## 1. Introduction

The origin of vertical vorticity in tornadoes is one of the most critical questions about tornadogenesis. A widely used approach to address this problem, both in models and dual-Doppler analyses, is the backward integration of trajectories initialized within the near-surface vortex. Based on these trajectories, vorticity or circulation budgets following individual parcels may be computed. However, these budgets—and the inferred sources of vorticity—critically depend on the accuracy of trajectories determined by the backward integrations. Herein, we find that the analysis becomes increasingly contaminated with trajectories that originate near the surface from the inflow side of the rear-flank gust front as the resolution of the velocity field is decreased. These “inflow trajectories” are quite prevalent in the literature (Rotunno and Klemp 1985; Davies-Jones and Brooks 1993; Wicker and Wilhelmson 1995; Adlerman et al. 1999; Atkins and St. Laurent 2009; Mashiko et al. 2009; Noda and Niino 2010). Herein we demonstrate that these may result from errors associated with the linear spatiotemporal interpolation of velocities in time-dependent and strongly curved, confluent flow.

To assess trajectory sensitivities in near-surface mesocyclones, we compare forward trajectories calculated on the time step during numerical simulations, with backward integrated trajectories for model output of varying spatiotemporal resolution. In section 2 we describe the simulations and the different methods of calculating trajectories. Section 3 is devoted to the results, which are discussed in section 4. Conclusions are offered in section 5.

## 2. Experimental setup

Two supercell simulations were carried out, using the Bryan Cloud Model 1 (CM1; Bryan and Fritsch 2002), version 14. Each simulation was initialized with a warm bubble in a homogeneous base-state environment using the Gilmore et al. (2004) version of the single-moment Lin et al. (1983) microphysics scheme. The model equations are discretized on a C-grid (Arakawa and Lamb 1977), with a horizontal grid spacing of 250 m in a 128 × 128 × 20 km^{3} domain. The stretched vertical grid extends from 50 m to 19.875 km AGL for scalar, *u* and *υ* grid points and from the surface to 20 km AGL for *w* grid points, with a vertical grid spacing ranging from 100 to 250 m, increasing with height. A zero-gradient condition is applied to the horizontal velocity beneath the lowest model level, and zero vertical velocity is enforced at the surface. Base states were provided by the Wicker (1996) sounding (hereafter “W96”) and by the Del City, Oklahoma, environment from 20 May 1977 (e.g., Klemp et al. 1981). The reason for showing two simulations is the different sensitivity of the backward trajectories to flow resolution in each case. The model was integrated forward for 5400 s using a 2.0- and 2.5-s time step for W96 and Del City, respectively.^{1} A constant grid motion was subtracted from the base-state flow to keep the storms near the domain center. During the simulations, intense vorticity maxima (>0.03 s^{−1}) developed at the lowest model level beneath the main updraft. The development of these circulations proceeds rather unsteadily, tied to successive surges of horizontal momentum that emanate from the main downdraft. Each of these surges is associated with vertical vorticity that may give rise to a compact vortex. For our analysis, we chose the strongest and deepest vortex that developed during the simulation.

Figure 1 shows horizontal cross sections of the two simulated storms at the time the selected vorticity centers at the surface have evolved to considerable strength. Several traits of real-world supercells are simulated, such as a hook echo, a broad updraft, and downdrafts north and west of the main updraft.

### a. CM1 trajectories

A useful feature of CM1 is its ability to calculate the location of fluid parcels during run time using velocity data on the staggered C-grid at every time step. Parcel velocities are obtained by trilinear interpolation and a linear Euler scheme is used to integrate the trajectories forward.

To obtain the trajectories of interest, approximately 2 000 000 parcels were released 18 and 11 min prior to vortex genesis in the W96 and Del City simulations, respectively, in a 15 × 15 × 4 km^{3} (W96) and 8 × 8 × 4 km^{3} (Del City) box surrounding the incipient vortex. Subsequently, the parcel trajectories were integrated forward within CM1, but only those trajectories passing through the lowest 100 m of the vorticity maximum (i.e., the near-surface mesocyclone) were considered for further analysis. Several box locations and sizes, as well as parcel densities, were tested to ensure robustness of the results. The results shown are based on initial parcel densities of 4000 km^{−3} in the W96 simulation and about 14 000 km^{−3} in the Del City simulation (this difference is owed to the much smaller vorticity maximum in the Del City case compared to W96, making it less likely for trajectories to find their way into the vorticity center). This analysis yielded a total of 759 trajectories in the near-surface vortex in the W96 simulation and 454 trajectories in the Del City case. In the following, we will refer to these forward-integrated trajectories as “CM1 trajectories.”

