# Vertical Velocity in the Gray Zone

## Abstract

We describe how convective vertical velocities
vary in the “gray zone” of horizontal resolution, using both hydrostatic and nonhydrostatic versions of GFDL's *FV*^{3} dynamical core, as well as analytical solutions to the equations of motion. We derive a simple criterion (based on parcel geometries) for a model to resolve convection, and find that
resolution can be required for convergence of
. We also find, both numerically and analytically, that hydrostatic systems overestimate
, by a factor of 2–3 in the convection-resolving regime. This overestimation is simply understood in terms of the “effective buoyancy pressure” of Jeevanjee and Romps (2015, 2016).

## Key Points

- Horizontal resolutions of can be required for convergence of convective vertical velocities
- Hydrostatic systems overestimate , by a factor of 2–3 at fine resolution
- This overestimation can be simply understood using the notion of “effective buoyancy pressure”

## 1 Introduction

As computer power increases, so does the maximum resolution of atmospheric models. Indeed, it is now possible to run global climate simulations at subdegree horizontal resolution (Noda et al., 2014; Wehner et al., 2014), and shorter-term numerical weather prediction simulations at resolution (Kain et al., 2008; Lean et al., 2008; VandenBerg et al., 2014). Such resolutions lie outside both the hydrostatic regime, wherein resolved-scale convective vertical velocities are negligible, as well as the convection-resolving regime, wherein converge to constant, realistic values.

This no-man's land of horizontal resolution is sometimes known as the “gray zone” (Figure 1). In the gray zone, the vertical transports of heat, moisture, and other tracers by the resolved-scale motion can be neither neglected nor relied upon, and are expected to vary strongly as a function of resolution. Indeed, the structure of atmospheric circulations at a variety of scales, as well as the utility of convective parameterizations, are known to vary significantly within the gray zone (Prein, 2015; Wehner et al., 2014; Weisman et al., 1997, hereafter W97). This raises difficult and important questions about when to turn convective parameterizations off, and/or how to make them “scale-aware.” Since these issues are all tied to , and since is relevant for both climate forcing and sensitivity (Donner et al., 2016), it seems worthwhile to address some basic questions about the behavior of in the gray zone:

- For a given dynamical phenomenon, at what resolution should one expect the associated to converge? In other words, where is the inner edge of the gray zone?
- If converges at fine resolutions and is negligible at coarse resolutions, what kind of curve interpolates between these two regimes?
- If hydrostatic models are used in the gray zone, how might their behavior differ from their nonhydrostatic counterparts, and how can we understand such differences?

Questions 1 and 2 are depicted schematically by the question marks in Figure 1.

This paper seeks to address these questions through both analytical solutions to the relevant equations of motion as well as cloud-resolving numerical simulations, run with an atmospheric model equipped with both hydrostatic and nonhydrostatic solvers. Of course, similar approaches have been taken by previous authors (W97; Kato & Saito, 1995; Morrison, 2016b; Orlanski, 1981; Pauluis & Garner, 2006), so we must build on these prior studies in meaningful ways. We do so by first taking advantage of the aforementioned computer power to explore much finer resolutions (
) than these previous studies (
km), ideally probing down to the inner edge of the gray zone. Second, we use a postprocessing algorithm to diagnose the diameter D and vertical extent (or height) *H* of individual convecting parcels in our simulations, to explicitly make the connection between parcel geometry and resolution. Third, we apply the recently developed “effective buoyancy” formalism (Davies-Jones, 2003; Jeevanjee & Romps, 2015) to develop scaling laws for both hydrostatic and nonhydrostatic vertical velocities, and we compare these to previously published scalings. Finally, we will reemphasize the perhaps counterintuitive fact that hydrostatic systems *over*estimate
. This fact is noted in the references listed above, but perhaps deserves fresh emphasis as computer power allows global hydrostatic models to be run in the gray zone. We will also give intuition for this overestimation using the “buoyancy pressure” framework of Jeevanjee and Romps (2015, 2016; hereafter JR15 and JR16, respectively).

