Multiscale Interactive Processes Underlying the Heavy Rainstorm Associated with a Landfalling Atmospheric River

: The heavy precipitation in Northern California—brought about by a landfalling atmospheric river (AR) on 25–27 February 2019—is investigated for an understanding of the underlying dynamical processes. By the peaks in hourly accumulation, this rainstorm can be divided into two stages (Stage I and Stage II). Using a recently developed multiscale analysis methodology, i.e., multiscale window transform (MWT), and the MWT-based theory of canonical transfer, the original ﬁelds are reconstructed onto three scale windows, namely, the background ﬂow, synoptic-scale and mesoscale windows, and the interactions among them are henceforth investigated. In both stages, the development of the precipitation is attributed to a vigorous buoyancy conversion and latent heating, and besides, the instability of the background ﬂow. In Stage I, the instability is baroclinic, while in Stage II, it is barotropic. Interestingly, in Stage I, the mesoscale kinetic energy is transferred to the background ﬂow where it is stored, and is released back in Stage II to the mesoscale window again, triggering intense precipitation.


Introduction
As an elongated and transient plume of strong horizontal water vapor transport, atmospheric rivers (ARs) not only are essential to the global water cycle [1,2], but also play an important role in the occurrence of extreme precipitation and hydrological hazards [3][4][5]. Particularly, landfalling ARs frequently give rise to extreme rainfall and flash flooding when they meet the topography [6][7][8][9][10][11][12]. Due to the linkage to different natural hazards, ARs have received more and more attention in recent years [13][14][15][16][17].
From the perspective of water supply, most ARs are beneficial because they can supply water vapor to alleviate the drought. For instance, Kim [18] found that ARs can account for over 70% of the winter precipitation in western United States. Dettinger et al. [3] reported that ARs may make contributions to more than 50% of the annual runoff over the west coast of North America. In other regions, similar results are obtained. However, in terms of hazardous weather, a few extreme ARs are damaging, incurring extreme rainfall. Numerous studies have demonstrated that the frequency of ARs is highly correlated to that of extreme precipitation events or flooding [3,4,6,10,19,20]. In Western Europe (e.g., Britain and Germany), ARs can increase the occurrence of flooding events by 40%, even up to 80% in some areas [4]. In East Asia, Kamae et al. [19] concluded that 20~90% of extreme rainfall events are associated with ARs during spring, summer and autumn. Kim et al. [11] further stated that the relationship between ARs and precipitation varies with seasons and regions.

Data
We utilized for this study the high-resolution data from the ERA5 reanalysis sets [26], with a temporal and horizontal resolution of 1 h and 0.25 • , respectively. The variables include geopotential (φ), temperature (T), three-dimensional wind vector (u, v, ω) and specific humidity (q), extending from 10 • N to 60 • N, from 180 • W to 100 • W. Vertically, there are 25 levels from 1000 to 50 hPa, with an interval of 25 hPa under 750 hPa and 50 hPa above 750 hPa. The temporal coverage is from 14 January to 9 April 2019.

