## Abstract

In favorable atmospheric conditions, fires can produce pyrocumulonimbus cloud (pyroCb) in the form of deep convective columns resembling conventional thunderstorms, which may be accompanied by strong inflow, dangerous downbursts, and lightning strikes that can produce dangerous changes in fire behavior. PyroCb formation conditions are not well understood and are difficult to forecast. This paper presents a theoretical study of the thermodynamics of fire plumes to better understand the influence of a range of factors on plume condensation. Plume gases are considered to be undiluted at the fire source and approach 100% dilution at the plume top (neutral buoyancy). Plume condensation height changes are considered for this full range of dilution and for a given set of factors that include environmental temperature and humidity, fire temperature, and fire-moisture-to-heat ratios. The condensation heights are calculated and plotted as saturation point (SP) curves on thermodynamic diagrams. The position and slope of the SP curves provide insight into how plume condensation is affected by the environment thermodynamics and ratios of fire heat to moisture production. Plume temperature traces from large-eddy model simulations added to the diagrams provide additional insight into plume condensation heights and plume buoyancy at condensation. SP curves added to a mixed layer lifting condensation level on standard thermodynamic diagrams can be used to identify the minimum plume condensation height and buoyancy required for deep, moist, free convection to develop, which will aid pyroCb prediction.

## 1. Introduction

Pyrocumulus (pyroCu) and pyrocumulonimbus (pyroCb) clouds are produced by intense heating of air from fire or volcanic activity that leads to ascent and subsequent condensation when the rising air becomes saturated due to cooling from adiabatic expansion. PyroCu clouds are relatively common and can form as small clouds above small fire plumes or individual fire plume puffs. Alternatively, in large fires with an intense convection column, the cloud may resemble towering cumulonimbus (Cb) with updrafts that penetrate into the stratosphere (e.g., pyroCb; Fromm and Servranckx 2003; Mitchell et al. 2006; Fromm et al. 2006; see also the review paper by Fromm et al. 2010). There is abundant anecdotal evidence to suggest that the presence of pyroCb activity can have a significant impact on fire behavior, including (i) the amplification of burn and spread rates (Fromm et al. 2006; Trentmann et al. 2006; Rosenfeld et al. 2007; Fromm et al. 2012), (ii) enhanced spotting due to larger and more intense plumes (e.g., Koo et al. 2010), and (iii) ignition of new fires by pyroCb lightning (Dowdy et al. 2017).

PyroCb events are very difficult to forecast. Ideal pyroCb conditions are similar to ideal thunderstorm conditions but with a dry, rather than moist, boundary layer, which favors high fire intensity and spread rates. A typical example is the classic inverted-V sounding on a thermodynamic diagram, which is widely recognized to favor severe weather (e.g., Beebe 1955; Wakimoto 1985). The inverted-V sounding (Fig. 1) represents a dry, well-mixed lower layer overlaid by a moist midtroposphere^{1} and is present in many pyroCb studies (e.g., Goens and Andrews 1998; Trentmann et al. 2006; Rosenfeld et al. 2007; Cunningham and Reeder 2009; Fromm et al. 2012; Johnson et al. 2014; Peterson et al. 2015; Lareau and Clements 2016; Peterson et al. 2017). While the inverted-V environment is readily identified from thermodynamic soundings, not all fires in these conditions produce pyroCb or even pyroCu (referred to collectively as pyroCu/Cb), since fire plumes typically lose buoyancy from entrainment of cooler and potentially drier air before they cool sufficiently for condensation to occur. Perhaps one of the biggest challenges to forecasters is to predict how favorable the environmental conditions are for plume entrainment [e.g., environmental turbulence and strong background winds can increase the plume turbulence, which accelerates the entrainment; Thurston et al. (2017)], since it has such a large impact on whether or not the plume will condense. Predicting entrainment is a difficult problem that is beyond the scope of this paper. Instead, we focus on understanding how the environment and fire properties influence plume condensation heights for a broad range of entrainment rates. This enables estimates of the minimum height and buoyancy a plume parcel must achieve for condensation or free convection to occur. The study also provides valuable insight into the amount of dilution required for plumes to condense at realistic heights and contributes to the debate on the relative importance of fire moisture to environment moisture on plume behavior (e.g., Potter 2005; Luderer et al. 2009, hereafter LTA09).

In the next section, a conceptual model is introduced to illustrate and justify assumptions that underpin a simple thermodynamic model (described in section 3) that calculates plume condensation levels. Results are presented in section 4, their implications for understanding and forecasting pyroCu/Cb are discussed in section 5, and the paper is summarized in section 6.

## 2. Plume conceptual model

The conceptual model of plume thermodynamics is based on the assumption that fire plumes initially comprise pure combustion gases (containing the fire heat and moisture) that mix with environment air as the hot gases rise. The model assumes the plume entrains air from an inverted-V environment of constant specific humidity (*q*_{env}) and constant potential temperature (*θ*_{env}). Initially, the plume consists of pure combustion gas of a specified specific humidity (*q*_{fire}) and potential temperature (*θ*_{fire}), with the concentration of combustion gas within individual plume elements (PEs) becoming increasingly diluted as more and more environment air is entrained. This is illustrated schematically in Fig. 2a, in which three hypothetical PEs are depicted rising from a fire. The PE shading becomes lighter with height to indicate increased dilution as they ascend, and the PE size increases to indicate both adiabatic expansion and expansion due to entrained environment air. The three PE paths show three possible plume condensation outcomes. PE1 rises rapidly in the center of the plume and condenses with enough buoyancy for free convection to occur. PE2 is more exposed to entrainment and loses buoyancy more rapidly, but still rises sufficiently for some cloud to form. PE3 experiences even more entrainment, and it loses buoyancy before it rises sufficiently for any condensation to occur. A better characterization of real turbulent plumes would include many more PEs of greatly varying rates of dilution (and thus buoyancy), with a dynamic mix and distribution of PEs that vary considerably in space and time.

A hypothetical temperature path for PE1 has been added to a thermodynamic diagram (Fig. 2b, yellow line), along with markers to illustrate the dilution phases and lines of constant potential temperature and specific humidity to illustrate the height to which each PE would need to rise (adiabatically) before it would condense [i.e., the PE saturation points (SPs)]. The blue line joins all the saturation points into an SP curve.^{2} In this scenario, the PE temperature curve intersects the SP curve, which means condensation occurs, and its position along the SP curve shows that it is significantly warmer than the mixed layer environment potential temperature (*θ*_{env}) indicated by the black line at 303 K. For comparison, hypothetical temperature curves have been added for PEs 2 and 3 in Fig. 2c, which zooms in on the lower troposphere. PE2 also intersects the SP curve, but it is more diluted and, thus, cooler than PE1 at condensation.^{3} PE3, on the other hand, intersects the mixed layer *θ*_{env} line, at which point it has become so diluted that its temperature is indistinguishable from the environment, and it is no longer positively buoyant.