Our focus here will be on the effects of horizontal resolution on parcel aspect ratio *D*/*H* and hence vertical acceleration, as opposed to the effects of horizontal resolution on turbulence and entrainment (e.g., Bryan et al., 2003; Bryan & Morrison, 2011; Lebo & Morrison, 2015; Wyngaard, 2004). These effects are likely related, though, as Lebo and Morrison (2015) find that the onset of resolved turbulent mixing *and* the convergence of vertical motion coincide at
. This gives a tentative answer to question 1 above, but the answer is purely empirical. A further aim of this paper will be to develop a criterion for resolving convection which reproduces this result (section 3), and explains it in terms of other phenomenological parameters of our simulations.

## 2 Simulations

We begin by probing the gray zone numerically. This task requires an atmospheric model which can run with both hydrostatic and nonhydrostatic solvers while keeping other model components fixed (such as the horizontal advection scheme and physics parameterizations). To that end, we employ a dynamical core with just such a capability, GFDL's *FV*^{3} (Finite-Volume Cubed-Sphere Dynamical Core) (Harris & Lin, 2013; Lin, 2004). Global nonhydrostatic convection-allowing simulations with *FV*^{3}-based models have been done for some time, with both GFDL's HiRAM (https://www.gfdl.noaa.gov/visualizations-mesoscale-dynamics/) and NASA GEOS (Putman & Suarez, 2011), but this paper marks the debut of *FV*^{3} in subkilometer, doubly periodic cloud-resolving simulations. We will confirm its suitability for this application below.

As for the model configuration and physics, our guiding principle is to avoid inessential complexity insofar as possible (Jeevanjee et al., 2017). Thus, we choose to run simple doubly periodic radiative-convective equilibrium (RCE) simulations over a fixed sea surface temperature of 300 K, using bulk aerodynamic surface fluxes with a fixed drag coefficient of
, along with an imposed gustiness of 5 m/s to obtain the small air-sea temperature difference typical of the tropics. Radiative cooling is noninteractive and fixed at 1 K/d between the surface and 150 hPa, above which temperatures are relaxed to a stratospheric target of 200 K over a timescale of 5 days. Above 100 hPa, we introduce a Rayleigh drag on the horizontal winds with a 1/2 day timescale to act as a sponge for upwelling gravity waves. No boundary layer or subgrid turbulence schemes are used, though small amounts of vorticity and divergence damping are used to stabilize the model and reduce noise. Microphysical transformations are performed by the six-category GFDL microphysics scheme (Chen & Lin, 2013). The vertical discretization is Lagrangian (Lin, 2004) with 151 levels, with spacings ranging smoothly from 4 hPa near the surface to 8 hPa in the midtroposphere, and back down to 4 hPa near the model top at 76 hPa. The horizontal grid has 96 points in both *x* and *y*.

Simulations were run at resolutions spanning *dx* = 0.0625 – 16 km, varying by factors of two, in both hydrostatic and nonhydrostatic mode. We initially spun up a nonhydrostatic *dx* = 2 km simulation for 480 days to attain full equilibration (i.e., no discernible temperatures or moisture trends) throughout the whole domain (including the stratosphere), and then branched all other runs off this run, running for at least 60 days to allow adjustment to different resolutions or the hydrostatic solver. Three-dimensional snapshots were saved daily from these runs, and the last 20 days of such output were fed into an algorithm which identified convecting parcels and diagnosed their diameter *D*, height *H*, and vertical velocity
. The algorithm identifies parcels as maxima in *w*, then identifies the associated maxima in Archimedean buoyancy *B*, and then identifies parcel dimensions *D* and *H* via a *B* threshold (see Appendix Appendix C for details). Averaging over all tracked parcels then yields characteristic values of *D*, *H*, and
for a given simulation, which are used in our analysis below.

