1. Introduction
Recent literature in the study of simulated severe convection has made frequent use of parcel trajectory analysis (e.g., Rotunno and Klemp 1985; Wicker and Wilhelmson 1995; Adlerman et al. 1999; Dahl et al. 2012; Naylor et al. 2012; Beck and Weiss 2013; Markowski and Richardson 2014; Dahl et al. 2014; Markowski et al. 2014; Schenkman et al. 2014; Coffer and Parker 2015; Dahl 2015; Davenport and Parker 2015; Dawson et al. 2016; Rotunno et al. 2017). Such analyses have been used to great effect in cloud-scale models in order to determine the source regions of parcels, study the evolution of model variables (e.g., potential temperature, vorticity, specific humidity, etc.) along parcel trajectories, produce budgets to determine how and why such attributes evolve when and where they do, and track material circuits (i.e., to analyze circulation evolution). While analyses of simulated parcels have undoubtedly been useful in further understanding the dynamics and thermodynamics of storm-scale processes, parcels that descend below the lowest scalar model level of the commonly employed Lorenz vertical grid (Lorenz 1960; Fig. 1) can be problematic in simulations that utilize the free-slip boundary condition, which is still used in state-of-the-art supercell simulations (e.g., Orf et al. 2017). This is because the free-slip boundary condition does not specify how horizontal velocities should be extrapolated from the model grid to parcels that are beneath the lowest scalar model level. This is a vexing problem, because it is common for parcels that feed a powerful low-level vorticity maximum to travel very near to the ground (indeed, often below the lowest scalar model level) before merging with the vorticity maximum.
Depiction of the Lorenz vertical grid. The solid (dashed) lines represent where vertical velocity (scalars and horizontal velocity) are defined. The lowest scalar model level as defined in this paper is located at 
Citation: Monthly Weather Review 146, 5; 10.1175/MWR-D-17-0190.1
Depiction of the Lorenz vertical grid. The solid (dashed) lines represent where vertical velocity (scalars and horizontal velocity) are defined. The lowest scalar model level as defined in this paper is located at
Citation: Monthly Weather Review 146, 5; 10.1175/MWR-D-17-0190.1
Depiction of the Lorenz vertical grid. The solid (dashed) lines represent where vertical velocity (scalars and horizontal velocity) are defined. The lowest scalar model level as defined in this paper is located at
Citation: Monthly Weather Review 146, 5; 10.1175/MWR-D-17-0190.1
Parcels whose trajectories are located beneath the lowest scalar model level1 require separate treatment from those parcels whose trajectories remain above the lowest scalar model level. While a parcel remains above the lowest scalar model level, model variables—both dynamic and thermodynamic—can be interpolated to its location directly from the model grid. Such an interpolation cannot be used for a parcel below the lowest scalar model level in a free-slip simulation because there are no meaningful model data below the surface of the model from which to interpolate. A decision must therefore be made on how to assign such variables to parcels that descend below the lowest model scalar level. Historically, a zero-order extrapolation from the lowest scalar model level was used to determine horizontal velocities and other scalar information (e.g., Schenkman et al. 2014). Only recently, free-slip models have begun to use higher-order extrapolation schemes to determine how parcels beneath the lowest scalar model level behave. However, the impact of using such extrapolations (both zero order and higher order) to determine parcel information below the lowest scalar model level have not yet been closely documented in the literature, though it has been anticipated to cause the unphysical evolution of the parcel location and both thermodynamic and kinematic attributes along trajectories below the lowest scalar model level (Dahl et al. 2014). This study aims to provide some clarity by analyzing trajectories that are beneath the lowest scalar model level and that have been subject to several different methods of determining model variables from the grid to the parcel location. In particular, we aim to identify any physical inconsistencies along the trajectories that descend below the lowest scalar model level. Section 2 outlines the design of the simulations used in this experiment, section 3 discusses the results of the experiment, and section 4 offers the conclusions of this study.