There is an infinite set of possible temperature pathways from the fire source (representing the entire mix of PEs within the plume and for the full gamut of plume types: large and small, upright or bent over, highly turbulent or nearly laminar, and strongly or weakly rotating) that ultimately intersect somewhere on the SP curve or the *θ*_{env} line. Thus, these lines make up the possible PE temperature path termination points, and it is these lines that the thermodynamic plume model focuses on. All possible PEs for a given thermodynamic scenario (*θ*_{env}, *q*_{env}, *θ*_{fire}, and *q*_{fire}) commence on the same dilution pathway from pure combustion gas (zero dilution) to almost pure environment air (almost complete dilution), regardless of whether they arise from a candle flame or wildfire. The only difference is the pathway termination points of condensing PEs.

It follows that the SP curve and *θ*_{env} line are fixed for a given set of fire and environment thermodynamic properties. A range of scenarios is investigated with a broad parameter space of flame temperatures and fire moisture production to account for fire property uncertainties.

## 3. Plume thermodynamic model

A number of physically reasonable assumptions are made to simplify the conceptual plume model, which includes a focus only on the plume thermodynamics up to the plume condensation level (i.e., the height at which a rising plume begins to condense), but excludes the thermodynamics of plume condensation.^{4} This height can be determined from the SP curve plotted on a thermodynamic diagram. Thermodynamic variables at the saturation point are hereafter labeled with the subscript “SP.” The assumptions made include the following:

The plumes develop in a well-mixed (homogeneous) boundary layer of constant potential temperature (

*θ*) and constant specific humidity (*q*).This constant boundary layer

*θ*and*q*is maintained up to the SP.Environment

*θ*and*q*are entrained equally into PEs.All radiative heat losses occur in the flaming zone prior to the commencement of entrainment.

The inverted-V profile of Fig. 1 is roughly consistent with the first assumption; the environmental temperature is dry adiabatic, and the moisture is nearly constant. The second assumption implies that the upper-boundary layer will be supersaturated in the layer between the environment lifting condensation level (ELCL; the apex of the inverted V in Fig. 1) and the plume SP for all SPs that are more elevated than the ELCL. This means PEs will entrain supersaturated air above the ELCL, but we expect associated errors will be relatively small since, according to large-eddy model (LEM) simulations presented in the next section, the majority of the entrainment occurs below this level. The third assumption states the PE temperature and moisture are diluted at exactly the same rate (consistent with the properties of turbulent mixing), which requires the radiation heat losses to be considered separately (assumption 4).

### a. Plume temperature and moisture

From the assumptions described above, the potential temperature and specific humidity of a PE are expressed as

and

where the subscripts pl, env, and fire refer to the PE, environment (i.e., the well-mixed boundary layer), and fire (combustion gas), and *α* is the PE dilution fraction. All variables and constants introduced in this subsection are described in Table 1. The environment quantities *θ*_{env} and *q*_{env} are specified for each scenario, and *α* is varied to represent a range of plume dilution amounts from pure combustion gases (*α* = 0, red shading in Fig. 2a) to pure environment (*α* = 1, white shading in Fig. 2a).

The combustion equation (Ward 2001), used by both Potter (2005) and LTA09, shows the mass ratio of air to hydrocarbon consumed is about 6 to 1. The temperature of the air–fuel mix is approximated to the temperature of the air, and the air temperature is assumed to be similar to that of the environment. While this underestimates the air–fuel mix temperature and, thus, underestimates *θ*_{fire}, the wide range of fire temperatures ensures the results span the range of uncertainty, and we show later that it has no impact on our conclusions regarding plume behavior.

The quantity *θ*_{fire} is expressed as a multiplier (*γ*) of the environment potential temperature,

The *γ* values are chosen to let *θ*_{fire} encompass the range of flame temperatures observed in wildfires (i.e., about 600 K at the flame tips and up to 1500 K at the flame base; e.g., Wotton et al. 2012 ^{5}). Thus, for *θ*_{env} = 300 K, *γ* ranges from 2 to 5. The fire adds a temperature increment (Δ*T*_{f}) to the air–fuel potential temperature (assumed to be *θ*_{env}) to produce a combustion gas of potential temperature *θ*_{fire}. To be consistent with assumption 4 above, in which radiative heat losses occur prior to the commencement of entrainment, we let *δ* be the fraction of heat lost to radiation and include only the remaining heat in the temperature increment. Thus,

The approximation in Eq. (4) is reasonable even for fires at relatively high elevations^{6} because the conclusions drawn from the model are insensitive to *γ* and *δ* (section 4).

The ratio of combustion moisture to potential temperature increment released by the burning fuel is specified following LTA09:

LTA09 proposed a “realistic” dry-to-moist fire range of *φ* ~ 3 to 15 × 10^{−5} kg kg^{−1} K^{−1}. The dry (moist) extreme assumes 0% (80%) fuel moisture and 0% (50%) energy loss to radiation. We also consider more extreme values ranging from *φ* = 0, a completely dry fire (e.g., some of the idealized LEM experiments discussed later), to *φ* ~ 100 × 10^{−5} kg kg^{−1} K^{−1} [Potter’s (2005) thought experiment].

The combustion moisture increment per unit mass of fuel burned is calculated using Eqs. (4) and (5):

with an additional moisture contribution from the air consumed in combustion, gives the moisture released per unit mass of combustion gas (assuming a 6 to 1 air-to-fuel mass ratio; Ward 2001):

and

Equations (8) and (9) show that there is overlap in the parameter space when incorporating the full range of *α*, *δ*, and *γ* (e.g., a diluted, hot fire could have the same *θ*_{pl} as a less diluted, cooler fire). The overlap can almost be eliminated when these terms are replaced with a buoyancy-like parameter [e.g., Eq. (3) of Smith et al. 2005]:

For the *α*, *δ*, and *γ* ranges considered in this paper, *β* varies from 0 to 4, although we expect values near realistic condensation levels to be less than 10^{−1}. Expressed in terms of *β*, Eqs. (8) and (9) become

and

The presence of *α* in Eq. (12) shows the parameter space overlap is not quite exact. However, the deviation from exact overlap is quite small^{7} and not visible to the eye when plotted on conventional thermodynamic diagrams (not shown). For realistic plume condensation levels, *α* ~ 1, and

which reduces the number of model parameters from six to four.

