1. Introduction
The skill of a numerical weather prediction (NWP) forecast is dependent on the flow regime (e.g., Palmer 1988), all else being equal. In chaotic flow, even the smallest change in initial conditions (ICs) can grow quasi exponentially, and ultimately yields a horizon of predictability relative to the scale of motion (Lorenz 1969; Palmer et al. 2014). Locally, predictability is destroyed quickly by moist convection (Zhang et al. 2002), potentially through the mechanism of tipping points (e.g., parcels reaching the level of free convection) that accentuate sensitivity to ICs (the so-called butterfly effect; Lorenz 1995). Furthermore, as limited-area-model forecasts are run at a higher resolution, the ratio of the vertical depth to the horizontal grid spacing of the model becomes larger: the elliptical dimension increases (Schertzer and Lovejoy 1985). In this circumstance, error growth is not confined to a quasi-2D field, and has a relative greater degree of freedom in the vertical. As such, the greater the elliptical dimension, the more detrimental small errors in time and space become to the forecast (Tennekes 1978).
Nonlinear dynamical systems across many disciplines may demonstrate extreme sensitivity to ICs, giving rise to tipping points (e.g., Gleick 1987; Coffer and Parker 2018). For example, consider an air parcel almost at its level of free convection. At this tipping point, an infinitesimal change in ICs of the parcel’s potential temperature or specific humidity is enough to nudge the atmosphere from one regime (no thunderstorm) to another (thunderstorm). Thunderstorm development then triggers phenomena such as gravity waves (Curry and Murty 1974), surface pressure changes (Markowski and Richardson 2010, 218–220), and upscale growth through cell mergers (e.g., Burke and Schultz 2004). The “no thunderstorm–thunderstorm” bifurcation results in a binary choice of solutions, or basins of attraction separated by a sharp divide. This is analogous to standing on a mountain peak along the Continental Divide of the North American Rockies and pouring water on one side. A small change in position will mean the water will end up in a different ocean. We can likewise conceptualize thunderstorm development in a “landscape” where basins of attraction (like watersheds; Grebogi et al. 1987; Nusse and Yorke 1996) are the thunderstorm’s mode, intensity, or any characteristic; this approach is demonstrated herein.
The application of theoretical predictability to operational NWP has yielded great insight into potential forecast-skill limits: for example, those associated with low-frequency variability (Palmer 1988), extratropical cyclones (Zhang et al. 2002; McMurdie and Ancell 2014), mesoscale convective systems (Wandishin et al. 2008, 2010; Rodwell et al. 2013; Durran and Weyn 2016; Lillo and Parsons 2017), supercells (Cintineo and Stensrud 2013; Flora et al. 2018), tornadogenesis (Zhang et al. 2016; Coffer et al. 2017), cloud-resolving models (Hohenegger et al. 2006), and so on. An estimate of maximum skill for a given phenomenon or regime, while impossible to diagnose precisely, can normalize probabilistic estimates for forecasters. A large ensemble spread in a less predictable regime may not be poor performance from the modeling system itself, but a fair reflection of the uncertainty given observational error. In fact, if forecasters and researchers can identify the most unpredictable scenarios, where the most value is gained from small improvements in IC estimates (e.g., in situ observations, satellites, radar assimilation), targeted observations (e.g., Ancell and Hakim 2007), or stochastic perturbations (e.g., Buizza et al. 1999), forecasts could be substantially improved if the model is nudged into the correct “watershed” through adjustment of the forecast’s trajectory through phase space (as in Leroy and Rodwell 2014; Berner et al. 2017). Of course, the correct watershed is not known a priori, but a sufficient sampling of phase space by the ensemble members is defined by this exploration of potential basins of attraction.
Hence, the variation of this theoretical predictability is sensitive to the large-scale environment. Small scales may or may not inherit extended predictability from larger scales (Durran and Weyn 2016), but large-scale flow does constrain the convective evolution of a thunderstorm (Weisman and Klemp 1982, hereafter WK82). In WK82, the authors tested the sensitivity of thunderstorm evolution to one definition of buoyancy (related to surface mixing ratio
The increase in computer power since WK82 now allows greater granularity with which to test sensitivity of thunderstorm evolution to buoyancy and shear. Herein, the methodology of WK82 is extended to construct a “landscape” of attraction basins as a function of atmospheric-profile ICs (outlined in section 2). In section 3, various storm diagnostics are analyzed for their variation in buoyancy–shear (phase) space, and the implications on predictability are synthesized and summarized in sections 4 and 5, respectively.
