1. Introduction
Atmospheric blocking is a nearly stationary process that lasts for more than a week during which the zonal jet stream is interrupted and split into two branches. Although difficult to define in an exact way (Lejenäs and Økland 1983), it is generally characterized by 1) a double-jet system extending over at least 45° of longitude and 2) a pattern persistent over one region for 5–30 days (e.g., Berggren et al. 1949; Rex 1950b; Austin 1980; Treidl et al. 1981; Charney et al. 1981; Tibaldi and Molteni 1990). It causes the affected regions to have the same weather for an extended period of time, typically with more precipitation in the south and less at higher latitudes (Trigo et al. 2004; Sousa et al. 2016). Its influence on the ambient weather and climate has been extensively investigated (e.g., Namias 1947; Berggren et al. 1949; Rex 1950a,b); this has made it one of the most studied processes in the atmospheric sciences.
As we know, the typical lifetime of synoptic eddies is no more than one week, while a substantial proportion of blocking episodes last for much longer. Besides, synoptic eddies always propagate eastward in the westerlies, while blockings are almost stationary and even retrograde. What is more, blockings have relative large amplitudes. These imply that nonlinearity may be important. Nonlinear theoretical models have been proposed (Charney and DeVore 1979; McWilliams 1980; Shutts 1983; Yamazaki and Itoh 2013; Luo et al. 2014), among which are, particularly, the model of equivalent modons (McWilliams 1980), the soliton model (Huang et al. 2007; Lou and Huang 2017), and the multiscale interaction model (Luo et al. 2014). While indeed these theories have been successful in their respective contexts, what nature really chooses is not quite clear. For this reason, many other scientists choose to diagnose the processes for an understanding of them. This includes the diagnoses of momentum flux (Green 1977; Austin 1980), heat flux (Austin 1980; Illari and Marshall 1983), potential vorticity (Illari 1984), and geopotential height (Tsou and Smith 1990), to name a few. In this study, we focus on the diagnostics of multiscale energetics.
Ever since Lorenz (1955) introduced the concept of energy cycle, multiscale energetic diagnosis has become a powerful tool for atmospheric research. For the blocking study, this has been extensively used. For example, Holopainen and Fortelius (1987) discuss the energy transfer between the high-frequency (less than 6 days) eddies and the 10-day mean flow during blocking and identify an enhanced transfer of eddy kinetic energy (KE) to the mean flow over the storm tracks in the blocking case. Hansen and Chen (1982) investigate two blockings using spectral energetics analysis, one occurring over Atlantic and another over Pacific, and find that the nonlinear interaction between the cyclone-scale and planetary-scale waves is essential to the Atlantic blocking, while baroclinic amplification plays the most important role in the formation of the Pacific one. Hansen and Sutera (1984) demonstrate that the critical difference in energetics between blocking and nonblocking periods lies in the nonlinear interaction term. However, zonal spectral analyses are inconclusive because they cannot distinguish between processes near to or far from the block. For this reason, Fournier (2002, 2003) uses orthogonal wavelet analysis to investigate the local information of KE transfer between different scales during a blocking process and finds that the KE transfer is upscale from the eddies in the upstream of the blocking and downscale in the downstream.
Since the center of a blocking may be on the move and its strength changes with time, information local in both space and time is needed to explain its evolution. Among the methods as we mention above in previous studies, only the orthonormal wavelet analysis can give local information. However, given its multiresolution nature, in a discrete wavelet domain how the moving pattern in space–time can be handled is still unclear. Besides, a blocking generally involves many different scales. A single wavelet scale is not enough to describe it; to remedy, Fournier (2005) sums over some designated individual scales to construct a physically meaningful mode that bears resemblance to blocking. But even so, if the block moves significantly then this reconstruction is still unsatisfactory. The underlying problem is that “moving” as a concept in a continuous space–time implies spurious changes in a discrete orthonormal wavelet domain [cf. Fournier (2000), his Eq. (D4)].
