An Idealized Physical Model for the Severe Convective Storm

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J OURNALOFTHE A TMOSPHERIC S CIENCES

An idealized physical model for the severe convective storm environmental sounding Accepted in Journal of the Atmospheric Sciences 2020-10-26, there may still be copy-editing errors

DANIEL R.CHAVAS*

Purdue University, Department of Earth, Atmospheric, and Planetary Sciences, West Lafayette, IN

DANIEL T. DAWSON II Purdue University, Department of Earth, Atmospheric, and Planetary Sciences, West Lafayette, IN

ABSTRACT

This work develops a theoretical model for steady thermodynamic and kinematic profiles for severe convective storm environments, building off of the two-layer static energy framework developed in Agard and Emanuel (2017). The model is phrased in terms of static energy, and it allows for independent variation of the boundary layer and free troposphere separated by a capping inversion. An algorithm is presented to apply the model to generate a sounding for numerical simulations of severe convective storms, and the model is compared and contrasted with that of Weisman and Klemp. The model is then fit to a case-study sounding associated with the 3 May 1999 tornado outbreak, and its potential utility is demonstrated via idealized numerical simulation experiments. A long-lived supercell is successfully simulated with the historical sounding but not the analogous theoretical sounding. Two types of example experiments are then performed that do simulate a long-lived supercell: 1) a semi-theoretical experiment in which a portion of the theoretical sounding is modified to match the real sounding (low-level moisture); 2) a fully-theoretical experiment in which a model physical parameter is modified (free-tropospheric relative humidity). Overall, the construction of this minimal model is flexible and amenable to additional modifications as needed. The model offers a novel framework that may be useful for testing how severe convective storms depend on the vertical structure of the hydrostatic environment, as well as for linking variability in these environments to the physical processes that produce them within the climate system.

1. Introduction A principal focus of SCS research is supercells, which produce the majority of SCS-related hazardous weather, While substantial advances have been made in the un- particularly significant tornadoes (Duda and Gallus 2013). derstanding and prediction of severe convective storms Past work has demonstrated that supercells are associated (SCS), operational predictability remains limited and thus with large magnitudes of CAPE and 0-6km bulk verti- substantial risks to life and property persist. From 1995- cal wind shear (Weisman and Klemp 1982, 1984; Tippett 2015, tornadoes caused 1710 deaths in the U.S., ranking et al. 2015), with the latter being a better discriminator as the third-deadliest weather phenomenon (NOAA). Our between supercell and non-supercell environments than ability to predict these weather risks in the current or fu- the former (Thompson et al. 2003, 2007). Additionally, ture climate depends crucially on a physical understand- the strength of the low-level storm-relative flow, which ing of the dependence of SCS events on their larger-scale arXiv:2004.11636v3 [physics.ao-ph] 28 Oct 2020 environment. Forecasting and research applications have is correlated with 0-6km bulk shear magnitude (Warren largely focused on bulk (i.e. vertically-integrated) ther- et al. 2017), has been identified as a discriminator be- modynamic and kinematic parameters as part of the suc- tween supercell and non-supercell environments (Droege- cessful “ingredients-based” framework for SCS environ- meier et al. 1993; Peters et al. 2019b, 2020a,b). Hence, ment diagnosis and forecasting (Doswell III et al. 1996; the product of CAPE and 0-6km bulk shear is commonly Doswell 2001; Tippett et al. 2015). used as an environmental proxy for potential SCS activity (Brooks et al. 2003; Gensini and Ashley 2011; Seeley and Romps 2015). Significant tornado events are further *Corresponding author address: Daniel R. Chavas, Purdue Univer- linked to high magnitudes of low-level storm-relative en- sity, Department of Earth, Atmospheric, and Planetary Sciences, 550 vironmental helicity (SRH) and low values of the lifting Stadium Mall Drive HAMP 3221, West Lafayette, IN 47907. condensation level (LCL) (Brooks et al. 1994; Rasmussen E-mail: [email protected] and Blanchard 1998; Thompson et al. 2003, 2004), which

Generated using v4.3.2 of the AMS LATEX template 1 2 J OURNALOFTHE A TMOSPHERIC S CIENCES are combined with CAPE and 0-6 km shear in the Signifi- CAPE and bulk shear, respectively (McCaul Jr and Weis- cant Tornado Parameter (STP) for the forecasting of strong man 2001; Kirkpatrick et al. 2009; Guarriello et al. 2018; tornadoes (Thompson et al. 2004). This ingredients-based Brown and Nowotarski 2019); and 3) variations in param- approach using bulk parameters can also provide mean- eters independent of bulk parameters, particularly free- ingful insight into the spatial and temporal distribution of tropospheric moisture (Gilmore and Wicker 1998; James SCS activity (Gensini and Ashley 2011; Rasmussen and et al. 2006; James and Markowski 2010; McCaul and Co- Houze Jr 2016; Li et al. 2020), including long-term spa- hen 2004; Honda and Kawano 2015). From these and re- tial shifts in tornado activity (Agee et al. 2016; Gensini lated studies, an improved understanding of how higher- and Brooks 2018). Moreover, these bulk parameter prox- order vertical variability of SCS environmental profiles ies have been used to estimate changes in severe weather is slowly emerging. This seminal work using idealized and tornado risk under future climate change (Trapp et al. sounding models to test sensitivities of convective evolu- 2007, 2009; Diffenbaugh et al. 2013; Seeley and Romps tion represents the foundation that we build off of in this 2015). study. Nevertheless, details of the vertical thermodynamic and While the WK thermodynamic sounding has been un- shear profiles not captured by bulk parameters are likely deniably useful in advancing our understanding of basic to play important roles in storm evolution. Which details storm dynamics over the past few decades, its construc- within a particular sounding actually matter for the evolution is somewhat ad hoc – its structure is composed of tion of a severe convective storm? The lack of understand- simple parametric equations for the tropospheric profile ing of the effects of such higher-order variability is likely of potential temperature and of relative humidity whose an important contributor to reduced SCS predictability vertical variations are motivated on practical, rather than on daily and sub-daily time scales (e.g, Elmore et al. physical, grounds to be broadly representative of the range 2002a,b; Cintineo and Stensrud 2013). Moreover, bulk of observed soundings associated with severe weather. An proxy statistical relationships trained on canonical high ideal alternative is a model for the environmental sounding CAPE and high bulk shear environments may be inap- that is defined by the physics of how these environments propriately applied to non-canonical environments, such are generated in the first place within the climate system, as ones with high bulk shear yet relatively low CAPE in and whose parameters directly represent key aspects of the which quasi-linear convective systems are common (Sher- vertical structure of the sounding (e.g. the strength of a burn and Parker 2014). Finally, because bulk proxies capping inversion). Recently, Agard and Emanuel (2017, are necessarily validated only against a relatively short hereafter AE17) developed the first theoretical model for historical record, their application to future climates is the time-dependent one-dimensional vertical thermody- not only uncertain but potentially misleading if the cho- namic state associated with severe weather environments sen proxies do not correctly scale with actual SCS risk across climate states (Trapp et al. 2011; Gensini and Mote on a diurnal timescale. AE17 employs a two-layer model 2015; Hoogewind et al. 2017; Trapp and Hoogewind 2016; for the atmosphere in which the boundary layer and free Trapp et al. 2019). The above issues indicate the need for troposphere may be varied independently. This state aligns a deeper physical understanding of the role of the vertical with the archetypal conceptual model of the generation thermodynamic and kinematic structure for fixed values of of high-CAPE environments east of the Rocky Mountains a given bulk proxy. (Carlson and Ludlam 1968; Benjamin and Carlson 1986; Because these bulk parameters are by definition ver- Benjamin 1986; Doswell 2001). A schematic of this set- tically integrated measures of the environment, two en- up is provided in Figure 1, in which warm, moist low- vironments can yield the same bulk value despite hav- level air originating from the Gulf of Mexico to the south ing very different vertical thermodynamic and kinematic lies beneath dry well-mixed air that is advected eastward structures (McCaul Jr and Weisman 2001; Peters et al. off the elevated terrain to the west. AE17 used this two- 2020a). Weisman and Klemp (1984) were among the layer model framework to demonstrate analytically that first to investigate how SCS morphology and evolu- peak CAPE is expected to increase with surface warming. tion depend on vertical environmental structure using a In principle, the AE17 theoretical model could be used cloud-resolving numerical model (CRM). Central to their to specify a steady environmental sounding for use in nu- methodology was a parametric model of the vertical ther- merical simulation experiments. This could further allow modynamic profile (Weisman and Klemp 1982, hereafter tests of fundamental SCS sensitivities to the external phys- WK). Since then, many idealized CRM studies have used ical parameters that specify the background state while the WK profile to investigate different aspects of SCS and holding bulk parameters (e.g. CAPE) fixed. However, this their environments. These include three categories of ex- model has yet to be phrased in a way that it can directly periments: 1) parameter sweep studies varying CAPE and define a sounding for use in an idealized CRM, nor has it shear (Kirkpatrick et al. 2011; Lawson 2019); 2) the ver- been applied in SCS numerical simulations. Thus, there is tical distribution of buoyancy or shear at fixed values of a significant opportunity to apply this physical model for J OURNALOFTHE A TMOSPHERIC S CIENCES 3