2. Methodology
a. Numerical model configuration
All simulations were performed with the Weather Research and Forecasting (WRF) Model (Powers et al. 2017), version 3.7, in idealized mode (em_quarter_ss). However, all vertical profiles herein imply a straight hodograph (i.e., Fig. 1), rather than one that is curved (as in the eponymous, default quarter-circle hodograph). Two matrices of runs were generated: one using the NSSL two-moment microphysics (Mansell et al. 2010), and the other with Morrison two moment (Morrison et al. 2009), to test for sensitivity of the results to microphysical scheme. In the absence of a cumulus parameterization, error from the microphysics parameterization is expected to dominate other parameterized processes (Jankov et al. 2005). Two two-moment schemes were chosen because of their increased realism over single-moment schemes (Igel et al. 2015). All other model options were left as default. Herein, we use an otherwise-identical model setup for each microphysics scheme to primarily detect sensitivity to ICs, rather than model error, as is the scope of the present study. However, this is not to diminish the importance of model uncertainty on thunderstorm forecasts (Stensrud et al. 2000; Lawson and Gallus 2016a), or errors stemming from truncation (Berner et al. 2012), numerical diffusion (Knievel et al. 2007), etc.
Skew T–logp graph showing atmospheric profiles of humidity, temperature, and wind speed and how these variables change across phase space.
Citation: Journal of the Atmospheric Sciences 76, 9; 10.1175/JAS-D-18-0218.1
Definitions of low, middle, high, and extreme levels of CAPE, qυ0, and Us used in the manuscript. The qυ0 ranges are estimated from the CAPE– qυ0 relationship (see y axes of Fig. 4, etc.). Informed by U.S. climatology for severe and tornadic weather events in Grams et al. (2012, their Figs. 14a, 15a, 16a, 17a).
Each idealized simulation was run for 2 h, though 50 runs (6%; mainly in the extreme-
The present phase-space exploration differs from WK82 primarily by increasing the granularity within buoyancy–shear space. WK82 used data from 60 simulations with 1.0 g kg−1 steps in buoyancy and 5 m s−1 steps in shear. Their simulations used 2-km horizontal grid spacing using the Klemp–Wilhelmson model with Chen modifications, and one microphysics parameterization (a Kessler-like warm-rain scheme). In contrast, 780 runs per microphysics scheme were performed herein, with two schemes; environmental profiles were varied in steps of 0.2 g kg−1 in
b. Predictability and uncertainty
At this point, we differentiate between two concepts of predictability; however, the sense of predictability is not always explicitly stated in the following sections when the implication applies to both senses. First, a point-by-point numerical verification assesses traditional or gridpoint predictability, the form primarily discussed by Lorenz (1963). In this framework, predictability is lost at a given point in time when the error between a pair of similar forecasts approaches the climatological error: the so-called predictability horizon (Palmer et al. 2014). Forecast evaluation metrics typically used on the synoptic scale, such as root-mean-square error, verify smooth fields with no explicit tolerance for temporal and spatial phase errors. However, such traditional scores may grossly underestimate the predictability horizon for events on smaller scales (e.g., Reinecke and Durran 2009), as error saturation is reached quickly, and phase errors of objects (e.g., thunderstorms) grow larger as a fraction of the object’s size. The implication of gridpoint error saturation is that all useful prognostic information has been extracted as the horizon is reached, when in fact useful information is still available to the forecaster (contingent on relevant filtering and/or statistical processing, either explicitly, or implicitly performed as part of the forecast process).
In response, recent verification scores and schemes have been developed that include tolerance for temporal and/or spatial phase errors (such as the fractions skill score; Roberts and Lean 2008), or that reduce a heterogeneous field to objects, such as the structure–amplitude–location method (Wernli et al. 2008) or the methodology recently applied in Skinner et al. (2018). These schemes avoid a double penalty in time and space for fields associated with discrete phenomena (Gilleland et al. 2009, 2010), and address a predictability that has been extended through filtering (as reviewed in Williams 1997). This is referred to herein as object-based predictability.
Schematic diagram outlining the heat map and uncertainty figures in the present paper for a given microphysics parameterization scheme. This methodology is repeated for each nine-pixel window in the phase-space matrix. Before plotting, the σ-uncertainty field is smoothed, or SAL-component standard deviation is computed, as discussed in text.