We hence need a methodology that is capable of handling the blocking process, which is in nature highly nonlinear, multiscale interactive, and localized in space and time. The localized multiscale energy and vorticity analysis (MS-EVA) recently developed by Liang and Robinson (2005, 2007) is just such a methodology. MS-EVA is based on a new functional analysis apparatus called multiscale window transform (MWT), which may be simply understood as an orthogonal decomposition according to scale ranges, while has the local information retained. Our purpose here is, using the MS-EVA and the MS-EVA-based theories, to investigate the dynamical processes underlying the evolution—including the onset, development, and decay—of the blockings. Some studies (Nakamura et al. 1997; Schneidereit et al. 2012; Park et al. 2015; Pelly and Hoskins 2003; Colucci 1985) show that both synoptic and low-frequency processes play important roles in the blocking evolution, so a three-scale separation will be performed. We will focus specifically on the blockings over the Atlantic Ocean, since the Atlantic is found to be one of the preferred blocking regions (Lejenäs and Økland 1983). In the following we will first give a brief introduction of MWT, MS-EVA, and the MS-EVA-based theory of barotropic and baroclinic instabilities. Section 3 is devoted to a description of the composited blocking and the reconstructed signals on the multiscale windows, and sections 4 and 5 provide an analysis of the resulting multiscale energetics. Some issues are discussed in section 6. The whole study is concluded in section 7.
2. Data and method
a. Data
The ERA-40 (http://apps.ecmwf.int/datasets/data/era40-daily/levtype=pl/) dataset (Uppala et al. 2005) will be used for our study. It includes temperature (T), wind components (u, υ, ω), and geopotential (Φ) from the European Centre for Medium-Range Weather Forecasts (ECMWF). For our purpose, we choose a time resolution of 6 h and a spatial resolution of 2.5° × 2.5°. The spatial domain covers the zonal circle between 30° and 87.5°N and 15 standard p levels from 1000 to 50 hPa. As it is required in the analysis that the number of the time steps should be a power of 2, the data period is chosen to be 1 September 1957–21 May 2002, which results in series with 216 time steps.
b. Blocking index
c. MWT, canonical transfer, and localized MS-EVA
The methodology for this study is the localized MS-EVA (Liang and Robinson 2005) [see Liang (2016) for a systematic introduction]. It is based on a new functional analysis tool (MWT; Liang and Anderson 2007), with which one can decompose a function space into a direct sum of orthogonal subspaces, each with an exclusive range of scales, while preserving its local properties. Each subspace is spanned by wavelet bases (can be any orthonormal bases; e.g., Fournier 2002) with the designated scales and is termed a scale window, or simply a window. In this study, we need three-scale windows, a basic-flow window, and a blocking window, plus a window of synoptic eddies. For convenience, write them as 
Given a series u(t), take the MWT of u and we obtain two types of quantities: the transform coefficients, 
Multiscale energetics for the blocking circulation.
Note that all the energetic terms in Eqs. (1) and (2) are localized both in space and in time; in other words, they are all four-dimensional field variables, distinguished notably from the classical Reynolds decomposition or Fourier analysis-based formalisms in which localization is lost in at least one dimension of space–time to achieve the scale decomposition. Processes localized in space and time are thus naturally embedded here. A schematic of the energy flow is shown in Fig. 1.
The energy processes in a three-scale window decomposition. The numbers 0, 1, and 2 indicate the basic-flow window, blocking window, and synoptic window, respectively. The symbols are referred to in Table 1.
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0295.1
The energy processes in a three-scale window decomposition. The numbers 0, 1, and 2 indicate the basic-flow window, blocking window, and synoptic window, respectively. The symbols are referred to in Table 1.
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0295.1
The energy processes in a three-scale window decomposition. The numbers 0, 1, and 2 indicate the basic-flow window, blocking window, and synoptic window, respectively. The symbols are referred to in Table 1.
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0295.1
In Eqs. (1) and (2), the canonical transfer terms are 
a flow is locally unstable if
for an unstable system, if
for an unstable system, if
if both
3. Blocking composition and multiscale geopotential reconstruction
a. Blocking composition
Considering that a blocking must be stationary at one longitude and extend to a relatively large region, only the episodes that meet the TM index criterion in at least a longitudinal range of 12.5°, as suggested by Barriopedro et al. (2006), and simultaneously last for more than 4 days (Tibaldi and Molteni 1990; Pelly and Hoskins 2003), are regarded as blocking. The blocking characteristics change with season, and Barriopedro et al. (2006) find that the long-lasting blocking episodes with larger extension and intensity prevail in winter over the ocean. So only the events occurring in December, January, and February (DJF) are analyzed in this study. The result is shown in Fig. 2, from which we see that most blocking episodes (45 in total) exist at 7.5°W; it is found that the mechanisms controlling the blockings are different for different longitudes; we hence only focus on this longitude for the composition.
