## 1. Introduction and background

Convective initiation is an important and well-studied topic in atmospheric science with broad application. Theories of convective initiation usually take into account the vertical profile of temperature and humidity and various conceptions of parcel or layer lifting. To set the stage for the later discussion, existing approaches will be briefly reviewed.

### a. Parcel theories of convective initiation

*δ*oscillates or grows exponentially with time

*t*with a real or imaginary frequency

*α*:

*θ*is potential temperature and

*θ*

_{es}is saturated equivalent potential temperature. The sign of

*N*

^{2}determines whether the parcel oscillates about its level of neutral buoyancy (

*N*

^{2}> 0) or accelerates vertically (

*N*

^{2}< 0) since

*α*can be real or imaginary. While (1) is generally used to understand atmospheric stability, in order to apply it to convective initiation, an immediate problem is the unknown initial displacement

*δ*(

*t*= 0).

Nonmixing parcel theory is readily extended to finite displacements through complex layered atmospheres using simple numerical or graphical calculations. For example, a parcel may need to be lifted above a stable layer and into an unstable layer or from an unsaturated to a saturated state to initiate convection. A variety of sounding indices are calculated by integrating along the vertical profile. Convective available potential energy (CAPE), convective inhibition (CIN), and lifted index (LI) are especially useful measures of instability (Emanuel 1994; Blanchard 1998; Riemann-Campe et al. 2009).

The utility of parcel theories can be improved if some account of mixing with the environment is included. When ambient air is entrained into a rising parcel, the conservation of moist static energy and water mass can be used to compute how the parcel will modify its temperature, condensed water, and buoyancy. Generally, entrainment cools a cloudy parcel by evaporating cloud water. Conceptions of entrainment into both “bubble” and “plume” geometries are well developed (e.g., Morton et al. 1956; Raymond and Blyth 1986; Kain and Fritsch 1990; Nie and Kuang 2012; Sherwood et al. 2013; de Rooy et al. 2013). In the present study, we use the term plume to describe convection but it is not meant to distinguish between either concept.

### b. Coupling updrafts and downdrafts

*w*and frequency

*α*:

*k*,

*l*, and

*m*are the horizontal and vertical wavenumbers in the

*x*,

*y*, and

*z*directions respectively. If the layer is all dry or all saturated, the dry or moist

*N*

^{2}is used in (4). Again, the sign of

*N*

^{2}determines whether the vertical velocities will oscillate or grow in time. According to (4), cells that are wide and shallow (i.e.,

*m*

^{2}≫

*k*

^{2}+

*l*

^{2}) will grow in amplitude much more slowly due to the “nonhydrostatic” or “added mass” effect related to compensating motion. As before, to apply (3) to convective initiation, knowledge of the initial amplitude

*A*

_{c}) while the downdrafts (dry area is denoted

*A*

_{d}) are not. If we define the strength of the convection as the difference between the speed of the updraft and downdraft, Δ

*w*=

*w*

_{c}−

*w*

_{d}, this quantity grows exponentially with a frequency

*α*:

*A*

_{c}/

*A*

_{d}is sufficiently small, the slice stability

*A*

_{c}/

*A*

_{d}) increases as the downdrafts are stronger. While the slice method does not account for nonhydrostatic effects on cloud dynamics, it does include the strong stabilizing effect of dry descent between the moist updrafts. As we have seen before in (1) and (3), (5) does not provide any information about the strength of the initial velocity perturbations Δ

*w*(

*t*= 0).

### c. Lifting of a layer

A final group of convective initiation theories involves lifting a layer instead of a parcel. For orographic convection, it is much easier to imagine the terrain slope causing a whole layer to rise instead of a single parcel. Traditionally, layer lifting is used to define “potential instability.” A layer is said to be potentially unstable if it has a *θ*_{e} that decreases with height (Saucier 1955, 76–78; Kreitzberg and Perkey 1976). If the layer is brought to saturation by lifting, the *N*^{2} values in (2) and (4) become

A basic problem with applying all the above theories to convective initiation is that they leave unspecified the source of the initial perturbations (i.e., the seeds of the convection). A frequency is useful, but without knowing the initial perturbations that will oscillate or grow, it is difficult to know how a convective environment will evolve. To clarify this issue we focus on the initial inhomogeneities in the layer before it is lifted and before plume convection is initiated.

According to KS09 there are two primary sources of layer inhomogeneity: preexisting scattered clouds and random humidity fluctuations in an unsaturated layer. The response of the scattered clouds to layer lifting is treated theoretically by KS09 using an extension of the slice method. The response of the unsaturated humidity “seeds” to layer lifting is the focus of the present paper. If the scale and amplitude of the initial humidity fluctuations are known, the buoyancy inhomogeneities within the lifted layer can be computed, giving a more complete and quantitative model of convective initiation (Kirshbaum and Grant 2012).

### d. Woodcock’s LCL theory

The first proposal that ambient humidity variations play a role in orographic convection was by Woodcock (1960, hereafter W60) in regard to precipitation over Hawaii. He observed that “as the lower air orographically ascends, the moist parcels will be the first to reach the condensation lifting level, and hence will be the first to experience the additional buoyancy as latent heat is released” (our Fig. 1, W60). Both KS09 and W60 consider this mechanism for convective initiation in a tropical region with a conditionally unstable atmosphere. The atmospheric lapse rate is important as it provides instability to saturated air parcels but remains stable for unsaturated parcels thus presenting a mechanism for the creation of differential buoyancy and convective initiation described in depth in section 3. This study will expand the variable lifting condensation level (LCL) ideas built in W60 and KS09 and apply them to observations from the Dominica Experiment (DOMEX) field campaign.

