• Adlerman, E. J., K. K. Droegemeier, and R. Davies-Jones, 1999: A numerical simulation of cyclic mesocyclogenesis. J. Atmos. Sci., 56, 20452069, https://doi.org/10.1175/1520-0469(1999)056<2045:ANSOCM>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Beck, J., and C. Weiss, 2013: An assessment of low-level baroclinity and vorticity within a simulated supercell. Mon. Wea. Rev., 141, 649669, https://doi.org/10.1175/MWR-D-11-00115.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Bryan, G. H., and J. M. Fritsch, 2002: A benchmark simulation for moist nonhydrostatic numerical models. Mon. Wea. Rev., 130, 29172928, https://doi.org/10.1175/1520-0493(2002)130<2917:ABSFMN>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • 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, 19451969, https://doi.org/10.1175/MWR-D-14-00328.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Dahl, J. M. L., 2015: Near-ground rotation in simulated supercells: On the robustness of the baroclinic mechanism. Mon. Wea. Rev., 143, 49294942, https://doi.org/10.1175/MWR-D-15-0115.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • 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, 29592966, https://doi.org/10.1175/MWR-D-12-00131.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • 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, 30273051, https://doi.org/10.1175/JAS-D-13-0123.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Davenport, C. E., and M. D. Parker, 2015: Impact of environmental heterogeneity on the dynamics of a dissipating supercell thunderstorm. Mon. Wea. Rev., 143, 42444277, https://doi.org/10.1175/MWR-D-15-0072.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • 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, 10391061, https://doi.org/10.1175/JAS-D-15-0114.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Lorenz, E. N., 1960: Energy and numerical weather prediction. Tellus, 12, 364373, 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, 243275, https://doi.org/10.1175/JAS-D-13-0159.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Markowski, P. M., Y. P. Richardson, and G. Bryan, 2014: The origins of vortex sheets in a simulated supercell thunderstorm. Mon. Wea. Rev., 142, 39443954, https://doi.org/10.1175/MWR-D-14-00162.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • 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, 25752589, https://doi.org/10.1175/MWR-D-11-00200.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • 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, 4568, https://doi.org/10.1175/BAMS-D-15-00073.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Parker, M. D., and J. M. L. Dahl, 2015: Production of near-surface vertical vorticity by idealized downdrafts. Mon. Wea. Rev., 143, 27952816, https://doi.org/10.1175/MWR-D-14-00310.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • 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.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Rotunno, R., P. M. Markowski, and G. H. Bryan, 2017: “Near ground” vertical vorticity in supercell thunderstorm models. J. Atmos. Sci., 74, 17571766, https://doi.org/10.1175/JAS-D-16-0288.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • 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, 130154, https://doi.org/10.1175/JAS-D-13-073.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • 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, 26752703, https://doi.org/10.1175/1520-0469(1995)052<2675:SAAOTD>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • View in gallery

    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 , where is the model vertical grid spacing.

  • View in gallery

    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.

  • View in gallery

    Vertical cross section through the center ( km) of the 0EL simulation at (a) t = 0, (b) t = 10, and (c) t = 20 min, with the perturbation θ as the fill. (d) A horizontal cross section at min and m is also shown, with the shadow of the heat sink represented by a dashed blue circle and the location of the horizontal cross sections in (a)–(c) shown as a black dashed line. Note how despite the passage of time, all regions behind the gust front remain largely steady state.

  • View in gallery

    An AOS parcel trajectory projected upon the shaded θ at m of the 0EL simulation, with the silhouette of the AOS parcel initial position shown as a purple diamond. The AOS trajectory lies entirely within the axis of symmetry of the outflow. Thus, the parcel has no motion in the y direction whatsoever throughout its trajectory.

  • View in gallery

    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.

  • View in gallery

    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.

  • View in gallery

    Vertical cross section of θ at from the (a) 2EL and (b) 0EH simulation, showing the location of the AOS trajectory. The dashed lines represent the lowest scalar model level. All of the θ fields are from min. The θ field below the lowest scalar model level is extrapolated using a second-order extrapolation scheme identical to what the model uses to calculate the θ evolution along trajectories in (a).

  • View in gallery

    (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).

  • View in gallery

    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).

  • View in gallery

    (a) The circles are locations of parcels from the 0EL simulation that spent at least some time beneath the lowest scalar model level at min projected upon m θ perturbation; the circle fill colors represent the absolute value of the distance to the location of a twin parcel from the 2EL simulation at min and the black line indicates where the twin parcel from the 2EL simulation is located. (b) As in (a), but for the maximum absolute value of the difference in ζ along twin parcel trajectories from the 0EL and 2EL simulations.