### b. Plume condensation height calculation

The height of condensation or saturation point pressure (*P*_{SP}), for all hypothetical PEs matching the range of parameters introduced above is determined from each corresponding *θ*_{pl}–*q*_{pl} pair. All of the thermodynamic constants featured in the following equations are listed in Table 2, and additional thermodynamic variables are listed in Table 3. The saturation point temperature (*T*_{SP}) for a plume air parcel can be determined from the vapor pressure (*e*_{pl}) and temperature (*T*_{pl}) of the parcel at any pressure level, using Bolton’s [1980, Eq. (21)] formula for the lifting condensation level temperature:

For convenience, we evaluate *T*_{pl} and *e*_{pl} at the surface where *P*_{s} = *P*_{0} = 1000 hPa and *T*_{pl} = *θ*_{pl}. Using Emanuel’s (1994) Eq. (4.1.4), the surface *e*_{pl} is given by

The *P*_{SP} is found by inverting the potential temperature equation to find the pressure at which *T*_{SP} has the same potential temperature as *θ*_{pl}:

where

[Emanuel 1994, Eq. (4.2.11)], and *r*_{pl} = *q*_{pl}/(1 − *q*_{pl}) is the plume mixing ratio. For most practical purposes (small *β*; see section 5), the approximate form of Eq. (17) is good, and Bolton’s formula [Eq. (14)] is very accurate. The more exact form of Eq. (17) is included here, since it becomes important at the extreme ends of the SP curves (large *β*), where undiluted PEs can be quite moist.

### c. Diagnostic parameters

Useful diagnostic parameters can be generated that provide insight into plume behavior above the SP that are constructed from thermodynamic quantities at the SP. These include the difference in height (or pressure level) between the SP and the ELCL (Δ*P*):

Here, *P*_{ELCL} is just *P*_{SP} when *α* = 1 (100% dilution, infinitesimal quantities of combustion gas) or *β* = 0 (zero plume buoyancy). The Δ*P* is useful for understanding the height at which a fire plume will condense and for estimating fireCAPE (e.g., Potter 2005). FireCAPE is the convective available potential energy (CAPE) of a hypothetical parcel of plume air, which is simply regular CAPE with additional increments of *θ* (Δ*θ*_{FC}) and *q* (Δ*q*_{FC}) that represent the plume values at the condensation height (SP) in excess of the environment,

and

For small *β*, *α* ~ 1 and Eq. (20) reduces to

FireCAPE is separated into the sum of regular CAPE plus the extra CAPE that results from nonzero Δ*θ*_{FC} and Δ*q*_{FC},^{8} and we label this fire contribution pyroCAPE (PC). The concept is illustrated on a thermodynamic diagram for our hypothetical warm environment scenario for one value of Δ*θ*_{FC} and three values of Δ*q*_{FC} (Fig. 3). The three values of pyroCAPE are represented by the area between the environment *θ*_{e} line (pale blue, large, dashed line) and the three plume *θ*_{e} lines (red, blue, and green, large, dashed lines), with the upper pyroCAPE area boundary defined by the environment temperature (solid black line). The lower boundary could be closed in a number of ways, with minimal sensitivity to the resulting pyroCAPE value. A line of constant temperature passing from the plume SP to the environment temperature or *θ*_{e} line enables a simplification to the pyroCAPE equation introduced below [Eq. (25)]. The three values of Δ*q*_{FC} represent three different fire scenarios with three corresponding values of Δ*P*. Note the difference in height of the three SPs, compared with the ELCL.

PyroCAPE is a simple but useful quantity that requires no detailed information about the environmental temperature profile above the ELCL (required for CAPE or fireCAPE calculations). A good estimate of pyroCAPE can be made from Eqs. (1) and (3) of Bister and Emanuel (2002), using an estimate of the equilibrium-level temperature (*T*_{EL}) and making two assumptions: (i) the ELCL temperature is similar to the plume SP temperature (*T*_{LCL} ~ *T*_{SP}), and (ii) *T*_{EL} is similar for the hypothetical environment-only and plume parcels (which is a good approximation for convection reaching the tropopause):

Here, Δ ln*θ*_{e} is the difference in the natural logarithm of the plume equivalent potential temperature at the SP and the natural logarithm of the environment equivalent potential temperature at the ELCL:

and

For (*θ*_{e})_{env} and for (*θ*_{e})_{pl}, *θ*, *r*, and *T* are evaluated at *P*_{LCL} and *P*_{SP}, respectively. Here, we have used Emanuel’s (1994) Eq. (4.5.14) to define *θ*_{e}, in which relative humidity is 100%, and the cloud liquid water is negligible since the plume has only just begun to condense. We show in appendix A that for small *β*, pyroCAPE can be approximated by

### d. Large-eddy model

Results from a selection of large-eddy model (LEM simulations of pyroCu and pyroCb (Thurston et al. 2015, 2016) are introduced in the next section. The simulations are performed using version 2.4 of the Met Office LEM (Gray et al. 2001), a high-resolution cloud-resolving model that solves a quasi-Boussinesq, anelastic equation set on a three-dimensional Cartesian grid. Subgrid stresses are parameterized using a stability-dependent version of the Smagorinsky–Lilly scheme, described by Brown et al. (1994). Moist processes are represented by the three-phase microphysics scheme of Swann (1998). The simulations were performed without any radiation parameterization, and the Coriolis force is neglected. A 12.8-km horizontal square grid with 50-m grid spacing was employed. The vertical grid is stretched with 20-m grid spacing near the surface, increasing to 50 m at a height of 1.5 km and remaining a constant 50 m to a height of 10 km, with further stretching to 60 m at the model top of 12.7 km. The lateral boundary conditions are doubly periodic. A free-slip condition is applied to the upper boundary with Newtonian damping in the upper 2.5 km, and a nonslip condition is applied to the lower boundary with a 0.05-m roughness length.

The simulations have zero background wind and an inverted-V boundary layer profile, with the ELCL about 4 km above the surface (section 4c). The fire is represented by a 250-m radius circle of surface heat and moisture fluxes of varying ratios of moisture to heat. The maximum surface heat flux used is 30 kW m^{−2}, which is very similar to the maximum above-surface heat fluxes measured for a grass fire during the FireFlux experiment (Clements et al. 2007). It amounts to a total heat flux of about 5.9 GW over the 250-m radius circle. These numbers are comparable to recent estimates of 87 kW m^{−2} and 2.7 GW (assuming a 100-m radius fire) for the El Portal fire (July 2014, near Yosemite National Park; Lareau and Clements 2017). Table 5 lists the parameters that define the atmospheric profile and the heat and moisture fluxes.

## 4. Results

The variation in height of PE condensation for the combination of parameters introduced above is illustrated with SP curves on thermodynamic diagrams in Fig. 4 for the six scenarios listed in Table 4. The solid black *θ*_{env} and *q*_{env} lines make up the inverted-V boundary layers of the two environments, with their apexes at the ELCL, and the SP curves represent 0 ≤ *β*_{SP} ≤ 1.^{9} For example, the black dashed lines represent the PE *θ*_{pl} and *q*_{pl} with *β*_{SP} = 0.05 for the moist fire in the warm environment, in which Δ*θ*_{FC} ~ 15 K, and Δ*q*_{FC} ~ 2.3 × 10^{−3} kg kg^{−1}.

The two environments depicted in Fig. 4 include one warm and dry (surface conditions of 30°C and 19% relative humidity) and the other cold and relatively humid (surface conditions of −2°C, 61% relative humidity, representative of the Flatanger fire in Norway, January 2014, that destroyed nearly 100 houses; e.g., Gabbert 2014). The SP curves representing three *φ* values are included: LTA09’s dry (red) and moist (blue) extremes and a value representing Potter’s (2005) fireCAPE thought experiment (green).