Number of blocking episodes during the study period vs longitude.
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0295.1
Number of blocking episodes during the study period vs longitude.
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0295.1
Number of blocking episodes during the study period vs longitude.
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0295.1
Since GHGS and GSGN are the geopotential height gradients of a blocking in its southern part and northern part, the difference of these two can be used to measure the blocking strength. With this, we pick the strongest day of every blocking episode, average the geopotential on all the strongest days, and take the average as the composite geopotential for blocking day 0. Likewise, we may obtain the composite geopotential for blocking day …, −6, −4, 0, 4, 6, …. The output of the composition is shown in Fig. 3. It is clear that the blocking commences on day −4, intensifies on day −2, culminates on day 0, and weakens and decays on days 2 and 4. This is the typical omega-type blocking. The stationary feature is also clear, as the center of the high almost stays in the same region during the evolution. The composite blocking lasts about 10 days, consistent with the typical lifetime.
The composite blocking over the Atlantic from (top left to bottom right) day −10 to day 8. Contoured is the geopotential (m2 s−2) at 300 hPa.
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0295.1
The composite blocking over the Atlantic from (top left to bottom right) day −10 to day 8. Contoured is the geopotential (m2 s−2) at 300 hPa.
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0295.1
The composite blocking over the Atlantic from (top left to bottom right) day −10 to day 8. Contoured is the geopotential (m2 s−2) at 300 hPa.
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0295.1
b. Multiscale window reconstruction of the 300-hPa geopotential field
As shown by Sawyer (1970), the maximum amplitude of the 15–60-day low-frequency disturbances corresponds well with the preferred location of blocking. We hence focus on the 8–64-day scale to include this scale range, or scale window, and exclude synoptic eddies with typical lifetime less than about one week in this study. We use the MWT developed by Liang and Anderson (2007) to reconstruct the geopotential on three different scale windows—namely, a basic-flow window containing scales above 64 days, a blocking window containing scales from 8 to 64 days, and a synoptic window with eddies of scales less than 8 days.
With the filtered fields, we repeat the composition as in the preceding section. The composite basic-flow-scale and blocking-scale geopotential are shown in Figs. 4a and 4b. The basic flow shows as a ridge with a southwest–northeast tilting; it does not change much with time. On the blocking scale, a clear high cell is found and it propagates westward during the period. As the basic flow rarely varies and the synoptic eddies have small magnitudes, we can safely say that it is the blocking-scale window that contributes most to the geopotential variability during the blocking. We hence focus on the analysis on this scale window.
The composite 300-hPa geopotential on (a) the basic-flow-scale window and (b) the blocking-scale window from day −14 to day 16 (m2 s−2), and (c) the Hovmöller diagram of the above-blocking-scale geopotential at 60°N.
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0295.1
The composite 300-hPa geopotential on (a) the basic-flow-scale window and (b) the blocking-scale window from day −14 to day 16 (m2 s−2), and (c) the Hovmöller diagram of the above-blocking-scale geopotential at 60°N.
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0295.1
The composite 300-hPa geopotential on (a) the basic-flow-scale window and (b) the blocking-scale window from day −14 to day 16 (m2 s−2), and (c) the Hovmöller diagram of the above-blocking-scale geopotential at 60°N.
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0295.1
The high cell on the blocking-scale window originates on day −12 over middle Eurasia, with a center at about 60°–110°E. After generation, it propagates westward, as is clear in the Hovmöller diagram of the 300-hPa geopotential on the blocking window (Fig. 4c). Overall, the lifetime of this high cell is about 24 days. A similar signal has also been identified over Euro-Atlantic by Michelangeli and Vautard (1998) through compositing the truncated planetary-scale anomalies before the onset of blocking and over North Pacific by Kushnir (1987) and Branstator (1987) through projecting the geopotential into its first principal component.
