Abstract

Accurate measurements of the convective inhibition (CIN) associated with capping inversions are critical to forecasts of deep convection initiation. The goal of this work is to determine the sounding characteristics most vulnerable to CIN errors arising from hysteresis associated with sensor response and ascent rate of profiling systems. This examination uses 5058 steady-state analytic soundings prescribed using three free parameters that control inversion depth, static stability, and moisture content. A theoretical well-aspirated first-order sensor mounted on a platform that does not disturb its environment is “flown” in these soundings. Sounding characteristics that result in the largest relative CIN errors are also the characteristics that result in the smallest CIN. Because they are more likely to support deep convection initiation, it is particularly critical that environments with small CIN are represented accurately. The relationship between relative CIN error and CIN exists because sounding characteristics that contribute to large CIN do not proportionally increase the CIN error. Analysis also considers CIN intervals with (operationally important) CIN on the threshold between environments that will and will not support deep convection initiation. For these soundings, CIN error is found to be largest for deep, dry inversions characterized by small static stability.

1. Introduction

In the central United States, the canonical capped sounding has an elevated mixed layer (EML; Carlson and Ludlam 1968; Carlson et al. 1983; Banacos and Ekster 2010). The initiation of deep convection depends in part on the strength of the capping inversion. While the presence of weak convective inhibition (CIN) can aid in the severity of storms by limiting the coverage of convection [e.g., the Fawbush and Miller (1954) type 1 or Johns and Doswell (1992) “loaded gun” soundings], the presence of sufficiently strong CIN will prevent deep convection initiation (DCI). In prior decades, operational assessment of capping inversion strength was limited to rawinsonde launches, thus requiring forecasters to estimate inversion strength both spatially (in between sounding sites) and temporally, based on thermal advection, large-scale ascent/descent, with specific methods detailed in Johns and Doswell (1992). One such method utilizes the forecast 700 mb temperature field, where DCI is not expected in areas with T700 > 12°C (Johns and Doswell 1992); however, the profile of temperature in the lower atmosphere and calculated CIN provides a broader perspective on the potential for DCI since the altitude of the capping inversion can vary (Bunkers et al. 2010). A substantial increase in the number of high-resolution weather forecast models [including convection-allowing models (CAMs)] has provided a wealth of information to aid in forecasts for DCI. However, model-forecast CIN can have a fair amount of noise, and errors in model-forecast low-level moisture can lead to substantial error in CIN forecasts (Moller 2001). In fact, uncertainty regarding the distribution of moisture in the boundary layer was one of the motivating factors for the IHOP campaign (Weckwerth and Parsons 2006). This low-level moisture uncertainty manifests itself in forecast CIN errors as early as model initialization (Bunkers et al. 2010). While the importance of accuracy in representation of moisture is clear, both Bunkers et al. (2010) and Gallo et al. (2017) also emphasize the importance of accurately representing vertical profiles of temperature to address differences between model-forecast and observed vertical temperature profiles and CIN.

The need to improve observed and CAM-forecasted CIN could be addressed by utilizing unnamed aircraft systems (UAS) to profile the lower atmosphere and assess the strength and mesoscale variability of capping inversions. UAS are well suited to this task, and have been increasingly used for thermodynamic sampling of the lower atmosphere in recent years (e.g., Houston et al. 2012; Kiefer et al. 2012; van den Kroonenberg et al. 2012; Reineman et al. 2013; Knuth and Cassano 2014; Riganti and Houston 2017; Lee et al. 2019; Bailey et al. 2019; de Boer et al. 2020). An observing system simulation experiment performed by Chilson et al. (2019) demonstrated that forecasts of boundary layer structure and DCI could be improved by assimilating data obtained using rotary-wing UAS. As with any data assimilation, proper characterization of instrument error is critical to producing an improved forecast. Examples of error sources include sensor placement relative to propellers (Greene et al. 2018; Islam et al. 2019), sensor exposure to solar radiation (Islam et al. 2019), and hysteresis due to aircraft ascent rate and sensor time response (Houston and Keeler 2018, hereafter HK18).

In HK18 the authors estimated the impact of sensor response and aircraft airspeed on the accuracy of in situ temperature observations by simulating UAS flights through large-eddy simulations (LESs). Error characteristics were estimated for multirotor UAS profiles through the convective boundary layer (CBL), and fixed-wing UAS flights across airmass boundaries. Errors in data sampled from the CBL flights scaled directly with sensor response time and airspeed (i.e., largest errors were for rapid ascent with a slow time response sensor, and smallest errors were for slow ascent with a fast time response sensor). It was found that maximum errors are smaller than typical sensor accuracies of ±0.5 K provided the ascent rate was less than ~10 m s−1 and sensor response time is less than ~5 s. Particular care is needed when sampling superadiabatic layers, since observation errors smeared-out the layer and replaced it with an artificial inversion. Overall, it was found that instantaneous errors resulting from sensor response and ascent rate are attributable to the background lapse rate, and are largely independent of temperature perturbations within individual thermals.

