effect of the canonical transfer from the mean-scale APE reservoir through baroclinic instability, the SSW-scale APE transport, and the buoyancy conversion on the SSW-scale window. This is similar to the first minor warming, except for their different relative strengths and a reversed canonical transfer between the SSW- and synoptic-scale windows. Here 
Followed by the rapid warming stage is the stage of maintenance (Fig. 11d). Compared to the rapid warming stage, the energetic scenario is completely different. Now the balance of the SSW-scale APE is among the following four terms: 
If we trace further the origin of the SSW-scale KE, we will find that it has two major sources. Recall that the SSW-scale KE has stored a part of the converted SSW-scale APE earlier on, so the first source is actually the SSW-scale APE itself. The second major source is the kinetic energy reservoir on the mean-scale window. From Fig. 11d it is clear that, in this stage, there is a very strong barotropic instability that extracts the energy from K0 to fuel K1, which is then instantaneously converted to A1. By the numbers given in the figure, the second source is by far the most important. Since K0 is mainly supplied through 
The final stage is the decaying stage. For example, on 8 February, the energetics are very similar to that in Fig. 11e, but the canonical transfers and buoyancy conversions are all nearly zero.
3) Horizontal distributions of the interscale transfer and buoyancy conversion
One of the advantages of the MS-EVA is that it can reveal the spatiotemporal structure of the energetics. Here we integrate the transfers 
(a) The canonical transfer of APE (109 W) from the mean window to the SSW-scale window on a typical day of the rapid warming stage (4 Jan). (b) As in (a), but for KE on a day of the maintenance stage (17 Jan). (c),(d) The buoyancy conversions on the above two days, respectively.
Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-17-0002.1
(a) The canonical transfer of APE (109 W) from the mean window to the SSW-scale window on a typical day of the rapid warming stage (4 Jan). (b) As in (a), but for KE on a day of the maintenance stage (17 Jan). (c),(d) The buoyancy conversions on the above two days, respectively.
Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-17-0002.1
(a) The canonical transfer of APE (109 W) from the mean window to the SSW-scale window on a typical day of the rapid warming stage (4 Jan). (b) As in (a), but for KE on a day of the maintenance stage (17 Jan). (c),(d) The buoyancy conversions on the above two days, respectively.
Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-17-0002.1
In this figure, 4 January is a typical day when 
4) More about the pressure work and energy transports on the SSW window
With the analysis above, we have seen that b1, 
To summarize, the major SSW in December 2012–January 2013 may be divided into three stages. In the rapid warming stage, because of the second precursor stratospheric warming in 20–28 December, part of the energy is stored in A1, plus the poleward heat transport and the canonical transfers through the baroclinic instabilities in the polar region, which cause A1, and hence the SSW-scale temperature, to grow explosively. A part of the increased A1 is converted into K1, the SSW-scale KE, and hence causes the polar stratospheric circulation to change, resulting in a weak westerly and a strengthened easterly. This makes the rapid warming stage. In the second stage, the system acquires K1 via barotropic instability. This together with the energy stored in K1 earlier on is converted back to A1 through the positive buoyancy conversion over Greenland and East Siberia. The buoyancy conversion collaborates with the canonical baroclinic transfer from the mean-scale window (through baroclinic instability) to increase A1, maintaining the warming to an appreciable extent. [We remark that this stage dependence of dynamical processes has also been observed in other phenomena, such as atmospheric blockings (Ma and Liang 2017).] It is important to note that positive buoyancy conversion does not always exist throughout the polar region, while the warming is much more uniformly distributed, though originally it appears only over the Eurasian continent. So how is energy transported from one place to another place? To see this, we draw in Fig. 13 the horizontal vectors of the SSW-scale KE and APE transports. Clearly, after the warming starts, both 
The horizontal SSW-scale (top) KE and (bottom) APE flux on 17 Jan 2013.
Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-17-0002.1
The horizontal SSW-scale (top) KE and (bottom) APE flux on 17 Jan 2013.
Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-17-0002.1
The horizontal SSW-scale (top) KE and (bottom) APE flux on 17 Jan 2013.
Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-17-0002.1
7. Conclusions
The December 2012–January 2013 sudden stratospheric warming (SSW) lasts for an extraordinarily long time. It is special in that, prior to the major warming, there exist two minor warmings in December 2012; moreover, the polar vortex is not only displaced, but also split. This study uses a recently developed tool, multiscale window transform (MWT), and the MWT-based localized multiscale energy and vorticity analysis (MS-EVA). The fields are reconstructed on three orthogonal subspaces or scale windows [i.e., mean window, sudden warming window (or SSW window), and synoptic-scale window]. The warming event is much clearer in the SSW-scale reconstructions than in the original fields. Particularly, on the SSW-scale window, the temperature evolution appears as an almost solitary warming center around the North Pole, in contrast to the dipolar or multipolar pattern in the original temperature maps.
We denote the multiscale available potential energy (APE) and kinetic energy (KE) as, respectively, 
The strong poleward heat flux is also seen in other energetics studies, such as Julian and Labitzke (1965). Note that sometimes a meridional eddy heat flux may result from an upward-propagating planetary wave. But in this case this flux cannot be due to mechanisms that originated in the troposphere. We examined the vertical integrals of the energetics from 100 to 10 hPa and found that the vertical component of the pressure work is small and, moreover, is downward. That is to say, the heat flux can only be due to processes within the stratosphere, among which meridional advection must play a role.