Citation: Journal of the Atmospheric Sciences 76, 9; 10.1175/JAS-D-18-0218.1
To complement this energy-based approach, object-based predictability is measured by computing the standard deviation of structure–amplitude–location (SAL) components (Wernli et al. 2008), which has been adapted for radar reflectivity (Lawson and Gallus 2016b). In short, SAL identifies objects in two fields; the two fields are often simulation-versus-observation comparisons, but herein we analyze adjacent idealized simulations only. The SAL method decomposes differences between these fields into three components, first by masking below a given threshold (here, 5 dBZ), and only identifying objects that surpass a pixel threshold (here, five pixels, which is approximately the minimum effective resolution). Results for real-world verification are sensitive to choice of threshold and footprint (Lawson and Gallus 2016b). However, in the present paper, the knowledge that a maximum of one storm can be generated reduces much of the spurious signal (Wernli et al. 2008), and preliminary results were not substantially affected by the choice of settings. The SAL components are
structure S (between −2 and +2), which considers the object size and peakedness/flatness, such that a positive S value for a given simulation may indicate a flatter (i.e., more stratiform) character of composite reflectivity compared to the control;
amplitude A (between −2 and +2), which measures a normalized domainwide accumulation, such that a positive A value indicates a positive reflectivity bias; and
location L (between 0 and +2), a sum of Euclidean differences between 1) object centers of mass, weighted by their integrated quantity, and 2) domainwide center of mass. Hence, a larger L component represents a larger distance error between in identified objects. No object matching is required, in contrast to the methodology of Skinner et al. (2018) and others.
SAL is deployed herein to evaluate reflectivity at 1 km above ground level (AGL) between a given simulation and the eight most similar initial-profile simulations (i.e., the eight cells surrounding the given location in the matrix of phase space). As with DKE, the runs around the perimeter of phase space are neglected, as a full nine-pixel computation cannot be performed, and are represented by black pixels in the heat map. After SAL is computed (Fig. 2), the standard deviation of each SAL component’s phase space is applied (over the same 3 × 3 windows, with no smoothing). In contrast to other figures of uncertainty, we show this standard deviation as the diagnostic variable (i.e., the pixel colors), to highlight potential tipping points in phase space. The standard deviation of SAL is used (rather than mean values) to highlight the variation of component differences between a given cell (a control) and its neighbors in phase space. Use of the mean would smooth out cancelling negative and positive errors, and mistakenly emphasize regions of consistently high (but similar types of) error.
The uncertainty of other diagnostic fields, shown in heat-map figures herein, is computed with a moving 3 × 3–window standard deviation operation. This standard deviation field is then passed through a two-sigma Gaussian smoother to remove small variations (i.e., noise) on the order of differences between experiments with different microphysics schemes. The resulting field is hereby termed σ uncertainty, where a subscript will indicate the field in question (e.g.,
c. Other diagnostics
Storm speed is estimated for a given time by first identifying the 500-hPa updraft maximum 10 min before and after the given time. A storm is deemed present if the location of this updraft corresponds to simulated composite reflectivity greater than 25 dBZ. If no storm is detected, this cell is ignored (and represented by a black pixel in heat maps); otherwise, the distance traveled is computed via the Pythagorean relationship, and converted to meters per second.
3. Results: Diagnostic heat maps
In the following sections, heat maps are presented in the manner outlined schematically in Fig. 2, with the σ uncertainty of each field contoured in black (other than Figs. 12 and 15). As each simulation domain is symmetrical along the y-axis center, only one-half of the domain data were considered during statistical processing to reduce the computational expense, as in WK82. The differences in diagnostics stemming from using Morrison or NSSL two-moment microphysics scheme were close to universally negligible. Almost all simulation differences were within 5% of the pair’s mean; for example, differences in 60-min
Differences in 60-min storm strength (also shown in Fig. 11b) between the two microphysics schemes in (a) absolute and (b) relative terms of the interscheme mean for that cell. Black pixels in (a) mark simulations that crashed early in either microphysics scheme. The y axis in (b) indicates histogram-bin count (percentage categories as labeled on the x axis).
