## 1. Introduction

The investigation of convective storm dynamics with multiple-Doppler radar and in situ storm observations requires methods to infer unmeasured fields related to the dynamically evolving internal storm processes (Hane et al. 1988). Two conventional methods for deriving the buoyancy field from observed wind fields in storms are dynamic retrieval (e.g., Gal-Chen 1978; Hane et al. 1981; Brandes 1984; Roux 1985) and microphysical continuity retrieval (e.g., Rutledge and Hobbs 1984; Ziegler 1985, 1988; Marecal et al. 1993). The present study develops a new buoyancy retrieval method, termed diabatic Lagrangian analysis (DLA), which avoids several limitations of conventional buoyancy retrievals (described below) while retaining both an accurate advection principle and explicit diabatic forcing proceeding from radar-inferred precipitation and microphysical processes.

The newly developed DLA method extends the Lagrangian analysis technique of Ziegler et al. (2007, hereafter Z07) by including local conservation of potential temperature, water vapor, and cloud water associated with microphysical processes that act in a time-dependent parcel-following Lagrangian reference frame. Whereas microphysical retrieval is predicated on the solution of a system of 3D Eulerian frame parabolic partial differential continuity equations for heat and water substance requiring prescribed 3D initial and boundary condition fields, DLA is instead based on an equivalent system of first-order ordinary differential equations with a Lagrangian initial condition prescribed at each parcel's initiation point either in a storm's environment or at the location of an in situ observation. The four-dimensional (4D) vector airflow field is only used to define air trajectories, thus avoiding high-order spatial differencing that under certain circumstances may lead to the amplification of wind analysis errors in dynamic buoyancy retrievals (e.g., Hane et al. 1988; Majcen et al. 2008). Another advantage of DLA is the ability to diagnose variables at boundaries (e.g., the surface) via advection and source/sink terms, in contrast to 3D dynamic buoyancy retrievals that typically impose Neumann conditions in bounding planes and thus do not obtain a solution at boundary points. An added general benefit of DLA is that a parcel's saturation point may optionally be initialized from a closely neighboring in situ observation given sufficient sampling density (Z07). Alternatively, selected in situ observations may optionally be used to validate individual trajectories.

In further contrast to microphysical retrieval, the DLA method approximates the precipitation size distributions from radar-measured quantities such as reflectivity (as detailed in sections 2c–e). The ongoing polarimetric upgrade of the Weather Surveillance Radar-1988 Doppler (WSR-88D) network (Crum et al. 1998; Saffle et al. 2006) promises to extend inferred information about precipitation beyond habit and phase (e.g., Straka et al. 2000; Zrnic et al. 2001) to include the provision of rain, graupel, and hail particle contents (e.g., Zhang et al. 2001; Brandes et al. 2006) and rainfall rate (Vasiloff 2012). It is hoped that future polarimetric radar studies will overcome the constraints of “multidimensionality that obfuscates … progress” (Zrnic et al. 2001, p. 907) to derive robust functions that relate *Z _{H}*,

*Z*

_{DR}, and other radar moments to the individual total concentrations and mixing ratios comprising various mixtures of rain and precipitating ice. Since storm-scale time-dependent polarimetric datasets covering an entire storm volume are not presently available in all cases and since mixed-phase precipitation particle distributions are difficult to deconvolve using present methods, a relatively simple approach is employed in the present study to partition the precipitation into rain and graupel components based on reflectivity only. The development of a procedure for incorporating polarimetric data from WSR-88D and mobile ground-based radars into the DLA is an important objective of future research.

This paper develops the DLA algorithm and demonstrates its performance via an observing system simulation experiment (OSSE). The known wind, reflectivity, and buoyancy fields for the OSSE are provided by a simulation of the 22 May 1981 Binger, Oklahoma, supercell storm (Ziegler et al. 2010, hereafter Z10). The 3D dynamical Binger storm simulation was conducted with the Straka Atmospheric Model, which includes one-moment parameterizations of warm- and cold-cloud microphysics (Straka and Mansell 2005, hereafter SM05; Z10). Other results obtained by applying the DLA via a detailed multiple-Doppler radar analysis of the 9 June 2009 Greensburg, Kansas, supercell storm [observed during the second Verification of the Origins of Rotation in Tornadoes Experiment (VORTEX2)] are discussed in Part II of this paper (Ziegler 2013, hereafter Z13).

