a. DLA algorithm

The first DLA step is to calculate backward air trajectories from all grid points in the Lagrangian domain at a given analysis time following the method described in section 2b. The calculation of trajectories utilizes wind fields obtained either from a storm simulation (section 3) or from dual- and multiple-Doppler wind syntheses (Z13). The wind fields are obtained within a fixed analysis domain that contains the moving storm to facilitate the calculation of ground-relative trajectories. In all cases, the storm core is contained well within the wind analysis domain. The algorithm verifies that each backward gridpoint trajectory has terminated in the storm environment via one or more of the following robust conditional tests: (i) the number of steps N > 76 and Z_H < 0 dBZ; (ii) N > 76 and w < 0.5 m s⁻¹ for at least five consecutive steps; or (iii) the backward trajectory passes through a lateral boundary.

The second DLA step consists of the forward integration of the system of ODEs to determine the evolution of θ, q_υ, and q_c along each trajectory that has been verified to initiate in the storm environment. The ODEs are all written in the form (e.g., Klemp and Wilhelmson 1978)

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where φ = (θ, q_υ, q_c); ϕ = (θ, q_υ); and M_φ, D_φ, and F_ϕ refer to microphysical terms (section 2g), a bulk parcel damping term (section 2h), and a mesoscale surface flux term (section 2i), respectively. Thermodynamic calculations assume an ideal gas with total pressure p(x, y, z) = p_B(z) + p′(x, y, z) ≅ p_B(z), where p_B and p′ are the base state and perturbation pressures (millibars), respectively; p′ is assumed to be negligible to a reasonable approximation (Wilhelmson and Ogura 1972) and air density ρ_a = 1.0 × 10⁵[(p_B/1000.0)^0.2854]^2.509/(287.04θ). The Lagrangian values of θ, q_υ, and p_B at the initial point and time for forward integration are interpolated from either a 3D mesoscale analysis (Z13) or an input environmental sounding to prescribe the saturation point (Betts 1984), while q_c is initialized to zero. The local Lagrangian raindrop and graupel size distributions at each time step are computed from spatially and temporally interpolated Z_H analysis values (as detailed in sections 2c–f), while p_B is interpolated from the sounding or mesoscale analysis to the Lagrangian point. The presently included microphysical processes are cloud condensation and evaporation, rain evaporation, rain collection and graupel accretion of cloud, graupel sublimation, and graupel melting [see the supplemental material in Gilmore et al. (2004, hereafter G04)]. A time step Δt = 20 s is selected for the forward and backward Lagrangian integrations in the current study to qualitatively minimize path-integrated truncation error, as defined by the ability to return to almost exactly any given initial point via a backward–forward trajectory sequence. The chosen Δt combined with typical 3D wind speeds effectively restricts a maximum parcel displacement to be smaller than the horizontal and vertical minimum resolvable wavelengths in the (filtered) wind analyses.

The family of endpoints of each forward Lagrangian integration corresponds to the grid points from which each backward trajectory originated. Thus, the DLA is completed by gathering the 3D fields composed of the ending Lagrangian values from the set of all gridpoint trajectories. At any grid point whose trajectory cannot be initialized or integrated, the resulting missing value is hole filled from surrounding nonmissing grid points. The output 3D analysis fields are lightly smoothed with a horizontal nine-point elliptic low-pass filter to suppress any poorly resolved small-scale (~2–3Δ) variations between neighboring grid points at the analysis time.

b. Trajectory calculations

Each time step of a backward gridpoint trajectory is calculated with three iterations of a first-order predictor corrector scheme as in Z07. Combined with the small chosen Δt value, the predictor corrector scheme increases the accuracy of trajectories in strongly curved flows such as supercell low-level mesocyclones. The values of u, υ, w, and Z_H at each iteration of a given time step are trilinearly interpolated in space from the eight nodes of the grid cell that contains the Lagrangian point. Local time trends of storm structure are accounted for by linearly interpolating from the spatially interpolated neighboring analysis times to the intermediate time of the Lagrangian point. A unique feature of the interpolation procedure is its treatment of a nonzero constant horizontal storm motion by advecting the x and y coordinates of the analysis grid in a time-to-space sense and bilinearly interpolating the advected radar-analyzed fields to the grid nodes prior to the trilinear spatial and linear temporal interpolation steps.

The calculation of backward trajectories from surface grid points requires allowance for possible vertical displacements driven by near-surface convective downdrafts. Applying the kinematic lower boundary condition w = 0 exactly would result in trapped surface trajectories and a trivial solution in which temperature and moisture values trend asymptotically toward an inflow parcel's wet-bulb temperature. The pure wet-bulb effect, which is contrary to surface storm observations (e.g., Z13) and the observed scenario described by Betts (1984), is mitigated by applying a small surface downdraft adjustment in precipitation and initiating nominal backward surface trajectories at a prescribed offset height H₀ < 50 m AGL (Table 1).

Table 1.

