1. Introduction
The precipitation along a narrow cold frontal rainband (NCFR) that propagates into an environment of weak instability or neutral stability is often broken up into segments of intense precipitation called precipitation cores (PCs) separated by gaps of weaker rainfall, as first noted by James and Browning (1979) and Hobbs and Biswas (1979). The PCs are typically aligned at a clockwise angle with respect to the synoptic-scale cold front (e.g., Hobbs and Persson 1982; Parsons and Hobbs 1983a; Wakimoto and Bosart 2000). Horizontal shear instabilities (HSIs) along the leading edge of a cold front or their coupling with a convective instability have frequently been proposed as mechanisms that lead to the alongfront variability (AFV) of NCFRs (e.g., Matejka 1980; Hobbs and Persson 1982; Moore 1985, Carbone 1982; Parsons and Hobbs 1983b; Wakimoto and Bosart 2000; Smart and Browning 2009).
Numerous modeling studies indicate that the ambient vertical wind shear causes strong influence on the structure and evolution of squall lines (e.g., Hane 1973; Thorpe et al. 1982; Rotunno et al. 1988, hereafter RKW; Weisman et al. 1988; Fovell and Ogura 1989; Robe and Emanuel 2001). RKW found that the ability of the cold outflow from an old cell to lift environmental boundary layer air to its level of free convection (LFC) is enhanced with low-level shear, and they developed a widely cited theory that explains the squall-line strength and longevity in terms of a relative horizontal vorticity balance between the circulation associated with the cold pool and that associated with the environmental wind shear (often called RKW theory). Modeling and observational studies have also identified the importance of the low-level vertical shear in modulating the structure of NCFRs. Parsons (1992) showed, in a series of two-dimensional numerical simulations, that the dependence of the neutrally buoyant frontal updraft associated with an NCFR on vertical shear is consistent with that predicted by RKW theory. Jorgensen et al. (2003) investigated the variation of the precipitation and updraft structure of a strong NCFR using Doppler radar data. They found that variations in the strength of the low-level vertical shear of ambient flow caused a systematic variation of the updraft tilt along the length of PCs and in gap regions that is consistent with RKW theory. However, most of the previous studies on AFVs of NCFRs were case studies, and no attempts have been made to systematically investigate the effect of ambient vertical shear through idealized numerical simulations.
Kawashima (2007, hereafter K07) recently investigated the mechanism responsible for core-gap structures of precipitation along NCFRs through a series of idealized simulations of cold fronts forced by shearing deformation. He attributed the formation of PCs to wavelike disturbances within slanted frontal updrafts, which derive their kinetic energy mainly from the kinetic energy of the vertical shear flow and the buoyancy. It was also shown in K07 that the structure of PCs changes from those oriented clockwise to the general orientation of the surface cold front (SCF) to those oriented counterclockwise to the SCF as convective available potential energy (CAPE) in the prefrontal environment increases. However, all simulations in K07 were initialized using an ambient cross-front wind with a rather weak low-level vertical shear, which caused mean frontal updrafts to lean significantly back over the cold air. This raises a question about the generality and applicability of the mechanism proposed by K07, since typical environments of cold fronts are strongly baroclinic and therefore strongly sheared (e.g., Houze et al. 1976). In fact, some discrepancies were noted by K07 between observations and his simulation results. For example, Doppler radar observations of typical PCs that were oriented clockwise to the SCF revealed that the PCs were related systematically to the distortions of the SCF that would be a manifestation of HSI (e.g., Hobbs and Persson 1982; Wakimoto and Bosart 2000). On the other hand, the distortion of the SCF was weak and characteristic features of HSI were not recognized in K07’s simulations that produced such PCs. Furthermore, while Parsons and Hobbs (1983a) reported a case in which typical PCs developed along an NCFR whose prefrontal environment was absolutely stable (see their Figs. 10 and 11), a certain amount of CAPE was required for the formation of typical PCs in K07’s simulations.
Lee and Wilhelmson (1997a, hereafter LW97a) investigated misocyclone development along numerically simulated thunderstorm outflow boundary. They noted that the growth of HSIs that could eventually form misocyclones is either delayed or completely absent as the low-level environmental vertical shear normal to the cold outflow leading edge is reduced from an optimal value. Although the horizontal shear zones associated with cold outflows simulated by LW97a were much narrower compared with that for typical NCFRs, their results suggest that the weak ambient vertical shear condition adopted in K07 was not conducive to the growth of HSIs along NCFRs.
The objective of this paper is to systematically investigate the sensitivity of AFV of NCFRs to variations in the environmental characteristics through a series of idealized simulations. The environmental characteristics studied include the ambient vertical shear of the cross-front wind, the atmospheric stability, and the strength of the leading edge horizontal wind shear. It is shown that core-gap structures of precipitation occur because of the development of an HSI wave along NCFRs under a wide range of environmental conditions. This paper focuses primarily on this type of AFV of NCFRs, which was not addressed in K07 but appears to be a more common type of AFV of NCFRs.
The outline of this paper is as follows. The model utilized in this study and the experimental design are described in section 2. In section 3, the impacts of variations in the vertical shear of ambient cross-front winds and the stability of the stratification on the AFVs of NCFRs are investigated. Section 4 offers an explanation concerning the suppression of HSI waves along NCFRs under weak ambient vertical shear conditions. Section 5 examines the impacts of variations in horizontal shear strength on HSI waves along NCFRs and discusses processes that account for a distinct relationship between the initial wavelength of the HSI wave and the horizontal shear strength identified in the simulations. Finally, the results are summarized in section 6.
2. Model description and experimental design
a. Model configuration
The model used in the present study is derived from the nonhydrostatic, moist shear–frontogenesis model described in K07. Several substantial modifications are made to systematically investigate the sensitivity of the AFV of an NCFR to variations in parameters of interest.
To investigate the sensitivity of the AFV of NCFR to variation in a parameter, it is desirable to keep other parameters unchanged. However, the use of the shear–frontogenesis model makes it difficult. For example, the variation of the ambient vertical shear of the cross-front wind entails the variation of the large-scale along-frontal gradient of temperature that could be advected by the alongfront wind through thermal wind relation. Thus, a cold front simulated by the shear–frontogenesis model tends to be more intense and the prefrontal environment tends to be more unstable as the vertical shear increases. The horizontal shear at the leading edge of cold front can be varied by varying the Coriolis parameter in the model equations. However, the variation of the Coriolis parameter also entails the variation of the along-frontal gradient of temperature.