## 2. Diabatic Lagrangian analysis

The DLA method predicts the 3D fields of potential temperature *θ*, water vapor mixing ratio *q*_{υ}, and cloud water mixing ratio *q _{c}* at a chosen analysis time by integrating a set of time-dependent ordinary differential equations (ODEs) for the predictive variables along air trajectories that terminate at grid points comprising the analysis domain (i.e., one trajectory per grid point). As described in more detail in section 2a, the ODEs combine Lagrangian transport with rate terms describing the evolutions of

*θ*,

*q*

_{υ}, and

*q*via parameterizations of selected warm- and cold-cloud microphysical processes and a simplified representation of mixing effects. The method for calculating trajectories from time-dependent 3D fields of the west–east (

_{c}*u*), south–north (

*υ*), and vertical (

*w*) wind components is described in section 2b.

To calculate microphysical processes at the individual Lagrangian points, the DLA combines the predicted Lagrangian values of *θ*, *q*_{υ}, and *q _{c}* with diagnosed values of rain and graupel total concentration and mixing ratio (i.e.,

*N*,

_{r}*N*,

_{g}*q*, and

_{r}*q*, respectively) that are approximated from radar reflectivity

_{g}*Z*(dB

_{H}*Z*) as detailed in sections 2c–f. Small high-density hail is implicitly included within the graupel category, while the fractional contribution of large hail to the total graupel–hail mass and number concentration is expected to be relatively small and thus negligible for present purposes. Although the

*N*,

_{r}*N*,

_{g}*q*, and

_{r}*q*values interact within the aforementioned microphysical terms, they do not affect each other since they are diagnosed rather than predicted. The parameterizations of microphysics, Lagrangian damping (which approximates mixing effects), and surface fluxes are described in sections 2g–i, respectively. A time series of 3D gridded analysis fields of the

_{g}*u*,

*υ*, and

*w*wind components (m s

^{−1}) and reflectivity

*Z*(dB

_{H}*Z*) provides Lagrangian data via temporal–spatial interpolation to calculate trajectories and diagnose precipitation variables.

### 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 dB

*Z*; (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.

*θ*,

*q*

_{υ}, and

*q*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)where

_{c}*φ*= (

*θ*,

*q*

_{υ},

*q*);

_{c}*ϕ*= (

*θ*,

*q*

_{υ}); and

*M*,

_{φ}*D*, and

_{φ}*F*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

_{ϕ}*p*(

*x*,

*y*,

*z*) =

*p*(

_{B}*z*) +

*p*′(

*x*,

*y*,

*z*) ≅

*p*(

_{B}*z*), where

*p*and

_{B}*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*= 1.0 × 10

_{a}^{5}[(

*p*/1000.0)

_{B}^{0.2854}]

^{2.509}/(287.04

*θ*). The Lagrangian values of

*θ*,

*q*

_{υ}, and

*p*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

_{B}*q*is initialized to zero. The local Lagrangian raindrop and graupel size distributions at each time step are computed from spatially and temporally interpolated

_{c}*Z*analysis values (as detailed in sections 2c–f), while

_{H}*p*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 Δ

_{B}*t*= 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 Δ*t* 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 *H*_{0} < 50 m AGL (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.

*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 expressionif

*w*

_{k}_{=2}< 0; otherwise,

*w*

_{sfc}= 0 if

*w*

_{k}_{=2}≥ 0. In Eq. (2), the reflectivity scale

*Z** is expressed aswhere 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*in broad accord with Goff's tall tower outflow analysis (Fig. 1).

_{H}### c. Bulk precipitation size distribution parameters

*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)where

*n*

_{0x}is the intercept parameter and

*x*=

*r*or

*g*),

*ρ*is air density (kg m

_{a}^{−3}), while

*ρ*is the density of either rainwater

_{x}*ρ*or graupel

_{w}*ρ*. Rain parameters are related to the equivalent rain reflectivity

_{g}*Z*

_{er}(mm

^{6}m

^{−3}) by the expressionwhere Γ is the complete gamma function. Following Ferrier (1994) and substituting the definitions of the distribution moments, the equivalent graupel reflectivity

*Z*

_{eg}(mm

^{6}m

^{−3}) is related to the graupel distribution parameters by the expressionThe values of the coefficients

*C*,

_{r}*α*, and

_{r}*C*are listed in Table 2.

_{g}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.)