FIG. 1: Conceptual diagram of how an environment with large CAPE is generated east of the Rocky Mountains, in a static energy framework following Agard and Emanuel (2017). the thermodynamic environment to modern SCS numer- Our paper is split into two parts: theory and numerical ical simulation experiments. Doing so could allow care- simulation. Section 2 develops our theoretical sounding ful testing of how smaller-scale SCS morphology depends model and motivates the use of static energy in lieu of po- on complex variability in the vertical structure. Further- tential temperature as the base thermodynamic variable. more, given that the SCS environment represents a hydro- An algorithm is then presented to put the model into prac- static background state, this model could also be used to tice, and an example comparison with the Weisman and directly link variability in SCS soundings to the energet- Klemp thermodynamic model is provided to discuss sim- ics of the large-scale hydrostatic atmosphere, which is the ilarities, differences, and benefits of our framework. Sec- focus of modern climate physics. Such physical linkages tion 3 presents an application of how our model can be fit from climate to mesoscale to storm-scale are critical for to a real-data sounding associated with an observed SCS understanding both fundamental SCS environmental de- event: the 3 May 1999 tornado outbreak, which was the pendencies as well as how SCS activity may change in a largest outbreak in Oklahoma recorded history and pro- future climate. duced multiple supercells and long-track tornadoes. We To fill this gap, this work seeks to extend AE17 to de- use this idealized sounding to demonstrate the model’s po- velop a novel theoretical model for a complete, steady SCS tential experimental utility via illustrative sensitivity tests thermodynamic and kinematic sounding for use in numer- of variability in vertical structure at fixed CAPE and bulk ical simulation experiments. The present work focuses on shear. Finally, Section 4 provides a summary of the model how our model is constructed and provides an illustrative and how it may be useful for future SCS research. example of how it can be used as a theoretical foundation for both observationally motivated sensitivity testing and controlled experimentation. Thus, the specific outcomes of our simulation examples shown here are not intended 2. Theoretical model for SCS environmental sounding to demonstrate robust sensitivities. Moreover, the way we We begin by reviewing the framework of AE17 and dis- apply our model is by no means the only approach; it is cuss the benefits of static energy in lieu of potential tem- simply a relatively straightforward one. We hope that as perature for defining a hydrostatic SCS background state. the model is put into use in future research it may evolve Next, we develop our theoretical sounding model and an further, or perhaps it will be applied in different ways for algorithm to apply it to generate an SCS sounding. Fi- different types of experiments. This type of comprehen- nally, we provide an example comparison with the pre- sive experimentation and in-depth analysis are left for fu- vailing sounding model (WK) to discuss similarities, dif- ture work. ferences, and benefits of our model framework. 4 J OURNALOFTHE A TMOSPHERIC S CIENCES a. Foundation: AE17 model LFC, i.e. The AE17 model provides a useful foundation for gen- Tp,s f c CIN ∼ (DBL − DFT )ln (4) erating physics-based thermodynamic environments with TLFC high CAPE amenable to SCS numerical simulation experiments. Specifically, AE17 defines the diurnal evolution of where Tp,s f c is the parcel temperature at the surface. In this environment with a time-dependent two-layer model practice, the CIN magnitude is more strongly sensitive to for dry and moist static energies. Their idealized model neglect of moisture due to both virtual temperature effects begins from an initial state with constant moist static en- and the effect of moisture on the height of the LCL and ergy, where the free troposphere is dry and the boundary hence the temperature of the LFC; such errors are larger layer is cooler and moist; this creates convective inhibition for CIN since the temperature difference across the CIN (a capping inversion). Energy is then input into the surface layer is relatively small compared to that across the CAPE at a constant rate to represent daytime solar heating, which layer. gradually generates CAPE. AE17 used this model to test Thus, a key benefit of the AE17 modeling framework is the dependence of peak CAPE on temperature on diurnal that CAPE and CIN may be directly modulated by vary- timescales. ing the limited number of model physical parameters. In Neglecting liquid/solid phases of water, moist static en- particular, the model explicitly incorporates an externally- ergy per unit mass M is given by: defined capping inversion into the sounding. Note that a similar two-layer slab model framework is presented us- M = CpT + Lvr + gz (1) ing potential temperature as the thermodynamic variable for understanding diurnal variability in general in Stull where T is temperature, r is the water vapor mixing ratio, (2012). and z is geopotential height. The quantities Cp, Lv, and g are the specific heat of air, the latent heat of vaporization b. Why static energy instead of potential temperature? of water, and the acceleration due to gravity, respectively, and all may be approximated as constants. Hence, moist While static energies are not commonly employed in static energy is a linear combination of temperature (sen- the severe weather literature, in a hydrostatic atmosphere sible heat), moisture (latent heat), and altitude (potential their vertical structures are dynamically equivalent to that energy). Dry static energy, D, is the same as M but taking of their potential temperature counterparts. For example, r = 0, i.e. Figure 2a displays a Skew-T plot for a proximity sounding from a simulation of the 3 May 1999 tornado outbreak D = CpT + gz (2) from Dawson et al. (2010) (3MAY99; analyzed in detail in The AE17 model defines a thermodynamic state com- Section 3). Figure 2b-c compares the vertical profiles of prised of a free troposphere (FT) layer with constant dry and moist static energy and dry and moist (equivalent) dry static energy, DFT , overlying a boundary layer (BL) potential temperature for our observational case. The ab- with constant moist static energy, MBL. The boundary solute values of these two quantities map onto one another layer has depth HBL. As noted in AE17 (their Eq. 37), non-linearly, but their vertical variations are very similar. CAPE scales approximately with the difference between Why do potential temperature and static energy map the boundary-layer moist static energy and the dry static onto one another in this way? Here we demonstrate their energy, MBL −DFT , multiplied by the difference in the nat- relationship for the dry case; the logic extends to the ural logarithm of temperatures between the level of free moist case but is significantly more complicated analyti- convection (LFC) and level of neutral buoyancy (LNB), cally (Emanuel 2004; Bryan 2008; Romps 2015). We be- given by gin from the First Law of Thermodynamics for an ideal gas, given by TLFC CAPE ∼ (MBL − DFT )ln (3) CpdT = dq + αdP (5) TLNB 1 where dq is the external specific heating, α = ρ is the Meanwhile, the difference in dry static energies between specific volume, and P is air pressure. We then consider an the base of the free troposphere and the boundary layer, adiabatic process (such as an air parcel ascending through DFT − DBL, represents a temperature jump moving up- an atmospheric column): dq = 0. This yields wards and hence a capping inversion. Note that this scaling neglects the effects of water vapor on buoyancy (i.e. CpdT = αdP (6) virtual temperature effects), which will modify the true CAPE. Though not explicitly stated in AE17, the convec- From Eq. (6), dry static energy requires making the tive inhibition (CIN) follows a scaling with similar form assumption of hydrostatic balance, as for CAPE, except taking the dry static energy differ- dP α = −g (7) ence across the layer bounded by the parcel level and the dz J OURNALOFTHE A TMOSPHERIC S CIENCES 5

