Volume 9, Issue 6 p. 2304-2316
Research Article
Open Access

Vertical Velocity in the Gray Zone

Nadir Jeevanjee

Corresponding Author

Nadir Jeevanjee

Program in Atmosphere and Ocean Sciences, Princeton University, Princeton, NJ, USA

Geophysical Fluid Dynamics Laboratory, Princeton, NJ, USA

Correspondence to: N. Jeevanjee, [email protected]Search for more papers by this author
First published: 07 September 2017
Citations: 60


We describe how convective vertical velocities urn:x-wiley:19422466:media:jame20477:jame20477-math-0001 vary in the “gray zone” of horizontal resolution, using both hydrostatic and nonhydrostatic versions of GFDL's FV3 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 urn:x-wiley:19422466:media:jame20477:jame20477-math-0002 resolution can be required for convergence of urn:x-wiley:19422466:media:jame20477:jame20477-math-0003. We also find, both numerically and analytically, that hydrostatic systems overestimate urn:x-wiley:19422466:media:jame20477:jame20477-math-0004, 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 urn:x-wiley:19422466:media:jame20477:jame20477-math-0005 can be required for convergence of convective vertical velocities urn:x-wiley:19422466:media:jame20477:jame20477-math-0006
  • Hydrostatic systems overestimate urn:x-wiley:19422466:media:jame20477:jame20477-math-0007, 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 urn:x-wiley:19422466:media:jame20477:jame20477-math-0008 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 urn:x-wiley:19422466:media:jame20477:jame20477-math-0009 are negligible, as well as the convection-resolving regime, wherein urn:x-wiley:19422466:media:jame20477:jame20477-math-0010 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 urn:x-wiley:19422466:media:jame20477:jame20477-math-0011, and since urn:x-wiley:19422466:media:jame20477:jame20477-math-0012 is relevant for both climate forcing and sensitivity (Donner et al., 2016), it seems worthwhile to address some basic questions about the behavior of urn:x-wiley:19422466:media:jame20477:jame20477-math-0013 in the gray zone:

Details are in the caption following the image

Schematic depiction of the gray zone. At some unknown fine resolution, we expect convective vertical velocities urn:x-wiley:19422466:media:jame20477:jame20477-math-0014 to converge to a realistic typical value, and at very coarse resolution we expect urn:x-wiley:19422466:media:jame20477:jame20477-math-0015 to become negligible, but it is unclear how to interpolate between these two regimes.

  1. For a given dynamical phenomenon, at what resolution should one expect the associated urn:x-wiley:19422466:media:jame20477:jame20477-math-0016 to converge? In other words, where is the inner edge of the gray zone?
  2. If urn:x-wiley:19422466:media:jame20477:jame20477-math-0017 converges at fine resolutions and is negligible at coarse resolutions, what kind of curve interpolates between these two regimes?
  3. 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 ( urn:x-wiley:19422466:media:jame20477:jame20477-math-0018) than these previous studies ( urn:x-wiley:19422466:media:jame20477:jame20477-math-0019 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 overestimate urn:x-wiley:19422466:media:jame20477:jame20477-math-0020. 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 urn:x-wiley:19422466:media:jame20477:jame20477-math-0021. 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 FV3 (Finite-Volume Cubed-Sphere Dynamical Core) (Harris & Lin, 2013; Lin, 2004). Global nonhydrostatic convection-allowing simulations with FV3-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 FV3 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 urn:x-wiley:19422466:media:jame20477:jame20477-math-0022, 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 urn:x-wiley:19422466:media:jame20477:jame20477-math-0023. 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 urn:x-wiley:19422466:media:jame20477:jame20477-math-0024 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 FV3 dynamical core and GFDL microphysics can indeed handle the transition from coarser-resolution general circulation calculations to subkilometer cloud-resolving calculations, xp 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 xp 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).

