1. Introduction
The powerful tool of trajectory analysis has found widespread application in the environmental sciences. When coupled with vorticity and thermal energy budgets, trajectory analyses have provided insights into convective storm dynamics, including longstanding problems on the origins of low-level rotation in mesocyclones and tornadoes. Such analyses have been constructed from multiple-Doppler radar wind observations (Brandes 1981, 1984; Johnson et al. 1987; Wakimoto et al. 1998; Ziegler et al. 2001; Bluestein and Gaddy 2001; Dowell and Bluestein 2002; Marquis et al. 2008, 2012, 2014; Frame et al. 2009; Kosiba et al. 2013) and from numerically simulated storm data (Rotunno and Klemp 1985; Davies-Jones and Brooks 1993; Adlerman et al. 1999; Dawson et al. 2010; Dahl et al. 2012, 2014; Schenkman et al. 2012, 2014; Naylor and Gilmore 2014). Most recently, trajectory analyses have been applied to ensemble Kalman filter wind retrievals of supercell thunderstorms observed by mobile radar (Marquis et al. 2012, 2014; Potvin et al. 2013). Trajectories have been used to study atmospheric dispersion and to determine the source regions of pollutants (Stohl and Kromp-Kolb 1994; Stohl 1996, 1998; Brankov et al. 1998; Buchanan et al. 2002; Sturman and Zawar-Reza 2002; Park et al. 2007; Baker 2010) and of spores and pollen allergens (Hjelmroos 1991; Rantio-Lehtimäki 1994; Stach et al. 2007; Smith et al. 2008; Belmonte et al. 2008), pollen from genetically modified crops (Van de Water et al. 2007; Kuparinen et al. 2007; Beckie and Hall 2008), and pollen from illegal cultivations (Cabezudo et al. 1997). Trajectories are used in synoptic and climatological analyses to elucidate prevailing airflow patterns and associated pathways for the transport of water vapor, chemical species, and aerosols (Miller 1981; Dayan 1986; Moody et al. 1995; D’Abreton and Tyson 1996; Salathé and Hartmann 1997; Kahl et al. 1997; Wernli and Davies 1997; Shadboldt et al. 2006; Engelstaedter et al. 2006; Knippertz and Wernli 2010). They can also be used to analyze and forecast the spread of contaminants from environmental disasters such as nuclear reactor breaches (Pöllänen et al. 1997; Povinec et al. 2013; Ioannidou et al. 2013) and offshore oil spills (Spaulding 1988; Price et al. 2006; Sotillo et al. 2008; Liu et al. 2011; Xu et al. 2013). Other oceanographic applications include tracking floating mines and fish larvae, providing guidance for rescue and recovery operations, improving the analysis of oceanic circulations, and revealing the origin and destination of water masses (Blanke and Raynaud 1997; Griffa et al. 2004; Breivik and Allen 2008; Melsom et al. 2012; Chu and Fan 2014).
The utility of a trajectory analysis depends, in part, on the sensitivity and accuracy of the computed trajectories, various aspects of which have been discussed by Kuo et al. (1985), Kahl and Samson (1986), Rolph and Draxler (1990), Doty and Perkey (1993), Seibert (1993), Stohl et al. (1995, 2001), Scheele et al. (1996), Stohl (1998), Stohl and Seibert (1998), and Dahl et al. (2012). Among the major sources of trajectory error are truncation errors associated with the numerical integration scheme, spatial and temporal interpolation errors in the estimation of 
Although a few trajectory models adopt a steady-state assumption or use quadratic interpolation, by far the most common time interpolation procedure is linear interpolation. In a comparison of the linear, quadratic, and nearest-neighbor (piecewise steady state) interpolation methods reported in Stohl et al. (1995), the nearest-neighbor approach yielded the largest errors, linear interpolation yielded the best results, and quadratic interpolation produced results that were similar to, but slightly worse than, linear interpolation. Rössler et al. (1992) also report only minor differences between uses of quadratic and linear interpolations. A common theme in the sensitivity/accuracy studies is that significant trajectory position errors can develop if the temporal resolution of the data is too coarse (Kuo et al. 1985; Rolph and Draxler 1990; Rössler et al. 1992; Doty and Perkey 1993; Stohl et al. 1995, 2001; Stohl 1998; Dahl et al. 2012). The establishment of criteria for minimal acceptable temporal resolutions does not appear to be straightforward; such resolutions appear to depend, in part, on the flow characteristics (Rolph and Draxler 1990; Stohl et al. 1995; Dahl et al. 2012).