Localized Multiscale Energetics Analysis
In the 1950s, Lorenz [27] derived the equations for the zonal-mean and eddy energy based on the Reynolds decomposition with respect to longitudes. This very successful and useful formalism, however, cannot reveal the zonally variable multiscale energetics. If the Reynolds decomposition with respect to time is performed, the resulting average and perturbation energies are invariant in time. To overcome the difficulty, during the past decades filtering has been widely utilized to fulfill the scale decomposition. In the literature, it is a common practice to write a multiscale energy simply as the square of the corresponding reconstructed (filtered) field (up to some constant factor). However, it is conceptually wrong to represent multiscale energy with filtered fields as clarified by Liang [24] since multiscale energy is a concept in phase space (e.g., the square of a Fourier coefficient with a Fourier transform) rather than in physical space. It is absolutely not equal to the square of the filtered/reconstructed variable. The phase space representation is related to its physical space counterpart through a renowned theorem called Parseval Relation [24,25].
To faithfully represent the time-varying multiscale energetics, Liang and Anderson [25] developed a new functional apparatus called multiscale window transform (MWT). With MWT, a function space is decomposed into a direct sum of several mutually orthogonal subspaces, each with an exclusive range of scales. Such a subspace is called a "scale window" [25].
For example, in a three-scale decomposition, u(t) can be decomposed as where u ∼0 (t), u ∼1 (t) and u ∼2 (t) stand for the low-pass, band-pass, and high-pass filtered component. Different from traditional filters, MWT can yield not only the filtered fields, but also the transform coefficients u ∼ n ( denotes any scale window, n stands for any time step), which allow for a representation of multiscale energies. By a theorem called Atmosphere 2022, 13, 29 3 of 15 "property of marginalization", Liang and Anderson [25] proved that the energy on scale window can be expressed as ( u ∼ n ) 2 (up to some factor). Note that this is not equal to the square of the filtered variable u ∼ (t).
In the framework of MWT, the kinetic energy (KE) and available potential energy (APE) on scale window at any time step n, denoted by K n and A n can be expressed as: where v h = (u, v) is the horizontal wind, T is the temperature departure from the vertical profile of the background temperature (averaged over time and area), is a constant which is proportional to the buoyancy frequency (g is the gravitational acceleration, c p is the specific heat at constant pressure, T is the background temperature, and L is the lapse rate). Based on the primitive equations, Liang [24] obtained the equations governing the kinetic energy (KE) and available potential energy (APE) on scale window ( = 0, 1, 2) (the subscript n is omitted for brevity): where v = (u, v, ω) is the 3D wind field, α is the specific volume. The operator (:) is defined such that (AB):(CD) = (A · C)(B · D) for two dyadic products AB and CD. Other notations are conventional. In Equation (4), −∇ · Q K is the spatial transport term of KE by advection on window , −∇ · Q P stands for the pressure work term, Γ K is a faithful representation of the transfer, i.e., canonical transfer, of KE to window from other windows, b denotes the buoyancy conversion between KE and APE on window , and F K is the frictional dissipation term, which is obtained by evaluating the residual in the balance. In Equation (5), −∇ · Q A is the spatial transport term of APE by advection on window , Γ A as above represents the canonical transfer of APE to window from other windows, and F A is the diabatic heating term.
As derived by Liang [24], the diabatic heating term F A is expressed as q net stands for the heating rate from all diabatic terms, including sensible heating, latent heating and radiation). To measure the latent heating from the phase transformation which is essential for precipitation, . q net is replaced with the latent heating rate Q L = −L v dq dt = −L v Fω following Shen et al. [28], Now the latent heating term F L can be calculated by where L v ≈ 2.5 × 10 6 J/kg is the specific heat of condensation, F = qT p L v R d −c p R w T c p R w T 2 +qL 2 v the condensation function, ω the vertical velocity, q specific humidity, R d ≈ 287.058 J/kg/K the specific gas constant for dry air, and R w ≈ 461.520 J/kg/K the specific constant for water vapor). Above we have introduced the concept of canonical transfer. It is an expression of energy transfer which is quite different from the traditional ones. As rigorously proved in Liang [24], a canonical transfer faithfully represents the inter-scale energy transfer because it automatically meets the criterion of energy conservation, while the traditional ones do not. Specifically, it satisfies This means that it is a process such that energy is only redistributed among the scale windows, and is conserved as a whole. Canonical transfer gains its name in that it has a Lie bracket form, reminiscent of the Poisson bracket in Hamiltonian dynamics. See Liang [24] for details.
As demonstrated by Liang and Robinson [29], the canonical transfer terms can be further analyzed to single out the transfers from one window, say, 1 , to another, say 2 . Notationally, we will henceforth write as Γ 1 → 2 . In a three-window framework as shown above, for example, Γ 0→2 So far, this theory has been widely applied to the studies of atmospheric and oceanic dynamics problems such as cold air outbreak [31], storm track [32], atmospheric blockings [33], tropical cyclogenesis [34], vortices over Tibet Plateau [35], Kuroshio Extension dynamics [36], to name a few. For more details, the reader is referred to Liang [24] and Liang and Anderson [25].

