On the Physics of High CAPE

Kerry Emanuel aLorenz Center, Massachusetts Institute of Technology, Cambridge, Massachusetts

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Abstract

Large values of convective available potential energy (CAPE) are an important ingredient for many severe convective storms, yet there has been comparatively little research on how, physically, such large values arise or why they take on the observed values and climatology. Here we build on recently published observational and theoretical work to construct a simple, one-dimensional coupled soil–atmosphere model of preconvective boundary layer growth, driven by a single diurnal cycle of prescribed net surface radiation. Based on this model and previously published research, we suggest that high CAPE (>∼1000 J kg−1) results when air masses that have been significantly modified by passage over dry, lightly vegetated soils are advected over moist and/or moderately vegetated soils and then exposed to surface solar heating. Several diurnal cycles may be needed to raise the moist static energy of the boundary layer to levels consistent with high CAPE. The production of CAPE and erosion of convective inhibition (CIN) are strongly affected by the potential temperature of the desert-modified air mass, the level of near-surface soil moisture (and root-zone soil moisture if significant vegetation is present), the type of soil, and the characteristics of the vegetation. Consequently, CAPE production and severe convective weather may be significantly affected by regional-scale land-use changes and by climate change.

Significance Statement

The energy available for severe convective storms depends strongly on the properties of the underlying soil and vegetation and the temperature of air masses formed over dry terrain upstream. This implies that the severity of convective storms can be strongly affected by changes in land use and by climate change.

© 2023 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Kerry Emanuel, emanuel@mit.edu

Abstract

Large values of convective available potential energy (CAPE) are an important ingredient for many severe convective storms, yet there has been comparatively little research on how, physically, such large values arise or why they take on the observed values and climatology. Here we build on recently published observational and theoretical work to construct a simple, one-dimensional coupled soil–atmosphere model of preconvective boundary layer growth, driven by a single diurnal cycle of prescribed net surface radiation. Based on this model and previously published research, we suggest that high CAPE (>∼1000 J kg−1) results when air masses that have been significantly modified by passage over dry, lightly vegetated soils are advected over moist and/or moderately vegetated soils and then exposed to surface solar heating. Several diurnal cycles may be needed to raise the moist static energy of the boundary layer to levels consistent with high CAPE. The production of CAPE and erosion of convective inhibition (CIN) are strongly affected by the potential temperature of the desert-modified air mass, the level of near-surface soil moisture (and root-zone soil moisture if significant vegetation is present), the type of soil, and the characteristics of the vegetation. Consequently, CAPE production and severe convective weather may be significantly affected by regional-scale land-use changes and by climate change.

Significance Statement

The energy available for severe convective storms depends strongly on the properties of the underlying soil and vegetation and the temperature of air masses formed over dry terrain upstream. This implies that the severity of convective storms can be strongly affected by changes in land use and by climate change.

© 2023 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Kerry Emanuel, emanuel@mit.edu

1. Introduction

Severe convective storms are a significant weather hazard. Globally, from 2000 through 2022, severe convective storms resulted in an average of about 400 fatalities per year and caused, on average, $15 billion in damages annually, in year 2000 U.S. dollars (EM-DAT 2022). Yet only in recent decades have we begun to explore how climate change may affect the incidence and severity of these storms.

Owing to their small size, short duration, and comparative rarity, direct reporting of severe convective storms tends to be temporally and spatially uneven. For this reason, most attempts to surmise trends in severe storms have relied on environmental proxies, most prominently low-level wind shear, CIN, and CAPE. A number of studies looked for trends in such proxies in reanalysis and/or rawinsonde data (e.g., Riemann-Campe et al. 2009; Tang et al. 2019; Taszarek et al. 2021), while others sought to detect proxy trends directly in global climate model output (e.g., Diffenbaugh et al. 2013; Lepore et al. 2021) or, more commonly, regional models driven by global model boundary conditions (Ye et al. 1998; Del Genio et al. 2007; Trapp and Hoogewind 2016; Trapp et al. 2007; Hoogewind et al. 2017; Rädler et al. 2019; Chen et al. 2020; Rasmussen et al. 2020; Glazer et al. 2021; Ashley et al. 2023). Almost all these studies show increasing trends of CAPE, particularly over middle- and high-latitude continents, both in reanalysis data and in historical simulations and future projections. Most, but not all these studies also detect increases in CIN. While, in general, low-level wind shear decreases as the climate warms, there is some indication that it may increase when and where CAPE is large (Diffenbaugh et al. 2013). A few studies (e.g., Trapp and Hoogewind 2016; Hoogewind et al. 2017; Rasmussen et al. 2020; Ashley et al. 2023) directly simulate severe convective storms using cloud-permitting models driven by boundary conditions from climate models. These also show increasing intensity of severe convective storms, and there is an indication that the incidence of weaker storms may decline. It should be noted, however, that even such cloud-permitting models do not resolve entraining eddies, which can have a large influence on the intensity of deep convection (Peters et al. 2023).

Notwithstanding the popularity of proxy trends, several studies of tornado activity in the United States suggest that the power and size of tornadoes has been increasing (e.g., Elsner et al. 2019).

Taken together, these studies suggest the possibility of increasing incidence of damaging convective storms. Here we seek to better understand the physical factors that are driving this increase, both from a general desire to increase our understanding of convective storms and from the practical question of identifying and addressing deficiencies in models. For example, a high CAPE bias in CMIP6-generation models has been identified (Chavas and Li 2022). In this paper, we confine ourselves to better understanding the thermodynamic factors, CAPE and CIN, which are important contributors to trends in severe convective storms identified in the aforementioned studies; we do not here address changes in wind shear, which is a very important factor in severe convection. Nor do we address the problem of convective triggering. Our study focuses on how CAPE accumulates regardless of whether convection is triggered; it does not address severe convective storms per se. We emphasize that severe convective storms can occur with relatively low values of CAPE (Schneider and Dean 2008), and that high values of CAPE do not necessarily lead to severe convection.