Rearranging this equation and subsituting into Eq. (6) For hydrostatic displacements, incremental changes in en- gives tropy are simply given by incremental changes in static CpdT = −gdz (8) energy, divided by temperature. This follows from the which can be written as the conservation equation basic thermodynamic relationship among entropy, energy, and temperature. Hence, vertical structures of entropy and dD = 0 (9) static energy are qualitatively similar but differ quantita- tively owing to variations in temperature with altitude. Fi- 1 where D is the dry static energy (Eq. (2)) . Thus, hydro- nally, we may link D to θ via Eqs. (13) and (17) to give static balance allows us to trade changes in pressure (i.e. pressure-volume work at constant pressure) with changes dsd dD in altitude (i.e. potential energy). d(lnθ) = = (18) C C T Meanwhile, from Eq. (6), potential temperature re- p p quires no new assumption. Instead, we reapply the Ideal Thus, changes in the natural logarithm of dry potential Gas Law, P = ρRdT, where Rd is the specific gas constant for dry air. Rearranging this and substituting gives temperature are related to changes in dry static energy, normalized by the sensible heat of the parcel. Cpd(lnT) = Rdd(lnP) (10) Eq. (18) is not very straightforward to interpret, which is the point: while potential temperature is practically use- which can be written as the conservation equation ful for translating entropy to a tangible temperature-like quantity, it does so by adding non-linearity to the prob- dsd = 0 (11) lem that makes it more complex analytically. Moisture where further exacerbates this problem via the equivalent poten- sd = CplnT − RdlnP (12) tial temperature (θe), which is itself a highly non-linear combination of potential temperature and moisture. In this is the dry entropy. Adding the constant RdlnP0, where P0 is a reference pressure, to both sides and rearranging yields way, then, θe is remarkably useful for combining together temperature, pressure, and moisture effects into a single sd + RdlnP0 = Cplnθ (13) quantity. The downside, though, is that it makes decon- structing its components – and the processes that control where each – much more complicated. Ultimately, while entropy Rd P Cp is better conserved than static energy for non-hydrostatic θ = T 0 (14) P displacements of an air parcel, such as in a thunderstorm, this more detailed accounting is not necessary for defining is the dry potential temperature. We can write θ as a hydrostatically-balanced state. sd +Rd lnP0 Meanwhile, static energy has practical benefits both Cp θ = e (15) for understanding mesoscale SCS dynamics (i.e. towards Thus, potential temperature is an alternative, non-linear smaller scales) and for linking SCS environments to cli- way to write entropy, in which entropy is modified by con- mate (i.e. towards larger scales). First, for SCS research, stants and then exponentiated. it is analytically simple to generate thermodynamic pro- How are adiabatic changes in dry entropy and dry static files for layers specified by dry static energy given that energy related? We start from the conservation of sd (Eq. static energy is a linear combination of temperature, alti- (12)) since this requires less stringent assumptions. We tude, and moisture. This enables precise testing of SCS use Eq. (2) to write an equation for differential changes in dependencies on specific aspects of the thermodynamic D as CpdT = dD − gdz and substitute to yield profile and makes it straightforward to incorporate additional modifications to the profile; an example comparison 1 Rd with the WK model is provided in Section 2e below. Fur- dsd = (dD − gdz) − dP (16) T P thermore, this framework defines the thermodynamic profile in terms of energy, which is the same physical quantity Reapplying hydrostatic balance, written as − Rd dP = g dz, P T as CAPE itself; this may have useful theoretical benefits. gives dD In the end, one may readily map the model sounding back ds = (17) into potential temperature space as needed (e.g. in the d T analysis of numerical simulations) in order to work with those variables that properly account for important nonhydrostatic processes. 1Note that Eq. (8) is readily rearranged to give the dry adiabatic lapse rate, which thus should formally be the “dry adiabatic hydrostatic Second, for climate research, an energy-based frame- lapse rate”. work offers the opportunity to directly link the hydrostatic 6 J OURNALOFTHE A TMOSPHERIC S CIENCES

FIG. 2: (a) Skew-T plot for example proximity sounding from historical simulation of the 05/03/99 tornado outbreak at 2300 UTC in SW Oklahoma; (b) vertical proﬁle of dry and moist static energies; (c) vertical proﬁle of dry and equivalent potential temperatures.

SCS sounding to the field of climate physics, whose prin- the sounding is physically and intuitively consistent with cipal focus is the energy budget of a hydrostatic atmo- the prevailing model for how severe convective storm en- sphere. This budget is composed of the transfers of environments are generated (Figure 1). A schematic of our ergy due to incoming and outgoing radiation at the top of sounding model, including both thermodynamic and kine- the atmosphere, surface energy fluxes, and internal trans- matic profiles, is shown in Figure 3. We explain the con- port of energy by atmospheric and oceanic circulations struction of each component next. (Lorenz 1955; Peixoto and Oort 1992). The partitioning of energy sources and sinks has been applied to un- 1) THERMODYNAMIC PROFILE derstand variability in the global-mean climate (Manabe We model the thermodynamic state (Figure 3a) begin- and Strickler 1964; Meehl 1984), horizontal variability in ning from the same two-layer tropospheric structure as climate (Budyko 1969; Sellers 1969; Cronin and Jansen AE17 described above: a boundary layer and a free tro- 2016; Shaw et al. 2018; Armour et al. 2019; Donohoe posphere. We then impose three additional useful modifi- et al. 2020), and the atmospheric response to global warm- cations to put the model into practice. ing (Rose et al. 2014; Roe et al. 2015; Siler et al. 2018). Partitioning between sensible and latent heat is relevant 1. We relax the assumption of constant dry static energy to SCS environments given that, for example, CAPE de- in the free troposphere (i.e. dry adiabatic lapse rate, pends on boundary layer moist static energy while CIN de- g Γd = C ) to allow for a constant rate of increase of pends on boundary layer dry static energy as noted above. p dry static energy with altitude, βFT . βFT sets the free- Thus, understanding how SCS activity changes with cli- tropospheric lapse rate: from the definition of D, we mate change requires an understanding of how the pro- may write β = dDFT = C dTFT + g, which may be cesses within the climate system alter the vertical distribu- FT dz p dz rearranged to give tion of dry and moist static energy in those hydrostatic environments that produce large values of CAPE. One great βFT example of this is Agard and Emanuel (2017) itself, which ΓFT = Γd − (19) Cp uses an energetic framework to develop a process-level, time-dependent theory that predicts a rapid increase in This is important given that free-tropospheric lapse peak diurnal CAPE with warming. rates are known to vary significantly in SCS environments (Blanchard 1998). True elevated mixed lay- c. Our model ers with dry-adiabatic lapse rates are not common through the depth of the troposphere. AE17 did not link their modeling framework for SCS environments to a real SCS sounding in order to be di- 2. Since AE17 does not specify a tropopause, we place rectly useful for SCS research. Our goal is to build off a simple dry isothermal “stratosphere” layer with of the AE17 framework to develop a model for a com- temperature Tt pp (Chavas and Emanuel 2014) at the plete, steady SCS sounding, i.e. joint thermodynamic and model top whose base altitude, Ht pp, represents the kinematic profiles. As described below, the model repre- tropopause altitude. Ht pp is defined simply by the sents a transition from predominantly southerly flow ad- height at which the environmental temperature pro- vecting moist air near the surface to predominantly west- file equals the tropopause temperature Tt pp. This erly flow advecting drier, well-mixed air aloft. In this way, temperature-based definition is desirable given that J OURNALOFTHE A TMOSPHERIC S CIENCES 7

