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    A series of CP-3 radar reflectivity PPI displays taken at an elevation angle of 0° and contoured at the 24-dBZ level on 8 Dec 1976. The warm-sector rainband, the cores of the narrow cold-frontal rainband, and the wide cold-frontal rainbands are cross hatched, blackened, and stippled, respectively. The position of the cold front at the surface (as located by the narrow cold-frontal rainband) is indicated. (a) 1351 UTC. The first WCFR (labeled 1 and denoted by a dashed line) is shown. (b) 1457 UTC. The first and second WCFRs (labeled 1 and 2, respectively) are shown. (c) 1523 UTC. The initial formation of the third WCFR (labeled 3) is shown. Adapted from Parsons and Hobbs (1983a).

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    Nested domains in the WRF simulation using 18-, 6-, and 2-km horizontal grid spacings and terrain of the model domain. The cross marks the location of Pt. Brown.

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    (a) Sea level pressure (solid contours every 5 hPa), potential temperature at 975 hPa (dashed contours every 2 K), wind vectors at 10 m above the surface, and surface precipitation rate (mm h−1, color shaded), and (b) geopotential height (solid contours every 50 m), potential temperature (dashed contours every 3 K) and wind vectors at 500 hPa, and frontogenesis function F averaged between 520 and 380 hPa (×10−9 K m−1 s−1, color shaded) for a portion of the 18-km grid domain at 1200 UTC 8 Dec 1976. The gray dashed line in (b) marks the position of the surface cold front. The line segment AA′ marks the position of the cross section in Fig. 4. The cross marks the location of Pt. Brown.

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    Vertical cross sections for the 18-km grid simulation along the line AA′ in Fig. 3. (a) Equivalent potential temperature θe (solid contours every 2 K) and horizontal winds (flag, feather, and half-feather represent 25, 5, and 2.5 m s−1 winds, respectively). (b) Vertical velocity (black contours every 0.05 m s−1 with zero contours omitted and negative values dashed), frontogenesis function F (×10−9 K m−1 s−1, color shaded), and potential temperature (thin gray contours every 2 K). The shaded areas in (a) indicate regions of potential instability (e/dz < 0). The thick solid line in each panel marks the dynamical tropopause defined as the 1.5-PVU isosurface. The mark “J” in each panel denotes the location of the upper-level jet axis. The thick blue line in (b) indicates the upper-level frontal zone.

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    Surface precipitation rate (mm h−1, color shaded) and horizontal wind vectors at 10 m above the surface for a portion of the 2-km grid domain at (a) 0900, (b) 1100, (c) 1230, (d) 1400, (e) 1530, and (f) 1630 UTC 8 Dec 1976 for CTRL. The lettered solid lines mark the positions of cross sections referred to in Figs. 8, 9, and 11. The cross marks the location of Pt. Brown. Precipitation features identified as the first, second, and third WCFRs are marked by dashed lines and labeled as WR1, WR2, and WR3, respectively. A hint of WR2 is marked by a dashed line and labeled as (WR2) in (c). Gray solid contours denote model terrain height at 250 and 1000 m. The long-dashed lines in (c) and (f) are explained in the text. Only a 390 km × 580 km portion of the domain is shown.

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    Snow mixing ratio at z = 3.5 km at (a) 1400, (b) 1530, and (c) 1630 UTC 8 Dec 1976 for CTRL (shaded contours). Precipitation features identified as the first, second, and third WCFRs are marked by dashed lines and labeled as WR1, WR2, and WR3, respectively. The lettered solid lines mark the positions of cross sections referred to in Figs. 8, 9, and 11. The cross marks the location of Pt. Brown. The thick solid line in each panel marks the position of the SCF. The long-dashed lines in (c) are explained in the text. Only a 390 km × 580 km portion of the domain is shown.

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    Hovmöller diagrams of (a) surface rainfall rate (shaded contours), (b) snow mixing ratio at z = 3.5 km (shaded contours), (c) vertical velocity at z = 5.0 km (shaded contours, unshaded dashed contours denote downdrafts of −0.04 and −0.08 m s−1), and (d) vertical velocity at z = 6.8 km (shaded contours, unshaded dashed contours denote downdrafts of −0.04 and −0.08 m s−1) for the northern part of the cold front in CTRL. Precipitation features identified as the first, second, and third WCFRs are marked by dashed lines and labeled as WR1, WR2, and WR3, respectively. The narrow cold-frontal rainband and the warm-sector rainband are labeled as NCFR and WSR, respectively. The updrafts responsible for WR1, WR2, and WR3 are marked by dashed lines and labeled as UP1, UP2, and UP3, respectively. The thick solid line in each panel marks the location of coastline. The SCF is located at x = 0.

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    Vertical cross sections of mixing ratios of rain (g kg−1, color shaded), snow (solid contours every 0.1 g kg−1), graupel (thick gray contours every 0.03 g kg−1), and equivalent potential temperature (thin gray contours every 2 K) along the lines (a) AA′, (b) BB′, (c) CC′, (d) DD′, (e) EE′, and (f) FF′ in Fig. 5. Fields represent an 80-km average (40 km to each side) in the direction normal to the cross section. Precipitation features identified as the first, second, and third WCFRs are labeled as WR1, WR2, and WR3, respectively. Hints of WR2 and WR3 in (c) are labeled as (WR2) and (WR3), respectively. The thick blue line marks the upper-level frontal zone. The SCF is located at x = 0.

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    As in Fig. 8, but for the SCF-relative cross-front velocity (black contours every 2 m s−1 with negative values dashed and zero contours thick), vertical velocity (color shaded), and diabatic cooling rate by melting (green contours at −0.1 and −1.0 K h−1). Positive (negative) values of the SCF-relative cross-front velocity indicate the northwesterly (southeasterly) flows directed toward the rear (front) of the SCF. Thick gray contours enclose statically unstable regions where the squared Brunt–Väisälä frequency N2 is negative. The enhanced frontal updrafts responsible for WR1, WR2, and WR3 are labeled as UP1, UP2, and UP3, respectively. The axis of UP2 at 1530 UTC is indicated by the white dashed line.

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    Equivalent potential temperature (thin shaded contours every 1 K), divergence of the cross-front wind (thick contours at ±3 × 10−4 and ±6 × 10−4 s−1 with negative values dashed), and wind vectors relative to the SCF at 1230 UTC 8 Dec 1976 for a part of the cross section in Fig. 9c. Rightward and upward unit vectors below the panel show the SCF-relative cross-front wind velocity and vertical velocity of 20 and 1.0 m s−1, respectively.

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    Vertical sections of the microphysical and dynamical structures of the southern part of the cold front in CTRL. (left) As in Fig. 8, but cross sections along the lines (a) GG′, (b) HH′, and (c) II′ in Fig. 5. (right) As in Fig. 9, but cross sections along the lines (d) GG′, (e) HH′, and (f) II′ in Fig. 5.

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    As in Fig. 5, but for (a)–(c) NOMELT-ALL and (d)–(f) NOMELT-POST at 0900, 1230, and 1530 UTC 8 Dec 1976. The lettered solid lines mark the positions of cross sections referred to in Figs. 14, 15, and 17. The precipitation features identified as WCFRs are marked by dashed lines.

  • View in gallery

    Hovmöller diagrams of surface rainfall rate (shaded contours) for (a) the northern part of the front in NOMELT-ALL, (b) the northern part of the front in NOMELT-POST, (c) the southern part of the front in NOMELT-ALL, and (d) the southern part of the front in NOMELT-POST. The thick solid line in each panel marks the location of coastline. The precipitation features identified as WCFRs are marked by dashed lines. The narrow cold-frontal rainband and the warm-sector rainband are labeled as NCFR and WSR, respectively.

  • View in gallery

    Vertical sections of the microphysical and dynamical structures of the northern part of the cold front in NOMELT-ALL. (left) As in Fig. 8, but cross sections along the lines (a) AA′, (b) BB′, and (c) CC′ in Fig. 12. (right) As in Fig. 9, but cross sections along the lines (d) AA′, (e) BB′, and (f) CC′ in Fig. 12. Green dashed contours (at −0.1 and −1.0 K h−1) indicate the simulated melting-induced cooling that is deactivated in the model thermodynamic equation. The cross in the right panels marks the point at which the melting layer intersects the cold-frontal zone.

  • View in gallery

    As in Fig. 14, but cross sections for the northern part of the cold front in NOMELT-POST along the lines (a),(d) FF′, (b),(e) GG′, and (c),(f) HH′ in Fig. 12. Green solid (dashed) contours (at −0.1 and −1.0 K h−1) indicate the simulated melting-induced cooling that is activated (deactivated) in the model thermodynamic equation.

  • View in gallery

    (a) Difference of equivalent potential temperature between NOMELT-ALL and NOMELT-POST (NOMELT-ALL minus NOMELT-POST, contours every 0.25 K with negative values dashed and zero contours thick), (b) the SCF-relative wind vectors and CAPE (contours every 2 J kg−1 starting from 2 J kg−1) for NOMELT-ALL, and (c) the SCF-relative wind vectors and CAPE (contours every 2 J kg−1 starting from 2 J kg−1) for NOMELT-POST at 1230 UTC 8 Dec 1976. Rightward and upward unit vectors below the panel show the SCF-relative cross-front wind velocity and vertical velocity of 10 and 0.5 m s−1, respectively. Light (dark) shading in (a) and (c) denotes the area where cooling by melting in NOMELT-POST is greater than 0.1 (1.0) K h−1.

  • View in gallery

    As in Fig. 14, but cross sections for the southern part of the cold front in NOMELT-ALL along the lines (a),(c) DD′ and (b),(d) EE′ in Fig. 12.

  • View in gallery

    Vertical cross sections at 1230 UTC 8 Dec 1976 for a part of the cross section in Fig. 9c. (a) Brunt–Väisälä frequency N for CTRL (×10−2 s−1, shaded contours). Unshaded regions enclosed by thick solid contours denote statically unstable regions where N2 is negative. (b) Differences of the vertical velocity w′ (color shaded) and wind vectors between CTRL and NOMELT-POST (CTRL minus NOMELT-POST). Rightward and upward unit vectors below the panel show the cross-front wind velocity difference u′ and the vertical velocity difference w′ of 2.0 and 0.2 m s−1, respectively. (c) Differences of the vertical velocity w′ (color shaded) and potential temperature θ′ (contours every 0.15 K with negative values dashed) between CTRL and NOMELT-POST (CTRL minus NOMELT-POST). Straight solid (dashed) lines indicate local θ′ maxima (minima). The thick solid line in each panel marks the level where the vertical shear of the cross-front wind is largest.

  • View in gallery

    (left) Time–height sections of (a) cross-front wind speed relative to the intersection of the frontal shear layer and the melting layer (contours every 2 m s−1 with negative values dashed and zero contours thick) and (b) Brunt–Väisälä frequency N (×10−2 s−1, shaded contours) from 0600 to 1700 UTC 8 Dec 1976 for NOMELT-POST. The profiles were constructed using composite vertical cross sections normal to the SCF every 10 min. (right) Time–height sections of (c) basic horizontal wind speed U (contours every 2 m s−1 with negative values dashed and zero contours thick) and (d) Brunt–Väisälä frequency N (×10−2 s−1, shaded contours) for simulations DRY-B1 and DRY-B2.

  • View in gallery

    Vertical velocity fields (contours every 0.01 m s−1 with negative values dashed and zero contours omitted) for (a) DRY-A1, (b) DRY-A2, (c) DRY-A3, and (d) DRY-A4 at t = 15 h. Cooling rates [=(θ0/cpT0)q] greater than 0.05 (0.5) K h−1 are lightly (darkly) shaded. The vertical profile of basic horizontal wind (U) for each case is also shown at the right.

  • View in gallery

    Schematic illustration of the superposition of gravity waves generated by two compact heat sinks that constitute the sloped cooling. The thin gray line indicates the axis of the sloped cooling. The gray filled circles, C1 and C2, represent compact heat sinks. The thick solid (dashed) lines mark lines of maximum (minimum) vertical velocity associated with gravity waves excited by C1, and the thin solid (dashed) lines mark lines of maximum (minimum) vertical velocity associated with gravity waves excited by C2.

  • View in gallery

    Vertical velocity fields (contours every 5 × 10−3 m s−1 with negative values dashed and zero contours omitted) for (a)–(c) DRYB1 and (d)–(f) DRY-B2 at 0900, 1230, and 1530 UTC 8 Dec 1976. Cooling rates [=(θ0/cpT0)q] greater than 0.05 (0.5) K h−1 are lightly (darkly) shaded. (right) Vertical profiles of basic horizontal wind (U) at 0900, 1230, and 1530 UTC are also shown. Gray lines in each panel indicate critical levels for stationary gravity waves.

