Editorial Type:
Article
Article Type:
Research Article

Development Processes of the Explosive Cyclones over the Northwest Pacific: Potential Vorticity Tendency Inversion

Joonsuk M. Kang aSchool of Earth and Environmental Sciences, Seoul National University, Seoul, South Korea

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Seok-Woo Son aSchool of Earth and Environmental Sciences, Seoul National University, Seoul, South Korea

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Online Publication:
02 Jun 2021
Print Publication:
01 Jun 2021
DOI:
https://doi.org/10.1175/JAS-D-20-0151.1
Page(s):
1913–1930
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Abstract

A novel method that quantitatively evaluates the development processes of extratropical cyclones is devised and applied to the explosive cyclones over the northwest Pacific in the cold season (October–April). By inverting the potential vorticity (PV) tendency equation, the contribution of dynamic and thermodynamic processes at different levels to explosive cyclone development is quantified. In terms of geostrophic vorticity tendency at 850 hPa, which is utilized to quantify cyclone development, the leading factors for the explosive cyclone intensification are upper-level PV advection by the mean zonal flow and the PV production from latent heating. However, explosive cyclones are also subject to hindrances from vertical and meridional PV advections. Quantitatively, the sum of thermodynamic contributions by the latent heating, vertical PV advection, and surface temperature tendency is about 1.6 times more important than the dynamical PV redistribution by horizontal advections on the explosive cyclone intensification. This result confirms the dominant role of thermodynamic processes in explosive cyclone development over the northwest Pacific. It turns out from further analysis that the interactions of lower-level anomalous flows are important for thermodynamic processes, whereas the advections by the upper-level mean flow are primary for dynamic processes.

© 2021 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Seok-Woo Son, seokwooson@snu.ac.kr

Abstract

A novel method that quantitatively evaluates the development processes of extratropical cyclones is devised and applied to the explosive cyclones over the northwest Pacific in the cold season (October–April). By inverting the potential vorticity (PV) tendency equation, the contribution of dynamic and thermodynamic processes at different levels to explosive cyclone development is quantified. In terms of geostrophic vorticity tendency at 850 hPa, which is utilized to quantify cyclone development, the leading factors for the explosive cyclone intensification are upper-level PV advection by the mean zonal flow and the PV production from latent heating. However, explosive cyclones are also subject to hindrances from vertical and meridional PV advections. Quantitatively, the sum of thermodynamic contributions by the latent heating, vertical PV advection, and surface temperature tendency is about 1.6 times more important than the dynamical PV redistribution by horizontal advections on the explosive cyclone intensification. This result confirms the dominant role of thermodynamic processes in explosive cyclone development over the northwest Pacific. It turns out from further analysis that the interactions of lower-level anomalous flows are important for thermodynamic processes, whereas the advections by the upper-level mean flow are primary for dynamic processes.

© 2021 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Seok-Woo Son, seokwooson@snu.ac.kr
Keywords: Atmosphere; Dynamics; Extratropical cyclones; Potential vorticity; Synoptic climatology

1. Introduction

The extratropical cyclones stand out as one of the most influential components of the midlatitude weather systems (Klawa and Ulbrich 2003; Hawcroft et al. 2012). The explosive cyclones (ECs), which are extratropical cyclones that deepen more than 24 hPa in 24 h at 60°N (Sanders and Gyakum 1980), are a prominent feature in this perspective, for the severe precipitation and wind gusts that accompany them (Bosart 1981; Gyakum 1983; Wernli et al. 2002). They preferentially occur over the northwest Pacific and North Atlantic in the cold season (October–April) (Sanders and Gyakum 1980; Roebber 1984), and are driven by various dynamical and thermodynamical processes.

The potential vorticity (PV) framework (Hoskins et al. 1985) serves as a powerful tool in explaining the diverse processes responsible for the development of ECs. The PV is particularly useful since it is a quantity conserved in an adiabatic-frictionless flow (Ertel 1942). The PV on the isobaric surface, q, is expressed as
q=−g⁡(f+ζ) ∂θ∂p+g⁡(∂υ∂p∂θ∂x−∂u∂p∂θ∂y),
where g is the gravitational acceleration, f is the planetary vorticity, ζ = ∂υ/∂x − ∂u/∂y is the relative vorticity, u and υ are the zonal and meridional winds, and θ is the potential temperature. Another advantage of PV comes from its invertibility. From PV anomalies in the free atmosphere and potential temperature anomalies at the surface, the three-dimensional potential temperature and nondivergent wind fields can be retrieved (Davis 1992).

In the PV framework, the processes associated with EC development can be linked with particular PV anomalies or surface potential temperature anomalies. For instance, the tropopause fold, or enhanced upper-level trough, which can strengthen the EC by promoting vertical motion and cyclonic circulation in the mid- to lower troposphere (Uccellini et al. 1985; Sanders 1986; Reader and Moore 1995), is commonly related to positive PV anomalies in the upper troposphere. Likewise, the effect of latent heating (LH) (Ahmadi-Givi et al. 2004; Fink et al. 2012), which strengthens cyclonic motion in the lower troposphere, is often attributed to positive PV anomalies in the lower troposphere. The effect of surface heat fluxes (Kuo et al. 1991; Hirata et al. 2015) can partly be linked to the warm temperature anomalies. The balanced cyclonic circulations, induced from these positive anomalies, act in a way that can amplify one another, particularly when aligned in a westward-tilted structure in the vertical. This structure is particularly conspicuous at the rapid deepening phase of ECs (Wang and Rogers 2001).

The piecewise PV inversion and above interpretation together function as technical tools for quantitative analyses of cyclone development. Utilizing these tools, Seiler (2019) examined more than 3000 intense cyclones in the Northern Hemisphere. The relative vorticity induced from each PV anomaly was compared to that of the cyclone. It is found that, at the maximum cyclone intensity, 34%, 43%, and 23% of the 850-hPa relative vorticity are contributed by the upper-tropospheric PV, lower-tropospheric PV, and surface potential temperature anomalies, respectively, with regional and seasonal dependency. This diagnostic analysis could readily be applied to the ECs to describe their intensity. However, considering the definition of EC which emphasizes temporal development, more focus is required on the prognostic perspective (i.e., change over time).

Recalling the PV conservation, the following PV tendency equation holds:
∂q∂t=−u⋅∇q+Q+F.
Here, u = (u, υ, ω) is the three-dimensional wind vector, ∇ = (∂/∂x, ∂/∂y, ∂/∂p), and Q and F are the PV changes from diabatic heating and friction. As explicated in Eq. (2), the effect of diabatic heating on PV distribution is prognostic. This addresses that there is a limitation when simply regarding the lower-level PV anomaly as indicative of the LH process. Büeler and Pfahl (2017) brought up this issue and devised a diagnostic method to better represent the LH process as an alternative solution. Nevertheless, a method capable of assessing the prognostic contribution of LH along with other dynamic factors, such as the Zwack–Okossi equation (Zwack and Okossi 1986) or pressure tendency equation (Fink et al. 2012), is still necessary in the PV framework.

To this end, the PV tendency equation can be considered as it well describes the physical processes related to EC development (e.g., Tamarin and Kaspi 2016). However, additional work is required on top of calculating Eq. (2) because PV tendency is not a direct measure of cyclone development as pressure deepening or vorticity increase rate. Besides, Eq. (2) alone does not incorporate the effect of PV tendencies of one level on other levels. To address these issues, PV tendency needs to be inverted, in analogy to diagnostic PV inversion (Davis and Emanuel 1991). This approach, which can be referred to as PV tendency inversion, makes it possible to quantify the wind changes from PV tendencies at different levels.

Together with introducing PV tendency inversion, this study utilizes it to investigate the EC development over the northwest Pacific during the cold season. Being one of the most prominent regions for EC occurrence, this is where strong surface fluxes from the Kuroshio Extension and intense upper-level jet provide a favorable condition for EC development in the winter (Chen et al. 1992; Yoshida and Asuma 2004; Zhang et al. 2017). In this regard, the PV tendency equation is inverted to quantitatively describe the dynamic and thermodynamic processes responsible for EC intensification. Additional inversions are also performed to evaluate the relative importance of upper- versus lower-tropospheric processes and mean versus anomalous flows. Note that, unlike Seiler (2019), a prognostic equation is utilized in this study.

The rest of the paper is organized as follows. Section 2 covers the data and methods. The inversion of the PV tendency equation is elaborated in section 3. The characteristics of ECs in the northwest Pacific are described in section 4. Section 5 scrutinizes the results of the inversion calculations, and section 6 is devoted to the summary and discussion on the results and method.

2. Data and method

a. Data

This study utilizes the 6-hourly ERA-Interim dataset for the period of 1979–2018 (Dee et al. 2011). Specifically, temperature, geopotential, horizontal winds, pressure velocity, sea level pressure (SLP), and specific humidity data interpolated onto a 1.5° × 1.5° latitude–longitude grid and 37 vertical levels (except for SLP) are used. From these variables, the PV is calculated as defined in Eq. (1). The second-order finite difference is used to approximate the partial differentials.

b. Statistical significance test

The anomalies discussed in this study are statistically tested by the two-tailed Student’s t test at the 95% confidence level, where the degree of freedom is set to the number of ECs analyzed. For variables in which normal distribution is not guaranteed, such as PV tendency budget and geostrophic vorticity tendency retrieved from inversion, a bootstrap resampling method (Efron and Tibshirani 1993) is alternatively used. In this case, the 95% confidence intervals are achieved by recalculating the composite mean 10 000 times with random resampling.

c. EC tracking and definition

The extratropical cyclones over the northwest Pacific are identified by applying the automated cyclone identifying and tracking algorithm (Wernli and Schwierz 2006; Sprenger et al. 2017) to the 6-hourly ERA-Interim SLP data in the Northern Hemispheric extratropics (25°–90°N). This method consists of identification, tracking, and filtering processes. In the cyclone identification step, the SLP minima are selected as cyclone candidates if the length of their outermost isobaric contour line is between 100 and 7500 km. To produce tracks from these minima, the tracking process estimates the cyclone position at the next time step (t + 6 h) by considering the direction and distance of cyclone migration between the previous (t − 6 h) and present (t) time steps. The SLP minimum that is closest to the estimated position is regarded as the cyclone position in the next time step if it is in a certain spatial range. Finally, the cyclones that last at least 48 h and travel at least 1000 km are selected, excluding short-lived or quasi-stationary cyclones. Tropical cyclones are discarded by confining the analysis domain to the extratropics (25°–90°N).