The results and conclusions presented in the remainder of this and the next section show that plume condensation is dependent on *β*_{SP}, rather than *α*, *δ*, or *γ* individually, which supports the notion that our conclusions are not sensitive to the largely unknown fire temperature.

### a. Does pyrocumulonimbus development require cores of undiluted combustion gases?

Most fires do not produce pyroCu/Cb because their plumes rapidly approach complete dilution (lose buoyancy) before they reach the condensation level (e.g., PE3 in Fig. 2). Furthermore, many plumes that do reach the condensation level, and produce small puffs of cloud, are too small and/or diluted for deep, buoyant cores (i.e., free convection) to develop (e.g., PE2 in Fig. 2). It follows that relatively large buoyant cores of somewhat less-diluted combustion gases (e.g., McRae et al. 2015) are necessary for pyroCu/Cb to develop (e.g., PE1 in Fig. 2). How dilute these cores need to be has not been quantified. In some cases, the language used to describe the necessary dilution could be interpreted as quite substantially less [e.g., “rates of dilution by ambient air…greatly reduced” (Taylor et al. 1973)] or even zero [e.g., “significant core of air unaltered by entrainment” (Potter 2005); “a lack of entrainment to the convection column” (Finney and McAllister 2011)]. Figure 4 can be used to cast some light on how diluted these buoyant plume cores need to be. Zero dilution is clearly extreme (top-right end of the blue and red curves), since it indicates plume condensation would need to occur deep in the stratosphere.^{10} We speculate that plume cloud base will typically occur within about 100 hPa of the ELCL;^{11} according to Fig. 4, this would require very significant levels of dilution (>90% and 95% for the moist and dry cool fires respectively, and considerably more for the hot fires; not shown).

The plume thermodynamic model results suggest the following:

PyroCb formation requires sufficiently large fires to maintain a core of buoyant plume gases that rise to the level of free convection in an environment that enables deep, moist convection.

To release the additional buoyant potential of condensational heating, the plume core requires substantial dilution to enable condensation to occur, but not so much dilution that it loses buoyancy.

### b. How does the thermodynamics of the boundary layer influence the plume saturation point?

Figure 4 shows the ELCL is more elevated and the SP curves steeper for the warm, compared with the cold, environment cases, both of which imply more elevated PE condensation for a given *φ* and nonzero *β*_{SP}. Equation (14) shows the saturation temperature decreases for warmer and drier air, which raises the condensation height [Eq. (16)]. This relationship can also be implied from Fig. 3. If the red solid line is considered to represent an incremental increase in *θ*_{env} (above the mixed layer *θ*_{env}; solid, pale blue line), and the colored, fine-dashed lines are considered to represent successive *q*_{env} increments, then the resulting ELCLs are raised and lowered for *θ*_{env} and *q*_{env} increments, respectively (indicated by the three labeled saturation points). The cold environment is also drier than the warm environment case depicted in Fig. 4, which demonstrates the *θ*_{env} difference dominates over the *q*_{env} difference in determining their relative ELCL heights. This is because the ELCL is 20 times more sensitive to fractional changes in *θ*_{env} than *q*_{env} ( appendix B), which suggests that for typical wildfire conditions, the plume SP is to first order a function of *θ*_{env} (i.e., becomes more elevated with increasing environment temperature). It follows that pyroCu/Cb formation in hot and dry conditions (elevated ELCLs) will require greater plume ascent than in cold conditions.

The relationship between the SP curve gradient and the environment can be demonstrated mathematically by deriving an equation for the change in pressure with change in *β*_{SP} that parallels the derivation of LTA09’s Eq. (8) and assumes *β*_{SP} is small:

The denominator is positive for all realistic values of *T*_{SP} (LTA09), and *L*_{υ}/*R*_{υ}*T*_{SP} varies by only a few percent in our range of scenarios. Thus, the SP curve slope is most sensitive to *φ*, *θ*_{env}, and *q*_{env}, with larger *φ* values producing flatter or negative sloping SP curves. For a given *φ*, Eq. (26) shows fractional changes in *θ*_{env} and *q*_{env} have equal magnitude but opposite sign impact on the SP curve slope. However, *q*_{env} varies proportionally by a greater magnitude than *θ*_{env} between the warm and cold environment scenarios, giving rise to flatter SP curves in the cold environment. Of note are the negative gradients of the green curves (Figs. 4 and 5), which confirm Potter’s (2005) finding that fires with *φ* = 100 × 10^{−5} kg kg^{−1} K^{−1} will produce plume condensation at lower heights than the LCL. A negative gradient is also possible for LTA09’s moist *φ*, with an indication of such behavior in the cold environment case of Fig. 4, where it can be seen that the blue curve tangent is approximately horizontal at the ELCL. (A similar diagram with reduced *q*_{env} does produce a negative gradient; not shown.)

In summary, for these *θ*_{env}/*q*_{env} ratios (and we expect for all typical wildfire environments), plume condensation heights should decrease with decreasing environment temperature, since decreasing *θ*_{env} decreases both the ELCL and the SP curve slope. This relationship implies that the changing thermodynamic environment over a fire ground that occurs with the passage of a cold front or sea breeze (especially if accompanied by moister air) should favor plume condensation by reducing the height to which the plume needs to rise for condensation to occur. This thermodynamic impact could be an additional reason why pyroCb often forms on such wind changes. Other reasons include increased “fire power” due to rapidly expanding head fires, as the wind shift transforms the longer flank fires to head fires, and reduced entrainment if the plume transitions from bent over to upright in response to a temporary increase in convergence and/or reduction in wind speed (e.g., Cunningham 2007; Peace et al. 2012, 2015a,b, 2016).

### c. Hypothetical plume paths

In the absence of plume observations, temperature and moisture profiles from four LEM plume simulations (section 3c; Table 5) are plotted in Fig. 5 for an environment similar to the warm scenario of Fig. 4. The temperature profiles are equivalent to the PE temperature paths depicted in Fig. 2 (yellow lines). Two values of sensible heat flux (Table 5) are used, which is equivalent to varying *γ*. The *φ* values include a completely dry fire (*φ* = 0; black) and LTA09’s driest and wettest realistic fires (*φ* = 3 × 10^{−5} kg kg^{−1} K^{−1} in red; *φ* = 15 × 10^{−5} kg kg^{−1} K^{−1} in blue). The SP curves for the three *φ* values illustrated in Fig. 4 have been repeated for comparison in Fig. 5. Note that the SP curve for a *φ* = 0 (i.e., completely dry) fire is just the upward continuation of the *q*_{env} line.