Details are in the caption following the image

xp slices of vertical velocity and Archimedean buoyancy from our nonhydrostatic, dx = 0.25 km simulation. This slice contains several parcels that were identified by the algorithm, and the diagnosed horizontal and vertical extent of the parcels are depicted via black boxes (some parcels have their B maxima in adjacent xp slices and so do not appear perfectly centered within their box). Note that the parcel extent is defined by the B field, not the w field, which typically has a larger vertical extent than the B field. Only part of the full xp domain is shown. nx denotes grid cell number in the x-direction.

3 Criterion for the Convection-Resolving Regime

We begin our analysis by addressing the first question from the introduction, namely: where is the inner edge of the gray zone? To answer this, we must first define the gray zone (for a given model configuration). We define it here, somewhat loosely, to be those horizontal resolutions at which a model's convective parameterizations might be credibly turned off, but at which resolved-scale convection is still 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
for fixed urn:x-wiley:19422466:media:jame20477:jame20477-math-0026. This is, of course, merely a definition; we must check whether a gray zone defined this way, with a corresponding urn:x-wiley:19422466:media:jame20477:jame20477-math-0027, exists for our simulations. This is done in Figure 3a, which plots the average parcel width (measured in grid cells) urn:x-wiley:19422466:media:jame20477:jame20477-math-0028 for both our hydrostatic and nonhydrostatic simulations. This plot shows that such a gray zone indeed exists for dx > 250 m, where urn:x-wiley:19422466:media:jame20477:jame20477-math-0029, and that urn:x-wiley:19422466:media:jame20477:jame20477-math-0030 for urn:x-wiley:19422466:media:jame20477:jame20477-math-0031.
Details are in the caption following the image

Average parcel properties, diagnosed using the algorithm discussed in Appendix Appendix C, as a function of resolution dx, for both hydrostatic and nonhydrostatic RCE simulations. (a) Parcel diameter measured in number of grid cells, urn:x-wiley:19422466:media:jame20477:jame20477-math-0032. This is fairly constant at urn:x-wiley:19422466:media:jame20477:jame20477-math-0033 for dx > 250 m, and starts to increase for urn:x-wiley:19422466:media:jame20477:jame20477-math-0034 m, marking the inner edge of the gray zone. (b) Parcel height H, which is fairly constant at roughly 500 m for urn:x-wiley:19422466:media:jame20477:jame20477-math-0035 m. (c) Aspect ratio D/H, which reaches the neighborhood of urn:x-wiley:19422466:media:jame20477:jame20477-math-0036 around urn:x-wiley:19422466:media:jame20477:jame20477-math-0037.

Why does the gray zone end at urn:x-wiley:19422466:media:jame20477:jame20477-math-0038? In the convection-resolving regime we have
which is just another way of saying that convection is no longer grid-limited. In the convection-resolving regime, we also expect that
since recent high-resolution simulations as well as older observations show that convecting parcels resemble spherical “thermals” (Hernandez-Deckers & Sherwood, 2016; Romps & Charn, 2015; Scorer, 1957; Woodward, 1959). This is confirmed in Figure 3c, which plots average parcel aspect ratio (where D/H is computed individually for each parcel before averaging) and shows that urn:x-wiley:19422466:media:jame20477:jame20477-math-0041 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 urn:x-wiley:19422466:media:jame20477:jame20477-math-0043 at fine resolution. Plugging urn:x-wiley:19422466:media:jame20477:jame20477-math-0044 and urn:x-wiley:19422466:media:jame20477:jame20477-math-0045 into equation 4 yields urn:x-wiley:19422466:media:jame20477:jame20477-math-0046 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 urn:x-wiley:19422466:media:jame20477:jame20477-math-0047 converge according to the same criterion. We thus plot average parcel urn:x-wiley:19422466:media:jame20477:jame20477-math-0048 for our simulations in Figure 4. This plot shows that for the nonhydrostatic solver, urn:x-wiley:19422466:media:jame20477:jame20477-math-0049 indeed converges around urn:x-wiley:19422466:media:jame20477:jame20477-math-0050, in agreement with Equation 4. Note that in the convection-resolving regime of urn:x-wiley:19422466:media:jame20477:jame20477-math-0051 we expect more turbulence with higher resolution, but the small variations of urn:x-wiley:19422466:media:jame20477:jame20477-math-0052 in this regime suggest that the impact of this on vertical velocities is not strong.