*ρ*is density (kg m

^{−3}),

*g*is gravity (m s

^{−2}), and a prime indicates a perturbation from the mean that is indicated by an overbar:

The conditional instability of the atmosphere is important in the generation of orographic convection but it does not enter into the calculation of buoyancy variance in a lifted layer. If the atmosphere is absolutely stable, the buoyancy variance generated by differences in LCL within a layer will quickly be eliminated by buoyancy sorting and adjustment. Any buoyancy perturbation present will oscillate and damp to zero. Parcels in the layer will move up or down a few tens of meters until the air density is constant on a horizontal plane. If the atmosphere is unstable, parcels rising out of the lifted layer will gain buoyancy and will accelerate faster, leading to plume formation. For a related but distinctly different usage of the term buoyancy sorting, see Telford (1975), Emanuel (1991), and Taylor and Baker (1991) among others.

This study presents a new theory following W60 for convective initiation that takes into account initial fluctuations in temperature *T*, water vapor mixing ratio *q*_{υ}, layer lifting by terrain, and the relationship between buoyancy and acceleration.

## 2. Observations

### a. DOMEX

Observational data from the field phase of the DOMEX project (Smith et al. 2012) are used to test a theory for convective initiation involving lifting an unsaturated layer with humidity fluctuations. DOMEX provides a unique opportunity as the first project in which lateral inhomogeneities were measured just prior to bulk lifting. The project centered on the Commonwealth of Dominica, a small island in the trade wind belt (15°N, 61°W) (Fig. 2). Its mountainous peaks extend above the LCL of incoming surface airflow (~600 m) but below the trade wind inversion (~2 km). Lifting of conditionally unstable air by the windward slopes initiates new convection and enhances already existing convection incoming with the trade wind flow (KS09; Smith et al. 2012). The background conditionally unstable environment (Fig. 3) produces buoyant clouds like those in Fig. 4. These conditions make Dominica an ideal place to study the triggering of moist convection.

A map of the Commonwealth of Dominica showing the locations and heights of the fixed flight tracks flown by the aircraft during the DOMEX field campaign. Adapted from Smith et al. (2012).

Citation: Journal of the Atmospheric Sciences 71, 12; 10.1175/JAS-D-14-0089.1

A map of the Commonwealth of Dominica showing the locations and heights of the fixed flight tracks flown by the aircraft during the DOMEX field campaign. Adapted from Smith et al. (2012).

Citation: Journal of the Atmospheric Sciences 71, 12; 10.1175/JAS-D-14-0089.1

A map of the Commonwealth of Dominica showing the locations and heights of the fixed flight tracks flown by the aircraft during the DOMEX field campaign. Adapted from Smith et al. (2012).

Citation: Journal of the Atmospheric Sciences 71, 12; 10.1175/JAS-D-14-0089.1

A typical example of the vertical structure of dry potential temperature (*θ*_{d}), equivalent potential temperature (*θ*_{e}), and saturated equivalent potential temperature (*θ*_{es}; Betts and Dugan 1973) from the aircraft on RF 13 showing the conditionally unstable nature of the environment. Adapted from Smith et al. (2012).

Citation: Journal of the Atmospheric Sciences 71, 12; 10.1175/JAS-D-14-0089.1

A typical example of the vertical structure of dry potential temperature (*θ*_{d}), equivalent potential temperature (*θ*_{e}), and saturated equivalent potential temperature (*θ*_{es}; Betts and Dugan 1973) from the aircraft on RF 13 showing the conditionally unstable nature of the environment. Adapted from Smith et al. (2012).

Citation: Journal of the Atmospheric Sciences 71, 12; 10.1175/JAS-D-14-0089.1

A typical example of the vertical structure of dry potential temperature (*θ*_{d}), equivalent potential temperature (*θ*_{e}), and saturated equivalent potential temperature (*θ*_{es}; Betts and Dugan 1973) from the aircraft on RF 13 showing the conditionally unstable nature of the environment. Adapted from Smith et al. (2012).

Citation: Journal of the Atmospheric Sciences 71, 12; 10.1175/JAS-D-14-0089.1

A photograph taken from the aircraft during DOMEX of the plumelike convection bubbling over Dominica’s windward slope. The air comes toward Dominica from the right side of the photo and is lifted by the terrain, whose coastline is just visible under the clouds in the bottom-right-hand corner of the photo. Upon orographic uplift, plume convection develops from the moist regions of the flow, defined herein as seeds.

Citation: Journal of the Atmospheric Sciences 71, 12; 10.1175/JAS-D-14-0089.1

A photograph taken from the aircraft during DOMEX of the plumelike convection bubbling over Dominica’s windward slope. The air comes toward Dominica from the right side of the photo and is lifted by the terrain, whose coastline is just visible under the clouds in the bottom-right-hand corner of the photo. Upon orographic uplift, plume convection develops from the moist regions of the flow, defined herein as seeds.

Citation: Journal of the Atmospheric Sciences 71, 12; 10.1175/JAS-D-14-0089.1

A photograph taken from the aircraft during DOMEX of the plumelike convection bubbling over Dominica’s windward slope. The air comes toward Dominica from the right side of the photo and is lifted by the terrain, whose coastline is just visible under the clouds in the bottom-right-hand corner of the photo. Upon orographic uplift, plume convection develops from the moist regions of the flow, defined herein as seeds.

Citation: Journal of the Atmospheric Sciences 71, 12; 10.1175/JAS-D-14-0089.1

The field project on Dominica lasted 6 weeks in April and May of 2011. During this period the University of Wyoming King Air aircraft flew 21 fixed-track research flights (RFs) outfitted with instruments for in situ and remote sensing measurements of the environment and clouds upstream, downstream, and over Dominica. The aircraft measured temperature, specific humidity, and vertical velocity among other variables. The repeated flight tracks and consistent trade wind environment created a dataset with statistical merit.

