Subtropical/polar jet influence on Plains and Southeast tornado outbreaks

Abstract

While extensive research consideration has been given to the Northern Hemispheric polar (PJ) and subtropical jet (STJ) streams, there have been fewer climatological studies relating these two jet types to tornado outbreaks. This study examines tornado outbreaks in two regions with substantial tornado risk, Plains Tornado Alley (PTA) and Southeast Tornado Alley (STA), and classifies the jet streak types associated with the outbreaks. Utilizing the Storm Prediction Center (SPC) tornado database and an objective jet identification scheme created from NCEP/NCAR Reanalysis 1 data, jet streaks were identified as STJ, PJ, merged (identified as STJ and PJ), superposed, or unidentified for a 30-year period between 1984 and 2013. Tornado outbreaks were categorized into different types based on these jet streak types. Results revealed STJ and PJ tornado outbreaks compose the majority of tornado outbreaks, as well as the most intense outbreaks, in both PTA and STA. STJ tornado outbreaks were found to be more common in PTA than in STA, while PJ outbreaks were more common in STA than in PTA. The study concludes by considering how a coupled jet structure may be important for tornado outbreaks.

Appendix

1.1 Weighted geographical centroid and weighted average angle

To create the plots in Figs. 4, 5, 6 and 7, calculations for average position and average angle were necessary. To represent average position, the geographical centroid between two or more points on earth were calculated under the assumption the earth is a perfect sphere. Taking the latitude and longitude of point n, one can convert to Cartesian coordinates and find the weighted average of the x, y, and z coordinates via the following:

$$ x_{\text{avg}} = \frac{{\sum\nolimits_{k = 1}^{n} {\left[ {R_{\text{E}} \cos (lat_{n} )\cos (lon_{n} )w_{n} } \right]} }}{{\sum\nolimits_{k = 1}^{n} {w_{n} } }}, $$

(2)

$$ y_{\text{avg}} = \frac{{\sum\nolimits_{k = 1}^{n} {\left[ {R_{\text{E}} \cos (lat_{n} )\sin (lon_{n} )w_{n} } \right]} }}{{\sum\nolimits_{k = 1}^{n} {w_{n} } }}, $$

(3)

$$ z_{\text{avg}} = \frac{{\sum\nolimits_{k = 1}^{n} {\left[ {R_{\text{E}} \sin (lat_{n} )w_{n} } \right]} }}{{\sum\nolimits_{k = 1}^{n} {w_{n} } }}, $$

(4)

where x_avg, y_avg, and z_avg are the weighted averages of the x, y, and z coordinates, lat_n is the latitude of point n in radians, lon_n is the longitude of point n in radians, R_E is the radius of the earth, and w_n is a weighting. If there is equal weighting between points, the weighting is equal to 1. The value of R_E is not important for the next calculation as long as it is positive and nonzero, and may be omitted. To convert from Cartesian coordinates back to spherical coordinates, the atan2(y, x) function is used:

$$ lat_{\text{cent}} = {\text{atan}}2(z_{\text{avg}} ,\sqrt {x_{\text{avg}}^{2} + y_{\text{avg}}^{2} } ) \cdot \frac{180}{\pi }, $$

(5)

$$ lon_{\text{cent}} = {\text{atan}}2(y_{\text{avg}} ,z_{\text{avg}} ) \cdot \frac{180}{\pi }, $$

(6)

where lat_cent and lon_cent are the latitude and longitude of the geographical centroid in degrees and the two-argument atan2 function uses the signs of both arguments in order to place the angle in the appropriate quadrant. While the atan2 function has a range of (− π, π], the one-argument arctangent function has a limited range of (− π/2, π/2) and thus cannot make a distinction between opposite angles such as π/4 and 3π/4.

The weighted average angle θ_avg in degrees was also calculated for two or more angles θ_n in radians using the atan2 function as follows:

$$ \theta_{avg} = {\text{atan2}}\left( {\sum\limits_{k = 1}^{n} {\left[ {{ \sin }\left( {\theta_{\text{n}} } \right)w_{n} } \right],\sum\limits_{k = 1}^{n} {\left[ {\cos \left( {\theta_{n} } \right)w_{n} } \right]} } } \right) \cdot \frac{180}{\pi }, $$

(7)

where w_n is a weighting. This formula is the result of converting each angle θ_n from polar to Cartesian coordinates, performing a weighted average of the Cartesian components of each point, and then converting back to polar coordinates. One may notice both the summation of weighted x and y components should be divided by n to calculate the actual weighted mean, but this has been omitted for the same reason R_E is able to be omitted in (2), (3), and (4); if the scaling of both arguments for the atan2 function is equal, it has no influence on the final result.