The inverted-V boundary layer in these simulations is 4 km deep (up to ~600 hPa), above which *θ* increases and *q* decreases (not shown). Turbulence has mixed some of this potentially warmer and drier air from above into the plume in a shallow layer at the boundary layer top, which is not incorporated in the plume model (hereafter, this layer is referred to as the downward-mixing layer). Accordingly, the following analysis focuses on the temperature lines below about 650 hPa, with visual extrapolation used to ascertain whether they intersect SP or *θ*_{env} curves. The plume contains a spectrum of PEs with the mean (solid) and maximum (dashed) temperature and moisture lines (for a given height measured over a period of 10 min across the width of the plume) plotted in Fig. 5. However, in order to distinguish between plume air and environment air in the model data, only air at least 1 K warmer than the environment was considered to be part of the plume.

The first two panels of Fig. 5 show temperature and moisture traces from simulations of varying heat-source intensities (*Q* = 5 and 30 kW m^{−2}), each with *φ* = 0, and thus, the plume moisture is identical to the environment moisture (except in the downward-mixing layer). The cooler fire mean and maximum temperature traces (Fig. 5a) are closest to the *θ*_{env} line, and they converge near 680 hPa (analogous to PE3 in Fig. 2), where the plume appears to terminate due to the plume air identification criterion of 1 K warmer than the environment. Extrapolation of the temperature traces to the relevant SP curve (an extension of the *q*_{env} line in Figs. 5a,b) suggests the hot fire has the potential to produce pyroCu/Cb because the mean and maximum temperature traces intersect the SP curves with nontrivial buoyancy (analogous to PE1 in Fig. 2). The temperature and moisture traces from the two simulations with fire moisture are depicted in Fig. 5c. The temperature traces are identical (except in the downward-mixing layer) and also identical to the dry, hot fire (Fig. 5b). While the moisture traces are clearly distinguishable, the mean plume moisture trace from the drier fire (solid red) barely deviates from *q*_{env}, except at the lowest model level, and even the maximum plume value (dashed red) is barely distinguishable from *q*_{env} above about 800 hPa, indicating that fire moisture has no discernible impact on pyroCu/Cb formation in these simulations.

The highest mean and maximum *β* values that appear in Fig. 5 are ~0.05 and ~0.18, respectively, at the lowest model level in the LEM, 10 m above the surface. These are the largest *β* values produced in any of the LEM simulations we investigated for this paper, and they support our expectation that *β* ≤ ~*O*(10^{−1}) for realistic plumes. At 1000 m, these values reduce to *β*_{mean} ~ 0.01 and *β*_{max} ~ 0.03, which suggests that for this heat source for any realistic wildfire scenario (with elevated condensation levels), *β* ≤ ~*O*(10^{−2}).

In Fig. 5a, the convergence of the mean and maximum plume temperature traces near 680 hPa occurs well below the ELCL (and any of the SP curves), and, as expected, minimal condensation occurred. A few parcels of buoyant air did occasionally reach the condensation height, producing short-lived puffs of shallow cloud. The hot fire mean and maximum plume temperature traces (Figs. 5b,c) remain more than 2 and 4 K warmer than the environment as they approach the SP curves, where simulations that show deep, moist convection with intense downbursts did develop (Thurston et al. 2015).

These examples support the suggestion in the introduction that if PE temperature traces intersect the *θ*_{env} trace, the PE reaches neutral buoyancy before cooling sufficiently for condensation to occur, and if the PE temperature trace instead intersects an SP curve, the plume condenses. From Fig. 5 and corresponding measurements (not shown), we can speculate on the potential for pyroCu formation for the two fires if they were located in environments with lower LCLs (e.g., higher *q*_{env} and/or lower *θ*_{env}). For example, we might expect the hot fire *β*_{SP} at 900 hPa to be about 3 times greater than that measured at 600 hPa (Fig. 5b), and the cool fire *β*_{SP} at 900 hPa (Fig. 5a) might be similar to the hot fire *β*_{SP} at 600 hPa.

Because of persistent entrainment during plume ascent, including plume capture of pockets of environmental air, we expect that at any given height and time, a cross-section of real plumes should contain PEs with buoyancy ranging from 0 ≤ *β* ≤ *β*_{max}. The difference between *β*_{mean} and *β*_{max} in the LEM simulations gives an indication of the buoyancy variability within these plumes at all levels and, importantly, near the condensation level. For the hot fire simulations (Figs. 5b,c), all of which produced deep, moist convection (7.5 km above the surface) with intense evaporatively cooled downbursts (Thurston et al. 2015), *β*_{SP,max} (~0.014) is about 3 times *β*_{SP,mean} (~0.005) and Δ*θ*_{FC,max} ~ 4.3 K, Δ*θ*_{FC,mean} ~ 1.5 K [using Eq. (19) and *θ*_{env} = 310 K].

## 5. Discussion

The plume thermodynamic model is a function of the mixed layer environment (*θ*_{env} and *q*_{env}) and two parameters that represent the fire and the influence of the atmosphere on the plume (*φ* and *β*). The quantity *φ* incorporates fuel type, fuel moisture, and fire intensity, and *β* incorporates fire size and intensity and any atmospheric conditions that affect plume entrainment. We label these pairs of parameters “environment thermodynamic” and “fire atmosphere” properties, respectively. Forecasting plume condensation and predicting pyroCb formation would be greatly simplified if these four parameters could be predicted or observed in near real time.

### a. Identifying the environment–thermodynamic and fire–atmosphere properties

The environment–thermodynamic properties can be measured from standard meteorological observations, if the environment is well mixed or otherwise approximated. The first fire–atmosphere property (*φ*) can, in theory, be determined from these same environment variables plus a single measure of the plume thermodynamics (*θ*_{pl} and *q*_{pl}), at any level in which the radiative heat losses are small (i.e., above the flaming zone), using Eqs. (10)–(12):

For the range of parameters considered, the square-bracketed term is small, compared to LTA09’s realistic *φ* range, and can be ignored. Alternatively, extreme values of the term could be used as error bars, based on estimated realistic ranges of each parameter, or if no in-plume observations are available, LTA09’s *φ* range could be used (e.g., Figs. 4, 5).

In reality, *φ* is likely to vary spatially and with time (e.g., as the fire burns through variable fuels). Thus, an array of *φ* values could be calculated from multiple plume measurements, with a corresponding cluster of SP curves added to thermodynamic diagrams. These SP curves should, in theory, reside within the LTA09 extremes, depicted as blue and red SP curves in Figs. 4 and 5. However, if LTA09’s *φ* extremes are truly extreme, then real SP curves may occupy a much narrower range and could perhaps be approximated by a single curve, based on a predetermined value of *φ* valid for most fires.

It is not practical at present to observe plumes with enough spatial and temporal resolution to generate the PE temperature and moisture distributions depicted in Fig. 5, which would in theory enable direct prediction of pyroCu/Cb formation. However, the LEM simulations do provide valuable insight. They confirm that *θ*_{pl} and, hence, *β* decrease with height and that there is a spectrum of *β* values within the plume at any given height (section 4c). Furthermore, the width of the *β*_{SP} spectrum and position on the SP curve is likely to reflect the energy or vigor of any resulting pyroCu/Cb.