Characteristics of the Heavy Rainfall
As introduced in Section 1, this rainstorm was brought by a landfalling AR on [25][26][27] February 2019 in the Northern California. Figure 1a shows the 48-h accumulated rainfall during the period based on the hourly precipitation data from the ERA 5 dataset. The box marks the rainfall area (38.5-43.5 • N, 125-121 • W), over which the area averaging is taken. As displayed in Figure 1b, there are two peaks in the trend of the hourly rainfall, respectively from 10 UTC 25 February to 18  condensation function, the vertical velocity, specific humidity, J/kg/K the specific gas constant for dry air, and ≈ 461.520 J/kg/K the sp stant for water vapor). Above we have introduced the concept of canonical transfer. It is an exp energy transfer which is quite different from the traditional ones. As rigorously Liang [24], a canonical transfer faithfully represents the inter-scale energy t cause it automatically meets the criterion of energy conservation, while the ones do not. Specifically, it satisfies = 0 This means that it is a process such that energy is only redistributed amon windows, and is conserved as a whole. Canonical transfer gains its name in th Lie bracket form, reminiscent of the Poisson bracket in Hamiltonian dynamics [24] for details.
As demonstrated by Liang and Robinson [29], the canonical transfer ter further analyzed to single out the transfers from one window, say, , to an . Notationally, we will henceforth write as → . In a three-window fram shown above, for example, → ( → ) represents the canonical transfer of from the background-scale window to the mesoscale window. Likewise, represents the canonical transfer of KE (APE) from the synoptic-scale wind mesoscale window. Moreover, Liang and Robinson [30] proved that → ( sponds to the barotropic (baroclinic) instability in the classical sense, if the positive. So far, this theory has been widely applied to the studies of atmospheric a dynamics problems such as cold air outbreak [31], storm track [32], atmosph ings [33], tropical cyclogenesis [34], vortices over Tibet Plateau [35], Kuroshio dynamics [36], to name a few. For more details, the reader is referred to Lian Liang and Anderson [25].

Characteristics of the Heavy Rainfall
As introduced in Section 1, this rainstorm was brought by a landfalling A February 2019 in the Northern California. Figure 1a shows the 48-h accumulat during the period based on the hourly precipitation data from the ERA 5 datas marks the rainfall area (38.5-43.5° N, 125-121° W), over which the area av taken. As displayed in Figure 1b, there are two peaks in the trend of the hour respectively from 10 UTC 25 February to 18   To examine the circulation pattern associated with this extreme precipitation, we plotted horizontal maps of the 850-hPa geopotential (contours) and the integrated water vapor transport from 1000 to 300 hPa (IVT; shaded), as illustrated in Figure 2. Here, IVT is used to represent the AR, following the convention as used in previous studies [37,38]. At 06 UTC February 25, there are two cyclonic centers at 850 hPa, located at 130 • W (C1) and 150 • W (C2), respectively. During Stage I (Figure 2a-c), the landfalling AR at the west coast of Northern California is rather vigorous, with a maximum of IVT exceeding 700 kg m −1 s −1 . Afterwards, it is weakened (not shown). It revives during Stage II (Figure 2d-f), becoming much stronger than Stage I. It should be noted that the AR is accompanied by C1 during Stage I, whereas it is close to C2 during Stage II. To examine the circulation pattern associated with this extreme precipitation, we plotted horizontal maps of the 850-hPa geopotential (contours) and the integrated water vapor transport from 1000 to 300 hPa (IVT; shaded), as illustrated in Figure 2. Here, IVT is used to represent the AR, following the convention as used in previous studies [37,38]. At 06 UTC February 25, there are two cyclonic centers at 850 hPa, located at 130° W (C1) and 150° W (C2), respectively. During Stage I (Figure 2a-c), the landfalling AR at the west coast of Northern California is rather vigorous, with a maximum of IVT exceeding 700 kg m −1 s −1 . Afterwards, it is weakened (not shown). It revives during Stage II (Figure 2d-f), becoming much stronger than Stage I. It should be noted that the AR is accompanied by C1 during Stage I, whereas it is close to C2 during Stage II. The difference between Stage I and Stage II is further compared in terms of rainfall area. Figure 3 presents the 24-h accumulated rainfall distributions during Stage I and Stage II, with the areas respectively boxed. During Stage I, it is seen located to the north of 40° N (box in Figure 3a), whereas during Stage II, it is predominantly situated south of the latitude (box in Figure 3b).  The difference between Stage I and Stage II is further compared in terms of rainfall area. Figure 3 presents the 24-h accumulated rainfall distributions during Stage I and Stage II, with the areas respectively boxed. During Stage I, it is seen located to the north of 40 • N (box in Figure 3a), whereas during Stage II, it is predominantly situated south of the latitude (box in Figure 3b).

