## 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.

*q*, is expressed as

*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).

**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).

*t*is calculated in the unit of Bergeron (B) for each cyclone (Sanders and Gyakum 1980):

*φ*

_{t}is the latitude of the cyclone at time step

*t*. The maximum DR (DR

_{max}) and the corresponding time (

*t*

_{max}) are determined for each cyclone, and the cyclones that have DR

_{max}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

*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):

*L*, defined as

*f*

_{0}is the planetary vorticity at the center of each EC, ∇

_{p}= (∂/∂

*x*, ∂/∂

*y*, 0) is the horizontal gradient operator,

*R*

_{d}is the gas constant of dry air,

*c*

_{p}is the specific heat of dry air under constant pressure, and

*p*

_{s}= 1000 hPa is the surface pressure. The overbar denotes the zonal and time mean (e.g.,

*q*′and

### b. Application of PV tendency equation

*q*/∂

*t*= ∂

*q*′/∂

*t*):

*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

*N*pieces since

*L*is a linear operator:

*i*denotes the

*i*th partition of the PV tendency, when

*q*/∂

*t*)

_{i}, and they sum up to the total geopotential tendency in the domain when all partitions are taken into account.

*process-based partitioning*is introduced in this study. This is enabled by equating Eqs. (2) and (4), which leads to the following:

*q*/∂

*t*, serve as the partitions of ∂

*q*/∂

*t*. For example, regarding −

*u*∂

*q*/∂

*x*as (∂

*q*/∂

*t*)

_{1},

*χ*

_{1}=

*L*

^{−1}(−

*u*∂

*q*/∂

*x*) is the geopotential tendency due to zonal PV advection.

*Q*=

*Q*

_{LH}+

*Q*

_{RES}, to investigate the pivotal contribution of LH exclusively and explicitly among other diabatic processes. Focusing on the vertical distribution of LH,

*Q*

_{LH}is calculated as follows (Tamarin and Kaspi 2016):

*θ*

_{e}is the equivalent potential temperature, and

*γ*

_{d}and

*γ*

_{m}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

*Q*

_{RES}. Then

*F*

_{RES}=

*Q*

_{RES}+

*F*is the PV tendency from all nonconservative processes that are not explicitly assessable.

It is recognizable that *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 (*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

*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:

*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 *t*_{max}. 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.

Summary of the terms used in the basic set.

## 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.

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 *t*_{max}. 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.

(a) Monthly variation in the number of extratropical cyclones with respect to their DR_{max}. 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 *t*_{max}. The colored lines represent individual months. (c) As in (b), but for *ζ*_{g} at 850 hPa (solid lines; left axis) and its intensification rate (dashed lines; right axis). The *ζ*_{g} 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

(a) Monthly variation in the number of extratropical cyclones with respect to their DR_{max}. 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 *t*_{max}. The colored lines represent individual months. (c) As in (b), but for *ζ*_{g} at 850 hPa (solid lines; left axis) and its intensification rate (dashed lines; right axis). The *ζ*_{g} 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

(a) Monthly variation in the number of extratropical cyclones with respect to their DR_{max}. 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 *t*_{max}. The colored lines represent individual months. (c) As in (b), but for *ζ*_{g} at 850 hPa (solid lines; left axis) and its intensification rate (dashed lines; right axis). The *ζ*_{g} 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 *t*_{max}. 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 *t*_{max}, the average DR_{max} 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 (DR_{max} ≥ 2 B) shown in Fig. 2a.

The time evolution of geostrophic vorticity (*ζ*_{g}) at 850 hPa is also shown in Fig. 2c. Its intensification rate is defined as the 12-h difference of *ζ*_{g} along the EC track. During the 24 h of explosive deepening, *ζ*_{g} 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 *t*_{max}, where it is about 4.4 CVU (12 h)^{−1}. This justifies the use of *ζ*_{g} 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 *t*_{max} 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 *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.

(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 *t*_{max}. (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

(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 *t*_{max}. (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

(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 *t*_{max}. (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 *t*_{max}. 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 *t*_{max} as in Fig. 3c. The *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.

