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  • View in gallery

    An xz cross section of (center) B and (right) for Gaussian () and hyper-Gaussian () density perturbations of the form . (left) The shape of the density perturbations at the surface is shown, and the precise form is given in the text. Note that is only a fraction of B throughout most of both bubbles and that is nonzero above , where . Also note the double peak in for the broad distribution.

  • View in gallery

    Plot of B, w, , and in the boundary layer for a CRM snapshot. Note that incipient convection () is almost entirely collocated with the cold-pool boundaries visible in the B field and that the vertical acceleration there appears strongly dominated by . Note the stretched color bar for all accelerations. The black circle in the w plot marks the gust front whose transect is plotted in Fig. 3.

  • View in gallery

    Plot of υ, w, B, θ, , , , , and for cold-pool transect. The nascent plume at (x, y) = (38 km, 150 m), visible in the w field, lies just above the gust front evident in the u field. The plume has anomalously high θ, qυ, and B but is nonetheless triggered primarily by . Note again the drastic differences in the and B fields.

  • View in gallery

    Regression coefficients ri and rb (g m−4) as defined in (13), plotted as a function of δx and δy, respectively, for various z. This more objective and comprehensive test again shows that dominates over .

  • View in gallery

    Cartoon depicting the dominance of over as resulting primarily from the ratio of H to h, as in (17).

  • View in gallery

    Time series of and , computed as in (13) for various z but with , along with precipitation time series, for the shallow-to-deep simulation. Note that dominates before precipitation sets in, at which point takes over.

  • View in gallery

    Idealized vector fields and their advective tendencies. (a) A pure rotational flow and (b) its divergent advective tendency. (c) A pure (irrotational) strain flow and (d) its convergent advective tendency. The inertial pressure balances these tendencies by generating a low (high) at the origin in the case of pure rotation (strain).

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Effective Buoyancy, Inertial Pressure, and the Mechanical Generation of Boundary Layer Mass Flux by Cold Pools

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  • 1 Department of Physics, University of California, Berkeley, and Earth Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California
  • | 2 Department of Earth and Planetary Science, University of California, Berkeley, and Earth Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California
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Abstract

The Davies-Jones formulation of effective buoyancy is used to define inertial and buoyant components of vertical force and to develop an intuition for these components by considering simple cases. This decomposition is applied to the triggering of new boundary layer mass flux by cold pools in a cloud-resolving simulation of radiative–convective equilibrium (RCE). The triggering is found to be dominated by inertial forces, and this is explained by estimating the ratio of the inertial forcing to the buoyancy forcing, which scales as H/h, where H is the characteristic height of the initial downdraft and h is the characteristic height of the mature cold pool’s gust front. In a simulation of the transition from shallow to deep convection, the buoyancy forcing plays a dominant role in triggering mass flux in the shallow regime, but the force balance tips in favor of inertial forcing just as precipitation sets in, consistent with the RCE results.

Corresponding author address: Nadir Jeevanjee, Department of Physics, University of California, Berkeley, 366 LeConte Hall MC 7300, Berkeley, CA 94720. E-mail: jeevanje@berkeley.edu

Abstract

The Davies-Jones formulation of effective buoyancy is used to define inertial and buoyant components of vertical force and to develop an intuition for these components by considering simple cases. This decomposition is applied to the triggering of new boundary layer mass flux by cold pools in a cloud-resolving simulation of radiative–convective equilibrium (RCE). The triggering is found to be dominated by inertial forces, and this is explained by estimating the ratio of the inertial forcing to the buoyancy forcing, which scales as H/h, where H is the characteristic height of the initial downdraft and h is the characteristic height of the mature cold pool’s gust front. In a simulation of the transition from shallow to deep convection, the buoyancy forcing plays a dominant role in triggering mass flux in the shallow regime, but the force balance tips in favor of inertial forcing just as precipitation sets in, consistent with the RCE results.

