Tracking the Centre of Asymmetric Vortices Using Wind Velocity Vector Data Fields

Abstract

Tornados are a major hazard in many regions around the world and as such it is necessary to analyze them. However, such analyses require accurately tracking them first. Currently, there are gaps in the available vortex detection methods when processing a wind-field dataset to locate a series of points that are identifiable as the tornado centreline. This study proposes a novel solution that corrects for deficiencies in previous attempts to identify vortex centres when applied to tornado wind-fields, which would have otherwise led to identifying merely the region of the vortex, several potential centres requiring post-processing, or erroneously approximating the tornado centre. Additionally, this method combines the efficiency required to process large datasets of temporal and spatial wind velocity vector distributions with the accuracy needed to reliably calculate a specific line as a tornado centre. This method is compared to five other approaches commonly used for vortex identification in order to assess: (a) how accurately they identify the centre region, (b) how they handle extraneous vortices that are not of interest, and (c) their computational efficiency in processing a wind-field dataset. With the proposed method, it would be possible to plot a tornado path from formation to dissipation and perform analyses to understand the vortex characteristics with respect to this path without requiring extensive user-intervention.

Appendices

Appendix 1: Summary of Processing Algorithm

The centre of a vortex may be tracked with the proposed method by performing the following series of calculations on a two-dimensional, horizontal slice of data from a tornado dataset. Here, each layer is analyzed without consideration of the layers above or below (Fig. 6a). If not already available in the dataset, the vorticity, ${\varvec{\upomega}}$, is computed for the two-dimensional plane:

$${\varvec{\upomega}}=\nabla \times \mathbf{u}=\left(\frac{\partial {u}_{z}}{\partial y}-\frac{\partial {u}_{y}}{\partial z}\right)\widehat{\mathbf{i}}+\left(\frac{\partial {u}_{x}}{\partial z}-\frac{\partial {u}_{z}}{\partial x}\right)\widehat{\mathbf{j}}+\left(\frac{\partial {u}_{y}}{\partial x}-\frac{\partial {u}_{x}}{\partial y}\right)\widehat{\mathbf{k}}\boldsymbol{ },$$

where $\mathbf{u}={u}_{x}\widehat{\mathbf{i}}+{u}_{y}\widehat{\mathbf{j}}+{u}_{z}\widehat{\mathbf{k}}$ is the wind-field velocity vector where ${u}_{x}$, ${u}_{y}$, and ${u}_{z}$ are the wind speeds in the x-, y-, and z-directions, respectively, but only the $\widehat{\mathbf{k}}$ remains. Then, to ensure that the analysis does not proceed with too many datapoints for each step, a scale is determined based on the resolution and dimensions of the dataset using

$$ n = \frac{{\min \lfloor {n_{x} ,n_{y} } \rceil }}{c}, $$

where $n$ is the scaling-factor (rounded to the nearest integer) based on the size of the dataset, ${n}_{x}$ is the number of points in x, ${n}_{y}$ is the number of points in y, and $c$ is the scaling parameter (see Appendix 2).

The first $n$ largest curl points (Fig. 6b), [${x}_{\mathrm{c}},{y}_{\mathrm{c}}$], are spatially averaged using

$$\langle {x}_{\mathrm{c}}\rangle =\frac{{\sum }_{i=1}^{n}{x}_{\mathrm{c},i}}{n} ,$$

and

$$\langle {y}_{\mathrm{c}}\rangle =\frac{{\sum }_{i=1}^{n}{y}_{\mathrm{c},i}}{n} .$$

The standard deviation, $\sigma$, of these locations, in $x$ and $y$, respectively, is calculated using

$${\sigma }_{x}=\lceil\sqrt{\frac{\sum_{i=1}^{n}{\left({x}_{\mathrm{c}, i}-\langle {x}_{\mathrm{c}}\rangle \right)}^{2}}{n-1}} \rceil ,$$

and

$${\sigma }_{y}=\lceil\sqrt{\frac{\sum_{i=1}^{n}{\left({y}_{\mathrm{c},i}-\langle {y}_{\mathrm{c}}\rangle \right)}^{2}}{n-1}} \rceil .$$

The values are rounded up to accommodate that the data locations are not continuous and also to take a more conservative selection approach.

Then, a desired number of standard deviations, $s$, from the averaged position is selected such that all of the maximum curl locations outside of the bounds of the ellipse, drawn by the following curve, are excluded:

$$\frac{{\left({x}_{\mathrm{c},i}-\langle {x}_{\mathrm{c}}\rangle \right)}^{2}}{{\left({\sigma }_{x}\right)}^{2}}+\frac{{\left({y}_{\mathrm{c},i}-\langle {y}_{\mathrm{c}}\rangle \right)}^{2}}{{\left({\sigma }_{y}\right)}^{2}}\le s .$$