  • View in gallery

    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.

All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 15 15 7
PDF Downloads 8 8 7

Sensitivities of Parcel Trajectories beneath the Lowest Scalar Model Level of a Lorenz Vertical Grid

View More View Less
  • 1 Atmospheric Science Group, Department of Geosciences, Texas Tech University, Lubbock, Texas
© Get Permissions
Full access

Abstract

Parcel trajectory analysis has become commonplace in the study of simulated severe convection, particularly that which deals with the development and maintenance of near-ground vertical vorticity. However, there are a number of unsolved problems with analyzing simulated trajectories that exist near the ground. One of these unsolved problems is how to deal with parcels that pass beneath the lowest scalar model level. Using the CM1 model, which uses a Lorenz grid, the sensitivity of parcel characteristics such as location or potential temperature to the choice of common extrapolation methods is documented. Using potential temperature as an example, it is explained why unphysical tendencies of scalar variables along trajectories may arise once parcels descend beneath the lowest scalar model level. Given the poorly constrained flow (and scalar) fields beneath the lowest scalar model level, errors such as those documented here appear unavoidable when using free-slip boundary conditions.

Current affiliation: Department of Earth and Environmental Science, University of Michigan, Ann Arbor, Michigan.

© 2018 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Andrew Vande Guchte, avandegu@umich.edu

Abstract

Parcel trajectory analysis has become commonplace in the study of simulated severe convection, particularly that which deals with the development and maintenance of near-ground vertical vorticity. However, there are a number of unsolved problems with analyzing simulated trajectories that exist near the ground. One of these unsolved problems is how to deal with parcels that pass beneath the lowest scalar model level. Using the CM1 model, which uses a Lorenz grid, the sensitivity of parcel characteristics such as location or potential temperature to the choice of common extrapolation methods is documented. Using potential temperature as an example, it is explained why unphysical tendencies of scalar variables along trajectories may arise once parcels descend beneath the lowest scalar model level. Given the poorly constrained flow (and scalar) fields beneath the lowest scalar model level, errors such as those documented here appear unavoidable when using free-slip boundary conditions.

Current affiliation: Department of Earth and Environmental Science, University of Michigan, Ann Arbor, Michigan.

© 2018 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Andrew Vande Guchte, avandegu@umich.edu

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.

Fig. 1.
Fig. 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 , where is the model vertical grid spacing.

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 m. In one realization, the horizontal wind is extended directly from the lowest scalar model level to a parcel’s location below the lowest scalar model. With this version of the free-slip boundary it is therefore assumed that the horizontal wind is constant with height beneath the lowest scalar model level for the purposes of parcel trajectories, and will be referred to as the zero-order extrapolation (0E) condition henceforth. In the 0E simulations, potential temperature θ is also treated as constant with height beneath the lowest scalar model level for the purposes of parcel trajectories. A second realization of the free-slip boundary condition extrapolates the horizontal wind from the lowest three scalar model levels to the surface. This realization will be referred to as the second-order extrapolation (2E) condition. In the 2E simulations, θ is also extrapolated from the lowest three scalar model levels to the surface. Such implementations of the free-slip boundary condition only impact parcels that descend below the lowest scalar model level and do not affect the solution on the model grid.

Two different spatial resolutions were used. Simulations with a lower resolution utilized a horizontal grid spacing of 250 m and a stretched vertical grid with a vertical grid spacing near the surface of 100 m. Contrastingly, the higher-resolution simulations used a m and a stretched vertical grid with a m near the surface. The lowest scalar model level was at () m for the lower(higher) resolution simulations. A sixth-order advection scheme with a large time step of 0.2 s and sixth-order artificial diffusion was used (with the kdiff6 parameter set to 0.04). Subgrid-scale turbulence was parameterized using a 1.5-order closure.