To get a feel for these simulations (as well as the diagnostic algorithm), and to show that the *FV*^{3} dynamical core and GFDL microphysics can indeed handle the transition from coarser-resolution general circulation calculations to subkilometer cloud-resolving calculations, *x* – *p* slices of vertical velocity *w* and Archimedean buoyancy *B* from our nonhydrostatic, *dx* = 0.25 km simulation are shown in Figure 2. (An animation of the cloud field is given in Movie S1.) The features in these fields seem reasonable, being comparable to those found in simulations we have run with other cloud-resolving models, such as DAM (Romps, 2008, not shown). The slice in Figure 2 contains several parcels that were identified by the diagnostic algorithm, and their diagnosed horizontal and vertical extents are depicted by black boxes drawn around the parcels (some parcels have their *B* maxima in adjacent *x* – *p* slices and so do not appear perfectly centered within their box). Note that parcel *w* fields typically have a significantly larger vertical extent than their *B* fields, and that neither of these fields extend down to the ground, contravening the simple “plume” picture of convection prevalent in convective parameterizations (e.g., Arakawa & Schubert, 1974).

## 3 Criterion for the Convection-Resolving Regime

*grid-limited*, in the sense that the physical size of convecting parcels is not independent of

*dx*, but rather scales directly with it. Thus, in the gray zone the diameter

*D*of convecting parcels varies as

*dx*> 250 m, where , and that for .

*D*/

*H*is computed individually for each parcel before averaging) and shows that at high resolutions (it is unclear why

*D*/

*H*is closer to 0.5 at

*dx*= 0.0625 km for the hydrostatic simulation, though it may be related to the asymmetry between vertical and horizontal motion discussed in section 5). Putting facts (2) and (3) together yields the following inequality for the convection-resolving regime, which also demarcates the inner edge of the gray zone:

This simple inequality provides a tentative answer to question 1 posed in the introduction. To evaluate it, we need to know the vertical extent *H* of our parcels. This is plotted in Figure 3b, which shows that *H* has some resolution dependence, but asymptotes to roughly
at fine resolution. Plugging
and
into equation 4 yields
for the inner edge of the gray zone, in good agreement with Figure 3a as well as Lebo and Morrison (2015).

Equation 4 thus seems to capture the edge of the gray zone, as we have defined it. But, this equation is only relevant to the questions posed in the introduction and by Figure 1 if parcel vertical velocities converge according to the same criterion. We thus plot average parcel for our simulations in Figure 4. This plot shows that for the nonhydrostatic solver, indeed converges around , in agreement with Equation 4. Note that in the convection-resolving regime of we expect more turbulence with higher resolution, but the small variations of in this regime suggest that the impact of this on vertical velocities is not strong.

While the convergence of
at the resolution predicted by (4) is reassuring, the much more striking feature of Figure 4 is the *over*estimation of
by the hydrostatic solver, by a factor of 2–3 in the convection-resolving regime (see also W97). We also still lack understanding about the overall shape of the nonhydrostatic
curve and how it interpolates between the hydrostatic (
) and convection-resolving (
) limit. We investigate these questions analytically in the next section.

## 4 The Shape of

### 4.1 Hydrostatic and Nonhydrostatic Effective Buoyancy

_{,}we must understand the resolution-dependence of the forces that act on convecting parcels. Since convection is by definition a buoyancy-driven flow, we will focus here on the “effective buoyancy” , defined as (JR15)

This definition applies to both hydrostatic and nonhydrostatic systems, and gives the net vertical acceleration due to density anomalies. By setting the wind field , the effective buoyancy neglects the “dynamic” pressure gradients arising from advection of momentum or inertia (JR15; Markowski & Richardson, 2011, hereafter MR11). We will return to possible effects of this approximation in section 6.