  • View in gallery

    Schematic description of the processes involved in the formation and evolution of WCFRs discussed in this study. The darkest shading denotes the melting layer. Thin solid lines indicate the boundaries of the low-level cold-frontal zone. Thick dashed lines indicate the boundaries of the upper-level frontal zone. The dashed gray line indicates the critical level for gravity waves. The gravity wave updrafts (downdrafts) are enclosed by solid (dashed) contours and marked by solid (dashed) arrows. Open arrows above the upper-level frontal zone denote updrafts formed by the release of potential instability within the ascent forced by upper-level frontogenesis. Gray arrows denote airflows relative to the SCF, with thicker arrows representing stronger airflows. Shading denotes areas of locally large snow mixing ratio with darker shading representing larger snow mixing ratio. The spacing between the rain streaks is proportional to rain intensity. The vertical profile of the basic cross-front wind relative to the movement of the intersection of the low-level frontal zone and the melting layer is indicated on the right-hand side of each panel.

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The Role of Vertically Propagating Gravity Waves Forced by Melting-Induced Cooling in the Formation and Evolution of Wide Cold-Frontal Rainbands

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  • 1 Institute of Low Temperature Science, Hokkaido University, Sapporo, Japan
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Abstract

Realistic mesoscale model simulations using the Weather Research and Forecasting (WRF) Model and idealized dry simulations were used to study the mechanisms responsible for the formation and evolution of wide cold-frontal rainbands (WCFRs) associated with a wintertime cyclone that moved onto the Washington coast. The WRF simulation reproduced observed characteristics of three successively formed WCFRs, including their spacing and movement as well as the timing of the formation of two WCFRs behind the first. Sensitivity experiments showed that melting-induced cooling in the stratiform precipitation area behind the surface cold front was essential for the formation of the first and second WCFRs, whereas the third WCFR was formed by the release of potential instability within an ascent forced by upper-level frontogenesis. Enhanced frontal updrafts responsible for the first and second WCFRs were created by a superposition of a broad updraft caused by frontal dynamics and upward-propagating gravity waves generated by the melting-induced cooling. The dry simulations forced by specified cooling revealed specific mechanisms for the wave generation and the evolution of the first and second WCFRs. The gravity waves were generated at the intersection of the low-level frontal zone and the melting layer, where strong vertical shear of the cross-front wind and upshear-sloped cooling by melting cooperatively enhanced the wave generation. The formation of the second WCFR behind the first and subsequent dissipation of these WCFRs was attributed to the evolution of a wave pattern associated with the evolution of cross-front flow above the frontal zone.

Corresponding author address: Masayuki Kawashima, Institute of Low Temperature Science, Hokkaido University, West 8, North 19, Kita-ku, Sapporo 060-0819, Japan. E-mail: kawasima@lowtem.hokudai.ac.jp

Abstract

Realistic mesoscale model simulations using the Weather Research and Forecasting (WRF) Model and idealized dry simulations were used to study the mechanisms responsible for the formation and evolution of wide cold-frontal rainbands (WCFRs) associated with a wintertime cyclone that moved onto the Washington coast. The WRF simulation reproduced observed characteristics of three successively formed WCFRs, including their spacing and movement as well as the timing of the formation of two WCFRs behind the first. Sensitivity experiments showed that melting-induced cooling in the stratiform precipitation area behind the surface cold front was essential for the formation of the first and second WCFRs, whereas the third WCFR was formed by the release of potential instability within an ascent forced by upper-level frontogenesis. Enhanced frontal updrafts responsible for the first and second WCFRs were created by a superposition of a broad updraft caused by frontal dynamics and upward-propagating gravity waves generated by the melting-induced cooling. The dry simulations forced by specified cooling revealed specific mechanisms for the wave generation and the evolution of the first and second WCFRs. The gravity waves were generated at the intersection of the low-level frontal zone and the melting layer, where strong vertical shear of the cross-front wind and upshear-sloped cooling by melting cooperatively enhanced the wave generation. The formation of the second WCFR behind the first and subsequent dissipation of these WCFRs was attributed to the evolution of a wave pattern associated with the evolution of cross-front flow above the frontal zone.

Corresponding author address: Masayuki Kawashima, Institute of Low Temperature Science, Hokkaido University, West 8, North 19, Kita-ku, Sapporo 060-0819, Japan. E-mail: kawasima@lowtem.hokudai.ac.jp

1. Introduction

The organization of precipitation within extratropical cyclones has been extensively studied, and it has been reported that rainfall patterns are often organized in the form of mesoscale bands (e.g., Browning and Harrold 1969; Houze et al. 1976; Hobbs 1978; Matejka et al. 1980; Parsons and Hobbs 1983a; Browning 1986). Among the six types of mesoscale rainbands classified by Houze et al. (1976) and Hobbs (1978), wide cold-frontal rainbands (WCFRs) were identified as regions of enhanced stratiform precipitation within an envelope of basic stratiform precipitation associated with a cold front. Observations indicate that the active regions of upward motion associated with WCFRs are above the cold-frontal zone and that the movement of a WCFR is independent of a narrow cold-frontal rainband (NCFR) that moves with the surface cold front (SCF).

Several mechanisms have been proposed to explain the formation of WCFRs. The release of conditional symmetric instability (CSI) [e.g., Bennetts and Hoskins 1979; Emanuel 1979; and as reviewed by Schultz and Schumacher (1999)] has been frequently cited to explain the formation of WCFRs (e.g., Parsons and Hobbs 1983c; Lemaitre and Testud 1988; Lagouvardos et al. 1993). Theoretical and observational studies have also shown that frontogenesis in the presence of small moist symmetric stability or CSI can lead to mesoscale band formation (e.g., Emanuel 1985; Thorpe and Emanuel 1985; Sanders and Bosart 1985; Xu 1992; Nicosia and Grumm 1999; Novak et al. 2004), and several authors examined the combined role of frontogenesis and CSI in producing WCFRs (e.g., Knight and Hobbs 1988; Bénard et al. 1992; Fischer and Lalaurette 1995; Lemaitre et al. 2001).

The formation of WCFRs has also been investigated in relation to cold-frontal zones whose bases are situated above the surface. Upper-level cold fronts that form at or near the tropopause (e.g., Keyser and Shapiro 1986) have been shown to produce the lifting responsible for band formation (e.g., Martin et al. 1992; Han et al. 2009). An advancing mid- to lower-tropospheric frontal zone ahead of the surface cold or occluded front within an occluding cyclone, often referred to as a cold front aloft (Hobbs et al. 1996) or as an upper cold front (Browning and Monk 1982), has been related to WCFRs that move ahead of the surface cold or occluded front (e.g., Locatelli et al. 2002; Locatelli et al. 2005).

In the present study, mechanisms responsible for the formation and evolution of observed WCFRs associated with a wintertime cyclone are investigated by using realistic mesoscale model simulations and idealized dry simulations, focusing on the dynamical effects of melting-induced air cooling. Observational studies (e.g., Heymsfield 1979; Carbone 1982; Heffernan and Marwitz 1996; Stewart et al. 1996) and numerical studies (e.g., Szeto et al. 1988; Barth and Parsons 1996; Szeto and Stewart 1997) have shown that cooling by melting has significant dynamical effects on the airflow and precipitation structures in frontal systems. In particular, Szeto and Stewart (1997) found in their idealized simulations of frontogenesis that the cooling by melting is instrumental in forming banded updraft and precipitation features. The mesoscale model simulations in the present paper indicate that melting-induced cooling also plays an essential role in the formation of the WCFRs studied here. Dry simulations forced by specified cooling are conducted in order to identify the specific mechanisms responsible for the formation and evolution of the WCFRs.

A brief description of the case examined in this paper is given in the next section. Section 3 describes the configuration of the mesoscale model simulations. The results of mesoscale model simulations are presented in section 4. Section 5 presents a description of the dry model and the design of dry simulations. The results of the dry simulations are presented in section 6. Section 7 examines the factors controlling the movement and alongfront variability of WCFRs. Finally, a summary and conclusions are given in section 8.

2. Case description

The WCFRs that provide observational basis for this paper were associated with a cold front that moved onto the Washington coast on 8 December 1976. The mesoscale structure and evolution of several types of observed rainbands as well as the synoptic structure of the frontal system for this case are given by Parsons and Hobbs (1983a,b,c). Thus, only a brief summary of the observation is given here.

The cold front was associated with a young cyclone that formed in the North Pacific Ocean on 6 December 1976. On 8 December, the low pressure center of the cyclone moved into central British Columbia and the warm sector and the cold front of the cyclone passed over the University of Washington’s Cyclonic Extratropical Storms (CYCLES; Hobbs et al. 1980) observation network located on the Washington coast. The variety of mesoscale rainbands observed includes warm-sector rainbands (WSRs), an NCFR, three WCFRs, wavelike rainbands, and postfrontal rainbands (Parsons and Hobbs 1983a).

Figure 1 shows a series of radar reflectivity patterns observed as an SCF moved into the range of the National Center for Atmospheric Research (NCAR) CP-3 Doppler radar located at Pt. Brown, on the coast of Washington (46.9°N, 124.1°W). Six WSRs with alignments approximately parallel to the NCFR were observed as the cold front approached the Washington coast [see Fig. 2 of Parsons and Hobbs (1983c)]. Among these bands, one immediately ahead of the NCFR shown in Fig. 1a had the largest extent in the alongfront direction. The WSR moved into the warm sector.

Fig. 1.
Fig. 1.

A series of CP-3 radar reflectivity PPI displays taken at an elevation angle of 0° and contoured at the 24-dBZ level on 8 Dec 1976. The warm-sector rainband, the cores of the narrow cold-frontal rainband, and the wide cold-frontal rainbands are cross hatched, blackened, and stippled, respectively. The position of the cold front at the surface (as located by the narrow cold-frontal rainband) is indicated. (a) 1351 UTC. The first WCFR (labeled 1 and denoted by a dashed line) is shown. (b) 1457 UTC. The first and second WCFRs (labeled 1 and 2, respectively) are shown. (c) 1523 UTC. The initial formation of the third WCFR (labeled 3) is shown. Adapted from Parsons and Hobbs (1983a).

Citation: Journal of the Atmospheric Sciences 73, 7; 10.1175/JAS-D-15-0163.1

The NCFR was composed of 5-km-wide precipitation cores oriented at a clockwise angle with respect to the synoptic-scale cold front (e.g., Hobbs and Biswas 1979; James and Browning 1979). Doppler radar measurements revealed that the maximum updrafts in the precipitation cores were 3–7 m s−1 (Parsons and Hobbs 1983c).

The first WCFR with a width of ~30 km had already been established when the cold front moved into the range of the radar (Fig. 1a). As the first WCFR moved toward the SCF, a second WCFR with a width of 20–30 km began to form ~35 km behind the first (Fig. 1b). The process was repeated with the formation of a third WCFR ~30 km behind the second (Fig. 1c). These WCFRs moved at a speed a few meters per second faster than the SCF. However, the WCFRs did not move ahead of the SCF but instead dissipated over the SCF.

3. Mesoscale model and experimental design

a. Numerical model

Version 3.2 of the Advanced Research version of the Weather Research and Forecasting Model (ARW) (Skamarock et al. 2008) was employed to simulate the 8 December 1976 frontal rainbands. The model grids consisted of three two-way interactive nested domains shown in Fig. 2. The grid spacings (number of grid points) for these three domains were 18 km (151 × 151), 6 km (271 × 271), and 2 km (541 × 451), respectively. The model top was set at 50 hPa, and 51 stretched vertical levels with the maximum resolution in the boundary layer were employed in all domains.

Fig. 2.
Fig. 2.

Nested domains in the WRF simulation using 18-, 6-, and 2-km horizontal grid spacings and terrain of the model domain. The cross marks the location of Pt. Brown.

Citation: Journal of the Atmospheric Sciences 73, 7; 10.1175/JAS-D-15-0163.1

The principal physical schemes included the Mellor–Yamada–Janjić planetary boundary layer scheme (Janjić 2002), the Rapid Radiative Transfer Model (RRTM) longwave radiation scheme (Mlawer et al. 1997), the Dudhia shortwave radiation scheme (Dudhia 1989), the Noah land surface model (Chen and Dudhia 2001), and the new Thompson mixed-phase bulk microphysics parameterization scheme (Thompson et al. 2008) with double-moment rain for all domains. A comparison with simulations using other mixed-phase bulk microphysics schemes showed that amounts of cloud ice and graupel produced by the new Thompson scheme were less than those produced by other schemes, as reported in previous studies (e.g., Liu et al. 2011). However, the choice of the microphysics scheme did not have much effect on the basic surface precipitation pattern and the dynamical structure of WCFRs presented in this paper (not shown). The Kain–Fritsch cumulus parameterization scheme (Kain 2004) was used for the 18-km grid domain.