Parameters related to the perturbation surface-layer downdraft, damping, and surface flux parameterizations employed in the OSSE tests of the DLA. (See Z13 for an application of the surface flux parameterization.) The symbols and values of the various parameters are also listed. The grid level index k = 2 corresponds to the first wind and Lagrangian analysis level above the surface. The parameter L_m is the mixing length discussed in the text, while |w| is the magnitude of the vertical velocity. Surface layer refers to the lowest ~50 m layer AGL.

View Table

Uniform surface mixed-layer profiles of θ and q_υ are assumed within the lowest ~50 m AGL to facilitate validation with surface in situ measurements. A small downdraft perturbation is objectively added at the surface in storm-scale downdrafts with precipitation to crudely account for net downward transport of cold outflow air into the surface layer from higher in the storm (e.g., Betts 1984). The parameterized surface downdraft w_k=1 ≈ w_50m = w_sfc is computed from the expression

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if w_k=2 < 0; otherwise, w_sfc = 0 if w_k=2 ≥ 0. In Eq. (2), the reflectivity scale Z* is expressed as

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where the parameters are listed in Table 1. Direct vertical velocity measurements with a tall tower array of vertical anemometers in storm outflows reported by Goff (1976) indicated that characteristic peak outflow downdrafts in near-neutral environments could range from about −1 m s⁻¹ at ~50 m AGL to stronger than −2 m s⁻¹ at ~450 m AGL in precipitation, implying a nonlinear downdraft profile with an aspect ratio of w_mix0 = w_{50 m AGL}/w_{450 m AGL} ~0.5 (Table 1). Goff (1976) also showed that low-level outflow became quasi-horizontal as the outflow decayed, an effect represented in Eq. (3) by Z* → 0. The parameterized surface downdraft perturbation varies as a function of downdraft at k = 2 and surface Z_H in broad accord with Goff's tall tower outflow analysis (Fig. 1).

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Functional dependence of parameterized surface layer downdraft w_sfc on radar-analyzed downdraft at first grid level above surface (w_k=2 < 0) and surface Z_H following Eq. (1) and using the parameters listed in Table 1 as discussed in the text.

Citation: Journal of Atmospheric and Oceanic Technology 30, 10; 10.1175/JTECH-D-12-00194.1

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c. Bulk precipitation size distribution parameters

The DLA adopts inverse exponential distribution functions for rain (r) and graupel (g) following supplemental material describing an implementation of the Lin–Farley–Orville (LFO) microphysical parameterization by G04. The inverse exponential distribution is a function of diameter D and takes the form (G04)

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where n_0x is the intercept parameter and is the slope parameter (i.e., x = r or g), is the concentration-weighted mean diameter, and the MKS units are employed for all parameterized quantities. The total concentration is

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while the mixing ratio is

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and ρ_a is air density (kg m⁻³), while ρ_x is the density of either rainwater ρ_w or graupel ρ_g. Rain parameters are related to the equivalent rain reflectivity Z_er (mm⁶ m⁻³) by the expression

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where Γ is the complete gamma function. Following Ferrier (1994) and substituting the definitions of the distribution moments, the equivalent graupel reflectivity Z_eg (mm⁶ m⁻³) is related to the graupel distribution parameters by the expression

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The values of the coefficients C_r, α_r, and C_g are listed in Table 2.

Table 2.

List of parameters contained in diagnostic relationships for precipitation quantities in the DLA. The symbol and values of the various parameters are also listed. (See Z13 for an application of the time-varying N_g scale.)

View Table

d. Relation of precipitation content to radar reflectivity

Since the combined parameterized precipitation size distributions contain five variable parameters (n_0r, λ_r, n_0g, λ_g, and ρ_g), an equal number of 3D fields of independent radar measurands would be required to provide a well-posed Lagrangian analysis problem. Since the time history of only the radar-observed Z_H field may be available through the entire storm volume (e.g., Z13), appropriate closure functions must be established between Z_H and the various precipitation parameters. A key assumption that the equivalent radar reflectivity (mm⁶ mm⁻³) equals the sum of Z_er [Eq. (7)] and Z_eg [Eq. (8)] assists in closing the problem of estimating the precipitation parameters. It has long been understood that the independent variation of N_x and q_x is theoretically rather more accurate than diagnosing N_x from q_x via an assumed constant n_0x because some microphysical processes change only N_x or q_x, while others change both N_x and q_x (Ziegler 1984). However, a reasonably accurate diagnosed value of N_x may be obtained by altering the n_0x value in a storm model and picking the particular n_0x value that associates with the best overall simulated airflow, reflectivity, downdraft, diabatic cooling and precipitation loading, and cold-pool morphology in comparison to the equivalent observed storm (e.g., the constant n_0r value deduced and employed by Z10).