Citation: Journal of the Atmospheric Sciences 76, 9; 10.1175/JAS-D-18-0218.1
a. Wind and rotation fields
The qυ0–Us phase space for maximum near-surface (~200 m AGL; Fig. 4) wind suggests that wind speeds potentially dangerous to aircraft, structures, etc. (>25 m s−1; Fujita 1990) occur in high to extreme values of CAPE, as found by Grams et al. (2012). Further, within this subregion of phase space, there are pockets of subsevere near-surface wind associated with Us ~ 8 and ~22–30 m s−1. This intermittency across phase space is likely the fingerprint of chaos: nonlinear, abrupt changes in sensible weather from small, linear increases in shear. Two local maxima exist in the uncertainty field
Heat map in qυ0–Us phase space, where each colored pixel represents the maximum lowest-level wind speed at any point or time for that given simulation, for the Morrison microphysics scheme. The darker the color, the stronger the wind, as denoted by the legend (m s−1). Note the color scheme is normalized for each time’s extrema to accentuate areas of largest uncertainty per time. Black lines contour σ uncertainty, which measures sensitivity to small changes in the initial environmental profile, and is contoured every 0.2 m s−1. Local maxima and minima in σ uncertainty are denoted by X and N symbols, respectively.
Citation: Journal of the Atmospheric Sciences 76, 9; 10.1175/JAS-D-18-0218.1
When we inspect the diagnostics for maximum updraft (Fig. 5a) and downdraft (Fig. 5b) speeds, we see a U-shaped lobe of higher (>15 m s−1) updraft speed at larger qυ0 and Us values. In updraft data (Fig. 5a), the two σ-uncertainty maxima at 1) low- to midshear, low- to midbuoyancy, and 2) extreme-shear, low-buoyancy regimes are shifted toward lower-buoyancy phase space, when contrasted with
As in Fig. 4, but for (a) maximum updraft and (b) maximum downdraft (m s−1) at any level, point, or time. The σ-uncertainty field is contoured every 0.4 m s−1 in (a) and 0.05 m s−1 in (b).
Citation: Journal of the Atmospheric Sciences 76, 9; 10.1175/JAS-D-18-0218.1
Next shown in Fig. 6 are heat maps of low- and midlevel vertical relative vorticity (at ~925 and 500 hPa, respectively), hereby termed
As in Fig. 4, but for maximum relative vertical vorticity (×10−3 s−1) at (a) ~925 and (b) ~500 hPa for any point or time. The σ-uncertainty field is contoured every 0.05 × 10−3 s−1 in (a) and 0.025 × 10−3 s−1 in (b).
Citation: Journal of the Atmospheric Sciences 76, 9; 10.1175/JAS-D-18-0218.1
The maximum in updraft helicity across phase space, as a product of vertical motion and rotation, is likely to occur near the intersection of maxima in Figs. 5 and 6, where the author assumes that strong updrafts collocate with the mesocyclone. This relationship is found in Fig. 7b, where the largest
As in Fig. 4, but for maximum updraft helicity (m2 s−2) for any point or time in the (a) 2–5- and (b) 0–2-km layers. The σ-uncertainty field is contoured every 1 m2 s−2 in (a) and 0.1 × 10−3m2 s−2 in (b).
Citation: Journal of the Atmospheric Sciences 76, 9; 10.1175/JAS-D-18-0218.1
b. Precipitation
Figure 8 presents three precipitation-related products: First (Fig. 8a), cubic rainfall accumulation integrated over the entire half domain is presented. As no simulated storm exits their domain boundaries within the 2 h, this product can be interpreted as total precipitation from the thunderstorm in this time. As such, it is unsurprising that Fig. 8a displays a similar U-shaped pattern in both rainfall and contours of uncertainty
As in Fig. 4, but for (a) rainfall integrated over the entire half domain (103 m3), (b) maximum 2-h accumulated rainfall at any point (mm), and (c) maximum 2.5-min accumulation for any point and time (mm). The σ-uncertainty field is contoured every 40 × 103 m3 in (a), 0.2 mm in (b), and 0.05 mm in (c).