Details are in the caption following the image

Convective vertical velocities urn:x-wiley:19422466:media:jame20477:jame20477-math-0053 as a function of resolution dx, for both hydrostatic and nonhydrostatic RCE simulations. Note that urn:x-wiley:19422466:media:jame20477:jame20477-math-0054 values appear to converge for urn:x-wiley:19422466:media:jame20477:jame20477-math-0055, roughly consistent with equation (4). Note also that the hydrostatic solver indeed overestimates wc, by a factor of about 2–3 for urn:x-wiley:19422466:media:jame20477:jame20477-math-0056.

While the convergence of urn:x-wiley:19422466:media:jame20477:jame20477-math-0057 at the resolution predicted by (4) is reassuring, the much more striking feature of Figure 4 is the overestimation of urn:x-wiley:19422466:media:jame20477:jame20477-math-0058 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 urn:x-wiley:19422466:media:jame20477:jame20477-math-0059 curve and how it interpolates between the hydrostatic ( urn:x-wiley:19422466:media:jame20477:jame20477-math-0060) and convection-resolving ( urn:x-wiley:19422466:media:jame20477:jame20477-math-0061) limit. We investigate these questions analytically in the next section.

4 The Shape of urn:x-wiley:19422466:media:jame20477:jame20477-math-0062

4.1 Hydrostatic and Nonhydrostatic Effective Buoyancy

To understand the shape of urn:x-wiley:19422466:media:jame20477:jame20477-math-0063, 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” urn:x-wiley:19422466:media:jame20477:jame20477-math-0064, 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 urn:x-wiley:19422466:media:jame20477:jame20477-math-0066, 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 urn:x-wiley:19422466:media:jame20477:jame20477-math-0067 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 urn:x-wiley:19422466:media:jame20477:jame20477-math-0068 (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).

Instead, we employ the formalism of Das (1979), which splits the total pressure into the local hydrostatic pressure urn:x-wiley:19422466:media:jame20477:jame20477-math-0069 and a residual nonhydrostatic pressure urn:x-wiley:19422466:media:jame20477:jame20477-math-0070. Applying (5) within this formalism to a nonhydrostatic, Boussinesq fluid with reference density ρ0 yields the following equation for the nonhydrostatic effective buoyancy βnh (JR16):

Here B is given by the usual formula urn:x-wiley:19422466:media:jame20477:jame20477-math-0072, and the horizontal Laplacian is (crucially) urn:x-wiley:19422466:media:jame20477:jame20477-math-0073. We will present intuition for βnh 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 urn:x-wiley:19422466:media:jame20477:jame20477-math-0074, and this implies urn:x-wiley:19422466:media:jame20477:jame20477-math-0075 there as well (JR15). We also assume that w (and hence urn:x-wiley:19422466:media:jame20477:jame20477-math-0076) go to 0 as urn:x-wiley:19422466:media:jame20477:jame20477-math-0077.

What is the analog of equation 6 for a hydrostatic system? In the hydrostatic approximation, we know that w is given diagnostically from the mass continuity equation by (e.g., Adcroft et al., 2011, Figure 1.17)
where urn:x-wiley:19422466:media:jame20477:jame20477-math-0079 is the horizontal wind field. Taking the time-derivative of both sides, and applying the nonrotating hydrostatic momentum equation urn:x-wiley:19422466:media:jame20477:jame20477-math-0080 along with (5), gives

This is the equation for hydrostatic effective buoyancy. Note that applying urn:x-wiley:19422466:media:jame20477:jame20477-math-0082 to equation 8 yields equation 6, if one neglects the horizontal Laplacian (as is appropriate in the hydrostatic limit).