4. Energetics of the blocking signal
Kushnir (1987) investigates the evolution of the spatially averaged energy (over half of the Northern Hemisphere) of the 10–40-day wintertime low-frequency disturbances over Pacific Ocean. While it has led to understanding of the phenomenon, atmospheric blockings are local and on the move, which is beyond the capability of a spatially averaged analysis methodology. In this study, the MS-EVA is utilized to approach the problem. As introduced previously, MS-EVA will yield the multiscale energetics in a 4D fashion. We then track the high-pressure center and integrate the energetics around the center. Shown in the following are the deviations of the energy budget terms from their wintertime means (averaged over December–February during the periods of 1957–2002; the mean part is referred to in the appendix), considering that the high pressure cell is only one part of the signal on the 8–64-day window, and that the effect of other parts should be taken out. Kushnir (1987) adopts a similar procedure; he takes deviations of energy budget terms from their 10-yr means. Statistical significance should be tested for the obtained composite energetics. This turns out to be difficult, as the ensemble is small. Nonetheless, Fournier (2003) manages to develop a practical method to tackle the problem. We hence follow his approach to fulfill the test. Note in the following, when we say a value is “significant,” we mean “it is significant by the standard set in Fournier (2003)”; it may be different from the significance test in the usual sense.
a. KE evolution
To see how the blocking as a whole evolves, we average the KE energetics over the blocking region. It is found that the blocking-scale KE is almost located in the regions with positive blocking-scale geopotential (Fig. 5a presents such an example); the integration/averaging of the energetics is hence over a domain with the positive composite Φ~1 (i.e., Φ~1 ≥ 0 m2 s−2) throughout all the pressure levels. But, note that these regions may involve some processes other than the blocking. This is fixed by excluding the regions west of 30°E during the onset phase and those east of 30°E during the decay phase. The averaging domains defined by other standards (Φ~1 ≥ 100 m2 s−2 and Φ~1 ≥ 200 m2 s−2) have also been used to test the sensitivity of energetics calculation. The outcome (not shown) is not sensitive to the choice of averaging region domain. From Fig. 5b, the KE of the high cell begins to increase from blocking day −8, maximizes on day 0, then becomes on decline. Obviously, during the KE evolution, the energy balance varies. In the beginning, the system mainly gains KE from the pressure work (
(a) The blocking-scale KE (color shaded; m2 s−2) and geopotential (contoured; m2 s−2) on day 0 averaged over 45°–80°N. (b) Evolution of the blocking-scale KE (m2 s−2) and KE energetics (10−5 m2 s−3) averaged over the blocking high pressure cell: −∇⋅
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0295.1
(a) The blocking-scale KE (color shaded; m2 s−2) and geopotential (contoured; m2 s−2) on day 0 averaged over 45°–80°N. (b) Evolution of the blocking-scale KE (m2 s−2) and KE energetics (10−5 m2 s−3) averaged over the blocking high pressure cell: −∇⋅
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0295.1
(a) The blocking-scale KE (color shaded; m2 s−2) and geopotential (contoured; m2 s−2) on day 0 averaged over 45°–80°N. (b) Evolution of the blocking-scale KE (m2 s−2) and KE energetics (10−5 m2 s−3) averaged over the blocking high pressure cell: −∇⋅
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0295.1
Energetics averaged over the three distinct phases: (a) onset, (b) amplification, and (c) decay. The thickness of the arrows is proportional to the magnitude of every term (10−5 m2 s−3). The superscripts 0, 1, and 2 represent the basic-flow, blocking, and synoptic windows. The orange shading means that the corresponding term is significant by Fournier’s test.
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0295.1
Energetics averaged over the three distinct phases: (a) onset, (b) amplification, and (c) decay. The thickness of the arrows is proportional to the magnitude of every term (10−5 m2 s−3). The superscripts 0, 1, and 2 represent the basic-flow, blocking, and synoptic windows. The orange shading means that the corresponding term is significant by Fournier’s test.
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0295.1
Energetics averaged over the three distinct phases: (a) onset, (b) amplification, and (c) decay. The thickness of the arrows is proportional to the magnitude of every term (10−5 m2 s−3). The superscripts 0, 1, and 2 represent the basic-flow, blocking, and synoptic windows. The orange shading means that the corresponding term is significant by Fournier’s test.