2. Experimental design
This study used Cloud Model 1 (CM1; Bryan and Fritsch 2002) release 16 with two separate realizations of the free-slip boundary condition in order to assess how trajectories are affected below the lowest scalar model level. Both realizations use a zero-stress condition at 
Two different spatial resolutions were used. Simulations with a lower resolution utilized a horizontal grid spacing 
Results from three idealized simulations will be presented: high resolution with 0E treatment for parcels beneath the lowest scalar model level (0EH), low resolution with 0E treatment (0EL), and low resolution with 2E treatment (2EL). Each simulation contained no moisture and was initialized with a 300-K neutrally stable environment. A spherical heat sink was placed at the center of the horizontal model domain and at 1.4-km altitude, with a 1.4-km radius [i.e., horizontally centered in the model domain and extending from the surface to 2.8 km, as in Parker and Dahl (2015)] by adding a negative θ tendency to the model’s thermodynamic equation. The heat sink was the most intense at its center (
A conceptual diagram of the model design. The heat sink is depicted here as a blue orb and the initial 10 m s−1 wind is shown as green vectors. The lattice of parcels was initialized just westward of the heat sink such that they were advected into the heat sink whereupon they became negatively buoyant, descended, and then spread out toward the eastern edge of the domain as outflow along the surface.
Citation: Monthly Weather Review 146, 5; 10.1175/MWR-D-17-0190.1
A conceptual diagram of the model design. The heat sink is depicted here as a blue orb and the initial 10 m s−1 wind is shown as green vectors. The lattice of parcels was initialized just westward of the heat sink such that they were advected into the heat sink whereupon they became negatively buoyant, descended, and then spread out toward the eastern edge of the domain as outflow along the surface.
Citation: Monthly Weather Review 146, 5; 10.1175/MWR-D-17-0190.1
A conceptual diagram of the model design. The heat sink is depicted here as a blue orb and the initial 10 m s−1 wind is shown as green vectors. The lattice of parcels was initialized just westward of the heat sink such that they were advected into the heat sink whereupon they became negatively buoyant, descended, and then spread out toward the eastern edge of the domain as outflow along the surface.
Citation: Monthly Weather Review 146, 5; 10.1175/MWR-D-17-0190.1
3. Results and discussion
a. Potential temperature evolution along trajectories
Figure 3 shows that by 
Vertical cross section through the center (
Citation: Monthly Weather Review 146, 5; 10.1175/MWR-D-17-0190.1
Vertical cross section through the center (
Citation: Monthly Weather Review 146, 5; 10.1175/MWR-D-17-0190.1
Vertical cross section through the center (
Citation: Monthly Weather Review 146, 5; 10.1175/MWR-D-17-0190.1
An AOS parcel trajectory projected upon the shaded θ at 
Citation: Monthly Weather Review 146, 5; 10.1175/MWR-D-17-0190.1
An AOS parcel trajectory projected upon the shaded θ at
Citation: Monthly Weather Review 146, 5; 10.1175/MWR-D-17-0190.1
An AOS parcel trajectory projected upon the shaded θ at
Citation: Monthly Weather Review 146, 5; 10.1175/MWR-D-17-0190.1
The AOS parcels were initialized at 
Altitude vs time plot of the (a) AOS and (b) non-AOS parcel trajectory. Solid lines indicate the trajectory is from the 2EL simulation, and dashed lines indicate that the trajectory is from the 0EH simulation. Bold lines indicate that the parcel is beneath the lowest scalar model level.
Citation: Monthly Weather Review 146, 5; 10.1175/MWR-D-17-0190.1
Altitude vs time plot of the (a) AOS and (b) non-AOS parcel trajectory. Solid lines indicate the trajectory is from the 2EL simulation, and dashed lines indicate that the trajectory is from the 0EH simulation. Bold lines indicate that the parcel is beneath the lowest scalar model level.
Citation: Monthly Weather Review 146, 5; 10.1175/MWR-D-17-0190.1
Altitude vs time plot of the (a) AOS and (b) non-AOS parcel trajectory. Solid lines indicate the trajectory is from the 2EL simulation, and dashed lines indicate that the trajectory is from the 0EH simulation. Bold lines indicate that the parcel is beneath the lowest scalar model level.
Citation: Monthly Weather Review 146, 5; 10.1175/MWR-D-17-0190.1
Time series of θ along the (a) AOS and (b) non-AOS parcel trajectories. Bold lines indicate the parcel is below the lowest scalar model level. Both the 2EL (solid lines) and 0EH simulations (dashed lines) are shown.