the tropopause temperature is expected to remain The upper shear layer extends from the base of the free s fixed locally with warming in both the tropics and troposphere up to a fixed altitude, Htop. The layer is speci- midlatitudes (Seeley et al. 2019; Hartmann and Lar- fied with westerly shear. We allow this shear to be con- son 2002; Thompson et al. 2019). stant or linearly-decreasing with height, as shear is often concentrated at lower levels in convective storm envi- 3. Since AE17 does not specify moisture in the free tro- ronments, particularly those associated with tornadic su- pospheric layer, we incorporate the simplest option: percells (e.g., Esterheld and Giuliano 2008; Coffer and constant relative humidity, RH (z) = RH . FT FT,0 Parker 2015; Thompson et al. 2003; Coffer et al. 2019), These modifications enable a more realistic representa- i.e. tion of historical case soundings and also provide a direct means for testing variations in the thermodynamic profile ∂uFT (z) = cFT,1 + cFT,2(z − HBL) (23) at fixed CAPE. One experimental benefit of assuming con- ∂z stant BL dry and moist static energy is that CAPE is then ∂vFT = 0 (24) insensitive to the parcel level of origin within the bound- ∂z ary layer. Note that the 100-mb mixed-layer CAPE (ML- CAPE) is often used in forecasting because it accounts for where cFT,1 represents the zonal shear at the base of the potential boundary layer mixing by turbulence. If the as- upper shear layer and cFT,2 represents the rate of change of sumed mixed layer extends above the top of the bound- zonal shear with height. Eq. (23) represents the transition ary layer, the MLCAPE may be considerably less than the from a zonal shear magnitude of cFT,1 at the layer base s surface-based CAPE (SBCAPE) depending on the mag- (z = HBL) to cFT,1 +cFT,2(Htop −HBL) at the layer top (z = s nitude of moisture near the base of the free troposphere. Htop). The bulk vector shear across the upper shear layer Because the amount of mixing depends on many details of is thus: the environment and storm evolution, we choose to focus Z Hs top ∂uFT s 1 s 2 principally on SBCAPE in this work. Additional complex- ∆VFT = dz = cFT,1(Htop −HBL)+ cFT,2(Htop −HBL) ities that could be added to the model, such as allowing HBL ∂z 2 for variations in free-tropospheric relative humidity and (25) boundary layer moisture, are discussed below. Thus for a fixed value of bulk layer shear, a range of com- binations of (cFT,1,cFT,2) are possible. There are two sim- 2) KINEMATIC PROFILE ple limit cases to consider for the upper shear layer:

∂uFT A schematic of the model kinematic profile is shown in 1. Constant shear: cFT,2 = 0; the zonal shear ∂z (z) = Figure 3b. We propose a similar two-layer model for rep- cFT,1 is constant throughout the layer resenting the kinematic structure of the sounding that is physically consistent with the thermodynamic model. Our 2. Shear decreasing linearly to zero at the layer top: s model is similar to recent work idealizing the kinematic cFT,2 = −cFT,1/(Htop − HBL); the zonal shear (Eq. profile using L-shaped hodographs that are often seen in ∂uFT z−HBL (23)) reduces to (z) = cFT,1 1 − s . tornadic supercell environments (Esterheld and Giuliano ∂z Htop−HBL 2008; Beck and Weiss 2013; Sherburn and Parker 2015; s For the remainder of the shear profile (z > Htop), we Guarriello et al. 2018; Peters et al. 2020a). The model impose zero shear (i.e. constant wind vector). is comprised of a lower free-tropospheric layer superim- posed over a boundary layer with the same depth as the d. Practical implementation of model thermodynamic model (HBL). Each layer is assumed to have unidirectional shear, with the boundary layer defined Our objective is to use the model sounding in numerical relative to a specified surface wind, (us f c,vs f c). simulations. We define our model moving upwards from The boundary shear layer is specified with constant the surface, similar to how a sounding is obtained by an southerly shear, i.e. ascending radiosonde. ∂u BL = 0 (20) ∂z 1) THERMODYNAMIC PROFILE ∂vBL The most straightforward implementation of the ther- = cBL (21) ∂z modynamic model is as follows: c represents the constant meridional shear in the bound- BL 1. Calculate surface dry and moist static energy, D ary layer. The bulk vector shear across the boundary layer s f c M is thus: (Eq. (2)) and s f c (Eq. (1)), from input sur- H Z BL ∂vBL face pressure, temperature, and relative humidity ∆VBL = dz = cBLHBL (22) (P ,T ,RH ). 0 ∂z s f c s f c s f c 8 J OURNALOFTHE A TMOSPHERIC S CIENCES

FIG. 3: Schematic of model for (a) thermodynamic profile, and (b) shear profile. External parameters are colored red. The thermodynamic profile is an extension of the AE17 model. The kinematic model assumes constant southerly shear in the boundary-layer, constant or linearly-decreasing westerly shear in the free troposphere, and sets the boundary layer height equal to its value in the thermodynamic profile.

2. Calculate temperature in the boundary layer (z ≤ pressure at all altitudes and the mixing ratio in the 2 HBL) assuming constant dry static energy, DBL = free troposphere . Mixing ratio is calculated using: 1 (RH)e∗ Ds f c: TBL(z) = (DBL − gz). ∗ Cp r = ε P−(RH)e∗ , where e is the saturation vapor pressure and RH (z) = RH . 3. Calculate mixing ratio in the boundary layer (z ≤ FT FT,0 HBL) assuming constant moist static energy, MBL = 8. Impose a dry isothermal “stratosphere” (i.e. Ms f c. This translates simply to holding mixing ratio statically-stable) at the model top. This is done constant: rBL(z) = rs f c (well-mixed). Mixing ratios by setting the temperature to Tt pp and the mixing are capped such that relative humidity does not ex- ratio to zero at all altitudes where the predicted ceed 99% (note: this is performed in the final step free-tropospheric temperature from the previous step and thus reduces MBL at those levels). is less than the tropopause temperature, T < Tt pp. 4. Calculate dry static energy at the base of the free This algorithm specifies the thermodynamic model from the following eight external parameters: P , T , troposphere, defined as the first level above HBL: s f c s f c RH , H , ∆D, β , RH , and T . DFT,0 = DBL(HBL) + ∆D s f c BL FT FT,0 t pp