Citation: Journal of the Atmospheric Sciences 76, 9; 10.1175/JAS-D-18-0218.1
Last, maximum 2.5-min rainfall accumulation at any point and time—a proxy here for the storm’s instantaneous rainfall rate—shows in Fig. 8c that a far broader area of phase space supports heavy rainfall rates >2 mm per time step (every 2.5 min; or as an hourly rate, 48 mm h−1 or 1.89 in. h−1). The heaviest rates are in low- to midshear and high- to extreme-buoyancy environments, corresponding to the aforementioned pointwise accumulation maximum. This area of phase space is also associated with a maximum in
c. Storm speed, size, strength
Figure 9 presents the speed of detected convective cells at 30, 60, and 90 min of simulation time. The detection methodology is discussed in section 2, and instances where a storm was not detected are represented by a black pixel in Fig. 9. After 30 min (Fig. 9a), only low to midshear and high to extreme CAPE supports detectable cell development. This echoes WK82 (their Fig. 3), where storms decrease in updraft strength as
As in Fig. 4, but for thunderstorm speed (m s−1) at (a) 30, (b) 60, and (c) 90 min. The σ-uncertainty field is contoured every 0.2 m s−1 in (a), 0.3 m s−1 in (b), and 0.5 m s−1 in (c). Thunderstorms were detected if the location of the maximum 500-hPa updraft at a given time corresponded to simulated composite reflectivity greater than 25 dBZ; runs with no detected storms are represented here with black pixels.
Citation: Journal of the Atmospheric Sciences 76, 9; 10.1175/JAS-D-18-0218.1
Equivalent storm diameter is shown in Fig. 10 (the formulation is discussed in section 2). This heat map shows much uncertainty in storm size at 1 h (Fig. 10a), with storms detected4 at almost all regimes away from the lowest
As in Fig. 4, but for equivalent thunderstorm diameter (km) at (a) 60 and (b) 120 min. The σ-uncertainty field is here contoured every 0.4 km in (a) and 0.8 km in (b).
Citation: Journal of the Atmospheric Sciences 76, 9; 10.1175/JAS-D-18-0218.1
To summarize numerous diagnostic concepts in one product, storm strength
As in Fig. 4, but for storm strength (unitless) at (a) 30, (b) 60, and (c) 90 min. The σ-uncertainty field is here contoured every 0.005 in (a), 0.008 in (b), and 0.015 in (c).
Citation: Journal of the Atmospheric Sciences 76, 9; 10.1175/JAS-D-18-0218.1
The discrepancy between profiles with no storm detected in Fig. 9 and those with nonzero storm attributes (e.g., low-buoyancy, low-shear regimes in Figs. 11a,b) is caused by the reflectivity threshold in the storm-detection methodology, during storm-speed estimation. At the lowest
When animated for every output time (not shown), the high variability in low-shear regimes is represented by local maximum–minimum couplets of
4. Predictability synthesis
Contours of σ uncertainty have been shown in the previous section to highlight potential tipping points in phase space. We now turn to two mathematical quantities to determine regions of phase space with low predictability in both the gridpoint and object-based senses.
a. Object-based predictability (SAL)
The following heat maps show standard deviation of each SAL component generated from the simulated composite reflectivity fields. This quantity represents the uncertainty in the nature of object-based differences (see section 2). Again, there is little difference in SAL-component character for both Morrison and NSSL two-moment matrices. Both are presented in Fig. 12 for comparison, but as the differences are negligible, the following discussion arbitrarily addresses the Morrison microphysics scheme.
Heat maps showing uncertainty in SAL components at 2 h of simulation time for the (a)–(c) NSSL two-moment and (d)–(f) Morrison microphysics schemes. Panels show the standard deviation of (a),(d) structure, (b),(e) amplitude, and (c),(f) location components (unitless; the values are functions of domain size and not meaningful here, hence neglected). Red boxes indicate the four cells that are shown in the time series of maximum 500-hPa vertical motion in Fig. 14.
Citation: Journal of the Atmospheric Sciences 76, 9; 10.1175/JAS-D-18-0218.1
The structure (purple; Figs. 12a,d) and amplitude (green; Figs. 12b,e) components trace out U-shaped maxima, resembling the perimeter of the U-shaped lobe in previous heat-map figures. Away from this U-shaped maximum and the low-shear, low-buoyancy maximum in the bottom left, there is little variation in object structure S and domain-integrated reflectivity A. These results suggest that thunderstorm structure and strength, in an object-based sense, are less predictable along the U-shaped perimeter. In contrast, the standard deviation of location component L yields little systematic pattern, though highest variation occurs in the high-shear, mid- to high-buoyancy regime. The lack of U-shaped pattern in L, in contrast to S and A components, is related to the asystematic errors in storm speed (Fig. 9), and suggests that the predictability “field” of, say, storm speed is very different from that associated with other (wind, rainfall, etc.) diagnostics.