In regimes of deep convection over tropical oceans, CAPE defined by reversible lifting of air from near the top of the subcloud layer is observationally indistinguishable from zero (Xu and Emanuel 1989), and phenomena associated with strong convective updrafts, such as lightning, hail, and tornadoes, are rare. Such regimes are close to a state of deep convective quasi equilibrium and their mean states are close to neutral. Over land, the diurnal cycle of surface forcing is too fast for convective quasi equilibrium to apply, and CAPE can accumulate during daytime heating (Jones and Randall 2011). Here we focus on the origin of CAPE values that are larger than those typically achieved in ordinary diurnal convection over land, roughly 1000 J kg−1.

Our work is strongly guided by two recent studies. The first, by Agard and Emanuel (2017), developed a simple one-dimensional model of the growth of convective boundary layers over land, assuming surface energy balance at each time step and a step function increase of sunlight at dawn. This study showed that temperature, surface moisture availability, and surface wind speed are all important determinants of the diurnal evolution of CIN and CAPE, and that, all other things being equal, diurnal peak CAPE increases rapidly with system temperature.

The second study (Tuckman et al. 2023) used high-resolution North American Regional Reanalysis (NARR) data to track air samples backward in time from locations and times of high CAPE over the continental United States. This study showed that the largest increase in CAPE generally occurs on the day that peak CAPE was identified and that, as had been deduced in many previous studies, advection of hot, dry air in an elevated mixed layer formed over dry, usually elevated terrain, is important in forming the capping inversion and associated CIN that allowed the CAPE to build up. In quite a few cases, the moist, boundary layer air associated with the CAPE peaks had not been over water for many days, with the elevation of boundary layer moist static energy owing to one or a few diurnal cycles of enthalpy flux from the land surface. This suggests that the presence of a warm body of water, such as the Gulf of Mexico, may not be of direct importance in high CAPE episodes, in agreement with the recent study by Li et al. (2021). (On the other hand, it may play an indirect role in providing moisture for precipitation that affects local soil moisture.)

These two studies emphasize the importance of the land surface, including vegetation, which affects surface albedo and the flux of water from subsurface soil to the atmosphere. If the surface is too dry, then the daytime growth of the convective boundary layer is too fast, entraining low moist static energy from above and eliminating CIN before much CAPE has built up. If the surface is too moist, on the other hand, there may not be enough sensible heating to erode the CIN before sunset, and although CAPE can become quite large, it is difficult to release it.

The importance of soil properties has been emphasized in many previous studies of convection over land. For example, Clark and Arritt (1995) developed a single-column atmospheric model coupled to a one-dimensional soil model and showed, among other things, that vegetation enhances convective rainfall over land. Likewise, Betts et al. (1996) demonstrated that summer continental convective precipitation is sensitive to the initialization of soil moisture. Indeed, a variety of studies (e.g., Schär et al. 1999; Findell and Eltahir 2003; Barthlott and Kalthoff 2011; Schlemmer et al. 2012; Leutwyler et al. 2021) show that summertime precipitation on regional to continental scales is enhanced by wetter soils. There is also a large body of literature showing that on local to regional scales, inhomogeneities of soil properties, including moisture, albedo, and vegetation, have a strong effect on convective triggering, with a preference for triggering over drier (and therefore hotter) soils (Emori 1998; Taylor and Ellis 2006; Hohenegger et al. 2009; Osuri et al. 2017), especially in those models in which convection is explicitly simulated. Taylor and Ellis (2006) also demonstrated, using satellite-based passive microwave measurements, that, on small scales, convection is slow to develop over moist soils.

In contrast with these studies, our work is focused on the production of very high values of CAPE in the days and hours leading up to the outbreak of convection. We do not concern ourselves here with either the triggering of convection or the influence of soil properties on the amount or distribution of convection or convective precipitation; rather, we attempt to explain the climatology of exceptionally large values of CAPE. Of particular relevance to our work is that of Lanicci et al. (1987). They used a mesoscale model to explore the physical factors leading to high CAPE values over the U.S. Plains states in spring and identified two critical factors. First is the production of a capping inversion and associated large CIN by the advection of air modified by the high deserts of the southwest United States and northern Mexico. When the soils of northern Mexico were artificially moistened in the model, the capping inversion over the Plains was significantly weakened, so not much CAPE could build up before being released in moist convection. Second, the production of CAPE under the capping lid proved sensitive to soil conditions, particularly soil moisture.

A more recent study by Zhang (2022) examined the evolution of CAPE and CIN during “dry-downs,” periods between rainfall events. Using satellite-based measurements of soil moisture and atmospheric temperature and humidity profiles, Zhang (2022) shows that composite values of both CAPE and CIN build up during these periods where and when soils are relatively wet, but remain roughly constant or decline in dry regions and times.

In the following sections, we build on the seminal work of Lanicci et al. (1987) and the more recent work of Agard and Emanuel (2017), Zhang (2022), and Tuckman et al. (2023) by coupling the simple, one-dimensional model of the growth of a convective boundary layer of Agard and Emanuel (2017) with a simple, soil and vegetation model and using it to explore the physics of high CAPE production.

2. A simple model of diurnal CAPE production

a. The conceptual picture

Our conceptual picture of an idealized diurnal evolution of CAPE in illustrated in Fig. 1. We begin with a simple, deep, dry-adiabatic boundary layer presumed to have been formed over dry land or desert (Fig. 1a). This extends up to a particular altitude which, for this purpose of this study, is held fixed at 3 km. Above this altitude, potential temperature increases up to the tropopause. For simplicity, the water content of this air mass is set to zero.

Fig. 1.
Fig. 1.

