1. Introduction
The vertical velocities of cloud updrafts strongly affect aerosol activation rates (Abdul-Razzak et al. 1998), formation of hail (Danielsen et al. 1972), clear-air turbulence (Lane et al. 2012), aircraft hazard (Lane et al. 2003), lightning flash rates (Romps et al. 2014), tornado occurrence (Davies-Jones 1984), gravity wave generation (Fovell et al. 1992), the depth of convective overshooting (Wang 2007), and the convective moistening of the stratosphere (Grosvenor et al. 2007). Despite the importance of vertical velocities, the balance of forces giving rise to those motions is poorly understood. To make some headway on elucidating this balance of forces, we focus here on the quantum of moist convection: the cloud thermal. Our goal is to answer the following question: what is the dominant balance in the vertical momentum budget of mature cloud thermals?
One hypothesis is that the dominant balance in the vertical momentum budget of a mature cloud thermal is between buoyancy and acceleration 
The alternate (and opposite) hypothesis is that the dominant balance is between buoyancy and drag 
The slippery-thermal hypothesis is that the dominant balance in Eq. (3) is between terms 1 and 3, as indicated. The sticky-thermal hypothesis is that the dominant balance is between terms 2 and 3, as indicated. Evidence from large-eddy simulations indicates that the fourth term—acceleration by entrainment and detrainment—is far smaller than originally thought (Dawe and Austin 2011) and may be practically negligible (de Roode et al. 2012; Sherwood et al. 2013). Therefore, the question of whether mature cloud thermals are slippery or sticky boils down to a question of the magnitude of form drag and wave drag, as manifested in the pressure perturbation gradient force (term 2 above).
As a brief aside, it is worth emphasizing a simple, but important, point. The retarding force on a blob of fluid moving in the 
To assess the magnitude of the drag on cloud thermals, we study output from a high-resolution (100-m grid spacing) large-eddy simulation of radiative–convective equilibrium (RCE) over a 300-K ocean surface (section 2). An objective tracking algorithm is used to track cloud tops, and the thermals underneath those tops are identified from their azimuthally averaged streamfunctions (section 3). This produces 4852 snapshots of cloud thermals (of 715 unique cloud thermals), which reveal the dominant role for drag, supporting the sticky-thermal hypothesis (section 4). The results are then briefly summarized (section 5).
2. The large-eddy simulation
Simulations were run to RCE with Das Atmosphärische Modell (DAM; Romps 2008) on a square doubly periodic domain with a width of 32 km and a model top at 30 km. The time step adjusts automatically to satisfy the Courant–Friedrichs–Lewy (CFL) condition. The lower boundary was specified to be an ocean surface with a fixed temperature of 300 K, and surface fluxes were calculated using a bulk formula. Both shortwave and longwave radiation were calculated interactively using the Rapid Radiative Transfer Model (Clough et al. 2005; Iacono et al. 2008), and the top-of-atmosphere insolation was specified to be a constant diurnal average for the equator on 1 January.
A simulation with a 500-m horizontal spacing (and a vertical spacing of 50 m below 600 m and 500 m above 5 km) was started from an RCE sounding and run for 2 days to guarantee a fully convecting RCE state. The state at the end of that simulation was then interpolated to a grid with the same domain size (36 km × 36 km × 30 km) but with a 100-m horizontal spacing (and a vertical spacing of 50 m below 600 m and 100 m between 1100 m and 17 km). This was run for 20 h to ensure that the RCE state had adjusted to the new grid spacing. The last hour was run with a time step of 0.25 s (explained below) and with snapshots saved every minute (for a total of 60 snapshots).