(a).
To examine the circulation pattern associated with this extreme precipitation, we plotted horizontal maps of the 850-hPa geopotential (contours) and the integrated water vapor transport from 1000 to 300 hPa (IVT; shaded), as illustrated in Figure 2. Here, IVT is used to represent the AR, following the convention as used in previous studies [37,38]. At 06 UTC February 25, there are two cyclonic centers at 850 hPa, located at 130° W (C1) and 150° W (C2), respectively. During Stage I (Figure 2a-c), the landfalling AR at the west coast of Northern California is rather vigorous, with a maximum of IVT exceeding 700 kg m −1 s −1 . Afterwards, it is weakened (not shown). It revives during Stage II (Figure 2d-f), becoming much stronger than Stage I. It should be noted that the AR is accompanied by C1 during Stage I, whereas it is close to C2 during Stage II. The difference between Stage I and Stage II is further compared in terms of rainfall area. Figure 3 presents the 24-h accumulated rainfall distributions during Stage I and Stage II, with the areas respectively boxed. During Stage I, it is seen located to the north of 40° N (box in Figure 3a), whereas during Stage II, it is predominantly situated south of the latitude (box in Figure 3b).

Scale Decomposition
To investigate the multiscale interactions underlying the rainstorm, we firstly applied MWT to fulfill scale decomposition. Considering the heavy rain is attributed to the AR within a synoptic scale [1,39,40], two cutoff periods are required to demarcate three scale windows in this study, namely, the background flow window, the synoptic-scale window and the mesoscale window. For easy reference, = 0, 1, 2 is used to denote the three windows, respectively. The wavelet spectrum analysis [41] is employed to define these cutoff periods/scale levels, which mark the bounds for the windows. Since ARs with strong horizontal water vapor transport take place below 3 km [42], here, the 850-hPa specific humidity is used for spectral analysis. Figure 4b presents the wavelet power spectrum of the 850-hPa specific humidity (q) in Venado (38.5 • N, 123 • W), California, with a maximum precipitation of 500 mm within 48 h. One observation is that there are three dominant peaks of the spectra, corresponding to periods longer than 256 h, between 32 and 256 h, and shorter than 32 h. Based on this, the cutoff periods are set to be 32 and 256 h, respectively. That is to say that processes with periods longer than 256 h are defined as the background flow window, those with periods between 32 and 256 h are treated as the synoptic-scale window (ARs and extratropical cyclones are included), and those with periods shorter than 32 h are included in the mesoscale window, such as mesoscale processes associated with heavy rain.

Scale Decomposition
To investigate the multiscale interactions underlying the rainstorm, we firstly applied MWT to fulfill scale decomposition. Considering the heavy rain is attributed to the AR within a synoptic scale [1,39,40], two cutoff periods are required to demarcate three scale windows in this study, namely, the background flow window, the synoptic-scale window and the mesoscale window. For easy reference, = 0,1,2 is used to denote the three windows, respectively. The wavelet spectrum analysis [41] is employed to define these cutoff periods/scale levels, which mark the bounds for the windows. Since ARs with strong horizontal water vapor transport take place below 3 km [42], here, the 850-hPa specific humidity is used for spectral analysis. Figure 4b presents the wavelet power spectrum of the 850-hPa specific humidity ( ) in Venado (38.5° N, 123° W), California, with a maximum precipitation of 500 mm within 48 h. One observation is that there are three dominant peaks of the spectra, corresponding to periods longer than 256 h, between 32 and 256 h, and shorter than 32 h. Based on this, the cutoff periods are set to be 32 and 256 h, respectively. That is to say that processes with periods longer than 256 h are defined as the background flow window, those with periods between 32 and 256 h are treated as the synoptic-scale window (ARs and extratropical cyclones are included), and those with periods shorter than 32 h are included in the mesoscale window, such as mesoscale processes associated with heavy rain. With the MWT setting, the original total fields are reconstructed onto three scale windows. On the background flow window, there is a nonstationary high-level jet between 40° to 50° N at 300 hPa (not shown). Figure 5 depicts the maps of the synoptic-scale geopotential anomaly (contours) and specific humidity (shadings) at 850 hPa. One common observation is that there is a long and narrow belt of enhanced values of the synoptic-scale specific humidity ( ∼ ) between the cyclone and anticyclone, in accordance with the synoptic-scale pattern associated with ARs [7,38]. This indicates that ARs and extratropical cyclones or anticyclones are well separated from the original field through an MWT application. With the MWT setting, the original total fields are reconstructed onto three scale windows. On the background flow window, there is a nonstationary high-level jet between 40 • to 50 • N at 300 hPa (not shown). Figure 5 depicts the maps of the synoptic-scale geopotential anomaly (contours) and specific humidity (shadings) at 850 hPa. One common observation is that there is a long and narrow belt of enhanced values of the synoptic-scale specific humidity (q ∼1 ) between the cyclone and anticyclone, in accordance with the synoptic-scale pattern associated with ARs [7,38]. This indicates that ARs and extratropical cyclones or anticyclones are well separated from the original field through an MWT application. To determine whether signals of the rainstorm are effectively separated or not, we plotted maps of the mesoscale KE ( ) and reconstructed mesoscale vertical velocity (− ∼ ) averaged vertically from 900 hPa to 300 hPa, as shown in Figure 6b,d. One can see that where the hourly precipitation is strong, both KE and ascending motion on the mesoscale window are enhanced. This collocation justifies our scale decomposition, indicating that signals of the rainstorm can be extracted from the original field and represented with the KE and vertical velocity on the mesoscale window.