Three stages in the diurnal evolution of CAPE. (a) A desert-modified air mass with little moisture and a deep dry adiabatic boundary layer is advected over a moist soil. (b) Advection of cool, moist air from a different region and/or wind-driven boundary layer turbulence creates a moist but cool new shallow boundary layer just saturated at its top. Dewpoint is shown by dashed lines, and the temperature of a parcel lifted adiabatically is shown by the dotted curve. (c) Surface heating by solar radiation causes the now-convective boundary layer to deepen and moisten, though the moistening is at least partially offset by entrainment of dry air from above the boundary layer. CIN decreases and CAPE increases, with the proportion depending on surface moisture availability.

Citation: Journal of the Atmospheric Sciences 80, 11; 10.1175/JAS-D-23-0060.1

In the next step (Fig. 1b), this air mass is placed bodily over a moist and/or vegetated soil, conceptually at night, and turbulence generated by background winds is presumed to form a shallow mixed layer. Fluxes of moisture from the soil moisten this layer while sensible fluxes into the soil cool it. Neglecting heat storage in the soil, radiative cooling, and horizontal advection, the moist static energy of the boundary layer air would remain constant. The partitioning of temperature and moisture in this boundary layer is determined such that the well-mixed air within it has a prescribed relative humidity at the prescribed top of the boundary layer. The moist static energy of the boundary layer is also prescribed. (In reality, the cool morning mixed layer usually arises from advection of dry air over a preexisting moist layer and from nocturnal radiative cooling that lowers its moist static energy. Our idealization here aims at maximum simplicity.)

In the final, time-dependent step (Fig. 1c), the column from the second step is exposed to time-dependent solar radiation, and the turbulence at the top of the boundary layer is presumed to switch from mechanically to buoyantly driven at sunrise. Sensible fluxes from the underlying soil drive convective turbulence, leading to entrainment at the boundary layer top and growth of the boundary layer. Warming of the boundary layer by sensible heat fluxes at both the surface and boundary layer top erodes the negative buoyancy of lifted parcels and leads to a reduction in CIN. The boundary layer moistens by evaporation from the soil surface and transpiration from any vegetation that is present. On the other hand, entrainment of dry air from above the boundary layer slows this moistening and damps the increase in moist static energy. CAPE may eventually develop. The evolution of this idealized system ceases at sunset, or when the CIN disappears, whichever happens first. We do this for expedience, noting that, in reality, deep convection can be triggered before CIN vanishes and may not erupt even when it does.

It is straightforward to see from Fig. 1c that an absolute upper bound on the boundary layer moist static energy corresponds to boundary layer air achieving the dry static energy of the desert-modified air and being saturated at the surface. As shown by Agard and Emanuel (2017), for a given sounding above the boundary layer, this upper bound on CAPE will scale with the Clausius–Clapeyron equation applied to air saturated at surface pressure and temperature following the dry adiabat of the desert-modified air down to the surface.

In the following two sections we quantify this conceptual picture using a simple, one-dimensional coupled soil–atmosphere model.

b. Atmospheric model

The atmospheric model is that developed by Agard and Emanuel (2017), except that we allow for time-dependent solar radiation. For clarity, we repeat the model equations here.

We consider the growth of a shallow boundary layer into a deep, dry adiabatic layer with no moisture, characterized by equal dry and moist static energies of magnitude D0. The equations for dry and moist static energy, D and M, are
ρhdDdt=Fs+ρwe(D0D),
and
ρhdMdt=Fs+FL+ρwe(D0M),
where ρ is the air density, h is the (variable) depth of the boundary layer, we is the entrainment velocity at the top of the boundary layer, Fs is the surface sensible heat flux, and FL is the surface latent heat flux (including transpiration from plants).
We use Lilly’s (1968) formulation for the entrainment velocity:
ρwe=ρdhdt=AFsD0D,
where A is a dimensionless constant, which we take to be 0.2. (In this simple model, we have taken the surface buoyancy flux to be owing to the sensible heat flux alone.)
We use aerodynamic flux formulas for the surface sensible and latent fluxes:
Fs=ρCk|V|(DsD)
and
FL=Fsoil+Fveg,
where
Fsoil=ρCk|V|(MsMDs+D)(1veg)
and
Fveg=ρCk|V|hυ(M*MDs+D)veg,
where “veg” is the ground cover of vegetation (0 to 1), Ck is the turbulent exchange coefficient for sensible and latent heat (assumed equal), |V| is the near-surface wind speed, and Ds is the dry and static energy of air at the temperature of the surface soil. The effective moist static energy of the soil surface is defined as
MscpTs+αLυqs*+gz,
where α is a number between 0 and 1 that describes the degree of saturation of the soil surface and will be discussed in the soil section of this study. The flux coefficient from vegetation, hυ, is also defined in the next section. In (8), qs* is the saturation specific humidity of air at surface pressure and at the temperature of the surface soil. The quantity M* is given by (8) with α = 1.

c. Soil and vegetation model

We here use the simple soil model of Noilhan and Planton (1989). This substitutes a force-response formulation for a truly one-dimensional soil model. Besides the soil model parameters, the coupled model will also depend on the initial soil and vegetation conditions chosen. We assume that no precipitation occurs during the 12–24-h simulations we perform, and therefore neglect water droplets clinging to vegetation, which can be an important source of surface evaporation.