The *β*_{SP} spectrum will be influenced by numerous factors, many of which are not well observed, and their relative importance is not well understood. These factors include fire size and intensity; fuel properties; terrain; background winds including speed and direction variability with height and turbulence; and atmospheric temperature, moisture, and stability. Furthermore, nonlinear interactions among these factors in the vicinity of the fire can generate fire–atmosphere feedbacks that generate dangerous and unpredictable fire behavior (e.g., Rothermel 1991; Banta et al. 1992; Goens and Andrews 1998; Peace et al. 2015a,b, 2016; Dowdy et al. 2017). Predicting the *β*_{SP} spectrum is perhaps one of the greatest pyroCu/Cb forecasting challenges.

### b. What have we learned about β_{SP}?

While *β*_{SP} is at present largely unpredictable, this study offers insight into likely *β*_{SP} values for typical fire conditions and how these values might affect pyroCu/Cb development and behavior, including well above the SP [see pyroCAPE estimates; Eq. (25)].

The LEM simulations suggest *β*_{SP} is likely to be very small for typical wildfire plumes that develop in warm–hot conditions [e.g., *β*_{mean} and *β*_{max} values for all LEM simulations were *O*(10^{−2}–10^{−3}). With increasing background winds and turbulence, *β* further decreases with height as plumes tilt toward the horizontal plane (e.g., Briggs 1975; Weil 1988; Cunningham and Goodrick 2013; Thurston et al. 2013; Lareau and Clements 2017), and smaller or weaker fires will also have reduced *β* values (Briggs 1975), which supports the proposition that *β*_{SP} ≤ *O*(10^{−2}) for most real wildfire plumes.

Quantitative estimates of pyroCAPE with respect to *β*_{SP} are provided in appendix A. Substantial pyroCAPE values would be expected for the hot fire plume depicted in Figs. 5b and 5c: for example, 220–290 J kg^{−1} for *β*_{mean} = 0.004, increasing to 770–1010 J kg^{−1} for *β*_{max} = 0.014 for the two *φ* extremes. These *φ* extremes introduce a 24% difference in pyroCAPE, regardless of *β*_{SP} ( appendix A). The same fire in a lower LCL environment would have larger *β*_{SP} (implied from Fig. 5) and even larger pyroCAPE [Eq. (25)] with values doubling and tripling, respectively, for *P*_{SP} at 800 and 900 hPa.

### c. Forecasting applications

While it is not currently possible to predict or observe the *β*_{SP} spectra for a particular fire, we can estimate the minimum *β*_{SP} required for deep, moist, free convection (*β*_{DC}) for a specific thermodynamic environment and the height at which this occurs (*z*_{DC}). (Hereafter, we use DC to abbreviate deep, moist, free convection, for variable subscripts and compound acronyms.) These values can be used to determine a relative pyroCb threat (e.g., larger values indicate deeper and hotter plumes will be required for deep, moist, free convection to develop). Wildfires that form with a stable layer capping the mixed layer must produce PEs with additional buoyancy to breach this stable layer before free convection can develop. For cases with very weak stable layers, *β*_{DC} → 0, Δ*P* → 0, pyroCAPE → 0 [Eq. (25)], and knowledge of *φ* is not required to determine the pyroCb relative threat.

Another complicating factor of real environments is nonideal inverted-V mixed (or semi-mixed) layers. This affects how the ELCL is defined and the subsequent shapes and position of the SP curves. Here, we discuss various forms of lifting condensation levels (LCLs) and convective condensation levels (CCLs), including a few additional variants, to demonstrate which is most appropriate for pyroCb prediction. First, we note that for our idealized inverted-V mixed layer, all LCLs reduce to the ELCL, and if we cap this mixed layer with a moist neutral environment (i.e., constant *θ*_{e}), all CCL variants also reduce to the ELCL.

Each LCL and CCL variant is depicted in Fig. 6a for a morning radiosonde sounding at Williamtown [coastal location in New South Wales (NSW), Australia] a few hours before the development of two near-simultaneous pyroCb events in mountainous terrain about 100 km inland from Williamtown (Fromm et al. 2012). For this sounding, the surface-based LCL (SLCL; the height at which a lifted surface parcel condenses) is only slightly more elevated than the most unstable LCL (MULCL; the lowest condensation height of any lifted parcel), and both would be strongly negatively buoyant (about 10 K cooler than the environment at these levels). They both assume zero entrainment as the parcels rise and thus do not represent buoyant PEs very well. The mixed layer LCL (MLLCL) is an improvement because it effectively samples air from all layers up to condensation, as would be expected in an entraining plume. In Fig. 6a, the MLLCL is more elevated than the SLCL and MULCL and would be less negatively buoyant. The conventional CCL (the height a surface parcel would condense with positive buoyancy, given sufficient surface heating) is essentially just an extension of the surface specific humidity isopleth until it intersects the environment temperature trace. Here, we label it the SCCL to highlight its surface parcel origin. The SCCL height is between the MLLCL and SLCL, but its underlying zero entrainment assumption raises questions about its validity for representing plume behavior. A more appropriate CCL might be one defined by the extension of the mixed layer specific humidity line to the environment temperature trace, which we call the mixed layer CCL (MLCCL). In Fig. 6a, the MLCCL is more elevated than the MLLCL and by definition is just positively buoyant. However, in this example, an extension of a moist adiabat from the MLCCL suggests any moist convection that ensues would be quite shallow, since the moist adiabat would intersect the environment temperature trace only 100 hPa higher. Thus, to identify potential for deep, moist, free convection, we introduce another CCL variant, the DCMLCCL, which is an extension of the ML-specific humidity line to the moist adiabat that first intersects the environment temperature trace near the tropopause (hereafter, *θ*_{e,DC}; Fig. 6a).

In Fig. 6a, the DCMLCCL is considerably more elevated than the MLCCL and much warmer than the immediate environment, suggesting PEs would need to rise more than 5 km while maintaining considerable buoyancy for free convection to develop. Large buoyancy suggests any PEs at this point on the diagram will contain nontrivial concentrations of combustion gas, in which case the fire moisture impact on PE condensation should be considered (i.e., use the SP curve attached to an appropriate ELCL).

Consistent with the arguments above, we approximate *θ*_{env} and *q*_{env} with mixed layer values, which places the ELCL at the MLLCL. The intersection of the SP curve with the environment temperature trace marks the minimum position on the diagram for buoyant moist convection (analogous to the MLCCL), which we call the pyro CCL (PCCL). Continuing further along the SP curve to *θ*_{e,DC}, we define the deep, moist, free convection PCCL (DCPCCL, analogous to the DCMLCCL). In this example, DCPCCL is much lower than DCMLCCL, and the potential temperature at DCPCCL is about 3 K cooler than at DCMLCCL (cf. the yellow and green adiabats in Fig. 6a). This shows that ignoring fire moisture can lead to a nontrivial overprediction of both *z*_{DC} and *β*_{DC} and an underestimation of the minimum fire power required for pyroCb formation.