Dynamics Underlying the Extreme Precipitation
In Section 4, the synoptic systems (ARs, extratropical cyclones or anticyclones) and mesoscale processes associated with the rainstorm are appropriately decomposed from the original field. Based on this, we investigated the multiscale interaction behind the extreme AR-related precipitation, which are quantitatively expressed in terms of the ca- To determine whether signals of the rainstorm are effectively separated or not, we plotted maps of the mesoscale KE (K 2 ) and reconstructed mesoscale vertical velocity (−ω ∼2 ) averaged vertically from 900 hPa to 300 hPa, as shown in Figure 6b,d. One can see that where the hourly precipitation is strong, both KE and ascending motion on the mesoscale window are enhanced. This collocation justifies our scale decomposition, indicating that signals of the rainstorm can be extracted from the original field and represented with the KE and vertical velocity on the mesoscale window. To determine whether signals of the rainstorm are effectively separated or not, w plotted maps of the mesoscale KE ( ) and reconstructed mesoscale vertical velocity (− ∼ ) averaged vertically from 900 hPa to 300 hPa, as shown in Figure 6b,d. One can se that where the hourly precipitation is strong, both KE and ascending motion on th mesoscale window are enhanced. This collocation justifies our scale decomposition, in dicating that signals of the rainstorm can be extracted from the original field and repre sented with the KE and vertical velocity on the mesoscale window.

Dynamics Underlying the Extreme Precipitation
In Section 4, the synoptic systems (ARs, extratropical cyclones or anticyclones) and mesoscale processes associated with the rainstorm are appropriately decomposed from the original field. Based on this, we investigated the multiscale interaction behind th extreme AR-related precipitation, which are quantitatively expressed in terms of the ca

Dynamics Underlying the Extreme Precipitation
In Section 4, the synoptic systems (ARs, extratropical cyclones or anticyclones) and mesoscale processes associated with the rainstorm are appropriately decomposed from the original field. Based on this, we investigated the multiscale interaction behind the extreme  2). Over the ascending area, the atmosphere is hence barotropically stable, but baroclinically unstable, while over the descending area, it is baroclinically stable, but barotropically unstable. Another observation is that positive values of −b 2 are everywhere throughout the rainfall area, indicating that the mesoscale APE is converting to mesoscale KE throughout, despite the distinct instabilities.  2). Over the ascending ar the atmosphere is hence barotropically stable, but baroclinically unstable, while over descending area, it is baroclinically stable, but barotropically unstable. Another obser tion is that positive values of − are everywhere throughout the rainfall area, indic ing that the mesoscale APE is converting to mesoscale KE throughout, despite the disti instabilities. To take a look at vertical structures of these dynamic processes, we drew the z al-vertical sectional distributions averaged over the latitudinal band 42°-44° N, a To take a look at vertical structures of these dynamic processes, we drew the zonalvertical sectional distributions averaged over the latitudinal band 42 • -44 • N, and show them in Figure 8. At 10:00 UTC, 25 February the ascending motion (solid contours) develops to reach 400 hPa. The canonical transfer Γ 0→2 K is negative through the whole column within the ascending region, while positive values of Γ 0→2 A are located in the lower and middle troposphere (from 900 to 500 hPa). In the lower troposphere (below 700 hPa), there are positive values of Γ 1→2 K and −b 2 around the ascending area. That's to say, during Stage I, the precipitation develops due to the KE transfer from the synoptic-scale window and a buoyancy conversion in the lower troposphere, and baroclinic instability in the middle troposphere.
Atmosphere 2022, 13, x FOR PEER REVIEW 9 of 15 show them in Figure 8. At 10:00 UTC, 25 February the ascending motion (solid contours) develops to reach 400 hPa. The canonical transfer → is negative through the whole column within the ascending region, while positive values of → are located in the lower and middle troposphere (from 900 to 500 hPa). In the lower troposphere (below 700 hPa), there are positive values of → and − around the ascending area. That's to say, during Stage I, the precipitation develops due to the KE transfer from the synoptic-scale window and a buoyancy conversion in the lower troposphere, and baroclinic instability in the middle troposphere. To explore the role of latent heating, we calculated on the mesoscale window as introduced in Section 2.2. Figure 9a displays the maps of averaged from 900 to 300 hPa on 25 February at 10:00 UTC. The great positive values (shadings in Figure 9a) located near the Northern California correspond to the ascending motion (solid contours) in Stage I. To explore the role of latent heating, we calculated F 2 L on the mesoscale window as introduced in Section 2.2. Figure 9a displays the maps of F 2 L averaged from 900 to 300 hPa on 25 February at 10:00 UTC. The great positive values (shadings in Figure 9a To explore the role of latent heating, we calculated on the mesoscale window as introduced in Section 2.2. Figure 9a displays the maps of averaged from 900 to 300 hPa on 25 February at 10:00 UTC. The great positive values (shadings in Figure 9a) located near the Northern California correspond to the ascending motion (solid contours) in Stage I.