The three surface variables we must predict are the surface temperature Ts, the degree of surface soil saturation α, and the flux coefficient hυ governing transpiration. The parameter α, appearing in (8), is parameterized as
α={12[1cos(πqgqfl)]forqg<qfl1forqgqfl,
where qg is the surface volumetric soil water content, and qfl is the field capacity. We relate field capacity to soil type following Rawls et al. (1982) and Hakojärvi (2015).
The rate of transpiration is limited by the coefficient hυ in (5); this specified as a function of aerodynamic resistance Ra and vegetation surface resistance Rs:
hυ=RaRa+Rs,
where
Ra=1Ck|V|
and
Rs=RsminLAIF11F21F31F41.
Here Rsmin is a lower limit on surface resistance and LAI is the leaf area index, and F1 measures the influence of photosynthetically active radiation and is defined
F1=1+ff+Rsmin/Rsmax,
with
f=0.55SGSGL2LAI,
where SG is the incoming solar radiation and SGL is a limiting value that is taken to be 30 W m−2 for a forest and 100 W m−2 for a crop.
The factor F2 takes into account the effect of the water stress on the surface resistance; it varies between 0 and 1 when the mean volumetric water content, q2, varies between the wilting point volumetric water content, qwilt, and a critical value qcr of 0.75qsat:
F2={1,ifq2>qcrq2qwiltqcrqwilt,ifqwiltq2qcr0,ifq2<qwilt.
The factor F3 represents the effect of atmospheric humidity on surface resistance and is given by
F3=1γ[e*(Ts)ea],
where e* is the saturation vapor pressure and ea is the actual vapor pressure. The coefficient γ is an empirical parameter, dependent on the type of vegetation.
Finally, the factor F4 governs the air temperature dependence of the surface resistance and is given by
F4=10.0016(298.0Ta)2.
Now we need to know the surface and the total volumetric water contents qg and q2, and the surface soil temperature Ts, which we obtain from four time-dependent equations:
Tst=CT(FnetFsFL)πt0(TsT2),T2t=12t0(TsT2),qgt=C1ρwd1EsoilC22t0(qgqgeq),q2t=1ρwd2(Esoil+Eveg),
where Fnet is the net surface radiative flux, Esoil = Fsoil/Lυ, Eveg = Fveg/Lυ,
CT=11vegCG+vegCυ,
CV = 10−3 K m2 J−1, and CG is parameterized as
Cg=Cgsat(qsatq2)b/[2log(10)],
where qsat is the saturation water content, which equals the porosity, which we take from Geotechdata.info (2013), and Cgsat and b are given in Table 2 of Noilhan and Planton (1989) as a function of soil type. Note that the tendencies of soil moisture are negative definite because we are assuming no precipitation (but we will only be integrating over one diurnal cycle).
In (18), T2 is the soil layer temperature and d2 is the soil depth, while d1 is a normalization depth of 0.1 m and ρw is the density of liquid water. The quantity qgeq is given by
qgeq=q2qsata(q2qsat)p[1(q2qsat)8p],
with a and p given in in Table 2 of Noilhan and Planton (1989). Finally, the constants C1 and C2 are given by
C1=C1sat(qsatqg)b/2+1,
and
C2=C2ref(q2qsatq2+ql),
where ql is a small numerical value that caps C2. The constants C1sat and C2ref are provided as a function of soil type in Noilhan and Planton (1989).

3. Results

a. Solutions with surface energy balance

We begin by showing solutions that impose surface energy balance and specify the parameter α in (8), controlling surface water availability, as in Agard and Emanuel (2017). We depart from that work in specifying a somewhat realistic diurnal evolution of incoming solar radiation, rather than a step function. Specifically, we specify the net surface incoming radiation, including the infrared component, as
Fnet=F0[max(sin2πt1day,0)1π],
where F0 is the diurnal peak surface radiative flux. The factor of 1/π in (24) represents an assumed constant net outward infrared flux, such that the diurnally integrated surface radiative flux vanishes. This function is illustrated in Fig. 2.
Fig. 2.
Fig. 2.

Imposed net surface radiative flux over a 24-h period as given by (24). Note that the model integration begins at sunrise (0600 local time).

Citation: Journal of the Atmospheric Sciences 80, 11; 10.1175/JAS-D-23-0060.1

The present setup also differs from that in Agard and Emanuel (2017) in that the former imposed an arbitrary depression of surface temperature and dewpoint from the initial, dry-adiabatic desert air whereas we here hold the moist static energy constant and partition the latter into sensible and latent components such that the prescribed boundary layer top is just saturated. This is also a somewhat arbitrary assumption, and makes the initial boundary layer depth somewhat more important than it otherwise might be.

Finally, we impose a sub-dry-adiabatic lapse rate between 3 km altitude and the tropopause whereas Agard and Emanuel (2017) continued the dry adiabatic layer through the whole troposphere. This change does not affect the model integration (since the daytime PBL top never rises as far as 3 km) but does affect the magnitude of CAPE.

The system (1)(5) with (8) is numerically integrated as in Agard and Emanuel (2017), using a leapfrog scheme with an Asselin filter.

For the particular choice of α = 0.8 and |V| = 8 m s−1, Fig. 3 shows the diurnal evolutions of CAPE and CIN for potential temperatures of the initial desert air ranging from 294 to 310 K. While the growth of CAPE begins at approximately 1000 local time, almost independent of initial desert air potential temperature, the peak CAPE values and the local time at which CAPE peaks both increase rapidly with desert air temperature. The CIN also increases rapidly with desert air temperature, and in each case falls off with time, finally vanishing later and later in the day until, for desert air potential temperatures greater than about 307 K, the CIN has not vanished by sunset, leaving some CAPE stranded. Note that no CAPE appears for desert air potential temperatures less than about 296 K for this choice of moisture availability and surface wind speed. The decline in CAPE after reaching its peak is owing to entrainment of dry air as the boundary layer grows, and later in the day to declining surface radiation as well.

Fig. 3.
Fig. 3.

Diurnal evolution of (a) CAPE and (b) CIN for nine values of desert air potential temperature, ranging from 294 to 310 K. For these plots, the soil moisture availability parameter α is set to 0.8, while the surface wind speed |V| is 8 m s−1.