The regression relationships between Z_H and q_r, n_0g, and q_g have been derived from model output data at the mature stage of a simulated supercell storm (Z10). Choosing the optimal value n_0r = 8 × 10⁵ m⁻⁴ (Z10) usefully reduces the number of degrees of freedom of the radar microphysics diagnosis. The input regression values of Z_H are calculated directly from the model output rain and graupel contents via Eqs. (7) and (8) and the total equivalent reflectivity Z_eh = Z_er + Z_eg following (SM05) and Ferrier (1994). The regression data are obtained from model output interpolated to constant height levels spaced at a 500-m interval through the lowest 5 km of the simulated storm. The regression data are limited to grid points where vertical velocity w is less than the approximate bulk precipitation fall speed V_t ~ 5 m s⁻¹ (i.e., where w < W_min = 5 m s⁻¹ and w/V_t < 1), thus effectively restricting fitted precipitation samples to descending precipitation particle trajectories (Kessler 1969). Although the method may easily be extended to span the entire storm depth, at present the analysis is conveniently restricted to the lowest 5 km of the Binger simulation to focus attention on the inflow to the low-level cold pool, the main updraft, and the forward- and rear-flank downdrafts. Although beyond the scope of the present study, the current radar microphysics closure technique could be extended to include either storm simulations with more detailed microphysics or a combination of polarimetric radar and storm-penetrating aircraft (SPA) observations (e.g., Ziegler et al. 1991).

The closure relationships between Z_H and q_r, n_0g, and q_g have been obtained from the previously described model data by applying a Levenburg–Marquardt nonlinear least squares regression algorithm. Independently varying graupel distribution moments and bulk density are implicitly represented in the regression data via weighted averages of the combined low-, medium-, and high-density graupel and frozen drop categories governed by the 10-class bulk ice (10-ICE) scheme (SM05) that was employed in the storm simulation (Z10). Proceeding from the single-moment rain distribution model employed by Z10, their value of the dynamically optimal fixed rain intercept parameter n_0r is listed in Table 2. The empirical relationships for , n_0g, and versus Z_H take the following forms:

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where and are provisional maximum values of the respective distribution parameters outside updraft cores, and single-valued parameters are listed in Table 2. In Eqs. (9)–(11), the height z (km AGL) is adjusted to z* (km) = z + (3.9 − H_melt), where H_melt is the environmental melting level (km) obtained either from the storm proximity sounding in the simulation (Fig. 2) or from an observed storm inflow sounding (Z13). In other words, z = z* in the special case where the DLA is applied to the simulated Binger storm and H_melt = 3.9 km (e.g., section 3), while z ≠ z* for the more general case where H_melt ≠ 3.9 km (e.g., radar-observed storm in Z13). Since ongoing modeled precipitation processes vary strongly with Z_H and height relative to H_melt, scaling the 3D model output data in height and regressing against Z_H accounts in a bulk sense for storm-scale vertical gradients of precipitation quantities that are forced by sedimentation, melting, evaporation, and other microphysical processes (e.g., Ziegler 1988). The vertical profiles of the coefficients Z_0r(z*), Z_0g(z*), , , and are spline fitted from the height-dependent regressions and output to look up tables at a 100-m vertical resolution.

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Skew T–logp plot of the storm proximity sounding used to define the inflow environment of the simulated 22 May 1981 Binger storm (Z10) in the OSSE tests of the DLA. The inflow sounding was obtained by interpolating to a column located about 20 km upstream of the supercell updraft with respect to the surface storm-relative wind at (x, y) = (92, 45) km in the model domain and 4 h 40 min into the simulation. Altitudes H_melt and H_frz are described in the text and listed in Table 2. The 0–3-km storm-relative helicity (SRH0–3) is computed relative to the mean simulated storm motion (m s⁻¹) of (u, υ) = (9, 2).

Citation: Journal of Atmospheric and Oceanic Technology 30, 10; 10.1175/JTECH-D-12-00194.1

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Some observational evidence exists to partially support the model-based regression Eq. (9) that parameterizes the variation of q_r versus Z_H (e.g., Schuur et al. 2001, hereafter S01). Equation (9) is broadly consistent with measured q_r–Z_H pairs from the National Severe Storms Laboratory (NSSL) two-dimensional video distrometer (2DVD) in many storms (S01), in that the q_r–Z_H data points are concentrated along the regression curve for Z_H up to about 45 dBZ (Fig. 3a). However, the q_r–Z_H points are concentrated above the regression curve for 45 dBZ < Z_H ≤ 50 dBZ and have a rapidly diminishing frequency above 50 dBZ (Fig. 3a). The time trajectory of q_r–Z_H points in a supercell storm (i.e., as adapted from S01) is characterized by an early period with points below the regression curve, a later period with points arrayed along the regression curve, and a mature late stage of the rain event with points arrayed above the regression curve (Fig. 3b). An implied correlation between the occurrence of a q_r–Z_H point located above the regression curve and high observed wind speed (Figs. 3a,b) is consistent with the interpretation of S01 that a wind-related drop undercounting might at least partially explain the lower 2DVD-measured bulk rainfall in comparison to a proximate catchment gauge (T. Schuur 2012, personal communication). Adjusting the q_r–Z_H regression curve to fit the observed data by increasing ɛ_r in Eq. (9) by 66% has limited utility for Z_H > 50 dBZ (Figs. 3a,b). A wind-induced bias (if any) and increasing scatter would associate with a shifting of q_r–Z_H points somewhat toward lower Z_H and q_r values [i.e., to the left of Eq. (9) for Z_H > 45 dBZ].