Equations 6 and 8 only determine urn:x-wiley:19422466:media:jame20477:jame20477-math-0083 once horizontal density anomalies or an Archimedean buoyancy field are specified, so we now specify this source field. This source field should have not only a characteristic Archimedean buoyancy B0 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
The resulting effective buoyancy fields urn:x-wiley:19422466:media:jame20477:jame20477-math-0085 and urn:x-wiley:19422466:media:jame20477:jame20477-math-0086 will also be doubly periodic, and what we seek are their amplitudes βhyd and βnh, evaluated at urn:x-wiley:19422466:media:jame20477:jame20477-math-0087, as a function of D, H, and B0 (we overload the symbols βhyd and βnh here slightly). These are derived in Appendix Appendix A. We find that βhyd and βnh 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 urn:x-wiley:19422466:media:jame20477:jame20477-math-0090, and find it exactly equal to (10b). Also note that both expressions in (10) go to 0 in this limit, as expected. In the urn:x-wiley:19422466:media:jame20477:jame20477-math-0091 limit βnh approaches B0, also as expected, and βhyd 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 urn:x-wiley:19422466:media:jame20477:jame20477-math-0092

With expressions for urn:x-wiley:19422466:media:jame20477:jame20477-math-0093 in hand, we now need to relate urn:x-wiley:19422466:media:jame20477:jame20477-math-0094 to the convective vertical velocity urn:x-wiley:19422466:media:jame20477:jame20477-math-0095. The relationship between these two quantities is not entirely settled, though two paradigms exist. One, known as the “slippery thermal” paradigm (Sherwood et al., 2013), employs the work-energy theorem and assumes that (effective) buoyancy is the dominant force on convecting parcels. This paradigm relates urn:x-wiley:19422466:media:jame20477:jame20477-math-0096 and urn:x-wiley:19422466:media:jame20477:jame20477-math-0097 as
where urn:x-wiley:19422466:media:jame20477:jame20477-math-0099 gives the height range over which the effective buoyancy urn:x-wiley:19422466:media:jame20477:jame20477-math-0100 has acted.
The other paradigm, known as the “sticky thermal” paradigm (Romps & Charn, 2015), says that convecting parcels usually rise at a terminal velocity determined by a balance between (effective) buoyancy and drag:
where Cd is a drag coefficient, A is the horizontal area of the parcel, and V its volume.
Though these two paradigms are not physically consistent, they are equivalent for our purposes, in that both equations 11 and 12 imply that urn:x-wiley:19422466:media:jame20477:jame20477-math-0102. If we now define urn:x-wiley:19422466:media:jame20477:jame20477-math-0103 as the limit of nonhydrostatic urn:x-wiley:19422466:media:jame20477:jame20477-math-0104 as urn:x-wiley:19422466:media:jame20477:jame20477-math-0105, and recall that urn:x-wiley:19422466:media:jame20477:jame20477-math-0106 in this limit, we then obtain the following scaling law for urn:x-wiley:19422466:media:jame20477:jame20477-math-0107 as a function of D/H:

Here, urn:x-wiley:19422466:media:jame20477:jame20477-math-0109 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 ( urn:x-wiley:19422466:media:jame20477:jame20477-math-0110) through the gray zone to the hydrostatic regime ( urn:x-wiley:19422466:media:jame20477:jame20477-math-0111), where the hydrostatic and nonhydrostatic solutions agree. These two curves also bear a qualitative similarity to the simulated urn:x-wiley:19422466:media:jame20477:jame20477-math-0112 in Figure 4. We can test the qualitative accuracy of equation 13 by plotting the simulated urn:x-wiley:19422466:media:jame20477:jame20477-math-0113 against the corresponding D/H, as shown by the stars in Figure 5. (For the simulations, urn:x-wiley:19422466:media:jame20477:jame20477-math-0114 is determined by a least-squares fit of urn:x-wiley:19422466:media:jame20477:jame20477-math-0115 to the nonhydrostatic urn:x-wiley:19422466:media:jame20477:jame20477-math-0116.) The nonhydrostatic scaling does a good job overall of capturing the simulated urn:x-wiley:19422466:media:jame20477:jame20477-math-0117, 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.