Corresponding author address: Nadir Jeevanjee, Department of Physics, University of California, Berkeley, 366 LeConte Hall MC 7300, Berkeley, CA 94720. E-mail: jeevanje@berkeley.edu

1. Introduction

The initiation of convection is an outstanding and pressing issue in cloud dynamics. Clarification of the mechanisms involved is necessary for the construction of reliable parameterizations, and in particular for reliably “closing” mass flux schemes, which must diagnose cloud-base mass flux in terms of prognostic variables. Many current mass flux schemes, such as the Zhang–McFarlane scheme (Zhang and McFarlane 1995) currently employed in the Community Atmosphere Model (Neale et al. 2013), have closures that rely on uncertain convective time-scale parameters, to which the parent models exhibit considerable sensitivity (Qian et al. 2015; Mishra 2011; Mishra and Srinivasan 2010). Thus, a firm understanding of how convection is initiated is critical for trustworthy convective parameterizations and accurate simulations of global climate.

Though convection can take many forms (e.g., trade cumulus, squall lines, mesoscale convective systems) and is variously influenced by the large-scale environment (e.g., surface temperature gradients, wind shear, and large-scale vertical motion), the mass flux closure problem remains unsolved even in the simple case of unorganized radiative–convective equilibrium (RCE) over an ocean with uniform temperature. It is known, however, that in cloud-resolving model (CRM) studies of RCE, convection is preferentially triggered at cold-pool gust fronts, as demonstrated by Tompkins (2001, hereafter T01). Thus, a closer study of what happens at such gust fronts is necessary to understand how convection in RCE is generated.

That gust fronts in general can trigger convection (i.e., generate boundary layer plumes with significant vertical velocity) is well known, for example, from the study of squall lines (Weisman and Rotunno 2004) or midlatitude continental convection (Droegemeier and Wilhelmson 1985). In these cases it has generally been assumed that the triggering is dynamical in nature—that is, that it arises from horizontal convergence at the gust front. For oceanic RCE, however, T01 noted that the thermal recovery of mature cold pools, along with pronounced moisture anomalies at the gust front, yield a dramatic reduction in convective inhibition (CIN) and enhancement of convective available potential energy (CAPE) there, pointing to a strong thermodynamic role for cold pools in organizing convection. While neither CIN nor CAPE directly relate to the generation of boundary layer mass flux, the thermal recovery of the mature cold pools pointed out by T01, along with the virtual effect due to the moisture anomalies at the gust front, raise the possibility that there is a significant buoyant contribution to the initial triggering. Our main goal in this paper is to assess this possibility, by evaluating the relative roles of mechanical and thermodynamical forces in generating mass flux at cold-pool gust fronts in oceanic RCE. We will focus on how low-level (z = 300 m) mass flux is generated and leave aside for the time being the question of how that low-level mass flux relates to cloud-base mass flux. Answering the latter question will be critical for solving the mass flux closure problem discussed above, and our work here can be seen as a first step in that direction.

Assessing the relative roles of thermodynamical and mechanical accelerations will require us to define these quantities, so we propose here the following simple definitions. We define the vertical buoyant acceleration [or “effective buoyancy”; Davies-Jones (2003, hereafter DJ03)] to be the Lagrangian acceleration that would result if the wind fields were instantaneously zeroed out; that is,
e1
where is the wind field. Up to a factor of (a reference density profile), this will be our “thermodynamic force.” We analogously define the vertical inertial acceleration to be the Lagrangian vertical acceleration resulting from an instantaneous zeroing out of any horizontal density anomalies; that is,
e2
where ρ is the system density (including the weight of hydrometeors). Up to a factor of , this will be our “mechanical” or “dynamical” force, though it is really due entirely to inertia, as we will see.

We will show in the next section that and that and depend entirely on density and wind fields, respectively, and thus constitute a suitable decomposition of vertical force into thermodynamic and mechanical components. Mathematically equivalent decompositions are somewhat well known and have been considered by previous authors (e.g., DJ03; Xu and Randall 2001; Krueger et al. 1995b), but the definitions (1) and (2) are new. In addition to bearing a simple physical interpretation, these definitions also yield unambiguous boundary conditions for and , which are often imposed by hand (DJ03; Klemp and Rotunno 1983) and can be a source of ambiguity (Markowski and Richardson 2011, p. 29).

A central feature of the definition (1) is that includes both the Archimedean buoyancy B as well as the environmental response to the accelerations produced by B. We will see that can be very different in magnitude and spatial distribution than B, to the degree that B cannot always be considered a first approximation for . Thus, a proper determination of the character of convective triggering by cold pools depends crucially on considering rather than B, as advocated by Doswell and Markowski (2004).