### b. T and q_{υ} anticorrelations

The inhomogeneous tropical Atlantic atmosphere was carefully observed during DOMEX. An upstream flight track (“leg 1” in Fig. 2) located 30 km east of Dominica's long axis and 70 km long was flown at two altitudes. We use the lower altitude (L = low) flown at 300 m above mean sea level. The aircraft measured the perturbations in water vapor mixing ratio (*q*_{υ}; g kg^{−1}), and temperature (*T*; K) in the subcloud layer before the layer is orographically lifted. After removing the mean, *T* varies on average ~0.2 K and *q*_{υ} varies ~1 g kg^{−1} for all RFs. It was found that these upstream *T* and *q*_{υ} perturbations are strongly anticorrelated (as in Paluch and Lenschow 1991; Shinoda et al. 2009) with an average correlation coefficient of −0.78 when not influenced by rain (Fig. 5). Regions within the layer measured by the aircraft tend to be warm and dry, or cool and moist.

Correlation coefficient (CC) of *T* and *q*_{υ} on leg 1L for every RF. On RFs where the aircraft encountered liquid water (LW) below cloud base (gray), the CC is less negative and sometimes positive while RFs that did not encounter LW (black) have a more negative CC showing strong anticorrelation. The average for all RFs that did not encounter LW is −0.78.

Citation: Journal of the Atmospheric Sciences 71, 12; 10.1175/JAS-D-14-0089.1

Correlation coefficient (CC) of *T* and *q*_{υ} on leg 1L for every RF. On RFs where the aircraft encountered liquid water (LW) below cloud base (gray), the CC is less negative and sometimes positive while RFs that did not encounter LW (black) have a more negative CC showing strong anticorrelation. The average for all RFs that did not encounter LW is −0.78.

Citation: Journal of the Atmospheric Sciences 71, 12; 10.1175/JAS-D-14-0089.1

Correlation coefficient (CC) of *T* and *q*_{υ} on leg 1L for every RF. On RFs where the aircraft encountered liquid water (LW) below cloud base (gray), the CC is less negative and sometimes positive while RFs that did not encounter LW (black) have a more negative CC showing strong anticorrelation. The average for all RFs that did not encounter LW is −0.78.

Citation: Journal of the Atmospheric Sciences 71, 12; 10.1175/JAS-D-14-0089.1

The *T* measurements were made with a Rosemount reverse-flow thermometer having an accuracy of 0.5 K and a resolution of 0.006 K. Measurements of *q*_{υ} were derived from a LICOR instrument measuring partial pressure of water vapor using optical absorption having an accuracy of 1% and a resolution of 0.1 hPa. These instrument capabilities suggest that the measured variations are real. The dominant horizontal scales of *T* and *q*_{υ} perturbations were measured to be ~3–8 km (Smith et al. 2012). Unfortunately no measurements were made of the vertical extent of *T* and *q*_{υ} anomalies or their horizontal extent in the along-wind direction perpendicular to leg 1L.

*T*and

*q*

_{υ}anticorrelated, but their contributions to virtual temperature (

*T*

_{υ}, K),

*T*

_{υ}varies on average ~0.13 K. By having a nearly constant

*T*

_{υ}, the layer also has nearly constant buoyancy. We define the anomalously moist patches with above-average

*q*

_{υ}as seeds because they are passive in the upstream environment but upon orographic uplift by Dominica, they seed or initiate convection. A typical flight track (RF4) with

*T*and

*q*

_{υ}balanced such that a nearly constant

*T*

_{υ}is achieved is shown in Fig. 6. In Fig. 7 the

*T*and

*q*

_{υ}perturbations from Fig. 6 are plotted against each other. The slope of constant

*L*/

*C*

_{p}(K) are also included for reference. Notice how the perturbations in Fig. 7 scatter around a slope of constant

*T*

_{υ}rather than along

*L*/

*C*

_{p}. The slope

*L*/

*C*

_{p}is what one would expect as a result of cooling and moistening from evaporating rainfall (Paluch and Lenschow 1991; Smith et al. 2012). In Paluch and Lenschow (1991), they find anticorrelated

*T*and

*q*

_{υ}in the subcloud layer as well, but their anomalies fall along the

*L*/

*C*

_{p}line and are therefore buoyant and would quickly adjust.

(top) The *T* (red) and *q*_{υ} (blue) with the mean removed as measured by aircraft along leg 1L at 300-m height upstream of Dominica on RF4. (bottom) A repeated *T* (red) with *T*_{υ} (black) computed using (8). Note the reduced variance in *T*_{υ} as compared to *T*; the standard deviation of *T* is 0.06 K while the standard deviation of *T*_{υ} is only 0.04 K.

Citation: Journal of the Atmospheric Sciences 71, 12; 10.1175/JAS-D-14-0089.1

(top) The *T* (red) and *q*_{υ} (blue) with the mean removed as measured by aircraft along leg 1L at 300-m height upstream of Dominica on RF4. (bottom) A repeated *T* (red) with *T*_{υ} (black) computed using (8). Note the reduced variance in *T*_{υ} as compared to *T*; the standard deviation of *T* is 0.06 K while the standard deviation of *T*_{υ} is only 0.04 K.

Citation: Journal of the Atmospheric Sciences 71, 12; 10.1175/JAS-D-14-0089.1

(top) The *T* (red) and *q*_{υ} (blue) with the mean removed as measured by aircraft along leg 1L at 300-m height upstream of Dominica on RF4. (bottom) A repeated *T* (red) with *T*_{υ} (black) computed using (8). Note the reduced variance in *T*_{υ} as compared to *T*; the standard deviation of *T* is 0.06 K while the standard deviation of *T*_{υ} is only 0.04 K.

Citation: Journal of the Atmospheric Sciences 71, 12; 10.1175/JAS-D-14-0089.1

Scatter diagram of *T* and *q*_{υ} perturbations from the leg mean along leg 1L of RF4. The perturbations scatter along a line of constant virtual temperature *T*_{υ} rather than along *L*/*C*_{p}. Adapted from Smith et al. (2012).

Citation: Journal of the Atmospheric Sciences 71, 12; 10.1175/JAS-D-14-0089.1

Scatter diagram of *T* and *q*_{υ} perturbations from the leg mean along leg 1L of RF4. The perturbations scatter along a line of constant virtual temperature *T*_{υ} rather than along *L*/*C*_{p}. Adapted from Smith et al. (2012).