Details are in the caption following the image

Plots of urn:x-wiley:19422466:media:jame20477:jame20477-math-0118, where urn:x-wiley:19422466:media:jame20477:jame20477-math-0119 as a function of D/H is given by equation (10). The nonhydrostatic curve interpolates from the convection-resolving regime ( urn:x-wiley:19422466:media:jame20477:jame20477-math-0120) through the gray zone to the hydrostatic regime ( urn:x-wiley:19422466:media:jame20477:jame20477-math-0121), where the hydrostatic and nonhydrostatic solutions agree. Also shown are the simulated urn:x-wiley:19422466:media:jame20477:jame20477-math-0122, plotted against the diagnosed D/H ( urn:x-wiley:19422466:media:jame20477:jame20477-math-0123 is determined by a least-squares fit of urn:x-wiley:19422466:media:jame20477:jame20477-math-0124 to the nonhydrostatic urn:x-wiley:19422466:media:jame20477:jame20477-math-0125 points). The nonhydrostatic scaling does a good job overall of capturing the simulated urn:x-wiley:19422466:media:jame20477:jame20477-math-0126, while the hydrostatic scaling is somewhat of an overestimate. Note, however, that the hydrostatic scalings and simulations both overestimate their hydrostatic counterparts by a factor of 2–3 for urn:x-wiley:19422466:media:jame20477:jame20477-math-0127.

5 Intuition for urn:x-wiley:19422466:media:jame20477:jame20477-math-0128

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 urn:x-wiley:19422466:media:jame20477:jame20477-math-0129 is actually neglected in the vertical momentum equation, overestimate urn:x-wiley:19422466:media:jame20477:jame20477-math-0130? Note that it is possible to have urn:x-wiley:19422466:media:jame20477:jame20477-math-0131 (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).

To proceed we turn to the “effective buoyancy pressure,” which was defined in JR15 as simply the nonhydrostatic pressure field that results when urn:x-wiley:19422466:media:jame20477:jame20477-math-0132:
(note the analogy to equation 5). From this it can be shown (JR15) that urn:x-wiley:19422466:media:jame20477:jame20477-math-0134 obeys the equation
which just says that urn:x-wiley:19422466:media:jame20477:jame20477-math-0136 must generate a mass divergence which cancels that from urn:x-wiley:19422466:media:jame20477:jame20477-math-0137, which is the only gravitational term in the momentum equation in this formalism. By taking urn:x-wiley:19422466:media:jame20477:jame20477-math-0138 of equation 15 and comparing to (6), it is straightforward to show that
i.e., the nonhydrostatic effective buoyancy is simply the vertical gradient of the effective buoyancy pressure. Note that urn:x-wiley:19422466:media:jame20477:jame20477-math-0140 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 urn:x-wiley:19422466:media:jame20477:jame20477-math-0141, whose horizontal gradients force mass convergence into the parcel. This convergence must be compensated for by gradients of the countervailing nonhydrostatic effective buoyancy pressure urn:x-wiley:19422466:media:jame20477:jame20477-math-0142. Figure 6 shows that for a surface parcel, urn:x-wiley:19422466:media:jame20477:jame20477-math-0143 responds to horizontal convergence from urn:x-wiley:19422466:media:jame20477:jame20477-math-0144 by generating both horizontal and vertical divergence to compensate, and the vertical divergence is given essentially by βnh. 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 urn:x-wiley:19422466:media:jame20477:jame20477-math-0145 gradients force vertical acceleration via mass continuity. Also note that the divergence from βnh needs only to compensate for the net convergence in the horizontal, which exhibits a partial cancellation between urn:x-wiley:19422466:media:jame20477:jame20477-math-0146 and urn:x-wiley:19422466:media:jame20477:jame20477-math-0147.

Details are in the caption following the image

Cartoon of how density-driven vertical accelerations are generated in both the nonhydrostatic and hydrostatic equations of motion (right and left plots, respectively). In the nonhydrostatic formulation the vertical acceleration βnh need only compensate for the net horizontal convergence between urn:x-wiley:19422466:media:jame20477:jame20477-math-0148 and urn:x-wiley:19422466:media:jame20477:jame20477-math-0149, whereas in the hydrostatic approximation urn:x-wiley:19422466:media:jame20477:jame20477-math-0150 must compensate for the gross convergence from urn:x-wiley:19422466:media:jame20477:jame20477-math-0151, with no offset from urn:x-wiley:19422466:media:jame20477:jame20477-math-0152. Thus, urn:x-wiley:19422466:media:jame20477:jame20477-math-0153.