Initial and boundary conditions were obtained from the National Centers for Environmental Prediction (NCEP) and NCAR Reanalysis 1 data, with 2.5° horizontal and 6-h temporal resolution. The 18-km grid domain was initialized at 1200 UTC 7 December 1976 and was integrated forward in time for 30 h until 1800 UTC 8 December 1976. The 6- and 2-km grid domains were initiated 6 and 12 h later and were integrated for 24 and 18 h, respectively. Topography data with horizontal resolution of 10′ was used for all domains. Each parent domain was also run without inner nests in order to examine the resolution dependency of the results.

b. Sensitivity experiments

The effects of cooling by melting on the formation of WCFRs were examined by comparing the control simulation (CTRL) with two simulations referred to as NOMELT-ALL and NOMELT-POST. The melting-induced diabatic cooling was artificially removed throughout the model domain in NOMELT-ALL, whereas the melting-induced cooling was removed only in the precipitation area behind the NCFR in NOMELT-POST. The simulation NOMELT-ALL indicates that the melting-induced cooling has substantial effects on the formation of two of three WCFRs found in CTRL. The simulation NOMELT-POST was conducted to further demonstrate that the melting-induced cooling within the stratiform precipitation area behind the NCFR was directly responsible for the formation of those WCFRs.

c. Identification of WCFR

Observations indicate that the formation of a WCFR is related to the dynamical processes above the cold-frontal zone (e.g., Houze et al. 1976; Hobbs 1978; Matejka et al. 1980). A WCFR in this study was defined as a band-shaped region of locally enhanced precipitation within the basic stratiform precipitation area behind the NCFR, having its most active region of upward motion above the cold-frontal zone.

As shown later, the simulated stratiform rainfall behind the NCFR generally decreased with increasing distance from the NCFR, and the stratiform rainfall at the latitude of the observation decreased with time as a whole. No specific threshold values of rainfall rate or precipitation mixing ratios were used to define WCFRs in the present study to extract banded precipitation features within a spatially and temporally varying background precipitation. Because evaporation of rain occurred and shallow convective cells developed in the cold air mass below the cold-frontal zone, banded precipitation features that could be related to enhanced updrafts above the cold-frontal zone were better defined in the snow field than in the rain field. Thus, the identification of a WCFR was first made by examining the snow field above the cold-frontal zone.

The CTRL simulation produced several bands of enhanced snow mixing ratio above the cold-frontal zone (referred to simply as “snowbands”). A snowband or a segment of a snowband that can be identified as a WCFR in this paper must have a length of at least 150 km, and the snowband must accompany regions of locally enhanced surface rainfall along a majority of its length. The subjective criterion for the length of the band was made considering the radar observation in Fig. 1, which indicates that the observed WCFRs had lengths greater than ~150 km. The local maximum of the surface rainfall rate must be at least 50% greater than the local minimum of the rainfall immediately ahead of (i.e., on the NCFR side of) the maximum.

The inspection of the precipitation fields for the 2-km grid simulation of CTRL showed that three successively formed snowbands satisfied the criteria for WCFR. Herein the three WCFR will be referred to as the first, second, and third WCFRs or WR1, WR2, and WR3, respectively.

4. Results of WRF simulations

a. Synoptic environment

Figure 3 shows the simulated synoptic fields at 1200 UTC 8 December 1976. The 18-km grid simulation run without inner nests is examined here, because the output fields are smoother than those from the nested simulation and are therefore more suitable to investigate synoptic-scale variations. The SCF was identified from a large gradient of temperature at 975 hPa, a sharp change in the near-surface wind, and a well-defined trough in the sea level pressure (Fig. 3a). A rainband located at the SCF can be identified as an NCFR. On the other hand, a rainband behind the northern portion of the NCFR in Fig. 3a (42.5°–48°N) was identified as the first WCFR. Note that only the first WCFR was recognizable at the coarse resolution.

Fig. 3.
Fig. 3.

(a) Sea level pressure (solid contours every 5 hPa), potential temperature at 975 hPa (dashed contours every 2 K), wind vectors at 10 m above the surface, and surface precipitation rate (mm h−1, color shaded), and (b) geopotential height (solid contours every 50 m), potential temperature (dashed contours every 3 K) and wind vectors at 500 hPa, and frontogenesis function F averaged between 520 and 380 hPa (×10−9 K m−1 s−1, color shaded) for a portion of the 18-km grid domain at 1200 UTC 8 Dec 1976. The gray dashed line in (b) marks the position of the surface cold front. The line segment AA′ marks the position of the cross section in Fig. 4. The cross marks the location of Pt. Brown.

Citation: Journal of the Atmospheric Sciences 73, 7; 10.1175/JAS-D-15-0163.1

The 500-hPa synoptic fields (Fig. 3b) indicate a trough with a north-northeast–south-southwest-oriented axis to the west of the SCF. The color shading in the figure shows the frontogenesis function due to horizontal deformation (Petterssen 1936; Miller 1948) defined as
e1
Here the value of F averaged between 520 and 380 hPa is shown because a sloping region of frontogenesis associated with an upper-level front was identified within the layer. A region of frontogenesis was identified in the downstream of the trough axis between 43° and 51°N, where a confluent flow concentrated the temperature gradient. The upper-level trough intensified, and the trough and the region of frontogenesis approached the SCF as the time elapsed (not shown).

Figure 4 shows vertical cross sections for the 18-km grid simulation along the line AA′ in Fig. 3. In the equivalent potential temperature (θe) field, regions of potential instability were identified within the near-surface cold air mass behind the SCF, at ~850 hPa immediately ahead of the SCF, and within a rearward-sloping layer in the warm-sector air (Fig. 4a). The distribution of potentially unstable regions roughly agrees with that derived from serial rawinsondes given by Parsons and Hobbs (1983c, their Fig. 1). A potentially unstable region was also found ahead of the upper-level jet axis in the simulation (x ≈ −220 km, 400 hPa). Observed characteristic wind structures such as a low-level jet immediately ahead of the SCF, low-level westerlies behind the front, and the backing of the wind with height in the frontal zone, were also reproduced.

Fig. 4.
Fig. 4.

Vertical cross sections for the 18-km grid simulation along the line AA′ in Fig. 3. (a) Equivalent potential temperature θe (solid contours every 2 K) and horizontal winds (flag, feather, and half-feather represent 25, 5, and 2.5 m s−1 winds, respectively). (b) Vertical velocity (black contours every 0.05 m s−1 with zero contours omitted and negative values dashed), frontogenesis function F (×10−9 K m−1 s−1, color shaded), and potential temperature (thin gray contours every 2 K). The shaded areas in (a) indicate regions of potential instability (e/dz < 0). The thick solid line in each panel marks the dynamical tropopause defined as the 1.5-PVU isosurface. The mark “J” in each panel denotes the location of the upper-level jet axis. The thick blue line in (b) indicates the upper-level frontal zone.

Citation: Journal of the Atmospheric Sciences 73, 7; 10.1175/JAS-D-15-0163.1

The vertical velocity field in Fig. 4b shows a surface-based shallow updraft of ~0.4 m s−1 above the SCF (x = 0 km), forming the NCFR in Fig. 3a. Note that the features associated with the NCFR were poorly resolved at this coarse resolution and the simulated updraft was much weaker than that reported by Parsons and Hobbs (1983c) (3–7 m s−1). An updraft of ~0.3 m s−1 ahead of the SCF was responsible for the precipitation area preceding the NCFR. On the other hand, a deeper updraft of ~0.3 m s−1 above the cold-frontal zone (x = −50 km, 800–400 hPa) was responsible for the WCFR in Fig. 3a.

The potential temperature and frontogenesis fields indicate that the upper-level frontal zone was separated from the low-level frontal zone by a sloping layer of frontolysis. Such a separation of upper-level and low-level frontal zones was noted by Martin et al. (1992) and Han et al. (2009), who investigated the roles of upper-level cold fronts in producing frontal rainbands. The dynamical tropopause, which is defined here as the 1.5 potential vorticity unit (PVU; 1 PVU = 10−6 m2 s−1 K kg−1) isosurface, also exhibited a tropopause fold as a manifestation of the upper-level frontogenesis. Previous studies have shown that the effects of latent heating significantly modify transverse frontal circulations, making the ascending branch narrower and more intense than that in the absence of latent heating (e.g., Emanuel 1985; Thorpe and Emanuel 1985; Han et al. 2007). An updraft of ~0.15 m s−1 at x = −200 km and 460–340 hPa in Fig. 4b was located immediately ahead of the frontogenetic region, and latent heating due to deposition occurred within the updraft (not shown). It is suggested that the updraft was forced by the upper-level frontogenesis and was concentrated by the effects of latent heat release. As shown later, the upper-level frontal zone and the associated updraft moved eastward faster than the SCF, forming the third WCFR in the 2-km grid simulation.

b. Evolution of simulated rainbands

In this section, the data from the innermost 2-km grid domain in the triply nested simulation, which provided detailed structures of the frontal rainbands, is analyzed. The structure and evolution of the WCFRs described herein, however, were also reproduced essentially by the 6-km grid simulation that was run without the 2-km grid domain (not shown). Although radar observations to verify the simulation were only available for the limited area on the coast of Washington depicted in Fig. 1, the structure and evolution of WCFRs farther west and south of the radar observation area will also be examined.

Figure 5 shows the evolution of the surface precipitation field from 0900 to 1630 UTC 8 December 1976 in CTRL. As in the radar observation, the NCFR was composed of precipitation cores oriented at a clockwise angle with respect to the general orientation of the SCF. The simulated SCF passed over the location of Pt. Brown at 1430 UTC, which is about 40 min earlier than that reported by Parsons and Hobbs (1983c). A WSR corresponding to that shown in Fig. 1a had become well defined by 1100 UTC and subsequently moved eastward relative to the NCFR. Shorter WSRs were also present ahead of this main WSR at 1230 UTC (Fig. 5c).

Fig. 5.
Fig. 5.

Surface precipitation rate (mm h−1, color shaded) and horizontal wind vectors at 10 m above the surface for a portion of the 2-km grid domain at (a) 0900, (b) 1100, (c) 1230, (d) 1400, (e) 1530, and (f) 1630 UTC 8 Dec 1976 for CTRL. The lettered solid lines mark the positions of cross sections referred to in Figs. 8, 9, and 11. The cross marks the location of Pt. Brown. Precipitation features identified as the first, second, and third WCFRs are marked by dashed lines and labeled as WR1, WR2, and WR3, respectively. A hint of WR2 is marked by a dashed line and labeled as (WR2) in (c). Gray solid contours denote model terrain height at 250 and 1000 m. The long-dashed lines in (c) and (f) are explained in the text. Only a 390 km × 580 km portion of the domain is shown.

Citation: Journal of the Atmospheric Sciences 73, 7; 10.1175/JAS-D-15-0163.1

On average, the stratiform precipitation behind the NCFR decreased its cross-front extent during the time period depicted. The first WCFR (WR1) identified in Fig. 3a was apparent in the stratiform area at 0900 UTC (Fig. 5a). After 0900 UTC, the rainfall within WR1 intensified and the northern part of WR1 moved toward the NCFR. Between 1230 and 1400 UTC, the northern part of WR1 weakened substantially, and the second WCFR (WR2), which was shorter and narrower than WR1, developed behind WR1 (Figs. 5c,d). The third WCFR (WR3), with precipitation rate greater than 0.8 mm h−1, was also identified behind WR2 at 1400 UTC (Fig. 5d). A locally enhanced region of surface rainfall was also identified behind the southern part of WR1 at 1400 UTC (45°N, ~127°W). As shown later, both WR3 and this banded precipitation feature were associated with a snowband that extended along much of the length of the front depicted in Fig. 5. Thus, the banded precipitation feature behind the southern part of WR1 will be also referred to as WR3. The first and second WCFRs in the northern part of the domain became obscure as the SCF moved inland, whereas WR3 in the southern part was intensified (Figs. 5e,f).

Parsons and Hobbs (1983a,c) reported that two narrow postfrontal rainbands associated with low-level convection were observed ~160 km behind the SCF. In Figs. 5d–f, scattered convective cells behind WR3 were aligned roughly parallel to the cold front. As shown later, this precipitation feature was also associated with low-level convection within the potentially unstable cold air mass (cf. Fig. 4a) and can be regarded as a postfrontal rainband.