Citation: Journal of the Atmospheric Sciences 71, 12; 10.1175/JAS-D-14-0089.1

Scatter diagram of *T* and *q*_{υ} perturbations from the leg mean along leg 1L of RF4. The perturbations scatter along a line of constant virtual temperature *T*_{υ} rather than along *L*/*C*_{p}. Adapted from Smith et al. (2012).

Citation: Journal of the Atmospheric Sciences 71, 12; 10.1175/JAS-D-14-0089.1

In nature, complex convective processes create a variety of *T* and *q*_{υ} perturbations. The usual description involves turbulent entrainment of potentially warm dry air at the top of a mixed layer and evaporation of falling rain to create cool moist patches (Paluch and Lenschow 1991; Smith et al. 2012). Additional mechanisms include dry downdrafts adjacent to clouds to create warm dry air and detrainment of cloudy air to create cool moist air. The neutral buoyancy regions are likely produced through buoyancy sorting as previously discussed. While turbulent mixing may homogenize the atmosphere, new inhomogeneities are continuously added by trade wind convection as parcels traverse the tropical Atlantic. The horizontal scale of these inhomogeneities is likely influenced by the scale of the convection and subsequent lateral spreading during buoyancy adjustment. A discussion of the scale of seeds and the resulting convection is included in section 3e. We hypothesize that the seeds remain as passive tracers in the flow until they are activated through lifting (Kirshbaum and Grant 2012). Air parcels in rain shafts require some interval of time before they have completed their buoyancy sorting. Until this adjustment occurs, we would expect a different relationship between *T* and *q*_{υ} [e.g., *L*/*C*_{p} as in Paluch and Lenschow (1991)].

The water vapor influence on air density is especially strong in the warm moist tropics. We see further evidence of this in the aircraft-derived properties of the boundary layer on legs 1L and 2L at *z* = 300 m. Latent heat fluxes were mostly positive with an average correlation between *q*_{υ} and *w* for all RFs of 0.23. In contrast, most of the sensible heat fluxes were negative with an average correlation between *T* and *w* of −0.05. This difference in sign between latent and sensible heat fluxes has been previously noted over the tropical ocean in observations and numerical models (e.g., Nicholls and LeMone 1980; Siebesma et al. 2003; Smith et al. 2012) and implies that small-scale convection is driven locally by slight differences in the average molecular weight of the air and *q*_{υ} mixture. It also highlights the importance of including *q*_{υ} in calculations of density or buoyancy as in (8). Convection driven by “virtual effects” is termed “compositional” convection and differs from the usual “thermal” convection found in midlatitudes (Stevens 2005; Smith et al. 2012). The differences in convection between the tropics and higher latitudes are further discussed in section 4b.

## 3. Convective initiation theory

This study tests the idea that seeds, or anomalously moist regions, play a dominant role in initiating convection over Dominica. This potential is explored by extending the ideas of W60 and quantifying the effects of lifting a layer with humidity fluctuations.

### a. Buoyancy generation by layer lifting through the LCL

We hypothesize that seeds become activated by orographic uplift by Dominica in a two-stage process. As the upstream low-level air encounters the island, the first stage is layer lifting whereby the entire layer is uniformly elevated. Upon lifting, the initially moist parcels reach their LCL first and begin to generate positive buoyancy by latent heat release with respect to other drier parcels that have not yet reached saturation and are still decreasing in temperature by the dry adiabatic lapse rate (our Fig. 1 and in KS09). The second stage involves the breakup of the layer as buoyant parcels accelerate into the surrounding environment.

The first stage of seed activation is quantitatively modeled using a simple lifting method where each air parcel is lifted step by step reversible moist adiabatically. Every 1-Hz aircraft measurement (~90 m apart) along the 70-km-long leg 1L is considered a different air parcel. All air parcels were initially measured at 300-m height and are lifted vertically above flight level a distance of 1 km in 1-m increments. Variables including *T*, *q*_{υ}, *ρ*, pressure (*P*, Pa), and the liquid water mixing ratio (*q*_{l}, g kg^{−1}) are tracked and computed at each lifting step. Differential buoyancy begins to develop within the layer when it is lifted past the LCL of the parcels with the highest initial relative humidity.

The second stage of the seed activation process begins when parcels have gained enough buoyancy to rise out of their layer, and lift farther into the environment. Section 3b describes how acceleration is calculated from buoyancy after taking into account the effects of added mass. A threshold value of acceleration necessary to transition from layer lifting to plume convection is found. When the lifted layer passes the threshold, it has the potential to break up and initiate larger plumes of convection like in Fig. 4. We imagine this process to be partly comparable to Rayleigh–Taylor instability where a buoyant layer breaks into plume convection (Kull 1991). It is at this point in stage 2 that the importance of conditional instability enters the problem. An air parcel’s ability to leave the layer depends on its initial acceleration. Its subsequent buoyancy and acceleration depends on the deeper (i.e., outside the layer) environmental lapse rate. In a conditionally unstable environment with *dθ*_{es}/*dz* < 0, saturated parcels will be buoyant.

Unlike the convective initiation theories discussed in section 1, an advantage of this method is the specified initial conditions for (1), (3), and (5) [i.e., *δ*(*t* = 0), *w*(*t* = 0)]. Knowing the structure of horizontal perturbations and assuming a two stage process allows for explicit calculation of the initial perturbations that will develop into plume convection after lifting.

### b. Acceleration from buoyancy accounting for added mass

Buoyancy anomalies on various horizontal scales are generated as an inhomogeneous layer is uniformly lifted. While buoyancy (a force per unit mass) has the same units as acceleration (a rate of change of velocity; m s^{−2}), the two are not equivalent. To understand the relationship between buoyancy and acceleration, the added mass effect needs to be taken into consideration. For example, a broad shallow area of positively buoyant fluid will not accelerate as fast as a narrower deeper region of positively buoyant fluid due to the large amount of fluid the first must displace in order to accelerate, as illustrated in Fig. 8. The pressure field needed to divert ambient fluid around the parcel also acts to slow the parcel ascent. We account for this slowing using added mass (Batchelor 1967).