With this picture of convection in hand, JR16 showed that the vertical length scale over which urn:x-wiley:19422466:media:jame20477:jame20477-math-0154 declines from parcel center scales with D, which explains why urn:x-wiley:19422466:media:jame20477:jame20477-math-0155 decreases with increasing D/H (a similar explanation was given in Pauluis & Garner, 2006). Here, we use this picture to understand why urn:x-wiley:19422466:media:jame20477:jame20477-math-0156, by considering its hydrostatic analog, illustrated in the left plot of Figure 6. In the hydrostatic approximation, nonhydrostatic pressures such as urn:x-wiley:19422466:media:jame20477:jame20477-math-0157 are 0, and the vertical acceleration βhyd is then defined by the requirement that it enforce mass continuity, as in equation 8. In this case, however, βhyd must compensate for the gross convergence from urn:x-wiley:19422466:media:jame20477:jame20477-math-0158, with no offset from urn:x-wiley:19422466:media:jame20477:jame20477-math-0159. It then follows that
i.e., that the hydrostatic approximation overestimates vertical accelerations.

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

We summarize our results as follows:
  1. For a model with characteristic parcel height H and gray zone grid cell width urn:x-wiley:19422466:media:jame20477:jame20477-math-0161, the convection-resolving regime requires urn:x-wiley:19422466:media:jame20477:jame20477-math-0162. For our FV3 simulations, this translates to urn:x-wiley:19422466:media:jame20477:jame20477-math-0163 (Figures 3 and 4).
  2. Hydrostatic solvers seem to overestimate urn:x-wiley:19422466:media:jame20477:jame20477-math-0164 by a factor of 2–3 in the convection-resolving regime (Figure 4), a behavior which is captured by our proxies (Figure 5).
  3. This overestimation can be simply understood using the effective buoyancy formalism (Figure 6).

Although the hydrostatic system overestimates urn:x-wiley:19422466:media:jame20477:jame20477-math-0165, 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 urn:x-wiley:19422466:media:jame20477:jame20477-math-0166 seems to converge in Figure 4, the hydrostatic urn:x-wiley:19422466:media:jame20477:jame20477-math-0167 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 urn:x-wiley:19422466:media:jame20477:jame20477-math-0168 is much less sensitive to D/H than urn:x-wiley:19422466:media:jame20477:jame20477-math-0169 at urn:x-wiley:19422466:media:jame20477:jame20477-math-0170 (Figure 5). Further increases in computational power and hence horizontal resolution should allow for a demonstration of convergence in D/H and hence also urn:x-wiley:19422466:media:jame20477:jame20477-math-0171 for hydrostatic systems.

Also, equation 4 appears to explain why urn:x-wiley:19422466:media:jame20477:jame20477-math-0172 marks the inner edge of the gray zone (consistent with Lebo & Morrison, 2015), but only does so in terms of H and urn:x-wiley:19422466:media:jame20477:jame20477-math-0173. What sets the values of these numbers? We find urn:x-wiley:19422466:media:jame20477:jame20477-math-0174, 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 urn:x-wiley:19422466:media:jame20477:jame20477-math-0175 to be associated with the overall diffusivity of the dynamical variables, and hence to be highly model-dependent. Indeed, we were able to increase urn:x-wiley:19422466:media:jame20477:jame20477-math-0176 in FV3 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 urn:x-wiley:19422466:media:jame20477:jame20477-math-0177 for a given set of simulations.

Questions also remain regarding our simple scaling (13). This simple scaling ignores factors besides urn:x-wiley:19422466:media:jame20477:jame20477-math-0178 which may change with resolution. For instance, attempts to fit urn:x-wiley:19422466:media:jame20477:jame20477-math-0179 data from individual nonhydrostatic simulations to the “sticky” thermal model (12) yielded drag coefficients Cd 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 Cd 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.