The precipitation trailing the NCFR was enhanced over the windward slopes of elevated terrain near the Washington coast (Fig. 5f, 46.5°N, ~123.7°W), whereas the precipitation was reduced significantly downwind of the Olympic Mountains (Fig. 5f, 47.8°N, ~123°W). These simulated modifications of stratiform precipitation over topography are consistent with those reported by Parsons and Hobbs (1983b).

The multiple band structure behind the SCF seen in Figs. 5d–f was more clearly seen in Fig. 6, which shows the distribution of snow mixing ratio at z = 3.5 km. The three WCFRs in the northern part of the domain can be well distinguished from each other even over the Olympic Mountains (Figs. 6a,b). The snow mixing ratio behind the northern part of the SCF decreased and WR1 and WR2 had dissipated by 1630 UTC (Fig. 6c). Although WR3 was not clearly identified in Fig. 5, the snow field clearly showed a corresponding snowband that extended along much of the length of the front at 1630 UTC (Fig. 6c). Note that some bands of enhanced snow mixing ratio developed immediately behind WR3 (e.g., 48°N, ~126°W in Fig. 6a). These snowbands, however, were not identified as WCFRs, because significant sublimation of snow occurred at lower levels (not shown) and the bands did not produce local maxima in the surface rainfall.

Fig. 6.
Fig. 6.

Snow mixing ratio at z = 3.5 km at (a) 1400, (b) 1530, and (c) 1630 UTC 8 Dec 1976 for CTRL (shaded contours). Precipitation features identified as the first, second, and third WCFRs are marked by dashed lines and labeled as WR1, WR2, and WR3, respectively. The lettered solid lines mark the positions of cross sections referred to in Figs. 8, 9, and 11. The cross marks the location of Pt. Brown. The thick solid line in each panel marks the position of the SCF. The long-dashed lines in (c) are explained in the text. Only a 390 km × 580 km portion of the domain is shown.

Citation: Journal of the Atmospheric Sciences 73, 7; 10.1175/JAS-D-15-0163.1

c. Microphysical and dynamical structures

To investigate the microphysical and dynamical structures of the WCFRs, a series of vertical cross sections normal to the general orientation of the SCF was constructed by using the model output saved every 10 min for both northern and southern parts of the front. The locations of the several cross sections are indicated by lettered solid lines in Fig. 5. A frame of reference moving with the SCF was used to construct the vertical cross sections. The origin of abscissa (x = 0) was located where the SCF intersects the long-dashed line shown in Figs. 5c, 5f, and 6c. To remove unrepresentative small-scale perturbations in the displayed quantities, all variables were averaged over 80 km in the direction normal to the cross section, with 40 km to each side of the cross section.

Figure 7 shows Hovmöller diagrams obtained by using the variables in the vertical sections for the northern part of the front. The evolution and movement of the WCFRs are clearly seen in the surface rainfall rate and snow mixing ratio at z = 3.5 km (Figs. 7a,b). Local maxima of the surface rainfall and the snow mixing ratio had become evident immediately behind WR1 by 1200 UTC. These precipitation features are referred to as a hint of WR2 until 1310 UTC, because the local maximum of the surface rainfall was not strong enough to meet the intensity criterion for WCFRs given in section 3c. The second and third WCFRs had become established by 1310 and 1350 UTC, respectively, and moved toward the SCF. While WR1 and WR2 moved at almost the same speed of ~2 m s−1 as they approached the coast, WR3 moved slightly faster than WR1 and WR2 (~3 m s−1). The distribution of snow was closely related to the vertical velocity field (Figs. 7c,d), in which successively formed enhanced updrafts responsible for WR1, WR2, and WR3 were labeled UP1, UP2, and UP3, respectively. The updrafts UP1 and UP2 were clearly indicated at z = 5.0 km at the locations of WR1 and WR2, respectively (Fig. 7c). On the other hand, UP3 was located at higher levels above the upper-level frontal zone and was clearly identified at z = 6.8 km (Fig. 7d). Note that the ~100-km-wide updraft above the upper-level frontal zone in the 18-km grid simulation (x = −200 km and 460–340 hPa in Fig. 4b) took the form of a forward-moving group of smaller-scale updrafts including UP3 in Fig. 7d. The updrafts behind UP3 did not produce remarkable local maxima in the snow and rain fields below.

Fig. 7.
Fig. 7.

Hovmöller diagrams of (a) surface rainfall rate (shaded contours), (b) snow mixing ratio at z = 3.5 km (shaded contours), (c) vertical velocity at z = 5.0 km (shaded contours, unshaded dashed contours denote downdrafts of −0.04 and −0.08 m s−1), and (d) vertical velocity at z = 6.8 km (shaded contours, unshaded dashed contours denote downdrafts of −0.04 and −0.08 m s−1) for the northern part of the cold front in CTRL. Precipitation features identified as the first, second, and third WCFRs are marked by dashed lines and labeled as WR1, WR2, and WR3, respectively. The narrow cold-frontal rainband and the warm-sector rainband are labeled as NCFR and WSR, respectively. The updrafts responsible for WR1, WR2, and WR3 are marked by dashed lines and labeled as UP1, UP2, and UP3, respectively. The thick solid line in each panel marks the location of coastline. The SCF is located at x = 0.

Citation: Journal of the Atmospheric Sciences 73, 7; 10.1175/JAS-D-15-0163.1

The properties of the three simulated WCFRs obtained from an inspection of horizontal sections of precipitation fields and Fig. 7 are summarized in Table 1 along with the properties available from the observation by Parsons and Hobbs (1983a). The WRF control simulation reproduced the successive formation of two WCFRs behind the first observed as the cold front approached the Washington coast, and the timing of the formation of WR2 and WR3 agreed reasonably well with the observation. As in the observation, the three WCFRs moved at a speed a few meters per second faster than the SCF and dissipated successively in the vicinity of the SCF. Although the simulated spacing between WR2 and WR3 was somewhat larger than that observed, these consistencies between the observation and the simulation suggests that the simulation captured key physical processes involved in the formation and evolution of the WCFRs.

Table 1.

Characteristics of simulated WCFRs at the latitude of observation (~46.8°N). Dash indicates not defined. The velocity of the first WCFR represents a time average over a period of 3 h between 1130 and 1430 UTC 8 Dec 1976, whereas those for the second and third WCFR represent averages over the period of band duration. The maximum length of the first WCFR was obtained using the output of 6-km grid domain. Numbers in parentheses are derived from observations by Parsons and Hobbs (1983a).

Table 1.

Figure 8 shows microphysical structure of the cold front in vertical sections along the lines AA′–FF′ in Fig. 5. The upper-level cold-frontal zone characterized by strong horizontal and vertical gradients of θe was marked by a blue solid line in each panel. As WR1 moved forward between 0900 and 1230 UTC, the snow and rain mixing ratios for WR1 increased, whereas the width of WR1 decreased (Figs. 8a–c). At 1230 UTC, contours of the snow mixing ratio for WR1 exhibited a slight forward tilt with height and a hint of WR2 was indicated in the snow field about 40 km behind WR1 (Fig. 8c). Another local maximum in the snow mixing ratio, which subsequently evolved into WR3, was also indicated at x ≈ −175 km. By 1400 UTC, WR2 had become better defined and WR3 had developed into a ~5-km-deep zone of high concentration of snow (Fig. 8d). As the time further elapsed, WR1 and WR2 dissipated successively (Figs. 8e,f). On the other hand, WR3 retained its depth and intensity until 1630 UTC.

Fig. 8.
Fig. 8.

Vertical cross sections of mixing ratios of rain (g kg−1, color shaded), snow (solid contours every 0.1 g kg−1), graupel (thick gray contours every 0.03 g kg−1), and equivalent potential temperature (thin gray contours every 2 K) along the lines (a) AA′, (b) BB′, (c) CC′, (d) DD′, (e) EE′, and (f) FF′ in Fig. 5. Fields represent an 80-km average (40 km to each side) in the direction normal to the cross section. Precipitation features identified as the first, second, and third WCFRs are labeled as WR1, WR2, and WR3, respectively. Hints of WR2 and WR3 in (c) are labeled as (WR2) and (WR3), respectively. The thick blue line marks the upper-level frontal zone. The SCF is located at x = 0.

Citation: Journal of the Atmospheric Sciences 73, 7; 10.1175/JAS-D-15-0163.1

Figure 9 shows dynamical structure of the cold front in the same cross sections as in Fig. 8. The cross-front velocity in Fig. 9 is relative to the motion of the SCF, with positive (negative) values indicate relative northwesterly (southeasterly) flows. The total diabatic cooling due to the melting of snow and graupel, and statically unstable regions where the squared value of the Brunt–Väisälä frequency (N2) is negative, are also indicated. Here N2 in saturated region was calculated considering the effects of latent heat release in the same manner as that described by Durran and Klemp (1982), although in the present study the saturation mixing ratio with respect to ice and the latent heat of sublimation were used for the region above the freezing level.

Fig. 9.
Fig. 9.

As in Fig. 8, but for the SCF-relative cross-front velocity (black contours every 2 m s−1 with negative values dashed and zero contours thick), vertical velocity (color shaded), and diabatic cooling rate by melting (green contours at −0.1 and −1.0 K h−1). Positive (negative) values of the SCF-relative cross-front velocity indicate the northwesterly (southeasterly) flows directed toward the rear (front) of the SCF. Thick gray contours enclose statically unstable regions where the squared Brunt–Väisälä frequency N2 is negative. The enhanced frontal updrafts responsible for WR1, WR2, and WR3 are labeled as UP1, UP2, and UP3, respectively. The axis of UP2 at 1530 UTC is indicated by the white dashed line.

Citation: Journal of the Atmospheric Sciences 73, 7; 10.1175/JAS-D-15-0163.1

The SCF-relative cross-front velocity field at 0900 UTC showed a wedge of surface-based flow of cold air directed toward the SCF and a 4-km-deep SCF-relative rearward flow above (Fig. 9a). The vertical velocity field imposed on Fig. 9a indicates that upward motions were predominant within the SCF-relative rearward flow and that enhanced updrafts noted in Figs. 7c,d were embedded within the rearward flow. A strong, surface-based updraft at x = 0 was associated with the NCFR, whereas updrafts between x = 10 and 35 km were associated with the WSR. Statically unstable regions were identified at the bottoms of the updrafts associated with the WSR. Statically unstable regions were also found in the upper troposphere farther behind the SCF, whereas the SCF-relative rearward flow between x ≈ −120 and 0 km was free from static instability. The enhanced updraft UP1 was identified at x ≈ −80 km within the rearward flow, whereas upward motions were almost absent in the region between the SCF and UP1 (−70 < x < 0 km, z < 3.5 km in Fig. 9a).

A melting layer with a maximum cooling rate greater than 1 K h−1 had an upward slope from left to right and intersected the shear layer associated with the cold-frontal zone at x = −70 km and z = 1.7 km. The slope of the melting layer was locally enhanced in the vicinity of the intersection (x = −80 km, z = 1.5 km in Fig. 9a) because of the enhanced horizontal gradient of temperature across the cold-frontal zone. Note that a persistent downdraft was present at the location of the intersection (e.g., x = −80 km in Fig. 9a and x = −45 km in Fig. 9d), and the base of UP1 was located immediately above the downdraft.

The SCF-relative rearward flow increased its depth but decreased its intensity between 0900 and 1230 UTC. A forward tilt of UP1 with height was enhanced in that period, and UP2 had developed by 1230 UTC (Fig. 9c). Note that UP3 and updrafts behind UP3 were associated with statically unstable regions. This suggests that these small-scale updrafts were a result of the release of potential instability within the ascent forced by upper-level frontogenesis.

The existence of negative stability near the top of WR3 suggests the workings of a “seeder–feeder” precipitation process (Bergeron 1950), with the seeder component being the ice particles falling from the cloud-top generating cells and the feeder component being the moisture supply by the frontal ascent (e.g., Herzegh and Hobbs 1981; Rutledge and Hobbs 1983). The horizontal and vertical resolutions in the present simulation, however, were insufficient to resolve shallow generating cells and accompanying snow fallstreaks.

The upper part of the SCF-relative rearward flow decreased in depth and intensity after 1400 UTC (Figs. 9d–f). Although UP1 and UP2 weakened after 1400 UTC, UP3 retained its intensity as it moved forward (Figs. 9e,f). Updrafts behind UP3 did not produce remarkable rainbands, because sublimation of snow occurred within a SCF-relative midlevel forward flow beneath the upper-level frontal zone (−200 < x < 135 km, z ~ 5 km in Fig. 9e). The midlevel forward flow had increased its intensity and extent considerably by 1630 UTC. This flow subsequently overran the lower part of the SCF-relative rearward flow by 1800 UTC (not shown).