This setup is similar to that used by Romps and Kuang (2010) and Romps (2011), and it produces a similar state of deep convective RCE. Figure 1 shows various profiles from the simulation, starting with the virtual potential temperature 
Profiles of virtual potential temperature, relative humidity (with respect to liquid for 
Citation: Journal of the Atmospheric Sciences 72, 8; 10.1175/JAS-D-15-0042.1
Profiles of virtual potential temperature, relative humidity (with respect to liquid for
Citation: Journal of the Atmospheric Sciences 72, 8; 10.1175/JAS-D-15-0042.1
Profiles of virtual potential temperature, relative humidity (with respect to liquid for
Citation: Journal of the Atmospheric Sciences 72, 8; 10.1175/JAS-D-15-0042.1
3. Tracking thermals
For each three-dimensional snapshot, cloud tops are identified as a grid point at the local top (highest point within ±500 m in x and y) of ascending cloudy regions. Cloud tops are then connected in time to build up sequences of cloud-top trajectories. Although this procedure may be straightforward to implement by eye, we do not attempt that here. Instead, we have built a set of objective criteria and then written a piece of software to automatically identify cloud tops and connect them in time. This algorithm, described in the appendix, identifies 4852 cloud tops (715 unique cloud sequences) from the 1 h of LES output. On average, a cloud sequence follows a cloud top for 6 min. For each cloud top, an azimuthally averaged map is made for each prognostic variable around a vertical axis that passes through the cloud top. These two-dimensional r–z maps, along with information on how they are connected in time, constitute the data used in this study.
A sketch of the process used to identify a thermal’s boundary. For a cloud top in a cloud sequence, 
Citation: Journal of the Atmospheric Sciences 72, 8; 10.1175/JAS-D-15-0042.1
A sketch of the process used to identify a thermal’s boundary. For a cloud top in a cloud sequence,
Citation: Journal of the Atmospheric Sciences 72, 8; 10.1175/JAS-D-15-0042.1
A sketch of the process used to identify a thermal’s boundary. For a cloud top in a cloud sequence,
Citation: Journal of the Atmospheric Sciences 72, 8; 10.1175/JAS-D-15-0042.1
We refer to the region 
The right panel in Fig. 1 shows the mass flux of the good thermals. For a height interval 
4. Results
A plot of the thermals’ mean (three-dimensional volume weighted) Eulerian vertical velocity against their Lagrangian cloud-top vertical velocity. The fact that these points fall on a 1-to-1 line (with an 
Citation: Journal of the Atmospheric Sciences 72, 8; 10.1175/JAS-D-15-0042.1
A plot of the thermals’ mean (three-dimensional volume weighted) Eulerian vertical velocity against their Lagrangian cloud-top vertical velocity. The fact that these points fall on a 1-to-1 line (with an
Citation: Journal of the Atmospheric Sciences 72, 8; 10.1175/JAS-D-15-0042.1
A plot of the thermals’ mean (three-dimensional volume weighted) Eulerian vertical velocity against their Lagrangian cloud-top vertical velocity. The fact that these points fall on a 1-to-1 line (with an
Citation: Journal of the Atmospheric Sciences 72, 8; 10.1175/JAS-D-15-0042.1
With confidence in the procedure for identifying thermals, we can now proceed to examine the properties of these thermals in detail. The following subsections discuss the thermal structure, the momentum budget, and the internal circulation of the cloud thermals.
a. Thermal structure
Because of the discrete nature of the 100-m grid, 138 of the 1224 thermals with good masks have a volume exactly equal to the median volume. These thermals have a volume equal to a sphere with a diameter of 560 m. Figure 4 shows various mean properties of these median-volumed thermals (cloud condensate, vertical velocity, streamfunction, pressure perturbation, pressure perturbation gradient acceleration, and buoyancy).
For the 138 thermals with the median volume, the average r–z maps of cloud condensate, vertical velocity, streamfunction, pressure perturbation, pressure perturbation gradient acceleration, and buoyancy.
Citation: Journal of the Atmospheric Sciences 72, 8; 10.1175/JAS-D-15-0042.1
For the 138 thermals with the median volume, the average r–z maps of cloud condensate, vertical velocity, streamfunction, pressure perturbation, pressure perturbation gradient acceleration, and buoyancy.
Citation: Journal of the Atmospheric Sciences 72, 8; 10.1175/JAS-D-15-0042.1
For the 138 thermals with the median volume, the average r–z maps of cloud condensate, vertical velocity, streamfunction, pressure perturbation, pressure perturbation gradient acceleration, and buoyancy.
Citation: Journal of the Atmospheric Sciences 72, 8; 10.1175/JAS-D-15-0042.1
The cloud condensate is located just below the top of the cloud top, as expected. The vertical velocity has a maximum underneath the cloud top and weakly negative values at the periphery of the thermal, consistent with previous large-eddy simulations (Heus and Jonker 2008; Glenn and Krueger 2014). For the streamfunction, only positive values are plotted here; the negative values outside the thermal would saturate the color bar if plotted. By the procedure outlined in section 3, the thermal is identified as the region with a positive streamfunction. Therefore, the shape of the mean thermal can be identified here as the region with any red or pink color.