The effective buoyancy
is typically understood as a combination of the Archimedean buoyancy *B* along with a “buoyancy perturbation pressure” gradient which offsets *B* (MR11). This offset is almost total when
(Morrison, 2016a, hereafter M16), which is why global model grid cells typically do not convect efficiently and convection must be parameterized. The perturbation pressure formalism is not well-suited to analyze hydrostatic systems, however, because the perturbation pressure contains both hydrostatic and nonhydrostatic components (M16, MR11).

*local*hydrostatic pressure and a residual nonhydrostatic pressure . Applying (5) within this formalism to a nonhydrostatic, Boussinesq fluid with reference density

*ρ*

_{0}yields the following equation for the nonhydrostatic effective buoyancy

*β*

_{nh}(JR16):

Here *B* is given by the usual formula
, and the horizontal Laplacian is (crucially)
. We will present intuition for *β*_{nh} in section 5. Also note that (6) is incomplete without boundary conditions, which can be derived from boundary conditions on *w*. In this paper, we assume a surface at *z* = 0, at which
, and this implies
there as well (JR15). We also assume that *w* (and hence
) go to 0 as
.

*w*is given diagnostically from the mass continuity equation by (e.g., Adcroft et al., 2011, Figure 1.17)

This is the equation for hydrostatic effective buoyancy. Note that applying to equation 8 yields equation 6, if one neglects the horizontal Laplacian (as is appropriate in the hydrostatic limit).

*B*

_{0}and characteristic width

*D*, but also a characteristic height

*H*, since it is only in relation to

*H*that

*D*gets large (for an atmospheric model,

*H*is bounded by the roughly 10 km height of the troposphere, whereas

*D*can easily be an order of magnitude larger). Since we will be interested in parcels for which

*D*may be comparable to their height above the surface (in which case they will feel the effects of the surface boundary condition, JR16), and since the diagnostic equation 8 only makes sense in the presence of a lower boundary, we place our source field directly above the surface. Finally, for simultaneous tractability of both equations 6 and 8, we take the source field to be doubly periodic in both

*x*and

*y*. Putting these specifications together gives the Archimedean buoyancy source field

*β*

_{hyd}and

*β*

_{nh}, evaluated at , as a function of

*D*,

*H*, and

*B*

_{0}(we overload the symbols

*β*

_{hyd}and

*β*

_{nh}here slightly). These are derived in Appendix Appendix A. We find that

*β*

_{hyd}and

*β*

_{nh}are functions of

*D*and

*H*only in their ratio

*D*/

*H*, and are given by

As a quick check of these results, one can Taylor-expand (10a) to lowest order in the hydrostatic limit
, and find it exactly equal to (10b). Also note that both expressions in (10) go to 0 in this limit, as expected. In the
limit *β*_{nh} approaches *B*_{0}, also as expected, and *β*_{hyd} diverges. It is not clear if this latter behavior should actually be realized in hydrostatic models, however, because once a given phenomenon is well resolved *D* ceases to scale with *dx*, just as we find here (Figure 3a).

### 4.2 A Scaling for

*C*is a drag coefficient,

_{d}*A*is the horizontal area of the parcel, and

*V*its volume.

*D*/

*H*:

Here,
is evaluated using equations 10a and 10b, and the resulting scalings are plotted in Figure 5. The nonhydrostatic curve interpolates from the convection-resolving regime (
) through the gray zone to the hydrostatic regime (
), where the hydrostatic and nonhydrostatic solutions agree. These two curves also bear a qualitative similarity to the simulated
in Figure 4. We can test the qualitative accuracy of equation 13 by plotting the simulated
against the corresponding *D*/*H*, as shown by the stars in Figure 5. (For the simulations,
is determined by a least-squares fit of
to the nonhydrostatic
.) The nonhydrostatic scaling does a good job overall of capturing the simulated
, while the hydrostatic scaling is an overestimate but roughly captures the factor of 2–3 discrepancy in the convection-resolved regime found in Figure 4. These scalings perform comparably to previously published scaling laws, as shown in Appendix Appendix B.