Figure 10 shows the SCF-relative wind vectors and θe at 1230 UTC for a part of the cross section in Fig. 9c. The SCF-relative rearward flow above the cold-frontal zone can be divided into upper and lower parts. The lower part of the SCF-relative rearward flow immediately above the cold-frontal zone (x < 0 km, 1.3 < z < 2.5 km) was fed primarily by the prefrontal near-surface air below z = 1 km, whereas the upper part (x < 0 km, z > 3 km) was fed partly by outflows from the updrafts associated with the WSR at x = 40 km. The height of maximum divergence near the top of the NCFR updraft (z = 1.6 km) and the height of maximum convergence at the bases of the WSR updrafts (z = 1.7 km) agree fairly well with those shown in Fig. 4 of Parsons and Hobbs (1983a). The prefrontal near-surface air feeding the NCFR updraft was colder than the air feeding the WSR updrafts (x > 40 km, z = 1–2 km). Thus, a frontal structure was identified at the leading edge of the outflow from the NCFR updraft (x = 40 km and z = 1–2 km). This indicates that the updrafts associated with the WSR were triggered by the forced uplifting of the potentially unstable air by the colder outflow from the NCFR updraft.

Fig. 10.
Fig. 10.

Equivalent potential temperature (thin shaded contours every 1 K), divergence of the cross-front wind (thick contours at ±3 × 10−4 and ±6 × 10−4 s−1 with negative values dashed), and wind vectors relative to the SCF at 1230 UTC 8 Dec 1976 for a part of the cross section in Fig. 9c. Rightward and upward unit vectors below the panel show the SCF-relative cross-front wind velocity and vertical velocity of 20 and 1.0 m s−1, respectively.

Citation: Journal of the Atmospheric Sciences 73, 7; 10.1175/JAS-D-15-0163.1

An inspection of the horizontal and vertical mass fluxes using the data depicted in Fig. 9 revealed that the total horizontal mass flux of the SCF-relative rearward flow at the location 20 km behind the WSR decreased substantially from 15 × 103 to 2.5 × 103 kg s−1 between 1100 and 1530 UTC. On the other hand, the upward mass flux associated with the WSR updrafts at the height of the maximum flux (z ≈ 2.6 km) increased from 5.5 × 103 to 6.4 × 103 kg s−1 in that period. Thus, the weakening of the upper part of the SCF-relative rearward flow in Fig. 9 was not due to a weakening of WSR updrafts but due to the evolution of the larger-scale cross-front wind.

As suggested by Figs. 5 and 6, the structure and evolution of WCFRs in the southern part of the front were rather different from those in the northern part. Figure 11 shows the microphysical and dynamical structures of the front in vertical sections across the southern part of the front. In the southern part, WR1 was clearly separated from the NCFR throughout the period depicted (Figs. 11a–c). The upper-level frontal zone was closer to the SCF compared to that in the northern part (cf. Fig. 8). A region of enhanced snow mixing ratio that subsequently evolved into WR3 had formed above the frontal zone by 1100 UTC (Fig. 11a), about 1.5 h earlier than in the northern part. The midlevel SCF-relative forward flow beneath the upper-level frontal zone was stronger than that in the northern part throughout the period depicted (Figs. 11d–f). The SCF-relative rearward flow above the cold-frontal zone at 1230 UTC (Fig. 11e) was considerably shallower than that in the northern part (cf. Fig. 9c), and the flow retained its intensity until 1530 UTC (Fig. 11f). As in the northern part, UP1 tilted forward with height, and updrafts above the upper-level frontal zone were associated with statically unstable regions. A forward-sloping updraft corresponding to UP2, however, was absent between UP1 and UP3. The postfrontal rainband noted in Fig. 5 and associated shallow updraft were indicated at x = −185 km in Figs. 11b and 11e, respectively. The postfrontal rainband subsequently moved to x = −150 km by 1530 UTC (Figs. 11c,f).

Fig. 11.
Fig. 11.

Vertical sections of the microphysical and dynamical structures of the southern part of the cold front in CTRL. (left) As in Fig. 8, but cross sections along the lines (a) GG′, (b) HH′, and (c) II′ in Fig. 5. (right) As in Fig. 9, but cross sections along the lines (d) GG′, (e) HH′, and (f) II′ in Fig. 5.

Citation: Journal of the Atmospheric Sciences 73, 7; 10.1175/JAS-D-15-0163.1

d. Sensitivity experiments

The effects of cooling by melting on the formation of WCFRs were examined by comparing CTRL with experiments NOMELT-ALL and NOMELT-POST.

Figure 12 shows the surface precipitation fields at 0900, 1230, and 1530 UTC 8 December 1976 for these cases. In both cases, a WSR and an NCFR were simulated at almost the same locations as in CTRL. However, remarkable local rainfall maxima corresponding to WR1 and WR2 were not recognized in the stratiform area in the northern part of the domain at 0900 and 1230 UTC. In NOMELT-ALL, the surface rainfall around the NCFR decreased substantially between 1230 and 1530 UTC, which resulted in the formation of a rainband clearly separated from the NCFR in the southern part of the domain (Fig. 12c, 45°N, ~126°W). The surface precipitation in the rainband, however, was weaker and more widely distributed compared to WR1 in CTRL (cf. Fig. 5e). Though less significant compared to NOMELT-ALL, the surface rainfall for NOMELT-POST also showed locally enhanced regions in the southern part of the domain at 1530 UTC (Fig. 12f, 44.5°N, ~126.3°W). On the other hand, a narrow rainband had developed near the back edge of the precipitation area in both cases by 1530 UTC (Figs. 12c,f).

Fig. 12.
Fig. 12.

As in Fig. 5, but for (a)–(c) NOMELT-ALL and (d)–(f) NOMELT-POST at 0900, 1230, and 1530 UTC 8 Dec 1976. The lettered solid lines mark the positions of cross sections referred to in Figs. 14, 15, and 17. The precipitation features identified as WCFRs are marked by dashed lines.

Citation: Journal of the Atmospheric Sciences 73, 7; 10.1175/JAS-D-15-0163.1

Figure 13 shows Hovmöller diagrams of the surface rainfall rate in the northern and southern parts of the front for these cases, constructed in the same manner as in Fig. 7a. Though not shown for brevity, local maxima of the surface rainfall marked by dashed lines in Fig. 13 were associated with snowbands aloft and, therefore, were identified as WCFRs. In both cases, the rainfall rate behind the NCFR in the northern part decreased with increasing distance from the NCFR, and the width of the area with rainfall rate greater than 1 mm h−1 decreased roughly with time (Figs. 13a,b). A substantial decrease of the stratiform rainfall behind the NCFR occurred between 1430 and 1530 UTC in NOMELT-ALL (Fig. 13a), whereas the stratiform rainfall in NOMELT-POST retained its intensity (Fig. 13b). The rainbands near the back edge of the precipitation areas in Figs. 12c and 12f had formed by 1400 and 1430 UTC in NOMELT-ALL and NOMELT-POST, respectively, and the rainbands moved forward at a speed of about 3 m s−1 in both cases. The timing of band formation and the movement relative to the SCF were similar to those for WR3 in CTRL. Forward-moving signals of enhanced rainfall corresponding to WR1 and WR2 in CTRL (cf. Fig. 7a), however, were not recognized in these simulations.

Fig. 13.
Fig. 13.

Hovmöller diagrams of surface rainfall rate (shaded contours) for (a) the northern part of the front in NOMELT-ALL, (b) the northern part of the front in NOMELT-POST, (c) the southern part of the front in NOMELT-ALL, and (d) the southern part of the front in NOMELT-POST. The thick solid line in each panel marks the location of coastline. The precipitation features identified as WCFRs are marked by dashed lines. The narrow cold-frontal rainband and the warm-sector rainband are labeled as NCFR and WSR, respectively.

Citation: Journal of the Atmospheric Sciences 73, 7; 10.1175/JAS-D-15-0163.1

The Hovmöller diagrams for the southern part indicate that a well-defined local maximum of rainfall had developed at x = −60 km by 1330 UTC in NOMELT-ALL, as a consequence of the substantial decrease of the rainfall around the NCFR (Fig. 13c). A similar maximum in the rainfall was also identified in NOMELT-POST (Fig. 13d), but its formation lagged that in NOMELT-ALL by about 2 h. As in the northern part, a narrow rainband formed near the back edge of the surface precipitation area and the rainband moved forward at a speed of about 3 m s−1 in both cases.

Figure 14 shows vertical sections of the microphysical and dynamical structures in the northern part of the front for NOMELT-ALL. Although a local maximum of the snow mixing ratio was found behind the NCFR at 0900 UTC (Fig. 14a, x = −70 km) and 1230 UTC (Fig. 14b, x = −40 km), the rain mixing ratio below did not exhibit a corresponding remarkable local maximum. The WCFR noted earlier in Fig. 13a was associated with a 5-km-deep region of enhanced snow mixing ratio farther behind (Fig. 14c, x = −120 km) and had similar appearance to WR3 in CTRL (cf. Fig. 8e). As suggested by Figs. 12c and 13a, the snow and rain mixing ratios around the NCFR had decreased whereas those for the WSR had increased substantially by 1530 UTC.

Fig. 14.
Fig. 14.

Vertical sections of the microphysical and dynamical structures of the northern part of the cold front in NOMELT-ALL. (left) As in Fig. 8, but cross sections along the lines (a) AA′, (b) BB′, and (c) CC′ in Fig. 12. (right) As in Fig. 9, but cross sections along the lines (d) AA′, (e) BB′, and (f) CC′ in Fig. 12. Green dashed contours (at −0.1 and −1.0 K h−1) indicate the simulated melting-induced cooling that is deactivated in the model thermodynamic equation. The cross in the right panels marks the point at which the melting layer intersects the cold-frontal zone.

Citation: Journal of the Atmospheric Sciences 73, 7; 10.1175/JAS-D-15-0163.1

As in CTRL, the SCF-relative rearward flow above the low-level cold-frontal zone increased its depth but decreased its intensity between 0900 and 1230 UTC, then the flow decreased its depth (Figs. 14d–f). However, the upper part of the SCF-relative rearward flow was stronger and the midlevel SCF-relative forward flow beneath it was better defined compared to those in CTRL. The low-level cold-frontal zone had a more uniform slope than that in CTRL, and the height of the frontal zone behind the SCF increased with time. The effects of the rise in the frontal zone height will be discussed later in section 7.

The strong midlevel SCF-relative forward flow caused the significant decreases of the snow and rain mixing ratios as it overran the SCF by limiting the vertical development of updrafts above the SCF (Fig. 14f). The intensified SCF-relative forward flow ahead of the SCF enhanced updrafts at its leading edge (x ≈ 55 km and z = 1.5–2.5 km in Fig. 14f), thereby reinforcing the precipitation associated with the WSR.

Unlike in CTRL, forward-sloping enhanced updrafts were not identified in the vertical velocity field above the low-level cold-frontal zone. Instead, a broader and weaker enhanced frontal updraft was indicated above the local maximum of snow mixing ratio at 0900 and 1230 UTC (centered at x = −60 km in Fig. 14d and x = −40 km in Fig. 14e). Another enhanced updraft had formed above the upper-level frontal zone by 1530 UTC (x = −140 km in Fig. 14f), forming the WCFR noted in Fig. 14c. The updraft was associated with statically unstable region above the upper-level cold front.

Figure 15 is as in Fig. 14 but for NOMELT-POST. At 0900 and 1230 UTC, the basic microphysical and dynamical structures were similar to those in NOMELT-ALL, except that the upper part of the SCF-relative rearward flow and the midlevel SCF-relative forward flow beneath it were weaker. The microphysical and dynamical structures at 1530 UTC (Figs. 15c,f) were rather different from those in NOMELT-ALL. The leading edge of the midlevel SCF-relative forward flow was located well behind the SCF (at x = −70 km), and updrafts above the SCF were deeper than those in NOMELT-ALL (cf. Fig. 14f). As a consequence, mixing ratios of snow and rain immediately behind the NCFR were considerably larger than those in NOMELT-ALL (cf. Fig. 14c).

Fig. 15.
Fig. 15.

As in Fig. 14, but cross sections for the northern part of the cold front in NOMELT-POST along the lines (a),(d) FF′, (b),(e) GG′, and (c),(f) HH′ in Fig. 12. Green solid (dashed) contours (at −0.1 and −1.0 K h−1) indicate the simulated melting-induced cooling that is activated (deactivated) in the model thermodynamic equation.