We begin by using the definitions (1) and (2) to derive diagnostic Poisson equations for and . With a proper force decomposition in hand, we then run a CRM in RCE and diagnose and at cold-pool gust fronts and across the entire domain. From this, we derive evidence that is the primary driver behind tropical convective triggering and give a simple argument for why this should be. Our argument suggests that in the absence of cold pools, may dominate; we confirm this by running a shallow-to-deep convection simulation, during which the dominant forcing transitions from to as precipitation sets in and cold pools appear.

2. Buoyant and inertial accelerations

a. Vertical force decomposition

We begin by deriving diagnostic equations for and . Our starting point is the anelastic momentum equation
e3
where p is the pressure and g is the gravitational acceleration. There is no Coriolis term as we are considering equatorial, oceanic RCE. A common approach is to approximate (3) by introducing a reference pressure profile in hydrostatic balance with , along with accompanying perturbations and , which leads to
e4
where is the usual Archimedean buoyancy. Here, however, we follow the approach taken in DJ03 and Das (1979) and alternatively decompose the pressure field as
e5a
where
e5b
so that is the local hydrostatic pressure field and is the local nonhydrostatic pressure field. Plugging this into (3) yields
e6
where and we have an exact cancellation between the gravitational force and . The z component of (6) is simply
e7
so the Lagrangian vertical acceleration is given by times the nonhydrostatic vertical pressure gradient force . [This result also dates back to List and Lozowski (1970) and Das (1979).] To apply the definitions (1) and (2), we must diagnose . This can be achieved by taking the divergence of both sides of (6) and invoking anelastic mass continuity, which yields the Poisson equation
e8
where is the usual three-dimensional Laplacian and . Here, as in subsequent equations, we write the Laplacian term with a minus sign so that positive values on the right-hand side of the Poisson equation tend to generate positive responses in the solution (in this case, the solution for ). Applying to (8) yields
e9
where we used the definition (5b) of .
We can now combine the diagnostic equation in (9) with the result in (7) and the definition in (1) to obtain a Poisson equation for :
e10
Some remarks on this equation are in order. Note that it is horizontal density variations, as opposed to more general thermodynamic variations, that give rise to , which is why we refer to it as a “buoyant” acceleration rather than a more general “thermodynamic” acceleration. Also, up to the factor of , which makes our an acceleration rather than a force per volume, (10) is identical to (6) of DJ03, so we adopt his nomenclature and alternatively refer to as the “effective buoyancy.” Finally, an alternative diagnostic expression for the effective buoyancy can be obtained via (4); this is the approach used by most other authors (e.g., Doswell and Markowski 2004; Xu and Randall 2001; Krueger et al. 1995b), and we will compare and contrast these approaches in appendix A.
Next, we turn to the diagnosis of . Combining (9), (7), and (2) yields the desired expression:
e11
Note that is sourced by the advection of momentum, which is nothing but inertia, which is why we refer to as an “inertial” acceleration. This quantity has appeared in the literature many times before, most often as the z component of a “dynamic” pressure gradient (e.g., Markowski and Richardson 2011; Rotunno and Klemp 1985; Klemp and Rotunno 1983). It will be both computationally and conceptually expedient for us to also consider as the vertical component of a pressure gradient. We will refer to the corresponding pressure field as the “inertial pressure,” denoted , and we give the details of its definition, interpretation, and computation in appendix B.

Note that the determination of and via the Poisson equations (10) and (11) is incomplete without boundary conditions (BCs). We will be considering an atmosphere with rigid bottom () and top (), so that at and . Equation (7) then implies at heights 0 and . These BCs are unchanged by setting or , and so the definitions (1) and (2) yield the Dirichlet BCs at and .

Finally, we observe that by (9)(11), obeys the same Poisson equation as . Furthermore, both quantities have the same BCs. This tells us that , and (7) then implies
e12

Equation (12) is the desired decomposition of the vertical acceleration into buoyant and inertial components. Mathematically equivalent forms of (12) can be found elsewhere in the literature [e.g., Markowski and Richardson 2011, their (10.15); Krueger et al. 1995b; Xu and Randall 2001], and the derivation given here closely follows that given in DJ03 in many respects. The novel elements are the definitions (1) and (2), which yield unambiguous boundary conditions for and and give them a simple physical interpretation.

b. Contrasting Archimedean and effective buoyancies

Before describing our experiments and their results, let us get a feel for how works and how it differs from Archimedean buoyancy (some intuition for and is developed in appendix B). From (10), we see that is sourced by the horizontal Laplacian of ρ, so that buoyant accelerations tend to be strongest at local extrema of density (or, more generally, regions of ) rather than at density anomalies defined relative to an arbitrary reference value (as with B). This means that of a parcel is defined relative to its immediate surroundings, so that a very warm parcel surrounded by other warm air may accelerate less than a moderately warm parcel surrounded by cool air.