The remaining number of positions, $N$, will be used as the starting point of lines drawn perpendicular to the vector direction of the datapoint at each location (Fig. 6c). The location of each intersection, [${x}_{\mathrm{t}},{y}_{\mathrm{t}}$], between each line is computed using

$${x}_{\mathrm{t}}=\frac{{b}_{j}-{b}_{i}}{{m}_{i}-{m}_{j}} ,$$

and

$${y}_{\mathrm{t}}=\frac{{m}_{j}{b}_{i}-{m}_{i}{b}_{j}}{{m}_{j}-{m}_{i}} ,$$

where $b$ is the $y$-intercept of the lines with the domain ordinate, $m$ is the slope of the lines, and i and j are referring to the indexing of each line being compared such that i varies from 1 to $N-1$ and j from $i+1$ to N (Fig. 6d). The definition of $N$ is modified to encompass only the intersections that are located within the domain range. The location of [$\langle {x}_{\mathrm{t}}\rangle ,\langle {y}_{\mathrm{t}}\rangle$] is then computed using

$$\langle {x}_{\mathrm{t}}\rangle =\frac{{\sum }_{i=1}^{N}{x}_{\mathrm{t},i}}{N} ,$$

and

$$\langle {y}_{\mathrm{t}}\rangle =\frac{{\sum }_{i=1}^{N}{y}_{\mathrm{t},i}}{N} .$$

The standard deviation, $\sigma$, of these locations, in $x$ and $y$, respectively, is calculated using

$${\sigma }_{x \mathrm{t}}=\lceil\sqrt{\frac{\sum_{i=1}^{N}{\left({x}_{\mathrm{t}, i}-\langle {x}_{\mathrm{t}}\rangle \right)}^{2}}{N-1}} \rceil ,$$

and

$${\sigma }_{y \mathrm{t}}=\lceil\sqrt{\frac{\sum_{i=1}^{N}{\left({y}_{\mathrm{t},i}-\langle {y}_{\mathrm{t}}\rangle \right)}^{2}}{N-1}} \rceil .$$

Then, a desired number of standard deviations, $s$ (its value may differ from that previously used), from the averaged position is selected such that all of the intersections outside of the bounds of the ellipse, drawn by the following curve, are excluded:

$$\frac{{\left({x}_{\mathrm{t},i}-\langle {x}_{\mathrm{c}}\rangle \right)}^{2}}{{\left({\sigma }_{x \mathrm{t}}\right)}^{2}}+\frac{{\left({y}_{\mathrm{t},i}-\langle {y}_{\mathrm{t}}\rangle \right)}^{2}}{{\left({\sigma }_{y \mathrm{t}}\right)}^{2}}\le s .$$

From the remaining number of intersections, $I$, an average location is computed using

$$\langle {x}_{\mathrm{I}}\rangle =\frac{{\sum }_{i=1}^{I}{x}_{\mathrm{t},i}}{I} ,$$

and

$$\langle {y}_{\mathrm{I}}\rangle =\frac{{\sum }_{i=1}^{I}{y}_{\mathrm{t},i}}{I} .$$

This final location (Fig. 6e) should be a valid position and also sufficiently far from the domain edge such that the subsequent centre search does not attempt to collect data from outside of the domain. In the event that this does occur, the maximum $z$-vorticity or minimum pressure location can be substituted in place of [$\langle {x}_{\mathrm{I}}\rangle ,\langle {y}_{\mathrm{I}}\rangle$].

The nearest indexed position to [$\langle {x}_{\mathrm{I}}\rangle ,\langle {y}_{\mathrm{I}}\rangle$] is used as the centre of a grid of size $c$-by-$c$ that will be used to identify the tornado centre, defined where there is zero horizontal, storm-relative velocity. The location of minimum velocity of these points is identified and, unless it is actually the tornado centre, is then used as the centre of a new smaller grid of size three-by-three. Each possible combination of loop pattern is identified so that there can be four quadrants made up of four positions in the new grid, four quadrants of six positions in the new grid, and a large loop around the outside of the new grid (see Fig. 7). The values of the signs of the vectors, $S$, in each loop are summated so that only the loops where this sum is equal to zero are noted as being the possible location of the tornado centre:

$$\sum_{i=1}^{n}{S}_{i}=0 .$$

The sum of the magnitude of the tangential velocity vector, ${u}_{t}$, around each of these identified loops is calculated so that the one with the minimum $\left|{u}_{t}\right|$ sum is taken to be the one with the tornado centre (Fig. 6f):

$$\mathrm{min}\sum_{i=1}^{n}{\left|{u}_{t}\right|}_{i} .$$

If the smaller grid is still too large, the above steps are applied iteratively until the grid is sufficiently small. If there is no path successfully identified, then the above steps are repeated, but the previously identified minimum velocity vector is skipped. The number of repetitions should be limited to a total of $c$. Should this still fail to yield a path, the point used for the final averaged intersection location may be substituted as the tornado centre.