To motivate use of the frozen-turbulence constraint in time interpolation, consider a thought experiment in which the u field is identically zero and the υ field is an eastward-translating (U > 0, V = 0) top-hat function (Fig. 1). Data are available at 
Linear-in-time and advection-correction-based interpolations for a υ field in the form of a top hat that translates in the x direction with speed U. The υ field is shown (top) at two data input times t = 0 s and 
Citation: Journal of the Atmospheric Sciences 72, 11; 10.1175/JAS-D-15-0095.1
Advection correction procedures have been used extensively in hydrology and in mesoscale and radar meteorology—for example, in accumulated rainfall estimation (Fabry et al. 1994; Ciach et al. 1997; Anagnostou and Krajewski 1999; Tabary 2007; Villarini and Krajewski 2010; Nielsen et al. 2014) and in multiple–Doppler radar wind and vertical velocity analysis (Carbone 1982; Chong et al. 1983; Roberts and Wilson 1995; Dowell and Bluestein 1997, 2002; Cifelli et al. 2002; Liu et al. 2004; Wurman et al. 2007; Marquis et al. 2008, 2012, 2014; Frame et al. 2009; Kosiba and Wurman 2014). However, the use of advection correction to reduce errors in the temporal interpolation step in trajectory calculations, as advocated in our study, is a relatively new application. Examples of such an application are the dual-Doppler analyses of Marquis et al. (2008, 2012) and Frame et al. (2009) and the EnKF trajectory analyses of Marquis et al. (2012, 2014), all of which used relatively simple translation-analysis formulations such as calculating trajectories in a storm-relative reference frame.
The paper is arranged as follows. The technical details of the trajectory and advection correction procedures are provided in section 2. In section 3, analytical tests are used to verify that the numerical modules for these procedures are free of code errors. In section 4, the utility of the advection correction procedure is explored in tests with a high-resolution numerically simulated supercell storm dataset. A brief summary follows in section 5.
2. Methodology
a. Nature of the experiments
The purpose of our study is to demonstrate that replacing the standard linear time interpolation step in trajectory calculations with an advection correction procedure can effectively reduce trajectory position errors, especially in cases where the input data are of coarse temporal resolution, and to understand the potential weaknesses and strengths of the approach. Tests are conducted using output from a high-resolution numerical storm simulation because that provides a convenient framework to explore the nature of the analysis errors. However, the advection-corrected trajectory procedure itself may find wider applications than in numerical convective storm modeling, for example, in trajectory analyses using multiple–Doppler radar–derived wind fields or in some of the other applications noted in the introduction.
We believe the goals of the study can be accomplished most efficiently in a two-dimensional framework, since trajectory and advection correction modules are then simpler to implement and verify (e.g., against analytical solutions), and run times with the high-resolution simulated storm dataset are shorter and less memory intensive. Given our limited computational resources, the two-dimensional framework allows us to conduct a much more extensive and systematic exploration of the analysis procedure, including the threat of solution nonuniqueness for the pattern-translation components. Moreover, if advection correction does not add value to trajectory analysis in the two-dimensional framework, there is little point in extending it to three dimensions.