## 5 Intuition for

The scalings plotted in Figure 5 seem to roughly capture the simulated overestimation of vertical velocities by the hydrostatic solver. But, why does this overestimation occur? Why should the hydrostatic approximation, in which
is actually *neglected* in the vertical momentum equation, *over*estimate
? Note that it is possible to have
(Figure 5), so this overestimation cannot be explained by assuming that the hydrostatic approximation neglects the buoyancy perturbation pressure (which it does not, since the buoyancy perturbation pressure has both hydrostatic and nonhydrostatic components).

*the nonhydrostatic effective buoyancy is simply the vertical gradient of the effective buoyancy pressure*. Note that was dubbed the “buoyancy pressure”: in JR15, but we refer to it here as the “effective buoyancy pressure” to help distinguish it from the buoyancy perturbation pressure mentioned above.

Some intuition for equations 15 and 16 is given in the right plot of Figure 6, as follows: a positively buoyant parcel produces an anomalous hydrostatic pressure field
, whose horizontal gradients force mass convergence into the parcel. This convergence must be compensated for by gradients of the countervailing *non*hydrostatic effective buoyancy pressure
. Figure 6 shows that for a surface parcel,
responds to horizontal convergence from
by generating both horizontal and vertical divergence to compensate, and the vertical divergence is given essentially by *β*_{nh}. Although there is no Archimedean buoyancy in this picture, it is not as unfamiliar as it may seem, as this is just the typical way we understand the operation of a chimney: horizontal
gradients force vertical acceleration via mass continuity. Also note that the divergence from *β*_{nh} needs only to compensate for the *net* convergence in the horizontal, which exhibits a partial cancellation between
and
.

*D*, which explains why decreases with increasing

*D*/

*H*(a similar explanation was given in Pauluis & Garner, 2006). Here, we use this picture to understand why , by considering its hydrostatic analog, illustrated in the left plot of Figure 6. In the hydrostatic approximation, nonhydrostatic pressures such as are 0, and the vertical acceleration

*β*

_{hyd}is then defined by the requirement that it enforce mass continuity, as in equation 8. In this case, however,

*β*

_{hyd}must compensate for the

*gross*convergence from , with no offset from . It then follows that

The essential reason for this is as follows. In the anelastic or Boussinesq governing equations, mass continuity is enforced by nonhydrostatic pressure fields, which generate divergence in all dimensions. In the hydrostatic approximation, on the other hand, mass continuity is enforced by simply demanding that vertical divergence cancel any horizontal divergence (equation 7). Thus, in hydrostatic systems the entire burden of mass conservation falls on the *vertical* motion, which is then exaggerated relative to nonhydrostatic vertical motion.

## 6 Summary and Discussion

- For a model with characteristic parcel height
*H*and gray zone grid cell width , the convection-resolving regime requires . For our*FV*^{3}simulations, this translates to (Figures 3 and 4). - Hydrostatic solvers seem to overestimate by a factor of 2–3 in the convection-resolving regime (Figure 4), a behavior which is captured by our proxies (Figure 5).
- This overestimation can be simply understood using the effective buoyancy formalism (Figure 6).

Although the hydrostatic system overestimates
, it is surprising how fine the resolution can become before this effect becomes appreciable. Figure 4 shows that differences between hydrostatic and nonhydrostatic solvers are virtually undetectable at resolutions of *dx* = 2 km or coarser. While this number is almost certainly dependent on one's model and the phenomena under consideration (W97 put this transition at 8 km rather than 2 km), it does at least point to the possibility of using hydrostatic models in the gray zone without substantial error.

While it is hoped that the results here provide some guidance for simulating in the gray zone, many questions and caveats still remain. For instance, although nonhydrostatic
seems to converge in Figure 4, the hydrostatic
does not. While surprising, this is consistent with the fact that *D*/*H* is not converged for either system (Figure 3c), and with the fact that
is much less sensitive to *D*/*H* than
at
(Figure 5). Further increases in computational power and hence horizontal resolution should allow for a demonstration of convergence in *D*/*H* and hence also
for hydrostatic systems.