Stage II
Following the same procedure as above, we assessed the canonical transfers and buoyancy conversions for the processes during Stage II. Figure 10 shows spatial distributions of (a) Γ 0→2  Figure 11. We can find that the main body of the ascending motion (solid contours) around 125 • W develops upward to 300 hPa, slightly inclining to the west. In the middle troposphere (from 700 to 500 hPa), there are enhanced values of Γ 0→2 K and −b 2 , corresponding to strong ascending motion. This means that, during Stage II, it is a barotropic instability and a buoyancy conversion in the middle layer that facilitate the development of the precipitation.
Like Stage I, the positive values of F 2 L (shadings; Figure 9b) are consistent with strong ascending motion (solid contours) in Stage II, implying that there is a large amount of the latent heat release during the rainfall.

Energy Pathway
To understand the energy pathway for the rainstorm in question at two stages, we took volumetric averages of the energetics from 900 hPa to 400 hPa and over the rainfall areas respectively for Stage I (box in Figure 3a) and Stage II (box in Figure 3b). Then, the volume-averaged energetics were further averaged over time for the two respective stages. The resulting spatiotemporally averaged energetics are schematized in Figure 12, where an arrow stands for the direction of energy flow.

Energy Pathway
To understand the energy pathway for the rainstorm in question at two stage took volumetric averages of the energetics from 900 hPa to 400 hPa and over the ra areas respectively for Stage I (box in Figure 3a) and Stage II (box in Figure 3b). Then volume-averaged energetics were further averaged over time for the two respe stages. The resulting spatiotemporally averaged energetics are schematized in Figu where an arrow stands for the direction of energy flow.
As we can see, during Stage I, the mesoscale APE ( ) is dominantly fueled b latent heating . Another source of is baroclinic instability which transfers from the background flow to the mesoscale window. As for the mesoscale KE ( ) obtained via buoyancy conversion, and is partly returned to the background flow.
For Stage II, the mesoscale window gains twice as much kinetic energy ( ) from mesoscale APE reservoir via buoyancy conversion as it does from the background via barotropic instability. As for the APE balance, latent heating is the dominant sou the mesoscale APE ( ) which then is converted to KE via buoyancy conversion.

Comparison between Stage I and Stage II
The similarity between the precipitation processes in Stage I and Stage II is th both stages, latent heating and buoyancy conversion dominate in the energy bal This may be attributed to the huge amount of latent heat release during the rainfall e They can directly heat the atmosphere and produce APE, most of which is convert KE via buoyancy conversion.
The differences between the rainfall energetics in the two stages are also obv Although the background flow contributes in both stages to the precipitation thr instability, the dominant type of instability is different, with baroclinic instability o ring in Stage I while barotropic instability in Stage II. Considering the fact illustrate Figure 12a that KE in Stage I is not obtained from but lost to the background flow conclude that the mesoscale KE ( ) is transferred to the background flow in Stage I is stored in there until Stage II, when it is released back to the mesoscale window a triggering the heavy rainstorm. This may explain why the rainfall is slightly strong Stage II than in Stage I.