Citation: Journal of the Atmospheric Sciences 80, 11; 10.1175/JAS-D-23-0060.1

The dependence of peak CAPE on desert air temperature is illustrated in Fig. 4a. Peak CAPE rises nonlinearly with desert air temperature, as in Agard and Emanuel (2017), but assumes a more linear increase for desert air potential temperatures above about 304 K. As peak CAPE values occur later and later in the day, they are limited by decreasing sunlight. For this choice of parameters, the magnitude of CIN at the time of peak CAPE (Fig. 4b) is small (around 50 J kg−1) and roughly constant for desert air potential temperatures up to about 306 K; beyond that CIN rises rather fast, reaching almost 110 J kg−1 when θ0 = 310 K.

Fig. 4.
Fig. 4.

(a) Diurnal peak values of CAPE corresponding to the evolutions shown in Fig. 3, with α set to 0.8, and the surface wind speed |V| to 8 m s−1. (b) Values of CIN at the time of peak CAPE are shown.

Citation: Journal of the Atmospheric Sciences 80, 11; 10.1175/JAS-D-23-0060.1

The CAPE and CIN evolutions are sensitive to both surface wind speed and soil moisture availability, α. We show the CAPE dependencies in Figs. 5 and 6. CAPE does not develop for sufficiently small surface moisture availability or desert air potential temperature, and increases monotonically with both variables above a threshold curve in αθ0 space. Figure 6 shows that the dependence of diurnal peak CAPE on surface wind speed is nonmonotonic, with a local minimum at particular values of surface wind speed that decline with increasing θ0. CIN at the local time of peak CAPE (not shown here) is generally smaller than about 50 J kg−1 when θ0 is less than about 304 K, but can reach values greater than 100 J kg−1 for larger values of the desert air potential temperature.

Fig. 5.
Fig. 5.

Contours of diurnal peak CAPE (J kg−1) as a function of the moisture availability α, and the desert air potential temperature θ0, for a specified surface wind speed of 8 m s−1.

Citation: Journal of the Atmospheric Sciences 80, 11; 10.1175/JAS-D-23-0060.1

Fig. 6.
Fig. 6.

Contours of diurnal peak CAPE (J kg−1) as a function of surface wind speed and the desert air potential temperature θ0, for a specified surface moisture availability of 0.8.

Citation: Journal of the Atmospheric Sciences 80, 11; 10.1175/JAS-D-23-0060.1

The physics underlying the dependencies of CAPE and CIN on initial and environmental parameters is fairly straightforward. Referring back to Fig. 1b, higher desert air potential temperatures are associated with a larger drop in the temperature of the new boundary layer in which moist static energy is preserved but the top is assumed to be just saturated. This explains the larger initial CIN evident in Fig. 3b. The larger cap allows more moist static energy to accumulate in the boundary layer as enthalpy fluxes from the surface increase during the day. Were it not for the limited length of the day, CAPE would scale with the Clausius–Clapeyron equation (Agard and Emanuel 2017). This scaling still applies at desert air potential temperatures small enough that CIN vanishes before sunset. For larger values of θ0, CAPE is limited by the length of the day.

The nonlinear dependence of CAPE on imposed surface wind speed (Fig. 6) is more challenging to understand. Because surface energy balance is imposed, the net enthalpy flux from the surface must equal the (specified) incoming radiation and therefore is not affected by wind speed. However, the partitioning of this net flux into sensible and latent components does depend on wind through the aerodynamic flux formulas.

At very low winds speeds, the surface must become relatively hot to maintain the specified total enthalpy flux. But the hotter the surface, the more the surface enthalpy is dominated by the latent heat term, owing to the nonlinearity of Clausius–Clapeyron. In this limit most of the surface enthalpy flux will be in latent form, so dry, boundary layer convection is weak, and entrainment at the boundary layer top is therefore also weak. The weak entrainment and shallow boundary layer depth lead to large diurnal increases in moist static energy and therefore CAPE.

As the surface wind speed is increased, the surface itself cools and a larger proportion of the surface enthalpy flux is in sensible form. There is stronger dry convection and stronger boundary layer top entrainment, thus the increase in moist static energy and CAPE is slowed.

A fundamental time scale in the system is the ratio h/(Ck|V|). As wind speed increases, this time scale decreases, and although the proportion of the total surface heat flux that is in sensible form starts out the day large at the higher wind speeds, it also declines more rapidly in time as the boundary layer warms. The surface moisture flux increases to compensate, and typically by the midafternoon CAPE becomes larger than it would have been at lower wind speeds.

b. Coupled soil–atmosphere solutions

Coupling the simple atmospheric boundary layer model to the soil introduces two important effects on the diurnal development of the boundary layer. First, some of the solar radiation is used to heat the soil and the water within it as well as any vegetation that might be present; this has the effect of decreasing the turbulent enthalpy flux below what would be achieved with enforced surface energy balance. Second, the properties of the soil and vegetation can exert a strong control over the rate of evaporation of water. For example, if there is not much surface soil water at sunrise, there is little or no vegetation, and the water potential of the soil is small, the soil surface may dry out appreciably during the day, the Bowen ratio will increase as the day progresses, and CAPE will begin to decline as the more rapidly growing boundary layer entrains low moist static energy from the desert-modified air above. In this case, CAPE will be water limited rather than energy limited. In general, we expect CAPE to be noticeably affected by both the soil and vegetation properties.

Even the very simple soil model of Noilhan and Planton (1989) used here introduces a plethora of soil and vegetation parameters, including soil type, porosity, saturation volumetric water content, field capacity, wilting point (the water potential below which plants are unable to extract water), initial soil water content at the surface and just below the surface, leaf area index, water drops on vegetation, fractional cover of vegetation, and threshold radiation for photosynthesis. It is impractical to explore even a small fraction of the phase space of the coupled model, so we shall here explore only a few dimensions but provide the reader with software to conduct more thorough explorations.

In what follows, we force the model, as before, with the diurnal cycle of solar radiation shown in Fig. 2 and described by (24), with a noon peak of 700 W m−2.

We begin by showing a case in which the development of CAPE, commencing about noon, is curtailed in part by limited water. Figure 7 shows the time evolutions of various quantities in this special case, with the parameters given in Table 1.