We compute backward trajectories on an unstaggered horizontal (x, y) Cartesian analysis grid using low-altitude u and υ (but not w) data. Experiments focus on the temporal resolution of the input data and on variations of implementing the advection correction. In each experiment, the trajectories are compared with a set of reference (verification) trajectories obtained using the linear time interpolation procedure with very-high-temporal-resolution input data from the numerical model. However, since w data are not used in any of our trajectory analyses, the reference trajectories may be quite different from the “true” trajectories. Even very low-altitude trajectories have been shown to be strongly dependent on the w field in these types of numerical experiments (Dahl et al. 2012). Further differences between reference and true trajectories likely follow from the interpolation of the numerical data on the model’s staggered grid to the unstaggered analysis grid. Thus, while our current two-dimensional implementation provides a fair basis for comparing results from the linear time interpolation and advection correction procedures, the particular trajectories obtained in these tests should not be used to gain insights into convective-scale flow fields or dynamics.
In each experiment, we evaluate the position errors of air parcels launched backward in time from the analysis grid points at a common start time. The trajectories are determined by numerically integrating (1) using gridded u and υ fields to estimate the parcel velocity at the instantaneous parcel location. The gridded u and υ fields are obtained through one of two temporal interpolation procedures: standard linear interpolation (LI) or advection correction (AC). The basic AC procedure is described next.
b. Advection correction
The AC procedure removes the artificial stationary disappearance and emergence of analyzed features when linear time interpolation is used (as in Fig. 1) by accounting for the translation of features between successive data input times. The procedure is schematized in Fig. 2. Between two consecutive data input times is a succession of computational times.2 At each computational time we consider an analysis grid on which each grid point serves as a home point for a pair of virtual “particles.” From each home point, one particle is launched forward in time until the second data input time is reached, and the other particle is launched backward in time until the first data input time is reached. The particles translate with the pattern-translation components U and V estimated at the home point and time (estimation described in section 2c). More accurately, U and V are used in the forward translation, while −U and −V are used in the backward translation. The arrival locations of the particles at the data input times are noted and the u and υ values at those locations are estimated via bilinear interpolation of u and υ data from the neighboring grid points. The interpolated u and υ values are then linearly interpolated (in a Lagrangian sense) to the home-point location and time.
Schematic of the advection correction procedure. At each computational time, each analysis grid point serves as a home point (red dot) for a pair of virtual particles. One particle is launched forward in time with pattern-translation components U and V until the second data input time is reached, and the other particle is launched backward in time with pattern-translation components −U and −V until the first data input time is reached. The red line connecting the home point to the two arrival locations and times marks the translation path. The u and υ data at the data input times are bilinearly interpolated to the two particles’ arrival locations (blue dots) from neighboring grid points (blue arrows around blue dots). These interpolated u and υ values are then linearly interpolated to the home-point location and time.
Citation: Journal of the Atmospheric Sciences 72, 11; 10.1175/JAS-D-15-0095.1
c. Determining U and V
A word of caution is in order, however, in cases where gradients in the u field (or whatever variable is being tracked) are small over an entire subdomain box. In such cases the expressions in (7) become very sensitive to small changes in the u field, and the problem becomes ill posed.4 Essentially, the Gal-Chen procedure—and many other pattern-tracking algorithms—break down when there is no pattern to track.