The pressure perturbation has a dipole structure with high pressure at and above the cloud top and a larger region of low pressure underneath the cloud top. If the perturbations of pressure and density were in hydrostatic balance, then the pressure perturbation gradient acceleration and the buoyancy would be mirror images of each other. Of course, cloud thermals are not hydrostatic, so this is not the case. The core of the thermal is positively buoyant (with the most buoyant air just below the cloud top), and that buoyant core is surrounded by an egg-shaped shell of negative buoyancy. In contrast, the pressure perturbation gradient acceleration resembles a sandwich with a wide and flat region of downward acceleration capped above and below by upward acceleration.
A scatterplot of 
Citation: Journal of the Atmospheric Sciences 72, 8; 10.1175/JAS-D-15-0042.1
A scatterplot of
Citation: Journal of the Atmospheric Sciences 72, 8; 10.1175/JAS-D-15-0042.1
A scatterplot of
Citation: Journal of the Atmospheric Sciences 72, 8; 10.1175/JAS-D-15-0042.1
b. Momentum budget
Before we attempt to diagnose the momentum budget, it is useful to first look at the cloud-top trajectories. For each cloud sequence, the time series of cloud-top displacement is plotted as a thin black line in Fig. 6. When the streamfunction mask is bad, the trajectory is plotted with the cloud-top heights at those times omitted; this choice is made to be as consistent as possible with the parcel trajectories described below, which can only use data from good masks. Although they are difficult to see on this busy plot, the trajectories resemble a splay of straight lines. The average of all of these time series is shown as the thick black line, which is plotted up to the time when the number of thin black lines drops below 5. It, too, is a straight line, indicating the absence of any significant acceleration or deceleration of these thermals. Recall that none of the conditions used to select cloud tops had anything to do with the stage of the cloud’s life cycle; therefore, the linearity of these trajectories is truly remarkable. This linearity strongly suggests a balance of forces, which would cause the thermals to ascend with a terminal velocity.
Actual and theoretical cloud-top trajectories. The thin black lines are the time series of actual cloud-top heights for individual cloud sequences. The thin blue, red, and purple lines show the theoretical cloud-top trajectories using only terms b, 
Citation: Journal of the Atmospheric Sciences 72, 8; 10.1175/JAS-D-15-0042.1
Actual and theoretical cloud-top trajectories. The thin black lines are the time series of actual cloud-top heights for individual cloud sequences. The thin blue, red, and purple lines show the theoretical cloud-top trajectories using only terms b,
Citation: Journal of the Atmospheric Sciences 72, 8; 10.1175/JAS-D-15-0042.1
Actual and theoretical cloud-top trajectories. The thin black lines are the time series of actual cloud-top heights for individual cloud sequences. The thin blue, red, and purple lines show the theoretical cloud-top trajectories using only terms b,
Citation: Journal of the Atmospheric Sciences 72, 8; 10.1175/JAS-D-15-0042.1
To check this balance, we calculate theoretical time series of cloud-top heights using the thermodynamic variables inside the thermal masks. In particular, we integrate Eq. (3) twice to give height, but using only the second term on the right-hand side (buoyancy only; blue lines), or only the first term on the right-hand side (pressure perturbation gradient acceleration only; red lines), or the sum of the first two terms on the right-hand side (buoyancy plus pressure perturbation gradient acceleration; purple lines). The thin blue, red, and purple lines show the trajectories calculated for individual cloud sequences. The thick blue, red, and purple lines show the respective means up until when the number of thin lines drops below 5. Clearly, buoyancy alone (blue lines) would cause the thermals to accelerate upward much more quickly than observed. Similarly, the pressure perturbation alone (red lines) would cause the thermals to rapidly decelerate and descend, which is in stark contrast to the observed ascent. But, by combining these two terms (purple lines), we closely replicate the observed ascent. This demonstrates that the dominant balance in the momentum budget is between buoyancy and drag; cloud thermals are sticky. Note that this success is achieved despite our neglect of the entrainment–detrainment term. This agrees with previous findings that the entrainment drag is weak (Dawe and Austin 2011; de Roode et al. 2012; Sherwood et al. 2013).