As seen in Fig. 11, the U-shaped perimeter appears to discriminate between weakening and strengthening cellular convection. As in WK82, a wide variety of storm strengths can be seen across phase space. To demonstrate whether this U-shape acts as a tipping point, two contrasting thunderstorm life cycles are shown in Fig. 13, taken from two runs adjacent in phase space, and whose only difference in profile is 0.2 g kg−1 (~125 J kg−1 CAPE equivalent). The two runs lie on the perimeter of the U-shaped maximum, marked by a the two left-side pixels within the red box in Fig. 12. Differences between the supercells’ appearance in simulated composite reflectivity become apparent between 80- and 100-min simulation time (Fig. 13), and by 120 min, continued strengthening is particularly evident in the simulation with slightly higher buoyancy (Fig. 13b).
Life cycle of two simulated thunderstorms shown every 20 min in simulated composite reflectivity for the (a) Us = 34 m s−1, qυ0 = 10.8 g kg−1 and (b) Us = 34 m s−1, qυ0 = 11.0 g kg−1 simulations using Morrison microphysics (whose location in phase space is shown by the red box in Fig. 12). The 0- and 20-min panels are omitted because of simulated reflectivity less than 0 dBZ. Scales for both panels are identical; thunderstorm location is centered within each frame.
Citation: Journal of the Atmospheric Sciences 76, 9; 10.1175/JAS-D-18-0218.1
Similarly, Fig. 14 shows maximum vertical motion at model level 13 (around 500 hPa or 5.5 km AGL), for the same two simulations and two additional adjacent simulations that straddle the U-shaped maximum in Fig. 12 (again, at the location of the red box). The time series show the initial explosive growth of the storm cell around 20 min, a gradual decrease in intensity between 40 and 60 min, and then a bifurcation of solutions: the two simulations with slightly larger buoyancy generate pulsating and strengthening updrafts with a period of around 20 min, whereas the slightly lower buoyancy results in a dissipating of the updrafts with somewhat longer (30 min) period of pulsation.
Time series of maximum vertical motion at model level 13 (~500 hPa or ~5 km AGL) for the Morrison microphysics scheme. These four simulations, whose location is marked with a red box in Fig. 12, show a bifurcation between strengthening and dissipating supercells occurring around an hour after initialization. The black vertical line indicates an hour of simulation time. The dashed lines represent storms embedded in atmospheres containing ~100 J kg−1 less CAPE than those denoted by solid lines (which tend to dissipate after 1 h). Also, blue lines indicate storms embedded in flow with a 2 m s−1 weaker 0–6-km wind difference.
Citation: Journal of the Atmospheric Sciences 76, 9; 10.1175/JAS-D-18-0218.1
Keeping wind shear constant, and increasing buoyancy (i.e., moving in the positive y direction), the wavelength and amplitude of the oscillations decrease after 60 min (not shown). Keeping buoyancy constant and increasing or decreasing shear dampens the oscillation and chokes the updraft, moving into the basin of dissipation within phase space. Hence, the U-shaped maxima in S and A components of SAL (Fig. 12) delineate the two basins of attraction in buoyancy–shear space. Away from this perimeter, storm attributes diverge quickly, and very different 2-h forecasts of similar atmospheric states ensue: this is the fingerprint of chaotic growth.
b. Gridpoint predictability (DKE)
In contrast to SAL, DKE measures gridpoint-scale predictability destruction, with no tolerance of small storm-location differences. The unit–time derivative of uncertainty (dDKE) highlights regions in which predictability is being destroyed the fastest. Values of dDKE from the Morrison scheme are presented in Fig. 15. In the first 30 min (Fig. 15a), there is rapid DKE growth in the high-buoyancy, low-shear regime. This appears as a rippling pattern (when animated; not shown) that evokes the pattern observed in the storm strength diagnostic (Fig. 11). By 60 min (Fig. 15b), the highest dDKE is occurring in the high-buoyancy, high-shear regime, associated with the long-lasting supercells occurring here. As seen in the animation (not shown), waves of “predictability destruction” emanate from the area of maximum dDKE, most clearly at 90 min (Fig. 15c). These waves may represent the phase difference between adjacent cells in supercell growth and mesocyclonic cycling with different cycling periods (Burgess et al. 1982; Adlerman et al. 1999). Signals of this pulsation can be seen as sinusoidal waves in 500-hPa vertical motion (Fig. 14). However, the 20–30-min periods observed herein are somewhat shorter than those simulated in Adlerman and Droegemeier (2002, their Fig. 8). From their conclusions on grid spacing, it is also surprising that a 3-km grid could resolve such cycling, though recent work supports such a claim (Britt et al. 2018).