Since (10) is a Poisson equation, effective buoyancy is nonlocal: that is, localized extrema of density give rise to accelerations everywhere, even where . This is, of course, because any localized acceleration must (by mass continuity) be accompanied by compensating accelerations in the rest of the atmosphere. These compensating accelerations (see appendix A for their definition) often oppose the buoyancy field that gives rise to them (Markowski and Richardson 2011; Houze 1994), yielding a reduced net acceleration for air with significant B. For isolated regions of significant buoyancy, we thus expect that .

These aspects of are illustrated in Fig. 1, which shows xz cross sections of B and for two Gaussian bubbles of the form
eq1
for n = 2 and 4. Here, , , and , where is taken from our RCE simulations described below. We calculate B with respect to the horizontal average of ρ rather than ; the fact that such a choice is required, yet somewhat arbitrary, is yet another shortcoming of B (Doswell and Markowski 2004).
Fig. 1.
Fig. 1.

An xz cross section of (center) B and (right) for Gaussian () and hyper-Gaussian () density perturbations of the form . (left) The shape of the density perturbations at the surface is shown, and the precise form is given in the text. Note that is only a fraction of B throughout most of both bubbles and that is nonzero above , where . Also note the double peak in for the broad distribution.

Citation: Journal of the Atmospheric Sciences 72, 8; 10.1175/JAS-D-14-0349.1

Perhaps the most striking feature (for both values of n) of Fig. 1 is the degree to which , requiring a rather severe stretching of the color bar to render features of both fields visible. Another interesting feature of Fig. 1, particularly for , is that for , even though . As discussed above, this is because not only accelerates the bubble but also sets up the accompanying environmental circulation.

Finally, for , one can calculate that is a maximum near the “shoulder” of the density distribution at , and Fig. 1 shows a corresponding maximum there in the field as well. This is to be contrasted with the B field, where the maximum is found at , where ρ itself has a minimum. Again, this is because net thermodynamic accelerations are a function of how buoyant a parcel is relative to its immediate surroundings, and so when the peak of the density distribution is too broad, the parcels there feel less acceleration than their counterparts at the shoulder of the distribution. Thus, the spatial distribution of can indeed differ from that of B; we will see even more dramatic examples of this in the next section.

3. RCE simulations

With a preliminary understanding of in hand, we can proceed to investigate whether or dominates the triggering of deep convection by cold pools in CRM simulations of RCE. This section describes the numerical model used as well as our specific case setup. We also briefly describe our calculation of and ; this is discussed in greater detail in appendix B.

Our cloud-resolving simulations were performed with Das Atmosphärische Modell (DAM) (Romps 2008). DAM is a three-dimensional (3D), fully compressible, nonhydrostatic CRM, which employs the six-class Lin–Lord–Krueger microphysics scheme (Lin et al. 1983; Lord et al. 1984; Krueger et al. 1995a). Radiation is interactive and is calculated using the Rapid Radiative Transfer Model (Mlawer et al. 1997). We rely on implicit LES (Margolin et al. 2006) for subgrid-scale transport, and thus no explicit subgrid-scale turbulence scheme is used.

Our RCE simulations ran on a square doubly periodic domain of horizontal dimension , with a horizontal resolution of . The vertical grid stretches smoothly from 50-m resolution below 1200- to 100-m resolution above, up to the model top at 30 km. We ran with a fixed sea surface temperature of 300 K and calculated surface heat and moisture fluxes using a bulk aerodynamic formula.

For a first diagnosis of and in RCE, we spun up the model for 60 days on an L = 12-km, dx = 200-m domain, then used the vertical profiles from this run to initialize a 13-day run on an L = 51-km, dx = 200-m domain. This run was then restarted with dx = 100 m and run for one more day to iron out any artifacts from changing the resolution. All data in the next section are from the end of this run.