Citation: Journal of the Atmospheric Sciences 73, 7; 10.1175/JAS-D-15-0163.1

The differences in the SCF-relative airflow between NOMELT-ALL and NOMELT-POST can be attributed to a difference in the stability of prefrontal air. Figure 16 shows the difference of θe between the two simulations (NOMELT-ALL minus NOMELT-POST) and convective available potential energy (CAPE) and the SCF-relative wind vectors for both simulations at 1230 UTC. While cooling by melting with a maximum cooling rate of ~1 K h−1 occurred in the warm sector in NOMELT-POST (denoted by shading in Fig. 16a), melting-induced cooling was turned off throughout the model domain in NOMELT-ALL. As a result, the difference field of θe exhibited a warm anomaly with a maximum difference of 2 K located within the melting layer (20 < x < 85 km, z ~ 1.5 km in Fig. 16a). Small CAPE was present at the bases of the WSR updrafts in both cases (20 < x < 85 km, z ~ 1.5 km in Figs. 16b,c). However, the CAPE for NOMELT-ALL (~18 J kg−1) was larger than that for NOMELT-POST (~6 J kg−1) because of the presence of higher-θe air there. The more unstable stratification in NOMELT-POST made the WSR updrafts and the upper part of the SCF-relative rearward flow as well stronger than those in NOMELT-POST.

Fig. 16.
Fig. 16.

(a) Difference of equivalent potential temperature between NOMELT-ALL and NOMELT-POST (NOMELT-ALL minus NOMELT-POST, contours every 0.25 K with negative values dashed and zero contours thick), (b) the SCF-relative wind vectors and CAPE (contours every 2 J kg−1 starting from 2 J kg−1) for NOMELT-ALL, and (c) the SCF-relative wind vectors and CAPE (contours every 2 J kg−1 starting from 2 J kg−1) for NOMELT-POST at 1230 UTC 8 Dec 1976. Rightward and upward unit vectors below the panel show the SCF-relative cross-front wind velocity and vertical velocity of 10 and 0.5 m s−1, respectively. Light (dark) shading in (a) and (c) denotes the area where cooling by melting in NOMELT-POST is greater than 0.1 (1.0) K h−1.

Citation: Journal of the Atmospheric Sciences 73, 7; 10.1175/JAS-D-15-0163.1

The upper part of the sloping SCF-relative rearward flow in NOMELT-ALL was also warmer than that in NOMELT-POST (x < 0 km, z > 4 km in Fig. 16a), because the prefrontal air feeding it was warmer. As a result, the hydrostatic pressure immediately beneath the upper part of the rearward flow in NOMELT-ALL was lower than that in NOMELT-POST (not shown). This pressure deficit in NOMELT-ALL resulted in a forward acceleration of air from the rear, making the midlevel forward-directed flow stronger than in NOMELT-POST.

Figure 17 shows the microphysical and dynamical structures for the southern part of the front for NOMELT-ALL at 1230 and 1530 UTC. As in CTRL, the upper-level frontal zone in the southern part was closer to the SCF than that farther north. An enhanced updraft above the cold-frontal zone (x ~ −60 km, z ~ 6 km in Fig. 17c) was stronger than the corresponding enhanced updraft in the northern part (cf. Fig. 14e), probably because the upper-level front played a role to enhance the frontal updraft. As in the northern part, the snow and rain mixing ratios around the NCFR decreased significantly between 1230 and 1530 UTC as the midlevel SCF-relative forward flow overran the SCF. On the other hand, the updraft above the cold-frontal zone and associated rainfall at x ≈ −60 km retained their intensities. Thus, as noted in Fig. 13c, a clear local minimum of stratiform rainfall formed immediately behind the NCFR, and a local maximum of rainfall farther behind (x ≈ −60 km) became identifiable as a WCFR. This WCFR, however, was weak and broad compared to WR1 in CTRL (cf. Fig. 11c).

Fig. 17.
Fig. 17.

As in Fig. 14, but cross sections for the southern part of the cold front in NOMELT-ALL along the lines (a),(c) DD′ and (b),(d) EE′ in Fig. 12.

Citation: Journal of the Atmospheric Sciences 73, 7; 10.1175/JAS-D-15-0163.1

As noted in Fig. 13d, the formation of corresponding WCFR in NOMELT-POST lagged that in NOMELT-ALL by about 2 h. This is because the midlevel SCF-relative forward flow was weaker and its penetration into the prefrontal air was delayed (not shown).

In summary, the results of the sensitivity experiments showed the importance of the melting-induced cooling on the formation of the first and second WCFRs found in CTRL. In both sensitivity experiments, the enhanced updraft above the low-level cold-frontal zone was broader and weaker compared to the updrafts found in CTRL, and WCFRs were not identified in the surface rainfall field below the updraft in the northern part of the front. Although the prefrontal cooling by melting was found to have a considerable impact on the intensity of the SCF-relative flow, this indicates that the effects of cooling by melting in the stratiform precipitation area behind the NCFR were essential for the formation of WR1 and WR2. A WCFR clearly separated from the NCFR developed in the southern part of the front as the SCF-relative midlevel forward flow overran the SCF in both sensitivity experiments. The WCFR, however, was weak and broad compared to WR1 in CTRL and was identified only in the late simulation period. This indicates that the cooling by melting was also important for the formation of WR1 in the southern part of the front. On the other hand, an enhanced updraft above the upper-level front and an associated rainband similar to WR3 in CTRL formed in both sensitivity experiments. It is suggested that the third WCFR was a result of the release of potential instability within the ascent forced by upper-level frontogenesis.

e. Characteristics of the disturbance responsible for WR1 and WR2

Using the serial rawinsondes for this case, Parsons and Hobbs (1983c) showed that there was a layer in the region of the WCFRs that met the criteria for symmetric instability (see their Fig. 13), and suggested that the WCFRs might have been produced by the release of instability. The circulations that release CSI should be sloped in the same sense as the isosurfaces of saturated equivalent potential temperature θes (e.g., Bennetts and Hoskins 1979; Schultz and Schumacher 1999). On the other hand, the slopes of UP1 and UP2 in Fig. 9 were opposite to that of θe contours above the frontal zone where the air is saturated and therefore θe is equal to θes. Further, while the disturbances that release CSI should be advected by the environmental flow in which they are embedded, UP1 and UP2 moved forward within the SCF-relative rearward flow. These fundamental differences in the disturbance characteristics eliminate CSI from consideration as a candidate mechanism of the formation of WR1 and WR2 in the present simulation.

Szeto and Stewart (1997) noted in their idealized numerical simulation of frontogenesis that the cooling effects of melting are instrumental in forming a stepwise structure in the frontal zone near the 0°C level and associated banded updraft and precipitation features (see their Fig. 12). In CTRL, the base of UP1 was located in the vicinity of the region where the model frontal zone and the melting layer intersected, consistent with the results of Szeto and Stewart (1997).

Szeto et al. (1988) showed in their numerical study of mesoscale circulations forced by melting snow that gravity waves and a density current comprise the mesoscale circulations, and these disturbances can have significant dynamical effects on the environment remote from the precipitation region. As shown in Fig. 9, both UP1 and UP2 were located within a statically stable region and moved forward against the rearward-directed embedding flow. This and the significant forward tilts of UP1 and UP2 suggest that these updrafts were also associated with vertically propagating gravity waves [e.g., Gill (1982), chapter 6].

To see the dynamical impact of the melting-induced cooling in the present simulation, Fig. 18 presents the differences of the vertical velocity, wind vectors relative to the SCF, and potential temperature between CTRL and NOMELT-POST simulations (CTRL minus NOMELT-POST) along with the distribution of buoyancy frequency N for CTRL at 1230 UTC. The difference fields represent the impact of the melting-induced cooling in the stratiform precipitation area behind the NCFR. As noted earlier, the SCF-relative rearward flow, in which updrafts UP1 and UP2 were embedded, was stably stratified (Fig. 18a). The difference field of the vertical velocity w′ in Fig. 18b clearly exhibited a wave structure with significant forward phase tilt with height. Difference wind vectors imposed on Fig. 18b indicate that w′ and the difference cross-front velocity u′ were almost in phase. The difference field of potential temperature θ′ in Fig. 18c also exhibited a wave structure with forward phase tilt in the region where N and the amplitude of w′ were large. Note that local maxima (minima) of θ′ were shifted about 90° toward the forward side of the w′ maxima (minima). The quadrature relationship between w′ and θ′ with in-phase relationship between u′ and w′ indicates that the wavelike disturbance above the low-level frontal zone had properties of upward-propagating gravity waves.

Fig. 18.
Fig. 18.

Vertical cross sections at 1230 UTC 8 Dec 1976 for a part of the cross section in Fig. 9c. (a) Brunt–Väisälä frequency N for CTRL (×10−2 s−1, shaded contours). Unshaded regions enclosed by thick solid contours denote statically unstable regions where N2 is negative. (b) Differences of the vertical velocity w′ (color shaded) and wind vectors between CTRL and NOMELT-POST (CTRL minus NOMELT-POST). Rightward and upward unit vectors below the panel show the cross-front wind velocity difference u′ and the vertical velocity difference w′ of 2.0 and 0.2 m s−1, respectively. (c) Differences of the vertical velocity w′ (color shaded) and potential temperature θ′ (contours every 0.15 K with negative values dashed) between CTRL and NOMELT-POST (CTRL minus NOMELT-POST). Straight solid (dashed) lines indicate local θ′ maxima (minima). The thick solid line in each panel marks the level where the vertical shear of the cross-front wind is largest.

Citation: Journal of the Atmospheric Sciences 73, 7; 10.1175/JAS-D-15-0163.1

It has been reported that numerically generated, spurious gravity waves are produced in simulations of sloping baroclinic zones through the inconsistency between the horizontal and vertical resolution (e.g., Pecnick and Keyser 1989; Lindzen and Fox-Rabinovitz 1989; Persson and Warner 1991). Pecnick and Keyser (1989) and Persson and Warner (1991) showed that spurious gravity waves are produced when the sloping thermal structure is sufficiently strong and narrow and the model grid aspect ratio Δzx is larger than the slope of the front, where Δz and Δx are the vertical and horizontal grid intervals, respectively. Persson and Warner (1991) also showed that both the horizontal wavelength and the amplitudes of the spurious gravity waves are approximately proportional to the grid aspect ratio. The grid aspect ratio for the 2-km grid simulation is ~0.08 at the height of low-level frontal zone, whereas the slope of the low-level frontal zone was ~0.02. Thus, the 2-km grid simulation is conducive to generate spurious gravity waves.

However, the low-level cold-frontal zone in the present simulation was not sharp enough to generate significant spurious waves that can affect band formation, and the upward-propagating gravity waves in Fig. 18 can be distinguished from spurious gravity waves; both the amplitude and the horizontal wavelength of the waves remained almost unchanged in a test simulation in which the model vertical grid size was doubled (not shown), unlike the spurious gravity waves whose amplitude and horizontal wavelength vary in proportion to the vertical grid size.

On the other hand, the test simulation revealed that wavelike disturbances with horizontal wavelengths of ~10–15 km in the vertical velocity field immediately above the upper-level frontal zone (cf. Figs. 11f, 14f, 15d, and 17d) were spurious waves caused by the large grid aspect ratio in the upper levels (~0.3). The effects of spurious waves, however, were little reflected in the simulated precipitation fields, and the horizontal scales and the amplitudes of main updrafts above the upper-level frontal zone were almost unchanged in the test simulation (not shown).

The sensitivity experiments in the previous subsection showed that a broad enhanced frontal updraft was formed by frontal dynamics in the absence of cooling by melting. The analysis of model results above indicates that the melting-induced cooling in the stratiform precipitation area acted to superimpose upward-propagating gravity waves upon the broad updraft, thereby forming forward-sloping updrafts UP1 and UP2 that were strong and narrow enough to produce WCFRs that could be identified in the surface rainfall field. In the following sections, specific mechanisms for the generation of gravity waves and the evolution of the first and second WCFRs are further investigated using dry simulations forced by specified cooling.

It should be mentioned that the total diabatic heating due to ice deposition and condensation within the enhanced updrafts above the cold-frontal zone was naturally much larger than that for the melting-induced cooling. However, because the diabatic heating is proportional to the vertical velocity and the saturated air above the cold-frontal zone was stably stratified in the present case, the enhanced diabatic heating should be regarded as a result of the gravity wave response to other sources rather than as a source for the gravity waves. The diabatic heating due to ice deposition and condensation was not explicitly treated as a source for gravity waves, but its effects on the modulation of the wave response were included in the dry simulations by using the moist Brunt–Väisälä frequency obtained in the WRF simulation. The effects of diabatic cooling due to the subcloud layer evaporation of rain below the cold-frontal zone were not considered in the dry simulations either. This is because the cold-frontal shear layer should act as a critical level for the upward-propagating gravity waves generated by such cooling, and therefore the cooling should have little direct influence on the wave response above the frontal zone.