The author thanks Leo Donner and Isaac Held for guidance, Usama Anber for FV3 orientation, and S-J Lin and especially Lucas Harris for assistance in configuring FV3 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 urn:x-wiley:19422466:media:jame20477:jame20477-math-0180

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

    We begin with the hydrostatic case. The urn:x-wiley:19422466:media:jame20477:jame20477-math-0181 field associated with (9) for z < H is given by
    The hydrostatic effective buoyancy βhyd can be obtained by direct substitution of (A1) into equation 8 and integration. The result is
    To obtain the nonhydrostatic βnh, we make the ansatz
    and substitute into (6) to obtain
    (A3) This is a second-order, constant coefficient, linear ordinary differential equation which is amenable to solution by standard textbook methods (which include invoking continuity of urn:x-wiley:19422466:media:jame20477:jame20477-math-0186 and urn:x-wiley:19422466:media:jame20477:jame20477-math-0187 at z = H, which can be derived from (6)). The result for urn:x-wiley:19422466:media:jame20477:jame20477-math-0188 is
    With these solutions to equations 6 and 8 in hand, we now evaluate them at cylinder's center urn:x-wiley:19422466:media:jame20477:jame20477-math-0190 to obtain characteristic values for βnh and βhyd as a function of B0, D, and H:

    These are equation 10a in the main text.

    Appendix B: Comparison With Other Scalings

    Previous authors have developed vertical velocity scalings analogous to ours. We focus here on the scalings of M16 (equation (27)), Pauluis and Garner (2006) (hereafter PG06, equation 19), and W97 (equation 4), which in our notation are

    Here we have set the α parameter of M16 equal to urn:x-wiley:19422466:media:jame20477:jame20477-math-0196 (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 urn:x-wiley:19422466:media:jame20477:jame20477-math-0197 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 urn:x-wiley:19422466:media:jame20477:jame20477-math-0198 in this limit, giving it a slightly different shape that seems to be a poorer match to the FV3 data, for both the hydrostatic and nonhydrostatic cases. The W97 scaling slightly underestimates urn:x-wiley:19422466:media:jame20477:jame20477-math-0199 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.

    Details are in the caption following the image

    As in Figure 5, but for the scalings in equation (B1).

    Appendix C: Parcel Diagnostics

    To diagnose average values of D, H, and urn:x-wiley:19422466:media:jame20477:jame20477-math-0203 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.

    For a given run, we analyze daily snapshots from the last 20 days of the run. In a given snapshot, we locate parcels by finding the global maximum urn:x-wiley:19422466:media:jame20477:jame20477-math-0204 of 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 urn:x-wiley:19422466:media:jame20477:jame20477-math-0205 is not discarded we then record its horizontal position urn:x-wiley:19422466:media:jame20477:jame20477-math-0206, and search for the center of the corresponding density anomaly (typically located higher than urn:x-wiley:19422466:media:jame20477:jame20477-math-0207) by maximizing urn:x-wiley:19422466:media:jame20477:jame20477-math-0208 in the vertical index k. In searching for this urn:x-wiley:19422466:media:jame20477:jame20477-math-0209 we exclude boundary-layer points at pressures higher than urn:x-wiley:19422466:media:jame20477:jame20477-math-0210 hPa (to avoid surface-induced buoyancy maxima), and also exclude stratospheric points at pressure lower than urn:x-wiley:19422466:media:jame20477:jame20477-math-0211 hPa (to avoid gravity-wave related buoyancy maxima). To ensure that the buoyancy maximum urn:x-wiley:19422466:media:jame20477:jame20477-math-0212 at the resulting urn:x-wiley:19422466:media:jame20477:jame20477-math-0213 is indeed associated with urn:x-wiley:19422466:media:jame20477:jame20477-math-0214, we discard parcels where the distance between urn:x-wiley:19422466:media:jame20477:jame20477-math-0215 and urn:x-wiley:19422466:media:jame20477:jame20477-math-0216 is greater than 2000 m. With urn:x-wiley:19422466:media:jame20477:jame20477-math-0217 so located, we then impose a “cloud-core” condition of urn:x-wiley:19422466:media:jame20477:jame20477-math-0218 m/s, urn:x-wiley:19422466:media:jame20477:jame20477-math-0219 kg/kg (where urn:x-wiley:19422466:media:jame20477:jame20477-math-0220 is nonprecipitating condensate), and urn:x-wiley:19422466:media:jame20477:jame20477-math-0221, where the latter is resolution-dependent and is given by