While the morning, coastal Williamtown sounding illustrates very well variations in LCL and CCL, it is unclear how well it represents actual conditions at the two inland mountainous fire grounds later in the day. For example, the MLLCL would be more elevated and potentially warmer in the absence of the maritime cool and moist layer (below ~800 m; Fig. 6a), which would require PEs to rise somewhat higher for condensation to occur, but they would not need to be as buoyant. This thermodynamic model sensitivity highlights the importance of accurate fire ground observations.^{12}

To provide a sense of how these LCLs and CCLs can differ from case to case, a similar analysis is applied to a mobile sounding upwind of the Sir Ivan fire (about 100 km east of Dubbo, New South Wales, Australia) that produced pyroCb on 12 February 2017 (Fig. 6b). It shows a near-ideal inverted-V mixed layer capped by a stable layer. The near-ideal inverted V means the SLCL, MULCL, and MLLCL are almost identical, as are the SCCL and MLCCL. As in Fig. 6a, the PCCL is lower than the MLCCL, and again, any buoyant convection that would develop from PEs reaching these points would be shallow due to a second stable layer above. Similarly, deep, moist, free convection requires additional buoyancy to breach these stable layers, leading to a nontrivial separation in position of the DCPCCL and DCMLCCL and subsequent differences in *z*_{DC} and *β*_{DC} estimates.

These conclusions appear to be at odds with the analysis of a pyroconvection event presented in Fig. 9 of Lareau and Clements (2016; hereafter LC16), who showed the SCCL was a good estimate of the plume cloud base height. This figure is reproduced in Fig. 7 with SP curves added to their MLLCL. It also shows a nonideal inverted V capped by a very dry stable layer. In this case, any of the CCLs introduced in Fig. 6 would be a good indicator of plume cloud base because they are all very similar in this particular environment.^{13} In this case, the PCCL is also identical to the DCPCCL, which, for clarity, is indicated in Fig. 7 by dashed lines. In this figure, two SP curves have been added to LC16’s MLLCL to illustrate the DCPCCL variability associated with LTA09’s moist and dry fire extremes. Both lines are slightly lower than the observed cloud base and LC16’s diagnosed CCL. This is because the conventional mixed layer (surface-based atmospheric mixed layer, here below ~670 hPa) differs from an effective mixed layer that the plume experiences, since the plume ascending to these free-convection levels will also entrain air from the drier and warmer layer above. As a consequence, the plume mixed layer will be somewhat more elevated, as will the SP curves and associated P CCLs. This demonstrates that a nonstandard mixed layer LCL will need to be incorporated in cases such as this (i.e., progressively average each layer of *θ* and *q* until the saturation height is reached^{14}).

The close similarity of all CCLs in this case provides a good opportunity to estimate *β*_{DC} and an associated pyroCAPE. Using Eqs. (19) and (A5) and *θ*_{env} = 314 K, Δ*θ*_{FC} = 7.4 K (LC16, p. 4013), yields *β*_{DC} = 0.024 and PC = 1180 to 1650 J kg^{−1}. These pyroCAPE values are larger than LC16’s fireCAPE calculation of 910 J kg^{−1} for two obvious reasons: *T*_{SP} − *T*_{EL} ~ 45 K, 10% less than the 50 K assumed in Eq. (A5); and the mixed layer CAPE is negative (has large convective inhibition, plus an extension of the mixed layer moist adiabat shows negative area above 400 hPa), which ensures fireCAPE is less than pyroCAPE [Eq. (22)].

The *β*_{DC} estimate also enables our plume dilution claims for this case to be tested. Using the two extreme flame temperatures and radiative heat losses corresponding to *γ* = 2 to 5 and *δ* = 0.5 to 1 and Eq. (10) yields a dilution range of 95.2%–99.4%.

## 6. Summary

PyroCu/Cb formation is difficult to predict due to many unknown factors that influence the size, buoyancy, moisture content, and dilution rates of plumes. By assuming an ideal inverted-V thermodynamic environment, the plume thermodynamic model reduces these unknowns to just four parameters: the environment potential temperature (*θ*_{env}) and specific humidity (*q*_{env}), the fire moisture to potential temperature increment ratio (*φ*), and a plume buoyancy parameter (*β*). Only the first two parameters can be readily predicted and observed. The third and fourth parameters are the most difficult to predict since they are sensitive to all factors that influence plume structure and behavior (e.g., fire size, fire intensity, fuel moisture, fuel type, atmospheric temperature, humidity, stability, winds, turbulence, and terrain slope).

Fortunately, the study suggests that for typical wildfire conditions, the properties of the condensing plume can be determined to a very good approximation from the easily predictable and observable *θ*_{env} and *q*_{env}, and estimates of *φ*, and these properties can be used to aid pyroCu/Cb prediction. Specific conclusions are as follows:

PyroCb formation requires a core of buoyant plume gases that rises to the level of free convection in an environment that enables deep, moist convection. However, the plume core requires substantial dilution ~(95%–99%) to enable condensation to occur, but not so much dilution that it loses buoyancy.

The primary and secondary influences on plume condensation height are the ELCL and SP curve slope, respectively, which are both functions of

*θ*_{env}and*q*_{env}. For typical wildfire conditions, the*θ*_{env}/*q*_{env}ratio is such that ELCL is dominated by*θ*_{env}, and the SP curve slope is dominated by*q*_{env}, such that warmer environments would be expected to have higher ELCLs and steeper SP curves (for a given*φ*) than cooler environments. Thus, in general, plumes need to reach greater heights to condense in warmer environments than in cooler environments.The SP curve slope is also dependent on the fire moisture to heat ratio

*φ*(steeper for drier fires). This ratio can in theory be estimated from individual measurements of*θ*_{pl}and*q*_{pl}, given knowledge of*θ*_{env}and*q*_{env}.LEM simulations showed the plume contains PEs with a broad spectrum of buoyancies that decrease with height (consistent with dilution from entrainment). These simulations led us to speculate that

*β*_{SP}≤ ~*O*(10^{−2}) for typical wildfires, with corresponding pyroCAPE potential of up to 1000 J kg^{−1}.The buoyancy parameter representing a PE (

*β*) is a function of the initial temperature of the combustion gas, the heat lost to radiation, and the amount of dilution experienced by the PE, which are functions of the parameters*γ*,*δ*, and*α*, respectively [Eq. (10)]. Fortunately, plume condensation is dependent on*β*_{SP}and not the individual values of*γ*,*δ*, and*α*, which means knowledge of these parameter values is not necessary, and our conclusions are valid for all fires.It is not practical at present to observe the

*β*_{SP}spectrum within plumes, and it is virtually impossible to predict. Instead, we can determine for specific thermodynamic environments the minimum value of*β*_{SP}required for deep, moist, free convection to occur (*β*_{DC}) and the minimum height at which it must be present (*z*_{DC}), which can be used to provide an indication of pyroCb threat.In realistic wildfire scenarios, the mixed layer might only approximate an inverted V and may be capped by a stable layer. In such conditions, we suggest the most consistent and accurate way to estimate

*β*_{DC}and*z*_{DC}is to use a mixed layer*θ*and*q*to determine the ELCL and SP curve, with either a midrange estimate of*φ*or with two SP curves plotted representing the LTA09 extreme*φ*values. The quantities*β*_{DC}and*z*_{DC}can be found at the DCPCCL, which is at the intersection of the SP curve and the lowest moist adiabat that first intersects the temperature trace near the tropopause (*θ*_{e,DC}).