5. Calculate dry static energy in the free troposphere 2) KINEMATIC PROFILE (z > H ): D (z) = D + β (z − H ). This BL FT FT,0 FT BL The shear profile may be similarly defined moving up- quantity defines the free-tropospheric lapse rate, wards from the surface: Γ = Γ − βFT . FT d Cp 1. Define the input surface wind vector, (us f c,vs f c). 6. Calculate temperature in the free troposphere from D (z): T (z) = 1 (D (z) − gz). FT FT Cp FT 2Technically these two integrations should be repeated until they 7. Integrate hydrostatic balance (Eq. (7)) upwards from converge to account jointly for the hydrostatic pressure of the free tro- the surface pressure Ps f c to calculate the hydrostatic pospheric moisture overhead and the pressure dependence of the mixing ratio, though the errors are generally very small for Earth-like temperatures. J OURNALOFTHE A TMOSPHERIC S CIENCES 9

2. Calculate the boundary shear layer flow velocities the EML was once a well-mixed boundary layer it- (z ≤ HBL): uBL(z) = us f c, vBL(z) = vs f c + cBLz. self. Such a layer could be applied as an intermediate layer in the lower free-troposphere. 3. Calculate the upper shear layer flow velocities s (HBL < z ≤ Htop): uFT (z) = uBL(HBL) + cFT,1(z − • Height dependence of shear in the boundary layer. 1 2 Recent studies have found evidence that strong shear HBL) + cFT,2(z − HBL) , vFT (z) = vBL(HBL). 2 in the lowest few hundred m AGL is more closely s 4. Set flow velocities constant for z > Htop: u(z) = related to significant tornado occurrence in supercell s s uFT (Htop), v(z) = vFT (Htop) storms than the 0-1 km layer more commonly uti- lized in operational contexts (Markowski et al. 2003; This algorithm specifies the kinematic model from the Esterheld and Giuliano 2008; Coffer et al. 2019). following six external parameters: us f c, vs f c, cBL, cFT,1, s cFT,2, and Htop. HBL is defined in the thermodynamic e. Comparison with Weisman and Klemp sounding model. The WK sounding is characterized by simple, 3) SUMMARY AND ADDITIONAL POTENTIAL MODIFI- smoothly-varying analytic expressions for potential tem- CATIONS perature and relative humidity as a function of altitude. Potential temperature increases from a specified surface The above is a minimal theoretical model that contains value to a specified tropopause value and then increases the necessary ingredients for a viable environmental SCS exponentially above the tropopause, according to sounding – i.e. one with significant CAPE and vertical  5 wind shear, and relatively low CIN. We emphasize here  z 4 θs f c + (θt pp − θs f c) , if z ≤ zt pp that this does not guarantee that a given sounding spec- zt pp θ(z) = h i (26) ified by this model will produce any specific SCS out- g(z−zt pp) θt ppexp C T , if z > zt pp come, such as a long-lived supercell. In this way, then, p t pp the base model provides a natural starting point for test- The latter equation yields an isothermal layer above the ing how changes to the sounding affect SCS outcomes, as tropopause (shown analytically in Appendix A; cf. WK demonstrated in Section 3. Figure 1), which is identical to our model. Note that the We have incorporated a few additional types of com- tropopause is overspecified in this formulation – its height, plexity to better capture real-world soundings. Without temperature, and potential temperature are all input pa- question, there are numerous additional degrees of com- rameters. As a result, changing the value of Tt pp alone plexity that could be readily added to the model to test does not actually alter the tropopause in the same manner their significance. We highlight a few possible options in the profile itself; one must first solve for one parameter here: from the solution below the tropopause before specifying the solution above the tropopause. • Relaxing the constant moist static energy constraint Relative humidity decreases moving upwards according in the boundary layer to allow for representation of to a similar dependence on altitude moisture entrainment or detrainment from the free troposphere. Schultz and Askelson (2012) found that 5 3 z 4 significant tornadoes from discrete supercells were RH(z) = 1 − (27) more likely when the boundary layer was capped and 4 zt pp the θ (and hence moist static energy) was constant e and is set constant at 0.25 above the tropopause. Finally, or increased with height. Note that this will intro- a boundary layer that is well-mixed in moisture is created duce new variation in CAPE and CIN calculated for by imposing an upper-bound on the water vapor mixing parcels from different levels (or vertically-averaged) ratio, r , which reduces the RH at all levels where the within the BL. v,s f c initial rv value exceeds rv,s f c. The boundary layer is not • A water vapor or relative humidity lapse rate at the well-mixed in potential temperature. base of the free troposphere, to allow for a more grad- Figure 4 displays an example of our model thermo- ual moisture transition across the capping inversion. dynamic profile with comparison against WK. The input In our model, this transition is sharp. parameters for WK are defined directly from our model sounding. This comparison allows us to highlight similar- • Multiple free-tropospheric layers. For example, here ities and differences in model construction. The param- we have allowed free tropospheric moisture to vary eters for our model are: Ps f c = 1000 hPa, Ts f c = 300 K, independently of temperature (dry static energy), RHs f c = 0.7, HBL = 700m, ∆D = 3000J/kg, Tt pp = 220K, which is not characteristic of a true EML. A real ΓFT = 7.0K/km, and RHFT,0 = 0.7. The resulting param- EML would also have constant mixing ratio, since eters for WK are: θs f c = 300 K, θt pp = 340.6 K, zt pp = 10 J OURNALOFTHE A TMOSPHERIC S CIENCES

11.6km, and rv,s f c = 15.8g/kg; Tt pp and Ps f c are the same (both thermodynamic and kinematic) and its relationship as above. to key quantities, such as CAPE and CIN, via its physi- The thermal profiles are overall quite similar. Note that cal parameters. Finally, the use of static energy is consis- 5 the 4 exponent used in WK for the increase in potential tent with CAPE as an energy quantity as well as with the temperature with height yields a free-tropospheric lapse large-scale energetics of a hydrostatic atmosphere as noted rate that is relatively close to constant; our model im- earlier. poses this structure by definition. The principal difference is the existence of an explicit, sharp capping inversion in 3. Application to historical case: the 3 May 1999 tor- our model, which is a result of the two-layer tropospheric nado outbreak framework. Such a sharp inversion is not straightforward to produce in WK owing to its simpler construction (Nay- We next provide a demonstration of how our model may lor et al. 2012). be used to idealize an SCS environmental sounding associ- The relative humidity profiles are nearly identical ated with a real historical event. We then demonstrate the within their respective boundary layers. Note that the experimental utility of the model via illustrative sensitivity boundary layer depth in our model may be varied indepen- tests of variability in vertical structure in SCS numerical dently of rv,s f c, whereas in WK the two are intrinsically simulations. 5 linked. In the free troposphere, WK again imposes a 4 exponent, which yields an RH profile that decreases quasi- a. Numerical simulation description linearly with altitude. In contrast, our model simply as- Experiments are performed using the CM1 numerical sumes constant RH, though a linear decrease with altitude model (Bryan and Fritsch 2002) version 19. CM1 is a fully could readily be added. Neither choice is “correct” nor compressible nonhydrostatic computational model de- more physical than the other. Arguably the most logical signed for idealized simulations of mesoscale and smaller structure based on observations is a ”C-shaped” profile, as atmospheric phenomena. CM1 has been employed to relative humidity is generally high at in the boundary layer gain fundamental insight into a wide range of mesoscale and near the tropopause with a local minimum in the mid- phenomena in both the mid-latitudes and tropics, includ- dle free troposphere (Gettelman et al. 2006; Romps 2014). ing severe convective storms and tornadoes (Bryan et al. Ultimately, though, free tropospheric RH is poorly con- 2006; James and Markowski 2010; Naylor and Gilmore strained for SCS research given that CAPE for a boundary 2012; Orf et al. 2017; Dahl et al. 2012, 2014; Naylor layer parcel is relatively insensitive to free tropospheric and Gilmore 2014; Parker 2014; Dahl 2015; Markowski moisture. Hence it is left constant in our model for sim- 2016; Peters 2016; Peters et al. 2019a), supercells (James plicity, which may serve as a baseline for comparison with and Markowski 2010; Coffer and Parker 2015; Davenport more complex vertical structures. and Parker 2015; Nowotarski and Markowski 2016) and Overall, our model offers useful physical insight into convective squall lines (Bryan et al. 2006); tropical cy- the vertical structure of the WK model, whose paramet- clones (Bryan and Rotunno 2009; Chavas and Emanuel ric formulation was motivated by a practical need to rep- 2014; Davis 2015; Navarro and Hakim 2016; Naylor and resent real-world soundings. Our model more explicitly Schecter 2014; Bu et al. 2014; Peng et al. 2018); and the represents key aspects of this vertical structure: scaling of vertical velocity, precipitation extremes, and • A distinct boundary layer and free troposphere whose CAPE with climate in radiative-convective equilibrium properties (temperature and moisture) can be varied (Singh and O’Gorman 2013, 2014, 2015). independently; CM1 is particularly well-suited for this work for a number of reasons, including 1) it has demonstrated flexibil- • A capping inversion (as represented by the dry static ity across a range of scales and scientific questions; 2) energy jump between the two tropospheric layers); its excellent mass and momentum conservation properties; and 3) inclusion of various thermodynamic terms • A well-mixed boundary layer; often neglected in other numerical models (such as the • Direct specification of the free tropospheric lapse heat capacity of hydrometeors), which may be important 5 on convection-resolving scales. Moreover, CM1 uses a rate, in lieu of the arbitrary 4 power law increase in θ with height height-based vertical coordinate, which fits naturally with our static energy-based theoretical sounding framework. We note that there may be experimental applications for which the WK model is equally viable for defining a b. Experiments sounding or set of soundings. At a minimum, our model can provide clearer physical motivation for the structure The setup of the simulation domain, grid parameters, of any idealized sounding. Our model can otherwise of- and physical parameterizations closely follows that of fer more precise control over the structure of the sounding Dawson et al. (2019), though we neglect the Coriolis force J OURNALOFTHE A TMOSPHERIC S CIENCES 11