Around the leading edge of a mature, sharp cold front, where very large horizontal gradients in pressure, temperature, and wind exist within a few kilometers, the wind field is highly ageostrophic in the across- and even along-frontal directions (e.g., Parsons et al. 1987). In fact, an analysis of momentum equations for cold fronts simulated in K07 indicates that the Coriolis accelerations were smaller than advection terms and horizontal pressure gradient forces by approximately one order of magnitude around the leading edges (not shown). This indicates that the structure of the leading edge of a cold front can be simulated well without the Coriolis force. Thus, for the present study, the Coriolis effect is excluded and prescribed heat and moisture sinks are used to initialize cold fronts, relying upon findings that have shown that the structure of the leading edge of a cold front is similar to a that of a gravity current (e.g., Carbone 1982; Hobbs and Persson 1982; Parsons 1992). The exclusion of the Coriolis force allows us to vary parameters of interest almost independently, which makes the experiment simpler and easier to understand, though less realistic.
As in K07, the model is first integrated in its 2D (x, z) configuration for computational efficiency. Using the 2D model output at t = 12 h as the meridionally uniform initial condition, the model is then run in its 3D configuration for 8 h, within which time period AFVs are allowed to develop. The synoptic cold front is oriented in the north–south (y) direction. The experimental domain has dimensions of Lx = 180 km, Ly = 300 km, and Lz = 10 km. The east and west lateral boundaries are open, whereas the north and south boundaries are periodic. Rigid-lid top boundary conditions are applied together with a Rayleigh wave-absorbing layer over the uppermost 3 km. The model uses 180 grid points in the x direction, the central 120 of which comprise a fine mesh with a constant grid spacing of Δx = 500 m. A 60-km-wide stretch grid is placed on both sides of the fine mesh. The grid spacing in the y direction is constant at Δy = 500 m. Observations of NCFRs (e.g., Browning and Harrold 1970; Hobbs and Persson 1982; Wakimoto and Bosart 2000) indicate that the horizontal shear at the leading edge is on the order of 10−2 s−1 and the width of the wind shift zone is about 2–3 km. A horizontal grid spacing of 500 m is chosen because simulations with this grid size reproduce these characteristic values. The vertical grid stretches smoothly from Δz = 100 m in the lowest 1 km to 300 m at the top of the domain. A schematic of the 2D model configuration is shown in Fig. 1.
A schematic of the configuration used to initialize 2D cold fronts. The dashed line at the left represents the location of HS and MS, and the shading indicates the cold air created by HS and MS. Arrows pointed out of the figure (southward) indicate the direction of MF applied in the cold air. An arrow pointed into the figure denotes the direction of environmental alongfront wind (northward). The vertical profile of the initial cross-front wind is shown at the right. The model boundary conditions are also given.
Citation: Journal of the Atmospheric Sciences 68, 4; 10.1175/2010JAS3599.1
As in K07, a bulk aerodynamic surface friction parameterization similar to that employed by Wilhelmson and Chen (1982) is included to represent a semislip condition at the bottom. In this study, the initial wind field is assumed to be in balance with surface friction, and the drag is applied to perturbation velocities following Adlerman and Droegemeier (2002) to prevent significant modifications of the basic-state wind fields with time. The nondimensional drag coefficient Cd is set to 1.0 × 10−3 (over oceans). Subgrid-scale turbulence is parameterized using a Smagorinsky-type first-order closure (e.g., Lilly 1962). Fourth-order numerical damping is also applied in the horizontal, with the mixing coefficient set as 6.25 × 107 m4 s−1. All of the simulations reported herein are made with a Kessler-type warm-rain microphysics scheme for simplicity. The author performed several experiments with the ice microphysics scheme of Lin et al. (1983) and confirmed that the inclusion of ice-phase microphysics does not impact the basic conclusions drawn herein.
b. Soundings
Vertical profiles of θ (solid line) and θe (dashed line) for the S0, S1, and S2 soundings.
Citation: Journal of the Atmospheric Sciences 68, 4; 10.1175/2010JAS3599.1
Thermodynamic profile diagnosis for the S0, S1, and S2 soundings. Listed are the lapse rate of potential temperature between 0 and 2 km, the strength of the potential instability [defined as the difference between surface and minimum θe (Δθe)], the level of free convection (LFC), the level of neutral buoyancy (LNB), and the convective available potential energy (CAPE).
Because the frontal updrafts considered in this study are rather shallow, cross-front wind profiles with constant positive shears confined to below z = 2.5 km—a condition used in many numerical studies (e.g., RKW; Fovell and Ogura 1989; Parsons 1992)—are examined (Fig. 3). The difference between the zonal wind speeds at the surface and that at z = 2.5 km is denoted as US, and a series of experiments are conducted for US = 5, 10, 15, and 20 m s−1. Test simulations indicated that as US was increased beyond 20 m s−1, a frontal updraft leaned downshear with height and a well-defined cold outflow boundary was not established. The vertical shear of the low-level cross-front wind used in K07 is almost the same as that for the US = 5 m s−1 profile. By subtracting a constant speed from u0, simulated NCFRs are maintained within the fine mesh. Test simulations also indicated that the variation in the vertical shear of υ0 exerts a relatively small influence on the structure of an AFV compared with that of u0; thus, υ0 is not changed for all of the simulations presented herein. Considering the effect of surface friction, υ0 is decreased linearly from 22.5 m s−1 at z = 0.75 km to 15 m s−1 at z = 0 km.
Initial cross-front (solid line) and alongfront (broken line) wind profiles.
Citation: Journal of the Atmospheric Sciences 68, 4; 10.1175/2010JAS3599.1
c. Initialization of fronts
The name of a simulation is created by combining the vertical shear of the cross-front wind with the thermodynamic sounding; for example, U5-S0 refers to the simulation initialized with a US = 5 m s−1 cross-front wind and the S0 sounding.
3. Vertical shear experiment
a. Overview of two-dimensional simulations
In this section, the dependence of the AFV of an NCFR on the vertical shear of the ambient cross-front wind and the stability of the stratification is examined. First, the results of 2D runs are presented to show that the initialization method adopted in this study produces realistic structures of cold fronts.
Time series of (a) wmax (m s−1) and (b) WF (km) for simulations initialized with the S0 sounding.
Citation: Journal of the Atmospheric Sciences 68, 4; 10.1175/2010JAS3599.1
Vertical cross sections around the leading edge for U5-S0 and U20-S0 at t = 12 h are shown in Fig. 5. The U5-S0 case produces a highly sloped leading edge of cold air and an updraft that slopes significantly back over the cold air (Fig. 5a). The momentum forcing applied in the cold outflow creates a horizontal shear of υ (i.e., vertical vorticity) exceeding 8 × 10−3 s−1 at the leading edge (Fig. 5b). Because the frontal slope from the vertical is large, a significant local vertical shear exists at low levels in the region immediately behind the leading edge of cold air (Fig. 5c; x ∼ 74 km). As the ambient vertical shear increases, the cold air leading edge becomes steeper and the frontal updraft becomes stronger. The U20-S0 case produces an almost upright frontal updraft and a deeper horizontal shear zone compared with those in U5-S0 (Figs. 5d,e). The strong local vertical shear zone behind the leading edge is located at higher levels compared with that in U5-S0 (Fig. 5f). Unlike the U5-S0 case, the local vertical shear is weak in the region of large horizontal shear at low levels.