Following Gal-Chen’s (1982) recommendation, we implement an iterative version of this procedure. Since U and V are not known prior to the first evaluation of the integrals in (6), first guesses for U and V are required to start the procedure. Using these first-guess values, u and υ at every home point at every computational time is advection corrected following the procedure of section 2b. The advection-corrected u and υ fields are then used to evaluate the integrals in (6), which are then used in (7) to update U and V. The procedure then iterates, with each cycle consisting of an advection correction for u and υ, an evaluation of (6), and an evaluation of (7). The procedure is terminated after 20 iterations (although, in practice, the procedure often converges after just a few iterations), at which point the changes in U and V from the nineteenth iteration are noted. If the magnitudes of the changes in U and V are each less than a threshold value (
In a converged iterative solution, the integrals in (6) are, implicitly, functions of U and V, the “linear” algebraic equations in (5a) and (5b) are actually nonlinear, and solution uniqueness for U and V cannot be guaranteed. Indeed, as shown in Shapiro et al. (2010, their section 5), a wide class of advection correction/analysis procedures, including the Gal-Chen procedure, is potentially subject to nonuniqueness associated with temporal aliasing. Mindful of this threat, we ran advection correction experiments with different first-guess assignments for U and V, and in some cases explicitly used (4) to evaluate 
d. Trajectory calculations
The air parcel trajectories 
e. Error statistics
3. Analytical verification tests
Tests with a two-dimensional (x, y) analytical wind field that satisfied the frozen-turbulence constraints in (3a) and (3b) exactly were conducted to verify that there were no errors in the numerical codes for the Runge–Kutta integration, bilinear spatial interpolation, linear time interpolation, or advection correction procedures. It is well known that in two-dimensional incompressible flows, u and υ can be written in terms of a streamfunction 
The u field (m s−1) in the analytical verification tests.
Citation: Journal of the Atmospheric Sciences 72, 11; 10.1175/JAS-D-15-0095.1
Tests were conducted with 40 411 backward trajectories launched from all grid points within an analysis subdomain (rectangle) extending approximately from 
Zoomed-in view of sample trajectories from 5 min of backward integration. The trajectories originated in a subdomain extending approximately from x = 7.5 to 15 km and from y = 9 to 14 km, and tended to migrate (backward in time) toward the southwest. Trajectories are shown only for parcels launched every fifteenth grid point (in x and y).
Citation: Journal of the Atmospheric Sciences 72, 11; 10.1175/JAS-D-15-0095.1
The LI and AC experiments were run using data input time intervals Δt = 1, 5, 10, 15, 20, and 30 s (LI using 
For all data input time intervals and box sizes, the retrieved U and V values converged rapidly (three or four iterations) to the correct values. At the end of 20 iterations, the errors in U and V remained on the order of 
Nondimensional deviation of a selected air parcel’s perturbation streamfunction 
Citation: Journal of the Atmospheric Sciences 72, 11; 10.1175/JAS-D-15-0095.1
In the AC experiments, the 
The near-zero values of the nondimensional deviation 
4. Tests with a 30-m simulation of a supercell storm
a. Description of the dataset
Low-altitude wind data were sampled from the inner grid of a high-resolution numerically simulated tornadic-supercell-storm dataset described in a previous study (Orf et al. 2014). These data were generated using a three-dimensional nonhydrostatic cloud model, version 16 of Cloud Model 1 (CM1; Bryan and Morrison 2012). Nested within an outer grid of size 
In preparation for the trajectory experiments, 5 min of storm-relative u and υ data on the model’s C grid (from model times 6310 to 6610 s) were extracted from the z = 45-m level at 1-s intervals and interpolated to an unstaggered trajectory analysis grid with 30-m grid spacing. The wind fields at the start time of the backward integrations (t = 6610 s) are shown in Fig. 6. At this time the storm is in a mature phase, with a wind couplet suggestive of a large tornado5 centered at 
Cross section of (left) u and (right) υ (both in m s−1) sampled from the z = 45-m level of the simulated supercell dataset at model time t = 6610 s. Backward trajectories are launched from all grid points within the inset rectangle. The rectangle includes strong easterly winds (
Citation: Journal of the Atmospheric Sciences 72, 11; 10.1175/JAS-D-15-0095.1
b. Experiments and results
Since the model storm was relatively stationary (by design), a pair of U–V values close to zero would correctly characterize the storm motion, and an advection correction procedure that used such small values over the whole domain would behave like a linear time interpolation procedure. Accordingly, against the backdrop of near-zero storm motion, we viewed the use of local estimation procedures for U and V as essential if we were to show how advection correction could add value to trajectory analyses. In this study, local estimates of U–V pairs were computed on subdomain boxes, as in the verification tests. Use of U and V fields obtained from the continuous spatially variable procedure of Shapiro et al. (2010) will be tested in a future study.