As in Fig. 5, but for dDKE (the time derivative of DKE) for the Morrison microphysics scheme at (a) 30, (b) 60, and (c) 90 min of simulation time. Large dDKE values represent higher rates of gridpoint predictability destruction. Magnitudes of dDKE are not meaningful here, and neglected for clarity.
Citation: Journal of the Atmospheric Sciences 76, 9; 10.1175/JAS-D-18-0218.1
The simulations that support strong supercells in the extreme regimes of shear and buoyancy dominate dDKE magnitude; as a result, the DKE (not shown) and dDKE (Fig. 15) fields look similar beyond around 90 min (not shown). Given dDKE is a proxy for predictability loss, we find that the strongest storms are associated with the fastest traditional (gridpoint) predictability destruction. In some ways, this is expected: if the bifurcation were between no storms at all and supercells, then an ensemble forecast ran in an environment devoid of convection would be very predictable (low ensemble spread) compared to those with strong thunderstorms (high ensemble spread). However, it is generally a counterintuitive result, given that supercells are a more persistent phenomenon than, for example, single cells, and long suspected to contain mechanisms that suppress (object-based) predictability destruction (Lilly 1986). As the storm cycles and strengthens, gridpoint predictability is more efficiently destroyed through chaotic error growth. However, the object-based predictability may be extended in the manner discussed by Lilly (1986): a strong rotating supercell insulates the storm from simply detraining and dissipating. In other words: the thunderstorm’s mode and characteristics are more predictable than its gridpoint evaluation (with zero tolerance.) While Lilly’s ideas relating to helicity and suppression of chaotic error growth was later shown to be overly optimistic (Lilly 1990), his suspicion of extended supercell predictability was confirmed by Droegemeier and Levit (1993), and the mechanism that yields a longer object-based predictability estimate may be related to nonlinear feedback loops (Shen et al. 2018), or continuous propagation on the storm’s flank as a consequence of midlevel mesocyclogenesis and a strengthening low-level updraft [see discussion in Weisman and Rotunno (2000, p. 1454)].
The phase error that may be associated with the waves in dDKE would result in unfair punishment from not only traditional point-to-point verification metrics, but also neighborhood schemes that include windows of location but not time [a review and critique of neighborhood schemes can be found in Schwartz and Sobash (2017)]. Scores such as the extended fractions skill score, published independently by Schwartz et al. (2010) and Duc et al. (2013), apply the windowing over a number of adjacent time steps (as well as considering the ensemble member dimension), and are recommended. It follows that one could subset a climatology such that it were confined to that region of phase space, and use this to more appropriately estimate the upper bound of predictability. In the case of quasi-operational and experimental forecast systems, however (e.g., Wheatley et al. 2015), it begs the question of defining a model climatology with an insufficient sample of archived cases.
5. Conclusions
This study has presented theoretical predictability for idealized cellular convection across a high-granularity phase space of vertical wind shear and buoyancy. After an hour of simulation time, storms can be split into two groups, separated by a U-shaped bifurcation in phase space (Fig. 16). This bifurcation suggests two basins of attraction: one basin at higher buoyancy values, associated with storms that continue strengthening, and another associated with dissipating convection unsupported by the environment, at lower buoyancy values. While it is expected that more CAPE and shear available to a storm will usually increase its strength, this increase does not occur gradually, but abruptly around the tipping point. This suggests ensemble forecasts of isolated thunderstorms in the low- to midbuoyancy regime—for all shear values—may be associated with large uncertainty (i.e., spread) as an inherent character of the regime predictability. Two areas of phase space were frequently sensitive to small IC changes: 1) the low-buoyancy, high-shear regime and 2) low-shear regime at all buoyancies. Both were identified as regimes with less skillful forecasts made by the U.S. Storms Prediction Center (Herman et al. 2018).
Schematic of general results detailed herein. The green curve denotes the U-shaped maximum (e.g., Fig. 12a) that marks the boundary between the strengthening and dissipating solutions (black text). The region of phase space that follows this green curve (and the swath of green for all low-shear values) is associated with lower object-based predictability. In contrast, the orange shading denotes the region of phase space with the maximum dDKE, or traditional predictability destruction, which decreases as one moves away from the top-right corner.