Two-dimensional frontal fields at t = 12 h for (a)–(c) U5-S0 and (d)–(f) U20-S0. (a),(d) Front-relative winds in the plane of the cross section, θd (every 0.5 K with zero contour omitted; shaded below −0.5 and −1.5 K), and rainwater mixing ratio (contoured at 0.1, 0.2, and 0.4 g kg−1). (b),(e) Meridional velocity (gray contours, every 3 m s−1) and vertical vorticity (every 2 × 10−3 s−1 with zero contours omitted). (c),(f) Magnitude of vertical shear—[(∂u/∂z)2 + (∂υ/∂z)2]1/2; every 3 × 10−3 s−1 with zero contours omitted.
Citation: Journal of the Atmospheric Sciences 68, 4; 10.1175/2010JAS3599.1
It has been reported that spurious numerically generated gravity waves are produced in simulations of sloping baroclinic zones due to the inconsistency between the horizontal and vertical resolution (e.g., Pecnick and Keyser 1989; Lindzen and Fox-Rabinovitz 1989; Persson and Warner 1991; Snyder et al. 1993; Jorgensen et al. 2003; Iga et al. 2007). Persson and Warner (1991) showed that significant spurious gravity waves will appear if the sloping frontal structures created by a model are sufficiently strong and sufficiently narrow when measured in grid spaces and the grid aspect ratio Δz/Δx is larger than the slope of the frontal structures. In some places of the fronts in the present simulations, the grid aspect ratio (0.2 below z = 1 km) is larger than the slope of frontal structures, making possible spurious gravity waves. However, no evidence of spurious gravity waves is seen in the present simulations. This is probably because the horizontal resolution adopted in the present study is high enough to resolve the sloped frontal structures. Persson and Warner (1991) also argued that enhanced diffusion will increase the width of the front, thereby decreasing the amplitude of the spurious gravity waves. In the present simulations, parameterized horizontal and vertical diffusions tend to be large along the sloped frontal zone because of the presence of strong wind shear. This may also contribute to the suppression of spurious gravity waves in the present simulations.
Figure 6 depicts key frontal properties as functions of US for all of the simulations. For reference, the results of the US = 25 m s−1 trial simulations are also included. For all soundings, the increase in US results in an increase in wmax until US = 20 m s−1, but further addition of the shear results in a decrease in wmax (Fig. 6a). The response of the frontal updraft to the variation in the cross-front vertical shear is consistent with RKW and Parsons (1992), in that the updraft becomes stronger and more upright with the addition of ambient vertical shear until some optimal values, but further addition of the shear results in the decrease in the updraft strength and downshear tilt of the updraft. The maximum vertical vorticity ωzmax at the leading edge also increases with the addition of ambient vertical shear until US = 20 m s−1, but its variations with US and the stability are less significant compared with that for wmax (Fig. 6b). The depth of the horizontal shear zone generally increases with the addition of US associated with a steepening of the cold air leading edge (Fig. 6c).
Plots against US of (a) wmax, (b) ωzmax, and (c) the depth of the horizontal shear zone where the vertical vorticity exceeds 2 × 10−3 s−1. Variables shown are averaged over the last 8 h (from 12 to 20 h) of the 2D model integration. The results for the US = 25 m s−1 test simulations are also included.
Citation: Journal of the Atmospheric Sciences 68, 4; 10.1175/2010JAS3599.1
b. Overview of three-dimensional simulations
1) Stably stratified cases
Figure 7 shows the horizontal cross sections of the precipitation intensity normalized by the maximum precipitation intensity in the domain Rmax and the horizontal wind fields around the leading edge at t = 4 and 8 h for all of the simulations initialized with the S0 sounding. Hereafter, the simulation time indicated is that from the beginning of the 3D run. It can clearly be seen that the magnitude of the undulation of SCF increases as the vertical shear is increased. As seen from the comparison between t = 4 and 8 h, the wavelengths of the AFVs for U5-S0 and U10-S0 change little throughout the simulation period, whereas the AFVs for U15-S0 and U20-S0 show marked upscale growth. The weak undulation of the SCF and weak AFV of precipitation in U5-S0 are consistent with the results of simulation A3D in K07 (see his Fig. 17), for which the prefrontal thermodynamic sounding was also stable to the surface parcel ascent. In U10-S0 and U15-S0, the precipitation is maximized along segments of the SCF that are tilted clockwise from the north, where the perturbed cold front makes largest angle with the prefrontal winds and, therefore, the largest forced convergence occurs. The formation mechanism of PCs for these cases appears to be consistent with the early proposed mechanism whereby HSIs create PCs aligned at a clockwise angle with respect to the synoptic-scale cold front (e.g., Matejka 1980; Hobbs and Persson 1982; Parsons and Hobbs 1983b).
Horizontal cross sections of the precipitation intensity normalized by Rmax (shaded) and the core-relative horizontal winds at the lowest grid level (z = 50 m) at (top) 4 and (bottom) 8 h for simulations initialized with the S0 sounding. The thick dashed line in each panel denotes the location of the SCF. Only a 40 km × 120 km portion of the full domain is shown. Rmax (mm h−1) is indicated above each panel. Rectangles indicate the horizontal domains displayed in Fig. 8.
Citation: Journal of the Atmospheric Sciences 68, 4; 10.1175/2010JAS3599.1
The strongest vertical shear case, U20-S0, produced bow-shaped PCs near the middle of which the precipitation is maximized. As suggested by the relative location of the surface precipitation to the SCF, the tilt of the frontal updraft is upshear at the southern part of the PC, almost upright around the apex of the PC, and downshear at the northern part of the PC. The variation of the updraft tilt along the length of a PC is consistent with those reported in Jorgensen et al. (2003, see their Figs. 14 and 18), though in their case the reflectivity was maximized near the apex or slightly north of the apex of each bow-shaped element. While Jorgensen et al. (2003) suggested that the AFV of the vertical shear of the cross-front ambient flow would account for the variation of updraft tilt, in the present simulation the vertical shear of the prefrontal wind is uniform in the alongfront direction.
Horizontal sections of the vertical vorticity for the regions enclosed by rectangles in Fig. 7 indicate that the near-surface vertical vorticity tends to accumulate at segments of the SCF tilted counterclockwise from the north, where the precipitation intensity is relatively weak (top panels of Fig. 8). This vorticity pattern resembles that found in the early growth stages of shear-instability waves along perturbed vortex sheets associated with parallel shear flows (e.g., Batchelor 1967; Corcos and Sherman 1984), density current interfaces (e.g., LW97a), or the Burgers vortex layer in which the enhancement of vorticity by the spanwise (i.e., along the vorticity direction) strain is balanced by diffusion (e.g., Lin and Corcos 1984; Neu 1984). Such vortex sheets subsequently show highly nonlinear behavior characterized by vortex sheet roll-up and formation of almost circular vorticity patches. However, the patches of near-surface large vertical vorticity in the present case are highly elongated and remained tilted upshear with respect to the mean alongfront wind. This point will be further discussed later. In U10-S0, U15-S0, and U20-S0 the horizontal gradient of temperature is weak in the region where the precipitation is weak, consistent with the observations by James and Browning (1979), Hobbs and Persson (1982) and Wakimoto and Bosart (2000).