1.2 Plotting geographical coordinates on a Cartesian grid

To plot the location of each jet streak maximum on a Cartesian grid for Figs. 4, 5, 6 and 7, the geographical coordinates of the jet streaks were converted to polar coordinates, then to Cartesian coordinates, and scaled appropriately. To get to polar coordinates, the distance and angle between the outbreak centroid and jet streak maxima were calculated. Since the earth is assumed to be a perfect spear, the shortest distance between two points on a sphere must be calculated, known as the great-circle distance. To calculate the great-circle distance between the outbreak centroid and jet streak maxima, consider two points A and B on the surface of a sphere (Fig. 8). The blue arc between points A and B represents the great-circle distance. Without simplification, unit vector A can be expressed as $\left( {\cos (lat1)\cos (lon1), \, \sin (lat1)\cos (lon1), \, \sin (lat1)} \right)$ in Cartesian coordinates, and unit vector B can be expressed similarly in terms of lat2 and lon2. The central angle $\alpha$ between unit vectors A and B can be found by the dot product of these vectors: $\cos (\alpha ) = {\vec{\text{A}}} \cdot {\vec{\text{B}}} = \cos (lat1)\cos (lat2)\left[ {\cos (lon1)\cos (lon2) + \sin (lon1)\sin (lon2)} \right]$ + $\sin (lat1)\sin (lat2) = \cos (lat1)\cos (lat2)\cos (lon1 - lon2) + \sin (lat1)\sin (lat2)$. Thus, the great-circle distance is

$$ D = R \times \arccos (\sin (lat1)\sin (lat2) + \cos (lat1)\cos (lat2)\cos (lon1 - lon2), $$

(8)

where D is the great-circle distance in km, R is an estimate of earth’s ellipsoidal quadratic mean radius (6373 km), lat1 and lon1 are the latitude and longitude of the jet streak maximum converted from degrees to radians, and lat2 and lon2 are the latitude and longitude of the respective tornado outbreak centroid converted from degrees to radians.

Fig. 8

A geometric figure to aid the description of how great-circle distance and initial bearing is calculated, which is used in plotting Figs. 4, 5, 6 and 7

Due to the angle or bearing changing along the path of a great circle, the final bearing will differ from the initial bearing. To calculate the average bearing, first the initial bearing was calculated as:

$$ \varphi_{i} = \bmod \left( {{\text{atan2}}(y,x) \cdot \frac{180}{\pi }, \, 360} \right), $$

(9)

where $\varphi_{i}$ is the initial bearing, $y \, = {\text{ cos(lat2)}}\sin (lon2 \, - \, lon1)$, and $x = \cos (lat1)\sin (lat2) - \sin (lat1)\cos (lat2)\cos (lon2 - lon1)$. The function mod is the usual modulo operation, and the atan2 function is the same as the one previously used in Sect. 2e. To explain the calculation of initial bearing in more detail, consider two points A and B on a unit sphere (Fig. 8). To simplify the derivation, point A has been chosen to have no y-component and its corresponding unit vector A can be expressed as $\left( {\cos (lat1), \, 0, \, \sin (lat1)} \right)$ in Cartesian coordinates; unit vector B is $\left( {\cos (lat2)\cos (lon2 - lon1), \, \cos (lat2)\sin (lon2 - lon1), \, \sin (lat2)} \right)$, and unit vector k is $\left( {0,{ 0, 1}} \right)$ in Cartesian coordinates. The angle between the plane parallel to vectors A and k (parallel to the green ellipse in Fig. 8) and the plane parallel to vectors A and B (parallel to the blue ellipse in Fig. 8) is the initial bearing $\varphi_{i}$. Thus, $\varphi_{i}$ is also the angle between cross products k × A and B × A. With k × A resulting in vector $\left( {0, \, \cos (lat2), \, 0} \right)$ and B × A resulting in vector $\left( {\sin (lat1)\cos (lat2)\sin (lon2 - lon1),} \right.$ $\cos (lat1)\sin (lat2) - \sin (lat1)\cos (lat2)\cos (lon2 - lon1)$, $\left. { - \cos (lat1)\cos (lat2)\sin (lon2 - lon1)} \right)$, the tangent of the angle between vectors k × A and B × A is $\frac{{\sqrt {\left[ {\sin (lat1)\cos (lat2)\sin (lon2 - lon1)} \right]^{2} + \left[ { - \cos (lat1)\cos (lat2)\sin (lon2 - lon1)} \right]^{2} } }}{\cos (lat1)\sin (lat2) - \sin (lat1)\cos (lat2)\cos (lon2 - lon1)}$. The numerator can be simplified as $\sqrt {\cos^{2} (lat2)\sin^{2} (lon2 - lon1)} \cdot \sqrt {\sin^{2} (lat1) + \cos^{2} (lat1)}$, which is equal to $\cos (lat1)\sin (lon2 - lon1)$. In (9), arctan2 of this expression was used to calculate the initial bearing, and then the modulo function was used to yield a result between 0° and 360°. To calculate the final bearing $\varphi_{f}$, (9) was used, except lat1 and lon1 were, respectively, exchanged with lat2 and lon2, and the angle was reversed by adding 180°. The modulo function mod($\varphi_{f}$, 360) was used so $\varphi_{f}$ would be between 0° and 360°. To get the average bearing, $\varphi_{i}$ and $\varphi_{f}$ were averaged using (7), and this result was used as the angle between the jet streak maximum and tornado outbreak centroid. To plot on a Cartesian grid, both the calculated great-circle distance and average bearing were used to convert from polar coordinates to Cartesian coordinates, and distance was scaled appropriately.