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Regression curve fits of NSSL 2DVD and storm simulation model output data described in text. (a) The 1-min interval 2DVD rain mixing ratio vs Z_H (surface 2DVD) stratified according to wind speed for all sampled events (adapted from S01); (b) as in (a), but for the supercell event only (adapted from S01); (c) rain mixing ratio vs Z_H stratified according to collocated graupel mixing ratio (surface); (d) graupel mixing ratio vs Z_H (5 km); (e) graupel intercept parameter vs Z_H (5 km); and (f) spline-fitted height profile of regression coefficient Z_0g(z*). Individual 1-min 2DVD samples in (a) and (b) are plotted as green crosses for proximate Oklahoma Mesonet surface wind speed V < 5 m s⁻¹ and as red circles for V ≥ 5 m s⁻¹. Individual gridpoint q_r values in (c) are plotted as green crosses for q_g < 0.01 g kg⁻¹, as red circles for 0.01 ≤ q_g < 0.05 g kg⁻¹, and as blue circles for q_g ≥ 0.05 g kg⁻¹ (max surface q_g = 0.42 g kg⁻¹). The long-dashed black curve denotes the fitted rain curve based on model output, while the short-dashed black curve is the equivalent fit of the 2DVD output assuming the same Z_0r and S_0r coefficient values as obtained for the model fit. The fitted coefficient Z_0g = 44.19 in (d) is the spline function value of Z_0g(z*) at z* = 5 km in (f).

Citation: Journal of Atmospheric and Oceanic Technology 30, 10; 10.1175/JTECH-D-12-00194.1

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In contrast to the above 2DVD measurements, modeled q_r–Z_H data points are arrayed along the regression curve through the maximum Z_H values exceeding 60 dBZ in the simulated storm (Fig. 3c). The modeled q_r–Z_H points are stratified according to the magnitude of collocated q_g values, revealing that rain–graupel mixtures with increasing q_g are typified by increasing q_r values (Fig. 3c). The cumulative melting of the deep column of falling graupel particles and the initially large q_g values above z = H_melt necessarily result in large surface meltwater q_r values in high reflectivity. It is also noted that the scatter of the modeled q_r–Z_H points is due to varying mixtures of rain and graupel for given values of Z_H. Although there is more variance in the observed q_r–Z_H points than in the modeled q_r–Z_H points, it is difficult to assess how much of the difference is a result of effects of horizontal wind as opposed to microphysics. Because of the approximation of precipitation particle size distributions (PSDs) with inverse exponential functions and also the likely time–space variability of the fitted intercept parameters in observed storms, it is expected that much of the difference between the scatters of q_r–Z_H points between the 2DVD observations and the model are because of the limited range of assumed model PSDs.

The standard deviations of q_r and q_g about the regression Eqs. (9) and (11) each range from about 0.1 g kg⁻¹ at the surface (e.g., Fig. 3c) up to about 0.5 g kg⁻¹ at 5 km (e.g., Fig. 3d), while varying roughly linearly from the surface to 5 km. The standard deviation of n_0g about the regression Eq. (10) is about 0.3 × 10⁵ m⁻⁴ (i.e., ~10% of typical n_0g values) at 5 km (Fig. 3e) and increases slightly with decreasing altitude. The spline curve fit Z_0g(z*) profile (Fig. 3f) reflects the predominance of graupel melting below 3 km. Since the model-simulated ρ_g varied approximately linearly with height and only weakly with Z_H, ρ_g is prescribed by a linear variation between input values at the surface and 5 km (i.e., and in Table 2).

e. Estimation of precipitation parameters and moments

The values of q_r and q_g at a Lagrangian point are derived from their provisional values via Eqs. (9) and (11) with consideration of the vertical velocity field, the height of the −15°C level in the main updraft core (H_frz) and H_melt, n_0g as derived from Eq. (10), and the assumed n_0r value described in section 2d. Inspection of the storm simulation model output implies that rain recycling increases and q_g decreases as w increases from W_min to W_max = 20 m s⁻¹. The model output also reflects the increasing parameterized rate of heterogeneous nucleation of undercooled rain in the updraft to form graupel (e.g., G04; SM05) as z* increases from above H_melt to H_frz.

The estimated q_g at any point is determined from its provisional value via one of the following conditional expressions:

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The vertical velocity scale w* (m s⁻¹) = (w − W_min)/(W_max − W_min) in Eqs. (14) and (15), while the heterogeneous rain freezing coefficient C_frz = exp[−α_frz(z* − H_melt)/(H_frz − H_melt)] in Eqs. (15) and (16). To maintain consistency with q_g, the corresponding λ_g value is derived by either the expression

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where z* ≥ H_melt and q_g = , or via Eq. (6) if either z* ≤ H_melt or z* ≥ H_melt and q_g < . It is noted that the condition q_g = in Eq. (17) is equivalent to the condition Z_eg = Z_eh. Total concentration N_g is computed from Eq. (5).