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 (), and was tapered by a squared cosine function of the distance from the center of the heat sink such that at the edge of the heat sink no more cooling was imposed. A 10 m s−1 westerly wind was imposed at all grid points as the base state in order to shape the outflow to something that resembles the outflow of a thunderstorm [supercell outflow is known to have a nonzero storm-relative wind, Parker and Dahl (2015)]. For each simulation, a lattice of 384 000 parcels in a dense, uniform grid (with a parcel spacing of 50 m in all directions) was initialized after 10 min of simulation (henceforth this time will be referred to as min) and forward trajectories calculated within CM1 were performed for the next 30 min. The lattice of parcels was placed just upstream of the heat sink such that most of the initialized parcels were advected into the cooling region where they began to descend in the downdraft. This model design is illustrated conceptually in Fig. 2. To obtain potential temperature budgets along the trajectories, first the Eulerian tendencies from the numerical solver (artificial heat-sink forcing as well as numerical and subgrid-scale mixing and dissipative heating) were tested against, and found to be in close agreement with, the actual local tendencies. Then the individual forcing functions were interpolated trilinearly to the parcel’s location and the abovementioned boundary conditions were applied. The θ integration along the trajectory was performed using a trapezoidal scheme.

Fig. 2.
Fig. 2.

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 min of simulation time, the center (i.e., near km) of the simulated outflow approached a quasi–steady state. For this reason, it is convenient to analyze a parcel whose trajectory lies entirely along the km plane with no movement in the y direction, because such a parcel could have its entire trajectory analyzed with a two-dimensional cross section of the model grid at and at a single time where the flow was in a quasi–steady state. In other words, the overall structure of the outflow is nearly time independent by the time the parcels have descended and are moving along the ground toward positive x. Such a parcel is analyzed here in both the 2EL and 0EH simulations and will be referred to as the axis of symmetry (AOS) parcel. The AOS parcel was initialized at the exact same location in both the 2EL and 0EH simulations (parcels that are initialized at the same location and same simulation time in two different simulations will henceforth be referred to as twins), and is shown in Fig. 4. The 2EL and 0EH simulations were chosen for this analysis because they highlight the differences twin near-ground parcels experience when one descends below the lowest scalar model level and the other does not.

Fig. 3.
Fig. 3.

Vertical cross section through the center ( km) of the 0EL simulation at (a) t = 0, (b) t = 10, and (c) t = 20 min, with the perturbation θ as the fill. (d) A horizontal cross section at min and m is also shown, with the shadow of the heat sink represented by a dashed blue circle and the location of the horizontal cross sections in (a)–(c) shown as a black dashed line. Note how despite the passage of time, all regions behind the gust front remain largely steady state.

Citation: Monthly Weather Review 146, 5; 10.1175/MWR-D-17-0190.1

Fig. 4.
Fig. 4.

An AOS parcel trajectory projected upon the shaded θ at m of the 0EL simulation, with the silhouette of the AOS parcel initial position shown as a purple diamond. The AOS trajectory lies entirely within the axis of symmetry of the outflow. Thus, the parcel has no motion in the y direction whatsoever throughout its trajectory.

Citation: Monthly Weather Review 146, 5; 10.1175/MWR-D-17-0190.1

The AOS parcels were initialized at m, and they began descending as they entered the heat sink. Since the strength of the heat sink is identical in the low- and high-resolution simulations, the height profile of the parcels practically coincides in each case (Fig. 5). As shown in Fig. 6 the θ evolution likewise is nearly identical between the 2EL and 0EH AOS trajectories up until the point where 2EL parcel reaches the lowest scalar level, which for the AOS parcel happened at s (designated by the thicker blue lines in Fig. 6a), while the 0EH AOS parcel remained above the lowest scalar model level. Instantaneously, the 2EL AOS parcel began to warm, and continued to warm by more than 3 K over the next 400 s. The 0EH AOS parcel remained at 296 K until s, whereupon it warmed by 2.5 K over 30 s (this warming appears to be due to a hydraulic-jump-like feature that is absent in the low-resolution simulations). Figure 6b shows a similar θ evolution for another set of twin trajectories (referred to as the non-AOS trajectories, because they are not confined to the outflow axis of symmetry). The 2EL non-AOS parcel warmed by 2 K after it descended below the lowest scalar model level, whereas the 0EH non-AOS parcel never descended below its lowest scalar model level. This parcel remained relatively isentropic until eventually it warmed by about 2 K, which again is attributed to the hydraulic-jump-like feature. Indeed, consistent with the Eulerian θ field, the 2EL parcel does encounter increasing potential temperatures once it descends below the lowest scalar level (Fig. 7a), while the 0EH parcel (Fig. 7b) does not exhibit such a warming trend (until the hydraulic jump feature is reached near km). Based on these parcels (and many more we analyzed), it thus seems that parcels start warming once they descend through the lowest scalar model level.

Fig. 5.
Fig. 5.

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

Fig. 6.
Fig. 6.

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

Fig. 7.
Fig. 7.