Citation: Monthly Weather Review 146, 5; 10.1175/MWR-D-17-0190.1
Time series of θ along the (a) AOS and (b) non-AOS parcel trajectories. Bold lines indicate the parcel is below the lowest scalar model level. Both the 2EL (solid lines) and 0EH simulations (dashed lines) are shown.
Citation: Monthly Weather Review 146, 5; 10.1175/MWR-D-17-0190.1
Time series of θ along the (a) AOS and (b) non-AOS parcel trajectories. Bold lines indicate the parcel is below the lowest scalar model level. Both the 2EL (solid lines) and 0EH simulations (dashed lines) are shown.
Citation: Monthly Weather Review 146, 5; 10.1175/MWR-D-17-0190.1
Vertical cross section of θ at 
Citation: Monthly Weather Review 146, 5; 10.1175/MWR-D-17-0190.1
Vertical cross section of θ at
Citation: Monthly Weather Review 146, 5; 10.1175/MWR-D-17-0190.1
Vertical cross section of θ at
Citation: Monthly Weather Review 146, 5; 10.1175/MWR-D-17-0190.1
Why does the descent of a parcel through the lowest scalar level lead to warming along the trajectory in these simulations? Consider Fig. 7, which reveals a positive θ gradient in the positive x direction. There is also flow in the positive x direction, implying an advective cooling tendency. Although not shown here, this tendency is (nearly) balanced by subgrid-scale (SGS) and artificial mixing.2 From the perspective of a parcel above the lowest scalar level, the mixing terms cause a minor slippage between the trajectories and the isentropes, such that parcels experience weak diffusive warming once they leave the heat sink. Interestingly, while the Eulerian budgets are reconciled, the budgets along a trajectory may be violated as demonstrated in Fig. 8. Here the parcel experiences an extrapolated (first or higher order) potential temperature field close to the ground (Fig. 8a) and alternatively a 0E scenario in Fig. 8b. In this example, the wind is constant on the lowest model levels, so the extrapolated near-surface wind, and hence the trajectories, are identical in both cases. The parcels will experience different heating rates below the lowest scalar level because the slope of the isentropes is different near the surface in each scenario (which determines the degree of “slippage”). The heating rate is thus directly linked to the order of extrapolation of the potential temperature and horizontal wind fields. While a certain amount of mixing still contributes to a warming trend below the lowest scalar level via the extrapolation of diffusive and turbulent θ tendencies, this mixing cannot account for the warming encountered by the parcel for all choices of θ extrapolation. The rate of warming is determined by how frequently a parcel cuts through an isentrope, and this rate depends on the arbitrary choices about the degree of extrapolation. Using as an example the 0EL AOS parcel (Fig. 9), it can be seen that indeed the warming experienced below the lowest scalar level cannot be fully explained with the available forcing terms (in this case, SGS mixing, artificial mixing, and dissipative heating). In our experiments, the horizontal wind field as well as the θ tendencies were extrapolated consistently for different degrees of extrapolation, but the warming could never be reconciled with the Eulerian tendencies extrapolated to the parcel location. Such warming must thus be regarded unphysical because it mainly depends on the arbitrary choice of the order of extrapolation.
(a),(b) Conceptualization of the warming experienced by parcels that descend through the lowest scalar level. The thick black directed line segment represents the parcel trajectory and the contours represent isentropes. The horizontal wind (black arrows) is assumed to be constant on the lowest few model levels. In (a) the θ field is extrapolated downward beneath the lowest scalar level and in (b) the θ values remain constant beneath that level (0E scenario). As the trajectories in both cases are practically identical, it is the different slopes of the isentropes near the surface that lead to different warming rates along the trajectory in each scenario (indicated by the spatial separation of the gray vertical lines). In each case, the parcel’s heating rate increases once it descends below the lowest scalar level (black dashed line).
Citation: Monthly Weather Review 146, 5; 10.1175/MWR-D-17-0190.1
(a),(b) Conceptualization of the warming experienced by parcels that descend through the lowest scalar level. The thick black directed line segment represents the parcel trajectory and the contours represent isentropes. The horizontal wind (black arrows) is assumed to be constant on the lowest few model levels. In (a) the θ field is extrapolated downward beneath the lowest scalar level and in (b) the θ values remain constant beneath that level (0E scenario). As the trajectories in both cases are practically identical, it is the different slopes of the isentropes near the surface that lead to different warming rates along the trajectory in each scenario (indicated by the spatial separation of the gray vertical lines). In each case, the parcel’s heating rate increases once it descends below the lowest scalar level (black dashed line).