To define ECs, the deepening rate (DR) at a 6-hourly time step t is calculated in the unit of Bergeron (B) for each cyclone (Sanders and Gyakum 1980):
DRt=−SLPt+12h−SLPt−12h24 hPa×sin 60osin⁡ φt [B].
Here, φt is the latitude of the cyclone at time step t. The maximum DR (DRmax) and the corresponding time (tmax) are determined for each cyclone, and the cyclones that have DRmax greater than or equal to 1 B are set as ECs (Sanders and Gyakum 1980).

3. PV tendency equation and inversion

a. Linearization of PV

The PV, q, is a nonlinear function of winds and potential temperature, as shown in Eq. (1). However, to perform an inversion of the PV tendency equation, the PV anomaly, q′, is first approximated to a linear function of the geopotential height anomaly (ϕ′), analogous to Charney and Stern (1962):
q′≈qL′=L⁡(ϕ′).
The linear operator L, defined as
L≡−g ∂θ¯∂p⁡(1f0 ∇p2+fσ∂2∂p2),
calculates the linearized PV anomaly, qL′, from geopotential anomaly. Here, f0 is the planetary vorticity at the center of each EC, ∇p = (∂/∂x, ∂/∂y, 0) is the horizontal gradient operator, σ=−(Rdπ/p)⁡(∂θ¯/∂p) is the stability parameter, Rd is the gas constant of dry air, π=⁡(p/ps)Rd/cp is the Exner function, cp is the specific heat of dry air under constant pressure, and ps = 1000 hPa is the surface pressure. The overbar denotes the zonal and time mean (e.g., θ¯ is a function of latitude and pressure), whereas the prime represents the corresponding anomaly. The detailed derivation of Eq. (3) is delineated in appendix A. As shown in appendix A, q′and qL′ in Eq. (3) are quantitatively similar (see Fig. A1).

b. Application of PV tendency equation

By letting q′=qL′ from the closeness of the two variables, the time differentiation of Eq. (3) is performed as follows to focus on the prognostic perspective of EC development (note that ∂q/∂t = ∂q′/∂t):
∂q∂t=∂qL′∂t=L⁡(χ).
Here, L is the linear operator that calculates local PV tendency from geopotential tendency (χ ≡ ∂ϕ′/∂t). Similarly, χ is achievable from ∂q/∂t by the inverse calculation of Eq. (4) as
χ=L−1⁡(∂q∂t).
On top of this, the following equation holds when both tendencies are partitioned to N pieces since L is a linear operator:
χi=L−1[⁡(∂q∂t)i].
Here, the subscript i denotes the ith partition of the PV tendency, when χ=∑i=1Nχi and ∂q/∂t=∑i=1N⁡(∂q/∂t)i. With appropriate boundary conditions, the piecewise geopotential tendencies can be retrieved from each partitioned (∂q/∂t)i, and they sum up to the total geopotential tendency in the domain when all partitions are taken into account.
Among many possible partitions of PV tendency, a process-based partitioning is introduced in this study. This is enabled by equating Eqs. (2) and (4), which leads to the following:
L⁡(χ)=−u ∂q∂x−υ ∂q∂y−ω ∂q∂p+QLH+FRES.
The terms in the rhs of Eq. (5), which sum up to ∂q/∂t, serve as the partitions of ∂q/∂t. For example, regarding −u∂q/∂x as (∂q/∂t)1, χ1 = L−1(−u∂q/∂x) is the geopotential tendency due to zonal PV advection.
In Eq. (5), the PV tendency from diabatic heating is decomposed as Q = QLH + QRES, to investigate the pivotal contribution of LH exclusively and explicitly among other diabatic processes. Focusing on the vertical distribution of LH, QLH is calculated as follows (Tamarin and Kaspi 2016):
QLH=−g⁡(f+ζ) ∂θ˙LH∂p.
Here, θ˙LH is the latent heating, which is calculated as below (Emanuel et al. 1987):
θ˙LH=ω⁡(∂θ∂p−γmγdθθe∂θe∂p),
where θe is the equivalent potential temperature, and γd and γm stand for dry and moist adiabatic lapse rates, respectively. The PV tendency from diabatic heating other than LH, such as radiative heating, is included in QRES. Then FRES = QRES + F is the PV tendency from all nonconservative processes that are not explicitly assessable.

It is recognizable that qL′ resembles the quasigeostrophic (QG) PV anomaly. Similarly, L(χ) commonly appears in QG height tendency analyses (e.g., Nielsen-Gammon and Lefevre 1996). Despite the resemblance to QG dynamics, adopting QG form to the rhs of Eq. (5) reveals the limitations when vertical motions and diabatic heating are of great importance. Thus, L(χ) is considered only as a linear approximation of PV tendency. Note also that the terms related to static stability (−g∂θ¯/∂p and σ) vary latitudinally and vertically, unlike in QG framework where they are only a function of pressure. Comparison with the QG framework is discussed in more detail in the last section.

c. EC intensification versus propagation

Utilizing Eq. (2) provides the benefit of process-based analysis. However, ECs are not stationary but propagate during their rapid deepening. This makes relating ∂q/∂t directly to EC intensification difficult. It turns out that ∂q/∂t features an asymmetric dipole during development. While propagation leads to positive and negative PV tendencies aligned to its motion, EC intensification makes the magnitude of the former stronger than that of the latter. The method to exclude EC propagation effect, introduced in section 5c, is built upon this asymmetricity of the dipole tendency, where larger asymmetricity implies stronger intensification. As will be shown in section 5c, the propagation and intensification respectively account for approximately half of total EC development.

d. Devising the inversion calculations

To perform inversion calculation, boundary conditions are required. In the lateral boundaries, homogeneous Dirichlet boundary condition (χ = 0) is applied to all inversion calculations. At the top and bottom boundaries, either homogeneous or nonhomogeneous Neumann boundary conditions are used. The following equation describes the nonhomogeneous Neumann boundary condition at both boundaries:
⁡(∂χ∂p)sb=−Rdp⁡(∂T∂t)sb.
The subscript “sb” denotes that this condition is applied at the top and bottom surface boundaries, which are set to 175 and 975 hPa, one level interior from the outermost levels (see appendix B). This surface boundary condition is important since it acts as a sheet of PV tendency at the surface. In analogy to positive θ anomaly in the PV inversion (Bretherton 1966), the positive temperature tendency increases geostrophic vorticity above. Further details of the inversion are described in appendix B.

To test the contributions of various processes to EC development, inversion calculations are performed for each EC at its tmax. Each inversion is conducted with a different partition of PV tendency and corresponding boundary conditions. Two sets of inversion calculations, referred to as basic and additional sets, are performed for each EC. For the basic set with seven inversions (Table 1), a full inversion that inverts the total PV tendency in the domain with nonhomogeneous Neumann surface boundary condition is first performed. It is followed by five inversions that respectively use the PV advections in three directions and the nonconservative terms in Eq. (5). For these five inversions, a homogeneous Neumann surface boundary condition is used [i.e., rhs of Eq. (8) is set to zero], meaning that no forcing is acting on surface boundaries (Table 1). The last is conducted with nonhomogeneous surface boundary condition with zero interior PV tendency, completing a linear set of partitions. The additional set consists of inversions of the decomposed advection terms and is described with details in section 5d.

Table 1.

Summary of the terms used in the basic set.

Table 1.

4. Characteristics of northwest Pacific ECs

a. EC sampling

The spatial frequency of extratropical cyclones detected over the northwest Pacific is illustrated in Fig. 1a. The cyclone frequency represents the number of cyclones that pass inside a circle of 555 km radius from each grid point (e.g., Kang et al. 2020). The identified cyclone tracks are consistent with the well-known East Asian storm tracks (Lee et al. 2019). The maximum frequency is found around 40°N, 160°E, where more than 60 cyclones affect the region in a year.

Fig. 1.
Fig. 1.

The frequency of (a) extratropical cyclones and (b) explosive cyclones (ECs), and (c) explosive deepening location of ECs over the northwest Pacific (shading; units: number per year). The black box in (c) refers to the target domain, and the blue lines indicate the longitudinal range (135°–165°E) for the zonally averaged quantities described in section 4b.

Citation: Journal of the Atmospheric Sciences 78, 6; 10.1175/JAS-D-20-0151.1

The frequency of the ECs in the northwest Pacific is shown in Fig. 1b. Though less in number, overall track distribution coincides with that of all cyclones in Fig. 1a. The ECs are again most frequent over the open ocean east of Japan, where more than 20 ECs impact the region annually. The maximum deepening of ECs is observed most frequently at 40°N, 150°E (Fig. 1c). The distribution shown in Fig. 1c is largely similar to that reported in Iwao et al. (2012).

To analyze the development of ECs, the target domain is set as a 10° × 10° box centered at 40°N, 150°E (black box in Fig. 1c). The 728 cyclones that underwent maximum deepening in the target domain in 40 years (1979–2018) are sampled, and 310 among them are ECs.

Figure 2a illustrates the monthly variation in numbers of ECs and non-ECs, based on the date of tmax. As expected, ECs are frequent during the cold season, and their numbers peak both in December and March (Zhang et al. 2017). Since the analysis domain is subjectively set to gain large samples of ECs, there are more ECs than non-ECs from December to March. The five warm months (May–September), with a total of 11 ECs, are excluded from the analysis. As a result, 299 ECs from October to April are selected for the budget analyses.

Fig. 2.
Fig. 2.