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0295.1
1) Onset phase
In the onset phase, most of the KE comes from pressure work (Figs. 5b and 6a). Its time-mean zonal section distribution is shown in Fig. 7a. Positive pressure work occupies most of the central part of the high-pressure cell. The secondary contribution in this phase is the canonical transfer from the synoptic eddies (Figs. 5b and 6a); shown in Fig. 7b is its distribution. An observation is that canonical transfer from the synoptic scale takes effect mainly between 500 and 200 hPa, while pressure work exists through almost all the levels. In the interior of the cell, pressure work plays the major role; on the western and eastern sides, canonical transfer from the synoptic scale contributes positively to the blocking. Obviously, the two processes play a vital role in blocking onset.
(a) Time-mean (over the onset phase) zonal–vertical sectional distribution of the blocking-scale pressure work rate −∇⋅
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0295.1
(a) Time-mean (over the onset phase) zonal–vertical sectional distribution of the blocking-scale pressure work rate −∇⋅
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0295.1
(a) Time-mean (over the onset phase) zonal–vertical sectional distribution of the blocking-scale pressure work rate −∇⋅
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0295.1
2) Amplification phase
After the onset, the KE keeps increasing, particularly from day −4 to day 0. The KE budget in Fig. 5b clearly tells that the canonical KE transfer dominates this period; next to it are KE transport and pressure work (Fig. 5b). The spatial distribution of the canonical KE transfer from day −4 to day 0 is displayed in Fig. 8a; also displayed is the KE transport and pressure work (Figs. 8b and 8c). These processes are most active through 500–200 hPa. From the figure, the KE transfer from the synoptic-scale window takes place around 50°–25°W and 10°W–60°E, which are both positive. In contrast, the KE transport on the blocking-scale window can be positive and negative, with the positive center around 25°W–10°E and the negative center around 50°–25°W. For pressure work, it is essentially negative through the region 25°W–60°E. We hence have the following KE energetic scenario on the blocking window: on the western side of the high-pressure cell, the positive canonical transfer from the synoptic window is mainly balanced by a negative KE transport, while on the eastern side of and within the cell, the balance is between the canonical transfer, the KE transport, and the pressure work.
As in Fig. 7, except that the color shading represents (a) the canonical transfer from the synoptic-scale window to the blocking window (
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0295.1
As in Fig. 7, except that the color shading represents (a) the canonical transfer from the synoptic-scale window to the blocking window (
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0295.1
As in Fig. 7, except that the color shading represents (a) the canonical transfer from the synoptic-scale window to the blocking window (
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0295.1
3) Decay phase
In Fig. 4b, we observe a fast retrograding process of the signal in the decay phase after day 0. The center lies at around 7.5°W on day 0 and moves to 75°W on day 10. The evolution of the kinetic energetics shows that the development during this period is mainly governed by KE transport and buoyancy conversion (Figs. 5b and 6c). So we draw the meridionally averaged (over 45°–80°N) time-mean vertical–longitudinal distribution of these two terms in Fig. 9. The negative contributions of KE transport and buoyancy conversion are evident. The pattern not only tells that the KE transport weakens the cell but also implies that it may account for the cell’s retrograding. At the eastern edge, the KE transport is positive, while on the western side it is negative. This makes the wind stronger in the east and weaker in the west. The wind in the east is southward; its sudden acceleration leads to an imbalance in geostrophy and hence a surplus of westward force due to the Coriolis effect. Likewise, the sudden deceleration of the wind in the west results in a surplus of westward pressure gradient. Both effects may give rise to a westward migration of the system.
As in Fig. 7, but the shading is (a) KE transport (−∇⋅
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0295.1
As in Fig. 7, but the shading is (a) KE transport (−∇⋅
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0295.1
As in Fig. 7, but the shading is (a) KE transport (−∇⋅
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0295.1
b. APE evolution
An interesting phenomenon is that, during the blocking, temperature is abnormally high (e.g., Green 1977); it is hence necessary to check the APE evolution. We find that the APE related to this temperature anomaly concentrates beneath 300 hPa and in the region enclosed by the 200 m2 s−2 geopotential isoline on the blocking window (e.g., Fig. 10a). We choose this domain (Φ~1 ≥ 200 m2 s−2, 1000–300 hPa) to average the energetics. Averaging over other domains (e.g., Φ~1 ≥ 300 m2 s−2 and Φ~1 ≥ 400 m2 s−2) have also been done and the results (not shown) are similar. From Fig. 10b, the APE evolution experiences two phases, each corresponding to a maximum. In the first phase, it is mainly governed by the APE transfer from the basic-flow window to the blocking window 
As in Fig. 5, but for the blocking-scale APE (m2 s−2) and APE energetics (10−5 m2 s−3) for the high-pressure cell. Refer to the text for the averaging domain.