### b. Backward trajectories

The other set of trajectories was calculated “offline” using velocity data stored in model history files. The trajectory computations were done on both C- and A-grids (velocity data interpolated to scalar grid points) to test the sensitivity of the trajectories to averaging the velocity field at the grid scale. The parcel velocities were obtained by trilinear interpolation in space and linear interpolation in time. To initialize the backward trajectories, the final locations of those CM1 trajectories that became part of the low-level vortex served as initial conditions.

The accuracy of the trajectories (with respect to the true solution obtained from the continuous representation of the velocity field) depends on two aspects: (i) the resolution of the 4D velocity field used to calculate the trajectories and (ii) the accuracy of the numerical integration method employed to integrate the backward trajectories. Specifically, the trajectory calculation may be perfectly accurate, but if spatiotemporal gradients of the flow are misrepresented, unrealistic trajectories may result. In this study, the accuracy of the numerical integration method has been verified by comparing numerical trajectory integrations with analytical solutions in idealized flows and by using different integration time steps, different numerical schemes [Euler, second-order Runge–Kutta (RK2), and fourth-order Runge–Kutta] as well as independently developed computer codes. We can thus focus on the effects of the velocity field. The backward trajectories shown in the following are all calculated using an RK2 integration with an integration step of 2.0 s.

## 3. Results

Figure 2 shows the CM1 trajectories in the W96 (Fig. 2a) and Del City (Fig. 2b) simulations. All trajectories that end up in the near-surface circulation originate from the (height dependent) upstream direction of the main downdraft. Subsequently, they are entrained into this downdraft, descend to the ground, and move southward into the circulation. No trajectories originating from near the surface east or northeast of the storms’ mesocyclones (“inflow trajectories”) were found to enter the vorticity maxima.

The output intervals^{2} tested for the backward trajectory integration include output at every model time step (2.0 and 2.5 s, respectively), every 30, every 120, and every 300 s. The latter two intervals roughly correspond to the duration of volume scans for platforms typically used in tornado research [such as the Doppler-on-Wheels (DOW; Wurman et al. 1997) or the Weather Surveillance Radar-1988 Doppler (WSR-88D; Crum and Alberty 1993)].

Figure 3 shows the sensitivity of the backward trajectories to decreasing the temporal sampling rate of the flow on the C-grid for the W96 simulation. The number of inflow trajectories^{3} consistently increases as the temporal resolution of the velocity field is decreased. Quite importantly, a few inflow trajectories appear even if the same resolution is used as for the CM1 trajectories (i.e., velocity sampled every 2 s on the C-grid; Fig. 3a). Figure 4 displays the corresponding results for the Del City case, also showing an increase of near-surface backward trajectories that turn into the inflow sector as the temporal resolution is decreased.

Repeating the backward-trajectory analyses on the A-grid (not shown) reveals that the number of inflow trajectories increases compared to the C-grid analyses, especially for small output intervals.

## 4. Discussion

The temporal resolution of the flow has a large impact on the trajectories, which is related to nonlinear tendencies in the spatial velocity field (∂_{t}**v** ≠ const, where **v** is the velocity vector). To understand this effect in the present analysis, we consider a trajectory that is integrated backward in the W96 simulation using a 300-s output interval. Figure 5 shows the horizontal velocity field at the beginning and at the end of the 300-s interval (Figs. 5a,b), as well as in the middle of this interval (Fig. 5c). Also shown is the temporally interpolated flow at 4830 s, using the velocity fields from 4980 and 4680 s (Fig. 5d). As indicated by the box in Figs. 5c,d, there is a stronger easterly wind component west of the gust front in the interpolated flow than in the “naturally evolving” flow. It is this spurious easterly component that draws some of the backward trajectories eastward into the inflow sector. The source of this error is the artificial manner in which a feature in continuous motion in the naturally evolving flow moves when the time interpolation is performed. At a given time in the interpolation interval, the location of some feature is determined by the weighted average of the flow configurations at the beginning and end of the interpolation interval. Within this interval, the feature thus disappears at its original location while simultaneously emerging at its final location. This effect creates quite an unnatural flow evolution, which is directly reflected by the trajectories.