Also, equation 4 appears to explain why
marks the inner edge of the gray zone (consistent with Lebo & Morrison, 2015), but only does so in terms of *H* and
. What sets the values of these numbers? We find
, consistent with some recent studies (Hernandez-Deckers & Sherwood, 2016; Romps & Charn, 2015), but no established theory for parcel height currently exists. Furthermore, one might expect
to be associated with the overall diffusivity of the dynamical variables, and hence to be highly model-dependent. Indeed, we were able to increase
in *FV*^{3} by a factor of 2–3 by simply increasing the divergence damping, which is a (hyper)-diffusion acting only on the divergent component of the flow (Zhao et al., 2012). Thus, while equation 4 may be diagnostically accurate, its predictive power may be limited by the need to diagnose *H* and
for a given set of simulations.

Questions also remain regarding our simple scaling (13). This simple scaling ignores factors besides
which may change with resolution. For instance, attempts to fit
data from individual nonhydrostatic simulations to the “sticky” thermal model (12) yielded drag coefficients *C _{d}* which varied by over an order of magnitude across our resolution range. These drag coefficients may be a manifestation of the dynamic pressure force, whose resolution dependence we have neglected. Such variations in

*C*do not render our scaling useless, but they do introduce error and hence limit the scaling's accuracy. Do such variations represent a real resolution-dependence of drag, or rather inaccuracies of the sticky thermal paradigm? Further work is needed to settle such questions.

_{d}## Acknowledgments

The author thanks Leo Donner and Isaac Held for guidance, Usama Anber for *FV*^{3} orientation, and S-J Lin and especially Lucas Harris for assistance in configuring *FV*^{3} to run at these relatively high resolutions. Thanks are also due to Leo Donner, Yi Ming, Lucas Harris, Steven Garner, and especially Jacob Seeley for commenting on drafts of this work. Special thanks are due to Ming Zhao for asking how effective buoyancy behaves in a hydrostatic model, which question initiated this study. Simulations were run on the Gaea supercomputer at Oak Ridge National Laboratory, Oak Ridge, Tennessee. The author is supported by the Visiting Scientist Program of the Princeton Atmosphere and Ocean Science program. Data and analysis scripts for this study (in R) are available at https://github.com/jeevanje/w_hydrostatic.

## Appendix A: Derivation of Analytical Solutions for

We seek here solutions to equations 6 and 8 for the source *B* field (9).

*z*<

*H*is given by

*β*

_{hyd}can be obtained by direct substitution of (A1) into equation 8 and integration. The result is

*β*

_{nh}, we make the ansatz

*z*=

*H*, which can be derived from (6)). The result for is

*β*

_{nh}and

*β*

_{hyd}as a function of

*B*

_{0},

*D*, and

*H*:

These are equation 10a in the main text.

## Appendix B: Comparison With Other Scalings

Here we have set the *α* parameter of M16 equal to
(right in the 0.5–1 range used in Morrison (2016b)). All three of these studies point out that the “1” in the denominator in their expressions comes from nonhydrostatic effects, so hydrostatic versions of these expressions can be obtained by simply omitting this “1.”

The plots of Figure B1 show the analogs of Figure 5, but for the above scalings (we recalculate
for each plot by least squares fit to the nonhydrostatic data, as in Figure 5). These scalings are qualitatively similar to each other and to equation 13, and the M16 and W97 scalings bear a particularly close relation to ours as all three vary as *H*/*D* in the hydrostatic limit. The PG06 scaling varies as
in this limit, giving it a slightly different shape that seems to be a poorer match to the *FV*^{3} data, for both the hydrostatic and nonhydrostatic cases. The W97 scaling slightly underestimates
throughout the gray zone but does well in capturing the hydrostatic overestimation in the convection-resolving regime, and the M16 scaling does quite well across the board.