Horizontal cross sections at t = 8 h for the regions enclosed by rectangles in Fig. 7. (a)–(d) ωz (3 × 10−3 s−1 contour interval with zero contours omitted) and θd (shaded below −0.5 and −1.5 K) at the lowest grid level (z = 50 m). (e)–(h) ωz (2 × 10−3 s−1 contour interval with zero contours omitted) at z = 550 m. Areas of updraft greater than 2 m s−1 are shaded in (e)–(h).
Citation: Journal of the Atmospheric Sciences 68, 4; 10.1175/2010JAS3599.1
The spatial distributions of the vertical vorticity at z = 550 m are rather different from those at z = 50 m. In the U5-S0 and U10-S0 cases (Figs. 8e,f), maxima in the vertical vorticity are found along the segments of the cold fronts tilted clockwise from the north, where frontal updrafts are also strong. This suggests that vertical stretching provides an important contribution to vorticity at this level. These vorticity maxima are also associated with the enhanced precipitation along the cold fronts, consistent with the observation by Wakimoto and Bosart (2000; see their Fig. 6). This is also true for the vorticity in the U15-S0 and U20-S0 cases (Figs. 8g,h); however, the vertical vorticity maxima found at the clefts of the cold air in Figs. 8c,d are still evident at this level.
2) Unstable cases
Figure 9 shows the precipitation intensities at t = 8 h for simulations initialized with the S1 and S2 soundings and Fig. 10 shows vertical vorticity at z = 50 m and vertical velocity at an indicated level for the regions enclosed by rectangles in Fig. 9. Mean vertical shear of y-averaged horizontal winds between z = 0.75 and 2.0 km are also shown by vectors for the U5-S1 and U5-S2 cases in Fig. 10. In these cases, velocity perturbations are large in this layer. As in the U5-S0 case, the undulation of the SCF is weak in U5-S1, but the precipitation along the leading edge breaks into regularly spaced PCs oriented clockwise from the north (Fig. 9a). These features are similar to those of the CNTL3D simulation in K07 (his Fig. 10). The dynamical structure of the AFV for U5-S1 shown in Fig. 10a is also similar to that of CNTL3D in K07. The AFV of near-surface vertical vorticity is rather weak, while the vertical velocity field at z = 1.52 km (above the frontal surface) exhibits a well-regulated, wavelike structure consisting of an alternating series of upward and downward motions.
As in Fig. 7, but for simulations initialized with the (top) S1 and (bottom) S2 soundings at t = 8 h. Rectangles indicate the horizontal domains displayed in Fig. 10.
Citation: Journal of the Atmospheric Sciences 68, 4; 10.1175/2010JAS3599.1
Horizontal cross sections of ωz at z = 50 m (3 × 10−3 s−1 interval with zero contours omitted) and w (gray-shade scale at the bottom of each panel) for the regions enclosed by rectangles in Fig. 9 for (a) U5-S1, (b) U5-S2, (c) U15-S2, and (d) U20-S2, showing w fields at (a),(b) z = 1.52 km and (c),(d) z = 550 m. Mean vertical shear of y-averaged horizontal winds between z = 0.75 and 2.0 km are also shown by vectors in (a) and (b).
Citation: Journal of the Atmospheric Sciences 68, 4; 10.1175/2010JAS3599.1
Weckwerth and Wakimoto (1992) documented a case in which intersections between a gust front and horizontal vortex lines associated with Kelvin–Helmholtz (K-H) waves with wave crests oriented nearly perpendicular to the gust front produced updraft maxima along the gust front. Similar intersections of horizontal vortex lines and a cold air leading edge were also reported for a nonprecipitating cold front studied by Friedrich et al. (2008). The phase lines of the vertical velocity perturbation for U5-S1 are oriented nearly perpendicular to the direction of the mean vertical shear. However, the wave is different from the simple K-H waves, in that the wave derives most of its energy from both the vertical shear and the buoyancy as shown in the next section.
In the U5-S2 case, PCs are oriented counterclockwise to the cold front and the SCF shows significant undulations (Fig. 9e). The vertical velocity field for this case shows elongated updrafts and downdrafts extending from the leading edge toward the northwest, and phase lines of the vertical velocity perturbations are oriented roughly parallel to the local vertical shear (Fig. 10b). These features are consistent with K07’s large CAPE simulations (his section 4), although the irregularity of the AFV is significant in the present case. Thus, the structure of the AFV and its dependence on the stability of the prefrontal environment in the US = 5 m s−1 simulations are generally consistent with K07, despite the differences in the frontal initialization method. The U10-S2 case (Fig. 9f) exhibits qualitatively similar AFVs to those found in U5-S2, although the PCs are less differentiated from each other.
In other cases, the convective-scale features of AFVs noted above are absent and the structures of the AFVs are basically the same as those found in the corresponding S0 simulations in Fig. 7. However, unlike the S0 sounding cases, the precipitation tends to be maximized north of the apex of each lobe of cold air in the U15-S2 and U20-S2 cases. The PCs for U20-S2 are similar in appearance to those reported by Jorgensen et al. (2003), although in this case precipitation is also maximized in regions immediately behind the clefts of cold air. Comparison of Figs. 8 and 10 for the US = 15 and 20 m s−1 cases indicates that the vertical vorticity at z = 50 m is more concentrated near the clefts of cold air in the S2 sounding cases than in the S0 sounding cases. Furthermore, while the maxima in vertical velocity in the S0 sounding cases are found along the segments of the cold front tilted clockwise from the north (Figs. 8g,h), those in the S2 sounding cases are found near the apex or north of the apex of each lobe of cold air (Figs. 10c,d).
Many studies have demonstrated that misocyclones along air mass boundaries distort the boundaries and produce local maxima of low-level convergence and vertical velocity in their vicinity, which often could be preferred regions to form clouds and/or radar reflectivity cores (e.g., Wilson et al. 1992; Atkins et al. 1995; Lee and Wilhelmson 1997b; Friedrich et al. 2005; Arnott et al. 2006; Murphey et al. 2006; Marquis et al. 2007; Wakimoto and Murphey 2009). Although well-defined vortices with quasi-circular vorticity contours do not develop in the present simulations, it is suggested that in the U15-S2 and U20-S2 cases the concentrated vertical vorticity distorted the horizontal wind field and enhanced the low-level convergence in the vicinity of the vorticity maxima, leading to formation of the precipitation maxima north of the bow apexes.