Vertical cross section of (a) *L*(*χ*), (b) ∂*q*/∂*t*, (c) −*u*∂*q*/∂*x*, (d) −*υ*∂*q*/∂*y*, (e) −*ω*∂*q*/∂*p*, and (f) *Q*_{LH} [shading; units: PVU (12 h)^{−1}] with respect to the EC center (red triangle) at *t*_{max} 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

Vertical cross section of (a) *L*(*χ*), (b) ∂*q*/∂*t*, (c) −*u*∂*q*/∂*x*, (d) −*υ*∂*q*/∂*y*, (e) −*ω*∂*q*/∂*p*, and (f) *Q*_{LH} [shading; units: PVU (12 h)^{−1}] with respect to the EC center (red triangle) at *t*_{max} 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

Vertical cross section of (a) *L*(*χ*), (b) ∂*q*/∂*t*, (c) −*u*∂*q*/∂*x*, (d) −*υ*∂*q*/∂*y*, (e) −*ω*∂*q*/∂*p*, and (f) *Q*_{LH} [shading; units: PVU (12 h)^{−1}] with respect to the EC center (red triangle) at *t*_{max} 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 *t*_{max}. 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.

Temperature tendency at 975 hPa [shading; units: K (12 h)^{−1}] with respect to the EC center (red triangle) at *t*_{max}. 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

Temperature tendency at 975 hPa [shading; units: K (12 h)^{−1}] with respect to the EC center (red triangle) at *t*_{max}. 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

Temperature tendency at 975 hPa [shading; units: K (12 h)^{−1}] with respect to the EC center (red triangle) at *t*_{max}. 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 [*ζ*_{t}) retrieved from the basic set is shown in Fig. 6. The observed *ζ*_{t} (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 *t*_{max}. It is strongly analogous to *ζ*_{t} 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 *t*_{max} + 6 h.

(a) *ζ*_{t} from reanalysis data [shading; units: CVU (12 h)^{−1}] with respect to the EC center (red triangle) at *t*_{max}. (b)–(g) As in (a), but for *ζ*_{t} achieved from the basic set. (h) As in (a), but for *ζ*_{t} achieved by the sum of six piecewise inversions including *F*_{RES}. 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) *ζ*_{t} are denoted in (b).

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

(a) *ζ*_{t} from reanalysis data [shading; units: CVU (12 h)^{−1}] with respect to the EC center (red triangle) at *t*_{max}. (b)–(g) As in (a), but for *ζ*_{t} achieved from the basic set. (h) As in (a), but for *ζ*_{t} achieved by the sum of six piecewise inversions including *F*_{RES}. 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) *ζ*_{t} are denoted in (b).

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

(a) *ζ*_{t} from reanalysis data [shading; units: CVU (12 h)^{−1}] with respect to the EC center (red triangle) at *t*_{max}. (b)–(g) As in (a), but for *ζ*_{t} achieved from the basic set. (h) As in (a), but for *ζ*_{t} achieved by the sum of six piecewise inversions including *F*_{RES}. 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) *ζ*_{t} 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 *ζ*_{t} [~14 CVU (12 h)^{−1}] southeast to the EC center (Fig. 6c). The positive *ζ*_{t} from the zonal advection south to the EC center is somewhat countered by the negative *ζ*_{t} from the meridional advection (−*υ*∂*q*/∂*y*) (Fig. 6d). The vertical advection (−*ω*∂*q*/∂*p*) brings a weak negative *ζ*_{t} near the EC center (Fig. 6e). The PV tendency from LH (*Q*_{LH}) results in positive *ζ*_{t} [~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 *Q*_{LH} to EC development can be deduced. From the positive temperature tendency at surface (Fig. 5), a weak positive *ζ*_{t} is caused east to the EC center (Fig. 6g). Undoubtedly, the *ζ*_{t} from the six piecewise inversions, i.e., Figs. 6c–g and inversion of *F*_{RES} (not shown) sum up to that from the full inversion (cf. Figs. 6b,h).

### c. Processes contributing to the EC intensification

*t*

_{max}− 6 h to

*t*

_{max}+ 6 h. To better quantify the results, a 6° × 6° box centered on the maximum

*ζ*

_{t}is first identified for each EC (red box in Fig. 6b). The average of

*ζ*

_{t}inside the box is considered as the value representing EC development during 12 h and is simply noted as

*ζ*

_{t}in Fig. 6b. The dipole tendency indicates cyclone migration from

*t*

_{max}− 6 h to

*t*

_{max}+ 6 h, whereas the asymmetry implies an intensification of cyclone. Keeping this in mind,

*ζ*

_{t}averaged inside the 6° × 6° box centered at its minimum (blue box in Fig. 6b). A purely migratory process (symmetric dipole

*ζ*

_{t}) would have

The ^{−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 ^{−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,

The ^{−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 *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 ^{−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 *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 ^{−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 *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 *Q*_{LH} (cyan), by 5.1 CVU (12 h)^{−1}. Second to *Q*_{LH}, 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}].