However, even if the forward and backward trajectories are calculated for the same, highly resolved flow, there is no guarantee that the results are identical (see the inflow trajectories in Fig. 3a, which are not present in Fig. 2a). From a purely mathematical perspective, it cannot be established which set of trajectories is preferable because *both* solutions are merely discretized approximations to the (unknown) true solution of the continuous flow. In each case, the velocity used to step the trajectory forward or backward relies on linear spatial interpolation within the numerical grid. This linear interpolation may cause errors in parcel velocity (cf. the continuous, natural flow that contains the true trajectories), especially in strongly curved flow. While both forward and backward trajectories experience these errors, amplification of these errors is asymmetric in confluent flow, as pointed out by A. Schenkman (2011, personal communication). In a confluent flow, location errors in backward trajectories tend to amplify, while they are damped in forward trajectories. Error growth in backward trajectories thus tends to be maximized within the highly confluent low-level circulation. These errors become sufficiently large to bifurcate some of these trajectories into the inflow sector. The opposite is true for forward trajectories. These are embedded in spatially well-resolved flow prior to entering the circulation, and possible location errors are damped owing to confluence. *It thus seems that forward trajectories are much less susceptible to errors near the vortex than backward trajectories*. The Del City storm features weaker gradients and consequently its backward trajectories are less sensitive to flow resolution.

Another indication that the forward trajectories might be more accurate than backward trajectories is the convergence of the backward trajectories toward CM1 trajectories as the spatiotemporal resolution of the velocity field is improved. This suggests that the CM1 trajectories might represent a limit of the backward trajectories for well-resolved flow (as implied above, “well resolved” may be different for forward and backward trajectories).

To show what ought to happen to inflow trajectories, we conducted an additional experiment in which we released a dense “curtain” of parcels between 5 and 50 m AGL in the inflow of the W96 supercell (Fig. 6). Parcel trajectories were obtained from forward integration within CM1. These trajectories support the notion that in the lowest 100 m the vortex is fed exclusively by outflow air. In this simulation, the inflow is effectively screened from the compact circulation by the gust front, with inflow parcels passing around the periphery of the vortex while ascending. Upward of several 100 m above the ground, however, these inflow parcels gain appreciable vertical vorticity by reorientation of ambient horizontal vorticity and thus increasingly do contribute to the mesocyclone (not shown).

## 5. Conclusions

We investigated the impact of the spatiotemporal resolution of the velocity field on backward trajectories feeding into the lowest 100 m AGL of simulated mesocyclones. Depending on the nature of spatial gradients within the storm and the time dependency of the flow, these trajectories may be quite sensitive to the spatiotemporal resolution of the velocity field. Location errors in backward trajectories are primarily manifest as “inflow trajectories” that enter the near-ground mesocyclone directly from inflow sector (without descending in downdraft). These erroneous trajectories may easily be mistaken as a source of air for the near-surface vortex. We speculate that backward trajectories are more susceptible than forward trajectories to errors in the confluent flow surrounding the near-surface circulation. Though obtained from simulations, these results very likely apply to dual-Doppler analyses as well. Our analysis suggests that error growth depends on the flow characteristics themselves; hence, no general recommendation can be given for what resolution of the velocity field is required to obtain accurate results. Given the nonlinearity and time dependence of the flow, it may be challenging to establish a quantitative estimate of the error growth, although it would be desirable to derive a theory for the minimal spatiotemporal resolution needed to calculate accurate trajectories.

We thank George Bryan for providing the CM1 model, and Paul Markowski, George Bryan, and Alexander Schenkman for discussions during the course of this work. We also thank the anonymous reviewers as well as the Convective Storms Group at NCSU for their comments on the manuscript. Funding was provided by NSF Grant ATM-0758509.

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^{2}

In the following, “output interval” always refers to the time between two CM1 history files containing the 3D velocity field. This interval is thus a measure of the temporal resolution of the velocity field.