The aim of the work presented here is to extend the results of HK18 by examining errors associated with in situ sampling of capping inversions. As in HK18, errors associated with sensor response and aircraft ascent rate are considered. However, unlike HK18, this work focuses on the sounding characteristics that yield errors due to platform/sensor characteristics and not on the platform/sensor characteristics themselves. Furthermore, instead of utilizing LES, steady-state analytic atmospheric profiles are used. The assumption of steady-state is justified by HK18 who find an insensitivity of errors in profiles to CBL heterogeneities. The steady-state assumption allows for the use of analytic profiles that enable the exploration of a much larger parameter space.

2. Methodology

a. Analytic steady-state atmospheric profiles

The atmospheric thermodynamic profiles used for this work are generated using three free parameters: inversion depth D, vertical gradient of potential temperature at the inversion base /dz|I, and water vapor mixing ratio at the inversion top qυ|D (Fig. 1). Thirty values of D are used ranging from 100 to 3000 m AGL (examples of variations in D are illustrated in Fig. 2a); 60 values of /dz|I are used, ranging from 0.002 to 0.03 K m−1 (Fig. 2b); and 11 values of qυ|D are used, ranging from 1 to 6 g kg−1 (Fig. 2c). Vertical profiles of temperature are prescribed using a constant θ of 300 K below zzML = 1000 m. Within the inversion (zML < z < zML + D), a linear profile of θ is imposed using /dz|I. Above the inversion and within the EML (Fig. 1), θ is constant with respect to height at a value of θ(zML + D). At the height where the relative humidity reaches 80%, the vertical gradient in potential temperature assumes a value of 0.003 K m−1 and a linear profile is prescribed through the depth of the troposphere. Water vapor mixing ratio is constant below zML at a value of 10 g kg−1. Within the inversion, qυ decreases linearly to qυ|D. Within the EML, qυ = qυ|D until the height where the relative humidity reaches 80%. Above this level, qυ is set to a value for which the relative humidity is 80%.

Fig. 1.

Example sounding illustrating the three free parameters that are used to generate analytical thermodynamic vertical profiles. Other sounding characteristics used to describe the profiles are also included: zML is the height of the top of the “mixed layer” and is set to 1000 m for this work, the inversion is the statically stable layer between the “mixed layer” and elevated mixed layer (EML), and the EML is the neutral layer above the inversion.

Fig. 1.

Example sounding illustrating the three free parameters that are used to generate analytical thermodynamic vertical profiles. Other sounding characteristics used to describe the profiles are also included: zML is the height of the top of the “mixed layer” and is set to 1000 m for this work, the inversion is the statically stable layer between the “mixed layer” and elevated mixed layer (EML), and the EML is the neutral layer above the inversion.

Fig. 2.

Example soundings illustrating the impact on the profiles of temperature and moisture from variations in (a) D, (b) /dz|I, and (c) qυ|D.

Fig. 2.

Example soundings illustrating the impact on the profiles of temperature and moisture from variations in (a) D, (b) /dz|I, and (c) qυ|D.

The free parameter values listed above yielded the generation of 18 810 soundings. However, soundings within this initial set were excluded from consideration if

  1. CIN < 5 or > 500 J kg−1, which removed 12 251 soundings (a description of the method used to calculate CIN appears in the  appendix);

  2. the level of free convection (LFC) is above the EML (removed 514 soundings);

  3. the relative humidity exceeds 95% within the inversion or exceeds 80% at the base of the EML (removed 218 soundings);

  4. the LFC is within the inversion (removed 769 soundings).

The resulting database contains 5058 analytic soundings. The soundings parameter space is illustrated in Fig. 3.

Fig. 3.

Heat maps of the number of analytic soundings used for this work for combinations of (a) /dz|I and D and (b) qυ|D and D. (c) Histogram of soundings as a function of CIN (using a bin size of 5 J kg−1).

Fig. 3.

Heat maps of the number of analytic soundings used for this work for combinations of (a) /dz|I and D and (b) qυ|D and D. (c) Histogram of soundings as a function of CIN (using a bin size of 5 J kg−1).

b. Simulated sensor

As in the work of HK18, the simulated sensor is assumed to have a first-order response, to be well aspirated, and to be mounted on a platform that does not disturb its environment. Importantly, only ascending sensors are used. This will impact the sign of the resulting errors.