A 2D schematic of the added mass effect. Pressure perturbations develop around large-scale buoyancy variations (see broad oval) result in smaller accelerations (*w*_{t}) than small-scale variations (see narrow oval) due to the lateral circulations that must develop to allow the large regions of positively buoyant air to rise. This effect is encapsulated in (10).

Citation: Journal of the Atmospheric Sciences 71, 12; 10.1175/JAS-D-14-0089.1

A 2D schematic of the added mass effect. Pressure perturbations develop around large-scale buoyancy variations (see broad oval) result in smaller accelerations (*w*_{t}) than small-scale variations (see narrow oval) due to the lateral circulations that must develop to allow the large regions of positively buoyant air to rise. This effect is encapsulated in (10).

Citation: Journal of the Atmospheric Sciences 71, 12; 10.1175/JAS-D-14-0089.1

A 2D schematic of the added mass effect. Pressure perturbations develop around large-scale buoyancy variations (see broad oval) result in smaller accelerations (*w*_{t}) than small-scale variations (see narrow oval) due to the lateral circulations that must develop to allow the large regions of positively buoyant air to rise. This effect is encapsulated in (10).

Citation: Journal of the Atmospheric Sciences 71, 12; 10.1175/JAS-D-14-0089.1

If the buoyant region is spherically shaped, the added mass effect increases the effective inertia of the region by 50% (e.g., Batchelor 1967; Romps and Kuang 2010; Sherwood et al. 2013). In the present study, which involves a variety of shapes and sizes of buoyant regions, we use an acceleration filter that reduces the vertical accelerations associated with broad shallow buoyancy variations.

*w*

_{t}) can be found through the following partial differential equation:

*ρ*′(

*y*)] is representative of a layer of depth

*H*with variable density in the horizontal dimensions (Fig. 8), embedded in an environment with uniform density. A relationship between buoyancy

*w*

_{t}) is

*y*direction, so there is uncertainty regarding the

*x*-direction structure of the density anomalies. Two assumptions regarding the

*x*dependence can be evaluated: (i) measured inhomogeneities are 3D and disk shaped such that

*k*=

*l*and (ii) measured inhomogeneities are 2D and elongated in the

*x*direction such that

*k*= 0. According to (10), short wavelength density or buoyancy anomalies in the limit of

*KH*≫ 1 accelerate quickly according to the following reduced gravity formulation for buoyancy, also given in (7):

*KH*< 1, lie far from the reduced gravity limit. It is at these larger horizontal scales that buoyancy cannot be assumed to be equivalent to acceleration and added mass is important to consider.

Long wavelength density anomalies accelerate more slowly due to the lateral circulations that must develop to allow the large parcel to rise (Fig. 8). An example of how the predicted acceleration *w*_{t} compares to buoyancy using (10) is shown in Fig. 9; note how large-scale variations are damped while small-scale variations are maintained. This relationship between buoyancy and acceleration acts as a high-pass filter in Fourier space and removes any nonphysical accelerations from density anomalies with large horizontal scales relative to the vertical scale *H*. In some cases, the regions of positive buoyancy are so large horizontally (e.g., 15 km) that if they were as deep as they are wide, they would reach the tropopause. Our assumption that they are shallow greatly decreases their acceleration. Sensitivity tests for the free parameters *H* and *k* are shown in section 3f but for the following analyses we assume *H* = 600 m, the approximate depth of the mixed layer, and *k* = *l* suggesting disklike density anomalies.

Buoyancy (blue) and predicted acceleration (*w*_{t}, red) after lifting RF4 leg 1L vertically 1 km. The only difference between the two curves is the application of the acceleration filter from (10) to the blue curve resulting in the red curve (*H* = 600 m and *k* = *l*). Note that large regions of buoyancy have smaller resulting acceleration due to the added mass effect.

Citation: Journal of the Atmospheric Sciences 71, 12; 10.1175/JAS-D-14-0089.1

Buoyancy (blue) and predicted acceleration (*w*_{t}, red) after lifting RF4 leg 1L vertically 1 km. The only difference between the two curves is the application of the acceleration filter from (10) to the blue curve resulting in the red curve (*H* = 600 m and *k* = *l*). Note that large regions of buoyancy have smaller resulting acceleration due to the added mass effect.

Citation: Journal of the Atmospheric Sciences 71, 12; 10.1175/JAS-D-14-0089.1

Buoyancy (blue) and predicted acceleration (*w*_{t}, red) after lifting RF4 leg 1L vertically 1 km. The only difference between the two curves is the application of the acceleration filter from (10) to the blue curve resulting in the red curve (*H* = 600 m and *k* = *l*). Note that large regions of buoyancy have smaller resulting acceleration due to the added mass effect.

Citation: Journal of the Atmospheric Sciences 71, 12; 10.1175/JAS-D-14-0089.1

### c. Development of differential acceleration

The lifting method and the relationship used to calculate acceleration from buoyancy in (10) are both steps toward determining how much acceleration an individual air parcel feels relative to the parcels that surround it. With enough acceleration, parcels will rise out of the lifted layer and plume convection will commence.

The standard deviation of vertical acceleration *σ*(*w*_{t}) is a useful measure for quantifying differential buoyancy. We view *σ*(*w*_{t}) as a statistical measure of the tendency of the lifted layer to break up and generate vertical velocities that lead to convection. At each lifting step, the data are filtered as described and the standard deviation is calculated in the usual manner. The growth in *σ*(*w*_{t}) for all RFs that did not encounter liquid water in the subcloud region is shown in Fig. 10. The initial value of *σ*(*w*_{t}) is highly dependent on the correlation of *T* and *q*_{υ} at flight level. If *T* and *q*_{υ} are strongly anticorrelated, the initial *σ*(*w*_{t}) will be small. As the layer is lifted and air parcels begin to reach their LCL, *σ*(*w*_{t}) begins to grow. As *σ*(*w*_{t}) grows, it becomes more likely that the uplifted layer will break up into plume convection. The value of *σ*(*w*_{t}) reaches a plateau when all the parcels have reached their LCL. Beyond this height, no further buoyancy or acceleration variance is generated.