We diagnose directly from CRM output via the Poisson equation (10). We compute slightly indirectly by first computing the inertial pressure from CRM output via (B2) and then using . We solve both Poisson equations by Fourier transforming in the horizontal, which yields systems of algebraic equations that can be solved by inverting a tridiagonal matrix. As discussed in section 2a, the fact that at model top and bottom yields Dirichlet boundary conditions for and Neumann boundary conditions for . The implementation of the latter is somewhat subtle, however, and requires some care; see appendix B for details.

4. RCE results

Plan views of the vertical velocity w at as well as the vertical accelerations B, , and at for a CRM snapshot are given in Fig. 2. Cold pools are clearly visible in the B field, and incipient convection at the cold-pool gust fronts is evident in the w field. Comparison of and suggests that this incipient convection is due primarily to rather than . Note the much reduced magnitude and differing spatial scales of relative to B, again requiring a severely stretched color bar; in this circumstance, B is not even a first approximation for . This drastic difference between and B is a result of the extreme aspect ratio of the cold pools as well as their proximity to the ground, where an boundary condition is enforced. Further work is needed to separately quantify these two effects and delineate the density distribution regimes over which B can be used as a proxy for .

Fig. 2.
Fig. 2.

Plot of B, w, , and in the boundary layer for a CRM snapshot. Note that incipient convection () is almost entirely collocated with the cold-pool boundaries visible in the B field and that the vertical acceleration there appears strongly dominated by . Note the stretched color bar for all accelerations. The black circle in the w plot marks the gust front whose transect is plotted in Fig. 3.

Citation: Journal of the Atmospheric Sciences 72, 8; 10.1175/JAS-D-14-0349.1

To further investigate the dominance of over , we take an yz transect through a particular cold-pool gust front from Fig. 2 and plot various quantities for this transect in Fig. 3. (This particular gust front is marked with a black circle in the w plot of Fig. 2.) We see a vigorous southward-moving cold pool with a gust front at as well as a nascent plume (w > 1 m s−1) just above at around . The warm, moist air feeding this plume from below is visible in the B, θ, qυ, and θe fields at (y, z) ≈ (38 km, 150 m), and the gust front and plume indeed exhibit anomalously high , as noted by T01. Despite such thermodynamic enhancement, however, the plume’s near-surface acceleration is strongly dominated by . Even when plotted on a stretched color scale, its is barely discernible and is much reduced relative to its Archimedean buoyancy, consistent with our results from section 2b.

Fig. 3.
Fig. 3.

Plot of υ, w, B, θ, , , , , and for cold-pool transect. The nascent plume at (x, y) = (38 km, 150 m), visible in the w field, lies just above the gust front evident in the u field. The plume has anomalously high θ, qυ, and B but is nonetheless triggered primarily by . Note again the drastic differences in the and B fields.

Citation: Journal of the Atmospheric Sciences 72, 8; 10.1175/JAS-D-14-0349.1

To quantitatively test the hypothesis that the spike is responsible for the nascent plume, we check if w of the nascent plume at , which is about 2.5 m s−1, is equal to where h is the height of the spike at the gust front. Taking and gives , which is consistent with the actual w.

We analyzed several other cold pools and came to similar conclusions. Still, a more comprehensive and objective test of the dominance of is needed. To that end, we identify “active” columns [defined here to be those with ], and for these regress and evaluated at , for various δx, δy, and z, against . This yields regression coefficients and which are least squares fits to
e13
where the intercepts Cb and Ci are negligible and ignored henceforth. We use rather than w because (by the work-energy theorem) a linear relationship with the forces is expected only for . We regress the forces on , rather than the other way around, because we want the regression coefficient to be directly proportional, rather than inversely proportional, to the magnitude of the forces. Maps of and as a function of and at various heights z are shown in Fig. 4. [The units and order of magnitude of the coefficients are given by .] These maps show clearly and objectively the dominance of over in generating new mass flux in the boundary layer and constitute our main numerical result.
Fig. 4.
Fig. 4.

Regression coefficients ri and rb (g m−4) as defined in (13), plotted as a function of δx and δy, respectively, for various z. This more objective and comprehensive test again shows that dominates over .