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0295.1
As in Fig. 5, but for the blocking-scale APE (m2 s−2) and APE energetics (10−5 m2 s−3) for the high-pressure cell. Refer to the text for the averaging domain.
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0295.1
As in Fig. 5, but for the blocking-scale APE (m2 s−2) and APE energetics (10−5 m2 s−3) for the high-pressure cell. Refer to the text for the averaging domain.
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0295.1
As in Fig. 6, but for the blocking-scale APE energetics for the two phases.
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0295.1
As in Fig. 6, but for the blocking-scale APE energetics for the two phases.
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0295.1
As in Fig. 6, but for the blocking-scale APE energetics for the two phases.
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0295.1
1) First maximum phase
The zonal–vertical distribution of the canonical APE transfer from the basic flow is shown in Fig. 12a. Clearly, this transfer occupies most of the lower region of the high-pressure cell. Because of its relation to heat transfer, it will give rise to temperature anomaly, which we draw in Fig. 12b. The positive temperature anomaly is obvious in the lower region. This agrees with the observation that a blocking usually has a warm core in the troposphere. According to Liang and Robinson (2007), 
(a) As in Fig. 7, except that the shading is the canonical APE transfer from the basic flow to the blocking (
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0295.1
(a) As in Fig. 7, except that the shading is the canonical APE transfer from the basic flow to the blocking (
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0295.1
(a) As in Fig. 7, except that the shading is the canonical APE transfer from the basic flow to the blocking (
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0295.1
2) Second maximum phase
In this phase, buoyancy conversion dominates the APE balance. Figure 13a is the vertical–longitudinal section distribution of the buoyancy conversion, which occupies most of the lower region of the cell. By definition, a positive buoyancy conversion means a loss of KE to APE. This implies warm air downward or cold air upward. To see this, we draw the vertical velocity ω in Fig. 13b. Clearly, in the region of the positive buoyancy conversion ω directs downward. A direct result is that the high-pressure cell in the troposphere must be warmer than the environment, which is indeed true (shown in Fig. 13c). Green (1977) suggests that the formation of blocking warm high is due to the descending of dry air from upper levels. Our analysis is consistent with this. So, in this phase, the warm core is maintained by the buoyancy conversion from the kinetic energy within the blocking window.
(a) As in Fig. 7, except that the shading is the conversion (−b1) from KE to APE in the second maximum phase. (b) As in (a), but the shading is vertical velocity (Pa s−1). (c) As in (a), but the shading is temperature (K).
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0295.1
(a) As in Fig. 7, except that the shading is the conversion (−b1) from KE to APE in the second maximum phase. (b) As in (a), but the shading is vertical velocity (Pa s−1). (c) As in (a), but the shading is temperature (K).
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0295.1
(a) As in Fig. 7, except that the shading is the conversion (−b1) from KE to APE in the second maximum phase. (b) As in (a), but the shading is vertical velocity (Pa s−1). (c) As in (a), but the shading is temperature (K).
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0295.1
5. Energetics on other scale windows
As is shown in Fig. 4b, the blocking high pressure cell is most active in the region 40°N–80°N, 45°W–45°E. We hence examine large-scale and synoptic-scale energetics averaged over this domain and throughout the vertical levels from day −12 to day 8. Shown in Figs. 14 and 15 are the averaged energetics departures from their respective time means; the means are referred to the appendix. By calculation, KE and APE are decreasing on both windows in this region. On the basic-flow window, as we know, the westerly is blocked when blocking occurs. This is why the large-scale KE is reduced, and indeed, the energetics show that the KE transport process is the most important factor. For APE, the buoyancy conversion accounts for its reduction. On the synoptic window, as shown in Fig. 15, KE is reduced mainly owing to the KE canonical transfer from the synoptic-scale window to the blocking-scale window, in agreement with the analysis in section 4a(2). Similarly, the upscale APE canonical transfer from the synoptic-scale window to basic-flow window causes the synoptic APE reduction.
Basic-flow energetics (with time mean removed) averaged from day −12 to day 8 over the region 40°N–80°N, 45°W–45°E throughout all vertical levels. The arrow thickness is proportional to the magnitude (10−5 m2 s−3). The shaded boxes are the statistically significant terms.