Vertical cross section of θ at from the (a) 2EL and (b) 0EH simulation, showing the location of the AOS trajectory. The dashed lines represent the lowest scalar model level. All of the θ fields are from min. The θ field below the lowest scalar model level is extrapolated using a second-order extrapolation scheme identical to what the model uses to calculate the θ evolution along trajectories in (a).

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.

Fig. 8.
Fig. 8.

(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

Fig. 9.
Fig. 9.

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 min. In some cases, the location difference between twin parcels was greater than 1600 m, and in many cases the difference was at least 800 m. Figure 10b shows that twin parcels with the smallest location difference from the 0EL and 2EL simulations experience the greatest differences in vertical vorticity ζ; on the order of 0.01 s−1 [a magnitude similar to the maximum near-ground ζ achieved by recent idealized supercell simulations, e.g., Rotunno et al. (2017)] at some point along their trajectory. The differences in location and ζ between the 0EL and 2EL simulations are purely due to the differences in how parcels are assigned horizontal velocities below the lowest scalar model level; the solution on the model grid is identical.

Fig. 10.
Fig. 10.

(a) The circles are locations of parcels from the 0EL simulation that spent at least some time beneath the lowest scalar model level at min projected upon m θ perturbation; the circle fill colors represent the absolute value of the distance to the location of a twin parcel from the 2EL simulation at min and the black line indicates where the twin parcel from the 2EL simulation is located. (b) As in (a), but for the maximum absolute value of the difference in ζ along twin parcel trajectories from the 0EL and 2EL simulations.

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).

Fig. 11.
Fig. 11.

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, 20452069, https://doi.org/10.1175/1520-0469(1999)056<2045:ANSOCM>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Beck, J., and C. Weiss, 2013: An assessment of low-level baroclinity and vorticity within a simulated supercell. Mon. Wea. Rev., 141, 649669, https://doi.org/10.1175/MWR-D-11-00115.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Bryan, G. H., and J. M. Fritsch, 2002: A benchmark simulation for moist nonhydrostatic numerical models. Mon. Wea. Rev., 130, 29172928, https://doi.org/10.1175/1520-0493(2002)130<2917:ABSFMN>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • 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, 19451969, https://doi.org/10.1175/MWR-D-14-00328.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Dahl, J. M. L., 2015: Near-ground rotation in simulated supercells: On the robustness of the baroclinic mechanism. Mon. Wea. Rev., 143, 49294942, https://doi.org/10.1175/MWR-D-15-0115.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • 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, 29592966, https://doi.org/10.1175/MWR-D-12-00131.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • 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, 30273051, https://doi.org/10.1175/JAS-D-13-0123.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Davenport, C. E., and M. D. Parker, 2015: Impact of environmental heterogeneity on the dynamics of a dissipating supercell thunderstorm. Mon. Wea. Rev., 143, 42444277, https://doi.org/10.1175/MWR-D-15-0072.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • 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, 10391061, https://doi.org/10.1175/JAS-D-15-0114.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Lorenz, E. N., 1960: Energy and numerical weather prediction. Tellus, 12, 364373, 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, 243275, https://doi.org/10.1175/JAS-D-13-0159.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Markowski, P. M., Y. P. Richardson, and G. Bryan, 2014: The origins of vortex sheets in a simulated supercell thunderstorm. Mon. Wea. Rev., 142, 39443954, https://doi.org/10.1175/MWR-D-14-00162.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • 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, 25752589, https://doi.org/10.1175/MWR-D-11-00200.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • 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, 4568, https://doi.org/10.1175/BAMS-D-15-00073.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Parker, M. D., and J. M. L. Dahl, 2015: Production of near-surface vertical vorticity by idealized downdrafts. Mon. Wea. Rev., 143, 27952816, https://doi.org/10.1175/MWR-D-14-00310.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • 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.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Rotunno, R., P. M. Markowski, and G. H. Bryan, 2017: “Near ground” vertical vorticity in supercell thunderstorm models. J. Atmos. Sci., 74, 17571766, https://doi.org/10.1175/JAS-D-16-0288.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • 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, 130154, https://doi.org/10.1175/JAS-D-13-073.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • 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, 26752703, https://doi.org/10.1175/1520-0469(1995)052<2675:SAAOTD>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
1

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, , T, etc.) are defined on a staggered vertical grid. Typically, this level is , where is the vertical grid spacing near the ground. The region beneath this level is still part of the model domain because the lowest grid cell extends from to the surface.

2

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.

Save