The qualitative difference of the structure of AFVs between weak and strong vertical shear cases initialized with unstable soundings can be explained by Fig. 11, which shows the distributions of moist absolutely unstable layers (MAULs; e.g., Bryan and Fritsch 2000; Bryan 2005) and θe for the 2D simulations initialized with the S2 sounding. Here, MAULs are identified in the model output where a grid point is saturated and the squared Brunt–Väisälä frequency for saturated air Nm2 calculated following Durran and Klemp (1982) is less than zero. The dependence of the horizontal and vertical extent of MAULs on the ambient vertical shear is consistent with that noted by Bryan (2005) for numerically simulated squall lines. The U5-S2 and U10-S2 cases, wherein the frontal updrafts are tilted strongly upshear, exhibit horizontally contiguous MAULs above the cold air. In such a situation, the strong local vertical shear above the cold air can align the convective updrafts in the MAUL in the shear direction, as discussed by Bryan et al. (2007). In contrast, MAULs are identified only in narrow regions immediately ahead of the cold air in U15-S2 and U20-S2, wherein the frontal updrafts are more upright. Convective-scale AFVs can hardly develop in such a situation because the residence time of air parcels within the MAUL becomes too short to allow significant convections to develop (e.g., Fuhrer and Schär 2005; Kirshbaum et al. 2007). Though not shown, U5-S1 also exhibits a MAUL above the cold air. However, the horizontal extent of the MAUL is small compared with those in U5-S2 and U10-S2, which probably prevented the formation of convective rolls aligned in the vertical shear direction.
θe (shaded) and front-relative winds in the plane of the cross section at t = 12 h of 2D configuration runs for (a) U5-S2, (b) U10-S2, (c) U15-S2, and (d) U20-S2. The solid contour encloses regions where Nm2 < 0.
Citation: Journal of the Atmospheric Sciences 68, 4; 10.1175/2010JAS3599.1
c. Energetics
Time evolutions of the domain-averaged HSP, VSP, and BP terms for each case are shown in Fig. 12. Note that each term has been divided by the domain-averaged EKE to clearly show the relative importance of each term during the early growth stage. The energetics for U5-S1 and U5-S2 cases are qualitatively similar to that for the CNTL3D and large CAPE simulations in K07, respectively (see his Figs. 14a and 21). In U5-S1 (Fig. 12e), the main energy sources are VSP and BP, and HSP is much smaller than these two terms throughout the simulation period. On the other hand, the contribution of HSP is also significant in U5-S2 (Fig. 12i). The time evolution of terms in the early simulation period for U10-S2 (Fig. 12j) is similar to that in U5-S2, but the term HSP becomes dominant beyond t = 3 h.
Time evolutions of the HSP (thick solid), VSP (dashed), and BP (thin solid) terms averaged over the whole calculation domain for each case. Each term is divided by the instantaneous value of the domain-averaged EKE and has a unit of 10−3 s−1.
Citation: Journal of the Atmospheric Sciences 68, 4; 10.1175/2010JAS3599.1
In all other cases, the main conversion is due to HSP, whereas VSP acts as a significant sink term. This indicates that the disturbance gains energy from the mean horizontal shear flow through horizontal Reynolds stresses, whereas simultaneously generated vertical Reynolds stresses feed the disturbance’s energy into the mean vertical shear flow. The magnitude of BP is small compared with that for HSP and VSP.
The time evolutions of the domain-averaged EKE for simulations initialized with the S0 sounding are displayed in Fig. 13 using a log-scaled ordinate. After a short period of initial adjustment, the EKE for each case grows exponentially at an almost constant growth rate (linear in Fig. 13). The comparison between Figs. 12 and 13 indicates that the magnitude of each energy term is relatively large in the exponential growth stage of EKE. All of the hydrodynamic fields (i.e., u′, υ′, w′, and θ′) for each case also grow at a common constant rate in that stage (not shown), indicating that the instability is essentially linear. Hereafter, AFVs for which the main energy source is HSP is referred to as the HSI-type AFVs or the HSI waves, although the instability characteristics are rather different from those for parallel shear flows, as shown later. It is evident in Fig. 13 that the growth rates and the amplitudes of the HSI waves decrease markedly with decreasing ambient vertical shear. The differences in growth rates between the cases are much larger than those expected from the differences in the vertical vorticity shown in Fig. 6b.
Time evolutions of the EKE averaged over the whole calculation domain for the simulations initialized with the S0 sounding. For reference, the EKEs for exponentially growing disturbances with constant growth rates of 4.5 × 10−4 and 8 × 10−5 s−1 are indicated by gray straight lines.
Citation: Journal of the Atmospheric Sciences 68, 4; 10.1175/2010JAS3599.1
The spatial distributions of the EKE and energy terms for U10-S0 in the linear instability phase (t = 3 h) are displayed in Fig. 14. The EKE is concentrated at the cold air leading edge (Fig. 14a), where the generation of EKE by HSP is also large due to the presence of strong horizontal shear (Fig. 14b). The depletion of EKE by VSP occurs in the region immediately behind the leading edge because the local vertical shears of both the cross-front and alongfront winds are large there (Fig. 14c). A small EKE generation by BP occurs at the leading edge (Fig. 14d), where the undulation of shear zone results in the generation of eddy potential energy that will be converted into EKE. Although energy flux convergence (EFC) and advection of EKE (ADV) do not have net contributions to the volume-averaged EKE, these terms act to transport the EKE upward from the region of large EKE generation by HSP (Figs. 14e,f). The distributions of energy terms for other HSI-type cases are basically the same as those in Fig. 14. However, as shown in Fig. 15a, the removal of EKE by VSP behind the leading edge in the U5-S0 case is somewhat larger and occurs at lower levels because the frontal slope from the vertical and the local vertical shear increase as the ambient vertical shear is reduced. In contrast, in the U20-S0 case, the removal of EKE by VSP behind the leading edge is smaller and is located at higher levels (Fig. 15b). Note also that a negative VSP region appears immediately ahead of the leading edge because of the presence of strong positive vertical shear of cross-front wind there (see Fig. 5f).
Spatial distributions of the EKE and energy terms divided by the maximum EKE in the domain for U10-S0 at t = 3 h: (a) normalized EKE, (b) HSP, (c) VSP, (d) BP, (e) EFC and energy flux vectors 
Citation: Journal of the Atmospheric Sciences 68, 4; 10.1175/2010JAS3599.1
Spatial distributions of the EKE and the VSP term divided by the maximum EKE in the domain for (a) U5-S0 at t = 5.5 h and (b) U20-S0 at t = 1.5 h. The normalized EKE is shaded at 0.2 and 0.6. The contour interval for VSP is 1 × 10−3 s−1 starting at ±0.5 × 10−3 s−1. Also,
Citation: Journal of the Atmospheric Sciences 68, 4; 10.1175/2010JAS3599.1
It should be mentioned that the specific vertical shear magnitude above which HSI-type AFVs occur in unstable sounding cases depends strongly on the temperature difference across the cold front because the structure of the uplifting of prefrontal unstable air depends strongly on the balance between the strength of cold air and the ambient vertical shear (e.g., RKW). However, the structural similarity of HSI-type AFVs with many previous observations and the strongly sheared environments typical of cold fronts (e.g., Houze et al. 1976) suggests that an HSI-type disturbance will be a more common type of AFV of NCFRs than those reported in K07. Thus, remaining discussions will be focused on HSI-type AFVs.