The *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 *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 *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*, *Q*_{LH}, and (−*R*_{d}/*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 *F*_{RES} (gray). The large contribution of *Q*_{LH} (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.

Summary of the terms used in the additional set.

### d. Decomposed advection terms

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* 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 *ζ*_{t} 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 *ζ*_{t} is induced broadly around the EC center by *ζ*_{t} about the EC center [*ζ*_{t} from *ζ*_{t} east to the EC center [*ζ*_{t} produced by *ζ*_{t} (Figs. 9c,g). In the lower troposphere, they induce quadrupole *ζ*_{t} that is almost out of phase of one another (Figs. 9d,h). In sum, they produce negative *ζ*_{t} 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 *ζ*_{t} 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.

As in Fig. 6, but for *ζ*_{t} achieved from the additional set.

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

As in Fig. 6, but for *ζ*_{t} achieved from the additional set.

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

As in Fig. 6, but for *ζ*_{t} achieved from the additional set.

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

Further quantifying, each ^{−1}. This demonstrates that EC intensification through ^{−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].

As in Fig. 8, but for *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

As in Fig. 8, but for *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

As in Fig. 8, but for *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 ^{−1} from the mean zonal PV advection (red; *Q*_{LH}) strengthens EC by 5.1 CVU (12 h)^{−1}. The nonlinear horizontal advections (light green; *R*_{d}/*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 (*ω*∂*q*/∂*p*) and −2.1 (green; ^{−1}, respectively.

As in Fig. 8, except the

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

As in Fig. 8, except the

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

As in Fig. 8, except the

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

Among the dynamical processes,

Horizontal composite of (a) −*u*∂*q*/∂*x*, (b) −*υ*∂*q*/∂*y*, (c) −*ω*∂*q*/∂*p*, (d) *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 *t*_{max}. 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

Horizontal composite of (a) −*u*∂*q*/∂*x*, (b) −*υ*∂*q*/∂*y*, (c) −*ω*∂*q*/∂*p*, (d) *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 *t*_{max}. 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

Horizontal composite of (a) −*u*∂*q*/∂*x*, (b) −*υ*∂*q*/∂*y*, (c) −*ω*∂*q*/∂*p*, (d) *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 *t*_{max}. 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

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

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

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 (

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 *t*_{max} − 6 h and *t*_{max} + 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 *F*_{RES} with computational errors, which makes the interpretation of *F*_{RES} vague. In the linear operator *L*(*χ*), *σ* is defined with monthly potential temperature climatology (*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

*θ*/∂

*p*can be decomposed into

*ζ*∂

*θ*′/∂

*p*(Sprenger 2007), Eq. (A1) is approximated as follows:

*θ*′ = −[

*p*/(

*R*

_{d}

*π*)](∂

*ϕ*′/∂

*p*) from the hydrostatic balance. Then

*p*/(

*R*

_{d}

*π*) is constant in the vertical, Eq. (A3) can be expressed as

*q*

_{L}, and it follows that

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 *q*′ is quantitatively similar to the approximated

Vertical cross section of (a) *q*′from Eq. (1) and (b) *t*_{max}.

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

Vertical cross section of (a) *q*′from Eq. (1) and (b) *t*_{max}.

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

Vertical cross section of (a) *q*′from Eq. (1) and (b) *t*_{max}.

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

*q*and

*q*

_{L}(also

*q*′ and

*q*

_{L}can be substituted into Eq. (5) instead of

*q*as

*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 *t*_{max}. 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 *t*_{max} are 1–2–1 filtered in time [i.e., *Q*_{LH} term at *t*_{max} is also calculated as the average of *Q*_{LH} at *t*_{max} and *t*_{max} + 6 h, i.e.,

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 *n*th iteration if |[*χ*]^{n} − [*χ*]^{n−1}| < 10^{−7} m^{2} 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 (^{−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 (^{−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 (

## APPENDIX D

### QG Height Tendency Equation

**V**

_{g}= (

*u*

_{g},

*υ*

_{g}) is the geostrophic wind vector and

*F*

_{QG}is the effect of friction. The effect of LH (

*Q*

_{LH,QG}) can be expressed as follows:

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 *t*_{max}.

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

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 *t*_{max}.

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

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 *t*_{max}.

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

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