(a) Monthly variation in the number of extratropical cyclones with respect to their DRmax. The red and white numbers denote the number of ECs and non-ECs each month. (b) Change in central pressure (solid lines; left axis) and DR (dashed lines; right axis) with respect to tmax. The colored lines represent individual months. (c) As in (b), but for ζg at 850 hPa (solid lines; left axis) and its intensification rate (dashed lines; right axis). The ζg is defined as the average of the geostrophic vorticity within the 6° × 6° box around the EC center.

Citation: Journal of the Atmospheric Sciences 78, 6; 10.1175/JAS-D-20-0151.1

Figure 2b illustrates the time evolution of the central pressure of 299 ECs with respect to tmax. On average, their pressure falls from 1002 to 980 hPa along their maximum deepening (note that 1 B is 17.8 hPa day−1 at 40°N). At tmax, the average DRmax of ECs reaches about 1.5 B. It is slightly larger from December to March than during other seasons, consistent with the monthly variation of strong ECs (DRmax ≥ 2 B) shown in Fig. 2a.

The time evolution of geostrophic vorticity (ζg) at 850 hPa is also shown in Fig. 2c. Its intensification rate is defined as the 12-h difference of ζg along the EC track. During the 24 h of explosive deepening, ζg strengthens more than threefold, from 3.8 cyclonic vorticity units (1 CVU: 1 × 10−5 s−1) to 11.6 CVU. The intensification rate also peaks at tmax, where it is about 4.4 CVU (12 h)−1. This justifies the use of ζg tendency as a variable to diagnose EC development (e.g., Yoshida and Asuma 2004).

b. Synoptic structure of the northwest Pacific ECs

The synoptic structures of the sampled ECs at tmax are explored prior to the inversion, and the associated anomalies are shown in Fig. 3. Here, the anomaly is defined as the deviation from the mean state, which is the monthly climatology that is zonally averaged over the domain of 135°–165°E (blue lines in Fig. 1c). The ECs accompany an enhanced upper-level trough, indicated by a strong 250-hPa PV anomaly west of the EC center (Fig. 3a). An intense PV anomaly is also observed at 850 hPa, partly related to the LH. This is also coherent with the conspicuous integrated water vapor transport (IVT) anomaly (Fig. 3a) over the warm sector (Fig. 3b), where the IVT is calculated as 1/g⁡(∫1000hPa300hPaSu dp)2+⁡(∫1000hPa300hPaSv dp)2, and S is the specific humidity. The EC center exhibits an SLP anomaly of −20 hPa, and accompanies anomalous cyclonic circulation in the lower-troposphere with a strong southerly (>20 m s−1) over the warm sector, following the sharp pressure gradient.

Fig. 3.
Fig. 3.

(a) PV anomalies at 250 hPa (shading; units: PVU) and 850 hPa (black contours; units: PVU), IVT anomalies (blue contours; units: kg m−1 s−1), and 250-hPa geopotential height (gray contours; 150-m interval) with respect to the EC center (red triangle) at tmax. (b) As in (a), but for SLP anomalies (shading; units: hPa), 850-hPa anomalous wind vectors (black vectors; units: m s−1), 1000-hPa temperature anomalies (red contours; for values larger than 3 K), and 1000-hPa isotherms (gray contours; 3-K interval). (c) Vertical cross section of PV (shading; units: PVU) and latent heating rate [red contours; units: K (12 h)−1]. The PV anomalies with absolute values greater than or equal to 0.1 PVU are contoured in gray, and those greater than 0.5 are contoured in black. The 2-PVU line is shown in white as a reference. In (a) and (b), only the anomalies that are statistically significant at the 95% confidence level, based on a two-tailed Student’s t test, are shown. In (c), statistically significant PV anomalies with absolute values greater than or equal to 0.1 PVU are dotted.

Citation: Journal of the Atmospheric Sciences 78, 6; 10.1175/JAS-D-20-0151.1

The vertical structure of the sampled ECs is illustrated in Fig. 3c. Here, the vertical cross section is made from the averaged value within a 15° latitude band centered at each EC at tmax. The upper-level PV anomaly intrudes down to about 600 hPa, exerting cyclonic circulation to the lower troposphere. Above the warm sector where moisture is transported (Figs. 3a,b), a strong LH of 8.0 K (12 h)−1 is observed around 600–700 hPa (Fig. 3c). A positive PV anomaly forms below this level, forming a westward-tilted PV structure from the surface to the tropopause [2 potential vorticity unit (PVU) surface denoted in white]. Above this level, PV is reduced by LH, and the upper-level ridge is enhanced (~−1.7 PVU), having a PV anomaly that is even larger in magnitude than compared to the trough (~1.4 PVU). Such PV distribution strengthens the PV gradient on the east of the upper-level trough.

5. Development processes of the northwest Pacific ECs

a. PV tendency budget

Figure 4 illustrates the vertical cross section of the first five terms in the basic set (Table 1) along with L(χ) at tmax as in Fig. 3c. The L⁡(χ)=∂qL′/∂t and ∂q/∂t bear a strong resemblance (Figs. 4a,b), indicating that the linearization of PV is applicable in the analysis. While the positive and negative dipolar tendencies in both figures indicate the migration of the westward-tilted PV structure, an amplification of the system is also inferred from the larger absolute values of positive tendency compared to negative tendency. It is notable that the propagations of upper- and lower-tropospheric PV occur in different length scales. The dipole tendency spans 40° zonally in the upper troposphere, whereas it is only found in ±10° about the EC center in the lower troposphere.

Fig. 4.
Fig. 4.

Vertical cross section of (a) L(χ), (b) ∂q/∂t, (c) −u∂q/∂x, (d) −υ∂q/∂y, (e) −ω∂q/∂p, and (f) QLH [shading; units: PVU (12 h)−1] with respect to the EC center (red triangle) at tmax as in Fig. 3c. In (c) and (d), the 600-hPa level is plotted in black dashed line, and absolute values greater than or equal to 2 PVU (12 h)−1 are contoured in 1-PVU interval. The values that are statistically significant at the 95% confidence level, based on the bootstrap resampling method, are dotted (near-zero values are not dotted for conciseness).

Citation: Journal of the Atmospheric Sciences 78, 6; 10.1175/JAS-D-20-0151.1

It is the zonal advection of PV that is most responsible for the positive PV tendency in the upper troposphere [~6 PVU (12 h)−1; Fig. 4c]. However, the positive tendency is substantially reduced by the meridional and vertical advections through which low-PV air flows into the same region [~−2 PVU (12 h)−1; Figs. 4d,e]. A dipole pattern of PV tendency is observed about the level of maximum LH (Fig. 4f). The PV production from LH is the most dominant contributor to the positive PV tendency in the lower troposphere [~0.6 PVU (12 h)−1]. The horizontal and vertical advections have negative effects on the lower-level PV tendency (Figs. 4c–e).

The temperature tendency at 975 hPa, which functions as a bottom boundary condition in this study, is shown in Fig. 5. As the warm sector in Fig. 3b migrates eastward, a positive temperature tendency ~5.4 K (12 h)−1] is found east to the EC center at tmax. A similar magnitude of negative tendency is also apparent [~−5.5 K (12 h)−1], hinting that the surface warm anomaly does not amplify during the 12 h. Though not shown, these temperature tendencies are mostly driven by meridional temperature advection (i.e., −υ∂T/∂y). The temperature at the top boundary is not shown since it has little impact on the near-surface geopotential tendency. Accordingly, the physical interpretation regarding the surface boundary conditions will hereafter be confined to the bottom temperature tendency.

Fig. 5.
Fig. 5.

Temperature tendency at 975 hPa [shading; units: K (12 h)−1] with respect to the EC center (red triangle) at tmax. The values that are statistically significant at the 95% confidence level, based on the bootstrap resampling method, are dotted (near-zero values are not dotted for conciseness). The temperature anomalies are also contoured in 1 K interval starting from 3 K.

Citation: Journal of the Atmospheric Sciences 78, 6; 10.1175/JAS-D-20-0151.1

The PV tendencies (Fig. 4) and temperature tendency (Fig. 5) shown above are forcings that derive EC development. To quantify the influences of these forcings distributed at different levels, the PV tendencies are inverted in the following section.

b. Inversion results

From the geopotential tendency achieved from each inversion, the geostrophic vorticity tendency at 850 hPa [ζt≡∂ζg/∂t=(1/f0)∇p2χ] is calculated. The geostrophic vorticity tendency (ζt) retrieved from the basic set is shown in Fig. 6. The observed ζt (Fig. 6a), which is calculated from the reanalysis data after applying a 1–2–1 filter in space (see appendix B), exhibits a maximum positive value of about 21 CVU (12 h)−1 on the east and slightly north of the EC center at tmax. It is strongly analogous to ζt achieved from the inversion of ∂q′/∂t (Fig. 6b) where a maximum tendency of about 20 CVU (12 h)−1 (white circle) is recorded. The location of this maximum is likely the position of the EC at tmax + 6 h.

Fig. 6.
Fig. 6.

(a) ζt from reanalysis data [shading; units: CVU (12 h)−1] with respect to the EC center (red triangle) at tmax. (b)–(g) As in (a), but for ζt achieved from the basic set. (h) As in (a), but for ζt achieved by the sum of six piecewise inversions including FRES. The values that are statistically significant at the 95% confidence level, based on the bootstrap resampling method, are dotted (near-zero values are not dotted for conciseness). The 6° × 6° box centered at the maximum (white circle, red box) and minimum (black circle, blue box) ζt are denoted in (b).