In each experiment, 40 411 backward trajectories were launched from all grid points within the rectangle shown in Fig. 6. The trajectories were integrated backward for 5 min using a computational time step of 0.1 s. The reference trajectories were generated from the LI procedure as applied to the 1-s input data. Trajectories computed using AC procedures with the same input data yielded nearly identical results (not shown). The sample reference trajectories shown in Fig. 7 reveal convergent flow structures (in a forward-in-time sense), which pose a challenging divergent flow scenario for accurate backward trajectory analysis (Dahl et al. 2012; Schenkman 2012, chapter 5).
Selected 5-min backward reference trajectories. Trajectories are launched from within the inset rectangle at model time t = 6610 s.
Citation: Journal of the Atmospheric Sciences 72, 11; 10.1175/JAS-D-15-0095.1
In this section, results from several variants of the AC procedure are compared to results from the LI procedure. The variant types (described below) are summarized in Table 1. All LI and AC experiments were conducted using data at time intervals 
Summary of experiments. Experiments were run using data of progressively coarser time resolution: Δt = 5, 10, 15, 20, and 30 s. AC experiments were run with U and V calculated on subdomain boxes of widths 1.2, 2.4, 3.3, and 4.38 km.
The individual displacement errors in the LI experiments, 
Individual displacement errors 
Citation: Journal of the Atmospheric Sciences 72, 11; 10.1175/JAS-D-15-0095.1
Two different first-guess options were tested for the iterative Gal-Chen procedure: experiments AC-GC1 and AC-GC9. In both experiments, convergence of the procedure was checked after 20 iterations, and U and V were set to zero in cases of nonconvergence.6 In AC-GC1, the first guesses for U and V were specified on each subdomain box as the box-averaged values of u and υ. As seen in Table 2, the incidences of nonconvergence in AC-GC1 were relatively infrequent, generally affecting just a few percent of the U–V pair estimates. However, there were trends for the incidences to increase with box size and data input time interval.
Incidences of nonconvergence of the iterative Gal-Chen procedure in AC-GC1 experiments run with different subdomain box sizes and data input time intervals. The incidences are reported as the number of times the procedure failed to converge divided by the number of times the procedure was run (number of boxes times the number of data input intervals in the 5-min window) and are expressed as a percentage.
In experiments AC-GC9, the Gal-Chen procedure was run on each subdomain box with nine first guesses for U and V, the permutations of 0, 20, and −20 m s−1 taken two at a time. For each first-guess pair, the converged U and V values and the advection-corrected u and υ fields were used to evaluate J from (4). The advection-corrected winds corresponding to the smallest J were then used in the trajectory analysis. Incidences of nonconvergence for all nine first guesses were very rare (typically much less than 1% of possible occurrences).
We also conducted an advection correction experiment AC-MW [mean wind (MW)] in which U and V were specified on each subdomain box as the box-averaged values of u and υ. In other words, the first-guess values in AC-GC1 were used to define U and V directly, in a noniterative, non-Gal-Chen procedure.
The MDEs from all experiments are shown in Fig. 9. The MDEs in the two Gal-Chen-based AC experiments, AC-GC1 and AC-GC9, were significantly less than in the corresponding LI experiments (typically 30%–50% less) for all 
MDEs vs data input time interval for the LI and AC experiments. In the AC experiments, U and V are computed on subdomain boxes of width (a) 1.2, (b) 2.4, (c) 3.3, and (d) 4.38 km. Errors are shown for LI (red curve), AC-GC9 (green curve), AC-GC1 (blue curve), and AC-MW (black curve).