The standard deviation of acceleration *σ*(*w*_{t}) produced as a layer is lifted. Each curve represents a different RF and shows the growth of *σ*(*w*_{t}) beginning at the measurement level (300 m) and continuing as the layer is lifted through the LCL. The straight black dashed line shows the threshold *σ*(*w*_{t}) for plume convection calculated from (12). The acceleration filter from (10) uses *H* = 600 m and *k* = *l*. Sensitivity tests for the thick black line (RF 12) are shown in Fig. 12 and the blue and red colored lines are distinguished for a description of curve differences in section 3c.

Citation: Journal of the Atmospheric Sciences 71, 12; 10.1175/JAS-D-14-0089.1

The standard deviation of acceleration *σ*(*w*_{t}) produced as a layer is lifted. Each curve represents a different RF and shows the growth of *σ*(*w*_{t}) beginning at the measurement level (300 m) and continuing as the layer is lifted through the LCL. The straight black dashed line shows the threshold *σ*(*w*_{t}) for plume convection calculated from (12). The acceleration filter from (10) uses *H* = 600 m and *k* = *l*. Sensitivity tests for the thick black line (RF 12) are shown in Fig. 12 and the blue and red colored lines are distinguished for a description of curve differences in section 3c.

Citation: Journal of the Atmospheric Sciences 71, 12; 10.1175/JAS-D-14-0089.1

The standard deviation of acceleration *σ*(*w*_{t}) produced as a layer is lifted. Each curve represents a different RF and shows the growth of *σ*(*w*_{t}) beginning at the measurement level (300 m) and continuing as the layer is lifted through the LCL. The straight black dashed line shows the threshold *σ*(*w*_{t}) for plume convection calculated from (12). The acceleration filter from (10) uses *H* = 600 m and *k* = *l*. Sensitivity tests for the thick black line (RF 12) are shown in Fig. 12 and the blue and red colored lines are distinguished for a description of curve differences in section 3c.

Citation: Journal of the Atmospheric Sciences 71, 12; 10.1175/JAS-D-14-0089.1

To gain a more complete understanding of the variations among flights, four curves and their properties are compared. We define Z1 as the height at which the *σ*(*w*_{t}) curves begin to increase and Z2 as the height at which the plateau is reached. The solid blue and dashed blue lines in Fig. 10 have significantly different initial *σ*(*w*_{t}) values and moderately different Z1 and Z2 values. The initial value of *σ*(*w*_{t}) in each case is related to the correlation coefficient (CC in Fig. 5) of *T* and *q*_{υ} before lifting (blue solid = −0.91 vs blue dashed = −0.52, respectively). A more negative CC leads to a *T*_{υ} horizontal profile with less variation and thus a smaller *σ*(*w*_{t}). The quantity Z1 is over 100 m larger for the solid blue curve as compared to the dashed blue curve due to the lower moisture content of the wettest parcels; the wettest parcels that make up the dashed blue curve reach their LCL first. The quantity Z2 is also over 100 m larger for the solid blue curve as compared to the dashed blue curve because both had similar *σ*(*q*_{υ}) before lifting making the growth of *σ*(*w*_{t}) within the layer comparable.

The solid and dashed red curves show two cases where the growth of *σ*(*w*_{t}) differs. While the two cases have similar wet parcels (i.e., Z1) the dashed red curve has dryer dry parcels such that Z2 is larger. Additionally, the dashed red curve has a larger initial *σ*(*q*_{υ}) before lifting than the solid red curve (i.e., 0.49 vs 0.31 g kg^{−1}, respectively) creating a larger spread in the height at which parcels reach their LCL.

### d. Threshold for plume convection

A method is needed to evaluate how much *σ*(*w*_{t}) is sufficient to generate convection. Here we define a threshold value of *σ*(*w*_{t}) needed to generate the aircraft measured *σ*(*w*) within the advective time constraint posed by the orography of Dominica.

The threshold is an evaluation criterion for the LCL theory. In general, a low threshold would be inconsistent with our convective initiation theory because the measured upwind variations are enough to initiate convection without lifting. A high threshold would also be inconsistent as not even lifting past the LCL is sufficient to generate enough differential buoyancy to match observations of plume convection.

*σ*(

*w*

_{t}) is defined as follows:

*σ*(

*w*), was measured by aircraft at 1.7-km height in the convection along leg 3 above the windward slopes of Dominica. The advective time scale

*t*

_{advec}in our problem is well defined. Parcels need time to produce buoyant plumes on Dominica’s windward side before being damped through descent on the lee side (Smith et al. 2012; Minder et al. 2013). The advective time scale is the distance traveled (

*L*) over the wind speed (

*U*):

*t*

_{advec}=

*LU*

^{−1}. Here

*L*is estimated to be 4 km, about a quarter of the island width, and the initiation of convection needs to occur early such that time is still available for convection to develop. The climatological average wind speed is ~7 m s

^{−1}, allowing less than 10 min for the lifted layer to generate plume convection.

Using an average *σ*(*w*) = 0.97 m s^{−1} from leg 3 on all RFs in (12), the threshold *σ*(*w*_{t}) needed for the transition to plume convection is estimated to be *σ*(*w*_{t}) = 0.97/600 = 0.0016 m s^{−2}. This value is included in Fig. 10 and shows that this threshold value is reached by the lifted layers between 500- and 900-m height. The layers begin at 300-m height thereby requiring 200–600 m of lifting above flight level, well within the lifting abilities provided by Dominica’s terrain, which ranges from 600 to 1400 m. The threshold for plume convection will be reached a majority of the time, though the particular value is somewhat arbitrary and varies from day to day based on wind speed and *σ*(*w*).