5. Experimental design for dry simulations

a. Model description

Dry simulations were performed using a 2D, nonhydrostatic, Boussinesq model. Although a cold-frontal zone represents a sloping, narrow zone of enhanced gradients of physical variables, the simulations presented here were initialized with horizontally uniform basic states for simplicity. The effect of rotation was also neglected because of a high characteristic Rossby number of the perturbation flow under consideration.

The model’s governing equations are written as
e2
e3
e4
e5
where U is the basic horizontal wind; u, w, p, and θ are the perturbations of horizontal wind speed, vertical wind speed, pressure, and potential temperature, respectively; ρ0, θ0, and T0 are the constant reference values of air density, potential temperature, and temperature, respectively; N is the Brunt–Väisälä frequency; g is the gravitational acceleration; cp is the specific heat at constant pressure; ν is the coefficient of Rayleigh damping in a sponge layer; and q is the diabatic cooling rate per unit mass (J kg s−1). The terms Du, Dw, and Dθ represent eddy diffusions for u, w, and θ, respectively. The parameter δNL was set to 0 (1) in linear (nonlinear) simulations. The horizontal and vertical eddy coefficients were set to 100 and 1 m2 s−2, respectively. In all simulations ρ0 was set to 1 kg m−3, and θ0 and T0 were set to 300 and 270 K, respectively.

The model domain was 1200 km wide (−600 ≤ x ≤ 600 km) and 30 km deep. Horizontal and vertical grid sizes were 500 and 50 m, respectively. A free-slip boundary condition was applied at the top and bottom boundaries and a radiation condition based on Orlanski (1976) was applied at the lateral boundaries. The damping coefficient ν in the sponge layer will be specified later. The horizontal derivative terms were approximated by the fourth-order centered difference scheme, while the vertical derivative terms by the second-order centered difference scheme. Fourth-order numerical diffusion was also applied in the horizontal, with the mixing coefficient set as 5 × 10−4 s−1. Leapfrog time differencing was used along with a Robert–Asselin time filter (Asselin 1972). The time step was 10 s, and the coefficient of the time filter was 0.1.

b. Experimental design

To investigate the specific mechanisms for the generation of gravity waves and the evolution of the first and second WCFRs in the WRF simulation, two sets of simulations were conducted using the dry model. The specification of each simulation is shown in Table 2.

Table 2.

Summary of the dry model configurations.

Table 2.

Previous studies indicate that the stable shear layers can augment wave generation in the presence of a thermal forcing (e.g., Beres et al. 2004; Bakas and Ioannou 2007) and that a tilted thermal forcing preferentially excite gravity waves with phase lines tilted in the same sense as the forcing (e.g., Pandya and Durran 1996; Kawashima 2003). As noted earlier in section 4c, the base of the updraft UP1 was persistently located above the intersection of the frontal zone and the melting layer, where both the vertical shear of the cross-front wind and the slope of the melting layer were largest. This suggests that the strong vertical shear of the cross-front wind and/or the enhanced slope of the cooling by melting at the intersection contributed to the wave generation.

The first set of the dry simulations, DRY-A1–DRY-A4, was conducted in order to examine the basic effects of the vertical shear and the slope of the cooling on the generation of gravity waves. In this set, N was set to be constant with height (N = 0.01 s−1) and nonlinear terms were deactivated (δNL = 0) for simplicity. The basic flow for DRY-A1 and DRY-A2 was specified to be constant with height (U = −5 m s−1). The basic flow for DRY-A3 and DRY-A4 had a concentrated vertical shear and is given analytically as
e6
where U0 = −5 m s−1, zc = 15 km, and h = 0.5 km. The basic flow reverses its direction at z = zc = 15 km, where the local Richardson number Ri [=N2/(dU/dz)2] takes the smallest value of 1.0. Thus, the basic flow is dynamically stable everywhere.
The diabatic cooling used in the first and second sets of simulations is specified to have the form
e7
where
e8
and
e9

In the above equations, q0 is the magnitude of the cooling; a and b are constants that control the scales of the cooling in the x and z directions, respectively; zf is the horizontally varying center height of the cooling; zc is the center height of the cooling at x = 0; and α, β, and γ are constants that control the slope of the cooling. The second term on the right-hand side of Eq. (8) represents the enhancement of the slope of the melting layer within the frontal zone.

As described later, the parameters that specify the cooling in the dry simulations were chosen based on the profile of the melting-induced cooling in the stratiform precipitation area in the NOMELT-POST simulation. The horizontal scale of the cooling, however, was set to be small (a = 10 km) and the slope of the cooling was set to be constant in the first set.

In the presence of vertical boundaries, outgoing waves from the cooling are reflected by the boundaries and the steady-state response to the cooling becomes strongly dependent on the distance between the cooling and boundaries. To properly assess the effects of the vertical shear and the slope of the cooling on the generation of gravity waves, both the lowermost and uppermost 8-km-deep layers were set as sponge layers in the first set. The damping coefficient ν was increased in the sponge layers from 0 s−1 at z = 8 and 22 km to 5 × 10−4 s−1 at z = 0 and 30 km according to a sine square function. Approximately 10 h of integration time was needed for the flow to become quasi steady. Here, output fields at the integration time of t = 15 h are examined.

The second set of simulations, DRY-B1 and DRY-B2, was conducted to demonstrate that the formation and evolution of WR1 and WR2 in the WRF CTRL simulation can be interpreted in terms of the gravity wave response to the melting-induced cooling in the stratiform precipitation area. This set used evolving, realistic profiles of U and N constructed based on the result of simulation NOMELT-POST. The result of NOMELT-POST was used because the physical variables for CTRL were already perturbed by the melting-induced cooling. The CTRL simulation suggests that the gravity waves were generated in the vicinity of the intersection of the frontal shear layer and the melting layer. As will be shown later, the first set of dry simulations also demonstrates that gravity wave generation should be locally enhanced at the intersection. Thus, the profiles taken at the location of the intersection (marked by the cross in Figs. 15d–f) were used to construct U and N.

Time–height cross sections of the cross-front wind relative to the movement of the intersection and the subsaturated or saturated Brunt–Väisälä frequency in NOMELT-POST are shown in Figs. 19a and 19b, respectively. As shown in Fig. 15, the height of the frontal zone steadily increased as the time elapsed. As a consequence, the intersection moved toward the SCF at an average speed of 1.7 m s−1 between 0900 and 1530 UTC. Note that critical levels where the cross-front velocity relative to the intersection becomes zero exist within the low-level frontal shear layer and near the top of the SCF-relative rearward flow (denoted by thick solid contours in Fig. 19a). The profiles of U and N were obtained by simplifying these profiles and are shown in Figs. 19c and 19d, respectively. For simplicity, the critical level and the level of maximum vertical shear within the low-level shear layer were set at z = 1.2 and 1.4 km, respectively.

Fig. 19.
Fig. 19.

(left) Time–height sections of (a) cross-front wind speed relative to the intersection of the frontal shear layer and the melting layer (contours every 2 m s−1 with negative values dashed and zero contours thick) and (b) Brunt–Väisälä frequency N (×10−2 s−1, shaded contours) from 0600 to 1700 UTC 8 Dec 1976 for NOMELT-POST. The profiles were constructed using composite vertical cross sections normal to the SCF every 10 min. (right) Time–height sections of (c) basic horizontal wind speed U (contours every 2 m s−1 with negative values dashed and zero contours thick) and (d) Brunt–Väisälä frequency N (×10−2 s−1, shaded contours) for simulations DRY-B1 and DRY-B2.

Citation: Journal of the Atmospheric Sciences 73, 7; 10.1175/JAS-D-15-0163.1

In the second set of simulations, the horizontal scale of the imposed cooling was set to be large (a = 50 km) and the slope of the cooling was locally enhanced at x ≈ 0 to mimic the melting-induced cooling. The locally enhanced melting-induced cooling associated with an established WCFR in CTRL will naturally induce a localized gravity wave response. To properly address the causal relationship between the melting-induced cooling and the formation of WCFRs, the parameters that specify the cooling were also determined based on the results of NOMELT-POST, and the effects of locally enhanced cooling associated with a WCFR were isolated. The analytical cooling approximate the properties of the melting-induced cooling in NOMELT-POST, with the thickness of the cooling of 0.7–0.9 km, the maximum cooling rate of 0.8–1.5 K h−1 (0.2–0.37 J kg s−1), the maximum slope of the cooling within the frontal zone of 0.016–0.035, the horizontal scale of the enhanced slope of the cooling of 20–50 km, and the slope of the cooling outside of the frontal zone of 0.003–0.006. The cooling was temporally held constant for simplicity. Thus, the location of the intersection of the cooling and the low-level shear layer remained unchanged at x = 0.

The simulation DRY-B1 was run in linear configurations (δNL = 0). The assumption of linearity is violated in the vicinity of the critical level, where a small perturbation in the horizontal velocity field will easily exceed the basic horizontal velocity. To address the nonlinear effect, the model was also run with the nonlinear terms retained (δNL = 1) in DRY-B2.

Only the uppermost 8 km of the domain served as a sponge layer in the second set. Thus, the downward-propagating waves generated by the cooling were reflected by the bottom boundary. The reflected waves, however, had little effect on the wave pattern above the low-level shear layer, because the waves were almost absorbed by the critical level at z = 1.2 km.

The first 20 h run of the dry model used fixed U and N profiles at 0600UTC in Figs. 19c and 19d, then the model was run for 11 h using the temporally varying profiles according to the rest of the time in Figs. 19c and 19d.

6. Results of dry simulations

a. Basic response to specified cooling

The steady-state vertical velocity field for each simulation in the first set is shown in Fig. 20. The response for the case with the uniform basic flow and the nonsloped cooling, DRY-A1 (Fig. 20a), consisted of repeated upward and downward motions with an upstream (downstream) phase tilt with height above (below) the cooling, indicating that vertically propagating gravity waves were generated by the cooling. In the simulation forced by the upstream-tilted cooling, DRY-A2 (Fig. 20b), upward-propagating waves were amplified, whereas the downward-propagating waves were reduced compared with those in DRY-A1. This indicates that a tilted thermal forcing enhances the waves with phase lines tilted in the same sense as the forcing, as suggested by previous studies (e.g., Pandya and Durran 1996; Kawashima 2003).

Fig. 20.
Fig. 20.

Vertical velocity fields (contours every 0.01 m s−1 with negative values dashed and zero contours omitted) for (a) DRY-A1, (b) DRY-A2, (c) DRY-A3, and (d) DRY-A4 at t = 15 h. Cooling rates [=(θ0/cpT0)q] greater than 0.05 (0.5) K h−1 are lightly (darkly) shaded. The vertical profile of basic horizontal wind (U) for each case is also shown at the right.

Citation: Journal of the Atmospheric Sciences 73, 7; 10.1175/JAS-D-15-0163.1

The vertical velocity amplitude in DRY-A3 (Fig. 20c), which utilized the sheared basic flow and the nonsloped cooling, was about twice as large as that in DRY-A1, indicating that the gravity wave generation can be enhanced by the presence of vertical shear. The vertical velocity perturbation was large and in phase with the cooling right at the critical level (z = zc = 15 km), because Eq. (5) predicts that the vertical velocity perturbation becomes directly proportional to the diabatic forcing in the absence of mean flow (cf. Lin 1987). The wave response was markedly enhanced in DRY-A4, which utilized the sheared basic flow and the upshear-tilted cooling (Fig. 20d), with the vertical velocity amplitude about 4 and 3 times larger than those in DRY-A2 and DRY-A3, respectively. This means that the gravity wave generation can be enhanced significantly by the combined effects of the vertical shear and the upshear tilt of the cooling.

The linear response to the specified cooling in Fig. 20 can be regarded as the superposition of gravity waves from a number of compact heat sinks that constitute the specified cooling. The enhanced wave response in DRY-A4 can be explained by Fig. 21, which schematically shows waves induced by two compact heat sinks, C1 and C2, placed on a sloped axis of the specified cooling. For brevity, only the response above the critical level is considered in Fig. 21.

Fig. 21.
Fig. 21.

Schematic illustration of the superposition of gravity waves generated by two compact heat sinks that constitute the sloped cooling. The thin gray line indicates the axis of the sloped cooling. The gray filled circles, C1 and C2, represent compact heat sinks. The thick solid (dashed) lines mark lines of maximum (minimum) vertical velocity associated with gravity waves excited by C1, and the thin solid (dashed) lines mark lines of maximum (minimum) vertical velocity associated with gravity waves excited by C2.