Fig. 7.
Fig. 7.

Time evolution of (a) CAPE and CIN, (b) static energies, (c) surface energy budget terms, and (d) soil moisture and surface moisture availability (α). In (a) the blue curve shows CAPE while CIN is shown in red. In (b) the solid green curve shows boundary layer moist static energy, the dashed green curve is soil surface moist static energy, the solid red curve is the soil surface dry static energy, the dashed red curve is deeper soil dry static energy, the blue curve is boundary layer dry static energy, and the dashed blue line is the desert air dry static energy. In (c) the blue curve represents the specified net surface radiation and the solid red curve is net surface latent heat flux. (The dashed and dotted red curves show the contributions from evaporation from soil and transpiration, respectively.) The yellow curve is the sensible heat flux from the soil, and the magenta curve shows heat storage in the soil. In (d), the dashed green line is the fraction α of evaporation available from a water surface and the other curves show volumetric water content, according to the legend. The model parameters are listed in Table 1.

Citation: Journal of the Atmospheric Sciences 80, 11; 10.1175/JAS-D-23-0060.1

Table 1.

Model parameters.

Table 1.

In this example, CAPE develops rapidly, beginning just before noon, and reaches a peak value of about 4000 J kg−1 at around 1430 local time, before gently declining to around 2800 J kg−1 at sunset. CIN starts out at about 500 J kg−1 at midday and vanishes just before sunset. (We stop all integrations when CIN = 0, at which time deep convection is presumed to break out.) The boundary layer moist static energy (green curve in Fig. 7b) declines from sunrise until around 0730 local time, when the net surface radiation becomes positive and increases until about 1400 local time, after which it is flat. The boundary layer is deepening fairly rapidly at this point (not shown here), entraining low moist static energy from the desert air aloft; this offsets the surface enthalpy flux. At the same time, a nontrivial amount of enthalpy is diffusing from the surface soil down into the deeper soil (magenta curve in Fig. 7c). Both plants and bare soil are contributing to the evaporative flux (Fig. 7c). Note that the surface soil moisture at first increases, because the deeper soil in this case starts with a larger volumetric water content and thus water diffuses upward. But after about 0900 local time, surface evaporation overwhelms this diffusion, and surface water content decreases, raising the Bowen ratio, leading to faster boundary layer growth and thus greater entrainment of low moist static energy air. The surface water availability (dashed green line in Fig. 7d) varies considerably through the day. (Remember that we held it fixed in the uncoupled calculations presented earlier.)

The picture here is one of surprising complexity, given that the model is about as simple as we dare make it. Surface water availability, heat uptake by the soil, and boundary layer physics all play important roles in the evolution of CAPE.

Figure 8 shows the evolutions of the single quantity CAPE varying only the type of soil; all other parameters are the same as those used in Fig. 7 (see Table 1).

Fig. 8.
Fig. 8.

Evolution of CAPE for five different soil types. All other parameters are as in Fig. 7 (see Table 1).

Citation: Journal of the Atmospheric Sciences 80, 11; 10.1175/JAS-D-23-0060.1

CAPE is exquisitely sensitive to soil type, at least given the values of the other parameters. For soils with relatively low capillary conductivity (e.g., clay, and clay loam), growth of CAPE can be severely limited by surface water availability. This is demonstrated, for clay loam soil, in Fig. 9, which shows the volumetric water contents and surface water availability α as a function of time. At first, the surface soil water content actually rises owing to dew deposition. The surface water availability α rises for a while. Then, when the sun is high enough, water evaporates from the surface and is not replaced fast enough from below. After about 0900 local time, the surface water suffers a catastrophic loss in a positive feedback loop, where in the dryer the boundary layer, the greater the surface evaporation until the surface water is depleted and later the root zone water potential falls below the wilting point. Thereafter, the surface dries rapidly and heats up, driving strong boundary layer convection, thus entraining higher dry static energy from aloft, and accelerating the drying. Consequently, very little CAPE builds up and what little there is vanishes after a few hours.

Fig. 9.
Fig. 9.

Evolution of volumetric soil moisture as in Fig. 7d but for the case of clay loam soil. Note that the dashed green line is the surface water availability α rather than a volumetric water content.

Citation: Journal of the Atmospheric Sciences 80, 11; 10.1175/JAS-D-23-0060.1

At the other extreme, the porosity of sand allows surface soil water to be replaced more rapidly from the soil below.

In highly vegetated areas, the development of CAPE and CIN can be sensitive to soil moisture in the root zone and the properties of the vegetation. Figure 10 shows the evolution of CAPE for three different initial saturation fractions of the subsurface soil. (The evolutions of CAPE for saturation fractions below 20% are identical to the 20% curve; likewise, the evolutions for saturation fractions above 50% are identical to the 50% curve.)

Fig. 10.
Fig. 10.

Evolution of CAPE over a fully vegetated surface for three values of the subsurface soil saturation fraction. Other parameter values are as listed in Table 1. CAPE for subsurface soil saturation fraction less than 20% is zero; likewise, values above 50% are identical to the 50% curve.

Citation: Journal of the Atmospheric Sciences 80, 11; 10.1175/JAS-D-23-0060.1

If the subsurface soil moisture content is less than the wilting point, transpiration is diminished and evaporation into the daytime boundary layer is insufficient to build up much CAPE. On the other hand, large volumetric water content of the subsurface soil can make transpiration from a fully vegetated surface almost as large as that from a plane surface of water at the same temperature. There is, however, large sensitivity to subsurface soil moisture in between these limits.