Citation: Journal of the Atmospheric Sciences 78, 6; 10.1175/JAS-D-20-0151.1

Among the processes that affect the PV tendency, the zonal advection (−u∂q/∂x) is responsible for the positive ζt [~14 CVU (12 h)−1] southeast to the EC center (Fig. 6c). The positive ζt from the zonal advection south to the EC center is somewhat countered by the negative ζt from the meridional advection (−υ∂q/∂y) (Fig. 6d). The vertical advection (−ω∂q/∂p) brings a weak negative ζt near the EC center (Fig. 6e). The PV tendency from LH (QLH) results in positive ζt [~11 CVU (12 h)−1] broadly to the northeast of the EC center (Fig. 6f). Since this positive tendency is mostly in phase with that shown in Fig. 6b, the significant contribution of QLH to EC development can be deduced. From the positive temperature tendency at surface (Fig. 5), a weak positive ζt is caused east to the EC center (Fig. 6g). Undoubtedly, the ζt from the six piecewise inversions, i.e., Figs. 6c–g and inversion of FRES (not shown) sum up to that from the full inversion (cf. Figs. 6b,h).

c. Processes contributing to the EC intensification

The results above provide a useful baseline on how both intensity and position of EC change from tmax − 6 h to tmax + 6 h. To better quantify the results, a 6° × 6° box centered on the maximum ζt is first identified for each EC (red box in Fig. 6b). The average of ζt inside the box is considered as the value representing EC development during 12 h and is simply noted as ζtmax. However, ζtmax is not directly related to cyclone intensification as addressed in section 3c. It can be decomposed as
ζtmax=ζtint+ζtpro,
where ζtint and ζtpro respectively are the vorticity tendencies from EC intensification and propagation. The former represents the change of EC intensity during the maximum deepening. Focusing on the intensification and factors responsible for it, ζtint is isolated from ζtmax using the method described below (other possible methods are also discussed in section 6).
The starting point for the method is the asymmetric dipole pattern of ζt in Fig. 6b. The dipole tendency indicates cyclone migration from tmax − 6 h to tmax + 6 h, whereas the asymmetry implies an intensification of cyclone. Keeping this in mind, ζtint can be defined as
ζtint=ζtmin+ζtmax.
Here, ζtmin is ζt averaged inside the 6° × 6° box centered at its minimum (blue box in Fig. 6b). A purely migratory process (symmetric dipole ζt) would have ζtmin and ζtmax that sum up to zero.

The ζtmax, ζtint, and ζtpro from reanalysis, full inversion, and the sum of piecewise inversions are shown in Fig. 7. From the reanalysis, ζtmax (purple) is about 10.0 CVU (12 h)−1 while that from the full inversion is about 9.3 CVU (12 h)−1, indicating that our method underestimates the observed state by 7%. The ζtmax from the sum of piecewise inversions differs with ζtmax from the full inversion only by 0.2%, proving that the linearity is well guaranteed. The ζtint (blue), which corresponds to the EC intensification in 12 h, is 4.8 CVU (12 h)−1 in the reanalysis. This tendency is well captured from full inversion [4.6 CVU (12 h)−1] and the sum of piecewise inversions [4.6 CVU (12 h)−1]. Most importantly, this value is close to the Lagrangian vorticity tendency in Fig. 2c [4.4 CVU (12 h)−1]. In any case, ζtpro (light blue) accounts for about a half of ζtmax.

Fig. 7.
Fig. 7.

The ζtmax, ζtint, and ζtpro [bars; units: CVU (12 h)−1] from reanalysis, full inversion, and sum of piecewise inversions with 95% confidence intervals calculated from the bootstrap resampling method. The values denoted to each bar refer to the relative contribution (%) to the ζtmax from ∂q/∂t. The numbers in the parenthesis refer to the confidence intervals (%).

Citation: Journal of the Atmospheric Sciences 78, 6; 10.1175/JAS-D-20-0151.1

Noting that the calculated ζtint well quantifies the EC intensity change, the factors responsible for EC intensification are evaluated by defining red and blue boxes for each EC, and then calculating ζtint for all inversions and the sum of piecewise inversions. The ζtint calculated from the basic set are shown in Fig. 8. The largest contribution to the intensification is made from QLH (cyan), by 5.1 CVU (12 h)−1. Second to QLH, EC is strengthened through the zonal advection (brown) by 4.6 CVU (12 h)−1. The rest terms in the set counter EC intensification. The meridional and vertical advections (dark green and yellow) weaken EC by −2.5 and −1.4 CVU (12 h)−1, respectively. The temperature tendency at the surface (magenta) also hinders EC intensification by a small factor [−0.2 CVU (12 h)−1].

Fig. 8.
Fig. 8.

The ζtint from basic set with 95% confidence intervals calculated from the bootstrap resampling method. The values denoted to each bar refer to the relative contribution (%) to the ζtint from ∂q/∂t. The numbers in the parenthesis refer to the confidence intervals (%).

Citation: Journal of the Atmospheric Sciences 78, 6; 10.1175/JAS-D-20-0151.1

The results shown in Fig. 8 suggest that the dynamic (−u∂q/∂x and −υ∂q/∂y) and thermodynamic [−ω∂q/∂p, QLH, and (−Rd/p)(∂T/∂t)sb] processes respectively account for 46.2% and 75.2% of EC intensification, highlighting the importance of the thermodynamic processes in the rapid intensification. Note that their sum is greater than 100% because of negative contributions from nonexplicit term FRES (gray). The large contribution of QLH (cyan) suggests that the lower-tropospheric PV production by LH is a leading factor for EC intensification, as emphasized in previous studies across various frameworks (Wernli et al. 2002; Ahmadi-Givi et al. 2004; Fink et al. 2012). Here, the vertical advection is considered as a thermodynamic process because it is mostly associated with the vertical variation of static stability. The surface temperature change (magenta), however, does not significantly contribute to the EC intensification. This is inferable from Fig. 5, where the warm anomaly does not amplify and only migrate in phase with the EC (see also Fig. 6g).

Among the dynamic processes, the most important contributor to the EC intensification is the zonal PV advection. The horizontal PV advections, however, are a collective representation of nonlinear interactions and different propagations of upper- and lower-level PV anomalies by the mean flow, which may differ in their extent of contributions to the EC intensification. To examine dynamic processes in more detail, inversions are extended to individual components of the PV advection (Table 2) in the following section.

Table 2.

Summary of the terms used in the additional set.

Table 2.

d. Decomposed advection terms

The advection terms in Eq. (5) are decomposed into mean and anomalous components as below to gain better insight into the role of PV advection in EC intensification:
−u ∂q∂x−υ ∂q∂y−ω ∂q∂p=−u¯ ∂q′∂x−u′ ∂q′∂x−υ ∂q¯∂y−υ ∂q′∂y−ω ∂q¯∂p−ω ∂q′∂p.
Here, the mean is defined as sector-mean (of 135°–165°E) monthly climatology, and anomaly is the deviation therefrom as in Fig. 3. As well as zonal advection, the meridional PV advection is decomposed to address the effect of nonlinear interactions. The decomposition of vertical advection is also considered to evaluate the influence of mean PV stratification. In Eq. (9), the mean and anomaly components of meridional and vertical winds are not separately considered because their mean components are close to zero.

With these decomposed advection terms, additional set of inversions are performed to evaluate the contribution of the mean flow compared to the nonlinear interactions of the anomalous flows (Table 2). The former corresponds to −u¯∂q′/∂x, −υ∂q¯/∂y, and −ω∂q¯/∂p, and the latter includes −u′∂q′/∂x, −υ∂q′/∂y, and −ω∂q′/∂p in Eq. (9). Additionally, the horizontal advection terms are divided into upper- and lower-tropospheric portions across 600 hPa, to quantify the effects of upper- and lower-level contributions to EC development. This level is chosen since upper-level PV tendencies extend down to 600-hPa level in Figs. 4c and 4d (black dashed line).

Figure 9 illustrates ζt calculated from additional set (see also Figs. C1 and C2 in appendix C for the spatial distribution of each advection term in the upper and lower troposphere). Positive ζt is induced broadly around the EC center by ⁡(−u¯∂q′/∂x)≤600hPa, representing the propagation of amplifying upper-level wave (Fig. 9a). The same process in the lower troposphere induces symmetric dipole ζt about the EC center [⁡(−u¯∂q′/∂x)>600hPa; Fig. 9b]. The ζt from ⁡(−υ∂q¯/∂y)≤600hPaindicates westward propagation of Rossby wave (Fig. 9e), while it is weaker and out of phase with that from ⁡(−u¯∂q′/∂x)≤600hPa (cf. Figs. 9a,e). This process in the lower troposphere only induces small negative ζt east to the EC center [⁡(−υ∂q¯/∂y)>600hPa; Fig. 9f]. The ζt produced by −ω∂q¯/∂p is negative in the broad region east to the EC center (Fig. 9i), but that from −ω∂q′/∂p counters it over the same region (Fig. 9j). The nonlinear advections (−u′∂q′/∂x and −υ∂q′/∂y) in the upper troposphere have minor effects on the overall ζt (Figs. 9c,g). In the lower troposphere, they induce quadrupole ζt that is almost out of phase of one another (Figs. 9d,h). In sum, they produce negative ζt to the southeast of EC center. Though upper-level PV tendencies (Fig. C1) are stronger compared those in lower-level (Fig. C2), the same does not apply when assessing the induced ζt in the lower troposphere (cf. left and right columns in Fig. 9). This evidences the necessity of inversion to weigh the influences of forcings distributed at different levels.

Fig. 9.
Fig. 9.

As in Fig. 6, but for ζt achieved from the additional set.

Citation: Journal of the Atmospheric Sciences 78, 6; 10.1175/JAS-D-20-0151.1

Further quantifying, each ζtint from the additional set is illustrated in Fig. 10. The most prominent positive contribution is from the zonal advection by the mean flow in the upper troposphere (left red), which intensifies the EC by 5.5 CVU (12 h)−1. This demonstrates that EC intensification through −u∂q/∂x (Fig. 8) is dominated by PV tendency due to the upper-level mean flow across a sharp PV gradient. The mean zonal advection in the lower troposphere (right red) does not contribute to EC strengthening. Though not shown, this process is the largest contributor to ζtpro. The nonlinear horizontal advections in the upper troposphere (left pink and left olive) produce a weak positive tendency in sum [0.7 CVU (12 h)−1], whereas those in the lower troposphere (right pink and right olive) weaken the EC [−1.7 CVU (12 h)−1]. Negative effect of −2.1 CVU (12 h)−1 is also made from the meridional advection of mean PV (green). Given that the meridional mean PV gradient is larger in the upper troposphere, the negative contribution from the upper level dominates. The climatological PV stratification strongly hinders EC development [−5.0 CVU (12 h)−1; dark yellow], although it is largely offset by the anomalous alteration to the stratification [3.7 CVU (12 h)−1; orange].