FIG. 4: Example of our model thermodynamic state (red) and comparison with the WK model (blue). WK input parameters are defined directly from our model sounding. and use free-slip lower boundary conditions only. Each of Name Details our simulation experiments is performed on a 200 km × 3MAY99 (Histor- Proximity sounding from a simulation of the 3 200 km × 20 km domain with a horizontal grid spacing ical) May 1999 tornado outbreak, from Dawson et al. (2010) of 250 m in an inner 100 × 100 km2 region and gradu- THEO Pure theoretical model fit to 3MAY99 ally stretched to 1 km at the lateral boundaries. The lat- MODHIST THEO with the low-level moisture set equal to eral boundary conditions are open radiative, while the top values from 3MAY99 historical event sounding and bottom boundaries are impermeable and free-slip. A (z ≤ 0.84 km) Rayleigh damping layer is located above 15 km with an MODTHEO THEO with enhanced constant free tropospheric inverse e-folding time of 1/300 s−1. The vertical grid has relative humidity (70%) 50 levels stretched from 20 m at the surface to ∼800 m at the domain top (20 km). The domain translates with TABLE 1: Soundings for our experiments. a constant [u, v] = [7.28, 8.78] m s−1 to keep the simulated storm near the center of the domain. Deep convection is initiated using the Naylor and Gilmore (2012) • Set Ps f c, Ts f c, RHs f c equal to the observed values. updraft nudging technique applied to an ellipsoidal region −1 with maximum w =10 m s and radii 10 km × 10 km × • Set HBL equal to the level of maximum RH. 1.5 km and centered at [x, y, z] = [100, 100, 1.5] km over the first 900 s of numerical model integration. The NSSL • Set ∆D equal to the difference between the mean dry triple-moment microphysics scheme (Mansell 2010; Daw- static energy in z ∈ (HBL, 3HBL] and the mean dry son et al. 2014) and a 1.5-order prognostic TKE turbu- static energy in z ≤ HBL. This captures the enhanced lence closure scheme (Deardorff 1980) is used. Finally, dry static energy at the base of the free troposphere. as is common in idealized CRM simulations of deep convection, no radiation or surface physics are included. All • Set Tt pp equal to the coldest temperature in the sound- simulations are run for 4 h. We perform simulation exper- ing, whose altitude defines Ht pp. iments using four soundings described in Table 1 to define the horizontally homogeneous initial environment. • Set ΓFT equal to the mean lapse rate in the layer We first perform a simulation with our example histori- z ∈ [HBL, HBL + 0.75(Ht pp − HBL)]. This average cal event sounding (3MAY99) from Dawson et al. (2010) avoids the top of the troposphere where lapse rates shown in Figure 2. We then perform simulations with our necessarily become more stable as they approach the model fit to 3MAY99 (THEO; fitting described in Section tropopause. 3c). Finally, we perform two experiments to illustrate distinct uses of the model: 1) experiment MODHIST, which • Set RHFT,0 equal to the free-tropospheric col- uses a semi-theoretical sounding that tests the inclusion W umn saturation fraction, W ∗ . The quantity W = of specific details of the real sounding into the theoreti- R ztop 0 1 R Ptop 0 ρqvdz = qvdP is the water vapor path cal model (here: low-level moisture); and 2) experiment zbot g Pbot ∗ MODTHEO, which uses a fully-theoretical sounding that and W is its saturation value (Bretherton et al. 2004; tests direct modifications of theoretical model parameters Camargo et al. 2014; Raymond et al. 2007). Each (here: free-tropospheric relative humidity). term is calculated within the free-tropospheric layer z ∈ (HBL,Ht pp). Saturation fraction is effectively identical to a mass-weighted relative humidity (see c. Fitting the model sounding Appendix B), and hence is also commonly called col- We fit our model thermodynamic profile to the historical umn relative humidity. This approach yields a sound- event sounding as follows: ing with nearly the same water vapor path as exists in 12 J OURNALOFTHE A TMOSPHERIC S CIENCES

−1 −1 the real sounding and thus avoids the addition of sig- constants are cBL = 0.0293 s , cFT,1 = 0.0139 s , and −6 −1 −1 nificant artificial sources or sinks of latent heat into cFT,2 = −5.367∗10 m s . Both soundings have sim- the column. ilar surface-based CAPE: 4711 J kg−1 for 3MAY99 and 4490 J kg−1 for THEO. Both soundings have identical 0-3 This algorithm will yield values of SBCAPE similar to the km bulk shear of 21 m s−1. historical event sounding. To compare the profiles in terms of standard mete- We fit our model kinematic profile to the sounding as orological variables, Figure 5d-f compare temperature, follows: mixing ratio, and relative humidity between THEO and • Set (us f c,vs f c) equal to the observed values. 3MAY99. The temperature structure is remarkably similar at all levels except near the top of the boundary layer