To demonstrate this balance in another way, Fig. 7 plots the thermal-mean pressure perturbation gradient acceleration against the thermal-mean buoyancy. Here, the mean b values from the 1224 thermals with good masks are divided into 20 quantiles, each containing 56 thermal snapshots. For each of those quantiles, a circle is plotted at the average b and average 
Drag vs buoyancy for the thermals with good masks. Each circle represents an average of 56 thermals from a 5% quantile of thermal-mean buoyancy.
Citation: Journal of the Atmospheric Sciences 72, 8; 10.1175/JAS-D-15-0042.1
Drag vs buoyancy for the thermals with good masks. Each circle represents an average of 56 thermals from a 5% quantile of thermal-mean buoyancy.
Citation: Journal of the Atmospheric Sciences 72, 8; 10.1175/JAS-D-15-0042.1
Drag vs buoyancy for the thermals with good masks. Each circle represents an average of 56 thermals from a 5% quantile of thermal-mean buoyancy.
Citation: Journal of the Atmospheric Sciences 72, 8; 10.1175/JAS-D-15-0042.1
Actual drag vs the drag-law prediction with 
Citation: Journal of the Atmospheric Sciences 72, 8; 10.1175/JAS-D-15-0042.1
Actual drag vs the drag-law prediction with
Citation: Journal of the Atmospheric Sciences 72, 8; 10.1175/JAS-D-15-0042.1
Actual drag vs the drag-law prediction with
Citation: Journal of the Atmospheric Sciences 72, 8; 10.1175/JAS-D-15-0042.1
It is important to note that care must be taken to extract the correct pressure field from the large-eddy simulation. DAM uses a split-time scheme in which acoustic modes are integrated with a small, but inexpensive, acoustic time step, and the rest of the dynamics is integrated with a large time step that is advanced using a third-order Runge–Kutta scheme (Romps 2008). The pressure field that is saved to the output is the pressure at the end of the final Runge–Kutta step. For a large time step that is too large, there are many intervening acoustic steps and the pressure field at the end of the large time step may not be representative of the mean pressure field experienced during the large time step. This difference manifests itself in a dependence of the inferred 
Care must be taken when extracting a pressure field from a large-eddy simulation with split-time integration. This plot of inferred drag coefficient vs the LES large time step shows convergence for time steps ≲1 s.
Citation: Journal of the Atmospheric Sciences 72, 8; 10.1175/JAS-D-15-0042.1
Care must be taken when extracting a pressure field from a large-eddy simulation with split-time integration. This plot of inferred drag coefficient vs the LES large time step shows convergence for time steps ≲1 s.
Citation: Journal of the Atmospheric Sciences 72, 8; 10.1175/JAS-D-15-0042.1
Care must be taken when extracting a pressure field from a large-eddy simulation with split-time integration. This plot of inferred drag coefficient vs the LES large time step shows convergence for time steps ≲1 s.
Citation: Journal of the Atmospheric Sciences 72, 8; 10.1175/JAS-D-15-0042.1
c. Internal circulation
The sticky behavior found here (i.e., 
(left) An average of 65 thermals with volumes in the 90th–95th percentiles. Contours show the streamfunction with an interval of 10 Mg s−1 ranging from −50 to 50 Mg s−1. Colors show the perturbation pressure. (right) As in (left), but for Hill’s vortex scaled to the same size and speed.
Citation: Journal of the Atmospheric Sciences 72, 8; 10.1175/JAS-D-15-0042.1
(left) An average of 65 thermals with volumes in the 90th–95th percentiles. Contours show the streamfunction with an interval of 10 Mg s−1 ranging from −50 to 50 Mg s−1. Colors show the perturbation pressure. (right) As in (left), but for Hill’s vortex scaled to the same size and speed.
Citation: Journal of the Atmospheric Sciences 72, 8; 10.1175/JAS-D-15-0042.1
(left) An average of 65 thermals with volumes in the 90th–95th percentiles. Contours show the streamfunction with an interval of 10 Mg s−1 ranging from −50 to 50 Mg s−1. Colors show the perturbation pressure. (right) As in (left), but for Hill’s vortex scaled to the same size and speed.
Citation: Journal of the Atmospheric Sciences 72, 8; 10.1175/JAS-D-15-0042.1
Note that the circulations are quite similar between the two. The contours in the cloud thermal are expected to be a bit washed out as a result of having averaged over 65 different thermals, all with slightly different shapes and sizes. Therefore, we should not read too much into the differences in the contours between the mean cloud thermal and Hill’s vortex; to the contrary, the agreement between the two circulations is quite remarkable.