Conclusions
Using a recently-developed localized multiscale energetics analysis tool, we investigated the dynamical processes underlying a heavy rainfall event in Northern ifornia associated with the landfalling atmospheric river (AR) during 25-27 Febr 2019. Based on the trend of the hourly precipitation, the lifecycle of the rainstorm As we can see, during Stage I, the mesoscale APE (A 2 ) is dominantly fueled by the latent heating F 2 L . Another source of A 2 is baroclinic instability which transfers APE from the background flow to the mesoscale window. As for the mesoscale KE (K 2 ), it is obtained via buoyancy conversion, and is partly returned to the background flow.
For Stage II, the mesoscale window gains twice as much kinetic energy (K 2 ) from the mesoscale APE reservoir via buoyancy conversion as it does from the background flow via barotropic instability. As for the APE balance, latent heating is the dominant source of the mesoscale APE (A 2 ) which then is converted to KE via buoyancy conversion.

Comparison between Stage I and Stage II
The similarity between the precipitation processes in Stage I and Stage II is that in both stages, latent heating and buoyancy conversion dominate in the energy balance. This may be attributed to the huge amount of latent heat release during the rainfall event. They can directly heat the atmosphere and produce APE, most of which is converted to KE via buoyancy conversion.
The differences between the rainfall energetics in the two stages are also obvious. Although the background flow contributes in both stages to the precipitation through instability, the dominant type of instability is different, with baroclinic instability occurring in Stage I while barotropic instability in Stage II. Considering the fact illustrated by Figure 12a that KE in Stage I is not obtained from but lost to the background flow, we conclude that the mesoscale KE (K 2 ) is transferred to the background flow in Stage I, and is stored in there until Stage II, when it is released back to the mesoscale window again, triggering the heavy rainstorm. This may explain why the rainfall is slightly stronger in Stage II than in Stage I.

Conclusions
Using a recently-developed localized multiscale energetics analysis tool, we have investigated the dynamical processes underlying a heavy rainfall event in Northern California associated with the landfalling atmospheric river (AR) during 25-27 February 2019. Based on the trend of the hourly precipitation, the lifecycle of the rainstorm is divided into two stages: Stage I (from 25 February at 00:00 UTC to 26 February at 00:00 UTC) and Stage II (from 26 February at 12:00 UTC to 27 February at 12:00 UTC). In Stage I, the rainfall is located to the north of 40 • N, whereas in Stage II, it is moved to the south of the latitude. Moreover, the circulation pattern changes as time moves from Stage I to Stage II: during Stage I, the AR is closely related to an extratropical cyclone, while it is associated with another cyclone during Stage II.
Application of the multiscale window transform (MWT) allows the original fields to be reconstructed onto three scale windows, i.e., the background flow window (periods > 256 h), the synoptic-scale window (periods of 32-256 h), and the mesoscale window (periods < 32 h). On the synoptic-scale window, the AR and extratropical cyclone or anticyclone are well captured; on the mesoscale window, the enhanced mesoscale kinetic energy (KE) and the ascending motion coincide with large values of the hourly precipitation exceeding 5 mm/h. By diagnosing the interactions between the mesoscale window and the other two windows, dynamic mechanisms underlying the rainstorm are revealed.
In both stages, the latent heating and buoyancy conversion are found to play an important role. This implies that the latent heat release can heat the atmosphere and directly produce APE which is mostly converted to KE via buoyancy conversion. Additionally, it is found that the instability of the background flow also contributes to the development of the precipitation which is baroclinic in Stage I whereas barotropic in Stage II. The energy pathway reveals a connection between the two stages of precipitation. During Stage I, a part of the mesoscale KE is inversely transferred to and stored within the background flow, and, as time moves on to Stage II, it is released back to the mesoscale window, triggering the extremely heavy rainstorm during Stage II. This implies that the energy transfer across different scales ahead of the rainfall event should receive more attention since it may be the key factor for the prediction of precipitation.
So far, the multiscale interaction underlying the rainfall associated with the designated landfalling AR has been explored. It would be of interest to quantify how much precipitation is related to the synoptic scale flow and how much comes from the meso-scale process. This issue, among others, is to be investigated in the future studies.