• Set c = ∂VBL equal to the average vector shear where THEO has a sharper capping inversion, indicating BL ∂z that in this case the use of a single free-tropospheric lapse magnitude in z < HBL. This matches the bulk total rate is quite reasonable. Meanwhile, clearly there are sig- shear magnitude between the surface and HBL and distributes this shear purely in the southerly direction. nificant vertical variations in free-tropospheric moisture in 3MAY99 that are not represented in our simple model, in- s • Set Htop to 3 km. This focuses on the low-level shear; cluding greater moisture in the lower free-troposphere and in the 3MAY99 sounding, most of the shear is con- less moisture in the middle free-troposphere. The role of fined to below 3 km (Figure 5c). these detailed variations could be tested in future experiments. • Set c = 2 ∂VFT , where ∂VFT equals the aver- FT,1 ∂z ∂z s age vector shear magnitude in HBL < z < Htop, and s set cFT,2 = −cFT,1/(Htop − HBL). This combination d. SCS simulation experiments matches the bulk total shear magnitude between HBL s We first perform a numerical simulation experiment us- and Htop and distributes this shear purely in the westerly direction, with a magnitude that decreases lin- ing the 3MAY99 sounding. Figure 6a displays time se- s early to zero at z = Htop (as noted above). ries of domain maximum vertical velocity and maximum vertical vorticity at 3km AGL and snapshots of surface This algorithm also matches the total bulk shear in the top simulated radar reflectivity at 1, 2, and 3 h into the sim- sounding across both layers (z ≤ Hs ). One potential ad- ulation. Our 3MAY99 simulation successfully produces a ditional kinematic constraint would be to fit the shear pro- long-lived supercell. Next, we perform a simulation exper- file to the storm-relative helicity. However, this requires iment using the theoretical sounding (THEO) and compare precise knowledge of the storm-motion vector, which is against 3MAY99, as shown in Figure 6b. While 3MAY99 a complex function of both the wind profile and inter- produces a long-lived supercell, THEO yields a short-lived nal storm processes such as cold pool propagation (e.g., convective cell that quickly dissipates after 1 hour despite Bunkers 2018). Nonetheless, we think this could be a having environments with similar SBCAPE and 0-3 km valuable addition that we leave for future work. bulk shear. Our approach is certainly not the only way to fit the Finally, we perform two demonstration experiments in model parameters. For example, while we fit the 0-3 km which we modify our THEO sounding to illustrate the ex- shear in this study, we note that the 0-6 km is the standard perimental utility of our model. The sounding used in shear layer for SCS forecasting (Doswell 2001). However, the model can be fit to any shear layer depending on the the first experiment (MODHIST) is ”semi-theoretical” in experimental purpose. that it is identical to THEO but in which rv is forced to Figure 5 displays the theoretical sounding (red, THEO) match 3MAY99 in the lowest 0.84 km (i.e. 2HBL). Thus it fit to our example historical event sounding (blue, demonstrates how a specific feature of a real-data sound- 3MAY99) following the algorithm described above. For ing may be incorporated into the model to test its impor- the THEO thermodynamic profile (Figure 5b), the bound- tance. The result is shown in Figure 7a. The experiment ary layer dry and moist static energies equal their re- with this slight modification now produces a long-lived spective 3MAY99 near-surface values. In the free tropo- supercell. The next experiment (MODTHEO) is “fully- sphere, the dry static energy jump is ∆D = 2095 J/kg; the theoretical” and demonstrates how physical parameters in boundary layer height is HBL = 0.42 km; the relative hu- the theoretical model can be directly varied to test their im- midity is RHFT,0 = 0.54; the free-tropospheric lapse rate portance. Experiment MODTHEO is identical to THEO is ΓFT = 7.34 K/km; and the tropopause temperature is but with the free-tropospheric relative humidity, RHFT,0, Tt pp = 211.25 K. For the THEO kinematic profile (Fig- enhanced to 70%. The result is shown in Figure 7b. This ure 5c), the surface flow vector equals the 3MAY99 value experiment also produces a long-lived supercell, similar to −1 of (us f c,vs f c) = (−2.64,5.83) m s . The shear profile both 3MAY99 and MODHIST. J OURNALOFTHE A TMOSPHERIC S CIENCES 13

FIG. 5: Comparison of soundings for 3MAY99 historical event (blue) and THEO (red). (a) THEO Skew-T; (b) dry and moist static energies; (c) wind shear; (d) temperature; (e) mixing ratio; (f) relative humidity. Both soundings have similar surface-based CAPE (3MAY99: 4711 J kg−1; THEO: 4490 J kg−1) and identical 0-3 km bulk shear (21 m s−1).

Overall, the results across our experiments suggest a updraft source parcels for the simulated storm are com- substantial sensitivity of convective evolution to the ver- ing from above the boundary layer, where the air is sim- tical structure of moisture in both the BL and FT. They are ply too dry and stable in THEO, such that they dilute the consistent with many past studies that highlight the im- unstable parcels coming from within the boundary layer. portant role of variability in the vertical thermodynamic This is in keeping with the lower magnitude of MLCAPE structure in governing storm dynamics (e.g., McCaul Jr and higher magnitude of MLCIN in THEO. Additionally, and Weisman 2001; McCaul and Cohen 2002; McCaul the greater free-tropospheric moisture in MODTHEO may et al. 2005; Cohen and McCaul 2007; Kirkpatrick et al. result in less dilution of updraft parcels throughout their 2009; James and Markowski 2010; Dawson et al. 2012; ascent such that they realize more of their CAPE, which Guarriello et al. 2018; Brown and Nowotarski 2019). For is consistent with the findings of James and Markowski example, we note that moisture at the base of the free tro- (2010). posphere varies across our experiments and hence yields Ultimately, though, our experiments are not intended to different values of MLCAPE and MLCIN. Owing to both demarcate robust sensitivities, nor can we can we cleanly the shallower moisture and the sharper cap in THEO, the attribute differences in qualitative behavior to any specific MLCAPE and CIN for THEO is 2447 J kg−1 and -47 J feature of the sounding (e.g. boundary layer vs. lower- kg−1, respectively, as compared with 3852 J kg−1 and - tropospheric vs. mid-tropospheric moisture) or to changes 6 J kg−1 for 3MAY99. For MODHIST, by replacing the in specific bulk parameters such as MLCAPE. Instead, our THEO low-level moisture profile with that in 3MAY99, results motivate how the model could be used as the ba- the MLCAPE increases to 3552 J kg−1 and the MLCIN sis for comprehensive testing of the role of these detailed decreases to -15 J kg−1, while the SBCAPE and SBCIN variations in vertical thermodynamic structure in the SCS are very similar to 3MAY99. For MODTHEO, despite outcome. Such an approach would require in-depth ex- having the same BL structure as THEO, the increase in perimentation via experimental ensembles and consider- free-tropospheric moisture causes the MLCAPE to in- ation of a range of carefully-defined soundings, whether crease to 3370 J kg−1 and the MLCIN to decrease to - semi-theoretical (akin to MODHIST) or fully-theoretical 19 J kg−1. Thus, one hypothesis for the failure to pro- (akin to MODTHEO). This effort lies beyond the scope of duce a long-lived supercell in THEO is that at least some this work. Here we focus simply on presenting the model 14 J OURNALOFTHE A TMOSPHERIC S CIENCES