### 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*, and

_{g}*ρ*), 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

_{g}*Z*field may be available through the entire storm volume (e.g., Z13), appropriate closure functions must be established between

_{H}*Z*and the various precipitation parameters. A key assumption that the equivalent radar reflectivity

_{H}^{6}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*and

_{x}*q*is theoretically rather more accurate than diagnosing

_{x}*N*from

_{x}*q*via an assumed constant

_{x}*n*

_{0x}because some microphysical processes change only

*N*or

_{x}*q*, while others change both

_{x}*N*and

_{x}*q*(Ziegler 1984). However, a reasonably accurate diagnosed value of

_{x}*N*may be obtained by altering the

_{x}*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*have been derived from model output data at the mature stage of a simulated supercell storm (Z10). Choosing the optimal value

_{g}*n*

_{0r}= 8 × 10

^{5}m

^{−4}(Z10) usefully reduces the number of degrees of freedom of the radar microphysics diagnosis. The input regression values of

*Z*are calculated directly from the model output rain and graupel contents via Eqs. (7) and (8) and the total equivalent reflectivity

_{H}*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*~ 5 m s

_{t}^{−1}(i.e., where

*w*<

*W*

_{min}= 5 m s

^{−1}and

*w*/

*V*< 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).

_{t}*Z*and

_{H}*q*,

_{r}*n*

_{0g}, and

*q*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

_{g}*n*

_{0r}is listed in Table 2. The empirical relationships for

*n*

_{0g}, and

*Z*take the following forms:where

_{H}*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*and height relative to

_{H}*H*

_{melt}, scaling the 3D model output data in height and regressing against

*Z*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

_{H}*Z*

_{0r}(

*z**),

*Z*

_{0g}(

*z**),

Some observational evidence exists to partially support the model-based regression Eq. (9) that parameterizes the variation of *q _{r}* versus

*Z*(e.g., Schuur et al. 2001, hereafter S01). Equation (9) is broadly consistent with measured

_{H}*q*–

_{r}*Z*pairs from the National Severe Storms Laboratory (NSSL) two-dimensional video distrometer (2DVD) in many storms (S01), in that the

_{H}*q*–

_{r}*Z*data points are concentrated along the regression curve for

_{H}*Z*up to about 45 dB

_{H}*Z*(Fig. 3a). However, the

*q*–

_{r}*Z*points are concentrated above the regression curve for 45 dB

_{H}*Z*<

*Z*≤ 50 dB

_{H}*Z*and have a rapidly diminishing frequency above 50 dB

*Z*(Fig. 3a). The time trajectory of

*q*–

_{r}*Z*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

_{H}*q*–

_{r}*Z*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

_{H}*q*–

_{r}*Z*regression curve to fit the observed data by increasing

_{H}*ɛ*in Eq. (9) by 66% has limited utility for

_{r}*Z*> 50 dB

_{H}*Z*(Figs. 3a,b). A wind-induced bias (if any) and increasing scatter would associate with a shifting of

*q*–

_{r}*Z*points somewhat toward lower

_{H}*Z*and

_{H}*q*values [i.e., to the left of Eq. (9) for

_{r}*Z*> 45 dB

_{H}*Z*].

In contrast to the above 2DVD measurements, modeled *q _{r}*–

*Z*data points are arrayed along the regression curve through the maximum

_{H}*Z*values exceeding 60 dB

_{H}*Z*in the simulated storm (Fig. 3c). The modeled

*q*–

_{r}*Z*points are stratified according to the magnitude of collocated

_{H}*q*values, revealing that rain–graupel mixtures with increasing

_{g}*q*are typified by increasing

_{g}*q*values (Fig. 3c). The cumulative melting of the deep column of falling graupel particles and the initially large

_{r}*q*values above

_{g}*z*=

*H*

_{melt}necessarily result in large surface meltwater

*q*values in high reflectivity. It is also noted that the scatter of the modeled

_{r}*q*–

_{r}*Z*points is due to varying mixtures of rain and graupel for given values of

_{H}*Z*. Although there is more variance in the observed

_{H}*q*–

_{r}*Z*points than in the modeled

_{H}*q*–

_{r}*Z*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

_{H}*q*–

_{r}*Z*points between the 2DVD observations and the model are because of the limited range of assumed model PSDs.

_{H}The standard deviations of *q _{r}* and

*q*about the regression Eqs. (9) and (11) each range from about 0.1 g kg

_{g}^{−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 × 10

^{5}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*varied approximately linearly with height and only weakly with

_{g}*Z*,

_{H}*ρ*is prescribed by a linear variation between input values at the surface and 5 km (i.e.,

_{g}### e. Estimation of precipitation parameters and moments

The values of *q _{r}* and

*q*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 (

_{g}*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*decreases as

_{g}*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}.

*q*at any point is determined from its provisional value

_{g}*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*, the corresponding

_{g}*λ*value is derived by either the expressionwhere

_{g}*z** ≥

*H*

_{melt}and

*q*=

_{g}*z** ≤

*H*

_{melt}or

*z** ≥

*H*

_{melt}and

*q*<

_{g}*q*=

_{g}*Z*

_{eg}=

*Z*

_{eh}. Total concentration

*N*is computed from Eq. (5).