Citation: Journal of the Atmospheric Sciences 72, 8; 10.1175/JAS-D-14-0349.1

5. Why does dominate?

The previous section presented anecdotal as well as systematic evidence that dominates over in triggering new low-level mass flux in a simulation of deeply convecting RCE. It remains, however, to gain some insight as to why this should be. In this section we will estimate the ratio of to at a cold-pool gust front by estimating the ratio of the relevant source terms in their respective Poisson equations. We justify this approach by noting that variations in and at the gust front occur over the same length scales and so .

First consider the source term for in (11). This can be rewritten as
e14
To analyze this, note that the scale of variations is much larger than those for the velocities, so we can neglect derivatives of . This leaves us with
e15
Let us evaluate this at a gust-front boundary, where U is a typical horizontal velocity of the front, W is a typical vertical velocity of a triggered updraft, h is a typical height of the front, and L is the length over which u and B transition from their cold-pool values to their ambient values. (From the surface level in Fig. 3, this is evidently the grid spacing , though we will see that our argument is resolution independent.) Then every i = 1 or 2 in (15) yields a factor of , and every factor of yields a factor of , which equals by continuity. The same is true for j. Also, contributes a factor of , and so we can estimate (15) as
eq2
Similarly, the source term for is given by
eq3
where is a characteristic density anomaly for a cold pool. Taking the ratio of our expressions for and gives
e16
where B is a characteristic magnitude of Archimedean buoyancy for the cold pool. To evaluate (16), we use the empirical observation (Hacker et al. 1996) that for a lock–release density current, the “total depth” Froude number is roughly equal to ½, independent of time and H.1 Here H is a characteristic height for the negatively buoyant downdraft that spawned the cold pool, and is the magnitude of the cold pool’s initial buoyancy (typically ). Plugging into (16) gives
e17
This is our main theoretical result. Taking typical values of , , , and gives a ratio of about 18, implying that, at the gust fronts, is larger than by about an order of magnitude. This corresponds roughly to what we see in Figs. 2 and 3. A cartoon of this result, emphasizing the ratio of H to h as a determining factor in the dominance of over , is given in Fig. 5.
Fig. 5.
Fig. 5.

Cartoon depicting the dominance of over as resulting primarily from the ratio of H to h, as in (17).

Citation: Journal of the Atmospheric Sciences 72, 8; 10.1175/JAS-D-14-0349.1

6. Shallow-to-deep simulation

Given that we have identified as the dominant force component in the generation of boundary layer mass flux in RCE, it is of interest to ask whether there are other settings in which , rather than , might dominate. One might expect that for shallow nonprecipitating convection, the absence of cold pools would mean that the triggering of convection is primarily buoyant in nature.

To test this, we run a shallow-to-deep CRM simulation similar to that of Kuang and Bretherton (2006), where we use the same model domain and grid spacing as for our RCE simulation above but initialize with a thermodynamic profile based on observations from the Barbados Oceanography and Meteorology Experiment (BOMEX). For heights between 0 and 3000 m we use the θ and qυ profiles given in the CRM intercomparison study of this case in Siebesma et al. (2003). We then simply (and somewhat crudely) extend the θ profile above 3000 m by linearly interpolating to a tropopause at and then to the model top at . We similarly extend the qυ profile via relative humidity (RH) by interpolating to and then to . These values roughly approximate those found in our RCE simulations. We fix the latent heat flux at and sensible heat flux at , in accordance with Siebesma et al. (2003). Note the contrast with other studies of the shallow-to-deep transition, such as Kuang and Bretherton (2006) and Böing et al. (2012), where the transition is forced by time-dependent surface fluxes. Here, we are less interested in a realistic transition between shallow and deep convection as we are in contrasting their convective triggering mechanisms, and we deem this simple simulation sufficient for that purpose.

We run this simulation for 2 days, saving 3-hourly snapshots. For each snapshot, we calculate and as in (13) but set for clarity. We thus get one value of and at each height for each snapshot, and time series of these (along with domain-mean precipitation) are presented in Fig. 6. Indeed, we see that dominates while precipitation is negligible and that takes over as precipitation sets in; this confirms our expectation and provides a clear hydrodynamical distinction between mass flux generation in the two regimes.

Fig. 6.
Fig. 6.