### e. Dominant scale of convection

In the upstream environment, the lag autocorrelation scale of the passive *q*_{υ} seeds is 3–8 km. The scale of the seeds is important because it will become the scale of the buoyancy variations upon uplift. However, the 3–8-km scale of the passive seeds is inconsistent with the scale of the resulting clouds over Dominica, which is on the order of 1–3 km. It is also inconsistent with the scale of the vertical velocity variations in the upstream environment (0.5–1 km). Taking into account the effects of added mass on buoyancy [using (10) with *H* = 600 m and *k* = *l*] changes the scale dramatically and brings the resulting scale of acceleration in line with the lag autocorrelation scale of the observed cloud liquid water content (Fig. 11). Acting as a high-pass filter in Fourier space, the large-scale variations in buoyancy are damped leaving smaller-scale accelerations. The resulting lag autocorrelation scale of acceleration is on average 1.5 km giving further confidence in the proposed method due to the comparability with the scale of the observed clouds. The inclusion of added mass in converting from buoyancy to acceleration is necessary for getting the scale right. The dominant scale for acceleration depends on the initial scale of buoyancy and the chosen layer depth *H*.

The lag autocorrelation scale of the predicted plume convection found from lifting and filtering (*H* = 600 m, *k* = *l*) leg 1L seeds. The average autocorrelation scale for all RFs is 1.5 km.

Citation: Journal of the Atmospheric Sciences 71, 12; 10.1175/JAS-D-14-0089.1

The lag autocorrelation scale of the predicted plume convection found from lifting and filtering (*H* = 600 m, *k* = *l*) leg 1L seeds. The average autocorrelation scale for all RFs is 1.5 km.

Citation: Journal of the Atmospheric Sciences 71, 12; 10.1175/JAS-D-14-0089.1

The lag autocorrelation scale of the predicted plume convection found from lifting and filtering (*H* = 600 m, *k* = *l*) leg 1L seeds. The average autocorrelation scale for all RFs is 1.5 km.

Citation: Journal of the Atmospheric Sciences 71, 12; 10.1175/JAS-D-14-0089.1

The horizontal scale of the upstream seeds has strong implications for numerical modeling of convection. In idealized numerical modeling of clouds, the environment is often initiated with random perturbations in temperature and/or moisture to break up the domain (e.g., Fuhrer and Schär 2005; Kirshbaum and Grant 2012; Minder et al. 2013; Nugent et al. 2014; KS09). These results confirm the idea that the scale of the perturbations could have an impact on the resulting scale of convection (Fuhrer and Schär 2005; Kirshbaum and Grant 2012). Kirshbaum and Grant (2012) found that larger seeds produce larger clouds that in turn experience less entrainment and precipitate more.

### f. Sensitivity tests

The free parameters in the acceleration filter in (10), including the layer depth *H* and streamwise wavenumber *k*, are tested with sensitivity experiments. In the previous sections, it has been assumed that the seeds exist in a layer of depth *H* = 600 m because this is approximately the height of the LCL and thus the depth of the mixed layer. However, because the vertical extent of the seeds was not measured, sensitivity tests are performed for *H* = 200 m, and 1 km. The prior sections also assume that *k* = *l*; the seeds are disk shaped and thus three-dimensional. Again, because the horizontal extent in the direction perpendicular to Leg 1L was not measured, it is also possible that *k* = 0 and the seeds are 2D and elongated in the along-wind direction. An example for a typical RF leg from Fig. 10 was chosen (RF 12) and sensitivity tests are shown in Fig. 12.

Sensitivity testing for the *H* parameter and 3D (*k* = *l*) vs 2D (*k* = 0). Tests are performed on the thick black line from Fig. 10 that corresponds to RF 12.

Citation: Journal of the Atmospheric Sciences 71, 12; 10.1175/JAS-D-14-0089.1

Sensitivity testing for the *H* parameter and 3D (*k* = *l*) vs 2D (*k* = 0). Tests are performed on the thick black line from Fig. 10 that corresponds to RF 12.

Citation: Journal of the Atmospheric Sciences 71, 12; 10.1175/JAS-D-14-0089.1

Sensitivity testing for the *H* parameter and 3D (*k* = *l*) vs 2D (*k* = 0). Tests are performed on the thick black line from Fig. 10 that corresponds to RF 12.

Citation: Journal of the Atmospheric Sciences 71, 12; 10.1175/JAS-D-14-0089.1

The sensitivity to the filter decreases with increasing *H*; the difference in the *σ*(*w*_{t}) between *H* = 1 km and *H* = 600 m is smaller than the difference in *σ*(*w*_{t}) between *H* = 600 m and *H* = 200 m. Choosing a smaller *H* creates a stronger filter since density anomalies with horizontal scale larger than the vertical scale are damped. Additionally, the reduction in vertical acceleration (added mass effect) is stronger in 2D (*k* = 0) than 3D (*k* = *l*) and thus choosing elongated seeds with *k* = 0 creates a stronger filter than disk-shaped 3D seeds with *k* = *l*. Disk-shaped seeds accelerate more easily because the displaced air can move out of the way in all horizontal directions. The threshold needed for plume convection is reached for almost all choices of *H* and *k*; in Fig. 12, only one case did not reach the threshold.

The analysis of scale (section 3e) is also sensitive to the layer depth *H* and the assumptions involving *k* and *l*. Smaller layer depths result in smaller lag autocorrelation length scales of acceleration.

## 4. Discussion

### a. Overview

Using a unique set of aircraft measurements, we evaluate a modified version of Woodcock’s hypothesis that moisture anomalies can initiate convection when orographically lifted (W60). Observations from the DOMEX project show regions of anticorrelated *T* and *q*_{υ} in the subcloud trade wind environment upstream of Dominica such that the layer had nearly constant *T*_{υ} and therefore uniform buoyancy. We hypothesize that the anomalous moist regions, or seeds, initiate convection when lifted in a layer due to their lower LCL and the conditionally unstable environment. With the use of a simple lifting method and accounting for added mass, the amount of differential acceleration produced through layer lifting is calculated. A threshold value of differential acceleration required for plume convection is quantified using observed vertical velocity variations and the relevant advection time. The predicted accelerations have magnitude and length scale consistent with observations.