Citation: Journal of the Atmospheric Sciences 72, 11; 10.1175/JAS-D-15-0095.1
Figure 9 also shows that the MDE improvements of the two Gal-Chen-based AC experiments over the LI experiments are on the order of 100 m. While these improvements appear to be small in an absolute sense, inspection of analysis-domain plots of 
Differenced individual displacement errors 
Citation: Journal of the Atmospheric Sciences 72, 11; 10.1175/JAS-D-15-0095.1
Histogram of 
Citation: Journal of the Atmospheric Sciences 72, 11; 10.1175/JAS-D-15-0095.1
The 
Differenced individual displacement errors 
Citation: Journal of the Atmospheric Sciences 72, 11; 10.1175/JAS-D-15-0095.1
c. Discussion
The two Gal-Chen-based AC experiments yielded results that were broadly superior to those obtained with linear time interpolation. Results from the Gal-Chen-based AC experiments also compared favorably to those from the mean-wind-based AC experiment. Between the two Gal-Chen-based experiments, use of a single first-guess U–V pair based on the mean-wind components (AC-GC1) outperformed use of multiple first guesses (AC-GC9). The fact that use of the mean wind for the first guess was desirable, but use of the mean wind for direct specification of the actual U and V was undesirable is somewhat paradoxical. We shed light on these and other results by examining the winds and cost function 
Closeup view of the u field (m s−1) in a 3.3-km-wide subdomain box in the RFD at model time t = 6550 s. Solid line at 
Citation: Journal of the Atmospheric Sciences 72, 11; 10.1175/JAS-D-15-0095.1
By examining contour plots of 
Contour plot of 
Citation: Journal of the Atmospheric Sciences 72, 11; 10.1175/JAS-D-15-0095.1
Closeup view of the Δt = 30-s 
Citation: Journal of the Atmospheric Sciences 72, 11; 10.1175/JAS-D-15-0095.1
Presumably the danger posed by multiminima in J(U, V) on analysis subdomains in the RFD for large 
Checkerboard plots of V (m s−1) from experiments AC-GC1 run on 1.2-km-wide subdomain boxes using data at time intervals (left) 5 and (right) 30 s. Solid black curve marks the 40-dBZ model-equivalent radar reflectivity contour.
Citation: Journal of the Atmospheric Sciences 72, 11; 10.1175/JAS-D-15-0095.1
Checkerboard plot of V (m s−1) from experiment AC-GC9 run on 1.2-km-wide subdomain boxes using data at time intervals of 
Citation: Journal of the Atmospheric Sciences 72, 11; 10.1175/JAS-D-15-0095.1
For completeness, we also show the U field retrieved in AC-GC1 using the 1.2-km-wide subdomain boxes (Fig. 18). The patterns obtained with Δt = 5 and 30 s are qualitatively similar, though with the less cohesive patterns in the Δt = 30-s experiment suggestive of a multiminima problem, as in the V field results. For the Δt = 5-s experiment, the U field in the northwestern half of the domain is fairly consistent with the u field (Fig. 6). However, there are significant differences between the U and u fields in the southeastern half of the domain, particularly in the region of strong inflow, where u is characterized by strong easterlies (
As in Fig. 16, but for U.
Citation: Journal of the Atmospheric Sciences 72, 11; 10.1175/JAS-D-15-0095.1
The J plots shown in Fig. 19 for a subdomain box in the inflow region reveals a single minimum in J near the origin (i.e., in the correct location) for both Δt = 5 and 30 s. Accordingly, the Gal-Chen procedure should yield the correct solution for any first-guess U–V pair, as long as the procedure converges. In contrast, since the box-averaged wind field (specifically, the u component) differs substantially from the pattern motion in this region, the AC-MW experiments would likely have large errors in this region. These disparate results are confirmed in Figs. 10 and 12.