Fig. 10.
Fig. 10.

As in Fig. 8, but for ζtint from the additional set. The values denoted for each bar refer to the relative contribution (%) to the ζtint from ∂q/∂t in Fig. 8. The left and right bars for each term represent the upper- and lower-tropospheric contributions, respectively.

Citation: Journal of the Atmospheric Sciences 78, 6; 10.1175/JAS-D-20-0151.1

Bringing together, the ζtint from basic and additional inversions are rearranged in Fig. 11 by the physical processes that they represent. The EC is intensified by 5.5 CVU (12 h)−1 from the mean zonal PV advection (red; −u¯∂q′/∂x). A pair of trough and ridge in the upper troposphere accounts mostly for this positive tendency. The LH (cyan; QLH) strengthens EC by 5.1 CVU (12 h)−1. The nonlinear horizontal advections (light green; −u′∂q′/∂x−υ∂q′/∂y) and temperature tendency at surface [magenta; (−Rd/p)(∂T/∂t)sb] weaken EC by −1.0 and −0.2 CVU (12 h)−1, respectively. Interestingly, the circulation induced by EC itself (ω and υ) counteracts with the climatological distribution of PV (∂q¯/∂p and ∂q¯/∂y), substantially hindering intensification of EC by −1.4 (yellow; −ω∂q/∂p) and −2.1 (green; −υ∂q¯/∂y) CVU (12 h)−1, respectively.

Fig. 11.
Fig. 11.

As in Fig. 8, except the ζtint from basic and additional sets are rearranged in terms of physical processes. See text for details.

Citation: Journal of the Atmospheric Sciences 78, 6; 10.1175/JAS-D-20-0151.1

Among the dynamical processes, −u¯∂q′/∂x has the largest contribution to the EC intensification. This contributes to EC intensification in two ways. First, due to a much larger zonal scale of advection in the upper level than in the lower level (see Figs. C1d and C2d), the propagating upper-level trough and ridge increase cyclonic vorticity over the EC center (Fig. 9a) with a zonal scale comparable to that of dipole tendency in Fig. 6b (i.e., the scale of cyclone migration). This indicates that differential advection can intensify EC, where analogies are found in QG frameworks when differential vorticity or temperature advections amplify vertical motion or geopotential tendency, respectively (Holton 2004). Second, the effect of −u¯∂q′/∂x is further strengthened by diabatic PV reduction over the upper-level ridge. This increases PV gradient on the east of the trough, resulting in stronger zonal PV advection over the EC center. It highlights the importance of two-way interaction between dynamic and thermodynamic processes in the EC intensification.

Fig. C1.
Fig. C1.

Horizontal composite of (a) −u∂q/∂x, (b) −υ∂q/∂y, (c) −ω∂q/∂p, (d) −u¯∂q′/∂x, (e) −υ∂q¯/∂y, (f) −ω∂q¯/∂p, (g) −u′∂q′/∂x, (h) −υ∂q′/∂y, and (i) −ω∂q′/∂p at 250 hPa as a representative of upper-tropospheric processes [shading; units: PVU (12 h)−1], with respect to EC center (red triangle) at tmax. In (a) and (d), the absolute values greater than or equal to 4 PVU (12 h)−1 are contoured with a 1 PVU (12 h)−1 interval. The values that are statistically significant at the 95% confidence level, based on the bootstrap resampling method, are dotted (near-zero values are not dotted for conciseness). In the first and second columns, PV anomalies at 250 hPa are also contoured in 0.4 PVU intervals starting from 0.6 PVU.

Citation: Journal of the Atmospheric Sciences 78, 6; 10.1175/JAS-D-20-0151.1

Fig. C2.
Fig. C2.

As in Fig. C1, but for 850 hPa as a representative of lower-tropospheric processes. In the first and second columns, PV anomalies at 850 hPa are also contoured in 0.2 PVU intervals starting from 0.2 PVU.

Citation: Journal of the Atmospheric Sciences 78, 6; 10.1175/JAS-D-20-0151.1

6. Summary and discussion

The current study examines the development processes of the northwest Pacific ECs in the cold season (October–April) by inverting the PV tendency equation. This is enabled by linearly approximating the PV anomaly, and combining its time derivative with the PV tendency equation. Then the terms in the PV tendency equation become piecewise PV tendencies, which are inverted to return respective geopotential tendencies. The 850-hPa geostrophic vorticity tendency, derived from the inverted geopotential tendency, is used in quantifying EC development.

It turns out that the PV tendency inversion reasonably reproduces the observed geostrophic vorticity tendency related to propagation and intensification of the EC, despite approximations. Focusing on the EC intensification, the contributing dynamic and thermodynamic processes are quantified. The upper-tropospheric PV advection by the mean zonal wind is the leading contributor to the intensification among dynamic processes. The nonlinear PV advections rather play a minor role in the EC intensification. Among the thermodynamic processes, the PV production from LH is the most prominent process that drives EC intensification. Unlike the leading contributor in the dynamic processes, this process occurs through anomalous moisture transport by anomalous flows in the lower troposphere. Collectively, the thermodynamic processes are 1.6 times more influential than the dynamic process.

The analyses presented in this study could be extended to any extratropical cyclone, since no assumption is made about cyclone intensity. Except for Seiler (2019), climatological quantification of the extratropical cyclone development in PV perspective has been obstructed due to the complicacy of nonlinear inversion and the computational burden of performing numerous inversions. While the linear operator in this study could be a remedy to the former, a slight modification of the method could also alleviate the latter with a trade-off of accuracy. For each sampled cyclone, the operator L would vary slightly. However, by neglecting this variation, the composite-mean geopotential tendency could be retrieved at once by inverting the composite-mean of PV tendency, instead of individual cases. This would reduce computation time by a factor of cyclone numbers, facilitating the climatological investigation of extratropical cyclone development processes.

The extratropical cyclone development in different climatological settings could also be evaluated by the method utilized in this study. This includes extratropical cyclone intensity in warming climate, which is yet unclear due to several countervailing factors (Shaw et al. 2016; Catto et al. 2019). However, there exists a consensus on the projected changes of the mean upper-level jet (u¯), meridional PV gradient (∂q¯/∂y), and static stability (−g∂θ¯/∂p). Since the influence of these variables to the cyclone development is directly computable, their impact on the extratropical cyclones in warm climate can be quantified.

The quantitative results could change with the method used. In the literature, the QG height tendency equation [Eq. (D1)] has been often used to evaluate the geopotential tendency (e.g., Hwang et al. 2020) and relative vorticity tendency. Taking the QG approach, however, substantially underestimates the LH effect as discussed in appendix D and shown in Fig. D1. This would deter quantitative analysis, particularly since a large fraction of EC development is driven by LH process. Another quantitative tool that is frequently used to determine EC development is the Zwack–Okossi equation (e.g., Yoshida and Asuma 2004). In this equation, the upper-level influence on the lower-level cyclone is represented through vertical integration. However, this introduces the difficulty of explaining the phase-shifted vertical interactions associated with EC development. In this regard, our method provides practical benefits as well as theoretical advantages by adhering to the PV framework.

Using a prognostic variable (PV tendency) to evaluate EC deepening could complicate the analysis since the effect of propagation has to be removed. Other than the method used here, there are alternative methods that could be used to isolate ζtint from local vorticity tendency. One is the quasi-Lagrangian framework (Sinclair and Revell 2000), and the other is the nine-point averaging (Lupo et al. 1992). In both methods, the vorticity tendency near the cyclone center (i.e., red triangle in Fig. 6b) is averaged. The former removes the effect of propagation by subtracting vorticity advection by translational velocity, and the latter excludes it by assuming it is zero near the cyclone center. Though the former is not applicable here since it disallows term-by-term analysis, qualitatively similar results are found when applying the latter (not shown). Nevertheless, our method may be preferable over the latter, since it well matches the actual Lagrangian tendency and provide additional information on propagation. Using the two boxes, each representing EC location at tmax − 6 h and tmax + 6 h, accounts for these benefits.

There are, however, a few caveats with respect to our method. The LH is calculated by assuming saturated condition (Emanuel et al. 1987) and excludes subgrid processes. Moist processes besides condensation (Attinger et al. 2019) and cloud radiative effects (Schäfer and Voigt 2018) are not quantified. The friction, which is also nonnegligible in the PV budget (Stoelinga 1996), is also not explicitly treated. These processes are collectively included in FRES with computational errors, which makes the interpretation of FRES vague. In the linear operator L(χ), σ is defined with monthly potential temperature climatology (θ¯). Although the magnitude of perturbed stability (|∂θ′/∂p|) is generally smaller than that of the mean stability (|∂θ¯/∂p|), they can be comparable near the surface. Since σ determines the vertical extent of the inverted solution, using mean stability could underestimate the contribution of the surface temperature tendency to the EC intensification, relating it more to propagation instead. The areas for spatial averaging are somewhat subjectively selected. They may need to be adjusted by considering the size of each cyclone.

Acknowledgments

The authors thank the four anonymous reviewers for their valuable comments. The authors also thank Heini Wernli and Michael Sprenger of ETH Zurich for providing the extratropical cyclone identifying and tracking algorithm. This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIT) (NRF2018R1A5A1024958).

Data availability statement

The ERA-Interim data are accessible at https://www.ecmwf.int/en/forecasts/dataset/ecmwf-reanalysis-interim, and the authors downloaded them from the ECMWF Data Server, using the ECMWF WebAPI.