The q_r value at any point is determined from its provisional value and the collocated derived q_g value by partitioning the measured total equivalent reflectivity Z_eh according to the quantity Z_er + Z_eg. For the case of w < W_min, q_r = is obtained from Eq. (9) while the rain slope parameter λ_r is derived from Eq. (6). For w ≥ W_min (in which case q_r ≤ and Z_er = Z_eh − Z_eg), λ_r is derived from Eq. (7) using the computed Z_er while q_r is computed from Eq. (6). Total concentration N_r is computed from Eq. (5).

The above algorithm, that partitions the observed Z_H into rain and graupel components via Eqs. (9)–(18), produces close agreement between the empirically derived and modeled fields of q_r and q_g (Fig. 4). The diagnosed q_r and q_g values differ slightly from their model output counterparts because of the mixed-phase nature of the modeled precipitation (i.e., rain–graupel) and scatter about the regression curve with Z_H (e.g., Figs. 3c,d). Peak values of the empirically derived q_r (Fig. 4a) and q_g (Fig. 4b) are up to about 0.5 g kg⁻¹ less than the modeled q_r and q_g values (Figs. 4c,d). The main supercell updraft recycles meltwater q_r from the hook echo on its southwest flank up into the subfreezing portion of the storm core, producing an elevated rain core above the bounded weak echo region (BWER) that contains little or no graupel at the relatively small undercoolings below 5 km in the updraft. Polarimetric radars routinely observe elevated volumes containing large values of differential reflectivity (Z_DR), known as Z_DR columns, which are caused by large unfrozen rain drops in the updraft region (e.g., Loney et al. 2002; Kumjian and Ryzhkov 2008; Schwarz and Burgess 2010). The DLA results imply that the Lagrangian analysis should be capable of approximating precipitation structures associated with polarimetric radar features such as Z_DR columns.

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Domain-relative vector airflow and precipitation mixing ratio fields from the (top) DLA and the (bottom) storm simulation model output data in a vertical southwest–northeast (SW–NE) cross section through the simulated updraft and storm core (see also cross section in Fig. 11 of Ziegler et al. 2010). (a) Lagrangian rain mixing ratio (g kg⁻¹); (b) Lagrangian graupel mixing ratio (g kg⁻¹); (c) simulated rain mixing ratio (g kg⁻¹); and (d) simulated graupel mixing ratio (g kg⁻¹). The black contours depict simulated Z_H (dBZ) that is used to determine precipitation-mixing ratios (as described in the text). Selected Z_H contours on the left side of each panel are omitted for clarity. The horizontal dashed red line is H_melt from the input sounding in Fig. 2, while the outer and inner dashed gray curves locate the updraft threshold values W_min = 5 m s⁻¹ and W_max = 20 m s⁻¹ (as listed in Table 1 and described in the text). The vertical dotted black line locates the intersection with the northwest–southeast (NW–SE) cross section shown in Fig. 5.

Citation: Journal of Atmospheric and Oceanic Technology 30, 10; 10.1175/JTECH-D-12-00194.1

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f. Optional scaling of graupel concentration with changes in storm intensity

The graupel concentration field may evolve in response to changing storm intensity as quantified by peak updraft strength and reflectivity in an analyzed storm. For example, decreasing updraft strength may lead to increasing concentrations of small graupel particles and decreasing reflectivity resulting from a combination of size sorting and precipitation fallout with reduced rime density from decreasing supercooled cloud water content. As another example, the formation of the first echo at midlevels of a newly initiated storm (e.g., Ziegler et al. 1991) may associate with transient high concentrations of small graupel particles whose size distribution is characterized by larger n_0g values than computed with Eq. (10) and illustrated in Fig. 3e. The regression Eqs. (9)–(11) do not presently incorporate evolution since they were obtained from the mature model storm state near the time of overall peak storm intensity. Modifying the regression relations to account for evolving storm intensity via an expanded assumed functional dependence on global variables such as peak reflectivity or updraft strength is a question of interest for future research.

Scaling the graupel concentration provides a simple means to adjust for potential changes of observed storm intensity. If the unscaled graupel concentration is , then the scaled concentration is computed as

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where α_N = α_N(t) is an input graupel concentration scaling parameter (Table 2). The other graupel distribution parameters are then obtained under the constraint of constant graupel reflectivity via the following steps:

1. compute graupel reflectivity

   

   

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2. compute adjusted

   

   

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3. compute adjusted

   

   

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4. compute λ_g from Eq. (6).