## Appendix C: Parcel Diagnostics

To diagnose average values of *D*, *H*, and
across parcels in a given simulation, we devised an algorithm to automatically detect convecting parcels in 3-D snapshot output and then diagnose these quantities individually for each parcel. Averaging over parcels within a given simulation then yields the data plotted in Figures 3, 5, and B1Figures B1 and C1. This appendix describes this diagnostic algorithm in detail.

*w*. This maximum is discarded if it lies within three grid cells of the horizontal boundaries (for ease of analysis), or if its pressure is lower than 250 hPa, since such maxima correspond to tall, rare plumes that are often comprised of multiple distinct parcels which are difficult to analyze and not representative of typical convecting parcels. If is not discarded we then record its horizontal position , and search for the center of the corresponding density anomaly (typically located higher than ) by maximizing in the vertical index

*k*. In searching for this we exclude boundary-layer points at pressures higher than hPa (to avoid surface-induced buoyancy maxima), and also exclude stratospheric points at pressure lower than hPa (to avoid gravity-wave related buoyancy maxima). To ensure that the buoyancy maximum at the resulting is indeed associated with , we discard parcels where the distance between and is greater than 2000 m. With so located, we then impose a “cloud-core” condition of m/s, kg/kg (where is nonprecipitating condensate), and , where the latter is resolution-dependent and is given by

This gives a factor of 4 increase in
that is linear in log *dx*, which is just how the average parcel
values behave across resolution (not shown). This functional form for
prevents changes in average
with resolution from unduly affecting our diagnoses. Any parcel not meeting the cloud-core condition is discarded.

If not discarded, the parcel is now considered to be based at
, and we diagnose its vertical extent by marching upwards and downwards from
until
. This yields indices
and
for parcel top and bottom. We then check that
and
, and discard the parcel if those conditions are not met. If they are, we proceed to diagnose the parcel's height as the sum of grid box heights from
to
. We then diagnose parcel diameter by, for each
, finding the largest contiguous range of *i*, including
, such that
. Multiplying the number of *i* values in this range by *dx* then gives an estimate of the parcel's diameter at level *k*. This can be repeated for all
, as well as by looking at parcel extent in *j* rather than *i*. A simple averaging of all these values then yields the parcel's diameter *D*. We set the parcel's
. We then “erase” this parcel from the 3-D *w* field by setting
NA for all *k* and all *i*, *j* within 3 of
and
. We then search for the next *w* maxima in this modified *w* field, and repeat the above processes. This algorithm repeats until 100 *w* maxima in a given snapshot have been identified and either discarded or had their properties fully diagnosed, or until the number of discarded parcels exceeds 25, whichever comes first. The next snapshot is then processed in the same manner. This procedure, while crude, is computationally efficient and also suffices to yield several hundred parcels for each simulation. Average values of all diagnostics for a given simulation are then computed as simple averages over all parcels from all snapshots.

To get a feel for the statistics of a given simulation, Figure C1 shows scatterplots of
versus *D* and
versus *H* for all 621 parcels identified from the nonhydrostatic, 2 km simulation. Note that *D* has a fairly symmetric distribution and an unremarkable spread, whereas *H* and *w* are highly skewed, with outliers roughly an order of magnitude larger than their means (the means of *H* and
being 750 m and 2.7 m/s, respectively). Most notably,
and *D* appear more or less uncorrelated, whereas there is a strong relationship between
and *H* (
). Though this might be expected from equation 10a, we cannot rule out feedbacks that elongate parcels which have higher
(for instance, because higher
implies higher vertical *w* gradients, and hence possibly more horizontal convergence and entrainment, adding mass and possibly height to the plume). Thus, the strong correlation between
and *H* should not be interpreted as causation from *H* to
, but should rather be regarded as an intriguing relationship worthy of further study.