In the next stage (11–25 January; i.e., the stage of maintenance) the mechanism for the warming is completely different; the SSW-scale APE is from the SSW-scale KE, or K1. Here, K1 includes three parts: 1) the previously converted energy stored in K1, 2) the energy newly acquired through pressure work, and, 3) most importantly, that released from the mean-scale window through a strong barotropic instability over Alaska. Since the mean-scale KE reservoir is mainly from the pressure work on this window and the pressure work has a large vertical component, the energy for the warming in this stage should be from the lower atmosphere.
In the decay stage, the energy flow takes a path similar to the maintenance stage, but now the canonical transfers and buoyancy conversions are all nearly turned off. Accordingly, the system gradually resumes its normal state.
To summarize, the above processes are schematized in Fig. 14. Of particular interest are the reversal of the buoyancy conversion and the appearance of the barotropic instability in the stage of the maintenance. Besides, the poleward SSW-scale heat flux and the upward pressure flux also distinguish the two stages. We remark that the dynamical scenario in the rapid warming stage is consistent with an intrinsic mechanism (e.g., the self-tuned resonance theory) but excludes the mechanism of upward planetary wave driving because the buoyancy conversion on the SSW-scale window is from APE to KE. This is in contrast to the stage of maintenance, when the scenario admits the classical theory of mean flow–wave interaction with the upward-propagating waves. This study provides for the first time, an example that the two completely different types of generating mechanisms proposed so far—i.e., the interaction with the upward-propagating waves (Charney and Drazin 1961; Matsuno 1970) and the intrinsic mechanisms, such as self-tuned resonance (Plumb 1981; McIntyre 1982; Dritschel and McIntyre 2008; Esler and Matthewman 2011; Matthewman and Esler 2011; Albers and Birner 2014)—might actually work together to drive the same event. Some mechanisms, such as the strong barotropic instability over Alaska–Canada and the backward conversion of the previously converted SSW-scale APE, among others, are also first seen. These results, though obtained with an individual case, may help to trace the origins of the SSWs and build up our knowledge of this important dynamical phenomenon.
Schematic of the major processes that lead to the January 2013 sudden stratospheric warming. (a) In the stage of rapid warming (28 Dec–10 Jan), the temperature rise is mainly due to a strong poleward heat flux and a canonical transfer through baroclinic instability, which extracts APE from the mean window. In the meantime, a portion of the SSW-scale APE is converted to the SSW-scale KE. (b) In the stage of maintenance (11–25 Jan), the previously converted energy is converted back; a strong barotropic instability transfers the mean-scale KE to the SSW-scale KE, which is also converted to the SSW-scale APE. The mean-scale KE is mostly brought upward from the troposphere by pressure work.
Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-17-0002.1
Schematic of the major processes that lead to the January 2013 sudden stratospheric warming. (a) In the stage of rapid warming (28 Dec–10 Jan), the temperature rise is mainly due to a strong poleward heat flux and a canonical transfer through baroclinic instability, which extracts APE from the mean window. In the meantime, a portion of the SSW-scale APE is converted to the SSW-scale KE. (b) In the stage of maintenance (11–25 Jan), the previously converted energy is converted back; a strong barotropic instability transfers the mean-scale KE to the SSW-scale KE, which is also converted to the SSW-scale APE. The mean-scale KE is mostly brought upward from the troposphere by pressure work.
Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-17-0002.1
Schematic of the major processes that lead to the January 2013 sudden stratospheric warming. (a) In the stage of rapid warming (28 Dec–10 Jan), the temperature rise is mainly due to a strong poleward heat flux and a canonical transfer through baroclinic instability, which extracts APE from the mean window. In the meantime, a portion of the SSW-scale APE is converted to the SSW-scale KE. (b) In the stage of maintenance (11–25 Jan), the previously converted energy is converted back; a strong barotropic instability transfers the mean-scale KE to the SSW-scale KE, which is also converted to the SSW-scale APE. The mean-scale KE is mostly brought upward from the troposphere by pressure work.
Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-17-0002.1
Several issues remain. First, one naturally may wonder how the above analysis could help to improve SSW prediction. A possible approach is, based on the energetic flow, to identify the important precursor regions; as an example, Garfinkel and Waugh (2014) suggested the importance of the North Pacific. Second, the 2012/13 SSW might not be representative in that it involves both vortex splits and vortex displacements, and that could be the reason why both mechanisms coexist in this single event. To gain a general understanding of the multiscale energetics underlying a typical SSW, an MS-EVA analysis of all the SSWs, followed by a composite analysis, is needed. These problems, among others, will be explored in future studies.
Acknowledgments
The comments and suggestions from three anonymous referees are sincerely appreciated. We thank ECMWF for making available the ERA-Interim product. This work was partially supported by the 2015 Jiangsu Program of Entrepreneurship and Innovation Group, by the National Science Foundation of China (NSFC) under Grant 41276032, by the National Program on Global Change and Air–Sea Interaction (GASI-IPOVAI-06), and by the Jiangsu Chair Professorship to X.S.L.
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