A first-order sensor is described using

 
dT^dt=T0T^τ,
(1)

where T^ is the observed temperature, T0 is the actual temperature, and τ is the sensor response time. For a fixed ascent rate w, (1) can be solved on a vertical grid with spacing Δz using

 
T^(k)=T^(k1)+Δzwτ[T0(k)T^(k1)],
(2)

where k − 1 indicates the previous (lower for ascent) grid point. For this work, (2) along with the similar expression for relative humidity are solved using fourth-order Runge–Kutta differencing.

Because the focus is on errors associated with sounding characteristics and not on platform/sensor characteristics and because fast ascent rate and slow sensor response have the same effect on the “observed” temperature and relative humidity, w and τ are not separately considered in this work (as done by HK18) but are instead combined into a single parameter, . Furthermore, since the primary focus of this work is not on the sensitivity of errors to , the value of is held fixed. The choice of values is informed by the capabilities of current and near-future profiling systems. Time constants for most of these systems range from less than a second to tens of seconds and ascent rates typically range from ~1–2 to ~10 m s−1. Koch et al. (2018) set a benchmark for sensor response identical to Jacob et al. (2018) and adopted by the NOAA UAS program office “as guidelines to follow in evaluation of potential UAS platforms for operational consideration” (Koch et al. 2018, p. 2267). This benchmark is τ < 5 s but ideally τ < 1 s. The nominal ascent rate for National Weather Service radiosondes is 5 m s−1 (NOAA 2010) and profiling UAS typically ascend at 3–5 m s−1 (e.g., Koch et al. 2018; Islam et al. 2019; Leuenberger et al. 2020). As such, in this work, values of 5 and 25 m are specifically considered.

3. Results

Across all soundings, the impact of ascent rate and sensor response is to increase the synthetic convective inhibition (CINs) relative to the actual convective inhibition (CINa) The largest absolute CIN error, CINs − CINa, for = 5 m is 6.5 J kg−1 and for = 25 m is 32.7 J kg−1 and is associated with a sounding characterized by a deep inversion and large actual CIN (476 J kg−1; Fig. 4a). The largest relative CIN error 100% × (CINs − CINa)/CINa, for = 5 m is 53.8% and for = 5 m is 269.4% and occurs for a sounding with a shallow inversion and small actual CIN (5.1 J kg−1; Fig. 4b).

Fig. 4.

Sounding corresponding to the (a) maximum absolute CIN error and (b) maximum relative CIN error.

Fig. 4.

Sounding corresponding to the (a) maximum absolute CIN error and (b) maximum relative CIN error.

In an effort to determine the characteristics of inversions that produce the largest errors due to ascent rate and sensor response, the sensitivity of relative CIN errors to the three free parameters used to prescribe the soundings used for this work will be examined. Considering all soundings, an inverse relationship between relative errors and /dz|I can be inferred (Fig. 5a) but becomes more apparent when D and qυ|D are held fixed (green curve in Fig. 5a). The relationships between relative errors and D (Fig. 5b) as well as qυ|D (Fig. 5c) are also inverse. It is possible to conclude from this analysis that soundings with small /dz|I, small D, and small qυ|D produce the largest relative CIN errors. These are also the conditions associated with the smallest CIN (Fig. 2; the sensitivity of CIN to qυ|D is manifested through the virtual temperature adjustment of the environmental profile); thus, soundings with less CIN are more vulnerable to large relative CIN errors. Moreover, the sensitivity of relative CIN error to the actual CIN is quasi logarithmic (Fig. 6) indicating that sounding characteristics that contribute to increases in CIN do not proportionally increase the CIN error. As an example, for /dz|I = 0.01 K m−1 and qυ|D = 6 g kg−1 increasing D from 100 m (Fig. 7a) to 600 m (Fig. 7b) increases the CIN by a factor of more than 28 (from 5.1 to 145.4 J kg−1) but only increases the depth over which errors in T are created (gray-shaded regions in Fig. 7 denoting the near-neutral layer within the EML below the LFC) by a factor of 2. This is because the addition of a 500-m isothermal layer (/dz|I = 0.01 K m−1) realized by increasing D does not add errors to the synthetic temperature but the associated 6-K increase in potential temperature within the inversion significantly increases CIN. As another example, if D is held fixed at 600 m and /dz|I is increased from 0.005 K m−1 (Fig. 7c) to 0.01 K m−1 (Fig. 7b), the error generation within the inversion of the larger CIN environment (Fig. 7b) is less than that of the smaller CIN environment (Fig. 7c) and the depth of the error generation layer within the EML is only a factor of 2.4 deeper in the larger CIN environment. However, CIN is 3.6 times larger. Ultimately, this analysis reveals that sounding characteristics associated with large CIN are not associated with large relative CIN error.