Plots of 
Citation: Journal of the Atmospheric Sciences 72, 11; 10.1175/JAS-D-15-0095.1
5. Summary
Our study suggests that judicious use of an advection correction procedure can reduce errors in trajectory calculations associated with the temporal interpolation step. We explored use of an iterative technique originally proposed by Gal-Chen to reduce errors in radar data analyses associated with nonsimultaneity of the data collection. Two first-guess options for the U–V pattern-translation components were tested: use of a single U–V pair defined by the local mean winds (winds averaged on analysis subdomain boxes) and use of an array of multiple first guesses for U and V. In tests using a high-resolution simulated supercell dataset, use of the Gal-Chen procedure with either first-guess option yielded superior results to the traditional linear time interpolation procedure for all data input time intervals and all subdomain box sizes considered. The Gal-Chen procedure also proved superior to directly using local mean-wind estimates as the pattern-translation components, although there were large areas of the RFD where this latter approach worked quite well.
However, despite the encouraging results, the specter of solution nonuniqueness (multiminima in the cost-function J underlying the estimation of U and V) hung over the Gal-Chen procedure when the time resolution was too coarse. The nonuniqueness was associated with temporal aliasing, a known side effect of the Gal-Chen procedure and related pattern-tracking algorithms (Shapiro et al. 2010). In the coarse-time-resolution tests, aliasing became significant in the RFD, where small-scale wavy disturbances moved with the prevailing winds. Moreover, in this region, the global minimum in J could not be relied on to represent the correct (physical) solution. Accordingly, in this region, use of a single first-guess U–V pair defined by the local mean-wind components was preferable to use of multiple first guesses. In contrast, analysis subdomains in the inflow region were characterized by unique solutions, and the Gal-Chen procedure yielded unique (and correct) solutions there regardless of the first-guess option. Use of a single first-guess U–V pair based on the local mean wind was therefore the overall preferred method. However, our recommendation to use such a first guess in general is tentative since our experiments were only performed with one dataset. Alternatively, in cases where a nonuniqueness threat has been identified (e.g., through analysis of J plots or use of multiple first guesses), one may forego the advection correction step and simply default to linear time interpolation.
Although the two-dimensional framework facilitated our exploration of advection correction in trajectory calculations, we recognize that three-dimensional procedures will be essential for most applications. The advection correction steps in section 2 can be extended to three dimensions by calculating U and V level by level and advection-correcting u and υ level by level. Alternatively, one may obtain U and V, and a vertical translation component W by minimizing a three-dimensional version of (4), as in Zhang and Gal-Chen (1996) (though with J based on u and/or υ instead of reflectivity). In this case, setting 
Acknowledgments
We thank Jim Marquis, David Parsons, and the anonymous reviewers for their helpful comments.
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Here we express the velocity of parcel 
The computational times are the same for the advection correction procedure and the subsequent Runge–Kutta integration used to calculate the trajectories.
If the frozen-turbulence concept is appropriate for a given flow field then it should be applicable to the vector velocity field and to the individual Cartesian velocity components. This can be checked by comparing the U and V values obtained from the frozen-turbulence constraint applied to the u field, to the corresponding values obtained from the υ field, and to the values obtained from a combination of u and υ [as in Gal-Chen’s (16)]. For the numerically simulated storm dataset considered in this study, the values of U and V obtained using the υ field were very similar to those obtained using the u field.
Theoretically, if u is constant over a subdomain box then all of the integrals in (6) vanish and (7) becomes indeterminate. In this case J is identically zero for any pair of U and V values.
This long-lived vortex persisted throughout the 5-min analysis time window with peak winds (>100 m s−1) exceeding the limits of the color bar scale. A smaller, weaker, and shorter-lived vortex is apparent approximately 4 km south of the primary vortex.
Included in our tally of nonconverged U–V pairs were cases where U or V actually did converge—but to unrealistically large values (>40 m s−1). Such cases were extremely rare, but when they occurred, we again set U and V to zero.