With its broader features, the pressure field is less affected by the averaging. Note that the thermal has an obvious fore–aft pressure gradient, which Hill’s vortex does not have. This dipole distribution of pressure in the cloud thermal is real and visible in individual cloud thermals. It differs markedly from the tripole distribution of pressure—with fore–aft symmetry—in Hill’s vortex. It is this lack of fore–aft symmetry that causes the drag on the cloud thermals.
What breaks the fore–aft symmetry in cloud thermals? One could certainly point a finger at turbulence: cloud thermals trail a turbulent, cloudy wake, which is distinctly different in character from the laminar, clear air above it. But, even in the absence of turbulence, it is impossible for a buoyant cloud thermal to have both a symmetric circulation and a symmetric pressure distribution like Hill’s vortex. The reason is buoyancy. The vertical acceleration is given by 
Finally, let us compare the ratio of cloud-top speed 
Maximum Eulerian cloud-core speed 
Citation: Journal of the Atmospheric Sciences 72, 8; 10.1175/JAS-D-15-0042.1
Maximum Eulerian cloud-core speed
Citation: Journal of the Atmospheric Sciences 72, 8; 10.1175/JAS-D-15-0042.1
Maximum Eulerian cloud-core speed
Citation: Journal of the Atmospheric Sciences 72, 8; 10.1175/JAS-D-15-0042.1
5. Summary
In summary, we have run an hour-long large-eddy simulation of maritime, tropical, deep convection and have automatically tracked 4852 cloud tops in that simulation. Azimuthally averaging around a vertical axis through the cloud top, we use the zero streamline in the cloud top’s reference frame to define the boundary of the cloud thermal (Fig. 2). These cloud thermals have unloaded virtual temperature anomalies and net buoyancies (Fig. 5) that are consistent with in situ measurements of convection over tropical oceans. The cloud tops are found to ascend with nearly constant vertical velocity, suggestive of a terminal rise speed. This conclusion is bolstered by the finding that the dominant balance in the thermals’ vertical momentum equation is between buoyancy and drag (Figs. 6 and 7). The standard drag law predicts this drag force well using a drag coefficient of 
Acknowledgments
This work was supported by the U.S. Department of Energy’s Atmospheric System Research, an Office of Science, Office of Biological and Environmental Research program under Contract DE-AC02-05CH11231, and by a grant from the Undergraduate Research Apprentice Program (URAP) at the University of California, Berkeley.
APPENDIX
Cloud-Tracking Algorithm
The first step is to take the three-dimensional snapshots, which are defined on the stretched vertical grid that is native to the simulations, and linearly interpolate them to a uniform 100-m vertical spacing. The stretched grid of the simulations has a spacing of 50 m below 600 m, a spacing of 100 m above 1100 m, and a smoothly transitioned spacing in between. The data on this stretched grid are interpolated to a uniform 100-m grid that exactly coincides with the 100-m levels above 1 km, so as to leave undistorted the data in the majority of the troposphere.
The data are on an Arakawa C grid with density, pressure, temperature, and mass fractions at the center of the grid boxes and velocity components on the faces. Let us denote the center of a grid box at time slice n by a 4-tuple of integers 
For a given cloud, the “cloud peak” is defined as the cloud top 
The next step is to track cloud peaks through time. For each cloud peak at time n, we consider a 1-km-wide square column centered on that cloud peak and search for higher cloud peaks at time 
the number N of cloud peaks in the set is greater than or equal to 4;
for some integer m, the set contains exactly one cloud peak for each time
for each
for each
this is the largest such set containing these cloud peaks.
To eliminate outlier clouds that seem to be ascending too rapidly, two final criteria are imposed. We define a “cloud-top velocity” to be a cloud’s cloud-peak ascent speed averaged over a 3-min interval (i.e., total vertical displacement during that interval, as calculated using the interpolated heights, divided by 3 min). Next, we define the “max velocity” associated with a cloud top located at 
every grid box between
a cloud-top velocity greater than the maximum of the cloud-core velocities associated with each of the cloud peaks during the associated 3-min interval or
a coefficient of variation of three adjacent vertical displacements that exceeds 0.5.
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