    This gives a factor of 4 increase in urn:x-wiley:19422466:media:jame20477:jame20477-math-0223 that is linear in log dx, which is just how the average parcel urn:x-wiley:19422466:media:jame20477:jame20477-math-0224 values behave across resolution (not shown). This functional form for urn:x-wiley:19422466:media:jame20477:jame20477-math-0225 prevents changes in average urn:x-wiley:19422466:media:jame20477:jame20477-math-0226 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 urn:x-wiley:19422466:media:jame20477:jame20477-math-0227, and we diagnose its vertical extent by marching upwards and downwards from urn:x-wiley:19422466:media:jame20477:jame20477-math-0228 until urn:x-wiley:19422466:media:jame20477:jame20477-math-0229. This yields indices urn:x-wiley:19422466:media:jame20477:jame20477-math-0230 and urn:x-wiley:19422466:media:jame20477:jame20477-math-0231 for parcel top and bottom. We then check that urn:x-wiley:19422466:media:jame20477:jame20477-math-0232 and urn:x-wiley:19422466:media:jame20477:jame20477-math-0233, 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 urn:x-wiley:19422466:media:jame20477:jame20477-math-0234 to urn:x-wiley:19422466:media:jame20477:jame20477-math-0235. We then diagnose parcel diameter by, for each urn:x-wiley:19422466:media:jame20477:jame20477-math-0236, finding the largest contiguous range of i, including urn:x-wiley:19422466:media:jame20477:jame20477-math-0237, such that urn:x-wiley:19422466:media:jame20477:jame20477-math-0238. 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 urn:x-wiley:19422466:media:jame20477:jame20477-math-0239, 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 urn:x-wiley:19422466:media:jame20477:jame20477-math-0240. We then “erase” this parcel from the 3-D w field by setting urn:x-wiley:19422466:media:jame20477:jame20477-math-0241 NA for all k and all i, j within 3 of urn:x-wiley:19422466:media:jame20477:jame20477-math-0242 and urn:x-wiley:19422466:media:jame20477:jame20477-math-0243. 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 urn:x-wiley:19422466:media:jame20477:jame20477-math-0244 versus D and urn:x-wiley:19422466:media:jame20477:jame20477-math-0245 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 urn:x-wiley:19422466:media:jame20477:jame20477-math-0246 being 750 m and 2.7 m/s, respectively). Most notably, urn:x-wiley:19422466:media:jame20477:jame20477-math-0247 and D appear more or less uncorrelated, whereas there is a strong relationship between urn:x-wiley:19422466:media:jame20477:jame20477-math-0248 and H ( urn:x-wiley:19422466:media:jame20477:jame20477-math-0249). Though this might be expected from equation 10a, we cannot rule out feedbacks that elongate parcels which have higher urn:x-wiley:19422466:media:jame20477:jame20477-math-0250 (for instance, because higher urn:x-wiley:19422466:media:jame20477:jame20477-math-0251 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 urn:x-wiley:19422466:media:jame20477:jame20477-math-0252 and H should not be interpreted as causation from H to urn:x-wiley:19422466:media:jame20477:jame20477-math-0253, but should rather be regarded as an intriguing relationship worthy of further study.

    Details are in the caption following the image

    Scatterplots of urn:x-wiley:19422466:media:jame20477:jame20477-math-0200 versus (left) D and (right) H for all parcels identified in the nonhydrostatic, dx = 2 km simulation. Note the relatively large spreads in urn:x-wiley:19422466:media:jame20477:jame20477-math-0201 and H, relative to D. Also note the high correlation between urn:x-wiley:19422466:media:jame20477:jame20477-math-0202 and H. See text for further discussion.