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0295.1
Basic-flow energetics (with time mean removed) averaged from day −12 to day 8 over the region 40°N–80°N, 45°W–45°E throughout all vertical levels. The arrow thickness is proportional to the magnitude (10−5 m2 s−3). The shaded boxes are the statistically significant terms.
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0295.1
Basic-flow energetics (with time mean removed) averaged from day −12 to day 8 over the region 40°N–80°N, 45°W–45°E throughout all vertical levels. The arrow thickness is proportional to the magnitude (10−5 m2 s−3). The shaded boxes are the statistically significant terms.
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0295.1
As in Fig. 14, but for the synoptic-scale window.
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0295.1
As in Fig. 14, but for the synoptic-scale window.
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0295.1
As in Fig. 14, but for the synoptic-scale window.
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0295.1
6. Discussion
The KE transfer from the synoptic-scale window is almost in phase with the high pressure cell on the blocking-scale window (Fig. 8a). This is in agreement with Robinson (1991), who shows in his theoretical study that the high-frequency forcing to wave 1 is more in phase with the low-frequency streamfunction, considering that the signal we discuss is almost of zonal wavenumber 1 (Fig. 4b). However, earlier on, Illari (1984) and Mullen (1987) find a quadrature phase relation between the eddy forcing and blocking. We note that the different meanings of “eddy forcing” between theirs and ours may account for this inconsistency. The eddy forcing in those previous studies is represented by a quadratic term in the vorticity equation, while here we are talking about energetics, with the terms involving the products of three perturbation fields (cf. Table 1). So, the eddy forcing in our study and that in theirs mean different things. In this sense, the canonical transfers should not be compared to the previously discussed eddy forcings. Actually, in some previous studies people already report possibly different relations. Except the study of Robinson (1991) mentioned above, Tsou and Smith (1990) find that, though the eddy forcing of synoptic-scale transients is one-quarter wavelength upstream of the blocking, the interaction between planetary and synoptic scales shows an in-phase relation with the blocking (cf. their Fig. 13). A related discussion based on PV flux diagnosis is referred to Arai and Mukougawa (2002).
7. Conclusions
Previously, the low-frequency disturbances and atmospheric blocking have been shown to be closely related (e.g., Namias 1947; Sawyer 1970; Michelangeli and Vautard 1998; Woollings et al. 2008); we hence investigate the dynamics on this range of scales in the hope of gaining an understanding of the blocking processes. For convenience, the scale range, or scale window as it is called, is termed the blocking-scale window. The scale windows above and below are for the synoptic eddies and lower variabilities and are called synoptic window and basic-flow window, respectively. Using a newly developed machinery—namely, multiscale window transform—we decompose the atmospheric fields over Atlantic into variabilities on the three windows and reconstruct the blocking process through compositing with the index proposed by Tibaldi and Molteni (1990) all the blocking episodes in winter (DJF) from September 1957 to May 2002. From the result, the blocking originates over Europe and behaves as a westward-retrograding signal. When the high-pressure cell starts to move toward the basic-flow ridge but still with a distance, blocking begins to emerge. When the cell travels over the basic-flow ridge, blocking develops, and decays when the cell leaves the ridge behind. This study diagnoses the dynamical processes that underlie the generation, development, and decay of the cell.
The highly localized nature and, particularly, the moving structure of the blocking requires a diagnostic methodology capable of handling processes that are highly nonlinear, multiscale interactive, and localized in space and time. The multiscale energy and vorticity analysis (MS-EVA) developed by Liang and Robinson (see Liang 2016) and the MS-EVA-based instability theory make such a methodology. Application of it reveals that, during different periods, the evolution of the high-pressure cell is controlled by different mechanisms. In the beginning, the KE balance is dominated by pressure work. In the amplification phase, the canonical KE transfer to the blocking window from the synoptic window is the major mechanism. In the decay phase, the KE transport and the buoyancy conversion from KE to APE on the blocking window are the key factors, where the former may account for the fast moving of the disturbances during this period.