The four corners of the loop found above are bilinearly interpolated for the position where ${u}_{x}=0$ and ${u}_{y}=0$ (see Fig. 10 a), unless no path is found in which case this interpolation step is omitted. These corner points are arranged such that $x\left(1\right)=x\left(3\right)$, $x\left(2\right)=x\left(4\right)$, $y\left(1\right)=y\left(2\right)$, and $y\left(3\right)=y\left(4\right)$ and such that $x\left(1\right)<x(2)$ and $y\left(1\right)>y(3)$ (Fig. 10b). The process of the bilinear interpolation is given for the first step in finding the interpolation of the x-position between points 1 and 2 by

$${x}_{12}=x\left(1\right)+\left[\frac{0-{u}_{y,1}}{{u}_{y,2}-{u}_{y,1}}\right]\left(x\left(2\right)-x\left(1\right)\right) ,$$

and 3 and 4 by

$${x}_{34}=x\left(3\right)+\left[\frac{0-{u}_{y,3}}{{u}_{y,4}-{u}_{y,3}}\right]\left(x\left(4\right)-x\left(3\right)\right) ,$$

for zero velocity in the y-direction, and then the interpolation of the y-position between points 1 and 3 by

$${y}_{13}=y\left(1\right)+\left[\frac{0-{u}_{x,1}}{{u}_{x,3}-{u}_{x,1}}\right]\left(y\left(3\right)-y\left(1\right)\right) ,$$

and 2 and 4 by

$${y}_{24}=y\left(2\right)+\left[\frac{0-{u}_{x,2}}{{u}_{x,4}-{u}_{x,2}}\right]\left(y\left(4\right)-y\left(2\right)\right) ,$$

for zero velocity in the x-direction (Fig. 10c). Next, the velocities of these interpolated points are calculated for use in the final interpolation of the position of zero tangential velocity that identifies the centre of the tornado. The velocity in the x-direction between points 1 and 2 is interpolated using

$${u}_{x,12}={u}_{x,1}+\left[\frac{{x}_{12}-x\left(1\right)}{x(2)-x(1)}\right]\left({u}_{x,2}-{u}_{x,1}\right) ,$$

then in the x-direction between points 3 and 4 using

$${u}_{x,34}={u}_{x,3}+\left[\frac{{x}_{34}-x\left(3\right)}{x(4)-x(3)}\right]\left({u}_{x,4}-{u}_{x,3}\right) ,$$

in the y-direction between points 1 and 3 using

$${u}_{y,13}={u}_{y,1}+\left[\frac{{y}_{13}-y\left(1\right)}{y(2)-y(1)}\right]\left({u}_{y,3}-{u}_{y,1}\right) ,$$

and finally, in the y-direction between points 2 and 4 using (Fig. 10d)

$${u}_{y,24}={u}_{y,2}+\left[\frac{{y}_{24}-y\left(2\right)}{y(4)-y(2)}\right]\left({u}_{y,4}-{u}_{y,2}\right) .$$

Fig. 10

Finding the tornado centre using artificial data for illustrative purposes as an example. a Schematic example of tornado dataset with vectors (shown as black arrows) on an isotropic grid (shown as dashed lines), b vector components ($x$ in green and $y$ in blue), c location of zero velocity (between $x$ components in green and $y$ components in blue), d interpolated vector components, e location of zero horizontal velocity (shown as red circle), and f tornado centre location

Hence, the x and y positions of zero tangential velocity between these calculated velocities can be interpolated to identify the centre of the tornado (Fig. 10e)

$$x={x}_{12}+\left[\frac{0-{u}_{x,12}}{{u}_{x,34}-{u}_{x,12}}\right]\left({x}_{34}-{x}_{12}\right) ,$$

and

$$y={y}_{13}+\left[\frac{0-{u}_{y,13}}{{u}_{y,24}-{u}_{y,13}}\right]\left({y}_{24}-{y}_{13}\right) ,$$

which concludes the searching portion of the algorithm (Fig. 10f).

Appendix 2: Comparison of Scale Factor, c

To test the effects of changing the number of points selected in the maximum curl identification process, c was varied from 1 to 10, as an integer only for simplicity, and, based on trial-and-error, was found to be an appropriate range. Although the overall effect of changing this value appears to be minimal, selecting an extreme value in this range will quickly deteriorate the accuracy of the centre detection capabilities of the proposed method and increase the chance of another smaller, less-defined vortex being included in the search (see Table

Table 3 Comparison of the number of times the programme failed to find a loop about a centre and/or interpolate for a centre using different c values

3).

An optimum value of c = 6 was determined for the current data based on the least number of errors and re-attempts, even though a non-integer value may be optimal. The paths determined for each different c-value can be seen in Fig. 

Fig. 11

Comparing the mode of identified tornado centre points, found by varying c, with the failed results of the different c values used. The data shown are taken relative to the simulation domain for all timesteps at height z = 15 m a.g.l

11. Adjusting c appears to only affect the searching algorithm negatively at either end of the range selected. This is because it begins to either exclude so many points that almost none are left for the analysis or, at the other extreme, include so many points that the selection criterion of maximum curl becomes meaningless. This scaling factor acts as a tool to eliminate the many data points identified as having a high curl but that only represent partially formed or small vortices and would otherwise skew the centre locating of the program. It should only need to be adjusted once by the user for a given dataset before proceeding with further analyses.

The searches performed neglected the results of the search from each preceding timestep, resulting in errors introduced by sweeping through the entire domain unnecessarily. Given the high spatial and temporal resolution of the dataset used in this paper, the search domain after the centre is successfully located the first time may be reduced to a very small region and carried through to each subsequent timestep processed.