The proposed convective initiation theory is testable in the Dominica case because the initial state is well observed. Our theory is incomplete with respect to the assumptions of uniform lifting, neglect of background turbulence, and the lack of 3D measurements of the moist seeds. Perfectly uniform lifting of a layer seldom occurs in nature because orography is inherently rugged. Convective initiation by nonuniform lifting is explored extensively in Kirshbaum et al. (2007a,b). If nonuniform lifting were applied to the DOMEX observations, we argue the initiation of convection would remain largely unchanged as long as the terrain variations have a larger scale than the seeds. The parcels that first reach their LCL would begin to generate buoyancy with respect to other drier parcels. Accounting for added mass and the development of differential buoyancy within the layer would still be important. Uniform lifting is a simplifying assumption in the analysis of the DOMEX observations. The key to the proposed convective initiation mechanism, regardless of the nature of lifting, is moisture anomalies and the differential LCL mechanism.

The second limitation is the neglect of background turbulence, which as noted in section 2b, homogenizes the atmosphere likely through eroding seed edges. Turbulence thereby acts as a low-pass filter since narrow perturbations will experience more relative entrainment. This would act to increase the favored scales for convection.

The final limitation is the lack of 3D seed measurements. High-quality measurements exist in one dimension, but the depth and width of the seeds remains largely unknown and sensitivity tests show a dependence on these parameters. Comparisons of the resulting convection with observations are good, however, and give confidence in the proposed method despite this limitation.

The proposed theory can be further tested observationally and numerically. Additional measurements of water vapor inhomogeneity in the lower troposphere will help to constrain seed dimensions and prevalence. The horizontal structure is especially important and, therefore, isolated balloon soundings are of little value for this purpose. Better tools may include airborne or surface-based remote sensing instruments capable of measuring *q*_{υ} inhomogeneities (e.g., Raman lidar, limb sounding satellite, or index of refraction radar measurements). Numerical simulations can also be done to further test the convective initiation theory and investigate the breakup of the lifted layer.

### b. Applicability to other climate zones

The triggering of orographic convection by moist seeds will occur in other climate zones if the seeds are present and the atmosphere is conditionally unstable. Humidity fluctuations are ubiquitous in the atmosphere because of boundary layer convection and a variety of cloud processes. In the tropics as we have seen, as the seeds adjust their buoyancy, a characteristic anticorrelation between *T* and *q*_{υ} develops. In higher-latitude cooler drier climates, where the molecular weight effect is mostly negligible, the moist seeds may have an imperceptible temperature signature. Constant density layers would have constant temperature. Still, in rapidly lifted air, the differential LCL seed mechanism may produce significant buoyancy variance.

The occurrence of conditional instability is more restrictive than the existence of seeds perhaps, but is still very widespread. Throughout the tropics, subtropics, monsoon, and midlatitude regions, conditional instability can develop in regions with strong surface water vapor fluxes or in overrunning situations where colder air masses override warm moist air masses. Midlatitude frontal cyclones may push an unstable atmosphere against a mountain range. This has been noted along the West Coast of the United States (e.g., Hobbs et al. 1975) and in the Alps (e.g., Smith et al. 2003; Rotunno and Houze 2007) among other locations.

An appropriate wind speed is needed for the triggering of orographic convection by moist seeds. A different mechanism, such as surface heating, may initiate convection if the wind speed is too low (Nugent et al. 2014). However, in hurricane-force winds, not enough time is allowed for orographic convection to develop and the seeder–feeder process dominates (Smith et al. 2009).

A fourth requirement is that the terrain lift the incoming air to the LCL where the differential buoyancy is generated. In the Dominica case, using air from the middle of the subcloud layer at *z* = 300 m, about 400 m of lifting is needed. In other regions, if the ambient air has a lower relative humidity, the hill is lower, or the airflow is diverted around the hill, orographic convection might not be generated.

The current study focuses on lifting by orography, but air can be lifted through numerous mechanisms including dynamical interaction with other air parcels. Lifting can occur at gust fronts, frontal boundaries, troughs, by cold pools, or in convergence zones that arise naturally through mesoscale weather patterns (Kingsmill 1995; Tompkins 2001; Kalthoff et al. 2009).

The present study describes a case of convective initiation via seeds within a layer lifted by orography in the tropics, but the idea of moisture anomalies acting as seeds of convection may be widely applicable.

## Acknowledgments

The authors would like to acknowledge funding from the NSF (Grant 0930356), and helpful discussions with Justin R. Minder, Campbell D. Watson, Daniel J. Kirshbaum, William R. Boos, Trude Storelvmo, and Mary-Louise Timmermans.

## APPENDIX

### Derivation of the Acceleration Filter

*H*with density fluctuations

*ρ*′(

*x*,

*y*) sandwiched between deep layers of homogeneous fluid. The inviscid linearized 3D nonhydrostatic Boussinesq Euler and continuity equations are used. Friction and viscosity are neglected as well as nonlinear advection terms since any distortion of the layer as it is lifted is also neglected. Variables include the three components of velocity (

*u*,

*υ*,

*w*) as well as mean and perturbation density

*P*), and gravity (

*g*). Subscripted letters indicate partial derivatives with respect to the given variable:

*w*

_{t}(

*x*,

*y*,

*z*):

*z*, the particular solution of (A6) is

*z*= ±

*H*/2 for the regions above and below the layer with inhomogeneities, as well as the layer itself:

*z*| ⇒ ∞, and (A9) within the seed layer. We assume that the velocity field is symmetric about

*z*= 0, so

*A*=

*D*and

*B*=

*C*and we impose continuity of vertical velocity (i.e.,

*z*= 0, the middle of the nonuniform seeded layer:

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