For the APE evolution, two phases may be distinguished: namely, the first maximum phase and the second maximum phase. In the first maximum phase, the APE evolution is dominated by canonical APE transfer from the basic flow to the blocking—namely, baroclinic instability according to the theory of Liang and Robinson (2007). In the second maximum phase, the buoyancy conversion from KE to APE on the blocking window is the most important factor. The vertical motion suggests that the buoyancy conversion is caused by the downward motion of warm air, in agreement with the hypothesis proposed by Green (1977). The baroclinic instability and the buoyancy conversion make the warm core for the blocking center.
All in all, the blocking is initiated collaboratively by the pressure work and the upscale canonical transfer from the synoptic eddies, but its further growth and decay are due to the upscale canonical transfer, KE transport within the blocking window, and the buoyancy conversion from KE to APE. This stage dependence of dynamical processes has already been observed in other phenomena, such as sudden stratospheric warming (Xu and Liang 2017). The corresponding warm core in the troposphere also has different maintaining mechanisms. In early times, it is maintained by a baroclinic instability, but in later times, the mechanism becomes the buoyancy conversion from the blocking-scale KE.
Acknowledgments
We are grateful to Aimé Fournier for important scientific discussions. Thanks are also due to an anonymous referee whose advice helped improve the manuscript. We are indebted to ECMWF for providing the ERA-40 data. Jiwang Ma thanks Yang Yang, Yuanbin Zhao, and Cheng Gong for many valuable discussions. The MS-EVA software can be downloaded (from http://www.ncoads.org/). This work was partially supported by the National Science Foundation of China (NSFC) under Grant 41276032, by the Jiangsu Provincial Government through the Jiangsu Chair Professorship to XSL and the 2015 Jiangsu Program of Entrepreneurship and Innovation Group, and by the State Oceanic Administration through the National Program on Global Change and Air-Sea Interaction (GASI-IPOVAI-06).
APPENDIX
Climatology of the Multiscale Energetics
Figures A1, A2, and A3 display the time-mean energetic cycles on the three-scale windows, respectively. The time mean is over all the wintertime of the data, and the spatial averaging is over 40°N–80°N, 45°W–45°E throughout all the vertical levels. Shaded are the terms significant by the significance test by Fournier (2003).
Time-mean energetics for the basic flow averaged over the region 40°N–80°N, 45°W–45°E through all the vertical levels. The arrow thickness is proportional to the magnitude (10−4 m2 s−3). The shaded boxes are the statistically significant terms.
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0295.1
Time-mean energetics for the basic flow averaged over the region 40°N–80°N, 45°W–45°E through all the vertical levels. The arrow thickness is proportional to the magnitude (10−4 m2 s−3). The shaded boxes are the statistically significant terms.
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0295.1
Time-mean energetics for the basic flow averaged over the region 40°N–80°N, 45°W–45°E through all the vertical levels. The arrow thickness is proportional to the magnitude (10−4 m2 s−3). The shaded boxes are the statistically significant terms.
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0295.1
As in Fig. A1, but for the blocking-scale window.
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0295.1
As in Fig. A1, but for the blocking-scale window.
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0295.1
As in Fig. A1, but for the blocking-scale window.
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0295.1
As in Fig. A1, but for the synoptic-scale window.
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0295.1
As in Fig. A1, but for the synoptic-scale window.
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0295.1
As in Fig. A1, but for the synoptic-scale window.
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0295.1
On the basic-flow window, obviously, the KE transported from other regions dominates the KE balance (Fig. A1a), while the internal processes in the atmosphere form a sink for the APE (Fig. A1b). On the blocking window, the upscale KE canonical transfer makes the most important source of the KE (Fig. A2a), but the APE is mainly gained through the transport processes (Fig. A2b). For the synoptic eddies, they gain KE mainly through the conversion from APE (Fig. A3a), while part of the APE is supplied from the basic flow through baroclinic instability and through transport from outside (Fig. A3b).
Comparing Figs. A1, A2, and A3, the scale interaction pathways are now clear. The basic-flow transfers APE to synoptic-scale eddies; and on the synoptic-scale window, APE is converted to KE; then, the synoptic-scale KE is transferred upscale to the basic-flow window and blocking-scale window. A similar diagram has been proposed in previous studies (e.g., Oort 1964) in a framework of mean-eddy decomposition. But their studies are conducted over a global domain and, hence, do not have local information and do not involve those mechanisms such as transport. In this study, we find, besides the scale interactions, the transport process is crucial on the basic-flow window: KE is transported into the blocking region while APE is transported out of it.
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