Citation: Monthly Weather Review 146, 5; 10.1175/MWR-D-17-0190.1
(a),(b) Conceptualization of the warming experienced by parcels that descend through the lowest scalar level. The thick black directed line segment represents the parcel trajectory and the contours represent isentropes. The horizontal wind (black arrows) is assumed to be constant on the lowest few model levels. In (a) the θ field is extrapolated downward beneath the lowest scalar level and in (b) the θ values remain constant beneath that level (0E scenario). As the trajectories in both cases are practically identical, it is the different slopes of the isentropes near the surface that lead to different warming rates along the trajectory in each scenario (indicated by the spatial separation of the gray vertical lines). In each case, the parcel’s heating rate increases once it descends below the lowest scalar level (black dashed line).
Citation: Monthly Weather Review 146, 5; 10.1175/MWR-D-17-0190.1
Time series of θ of the 0EL AOS parcel (red line; line thickness is increased while the parcel is below the lowest scalar level). The dashed red line represents the integrated forcing (which is mainly due to artificial mixing and to a smaller extent due to SGS turbulence).
Citation: Monthly Weather Review 146, 5; 10.1175/MWR-D-17-0190.1
Time series of θ of the 0EL AOS parcel (red line; line thickness is increased while the parcel is below the lowest scalar level). The dashed red line represents the integrated forcing (which is mainly due to artificial mixing and to a smaller extent due to SGS turbulence).
Citation: Monthly Weather Review 146, 5; 10.1175/MWR-D-17-0190.1
Time series of θ of the 0EL AOS parcel (red line; line thickness is increased while the parcel is below the lowest scalar level). The dashed red line represents the integrated forcing (which is mainly due to artificial mixing and to a smaller extent due to SGS turbulence).
Citation: Monthly Weather Review 146, 5; 10.1175/MWR-D-17-0190.1
So, the difference between the 0EH and 2EL parcels described in the beginning of this section is largely attributed to the fact that the parcels of the low-resolution simulations have their trajectories and characteristics extrapolated from the model grid beneath the lowest scalar level, whereas the parcels of the high-resolution simulation remain above the lowest scalar model level and are not subject to extrapolation.
b. Trajectory accuracy
Figure 10a depicts the difference in the location of 0EL parcels that descended beneath the lowest scalar model level and their twin from the 2EL simulation at 
(a) The circles are locations of parcels from the 0EL simulation that spent at least some time beneath the lowest scalar model level at 
Citation: Monthly Weather Review 146, 5; 10.1175/MWR-D-17-0190.1
(a) The circles are locations of parcels from the 0EL simulation that spent at least some time beneath the lowest scalar model level at
Citation: Monthly Weather Review 146, 5; 10.1175/MWR-D-17-0190.1
(a) The circles are locations of parcels from the 0EL simulation that spent at least some time beneath the lowest scalar model level at
Citation: Monthly Weather Review 146, 5; 10.1175/MWR-D-17-0190.1
The unphysical θ evolution in the 2EL simulation and the location differences between the 0EL and 2EL simulations are both a direct consequence of the arbitrary choice of how model variables are handled beneath the lowest scalar model level for the purposes of parcel analysis (i.e., what order of extrapolation is used to determine those model variables). While one can point out where a parcel evolves in an unphysical fashion (e.g., when the AOS parcel warms without any apparent physical forcing mechanism), it is difficult to determine whether a parcel’s trajectory location is accurate when it is beneath the lowest scalar model level. In this study, only two implementations of the free-slip boundary condition were examined (zero- and second-order extrapolation), which resulted in dramatic discrepancies between many twin trajectories (Fig. 10). However, there are conceivably many other ways to handle the horizontal wind profile beneath the lowest scalar model level for the purposes of trajectories in the lower free-slip boundary, each of which could result in a different trajectory solution. It is thus difficult to be sure if the trajectory accurately depicts a physically consistent flow. Indeed, Fig. 11 shows that there may even exist a sizable location discrepancy between a trajectory in the low-resolution simulation and its twin in the high-resolution simulation: while there exist differences in the location of the 2EL and 0EH parcel trajectories caused by the discrepancies in the resolved flow fields on the model grid, the difference in location increases dramatically after the 2EL parcel descends below the lowest scalar model level. This suggests that the second-order extrapolation utilized by the 2EL simulation does not accurately represent the trajectories of parcels that stay above the lowest scalar model level in the high-resolution simulations (0EH in this case).