The analyst could also scale q_g to account for changing storm intensity, although the latter option is beyond the scope of the present study and has not yet been implemented. If no information on evolving storm intensity is available, the analysis should assume the default value of α_N = 1.

g. Microphysical parameterization

The Lagrangian saturation adjustment applies ideas from the Eulerian frame modeling approach of Soong and Ogura (1973). The conservation of θ and q_υ depends on an air parcel's saturation ratio S = q_υ/q_υs, where q_υs is the saturation vapor mixing ratio with respect to water. If S* is the provisional value of S following displacement during a Lagrangian time step and if S* < 1, then S = S* and the parcel values of θ and q_υ are conserved following the 3D motion. On the other hand, if S* ≥ 1 the parcel is assumed to follow a water-saturated pseudoadiabatic process that may be defined by its equivalent potential temperature θ_e as characterized by its saturation point temperature (Betts and Dugan 1973; Bolton 1980). If vertical velocity w is positive, excess water vapor generated in the updraft requires condensation and increasing parcel q_c to maintain exact water saturation (S = 1). Conversely, if w < 0 and q_c > 0, the cloud water is evaporated into the parcel to maintain S = 1 until q_c = 0 (after which the downward motion is adiabatic in the absence of rain evaporation). To increase accuracy where q_c > 0, the net condensation or evaporation on the large time step is split into a series of small-step calculations with Δt_small = 4 s.

The parameterizations for rain collection and graupel accretion of cloud, rain evaporation and freezing, graupel sublimation, and graupel melting follow the modified LFO formulation used by G04. These parameterized local derivative expressions are applied as substantial derivatives held constant during a Lagrangian time step.

h. Lagrangian damping parameterization

Since the predicted scalar variables are Lagrangian (and thus gridded scalar fields are unavailable to calculate their spatial derivatives during integration), it is not presently feasible to compute the turbulent mixing terms by applying either a conventional turbulent kinetic energy (TKE)-based closure (e.g., Klemp and Wilhelmson 1978; Deardorff 1980) or a simpler local closure based on airflow deformation (e.g., Smagorinsky 1963). Instead, bulk mixing effects are replaced by a highly simplified bulk “Lagrangian damping” term for the present Lagrangian analysis. Adapting the one-dimensional (1D) mixing parameterization of Danielsen et al. (1972) to an equivalent 3D form [i.e., their Eqs. (1), (A3), and (A4)], the scalar damping term D_φ is expressed as

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where c_d is a constant dimensionless coefficient, V and L_d are velocity and length scales, respectively (defined below), φ_B = φ_B(x, y, z) is the base-state scalar value at the location of a Lagrangian point, and the exponential weighting term allows the damping to be optionally scaled by either unity (b = 0) or as a decreasing function of height (b > 0). The damping length and velocity scale parameterizations are paired via one of the following three forms depending on the Lagrangian vertical velocity value; that is,

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where |w| is the vertical velocity magnitude, and the damping length scale is prescribed for the cases of quasi-horizontal flow (i.e., |w| ≤ W₀), updraft (+), and downdraft (−). The coefficient C_d0 is determined by the expression

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where the precipitation mixing ratio q_p = q_r + q_g, , and the dependence on q_p is broadly consistent with the parameterized relationship between surface downdraft and Z_H described in section 2b. The velocity scale is applied for quasi-horizontal flow, where u_B and υ_B are the base-state horizontal wind components.

The experimentally derived Lagrangian damping coefficients (Table 1) produce damping rates via Eq. (22) that are broadly comparable to representative extreme values of the directly computed scalar mixing terms in the simulated storm, while also reducing RMS errors in the OSSE test described in section 3. The vertical scaling term in Eq. (22) offsets a strong ubiquitous tendency for undamped trajectories approaching the surface to become excessively cool and anomalously stratified as the vertical velocity and adiabatic warming tendency both decrease to zero in the presence of maintained diabatic cooling from melting or evaporating precipitation. The resultant damping effects assuming 0 < b < 1 are consistent with both the rather undiluted nature of the cores of high-speed storm updrafts at least through storm midlevels (Davies-Jones 1974) and the considerable dilution of negative buoyancy via lateral entrainment in precipitation-cooled low-level downdrafts (Gilmore and Wicker 1998). The damping term D_φ is set to zero if q_p < q₀ for surface grid points or q_p < q₁ for elevated grid points (q₀ and q₁ listed in Table 1).

i. Surface flux parameterization

In the general case of a quasi-steady mesoscale storm environment that is characterized by a horizontal wind V_h and a horizontal scalar gradient ∇ϕ (e.g., Z13), the local Lagrangian tendency of the surface ϕ field may be approximated by equating the surface flux to the horizontal advection . The bulk boundary layer (BL) flux term thus takes the general form

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where Eq. (27) is evaluated for z ≤ z_BL and q_hydro = q_c + q_r + q_g ≤ q₁ only and F_ϕ = 0 for z > z_BL. The values of the BL height z_BL and the vertical mixing coefficient b_F employed by Z13 are listed in Table 1. The simple surface flux parameterization allows the analyst to crudely control the effect of varying bulk vertical mixing strength by modulating the chosen b_F value. It is assumed that surface heating vanishes in local cloud or precipitation.