4. Mechanisms of HSI wave suppression
It is shown in the previous section that the growth rates and amplitudes of HSI-type AFVs decrease as the vertical shear of the ambient cross-front wind is reduced. This tendency is qualitatively consistent with the results of numerical simulations of nonsupercell tornadogenesis of LW97a (their Figs. 16 and 17). LW97a attributed the delay or the absence of misocyclone development in their negative vertical shear or unsheared cases to a lack of outflow nose structure, which is conducive to the lobe and cleft instabilities that trigger HSIs. However, the simulated leading edge lacks nose structure and therefore is free from lobe and cleft instabilities in all of the simulations presented herein. Furthermore, the difference in the growth rates of HSI waves as noted in Fig. 13 cannot be explained by the difference in the magnitude of initial perturbations. In this section, an explanation is offered concerning the significant vertical shear dependency of the HSI wave growth along NCFRs.
The instability of parallel shear flows is often interpreted in terms of the synergetic interaction between two vorticity edge waves that propagate in opposing directions on oppositely signed vorticity gradients (e.g., Bretherton 1966; Hoskins et al. 1985, Fig. 18; Bishop and Thorpe 1994, Fig. 1; Heifetz et al. 1999). The edge waves interact because the vorticity anomalies of one edge wave induce cross-stream velocity at the location of the other wave, and their interaction can result in phase-locking and mutual growth. The concept of edge wave interaction helps explain the significant dependence of HSI wave growth on ambient vertical shear.
Figure 15 suggests that a low-level wave on the western edge of the shear zone is strongly damped by VSP when the ambient vertical shear of cross-front wind is weak. Although a wave on the eastern edge of the shear zone is not directly damped by VSP in weak ambient vertical shear cases, the mutual amplification of low-level edge waves will be strongly inhibited in such a situation.
Figure 16 shows the spatial distributions of ADVy and TILTy as well as the deviations of ADVx and TILTx from their y averages (ADV′x and TILT′x) for U5-S0 at t = 5.5 h. Here, ADV′x and TILT′x are shown because ADVx and TILTx are almost uniform in the y direction when the amplitude of an HSI wave is small. In the horizontal sections, averages over the depth between z = 50 and 450 m are shown because the meridional advection of the vertical vorticity on the western edge of the shear zone is concentrated in that layer. The term ADVy (Figs. 16a,e) acts to accumulate ωz on both edges of the segments of perturbed shear zone rotated counterclockwise from the north (y ∼ 98 km). The accumulation of ωz creates a circulation tendency that further enhances the undulation of the shear zone. This effect is represented by ADV′x (Figs. 16c,g), which shows positive values on the western (eastern) edges of the segments of the shear zone that are displaced westward (eastward) from the mean locations of the vorticity edges. The growth of the vertical vorticity perturbation through the differential horizontal advection is consistent with the qualitative explanation of shear instability by Batchelor (1967). Because ωy is negative on the western side of shear zone, TILTy creates a negative (positive) vorticity tendency along the western edges of the segments of shear zone that are rotated counterclockwise (clockwise) from the north, where the y gradient of w is positive (negative) (Figs. 16b,f). In this case, ADVy on the western edge of the shear zone is almost completely offset by TILTy because ωy on the western side of the shear zone is large. On the other hand, TILT′x creates a negative (positive) vorticity tendency on the western edges of the segments of the shear zone that are displaced westward (eastward). The values of TILT′x are negatively correlated and almost in balance with those of ADV′x on the western edge of the shear zone (Figs. 16d,h). Thus, the growth of the western edge wave and, therefore, the mutual amplification of the vorticity edge waves are strongly suppressed in this case.
Spatial distributions of the vorticity tendency terms for U5-S0 at t = 5.5 h: (a) ADVy and core-relative horizontal wind vectors, (b) TILTy and horizontal vorticity vectors (ωx, ωy), (c) ADV′x and core-relative horizontal wind vectors, (d) TILT′x and horizontal vorticity vectors (ωx, ωy), and (e)–(h) vertical sections along the dashed lines in (a)–(d), respectively. Variables shown are averaged between z = 50 and 450 m in (a)–(d). Contour intervals are 0.25 × 10−7 s−2 for ADVy and TILTy and 0.5 × 10−7 s−2 for ADV′x and TILT′x with zero contours omitted. The shaded area in each panel represents the region where ωz exceeds 3 × 10−3 s−1. The zonal scale is exaggerated by 2.7 times in (a)–(d) to show the spatial relationship between the tendency terms and the vertical vorticity clearly.
Citation: Journal of the Atmospheric Sciences 68, 4; 10.1175/2010JAS3599.1
For comparison, Fig. 17 shows the distributions of the same terms for the U20-S0 case at t = 1.5 h, when the wave amplitude is almost the same as that in Fig. 16. The spatial distribution of each term on the western edge of the shear zone is qualitatively the same as in Fig. 16. However, ADVy and ADV′x on the western edge of the shear zone are only partially offset by TILTy and TILT′x, respectively, because the magnitudes of ωx and ωy in that region are considerably smaller than those in U5-S0. Thus, the mutual amplification of the vorticity edge waves is less inhibited than in U5-S0.
As in Fig. 16, but for U20-S0 at t = 1.5 h. Contour intervals are 0.5 × 10−7 s−2 for ADVy and TILTy and 1.0 × 10−7 s−2 for ADV′x and TILT′x with zero contours omitted. The shaded area in each panel represents the region where ωz exceeds 4 × 10−3 s−1.
Citation: Journal of the Atmospheric Sciences 68, 4; 10.1175/2010JAS3599.1
Although the local vertical shear of alongfront wind ∂
5. Horizontal shear experiments
a. Results
LW97a and Lee and Wilhelmson (2000) showed that strength of across-front horizontal shear plays a controlling role in the upscale progression of misocyclones along outflow boundaries. Since the horizontal shear across the cold front also varies from case to case, it is of interest to investigate the sensitivity of the AFVs of NCFRs to variation in horizontal shear strength. For this purpose, simulations presented in the previous sections are repeated for various values of CM ranging from 0.2 × 10−4 to 1.2 × 10−4 s−1. Although the horizontal shears in the CM = 0.2 × 10−4 and 0.4 × 10−4 s−1 cases are considerably weaker compared with those of typical cold fronts, these cases are run to demonstrate clearly the dependence of AFV on horizontal shear. On the other hand, test simulations utilizing CM values greater than 1.2 × 10−4 s−1 indicated that short-wavelength Kelvin–Helmholtz waves develop between the warm and cold air masses because of the enhancement of the local vertical shear, which strongly masks the HSI-type AFVs of interest. Thus, the results for such simulations are not presented here. This parameter study reveals a distinct relationship between the initial wavelength of HSI and the horizontal shear strength. It should be mentioned that in LW97a’s simulations the initial wavelengths of HSIs were determined largely by the scale of lobe and cleft gravitational instabilities, which were triggered by relatively strong thermal perturbations initially distributed along the outflow leading edges.