Fig. 5.

Distributions of relative CIN error as a function of (a) /dz|I, (b) D, and (c) qυ|D for all soundings (black dots). The green curves are the distributions of relative CIN error for (a) D = 600 m and qυ|D = 3 g kg−1, (b) /dz|I = 0.005 K m−1 and qυ|D = 3 g kg−1, and (c) /dz|I = 0.005 K m−1 and D = 600 m. The blue curve in (c) is the distribution of relative CIN error for /dz|I = 0.005 K m−1 and D = 400 m.

Fig. 5.

Distributions of relative CIN error as a function of (a) /dz|I, (b) D, and (c) qυ|D for all soundings (black dots). The green curves are the distributions of relative CIN error for (a) D = 600 m and qυ|D = 3 g kg−1, (b) /dz|I = 0.005 K m−1 and qυ|D = 3 g kg−1, and (c) /dz|I = 0.005 K m−1 and D = 600 m. The blue curve in (c) is the distribution of relative CIN error for /dz|I = 0.005 K m−1 and D = 400 m.

Fig. 6.

Scatterplot of relative CIN error as a function of CIN for all soundings. Inset illustrates the distribution in log–log space.

Fig. 6.

Scatterplot of relative CIN error as a function of CIN for all soundings. Inset illustrates the distribution in log–log space.

Fig. 7.

(left) Soundings illustrating the impact of (a),(b) changes in D and (b),(c) changes in /dz|I on (right) the vertical distribution of absolute CIN errors.

Fig. 7.

(left) Soundings illustrating the impact of (a),(b) changes in D and (b),(c) changes in /dz|I on (right) the vertical distribution of absolute CIN errors.

Given the strong dependence of relative CIN errors on CIN, it is instructive to assess the sounding characteristics that lead to CIN errors when controlling for CIN. Analysis will focus on soundings with CIN generally considered to be on the threshold between environments that will and will not support deep convection initiation; operationally, errors in the observed CIN for these environments are the most important to characterize. Soundings within three small CIN intervals (10–15, 30–35, and 50–55 J kg−1) will be considered as a way to control for the dependence of relative CIN errors on CIN while retaining enough soundings in each group to establish robust sensitivities.

Median relative CIN errors for = 5 m are 22.4%, 9.3%, and 6.1% for 10–15, 30–35, and 50–55 J kg−1, respectively (Fig. 8). For reference, the sensitivity of relative CIN errors for CIN = [10, 15] J kg−1 and ranging from 0.5 to 50 exhibit the expected direct relationship (Fig. 9) with a linear least squares fit to median relative CIN errors closely adhering to a 4.57 relationship (R2 > 0.99). Linear best fits to the distributions of relative CIN error versus /dz|I all have negative slopes (Fig. 10a). However, even though the statistical relationship is reasonably robust for the largest CIN interval (R2 = 0.35 for CIN = 50–55 J kg−1), Pearson correlation coefficients for the smaller intervals are low. Nevertheless, the consistency in best-fit slopes supports the conclusion that, when controlling for CIN, relative CIN errors are inversely proportional to /dz|I. Linear best fits to the distributions of relative CIN error versus D are all positive. Like the relationship between CIN error and /dz|I, statistical robustness of the relationship between CIN errors and D increases with CIN (Fig. 10b). But the consistent linear-fit slopes between the three distributions leads to the conclusion that, when controlling for CIN, a direct relationship is exhibited between D and relative CIN. This direct relationship contrasts with the inverse relationship exhibited between relative CIN errors and D for the entire parameter space (Fig. 5b). Correlations for qυ|D are small (R2 < 0.01) for all three CIN intervals (Fig. 10c); although, the linear best fit for the largest CIN interval is negative. In summary, when controlling for CIN, soundings that result in the largest relative CIN error are characterized by small /dz|I, large D, and small qυ|D. An example sounding from the CIN = [50, 55] J kg−1 interval is illustrated in Fig. 11.

Fig. 8.

Box-and-whisker plots of the relative CIN error for three small CIN intervals. Median values are indicated with the horizontal line in the middle of each box, boxes extend through the second and third interquartile ranges, and whiskers extend to the minimum and maximum values.