APPENDIX A

Assumptions in the Linearization of PV

The first step in the linearization introduced in section 3a is to remove the second term in the rhs of Eq. (1), by assuming that it is smaller than the first term:
q≈−g⁡(ζ+f) ∂θ∂p.
In Eq. (A1), ∂θ/∂p can be decomposed into ∂θ¯/∂p+∂θ′/∂p, where the perturbation term is smaller than the mean term (i.e., |∂θ¯/∂p|≫|∂θ′/∂p|). Neglecting the nonlinear term ζ∂θ′/∂p (Sprenger 2007), Eq. (A1) is approximated as follows:
q≈−gζ ∂θ¯∂p−gf ∂θ¯∂p−gf ∂θ′∂p.
Let θ′ = −[p/(Rdπ)](∂ϕ′/∂p) from the hydrostatic balance. Then
q≈−g ∂θ¯∂p[ζ+f−f∂θ¯∂p∂∂p⁡(pRdπ∂ϕ′∂p)].
By assuming that the wind is close to geostrophic and p/(Rdπ) is constant in the vertical, Eq. (A3) can be expressed as
q≈−g ∂θ¯∂p⁡(1f0 ∇p2ϕ+f+fσ∂2ϕ′∂p2),
where σ=−(Rdπ/p)⁡(∂θ¯/∂p). The rhs of Eq. (A4) is the linearized PV, qL, and it follows that
q′≈−g ∂θ¯∂p⁡(1f0∇p2ϕ′+fσ∂2ϕ′∂p2).
Note that rhs of Eq. (A5), qL′, is identical to that of Eq. (3).

The linearization process involves scale analyses, hydrostatic balance, and geostrophic assumption. The geostrophic assumption holds valid only for a system with a small Rossby number, and this may not necessarily be the case for each cyclone. However, when dealing with composites of multiple ECs, as in this study, the effects of ageostrophic components become somewhat obscured compared to the geostrophic components. Figure A1 illustrates the vertical cross sections of q′ from Eq. (1) and approximated qL′ in Eq. (3) or (A5) at the time of maximum EC deepening. Not surprisingly, the q′ is quantitatively similar to the approximated qL′. This similarity suggests that the linearized PV retains the essence of the circulation anomalies related to EC deepening, providing the advantages of linearity.

Fig. A1.
Fig. A1.

Vertical cross section of (a) q′from Eq. (1) and (b) qL′ from Eq. (3) (shading; units: PVU) with respect to the EC center (red triangle) at tmax.

Citation: Journal of the Atmospheric Sciences 78, 6; 10.1175/JAS-D-20-0151.1

Owing to the similarity between q and qL (also q′ and qL′), qL can be substituted into Eq. (5) instead of q as
L⁡(χ)=−u⋅∇qL+Q+F,
which is an alternative use of the linear approximation. Though not shown, the inversions based on Eq. (A6) point to the results similar to those shown in Fig. 8. In this approach, the effects of horizontal advections are larger than in Fig. 8, due to larger qL′ than q′ in the upper troposphere (Fig. A1). However, this approach is less accurate, since the inversion of the rhs is 10% larger than that of the lhs in Eq. (A6).

APPENDIX B

Details of the Inversion Calculation

Each inversion is carried out within a cubic domain, where the top and bottom of the domain is set to 150 and 1000 hPa, respectively. Horizontally, the domain spans ±30° zonally and ±15° meridionally about the EC center at tmax. The horizontal margin is sufficiently wide so that the choice of the lateral boundary condition has a negligible impact on the result.

Some minor manipulations are made to compute the advection and diabatic heating terms. Since the calculation would be prone to numerical phase error coming from finite differencing, the advecting winds at tmax are 1–2–1 filtered in time [i.e., −u⋅∇q=−(1/4)⁡(utmax−6h+2utmax+utmax+6h)⋅∇q]. The QLH term at tmax is also calculated as the average of QLH at tmax and tmax + 6 h, i.e., QLH=(1/2)⁡(QLH,tmax+QLH,tmax+6h), to best represent the amplifying system. To smooth out the shortwave noise arising from finite differencing, these terms are also 1–2–1 filtered in space.

The inversion is carried out using the successive overrelaxation method, with vertically and latitudinally dependent relaxation factor. The convergence of the numerical solution is determined at the nth iteration if |[χ]n − [χ]n−1| < 10−7 m2 s−3 is reached, where [⋅] denotes average over inversion domain.

APPENDIX C

Details of Upper- and Lower-Tropospheric PV Advections

The structures of PV advection terms in the additional set (Table 2) are shown in Figs. C1 and C2 for the upper and lower troposphere, respectively. In the upper troposphere (250 hPa), the positive tendency from zonal advection over the EC center (Fig. C1a) is the greatest among the advection terms (Figs. C1a–c). The intense zonal advection is mostly caused by the strong mean wind (u¯) that advects the PV anomalies over the EC center [~10 PVU (12 h)−1, Fig. C1d]. This strong positive tendency is also attributed to the asymmetry of the PV anomalies, where the zonal gradient is larger on the east of the maximum PV by the diabatic PV reduction at the ridge. The nonlinear zonal advection (−u′∂q′/∂x) has a negligible contribution to this positive tendency (Fig. C1g) and is largely canceled out with the nonlinear meridional advection (−υ∂q′/∂y) (Fig. C1h). Considering the climatological distribution of PV (∂q¯/∂y>0 and ∂q¯/∂p<0), the anomalous southerly wind by the trough and the ascending motion slightly east to the EC center brings negative tendency [~−2 PVU (12 h)−1, Figs. C1e,f]. These advections associated with climatological PV distribution dominate the negative tendencies produced by total meridional and vertical advections (Figs. C1b,c). To summarize, the mean flow has dominant influence on the PV tendency as compared to the anomalous interactions in the upper troposphere.

Figure C2a demonstrates the zonal advection of PV at 850 hPa. The positive PV tendency east to the EC center is again mostly explained by the zonal advection by the mean flow (Fig. C2d). This represents the zonal migration of the EC by the mean wind. The nonlinear horizontal advections (−u′∂q′/∂x and −υ∂q′/∂y) are out of phase and mostly cancel out as in the upper level [see also Fig. 7 in Tamarin and Kaspi (2016)], though a negative residual remains southeast to the EC center (Figs. C2g,h). Since the climatological meridional PV gradient is weaker in the lower troposphere, only a weak negative tendency is induced from the meridional advection of climatological PV (−υ∂q¯/∂y) (Fig. C2e). By the ascending motion near the EC center at 850 hPa, both climatologically low and anomalously high PV are advected from the lower levels, somewhat offsetting each other (Figs. C2f,i). In sum, the effect of the climatological PV distribution overwhelms that of anomalous distribution in the vertical (Fig. C2c).

APPENDIX D

QG Height Tendency Equation

It is more common for the QG height tendency equation to be used on the rhs of Eq. (5) instead of the PV tendency equation. The QG height tendency equation, scaled by −g∂θ¯/∂p, is formulated as following
L⁡(χ)=−g ∂θ¯∂p⁡([−Vg⋅∇p⁡(1f0 ∇p2ϕ+f)]+f0 ∂∂p{−Vg⋅∇p[1σ⁡(∂ϕ′∂p)]}+f0 ∂∂p⁡(θ˙∂θ¯∂p))+FQG,
where Vg = (ug, υg) is the geostrophic wind vector and FQG is the effect of friction. The effect of LH (QLH,QG) can be expressed as follows:
QLH,QG=−g ∂θ¯∂pf0∂∂p⁡(θ˙LH∂θ¯∂p).
It is important to note that Eq. (D2) differs from Eq. (7). This QG estimation typically underestimates LH, as exemplified in Fig. D1.
Fig. D1.
Fig. D1.

Vertical cross section of the rhs of (a) Eq. (7) and (b) Eq. (D2) [shading; units: PVU (12 h)−1] with respect to the EC center (red triangle) at tmax.

Citation: Journal of the Atmospheric Sciences 78, 6; 10.1175/JAS-D-20-0151.1

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    • Search Google Scholar
    • Export Citation

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Journal of the Atmospheric Sciences

  • Ahmadi-Givi, F., G. C. Craig, and R. S. Plant, 2004: The dynamics of a midlatitude cyclone with very strong latent-heat release. Quart. J. Roy. Meteor. Soc., 130, 295–323, https://doi.org/10.1256/qj.02.226.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Attinger, R., E. Spreitzer, M. Boettcher, R. Forbes, H. Wernli, and H. Joos, 2019: Quantifying the role of individual diabatic processes for the formation of PV anomalies in a North Pacific cyclone. Quart. J. Roy. Meteor. Soc., 145, 2454–2476, https://doi.org/10.1002/qj.3573.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Bosart, L. F., 1981: The Presidents’ Day snowstorm of 18–19 February 1979: A subsynoptic-scale event. Mon. Wea. Rev., 109, 1542–1566, https://doi.org/10.1175/1520-0493(1981)109<1542:TPDSOF>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Bretherton, F. P., 1966: Critical layer instability in baroclinic flows. Quart. J. Roy. Meteor. Soc., 92, 325–334, https://doi.org/10.1002/qj.49709239302.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Büeler, D., and S. Pfahl, 2017: Potential vorticity diagnostics to quantify effects of latent heating in extratropical cyclones. Part I: Methodology. J. Atmos. Sci., 74, 3567–3590, https://doi.org/10.1175/JAS-D-17-0041.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Catto, J. L., and Coauthors, 2019: The future of midlatitude cyclones. Curr. Climate Change Rep., 5, 407–420, https://doi.org/10.1007/s40641-019-00149-4.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Charney, J. G., and M. E. Stern, 1962: On the stability of internal baroclinic jets in a rotating atmosphere. J. Atmos. Sci., 19, 159–172, https://doi.org/10.1175/1520-0469(1962)019<0159:OTSOIB>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Chen, S.-J., Y.-H. Kuo, P.-Z. Zhang, and Q.-F. Bai, 1992: Climatology of explosive cyclones off the East Asian coast. Mon. Wea. Rev., 120, 3029–3035, https://doi.org/10.1175/1520-0493(1992)120<3029:COECOT>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Davis, C. A., 1992: Piecewise potential vorticity inversion. J. Atmos. Sci., 49, 1397–1411, https://doi.org/10.1175/1520-0469(1992)049<1397:PPVI>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Davis, C. A., and K. A. Emanuel, 1991: Potential vorticity diagnostics of cyclogenesis. Mon. Wea. Rev., 119, 1929–1953, https://doi.org/10.1175/1520-0493(1991)119<1929:PVDOC>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Dee, D. P., and Coauthors, 2011: The ERA-Interim reanalysis: Configuration and performance of the data assimilation system. Quart. J. Roy. Meteor. Soc., 137, 553–597, https://doi.org/10.1002/qj.828.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Efron, B., and R. J. Tibshirani, 1993: An Introduction to the Bootstrap. Chapman and Hall/CRC, 456 pp.