Figure 18 shows profiles of υ across the leading edge for 2D-configuration runs of U10-S0 using various CM values. The maximum horizontal shear and Reynolds number Re are also indicated. Here, Re is defined as Re = (Vmax − Vmin)WF/(2Km), where Km is the parameterized horizontal eddy-mixing coefficient at the location of maximum horizontal shear. Although the increase of CM from 0.2 × 10−4 to 1.2 × 10−4 s−1 entails the increase of Km from about 120 to 200 m2 s−1 (not shown), the qualitative profiles of υ and the 2D structures of the cold front remain virtually unchanged.
Horizontal profiles of υ at z = 0.15 km and t = 12 h for the U10-S0 horizontal shear experiments. Re and ωzmax (10−3 s−1) are indicated in the parentheses. The origin along the abscissa represents the location of ωzmax.
Citation: Journal of the Atmospheric Sciences 68, 4; 10.1175/2010JAS3599.1
Figure 19 shows the vertical vorticity field in the exponential growth stage of the EKE, whereas Fig. 20 shows the precipitation intensity at t = 8 h for U10-S0 simulations using various CM values. The growth of HSI waves is completely absent in the simulation using CM = 0.2 × 10−4 s−1. This is also true for U5-S0 simulations using CM = 0.2 × 10−4 and 0.4 × 10−4 s−1 (not shown), suggesting that the critical value of horizontal shear required for HSIs increases as US is reduced. It is evident in Fig. 19 that the initial wavelength of AFV increases with increasing horizontal shear, despite the fact that the widths of the horizontal shear zones differ little between the simulations. In this and U5-S0 cases, marked upscale growths of the HSI-type AFVs are absent regardless of the horizontal shear strength; thus, the wavelength of the AFV of precipitation at t = 8 h for each case is almost the same as that in the early growth stage of the HSI wave (Fig. 20). The initial wavelength of AFV also increases with increasing horizontal shear for the US = 15 and 20 m s−1 simulations. In these cases, however, the upscale growths of AFVs beyond the exponential growth stage of the HSI waves tend to be enhanced as the horizontal shear is increased, consistent with the results of LW97a and Lee and Wilhelmson (2000) (not shown).
Horizontal cross sections at z = 50 m showing the distributions of ωz during the exponential growth stage of the EKE for the U10-S0 horizontal shear experiments. The vorticity at t = 8 h is shown for the simulation using CM = 0.2 × 10−4 s−1, wherein the growth of HSI waves is completely absent. The contours of ωz are at 20% and 80% of ωzmax in the horizontal sections. The values of CM (10−4 s−1) and ωzmax (10−3 s−1) are indicated above each panel. The zonal scale is exaggerated 2 times to clearly show the AFV of ωz. Only a 20 km × 150 km portion of the full domain is shown.
Citation: Journal of the Atmospheric Sciences 68, 4; 10.1175/2010JAS3599.1
As in Fig. 7, but for the U10-S0 horizontal shear experiments at t = 8 h. Only a 40 km × 150 km portion of the full domain is shown.
Citation: Journal of the Atmospheric Sciences 68, 4; 10.1175/2010JAS3599.1
The statistics of the horizontal shear experiments are presented in Fig. 21, which plots the ratio of the mean initial wavelength of AFV λ to the effective width of horizontal shear zone WE as a function of the horizontal shear for various vertical shear simulations. Here, λ is calculated by dividing Ly by the total number of local vorticity maxima, and WE is defined as the average of WF at t = 0 h over a depth for which the horizontal average of the HSP in the subsequent exponential growth stage of the EKE exceeds 20% of its maximum value. The horizontal shear in Fig. 21 is the average of maximum horizontal shear over this depth. For all of the cases initialized with the stable S0 sounding (Fig. 21a), a stronger horizontal shear leads to a larger λ/WE, except in the weak horizontal shear range where a slight decrease of λ/WE with increasing horizontal shear is observed in U15-S0 and U20-S0 cases. A similar decreasing tendency for λ/WE with increasing horizontal shear is observed in unstable sounding cases (Fig. 21b) for a wider horizontal shear range (less than ∼6 × 10−3 s−1) compared with the stable sounding cases, but a similar increase of λ/WE with increasing horizontal shear occurs above that shear range as for the stable sounding cases. It should be mentioned here that the basic horizontal-shear dependency described above was also observed in experiments utilizing a constant Km.
Ratio of λ to WE as a function of the horizontal shear strength. (a) Simulations initialized with the S0 sounding and (b) simulations initialized with the S1 and S2 soundings. Data for U5-S0 utilizing CM = 0.2 × 10−4 and 0.4 × 10−4 s−1 and U10-S0 utilizing CM = 0.2 × 10−4 s−1 are not included because in each of these cases the growth of HSI waves was not observed.
Citation: Journal of the Atmospheric Sciences 68, 4; 10.1175/2010JAS3599.1
Numerical analyses of shear instabilities for viscid parallel flows by Betchov and Szewczyk (1963) indicate that the most unstable wavelength decreases with increasing Re if the Re is smaller than approximately 40–50. Above this Re, the instability becomes essentially insensitive to viscosity and the wavelength becomes almost constant as in the inviscid case. This is because the viscosity damps short waves more strongly than long waves, and the damping effect weakens as Re increases. The decrease of λ/WE with increasing horizontal shear observed in the weak horizontal shear range in Fig. 21, where Re is below 40, appears to be consistent with the results of Betchov and Szewczyk (1963). However, the remarkable increase of λ/WE with increasing horizontal shear above that range appears to be quite curious. Shear instability waves attenuate with distance from the shear zone, with e-folding scales proportional to the wavelengths (e.g., Drazin and Reid 1981). Thus, for the Burgers vortex layer, long waves will feel more strongly the effect of the convergent cross-shear flow which increases linearly with the distance from the shear zone. Lin and Corcos (1984) showed that the most unstable wavelength for the Burgers vortex layer becomes somewhat shorter than that in the unstrained case described in Betchov and Szewczyk (1963) especially in small Re range. As discussed by Neu (1984), this is because the uniform plain convergence will act to stabilize long waves more strongly than short waves, in contrast to the viscosity. However, the dependency of the most unstable wavelength on Re shown in Lin and Corcos (1984) is qualitatively the same as that for the unstrained case (i.e., the effect of convergence only partially offsets the decreasing tendency of the wavelength with increasing Re). Considering the results of Lin and Corcos (1984), it is unlikely that the decrease in the magnitude of horizontal convergence relative to that of horizontal shear is responsible for the increase of λ/WE with increasing horizontal shear observed in our simulations.
b. Effects of local vertical shear on scale selection
As shown in the previous section, the local vertical shear around the leading edge acts to suppress the growth of HSI waves. The local vertical shear increases with increasing distance from the location of maximum horizontal shear, associated with the increase of the frontal slope from the vertical (see Fig. 5). Thus, it is suggested that the local vertical shear will stabilize long waves more strongly than short waves because long waves are less attenuated in the cross-frontal direction and, therefore, experience the effect of the local vertical shear more strongly than short waves.