Many other sensitivities could be explored, including sensitivity to the properties of the vegetation. The reader is invited to explore these using the software linked to at the end of this article.

c. Stranded CAPE

For the case of sand soil in Fig. 8, CIN (not shown here) does not fall to zero before sunset (its sunset value is, in this case, 55 J kg−1). As a new, stable nocturnal boundary layer develops underneath the mixed layer, the “stranded CAPE” may nevertheless be released after sunset if, for example, strong enough dynamic lifting occurs to overcome the CIN. Otherwise, the mixed-layer moisture may survive until the next day if there are no large sinks, so that the next diurnal cycle may result in greater CAPE and vanishing CIN. Tuckman et al. (2023) followed boundary layer parcels through several diurnal cycles over land and found that in several cases, the boundary layer moist static energy trends upward, and the daily minimum CIN trends downward, with a diurnal cycle superimposed on both quantities.

4. Discussion and summary

The simple model presented here, together with the previous published results of Lanicci et al. (1987), Agard and Emanuel (2017), Zhang (2022), and Tuckman et al. (2023), strongly suggests that the development of large values of CAPE can occur when air masses with steep lapse rates, formed over dry terrain, are advected over moist and/or vegetated soils, whether or not there is an intervening moist layer of separate origin. The elevation of the dry terrain can further increase the potential temperature of the air mass, yielding even larger potential values of CAPE over moist land (Molnar and Emanuel 1999). The magnitudes of CIN and CAPE that develop during daylight hours strongly depend on the potential temperature of the desert air, surface wind speed, the temperature, moisture content and depth of the initial moist boundary layer, and the properties of the underlying moist surface, including the type of soil, its moisture content at and below the surface, and the fractional coverage and type of vegetation present. This suggests that forecasts of severe convective storm outbreaks could be strongly affected by the formulation of both boundary layer and soil physics, and initialization of upper soil properties, especially soil type and moisture content as well as vegetation and its seasonal state. The degree of urbanization may also be influential. Some degree of seasonal prediction of severe convective weather might be possible if there is sufficient seasonal predictability of upper soil properties and seasonal vegetation.

The present study also reiterates the finding of Agard and Emanuel (2017) that the upper bound on both CIN and CAPE scales with Clausius–Clapeyron. However, at high enough temperatures, CAPE is limited by the length of the day. The physics behind this dependence can be seen by referring to Fig. 1: The amount of CAPE that can built up until the CIN vanishes is proportional to the difference between the wet- and dry-bulb temperature of the desert air at the level of the boundary layer top. This difference in turn depends on the saturation specific humidity. As climate warms, depending to some extent on what happens to the distribution of soil moisture and vegetation, one might expect to see more intense but perhaps less frequent storms, and more cases of stranded CAPE because CIN may still be relatively large at sunset. Consequently, there may be more cases where CAPE is built up over more than one diurnal cycle before being released.

We emphasize again, though, that this study focuses strictly on the thermodynamic environments conducive to severe convection; we have not addressed the all-important kinematic environment known to be an essential ingredient of severe convection.

The physics of high CAPE described here could help explain the climatology of severe convective storms. In general, such storms should favor regions of moist soils downwind of dry, preferably elevated terrain, and regions in which strong low-level wind shear is common. But note that while, in general, CAPE increases with soil moisture (all other things being equal), so does CIN, so that severe convection might not develop over soils that are too moist. While large CAPE might also develop over relatively warm bodies of water downwind of dry terrain, the lack of strong diurnal heating would result in large values of CIN and so prevent deep convection from occurring.

We apply this reasoning to a climatological distribution of surface soil moisture as shown in Fig. 11. This climatology was created from NASA’s satellite-based Soil Moisture Active Passive (SMAP) program (Reichle et al. 2022) and extends from April of 2015 through March of 2021.1 Level 4 SMAP data have been used to construct this figure. The red ellipses in Fig. 11 are subjectively identified regions where moist soils are downwind of dry terrain.

Fig. 11.
Fig. 11.

Surface soil moisture measured by the Soil Moisture Active Passive (SMAP) project averaged over the period April 2015–March 2021 (Reichle et al. 2022). This is based on level 4 SMAP data. The red ellipses show areas of moist soils downwind from dry terrain, as described in the text.

Citation: Journal of the Atmospheric Sciences 80, 11; 10.1175/JAS-D-23-0060.1

We might compare these subjectively identified regions to climatologies of CAPE, such as the recent CAPE climatology by Taszarek et al. (2021), based on the European Centre for Medium-Range Forecasts Reanalysis 5 (ERA5). But high values of CAPE can be strongly biased in reanalyses, including ERA5 (Wang et al. 2021). Another approach is to compare to a climatology of some measure of severe convection. The drawback is that such measures also depend on vertical wind shear, not just CAPE. Here we choose to compare to a satellite-estimated global climatology of hail, using the mean Ku-band reflectivity in the mixed-phase layer from the Global Precipitation Measurement (GPM) mission Core Observatory satellites (Mroz et al. 2017). Using many soundings from the RELAMPAGO field project conducted in the Pampas region of Argentina, Schumacher et al. (2021) found that a particular product of wind shear and mixed-layer average CAPE was a reasonable discriminator of severe convective environments, and that in practice this discriminator rules out CAPE values below about 1000 J kg−1 (see their Fig. 7).

Figure 12 shows the percentage of Dual-Frequency Precipitation Radar (DPR) soundings that have hail, based on the mean Ku-band reflectivity in the mixed-phase layer.

Fig. 12.
Fig. 12.

Percentage of Dual-Frequency Precipitation Radar soundings from GPM satellites that contain hail, based on the mean Ku-band reflectivity in the mixed-phase layer. From Mroz et al. (2017).

Citation: Journal of the Atmospheric Sciences 80, 11; 10.1175/JAS-D-23-0060.1

There is at least a loose correspondence between frequent hail and moist regions that lie downwind of dry soils, as depicted qualitatively in Fig. 11.

Finally, our results imply that severe convective weather may be tangibly affected by sufficiently large-scale changes in land use, given the sensitivity of CAPE and CIN to soil properties including soil moisture, and to vegetation. It is quite possible that large-scale agriculture and urbanization has already affected the climatology of severe convection. The extent to which this may be true, and to which land use may change in response to climate change, should be the subject of future research.