    • Crossref
    • Export Citation

  • Emanuel, K. A., M. Fantini, and A. J. Thorpe, 1987: Baroclinic instability in an environment of small stability to slantwise moist convection. Part I: Two-dimensional models. J. Atmos. Sci., 44, 1559–1573, https://doi.org/10.1175/1520-0469(1987)044<1559:BIIAEO>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Ertel, H., 1942: Ein neuer hydrodynamischer Erhaltungssatz. Naturwissenschaften, 30, 543–544, https://doi.org/10.1007/BF01475602.

  • Fink, A. H., S. Pohle, J. G. Pinto, and P. Knippertz, 2012: Diagnosing the influence of diabatic processes on the explosive deepening of extratropical cyclones. Geophys. Res. Lett., 39, L07803, https://doi.org/10.1029/2012GL051025.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Gyakum, J. R., 1983: On the evolution of the QE II storm. Part I: Synoptic aspects. Mon. Wea. Rev., 111, 1137–1155, https://doi.org/10.1175/1520-0493(1983)111<1137:OTEOTI>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Hawcroft, M. K., L. C. Shaffrey, K. I. Hodges, and H. F. Dacre, 2012: How much Northern Hemisphere precipitation is associated with extratropical cyclones? Geophys. Res. Lett., 39, L24809, https://doi.org/10.1029/2012GL053866.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Hirata, H., R. Kawamura, M. Kato, and T. Shinoda, 2015: Influential role of moisture supply from the Kuroshio/Kuroshio Extension in the rapid development of an extratropical cyclone. Mon. Wea. Rev., 143, 4126–4144, https://doi.org/10.1175/MWR-D-15-0016.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Holton, J. R., 2004: An Introduction to Dynamic Meteorology. 4th ed. Elsevier Academic Press, 535 pp.

  • Hoskins, B. J., M. E. McIntyre, and A. W. Robertson, 1985: On the use and significance of isentropic potential vorticity maps. Quart. J. Roy. Meteor. Soc., 111, 877–946, https://doi.org/10.1002/qj.49711147002.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Hwang, J., P. Martineau, S. Son, T. Miyasaka, and H. Nakamura, 2020: The role of transient eddies in North Pacific blocking formation and its seasonality. J. Atmos. Sci., 77, 2453–2470, https://doi.org/10.1175/JAS-D-20-0011.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Iwao, K., M. Inatsu, and M. Kimoto, 2012: Recent changes in explosively developing extratropical cyclones over the winter northwestern Pacific. J. Climate, 25, 7282–7296, https://doi.org/10.1175/JCLI-D-11-00373.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Kang, J. M., J. Lee, S.-W. Son, J. Kim, and D. Chen, 2020: The rapid intensification of East Asian cyclones around the Korean Peninsula and their surface impacts. J. Geophys. Res. Atmos., 125, e2019JD031632, https://doi.org/10.1029/2019JD031632.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Klawa, M., and U. Ulbrich, 2003: A model for the estimation of storm losses and the identification of severe winter storms in Germany. Nat. Hazards Earth Syst. Sci., 3, 725–732, https://doi.org/10.5194/nhess-3-725-2003.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Kuo, Y.-H., R. J. Reed, and S. Low-Nam, 1991: Effects of surface energy fluxes during the early development and rapid intensification stages of seven explosive cyclones in the western Atlantic. Mon. Wea. Rev., 119, 457–476, https://doi.org/10.1175/1520-0493(1991)119<0457:EOSEFD>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Lee, J., S.-W. Son, H.-O. Cho, J. Kim, D.-H. Cha, J. R. Gyakum, and D. Chen, 2019: Extratropical cyclones over East Asia: Climatology, seasonal cycle, and long-term trend. Climate Dyn., 54, 1131–1144, https://doi.org/10.1007/s00382-019-05048-w.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Lupo, A. R., P. J. Smith, and P. Zwack, 1992: A diagnosis of the explosive development of two extratropical cyclones. Mon. Wea. Rev., 120, 1490–1523, https://doi.org/10.1175/1520-0493(1992)120<1490:ADOTED>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Nielsen-Gammon, J. W., and R. J. Lefevre, 1996: Piecewise tendency diagnosis of dynamical processes governing the development of an upper-tropospheric mobile trough. J. Atmos. Sci., 53, 3120–3142, https://doi.org/10.1175/1520-0469(1996)053<3120:PTDODP>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Reader, M. C., and G. W. K. Moore, 1995: Stratosphere–troposphere interactions associated with a case of explosive cyclogenesis in the Labrador Sea. Tellus, 47A, 849–863, https://doi.org/10.3402/tellusa.v47i5.11579.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Roebber, P. J., 1984: Statistical analysis and updated climatology of explosive cyclones. Mon. Wea. Rev., 112, 1577–1589, https://doi.org/10.1175/1520-0493(1984)112<1577:SAAUCO>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Sanders, F., 1986: Explosive cyclogenesis in the west-central North Atlantic Ocean, 1981–84. Part I: Composite structure and mean behavior. Mon. Wea. Rev., 114, 1781–1794, https://doi.org/10.1175/1520-0493(1986)114<1781:ECITWC>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Sanders, F., and J. R. Gyakum, 1980: Synoptic-dynamic climatology of the “bomb.” Mon. Wea. Rev., 108, 1589–1606, https://doi.org/10.1175/1520-0493(1980)108<1589:SDCOT>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Schäfer, S. A. K., and A. Voigt, 2018: Radiation weakens idealized midlatitude cyclones. Geophys. Res. Lett., 45, 2833–2841, https://doi.org/10.1002/2017GL076726.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Seiler, C., 2019: A climatological assessment of intense extratropical cyclones from the potential vorticity perspective. J. Climate, 32, 2369–2380, https://doi.org/10.1175/JCLI-D-18-0461.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Shaw, T. A., and Coauthors, 2016: Storm track processes and the opposing influences of climate change. Nat. Geosci., 9, 656–664, https://doi.org/10.1038/ngeo2783.

    • Crossref
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  • <sc>Fig</sc>. 1.
    View in gallery
    Fig. 1.

    The frequency of (a) extratropical cyclones and (b) explosive cyclones (ECs), and (c) explosive deepening location of ECs over the northwest Pacific (shading; units: number per year). The black box in (c) refers to the target domain, and the blue lines indicate the longitudinal range (135°–165°E) for the zonally averaged quantities described in section 4b.

  • <sc>Fig</sc>. 2.
    View in gallery
    Fig. 2.

    (a) Monthly variation in the number of extratropical cyclones with respect to their DRmax. The red and white numbers denote the number of ECs and non-ECs each month. (b) Change in central pressure (solid lines; left axis) and DR (dashed lines; right axis) with respect to tmax. The colored lines represent individual months. (c) As in (b), but for ζg at 850 hPa (solid lines; left axis) and its intensification rate (dashed lines; right axis). The ζg is defined as the average of the geostrophic vorticity within the 6° × 6° box around the EC center.

  • <sc>Fig</sc>. 3.
    View in gallery
    Fig. 3.

    (a) PV anomalies at 250 hPa (shading; units: PVU) and 850 hPa (black contours; units: PVU), IVT anomalies (blue contours; units: kg m−1 s−1), and 250-hPa geopotential height (gray contours; 150-m interval) with respect to the EC center (red triangle) at tmax. (b) As in (a), but for SLP anomalies (shading; units: hPa), 850-hPa anomalous wind vectors (black vectors; units: m s−1), 1000-hPa temperature anomalies (red contours; for values larger than 3 K), and 1000-hPa isotherms (gray contours; 3-K interval). (c) Vertical cross section of PV (shading; units: PVU) and latent heating rate [red contours; units: K (12 h)−1]. The PV anomalies with absolute values greater than or equal to 0.1 PVU are contoured in gray, and those greater than 0.5 are contoured in black. The 2-PVU line is shown in white as a reference. In (a) and (b), only the anomalies that are statistically significant at the 95% confidence level, based on a two-tailed Student’s t test, are shown. In (c), statistically significant PV anomalies with absolute values greater than or equal to 0.1 PVU are dotted.

  • <sc>Fig</sc>. 4.
    View in gallery
    Fig. 4.

    Vertical cross section of (a) L(χ), (b) ∂q/∂t, (c) −u∂q/∂x, (d) −υ∂q/∂y, (e) −ω∂q/∂p, and (f) QLH [shading; units: PVU (12 h)−1] with respect to the EC center (red triangle) at tmax as in Fig. 3c. In (c) and (d), the 600-hPa level is plotted in black dashed line, and absolute values greater than or equal to 2 PVU (12 h)−1 are contoured in 1-PVU interval. The values that are statistically significant at the 95% confidence level, based on the bootstrap resampling method, are dotted (near-zero values are not dotted for conciseness).

  • <sc>Fig</sc>. 5.
    View in gallery
    Fig. 5.

    Temperature tendency at 975 hPa [shading; units: K (12 h)−1] with respect to the EC center (red triangle) at tmax. The values that are statistically significant at the 95% confidence level, based on the bootstrap resampling method, are dotted (near-zero values are not dotted for conciseness). The temperature anomalies are also contoured in 1 K interval starting from 3 K.