a. OSSE results

The spatial cold pool and main updraft structures of the control (CNTL; for a complete list of test category acronyms see Table 3) analysis and the simulated storm are coherent and have very similar amplitudes (Fig. 5). Given virtual potential temperature θ_υ and its base-state value θ_υ0(z), the local cold-pool magnitude or buoyancy may be defined by Δθ_υ = θ_υ − θ_υ0(z). The intensity and depth of the forward-flank and rear-flank outflow boundaries and the buoyancy profile in the cold pool and main updraft (Figs. 5a,b versus Figs. 5e,f) are all in good agreement between the modeled and DLA output fields. A downward warm intrusion into the midlevel cold pool in the precipitation-filled downdraft centered at a horizontal distance of 10.5 km in the cross section and z = 2 km (Figs. 5b,f) reflects the offsetting tendencies of precipitation melting and evaporation versus adiabatic warming. The levels of zero virtual buoyancy and the cloud-base heights in the updraft core (Fig. 5b versus Fig. 5f and Fig. 5d versus Fig. 5h, respectively) are internally consistent with the ambient LCL and LFC (Fig. 2). An examination of the time series Lagrangian output data following in-cloud trajectories indicates that the cloud base on the downdraft portion is typically higher than the cloud base on the updraft portion if either rain collection or graupel accretion of cloud is nonzero along any portion of the trajectory. The DLA-derived water vapor mixing ratio values are somewhat larger in the outflow below z = 0.5 km at a horizontal location of 10.5 km than in the model (Fig. 5c versus Fig. 5g). The DLA-derived cloud mixing ratio values are up to about 1 g kg⁻¹ less than the model values in the updraft core at 5 km (Fig. 5d versus Fig. 5h).

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Comparison of output fields from the DLA and the simulated storm in the CNTL OSSE test. Fields of (a) virtual potential temperature deviation from the environment Δθ_υ′ (K), (b) vapor mixing ratio q_υ (g kg⁻¹), and (c) cloud water mixing ratio q_c (g kg⁻¹) from the DLA. (d)–(f) The storm simulation model output in a vertical NW–SE cross section through the simulated updraft and storm core. Vectors are domain-relative airflow in the plane. The black contours depict in (a)–(c) w (m s⁻¹) and (d)–(f) Z_H (dBZ). The black dashed curves and OF denote the locations of surface-based outflow boundaries in the simulated storm, while the black dot labeled S in (a) locates the inflow sounding shown in Fig. 2. The horizontal dashed red line is H_melt. The vertical dotted black line locates the intersection with the SW–NE cross section shown in Fig. 4.

Citation: Journal of Atmospheric and Oceanic Technology 30, 10; 10.1175/JTECH-D-12-00194.1

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The DLA-derived and simulated fields in the OSSE (CNTL) test display high linear correlations between modeled and analyzed variables (Fig. 6). The correlation coefficient R values are high (0.98, 0.97, and 0.92 in Figs. 6a–c, respectively), the root-mean-square error (RMSE) values are rather small (0.83 K, 0.56 g kg⁻¹, and 0.33 g kg⁻¹ in Figs. 6a–c, respectively), and rather small scatter and bias are indicated by the associated box-and-whiskers plots. The DLA produces slightly high-biased q_υ values in the range of 8 < q_υ < 10 g kg⁻¹ (Fig. 6b) owing to contributions of some grid points in the spreading of low-level precipitation downdraft (e.g., Fig. 5c versus Fig. 5g). Since q_c generally increases with height from vertical water vapor transport and condensation, the DLA produces some slightly low-biased q_c values exceeding 4 g kg⁻¹ in the narrow elevated main updraft core (see also Fig. 5d versus Fig. 5h). It is hypothesized that the slight positive q_υ bias is caused by approximating the heterogeneous storm environment by assuming the single inflow sounding (producing occasional downdraft trajectories that are initially too moist; also compare Fig. 5c versus Fig. 5g), while the negative q_c bias aloft is hypothesized to be caused by horizontal postanalysis spatial filtering. It is also hypothesized that biases caused by the differing discrete time integral forms of the Eulerian and Lagrangian transport terms may be mitigated by increasing the spatial and temporal resolution of the model and DLA.

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Scatterplot comparisons between the gridpoint outputs of the DLA and the simulated storm in the CNTL OSSE test. (a) θ_υ, (b) q_υ, and (c) q_c. The term R is the correlation coefficient for the indicated linear regression (black line).