To confirm the effect of local vertical shear proposed above and examine its dependency upon the horizontal shear strength, additional simulations are performed in which the wavelengths of AFV are specified by setting θpy(y) = 0.25 sin(2πy/λp) in (9). The wavelength λp is varied from 4 to 40 km in increments of 2 km, and the model meridional domain size is set to λp. The specification of the initial wavelength makes the growth rate of the HSI wave in the early growth stage almost constant with time, which makes it possible to construct a dispersion curve.
Figure 22 shows the growth rates and the magnitudes of the volume-averaged VSP relative to that of HSP (−VSP/HSP) for the U10-S0 simulations utilizing CM = 0.4 × 10−4, 0.8 × 10−4, and 1.2 × 10−4 s−1. The growth rate σ was obtained by dividing the average change rate of the EKE for the middle 20 min of its exponential growth stage in half {i.e., σ = (1/2)d[ln(EKE)]/dt}, for which period −VSP/HSP was also calculated. The magnitude of −VSP/HSP provides a measure of the effect of local vertical shear on suppressing HSI waves. The growth rate is maximized at longer wavelengths for stronger horizontal shear simulations (Fig. 22a), consistent with the results shown in Fig. 21. On the other hand, −VSP/HSP increases with increasing wavelength (Fig. 22b), supporting the assumption that the local vertical shear will stabilize long waves more strongly than short waves. Furthermore, the value of −VSP/HSP for a given wavelength decreases as CM increases, indicating that the stabilizing effect of local vertical shear decreases with increasing horizontal shear. This is because the magnitude of the local vertical shear of cross-front wind (∂
Plots against the wavelength of (a) growth rates (10−4 s−1) and (b) magnitudes of the domain-averaged VSP relative to that of HSP for the U10-S0 simulations utilizing CM = 0.4 × 10−4 (closed circle), 0.8 × 10−4 (cross), and 1.2 × 10−4 s−1 (open circle).
Citation: Journal of the Atmospheric Sciences 68, 4; 10.1175/2010JAS3599.1
Figure 23 shows the same variables as in Fig. 22 for the US = 20 m s−1, CM = 0.8 × 10−4 s−1 simulations initialized with the S0, S1, and S2 soundings. The peak of the growth rate curve shifts toward shorter wavelengths as the stratification becomes unstable (Fig. 23a), consistent with the stability dependency of λ/WE noted in Fig. 21. The value of −VSP/HSP for a given wavelength increases as the stratification becomes more unstable (Fig. 23b). The increase of −VSP/HSP stems from the increases in the perturbation vertical velocity w′ and the vertical Reynolds stresses. These results indicate that the stabilizing effect of the local vertical shear on long waves increases as the stratification becomes more unstable. Thus, as the stratification becomes more unstable, stronger horizontal shear will naturally be required for a marked increase of λ/WE to occur, as observed in Fig. 21.
As in Fig. 22, but for US = 20 m s−1, CM = 0.8 × 10−4 s−1 cases initialized with the S0 (closed circle), S1 (cross), and S2 (open circle) soundings.
Citation: Journal of the Atmospheric Sciences 68, 4; 10.1175/2010JAS3599.1
Though not shown, additional simulations in which the strengths of heat and moisture sinks were varied indicated that increasing the temperature difference across the cold front in stably stratified cases has similar effect on the initial wavelength of an HSI wave as reducing the stability of stratification. This is probably because the local vertical shear of the cross-front wind and the strength of the vertical Reynolds stress increase as the temperature difference across the cold front increases.
6. Summary and conclusions
The sensitivity of AFV of wintertime maritime NCFRs to variations in various environmental characteristics have been investigated through a series of idealized simulations. In simulations initialized with a weak ambient cross-frontal vertical shear, the structures of the AFVs of NCFRs and their dependence on the stability of the prefrontal thermal stratification were found to be consistent with those described in K07, despite the differences in the frontal initialization method. In these cases, the AFVs of precipitation along NCFRs are attributed to wavelike disturbances within slanted frontal updrafts, which derive their energy mainly from the kinetic energy of mean local vertical shear and buoyancy. When the prefrontal environment is slightly unstable, the wave phase lines are oriented nearly perpendicular to the direction of local vertical shear and PCs are aligned at a clockwise angle with respect to the cold front. As the prefrontal environment becomes sufficiently unstable, the wave phase lines become nearly parallel to the direction of local vertical shear and the PCs become oriented counterclockwise to the cold front.
On the other hand, the AFVs of NCFRs that are broadly consistent with previous observations occurred over a wider range of environmental conditions considered, including those of stable stratification, due to the development of HSI waves at the cold air leading edge. In relatively weak ambient vertical shear conditions, elongated PCs form along the segments of the SCF tilted clockwise from the general orientation of cold front, where the perturbed front makes largest angle with the low-level prefrontal wind and therefore the largest forced convergence occurs. On the other hand, in sufficiently strong vertical shear conditions, the precipitation along the NCFR tends to be organized into bow-shaped cores.
The growth rate and amplitude of the HSI wave decrease significantly as the ambient vertical shear of the cross-front wind is reduced, and the initial wavelength of the wave increases markedly with increasing horizontal shear. It is argued that the local vertical shear behind the leading edge causes these distinct dependencies of the HSI wave upon the vertical and horizontal shear strengths. The local vertical shear acts to damp the vorticity edge wave on the cold air side of the horizontal shear zone, thereby suppressing the growth of the HSI wave through the interaction of two vorticity edge waves. The growth rate and amplitude of the HSI wave become small in weak ambient vertical shear conditions as a result of the enhancement of the local vertical shear. The local vertical shear is also shown to stabilize long waves more strongly than short waves. The marked increase of the initial wavelength with increasing horizontal shear is attributed to the decrease in the magnitude of the cross-front component of the local vertical shear relative to that of the horizontal shear.
The application of the results of this study to observations must be done with caution, since the environmental conditions for real NCFRs are considerably more complex and the parameters varied independently in this study are related to each other in a real cold front. It is hoped, however, that the information provided in this paper will help to interpret observations and more realistic simulations of NCFRs. The applicability of the results obtained for NCFRs to other boundaries possessing marked horizontal shear such as drylines, gust fronts, and dry cold fronts is largely debatable since those boundaries have rather different environmental characteristics (e.g., static stability, surface friction). The author thinks, however, that the discussions made in this paper may also contribute to the understanding of wave development along other boundaries, since NCFRs and other boundaries have similarities in their basic kinematic and thermodynamic structures.
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 would like to thank two anonymous reviewers for their helpful comments.
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