A time series of the altitude of a 2EL parcel (red) and its twin 0EH (blue) parcel. The difference in the locations of the two parcels is also shown (black). A bold line indicates that the parcel is below the lowest scalar model level.
Citation: Monthly Weather Review 146, 5; 10.1175/MWR-D-17-0190.1
A time series of the altitude of a 2EL parcel (red) and its twin 0EH (blue) parcel. The difference in the locations of the two parcels is also shown (black). A bold line indicates that the parcel is below the lowest scalar model level.
Citation: Monthly Weather Review 146, 5; 10.1175/MWR-D-17-0190.1
A time series of the altitude of a 2EL parcel (red) and its twin 0EH (blue) parcel. The difference in the locations of the two parcels is also shown (black). A bold line indicates that the parcel is below the lowest scalar model level.
Citation: Monthly Weather Review 146, 5; 10.1175/MWR-D-17-0190.1
4. Conclusions
This study utilized idealized simulations to document how a modern free-slip model with a Lorenz vertical grid (viz., CM1) might handle trajectories that descend below the lowest scalar model level. It was found that physical inconsistencies can manifest along such trajectories in several ways. Simply extrapolating θ beneath the lowest scalar model level for the purposes of parcel trajectories can result in thermodynamically inconsistent evolution of θ along the trajectory. Furthermore, there is a large discrepancy between the trajectory solutions that result from different orders of extrapolation of horizontal wind to parcels below the lowest model level in free-slip simulations (in this paper, we used zero- and second-order extrapolation regimes), suggesting that there could be any number of different solutions if other methods of determining the horizontal wind below the lowest scalar model were utilized. Such inconsistencies and ambiguous trajectories make it difficult to interpret trajectories that descend below the lowest scalar model level. In particular, parameters that are sensitive to the thermal evolution of parcels—such as baroclinic vorticity generation—are questionable when using the standard extrapolation techniques herein. While in the future there may be a method to ameliorate the issues raised in this paper, they are currently unresolved. Because of this, in order to avoid the issues raised within this paper, it is recommended that parcels beneath the lowest scalar model level be removed from analyses if at all possible.
Acknowledgments
The authors would like to acknowledge Drs. George Bryan and Matt Parker for providing the CM1 model and the idealized format utilized in this paper, respectively. Furthermore, the lead author thanks Drs. Chris Weiss and Eric Bruning for their insightful comments on the thesis research from which this paper was written. We also would like to acknowledge the reviewers, one of whom (G. Bryan), provided especially helpful feedback via several offline discussions that led to an improved analysis.
REFERENCES
Adlerman, E. J., K. K. Droegemeier, and R. Davies-Jones, 1999: A numerical simulation of cyclic mesocyclogenesis. J. Atmos. Sci., 56, 2045–2069, https://doi.org/10.1175/1520-0469(1999)056<2045:ANSOCM>2.0.CO;2.
Beck, J., and C. Weiss, 2013: An assessment of low-level baroclinity and vorticity within a simulated supercell. Mon. Wea. Rev., 141, 649–669, https://doi.org/10.1175/MWR-D-11-00115.1.
Bryan, G. H., and J. M. Fritsch, 2002: A benchmark simulation for moist nonhydrostatic numerical models. Mon. Wea. Rev., 130, 2917–2928, https://doi.org/10.1175/1520-0493(2002)130<2917:ABSFMN>2.0.CO;2.
Coffer, B. E., and M. D. Parker, 2015: Impacts of increasing low-level shear on supercells during the early evening transition. Mon. Wea. Rev., 143, 1945–1969, https://doi.org/10.1175/MWR-D-14-00328.1.
Dahl, J. M. L., 2015: Near-ground rotation in simulated supercells: On the robustness of the baroclinic mechanism. Mon. Wea. Rev., 143, 4929–4942, https://doi.org/10.1175/MWR-D-15-0115.1.