_{g}The *q _{r}* value at any point is determined from its provisional value

*q*value by partitioning the measured total equivalent reflectivity

_{g}*Z*

_{eh}according to the quantity

*Z*

_{er}+

*Z*

_{eg}. For the case of

*w*<

*W*

_{min},

*q*=

_{r}*λ*is derived from Eq. (6). For

_{r}*w*≥

*W*

_{min}(in which case

*q*≤

_{r}*Z*

_{er}=

*Z*

_{eh}−

*Z*

_{eg}),

*λ*is derived from Eq. (7) using the computed

_{r}*Z*

_{er}while

*q*is computed from Eq. (6). Total concentration

_{r}*N*is computed from Eq. (5).

_{r}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*and

_{r}*q*(Fig. 4). The diagnosed

_{g}*q*and

_{r}*q*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

_{g}*Z*(e.g., Figs. 3c,d). Peak values of the empirically derived

_{H}*q*(Fig. 4a) and

_{r}*q*(Fig. 4b) are up to about 0.5 g kg

_{g}^{−1}less than the modeled

*q*and

_{r}*q*values (Figs. 4c,d). The main supercell updraft recycles meltwater

_{g}*q*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 (

_{r}*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.

### 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.

*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:

- compute graupel reflectivity
- compute adjusted
- compute adjusted
- compute
*λ*from Eq. (6)._{g}

*q*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

_{g}*α*= 1.

_{N}### 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*to maintain exact water saturation (

_{c}*S*= 1). Conversely, if

*w*< 0 and

*q*> 0, the cloud water is evaporated into the parcel to maintain

_{c}*S*= 1 until

*q*= 0 (after which the downward motion is adiabatic in the absence of rain evaporation). To increase accuracy where

_{c}*q*> 0, the net condensation or evaporation on the large time step is split into a series of small-step calculations with Δ

_{c}*t*

_{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

*D*

_{φ}is expressed aswhere

*c*is a constant dimensionless coefficient,

_{d}*V*and

*L*are velocity and length scales, respectively (defined below),

_{d}*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,where |

*w*| is the vertical velocity magnitude, and the damping length scale is prescribed for the cases of quasi-horizontal flow (i.e., |

*w*| ≤

*W*

_{0}), updraft (+), and downdraft (−). The coefficient

*C*

_{d}_{0}is determined by the expressionwhere the precipitation mixing ratio

*q*=

_{p}*q*+

_{r}*q*,

_{g}*q*is broadly consistent with the parameterized relationship between surface downdraft and

_{p}*Z*described in section 2b. The velocity scale

_{H}*u*and

_{B}*υ*are the base-state horizontal wind components.

_{B}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}*q*

_{0}for surface grid points or

*q*<

_{p}*q*

_{1}for elevated grid points (

*q*

_{0}and

*q*

_{1}listed in Table 1).

### i. Surface flux parameterization

**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

*z*≤

*z*

_{BL}and

*q*

_{hydro}=

*q*+

_{c}*q*+

_{r}*q*≤

_{g}*q*

_{1}only and

*F*= 0 for

_{ϕ}*z*>

*z*

_{BL}. The values of the BL height

*z*

_{BL}and the vertical mixing coefficient

*b*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

_{F}*b*value. It is assumed that surface heating vanishes in local cloud or precipitation.

_{F}## 3. OSSE application of DLA to a simulated supercell storm

The DLA was tested via a simple OSSE approach in which wind and reflectivity fields obtained from the simulated supercell storm (Z10) were input to compute trajectories, define the precipitation fields (e.g., Fig. 4), and compute the Lagrangian source and sink terms. The DLA of the simulated storm was performed 4 h 40 min after initialization, by which time the slowly evolving simulated mature supercell had achieved its overall peak intensity. The simulated domain-relative 3D wind and total precipitation reflectivity fields were output by the model at a 2-min interval for a period of 1 h 40 min beginning at 3 h in the simulation and ending at the time of the Lagrangian analysis. Although a heterogeneous mesoscale environment was prescribed in the simulation study, a single sounding that represented the supercell updraft's inflow (Fig. 2) and zero surface flux were assumed for the OSSE test since only a single proximity inflow sounding may be available in a typical observed storm case. It has been verified that 100% of the backward trajectories reach the storm environment based on the conditions described in section 2a. A series of tests (Table 3) were conducted to determine the sensitivity of the DLA to its chosen parameters.