Citation: Journal of the Atmospheric Sciences 73, 7; 10.1175/JAS-D-15-0163.1

In a steady state, the angle φ between the wave phase lines and the horizontal can be given by
e10
where k is the horizontal wavenumber of each heat sink. The plus and minus signs indicate upward and downward-propagating waves, respectively.

As in DRY-A1, both upward- and downward-propagating waves are excited by each heat sink, but the slopes of wave phase lines within the shear layer vary with height according to Eq. (10). Because the phase lines of waves from C2 have small slopes in the vicinity of the critical level, the phase difference between the upward-propagating waves from C1 and C2 can be small even if C1 and C2 are widely separated. This indicates that waves from many compact heat sinks that constitute the sloped cooling can be favorably superposed, thereby significantly reinforcing the total wave response. It should be mentioned that the optimal slope of the cooling for the generation of upward-propagating waves increases as the magnitude of the shear and/or the mean leftward wind speed increase.

The air motion forced directly by the sloped cooling in DRY-A4 was tilted against the shear, allowing downgradient Reynolds stress and therefore kinetic energy transfer from the basic shear flow (i.e., −uwU/∂z > 0). The enhanced wave response in DRY-A4 can be explained alternatively by this persistent energy transfer from the basic shear flow.

The results of the first set of simulations imply that gravity wave generation by the melting-induced cooling should be locally enhanced at the intersection of the melting layer and the frontal zone because of the combined effects of the largest vertical shear and the steepest slope of the melting layer there.

b. Simulations with evolving basic flow

Figure 22 shows vertical velocity fields for DRY-B1 and DRY-B2 at 0900, 1230, and 1530 UTC 8 December 1976. As shown in Fig. 19, the time evolution of N was less significant compared to that of U. Thus, the evolution of the response described below was caused mainly by the evolution of U.

Fig. 22.
Fig. 22.

Vertical velocity fields (contours every 5 × 10−3 m s−1 with negative values dashed and zero contours omitted) for (a)–(c) DRYB1 and (d)–(f) DRY-B2 at 0900, 1230, and 1530 UTC 8 Dec 1976. Cooling rates [=(θ0/cpT0)q] greater than 0.05 (0.5) K h−1 are lightly (darkly) shaded. (right) Vertical profiles of basic horizontal wind (U) at 0900, 1230, and 1530 UTC are also shown. Gray lines in each panel indicate critical levels for stationary gravity waves.

Citation: Journal of the Atmospheric Sciences 73, 7; 10.1175/JAS-D-15-0163.1

As expected from the results of the first set, in DRY-B1 vertically propagating waves with large vertical velocity perturbations were generated locally at the intersection of the imposed cooling and the low-level shear layer (x ~ 0 and z ~ 1.4 km). The upward-propagating waves were strongly attenuated as they passed through the upper critical level. The vertical velocity field at 0900 UTC (Fig. 22a) bore a qualitative resemblance to that in the WRF simulation (cf. Fig. 9a); a downdraft region immediately above the cooling at 0900 UTC (0 < x < 80 km in Fig. 22a) corresponds to the region of suppressed frontal upward motion (−70 < x < 0 km in Fig. 9a), whereas a sloping updraft above the intersection corresponds to the enhanced updraft UP1. As the basic leftward flow above the low-level shear layer increased in depth but decreased in intensity after 0900 UTC, the upper critical level was elevated and the vertical wavelength of the excited waves was reduced. As a result, another phase of upward motion had formed by 1230 UTC (x ~ 0 and z ~ 7 km in Fig. 22b). The vertical velocity field closely resembles the difference vertical velocity field between CTRL and NOMELT-POST at 1230 UTC shown in Fig. 18. The strongly tilted primary updraft and the secondary updraft correspond to UP1 and UP2 in Fig. 9c, respectively. Consistent with the WRF simulation, the primary and secondary wave updrafts had weakened considerably by 1530 UTC in association with decreases in the depth and strength of the leftward flow above the low-level shear layer (Fig. 22c).

In the nonlinear simulation, DRY-B2, oppositely propagating density currents developed in the early simulation period (not shown). This is because the low-level basic flow was weak and therefore subcritical relative to density current propagation (cf. Raymond and Rotunno 1989). The leading edges of the density currents and associated updrafts, however, had dissipated well before 0900 UTC. In this case, the low-level shear layer, defined by the total horizontal wind U + u, was shifted downward owing to the advection effect of the perturbation flow along the cooling. As a result, the intersection of the cooling and the shear layer, and therefore the induced wave pattern as well, slightly shifted leftward and downward compared with those in DRY-B1. The wave response, however, was essentially similar to that found in DRY-B1, indicating that the generation of gravity waves can be explained essentially by a linear process.

In summary, the second set of simulations reproduced the structure and evolution of the vertical velocity perturbations responsible for the first and second WCFRs in the WRF simulation. This indicates that the upward-propagating gravity waves responsible for the formation of the first and second WCFRs were generated at the intersection of the low-level frontal zone and the melting layer and that the evolution of these WCFRs can be attributed to the evolution of wave pattern associated with the evolution of cross-front flow above the frontal zone.

It should be mentioned that the horizontal wavelength of the gravity waves is determined by the horizontal scale of the intersection of melting layer and the frontal shear layer, which in turn depends on the depths and slopes of both the melting layer and the frontal shear layer. Thus, the width and spacing of WCFRs that form by this mechanism may differ from case to case. For example, the width and spacing of WCFRs will decrease as the horizontal gradient of temperature across the frontal zone become more intense.

7. Movement and alongfront variability of WCFRs

In this section, factors controlling the simulated movement and alongfront variability of WCFRs are discussed based on the generation mechanisms of WCFRs proposed above.

Since the gravity waves responsible for the WCFRs are generated at the intersection of the melting layer and the frontal shear layer, the orientation of the WCFRs should be approximately parallel to the orientation of the intersection, and the movement of the WCFR should be determined largely by the movement of the intersection. As shown in Figs. 14 and 15, the height of the low-level frontal zone steadily increased between 0900 and 1530 UTC in the absence of cooling by melting behind the NCFR, and the intersection moved about 40 km toward the SCF in that period. The forward movement of WR1 and WR2 in CTRL in that period (about 60 km) can be attributed mainly to the forward movement of the intersection associated with the increase of the frontal zone height, although the low-level frontal zone in CTRL was perturbed significantly by the melting-induced cooling. The decrease in the strength of the SCF-relative rearward flow may also have contributed to the forward movement of the WR1 and WR2 by enhancing the forward phase tilt of gravity waves and decreasing the rearward advection of snow particles relative to the SCF.

As shown in Fig. 5, the spacing between the NCFR and WR1 increased southward and WR2 was absent in the southern part of the front, although no observations were available for the verification of these simulated alongfront variabilities. In the present case, the simulated melting-layer height increased southward at a rate approximately 0.1 km (100 km)−1 because of the general southward increase of air temperature (not shown). Thus, as can be seen from the comparison of Figs. 14 and 17, the intersection of the melting layer and the frontal shear layer was located farther behind the SCF in the southern part of the front than that in the northern part in NOMELT-ALL. The southward increase of the spacing between the NCFR and WR1 in CTRL can be explained mainly by the southward increase of the spacing between the SCF and the intersection associated with the southward increase of the melting layer height.

The rainband WR2 was absent in the southern part, because the SCF-relative rearward flow above the cold-frontal zone remained shallow and the critical level absorption of the gravity waves occurred at a lower level than that farther north. Further, the upper-level frontal zone and UP3 were closer to UP1 than those farther north, which may also have exerted a negative influence on the formation of UP2.

The mechanisms of WCFR formation proposed in this paper require the presence of an elevated melting layer that intersects the frontal zone. While both observed and simulated WCFRs dissipated near the SCF in the present case, WCFRs formed behind the SCF often move ahead of the SCF (e.g., Parsons and Hobbs 1983a). It should be mentioned that the mechanisms alone cannot explain the movement of WCFRs ahead of the SCF.

8. Summary and conclusions

The mechanisms responsible for the formation and evolution of WCFRs associated with a wintertime cyclone that moved onto the Washington coast on 8 December 1976 were investigated with WRF simulations and idealized dry simulations. The WRF control simulation reproduced precipitation features observed as the cold front approached the Washington coast, including WSRs, an NCFR, and three successively formed WCFRs.

Sensitivity experiments showed that cooling by melting in the stratiform precipitation area behind the NCFR was essential for the formation of the first and second WCFRs, whereas the third WCFR was formed by the release of potential instability within an ascent forced by upper-level frontogenesis. It was also shown that enhanced frontal updrafts responsible for the first and second WCFRs were created by a superposition of a broad updraft caused by frontal dynamics and upward-propagating gravity waves generated by the melting-induced cooling. Dry simulations forced by specified cooling were conducted in order to identify the specific mechanisms responsible for the generation of gravity waves and the evolution of the first and second WCFRs.

The mechanisms responsible for the formation and evolution of the WCFRs in the northern part of the front are summarized schematically in Fig. 23. Upward-propagating gravity waves with large vertical velocity amplitudes were generated locally at the intersection of the frontal zone and the melting layer, where the strong vertical shear of the cross-front wind and the enhanced slope of the cooling by melting cooperatively enhanced the wave generation. The upward motion and the growth of snow were suppressed within the gravity wave downdraft ahead of the intersection, which resulted in the reduction of stratiform rainfall between the NCFR and the intersection. On the other hand, the growth of snow was enhanced within the updraft that was reinforced by the gravity wave updraft, forming a clear local maximum of stratiform rainfall that can be identified as WR1 (Fig. 23a).

Fig. 23.
Fig. 23.

Schematic description of the processes involved in the formation and evolution of WCFRs discussed in this study. The darkest shading denotes the melting layer. Thin solid lines indicate the boundaries of the low-level cold-frontal zone. Thick dashed lines indicate the boundaries of the upper-level frontal zone. The dashed gray line indicates the critical level for gravity waves. The gravity wave updrafts (downdrafts) are enclosed by solid (dashed) contours and marked by solid (dashed) arrows. Open arrows above the upper-level frontal zone denote updrafts formed by the release of potential instability within the ascent forced by upper-level frontogenesis. Gray arrows denote airflows relative to the SCF, with thicker arrows representing stronger airflows. Shading denotes areas of locally large snow mixing ratio with darker shading representing larger snow mixing ratio. The spacing between the rain streaks is proportional to rain intensity. The vertical profile of the basic cross-front wind relative to the movement of the intersection of the low-level frontal zone and the melting layer is indicated on the right-hand side of each panel.

Citation: Journal of the Atmospheric Sciences 73, 7; 10.1175/JAS-D-15-0163.1

The simulated evolution of the first and second WCFRs was controlled mainly by the evolution of the SCF-relative cross-front flow above the low-level cold-frontal zone. As the SCF-relative rearward flow increased its depth and decreased its intensity, the vertical wavelength of the gravity waves was reduced and the critical level for the gravity waves was elevated. As a consequence, another updraft phase of gravity waves became apparent within the SCF-relative rearward flow, and the associated enhanced updraft formed WR2 (Fig. 23b). Meanwhile, WR1 and WR2 moved toward the SCF in association with the forward movement of the intersection caused by the increase in the frontal zone height. At this stage, an enhanced updraft formed above the upper-level cold front as a result of the release of potential instability by upper-level frontogenesis.

The gravity wave updrafts weakened and WR1 and WR2 dissipated successively as the SCF-relative rearward flow decreased its depth and strength. On the other hand, the updraft above the upper-level front retained its intensity and formed WR3 (Fig. 23c). The second WCFR was absent in the southern part of the front because the SCF-relative rearward flow was shallower and the upper-level front was closer to WR1 compared to those in the northern part (not shown).

The mechanisms developed herein explain the formation, evolution, and movement of the observed WCFRs consistently in terms of the gravity wave response to the melting-induced cooling. Although the mechanisms were drawn from a numerical study of only a single event, preliminary results of additional numerical case studies by the author suggest that gravity waves forced by the melting-induced cooling also play an essential role in several other cases of WCFRs located behind the SCF. The results of this continuing numerical study will be reported in a future manuscript. The mechanisms developed herein should also be tested against observations of detailed atmospheric motions inside the areas enveloping frontal rainbands.

Acknowledgments

This work was supported by the Grant-in-Aid for Scientific Research of the Ministry of Education, Culture, Sports, Science and Technology of Japan under Grant 21540446. The author thanks three anonymous reviewers for their helpful comments.

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