Citation: Journal of Atmospheric and Oceanic Technology 30, 10; 10.1175/JTECH-D-12-00194.1

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Additional analysis tests (Table 3) demonstrate that increased error levels result from various suboptimal choices of included processes or treatment of the lower kinematic boundary condition in the DLA (Fig. 7). The θ_υ, q_υ, and q_c RMS error values of the CNTL analysis are small in relation to the maximum values in the updraft and minimum values in the simulated storm (Figs. 7a–c). In particular, the θ_υ error in the lowest 2 km (dominated by the storm's cold pool) is as small as 0.6 K in comparison to a maximum cold-pool deficit of Δθ_υ of less than −7 K. Analyses GMLT and RVAP have larger RMSE than CNTL in the lowest 2 km (Fig. 7d), owing to substantially weakened cold pools arising from reduced total diabatic cooling along downdraft trajectories. Conversely, analyses NGSC, NOLD, and NOCOL have larger RMSE than CNTL in the lowest 2 km (Fig. 7d), owing to anomalously stronger cold pools. Analysis WSFC displays the largest sensitivity of water vapor mixing ratio RMSE in the lowest 1 km (Fig. 7e), owing to the impact of surface vertical velocity on low-level trajectories and the spurious wet-bulb effect previously discussed in section 2b. The cloud mixing ratio RMSE is significantly elevated only in analysis NOCOL (Fig. 7f), due to the well-known predominance of cloud collection by precipitation (e.g., Ziegler 1988).

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Statistical comparisons between the gridpoint outputs of the DLA and the simulated storm in the CNTL and sensitivity OSSE tests. (left) Vertical profiles from the min and max model output values and the RMSE between model and CNTL (red curve) for (a) θ_υ, (b) q_υ, and (c) q_c. (right) Vertical profiles of the RMSE between the model and the various Lagrangian analysis sensitivity tests for (d) θ_υ, (e) q_υ, and (f) q_c. To increase the potential RMSE and deemphasize the storm's environment (which is rather poorly radar observed in actual storm cases), the volume in which the RMS deviations are computed is the union of the volumes of w ≥ 5 m s⁻¹ and Z_H ≥ 20 dBZ (i.e., the storm). The profiles of max and min values in (a)–(c) are derived in the volumes of w ≥ 5 m s⁻¹ (i.e., the updraft) and the storm, respectively.

Citation: Journal of Atmospheric and Oceanic Technology 30, 10; 10.1175/JTECH-D-12-00194.1

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b. Solenoidal forcing of horizontal vorticity via thermal buoyancy gradients

Horizontal vorticity that is solenoidally generated within the forward-flank baroclinic zone may subsequently be tilted and stretched in the main updraft, thus potentially contributing an important source of vertical vorticity to the developing low-level mesocyclone (e.g., Klemp 1987). Although the total solenoidal generation receives contributions from horizontal gradients of the fields of predicted θ_υ and diagnosed q_p, the current discussion concentrates on the thermal solenoid term to explore the possible impact of accumulated prediction errors on DLA-analyzed vorticity dynamics. The discussion in section 3a has documented the expected error of DLA-derived θ_υ values. However, inasmuch as the error of the difference of two random variables (RVs) increases with the sum of the errors of the individual RVs (Neter and Wasserman 1974), it is also important to check that the dynamically significant horizontal θ_υ gradients are consistent between the DLA and model.

The magnitude of the horizontal thermal–solenoidal vorticity tendency in the cold pool has been calculated from the DLA and model outputs via the expression (e.g., Buban et al. 2007)

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where g is the gravitational acceleration (m s⁻²). Since the low-level mesocyclone straddles the forward-flank baroclinic zone in the simulated storm, the solenoid is evaluated via Eq. (28) at any grid point in the lowest 1 km at which the modeled ∇_hθ_υ value is less than 0.2 K km⁻¹ and Z_H > 10 dBZ to increase the focus on the subset of vorticity tendency vectors that are oriented with a component toward the low-level mesocyclone (Fig. 8). The solenoidal component at a horizontal location of 14.5 km in the cross section and z = 0.5 km from the DLA output (Fig. 5b) is about (9.8/306)(1.5/500) ~ 10 × 10⁻⁵ s⁻², which is reasonably consistent with the modeled solenoidal term at the same location (Fig. 5f). The magnitude of the DLA-derived values is broadly comparable to modeled values for ≤ 10 × 10⁻⁵ s⁻² (Fig. 8). The more densely clustered data points near the origin appear to be somewhat better represented using a constrained linear regression through the origin (RTO; e.g., Eisenhauer 2003) than via ordinary regression (OLR; Fig. 8). At least a portion of the increasing negative bias of both the RTO- and OLR-obtained regression lines with increasing is consistent with an overall reduction of DLA-derived solenoidal values owing to a reduction of horizontal θ_υ gradients via smoothing from the horizontal low-pass filter. The more frequent weak-to-moderate values would be encountered more often than the rare rather large outlier values along individual air trajectories, hence the former would be expected to dominate the evolution of the time-integrated Lagrangian horizontal vorticity.

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Scatterplot comparison of the magnitude of solenoidal horizontal vorticity generation (dω_h/dt) between the gridpoint outputs of the DLA and the simulated storm in the CNTL OSSE test. The solid black line denotes the ordinary least squares (OLS) linear regression (with indicated fitting statistics in the upper left), while the dotted line is the fit obtained via constrained linear RTO. The RTO provides a slightly better fit than OLS for dω_h/dt < 10 s⁻² but also has larger RMSE (not shown).

Citation: Journal of Atmospheric and Oceanic Technology 30, 10; 10.1175/JTECH-D-12-00194.1

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