FIG. 6: Simulated supercell evolution associated with (a) historical-event sounding (3MAY99) shown in Figure 2 from the 3 May 1999 tornado outbreak; (b) theoretical sounding (THEO; red) fit to the 3MAY99 sounding, with 3MAY99 result (blue) repeated for comparison. Timeseries show peak vertical velocity (solid) and 3km AGL vertical vorticity (dash) at 3 km AGL, with snapshots of reflectivity (dBz, boxes). construction and demonstrating how it could be used to Here we have presented a simple physical model for the improve our understanding of the effects of any type of combined steady thermodynamic and kinematic profiles variability in a sounding in a simplified setting. associated with severe convective storm environments. The thermodynamic component of the model builds off of 4. Conclusions the two-layer static energy framework proposed by Agard and Emanuel (2017). The model superposes a boundary Severe convective storm activity depends not only on layer with constant moist and dry static energy and con- bulk parameters such as CAPE and lower-tropospheric stant southerly shear beneath a free tropospheric layer with shear but also on the detailed vertical structure of the dry static energy increasing linearly with height (allow- thermodynamic and kinematic profiles that can vary in- ing a sub-dry-adiabatic lapse rate), constant relative hu- dependently of those bulk parameters. Past simulation work has tested these dependencies using the Weisman midity, and pure westerly shear. A step-function increase and Klemp idealized thermodynamic profile model, whose in dry static energy, which represents a capping inversion simple parametric construction was motivated by practical that scales with convective inhibition, is imposed across utility for SCS research. A preferable alternative would be the boundary layer top. The model is topped off with a model whose structure is defined on physical grounds, a dry isothermal stratosphere that defines the tropopause and in a manner consistent with how such environments temperature. Overall, the thermodynamic and kinematic are generated within the climate system. Such a model components are mutually consistent, as they represent a could be a useful tool for understanding how SCS evo- transition from predominantly southerly flow advecting lution depends on the vertical structure of the hydrostatic warm, moist (i.e. high moist static energy) air near the background environment (i.e. towards smaller scales), as surface to predominantly westerly flow advecting warmer, well as how SCS environments depend on the process- dry (i.e. high dry static energy) air aloft. This static energy level energetics of the hydrostatic atmosphere (i.e. to- framework provides greater physical insight into the ad- wards larger scales). hoc structure of the Weisman and Klemp sounding while J OURNALOFTHE A TMOSPHERIC S CIENCES 15

FIG. 7: Simulated supercell evolution for two experiments modifying THEO. (a) MODHIST (red), whose sounding is identical to that of THEO except with rv forced to match 3MAY99 in the lowest 0.84 km. (b) MODTHEO (red), whose sounding is identical to that of THEO except with RHFT,0 enhanced to 70%. 3MAY99 evolution also shown (blue). In both experiments, a long-lived supercell emerges as was found with 3MAY99. Plot aesthetics as in Figure 6b. offering novel benefits, particularly the explicit represen- is constructed, and how it can be applied to a real-data tation of a capping inversion at the interface between the sounding for sensitivity testing and controlled experimen- boundary layer and the free troposphere, each of which tation. The specific outcomes of our simulation examples may be varied independently. shown here are not intended to demonstrate robust sensi- To demonstrate its experimental utility, we then pro- tivities. Such tests of any individual parameter or struc- vided an algorithm for creating a model sounding as well tural feature requires careful experimentation accounting as for fitting the model to a real-data sounding associated for key sensitivities in model design via ensemble simu- with the 3 May 1999 Oklahoma tornado outbreak. Using lations to ensure real, systematic variability and to falsify numerical simulation experiments, the real-data sounding alternative hypotheses. produces a long-lived supercell, whereas our theoretical Importantly, the specific construction of our theoretical sounding produces only a short-lived storm. We then model as presented here should not be interpreted as fi- demonstrate two specific types of experiments with our nal. Instead, this framework should be viewed as a flexible theoretical model that also simulate a long-lived super- minimal model sufficient to define a viable SCS sounding cell: 1) experiments using semi-theoretical soundings that – i.e. one with substantial CAPE and vertical wind shear, test the importance of specific features in real soundings and relatively low CIN. Note that this does not guarantee by incorporating them directly into the model (here we that any specific SCS type (e.g. supercell) will form for matched the low-level moisture); and 2) experiments us- a given set of model parameters. This is a natural base ing fully-theoretical soundings that test direct variations model that can be used to test hypotheses regarding SCS in the model’s physical parameters (here we enhanced the environmental dependencies. Experiments could vary ver- free-tropospheric relative humidity). These two types of tical thermodynamic structure at fixed CAPE or vertical experiments demonstrate the potential utility of our theo- kinematic structure at fixed bulk shear over different shear retical model for testing how SCS evolution depends on layer depths. Structural features may be added or mod- details of the vertical structure of a sounding. ified as needed; there is no single “correct” model. We This work has focused narrowly on presenting for the hope that future research testing the model and modifica- community the motivation for the model, how the model tions to it may identify other features that are essential to 16 J OURNALOFTHE A TMOSPHERIC S CIENCES

SCS morphology and evolution and thus may be incorpo- We may use the Ideal Gas Law and hydrostatic balance rated into this minimal model for practical applications to (Eq. (8)) to rewrite the log-pressure differential term understanding the diverse range of SCS types on Earth. dlnP = − g dz. Substituting in and rearranging yields Rd T Moreover, we note that this modeling framework may potentially be adaptable to other types of convective scenar- θ g dln = dz (A3) ios, such as nocturnal convection. T CpT Finally, the phrasing of the model in terms of moist static energy aligns neatly with the ﬁeld of climate Integrating both sides from the tropopause upwards yields physics, whose principal focus is the global and regional θ θ g Z z 1 energy budget of our hydrostatic atmosphere. Hence, the ln − ln t pp = dz0 (A4) model may provide a useful tool for understanding how T Tt pp Cp zt pp T and why SCS environments are produced within the climate system in the ﬁrst place. Understanding how SCS which can be rewritten as activity will change in a future climate state depends on Z z T g 1 0 understanding not only changes in bulk parameters such as θ = θt pp exp dz (A5) T C T CAPE but also changes in the vertical thermodynamic and t pp p zt pp kinematic structure within favorable SCS environments. Taking T(z) = Tt pp constant (i.e. isothermal) yields

g θ = θt ppexp (z − zt pp) (A6) Data Availability Statement. All data and code from this CpTt pp manuscript are available by emailing the corresponding author at [email protected]. which matches Eq. (26) above the tropopause.

Acknowledgments. The authors thank John Peters, APPENDIX B Matt Gilmore, and one anonymous reviewer for their useful feedback that helped improve this manuscript. This Saturation fraction vs. relative humidity work also benefited from discussions with Tim Cronin For Earth-like atmospheres in which the saturation va- and Mike Baldwin. Chavas was partially supported by por pressure is small compared to the total pressure, i.e. NSF grant 1648681 and NOAA grant NA16OAR4590208. e∗ P, one can show that the saturation fraction and rel- Dawson was partially supported by NOAA grants ative humidity of an air parcel are nearly identical. As a NA16OAR4590208 and NA18OAR4590313. The authors result, the two quantities will also be nearly identical for gratefully acknowledge the open-source Python commu- any mass-weighted layer average. nity, and particularly the authors and contributors to the Relative humidity is defined as Matplotlib (Hunter 2007) and MetPy (May et al. 2008 - e 2020) packages that were used to generate many of the RH = (B1) figures. Computational resources were provided by the e∗ Purdue RCAC Community Cluster program. Saturation fraction is defined as q SF = (B2) APPENDIX A q∗

Potential temperature expression for an isothermal These equations may be combined with the relations layer r Here we show how the potential temperature equation q = (B3) 1 + r above the tropopause in the WK sounding (second equation in Eq. (26)) yields an isothermal layer. This result is and e obtained by first taking the natural logarithm of the defini- r = ε (B4) tion of potential temperature (Eq. (14)) to yield P − e Rd R and their saturated counterparts, where ε = R = 0.622 lnθ = lnT − d (lnP − lnP ) (A1) v C 0 is the ratio of specific gas constants for dry air and water p vapor. The result may be written as Taking a differential yields ∗ ! 1 − (1−ε)e Rd SF = RH P (B5) dlnθ = dlnT − dlnP (A2) (1−ε)e Cp 1 − P J OURNALOFTHE A TMOSPHERIC S CIENCES 17

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