1

Ideally, we should use a climatology relevant to the severe convective storm season for each region, but such was not available at the time of this writing.

Acknowledgments.

The author thanks Dr. John Peters and two anonymous reviewers for exceptionally thorough and thoughtful reviews. He also thanks P. J. Tuckman for helpful comments. This research is part of the MIT Climate Grand Challenge on Weather and Climate Extremes. This research received support by the generosity of Eric and Wendy Schmidt by recommendation of Schmidt Futures as part of its Virtual Earth System Research Institute (VESRI).

Data availability statement.

A fast MATLAB program for solving the coupled PBL–soil model is available at http://texmex.mit.edu/pub/emanuel/CAPE/pbl_soil_evol_veg.m. This permits the reader to reproduce all the figures in this paper as well as the underlying data.

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  • Agard, V., and K. Emanuel, 2017: Clausius–Clapeyron scaling of peak CAPE in continental convective storm environments. J. Atmos. Sci., 74, 30433054, https://doi.org/10.1175/JAS-D-16-0352.1.

    • Search Google Scholar
    • Export Citation
  • Ashley, W. S., A. M. Haberlie, and V. A. Gensini, 2023: The future of supercells in the United States. Bull. Amer. Meteor. Soc., 104, E1E21, https://doi.org/10.1175/BAMS-D-22-0027.1.

    • Search Google Scholar
    • Export Citation
  • Barthlott, C., and N. Kalthoff, 2011: A numerical sensitivity study on the impact of soil moisture on convection-related parameters and convective precipitation over complex terrain. J. Atmos. Sci., 68, 29712987, https://doi.org/10.1175/JAS-D-11-027.1.

    • Search Google Scholar
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  • Betts, A. K., J. H. Ball, A. C. M. Beljaars, M. J. Miller, and P. A. Viterbo, 1996: The land surface-atmosphere interaction: A review based on observational and global modeling perspectives. J. Geophys. Res., 101, 72097225, https://doi.org/10.1029/95JD02135.

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  • Fig. 1.

    Three stages in the diurnal evolution of CAPE. (a) A desert-modified air mass with little moisture and a deep dry adiabatic boundary layer is advected over a moist soil. (b) Advection of cool, moist air from a different region and/or wind-driven boundary layer turbulence creates a moist but cool new shallow boundary layer just saturated at its top. Dewpoint is shown by dashed lines, and the temperature of a parcel lifted adiabatically is shown by the dotted curve. (c) Surface heating by solar radiation causes the now-convective boundary layer to deepen and moisten, though the moistening is at least partially offset by entrainment of dry air from above the boundary layer. CIN decreases and CAPE increases, with the proportion depending on surface moisture availability.

  • Fig. 2.

    Imposed net surface radiative flux over a 24-h period as given by (24). Note that the model integration begins at sunrise (0600 local time).

  • Fig. 3.

    Diurnal evolution of (a) CAPE and (b) CIN for nine values of desert air potential temperature, ranging from 294 to 310 K. For these plots, the soil moisture availability parameter α is set to 0.8, while the surface wind speed |V| is 8 m s−1.

  • Fig. 4.

    (a) Diurnal peak values of CAPE corresponding to the evolutions shown in Fig. 3, with α set to 0.8, and the surface wind speed |V| to 8 m s−1. (b) Values of CIN at the time of peak CAPE are shown.

  • Fig. 5.

    Contours of diurnal peak CAPE (J kg−1) as a function of the moisture availability α, and the desert air potential temperature θ0, for a specified surface wind speed of 8 m s−1.

  • Fig. 6.

    Contours of diurnal peak CAPE (J kg−1) as a function of surface wind speed and the desert air potential temperature θ0, for a specified surface moisture availability of 0.8.

  • Fig. 7.

    Time evolution of (a) CAPE and CIN, (b) static energies, (c) surface energy budget terms, and (d) soil moisture and surface moisture availability (α). In (a) the blue curve shows CAPE while CIN is shown in red. In (b) the solid green curve shows boundary layer moist static energy, the dashed green curve is soil surface moist static energy, the solid red curve is the soil surface dry static energy, the dashed red curve is deeper soil dry static energy, the blue curve is boundary layer dry static energy, and the dashed blue line is the desert air dry static energy. In (c) the blue curve represents the specified net surface radiation and the solid red curve is net surface latent heat flux. (The dashed and dotted red curves show the contributions from evaporation from soil and transpiration, respectively.) The yellow curve is the sensible heat flux from the soil, and the magenta curve shows heat storage in the soil. In (d), the dashed green line is the fraction α of evaporation available from a water surface and the other curves show volumetric water content, according to the legend. The model parameters are listed in Table 1.

  • Fig. 8.

    Evolution of CAPE for five different soil types. All other parameters are as in Fig. 7 (see Table 1).

  • Fig. 9.

    Evolution of volumetric soil moisture as in Fig. 7d but for the case of clay loam soil. Note that the dashed green line is the surface water availability α rather than a volumetric water content.

  • Fig. 10.

    Evolution of CAPE over a fully vegetated surface for three values of the subsurface soil saturation fraction. Other parameter values are as listed in Table 1. CAPE for subsurface soil saturation fraction less than 20% is zero; likewise, values above 50% are identical to the 50% curve.

  • Fig. 11.

    Surface soil moisture measured by the Soil Moisture Active Passive (SMAP) project averaged over the period April 2015–March 2021 (Reichle et al. 2022). This is based on level 4 SMAP data. The red ellipses show areas of moist soils downwind from dry terrain, as described in the text.

  • Fig. 12.

    Percentage of Dual-Frequency Precipitation Radar soundings from GPM satellites that contain hail, based on the mean Ku-band reflectivity in the mixed-phase layer. From Mroz et al. (2017).

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