  • <sc>Fig</sc>. 6.
    View in gallery
    Fig. 6.

    (a) ζt from reanalysis data [shading; units: CVU (12 h)−1] with respect to the EC center (red triangle) at tmax. (b)–(g) As in (a), but for ζt achieved from the basic set. (h) As in (a), but for ζt achieved by the sum of six piecewise inversions including FRES. The values that are statistically significant at the 95% confidence level, based on the bootstrap resampling method, are dotted (near-zero values are not dotted for conciseness). The 6° × 6° box centered at the maximum (white circle, red box) and minimum (black circle, blue box) ζt are denoted in (b).

  • <sc>Fig</sc>. 7.
    View in gallery
    Fig. 7.

    The ζtmax, ζtint, and ζtpro [bars; units: CVU (12 h)−1] from reanalysis, full inversion, and sum of piecewise inversions with 95% confidence intervals calculated from the bootstrap resampling method. The values denoted to each bar refer to the relative contribution (%) to the ζtmax from ∂q/∂t. The numbers in the parenthesis refer to the confidence intervals (%).

  • <sc>Fig</sc>. 8.
    View in gallery
    Fig. 8.

    The ζtint from basic set with 95% confidence intervals calculated from the bootstrap resampling method. The values denoted to each bar refer to the relative contribution (%) to the ζtint from ∂q/∂t. The numbers in the parenthesis refer to the confidence intervals (%).

  • <sc>Fig</sc>. 9.
    View in gallery
    Fig. 9.

    As in Fig. 6, but for ζt achieved from the additional set.

  • <sc>Fig</sc>. 10.
    View in gallery
    Fig. 10.

    As in Fig. 8, but for ζtint from the additional set. The values denoted for each bar refer to the relative contribution (%) to the ζtint from ∂q/∂t in Fig. 8. The left and right bars for each term represent the upper- and lower-tropospheric contributions, respectively.

  • <sc>Fig</sc>. 11.
    View in gallery
    Fig. 11.

    As in Fig. 8, except the ζtint from basic and additional sets are rearranged in terms of physical processes. See text for details.

  • <sc>Fig</sc>. C1.
    View in gallery
    Fig. C1.

    Horizontal composite of (a) −u∂q/∂x, (b) −υ∂q/∂y, (c) −ω∂q/∂p, (d) −u¯∂q′/∂x, (e) −υ∂q¯/∂y, (f) −ω∂q¯/∂p, (g) −u′∂q′/∂x, (h) −υ∂q′/∂y, and (i) −ω∂q′/∂p at 250 hPa as a representative of upper-tropospheric processes [shading; units: PVU (12 h)−1], with respect to EC center (red triangle) at tmax. In (a) and (d), the absolute values greater than or equal to 4 PVU (12 h)−1 are contoured with a 1 PVU (12 h)−1 interval. The values that are statistically significant at the 95% confidence level, based on the bootstrap resampling method, are dotted (near-zero values are not dotted for conciseness). In the first and second columns, PV anomalies at 250 hPa are also contoured in 0.4 PVU intervals starting from 0.6 PVU.

  • <sc>Fig</sc>. C2.
    View in gallery
    Fig. C2.

    As in Fig. C1, but for 850 hPa as a representative of lower-tropospheric processes. In the first and second columns, PV anomalies at 850 hPa are also contoured in 0.2 PVU intervals starting from 0.2 PVU.

  • <sc>Fig</sc>. A1.
    View in gallery
    Fig. A1.

    Vertical cross section of (a) q′from Eq. (1) and (b) qL′ from Eq. (3) (shading; units: PVU) with respect to the EC center (red triangle) at tmax.

  • <sc>Fig</sc>. D1.
    View in gallery
    Fig. D1.

    Vertical cross section of the rhs of (a) Eq. (7) and (b) Eq. (D2) [shading; units: PVU (12 h)−1] with respect to the EC center (red triangle) at tmax.

Fig. 1.

The frequency of (a) extratropical cyclones and (b) explosive cyclones (ECs), and (c) explosive deepening location of ECs over the northwest Pacific (shading; units: number per year). The black box in (c) refers to the target domain, and the blue lines indicate the longitudinal range (135°–165°E) for the zonally averaged quantities described in section 4b.
Fig. 2.

(a) Monthly variation in the number of extratropical cyclones with respect to their DRmax. The red and white numbers denote the number of ECs and non-ECs each month. (b) Change in central pressure (solid lines; left axis) and DR (dashed lines; right axis) with respect to tmax. The colored lines represent individual months. (c) As in (b), but for ζg at 850 hPa (solid lines; left axis) and its intensification rate (dashed lines; right axis). The ζg is defined as the average of the geostrophic vorticity within the 6° × 6° box around the EC center.
Fig. 3.

(a) PV anomalies at 250 hPa (shading; units: PVU) and 850 hPa (black contours; units: PVU), IVT anomalies (blue contours; units: kg m−1 s−1), and 250-hPa geopotential height (gray contours; 150-m interval) with respect to the EC center (red triangle) at tmax. (b) As in (a), but for SLP anomalies (shading; units: hPa), 850-hPa anomalous wind vectors (black vectors; units: m s−1), 1000-hPa temperature anomalies (red contours; for values larger than 3 K), and 1000-hPa isotherms (gray contours; 3-K interval). (c) Vertical cross section of PV (shading; units: PVU) and latent heating rate [red contours; units: K (12 h)−1]. The PV anomalies with absolute values greater than or equal to 0.1 PVU are contoured in gray, and those greater than 0.5 are contoured in black. The 2-PVU line is shown in white as a reference. In (a) and (b), only the anomalies that are statistically significant at the 95% confidence level, based on a two-tailed Student’s t test, are shown. In (c), statistically significant PV anomalies with absolute values greater than or equal to 0.1 PVU are dotted.
Fig. 4.

Vertical cross section of (a) L(χ), (b) ∂q/∂t, (c) −u∂q/∂x, (d) −υ∂q/∂y, (e) −ω∂q/∂p, and (f) QLH [shading; units: PVU (12 h)−1] with respect to the EC center (red triangle) at tmax as in Fig. 3c. In (c) and (d), the 600-hPa level is plotted in black dashed line, and absolute values greater than or equal to 2 PVU (12 h)−1 are contoured in 1-PVU interval. The values that are statistically significant at the 95% confidence level, based on the bootstrap resampling method, are dotted (near-zero values are not dotted for conciseness).
Fig. 5.

Temperature tendency at 975 hPa [shading; units: K (12 h)−1] with respect to the EC center (red triangle) at tmax. The values that are statistically significant at the 95% confidence level, based on the bootstrap resampling method, are dotted (near-zero values are not dotted for conciseness). The temperature anomalies are also contoured in 1 K interval starting from 3 K.
Fig. 6.

(a) ζt from reanalysis data [shading; units: CVU (12 h)−1] with respect to the EC center (red triangle) at tmax. (b)–(g) As in (a), but for ζt achieved from the basic set. (h) As in (a), but for ζt achieved by the sum of six piecewise inversions including FRES. The values that are statistically significant at the 95% confidence level, based on the bootstrap resampling method, are dotted (near-zero values are not dotted for conciseness). The 6° × 6° box centered at the maximum (white circle, red box) and minimum (black circle, blue box) ζt are denoted in (b).
Fig. 7.

The ζtmax, ζtint, and ζtpro [bars; units: CVU (12 h)−1] from reanalysis, full inversion, and sum of piecewise inversions with 95% confidence intervals calculated from the bootstrap resampling method. The values denoted to each bar refer to the relative contribution (%) to the ζtmax from ∂q/∂t. The numbers in the parenthesis refer to the confidence intervals (%).
Fig. 8.

The ζtint from basic set with 95% confidence intervals calculated from the bootstrap resampling method. The values denoted to each bar refer to the relative contribution (%) to the ζtint from ∂q/∂t. The numbers in the parenthesis refer to the confidence intervals (%).
Fig. 9.

As in Fig. 6, but for ζt achieved from the additional set.
Fig. 10.

As in Fig. 8, but for ζtint from the additional set. The values denoted for each bar refer to the relative contribution (%) to the ζtint from ∂q/∂t in Fig. 8. The left and right bars for each term represent the upper- and lower-tropospheric contributions, respectively.
Fig. 11.

As in Fig. 8, except the ζtint from basic and additional sets are rearranged in terms of physical processes. See text for details.
Fig. C1.

Horizontal composite of (a) −u∂q/∂x, (b) −υ∂q/∂y, (c) −ω∂q/∂p, (d) −u¯∂q′/∂x, (e) −υ∂q¯/∂y, (f) −ω∂q¯/∂p, (g) −u′∂q′/∂x, (h) −υ∂q′/∂y, and (i) −ω∂q′/∂p at 250 hPa as a representative of upper-tropospheric processes [shading; units: PVU (12 h)−1], with respect to EC center (red triangle) at tmax. In (a) and (d), the absolute values greater than or equal to 4 PVU (12 h)−1 are contoured with a 1 PVU (12 h)−1 interval. The values that are statistically significant at the 95% confidence level, based on the bootstrap resampling method, are dotted (near-zero values are not dotted for conciseness). In the first and second columns, PV anomalies at 250 hPa are also contoured in 0.4 PVU intervals starting from 0.6 PVU.
Fig. C2.

As in Fig. C1, but for 850 hPa as a representative of lower-tropospheric processes. In the first and second columns, PV anomalies at 850 hPa are also contoured in 0.2 PVU intervals starting from 0.2 PVU.
Fig. A1.

Vertical cross section of (a) q′from Eq. (1) and (b) qL′ from Eq. (3) (shading; units: PVU) with respect to the EC center (red triangle) at tmax.
Fig. D1.

Vertical cross section of the rhs of (a) Eq. (7) and (b) Eq. (D2) [shading; units: PVU (12 h)−1] with respect to the EC center (red triangle) at tmax.
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