Dahl, J. M. L., M. D. Parker, and L. J. Wicker, 2012: Uncertainties in trajectory calculations within near-surface mesocyclones of simulated supercells. Mon. Wea. Rev., 140, 2959–2966, https://doi.org/10.1175/MWR-D-12-00131.1.
Dahl, J. M. L., M. D. Parker, and L. J. Wicker, 2014: Imported and storm-generated near-ground vertical vorticity in a simulated supercell. J. Atmos. Sci., 71, 3027–3051, https://doi.org/10.1175/JAS-D-13-0123.1.
Davenport, C. E., and M. D. Parker, 2015: Impact of environmental heterogeneity on the dynamics of a dissipating supercell thunderstorm. Mon. Wea. Rev., 143, 4244–4277, https://doi.org/10.1175/MWR-D-15-0072.1.
Dawson, D. T., M. Xue, A. Shapiro, J. A. Milbrandt, and A. D. Schenkman, 2016: Sensitivity of real-data simulations of the 3 May 1999 Oklahoma City tornadic supercell and associated tornadoes to multimoment microphysics. Part II: Analysis of buoyancy and dynamic pressure forces in simulated tornado-like vortices. J. Atmos. Sci., 73, 1039–1061, https://doi.org/10.1175/JAS-D-15-0114.1.
Lorenz, E. N., 1960: Energy and numerical weather prediction. Tellus, 12, 364–373, https://doi.org/10.3402/tellusa.v12i4.9420.
Markowski, P. M., and Y. P. Richardson, 2014: The influence of environmental low-level shear and cold pools on tornadogenesis: Insights from idealized simulations. J. Atmos. Sci., 71, 243–275, https://doi.org/10.1175/JAS-D-13-0159.1.
Markowski, P. M., Y. P. Richardson, and G. Bryan, 2014: The origins of vortex sheets in a simulated supercell thunderstorm. Mon. Wea. Rev., 142, 3944–3954, https://doi.org/10.1175/MWR-D-14-00162.1.
Naylor, J., M. A. Askelson, and M. S. Gilmore, 2012: Influence of low-level thermodynamic structure on the downdraft properties of simulated supercells. Mon. Wea. Rev., 140, 2575–2589, https://doi.org/10.1175/MWR-D-11-00200.1.
Orf, L., R. Wilhelmson, B. Lee, C. Finley, and A. Houston, 2017: Evolution of a long-track violent tornado within a simulated supercell. Bull. Amer. Meteor. Soc., 98, 45–68, https://doi.org/10.1175/BAMS-D-15-00073.1.
Parker, M. D., and J. M. L. Dahl, 2015: Production of near-surface vertical vorticity by idealized downdrafts. Mon. Wea. Rev., 143, 2795–2816, https://doi.org/10.1175/MWR-D-14-00310.1.
Rotunno, R., and J. Klemp, 1985: On the rotation and propagation of simulated supercell thunderstorms. J. Atmos. Sci., 42, 271–292, https://doi.org/10.1175/1520-0469(1985)042<0271:OTRAPO>2.0.CO;2.
Rotunno, R., P. M. Markowski, and G. H. Bryan, 2017: “Near ground” vertical vorticity in supercell thunderstorm models. J. Atmos. Sci., 74, 1757–1766, https://doi.org/10.1175/JAS-D-16-0288.1.
Schenkman, A. D., M. Xue, and M. Hu, 2014: Tornadogenesis in a high-resolution simulation of the 8 May 2003 Oklahoma City supercell. J. Atmos. Sci., 71, 130–154, https://doi.org/10.1175/JAS-D-13-073.1.
Wicker, L. J., and R. B. Wilhelmson, 1995: Simulation and analysis of tornado development and decay within a three-dimensional supercell thunderstorm. J. Atmos. Sci., 52, 2675–2703, https://doi.org/10.1175/1520-0469(1995)052<2675:SAAOTD>2.0.CO;2.
For convenience, the nomenclature lowest scalar (model) level is used, which refers to the lowest level on a Lorenz vertical grid where u, υ, and scalar variables (such as θ, 
The Eulerian tendencies due to advection, mixing, and dissipative heating were compared to the actual local rate of change of θ at the lowest scalar level on a gridpoint basis, and the budgets were found to be in close agreement.