Sensitivity tests of the DLA. The CNTL case includes all source and sink terms, while the CNTL parameters are listed in Tables 1 and 2. The sensitivity tests all vary only one term or parameter, but otherwise are identical to CNTL.

### 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 Δ*θ*_{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).

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*values exceeding 4 g kg

_{c}^{−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*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.

_{c}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 Δ

*θ*

_{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).

### 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.

*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

**∇**

_{h}

*θ*value is less than 0.2 K km

_{υ}^{−1}and

*Z*> 10 dB

_{H}*Z*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

^{−5}s

^{−2}(Fig. 8). The more densely clustered

*θ*

_{υ}gradients via smoothing from the horizontal low-pass filter. The more frequent weak-to-moderate

## 4. Conclusions

The present study develops a new buoyancy retrieval method, termed diabatic Lagrangian analysis (DLA), which is predicated on the calculation of thermodynamic and microphysical tendencies along trajectories that are initialized in a storm's environment and terminate at analysis grid points at the nominal analysis time. The present version of the DLA predicts the evolution of potential temperature, water vapor mixing ratio, and cloud water mixing ratio along trajectories and gathers their values at trajectory endpoints (i.e., the analysis grid points) to obtain the 3D fields of the predicted variables in the Lagrangian analysis domain at any given analysis time. This fully time-dependent Lagrangian buoyancy retrieval avoids several limitations of conventional Eulerian frame buoyancy retrievals, while retaining both an accurate advection principle and explicit diabatic microphysical forcing. The DLA is also unique in proceeding from radar-diagnosed (as opposed to predicted) precipitation contents and microphysical processes. In principle, the DLA is designed to assimilate all available radar-observed and synthesized variable fields (e.g., including 3D time-dependent multiple-Doppler airflow and reflectivity as demonstrated in Z13).

The DLA has been tested with an observing system simulation experiment (OSSE) in which the surrogate input wind and reflectivity fields are obtained from a supercell storm simulation. Although varying local mixtures of rain and graupel contribute to the total reflectivity (thus complicating the diagnosis of separate rain and graupel contents from that single radar measurand), the OSSE nevertheless demonstrates that the DLA can accurately recover local values of modeled precipitation content via functional dependences on local reflectivity and vertical motion values. The OSSE also demonstrates that the DLA-predicted thermal and cloud variables accurately reproduce the known modeled values and reasonably reproduce the magnitude of the thermal–solenoidal horizontal vorticity generation term.

The main limitations of the present DLA version are its adoption of one-moment rain and graupel size distributions, the omission of additional hydrometeor categories, the approximation of mixing effects via a bulk damping parameterization, and its simple parameterization of near-surface downdrafts to allow BL air trajectories to penetrate down into the surface layer. Future plans include the incorporation of two-moment rain and graupel size distribution functions and the addition of bulk diagnostic precipitation categories for cloud ice, snow, and hail, subject to the strong constraining hypothesis that polarimetric radar measurements contain sufficient microphysical information that is independent of reflectivity to facilitate a well-determined analysis. Mixing effects may be explicitly represented via a suitable local turbulence closure provided that a scheme to calculate accurate spatial gradients from the instantaneous Lagrangian data fields can be implemented. Although turbulence measurements in the rainy surface layer are difficult to implement (M. LeMone and A. Betts 2012, personal communications), surface in situ measurements of velocity, temperature, and water vapor fluctuations in storms would potentially provide an observational basis to help improve the surface layer downdraft parameterization. Assuming sufficiently dense point measurements of state variables in an observed storm and its environment, the DLA may be applied following Z07 to initialize trajectories with proximate in situ measurements. A related future application of assimilating in situ data via the DLA is to study the BL and convection initiation (e.g., Z07; Buban et al. 2007; Richardson et al. 2009). Proceeding from a preliminary test application of DLA to a radar-observed storm (Z13), the DLA could employ a hybrid combination of DLA with dynamic pressure retrieval (e.g., Hane et al. 1988) to investigate the momentum and vorticity dynamics of storms.

## Acknowledgments

The author gratefully acknowledges helpful discussions with Ted Mansell and Lou Wicker concerning the Lagrangian model's microphysical parameterizations. Terry Schuur provided the author with the NSSL digital video distrometer data and interpreted its measurements. Insightful editorial comments provided by Adam Houston, Erik Rasmussen, and Matt Kumjian led to significant improvements of the revised manuscript, while Lou Wicker also provide numerous helpful editorial comments on an internal review of an earlier draft version. Support for the current project was provided under National Science Foundation Grants AGS-0130316, AGS-0638572, and AGS-0802717 and the National Severe Storms Laboratory Director's Discretionary Fund.

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