Citation: Journal of the Atmospheric Sciences 76, 9; 10.1175/JAS-D-18-0218.1
To further assess theoretical predictability, a distinction between two senses of predictability destruction was made: (i) one that is measured by DKE growth, and maximized in the largest storms at extreme buoyancy and shear levels; and (ii) another measured by object-based approaches, maximized near tipping points at the discriminator in phase space between strengthening and dissipating storms. The former sense is associated with traditional gridpoint-to-gridpoint forecast error (gridpoint predictability), whereas the latter is associated with error between cellular features, with tolerance for small phase errors (object-based predictability). The latter error is more intuitive when discussing convective-scale forecasts, and tallies with tolerances considered by human forecasters (i.e., when storm mode is more important than a precise location). As such, practical predictability estimates at the convective scale should mimic the forecaster’s mental filtering processes, lest they profoundly underestimate the utility remnant even after gridpoint error saturation has been reached.
Other findings include the following:
There is very little difference between the two double-moment microphysics schemes, which suggests that (i) results are likely applicable to other model configurations, and given that microphysics error is a primary source of model error, (ii) predictability is primarily a function of IC error at thunderstorm time scales.
A rippling pattern in phase space, seen in gridpoint-predictability destruction, may be interpreted as phase offsets of updraft oscillations whose periods change with shear and buoyancy.
As revealed by an object-based decomposition, the structure and amplitude (or strength) of the thunderstorm is profoundly sensitive to shear and buoyancy in regimes of phase space near the U-shaped perimeter; conversely, there is little pattern to storm-location or storm-speed uncertainty in phase space, suggesting that a strong tipping point may not exist for temporal and spatial phase errors in storm forecasts.
The maximum rainfall accumulation occurs at low shear and high to extreme CAPE values, whereas total storm rainfall is highest in extreme buoyancy and shear regimes.
Low-level updraft helicity, a proxy for tornadogenesis, is maximized at the highest
and .
In summary, these results should continue to motivate the use of ensemble forecasting systems at the convective scale. It is paramount that these ensemble systems are reliable in order to fully explore the phase space of extreme and rare events; as such, system designers should consider methods that further account for uncertainty, such as stochastic perturbations (Berner et al. 2017), probabilistic postprocessing (e.g., Bröcker and Smith 2008), and so on.
There are inherent shortcomings of this experiment, not least that idealized studies are not immediately applicable to the observed world. Results above must be interpreted within the scope of the experimental configuration: the prescription of a straight hodograph, the limitation of 3-km horizontal grid spacing, and a fixed parameterization suite (bar microphysics diversity). The imposition of a warm bubble also precludes any statement on convective initiation, cell mergers, and so on; more strongly forced phenomena, such as quasi-linear convective systems, may be partly sheltered from the IC sensitivity presented herein. Conversely, horizontally homogeneous lateral-boundary conditions do not introduce any strong forcing (upper-level troughs) events that may improve predictability in real-world dynamics (Anthes et al. 1985). Furthermore, model error is neglected here other than microphysics diversity. These caveats aside, the signal appears genuine: results herein corroborate observational climatologies, forecast-center verification studies, and the coarser matrices of phase space mapped by WK82.
Future work may repeat the above procedure for a range of sounding types, such as in Kirkpatrick et al. (2007, 2011), or investigate a link between the idealized literature with operational datasets from severe weather forecasting centers.
Acknowledgments
The author thanks two anonymous reviewers, Cody Kirkpatrick, and Robert Fovell for valuable contributions to the manuscript’s clarity and content; Ted Mansell and Harold Brooks for improving earlier versions of the manuscript; Corey Potvin, Timothy DelSole, Louis Wicker, Jeffrey Duda, Jack Kain, Nusrat Yussouf, and William Gallus for useful conversation; and Gerald Creager and Oklahoma University OSCER for computational assistance. The WRF runs contained herein were created, submitted, and data mined by open-source Python packages. These include numpy, matplotlib, scipy, netcdf-python, and the author’s open-source ensemble verification and creation (evac) codebase found online (https://github.com/johnrobertlawson/evac). Funding was provided by NOAA/Office of Oceanic and Atmospheric Research under NOAA–University of Oklahoma Cooperative Agreement NA11OAR4320072, U.S. Department of Commerce.
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