The last two points demonstrate the potential use of the thermodynamic model for pyroCu/Cb prediction. Indeed, the qualitative relationship between total fire power required for pyroCb formation and values of *β*_{DC} and *z*_{DC} can in theory be quantified by connecting a plume-rise model to the thermodynamic model.

## Acknowledgments

Thanks to Andrew Dowdy and Dean Sgarbossa for their manuscript review. Thanks also to Mika Peace, Claire Yeo, and Jason Sharples for discussions that prompted the initial investigation and to David Morgan for the use of the background image of a stylized pyroCb cloud in Fig. 2a. The manuscript also benefitted from insightful criticism from Mike Fromm and two anonymous reviewers.

### APPENDIX A

#### PyroCAPE Sensitivity to *β* and *φ*

Approximations valid for small *β* can be made to Eqs. (22)–(24) and provide useful insight into the sensitivity of pyroCAPE to *β* and *φ*. If we assume *C*_{p} ≫ *C*_{l}*r* and *r* ~ *q*, and *T*_{SP} ~ *T*_{LCL},

Using *α* ~ 1 for small *β*_{SP}, Eq. (12) is used to show

and using Eq. (11),

for small *β*. Then,

A rough estimate of pyroCAPE as a function of *β*_{SP} can be obtained by assuming *T*_{SP} − *T*_{EL} ~ 50 and considering LTA09’s range of *φ* values and the cold-to-warm range of *θ*_{env}, yielding

For *β*_{SP} = 0.01, Eq. (A5) yields pyroCAPE of ~500–700 J kg^{−1}. The true value of pyroCAPE could potentially double in a warmer and moister environment as both *T*_{SP} increases and *T*_{EL} decreases. Repeating for the maximum value of *β*_{SP} for the hot fire LEM simulation at 4000 m (*β*_{max} = 0.014, *θ*_{env} = 310, *T*_{SP} = 267, *T*_{SP} − *T*_{EL} = 50), pyroCAPE ranges from about 770 to 1010 J kg^{−1} for LTA09’s driest to moistest fires. Equivalent values of pyroCAPE for the mean plume (*β*_{mean} = 0.004) are about 220 and 290 J kg^{−1}.

A fractional difference in pyroCAPE for two values of *φ* can be estimated from Eq. (A4):

Here, the term in parentheses is about 35 × 10^{−5} K^{−1} for the values used in this paper, which yields a 24% difference in pyroCAPE for LTA09’s two extreme *φ* values.

### APPENDIX B

#### ELCL Sensitivity to *θ* and *q*

The sensitivity of the ELCL to variations in *θ*_{env} and *q*_{env} (evident in Fig. 3) can be quantified using Eq. (A8) of LTA09:

The subscript “ref” signifies reference values of pressure, *θ*, and *q* corresponding to a particular ELCL. Using the thermodynamic values given in Table 2 and *T*_{LCL} = 271 K, Eq. (B1) becomes

Equation (B2) shows that a fractional change in *q*_{env} (*θ*_{env}) produces a fractional change in ELCL pressure of the same (opposite) sign, but about 1/5 (4 times) the magnitude. Thus, ELCL pressure changes are 20 times more sensitive to fractional changes in *θ*_{env} than *q*_{env}.

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## Footnotes

^{a}

Current affiliation: Met Office, Exeter, United Kingdom.

© 2018 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

^{1}

The inverted-V profile also favors downburst development, when precipitation from the moist midtroposphere evaporates as it falls through the dry layer below. Intense, gusty downburst winds are difficult to predict and can come from completely different direction to the ambient flow, which can be very hazardous to fire crews (e.g., Rothermel 1991; Goens and Andrews 1998; Johnson et al. 2014; Potter and Hernandez 2017).

^{2}

The SP curve comprises the plume LCL temperature and pressure loci for all possible mixtures of combustion gas and environment air.

^{3}

The intersection positions of the hypothetical temperature curves (yellow) with the SP curve (blue) in Fig. 2c have been exaggerated for illustrative purposes. We speculate that condensing elements within typical wildfire plumes will occur much closer to the environment LCL (at the peak of the inverted V).

^{4}

Useful information about the condensing plume buoyancy can be derived from pre-condensation plume thermodynamics and environment thermodynamics at the condensation level (section 5b).

^{5}

Wotton et al. (2012) observed flame temperature ranges from about 600 to 1400 K in experimental forest fires. We chose an upper temperature of 1500 K to extend the parameter space to include potentially higher temperatures that might occur in very large and intense wildfires. This value matches observed temperatures for methane fires (Smith and Cox 1992 *)*.

^{6}

For example, at 4000-m elevation, the difference between Δ*θ* and Δ*T* is less than 15%.

^{7}

For the parameter space investigated, the deviation from exact overlap is greatest for the hot, dry fire in the warm environment and increases with increasing *β*. The maximum difference in *q*_{pl} for two pairs of α and *γ* corresponding to *β* = 0.05 (the warm scenario red dot in Fig. 3) is less than 0.05% (not shown).

^{8}

This expression of fireCAPE will require the “regular CAPE” to be negative in some cases.

^{9}

This is only part of the *β*_{SP} range represented by the entire parameter space considered (e.g., the maximum *β*_{SP} is 4, which corresponds to the hot and dry fire in which *γ* = 5, *α* = 0, and *δ* = 0).

^{10}

Combustion gases experiencing zero dilution for the hot fires would need to be raised to a level of 1.2 hPa (about 47 km above sea level) before condensation would occur.

^{11}

All our idealized plume simulations (not shown) condense near the ELCL, as does the plume reported in LC16. Additionally, the existence of competing arguments stating that plume cloud bases will be lower [Potter’s (2005) hypothesis] and higher (LTA09’s theory) than the ELCL suggest typical plume cloud bases are at least near the ELCL.

^{12}

Observations showed Wollemi-fire plume tops reaching 14 km above sea level for a period of almost an hour in the afternoon (Fromm et al. 2012). Assuming the sounding above ~1 km represents the true mixed layer, and the true neutral-buoyancy height is somewhat less than 14 km (i.e., there is overshooting of negatively buoyant PEs), the thermodynamic model suggests the most buoyant PEs may have been more than 10 K warmer than the surrounding environment just prior to condensing (not shown).

^{13}

The surface *q* is almost the same as the mixed layer *q*, which means SCCL and MLCCL are very similar, and because the next stable layer above is near the tropopause, these CCLs are identical to the DCMLCCL.

^{14}

We are currently experimenting with a weighted average to incorporate increased entrainment rates with height predicted by the Briggs (1975) plume model.