Fig. 8.

Box-and-whisker plots of the relative CIN error for three small CIN intervals. Median values are indicated with the horizontal line in the middle of each box, boxes extend through the second and third interquartile ranges, and whiskers extend to the minimum and maximum values.

Fig. 9.

Distribution of median relative CIN errors as a function of (squares) along with a least squares curve fit to the median values (black line) and the region spanned by maximum and minimum relative CIN (gray shading).

Fig. 9.

Distribution of median relative CIN errors as a function of (squares) along with a least squares curve fit to the median values (black line) and the region spanned by maximum and minimum relative CIN (gray shading).

Fig. 10.

Scatterplot of relative CIN errors for each of the three CIN intervals: 10–15 J kg−1 (blue), 30–35 J kg−1 (red), and 50–55 J kg−1 (black). Errors are plotted as a function of (a) /dz|I, (b) D, and (c) qυ|D. Linear fits to the distributions appear as broken lines with the associated equations and coefficients of determination annotated. Histograms of the number of points in each bin appear above each scatterplot color matched to each CIN interval.

Fig. 10.

Scatterplot of relative CIN errors for each of the three CIN intervals: 10–15 J kg−1 (blue), 30–35 J kg−1 (red), and 50–55 J kg−1 (black). Errors are plotted as a function of (a) /dz|I, (b) D, and (c) qυ|D. Linear fits to the distributions appear as broken lines with the associated equations and coefficients of determination annotated. Histograms of the number of points in each bin appear above each scatterplot color matched to each CIN interval.

Fig. 11.

Illustration of a sounding with “threshold” CIN (50.6 J kg−1 for this sounding) that is most prone to relative CIN error.

Fig. 11.

Illustration of a sounding with “threshold” CIN (50.6 J kg−1 for this sounding) that is most prone to relative CIN error.

4. Conclusions

Analysis has been presented focused on determining the sounding characteristics most prone to sampling errors due to hysteresis associated with the ascent rate and sensor response of profiling systems observing inversions. Synthetic vertical profiles of temperature and moisture are collected using more than 5000 analytic soundings generated by altering inversion depth and strength. The simulated sensor used to “collect” synthetic profiles is assumed to have a first-order response, to be well aspirated, and to be mounted on a platform that does not disturb its environment. Analysis shows that, across all soundings, the sounding characteristics that result in the largest relative CIN errors are also the characteristics that result in the smallest CIN: e.g., small /dz|I, small D, and small qυ|D. Because they are more likely to support deep convection initiation, it is particularly critical that environments with small CIN are represented accurately. The relationship between relative CIN error and CIN exists because sounding characteristics that contribute to large CIN, particularly increased /dz|I (more statically stable soundings) and increased D, do not proportionally increase the CIN error.

Analysis also focused on sounding characteristics associated with large relative CIN errors when controlling for CIN. Consideration is given to small CIN intervals with (operationally important) CIN on the threshold between environments that will and will not support deep convection initiation. For these environments, relative CIN error is found to be largest for deep, dry inversions characterized by small /dz|I (large lapse rates).

Acknowledgments

This work was supported by the National Science Foundation Grant OIA-1539070.

APPENDIX

Calculation of CIN

CIN is calculated for this work using the following procedure:

  1. For a given sounding, the initial parcel temperature and water vapor mixing ratio qυ are chosen (surface values are used).

  2. Parcel θ and qυ are held constant as the parcel is lifted.

  3. Upon saturation, parcels are lifted to the next height/pressure level (Δz = 1 m) assuming θ and qυ conservation (yielding supersaturation).

  4. Temperature T and qυ are corrected incrementally back to saturation (defined as |qυsqυ| ≤ 10−7 g kg−1, where qυs is the saturation water vapor mixing ratio) using the Tetens–Murray (Tetens 1930; Murray 1967) formula for saturation vapor pressure es,

     
    es=610.78exp[17.27(T273.15)T35.86],

    where es and pressure p are in pascals and temperature T is in kelvins, and the first law of thermodynamics,

     
    dθ=dqυLυθcpT,

    where Lυ = 2.5 × 106 J kg−1.

  5. Buoyancy is determined using the virtual temperatures of the parcel and environment.

  6. CIN is then calculated as the integrated negative buoyancy below the LFC.

As noted by Murray (1967) and Bolton (1980), the Tetens–Murray formula for calculating es is prone to error at low temperatures. However, across all soundings used for this work, the maximum relative es difference below 5000 m is 0.5%. Moreover, the mean CIN difference across all soundings is −0.1 ± 0.07 J kg−1 or −0.1%.

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