Time series of and , computed as in (13) for various z but with , along with precipitation time series, for the shallow-to-deep simulation. Note that dominates before precipitation sets in, at which point takes over.

Citation: Journal of the Atmospheric Sciences 72, 8; 10.1175/JAS-D-14-0349.1

7. Implications

We have used a carefully chosen formulation of the anelastic equations of motion to decompose vertical accelerations into inertial and buoyant components and have used the resulting decomposition to analyze the triggering of low-level mass flux by cold-pool gust fronts. This can be seen as a first step toward answering the question of how cloud-base mass flux is generated in the boundary layer of an atmosphere in deeply convecting RCE. Along the way, we have also developed some intuition for the inertial and buoyant accelerations and have addressed some ancillary questions that arise in their interpretation and computation.

The notion of effective buoyancy, though not new, has received relatively little attention. The physics that it embodies is well known, in that it is widely acknowledged in the literature that buoyant accelerations of parcels are reduced by back reaction from the environment and that this effect depends on the horizontal extent of the parcel, but these effects are rarely computed explicitly. Furthermore, widely used diagnostic quantities such as CAPE and CIN, which play central roles in various convective parameterizations (e.g., Zhang and McFarlane 1995; Bretherton et al. 2004), are based on easily calculated Archimedean buoyancy, rather than on the complete buoyant force.2 Since the results presented here (and in particular Figs. 1 and 2) suggest that Archimedean buoyancy can be highly inadequate in capturing buoyant acceleration, both in magnitude and spatial distribution, care must be taken in the quantitative application of such diagnostics. If a parcel’s CIN, for instance, is a poor estimate of the negative buoyant acceleration it experiences as it makes its way to cloud base, then there may be little theoretical justification for the CIN–TKE mass flux closures employed in, for example, Mapes (2000) and Bretherton et al. (2004). There is thus a need for a simple yet quantitatively reliable way to estimate the effective buoyancy of a parcel given some additional datum about its spatial dimensions and proximity to the ground.

The other component of vertical force, the inertial pressure gradient , seems to be more familiar than effective buoyancy, having in particular played a key role in the analysis of severe storms and tornadoes (e.g., Rotunno and Klemp 1985; Klemp and Rotunno 1983; Markowski and Richardson 2011). Despite this exposure though, remains slightly enigmatic. There seems to be little discussion in the literature of as an enforcer of mass continuity in the face of inertial motions and how this implies that strain and vorticity generate disturbances of opposing sign. Furthermore, the calculation of via (B2) raises issues of boundary-condition implementation and finite-difference interpolation that often go unmentioned but actually require rather exquisite care. We have endeavored to fill these gaps in the interpretation and calculation of in appendix B.

Finally, our result that the inertial acceleration dominates the low-level triggering of new mass flux provides a stepping stone to a more complete picture of mass flux generation in the boundary layer. Though we have not made a precise connection between the low-level mass flux investigated here and cloud-base mass flux, a positive correlation is to be expected, and thus our results lend preliminary support to convective parameterizations in which the inertial triggering of mass flux at cold-pool gust fronts takes center stage, as in Rio et al. (2013).

Of course, many details remain to be filled in. For instance, the origins of the anomalous moisture at the gust front remain uncertain. Surface fluxes and entrainment of environmental air are potential sources of both heat and moisture for the gust front, which may have already been significantly premoistened by evaporating precipitation, but a quantification of these various sources is still lacking. Also, although dominates over in the lower boundary layer, the transect in Fig. 3 (as well as other transects that we inspected) suggest that the force balance may shift as the plume rises and that may play a role in shepherding nascent plumes through the middle boundary layer and up to their lifting condensation levels. Quantifying this role would be necessary to complete the picture of cloud-base mass flux generation.

Apart from the generation of deep convective mass flux, there are other problems that might be fruitfully analyzed with the approaches taken here. It could be helpful to attempt a scaling estimate for as in (17), but for shallow convection, where turbulent motions instead of cold pools are responsible for generating the heterogeneity in density and wind that generate and . Also, applying the force decomposition in (12) to convection in the free troposphere could further refine our picture of the vertical momentum budget of thermals, as recently studied in, for example, Wang and Zhang (2014), Sherwood et al. (2013), and de Roode et al. (2012).

Finally, we note that as we were revising this paper, a similar study was published that also examines the relative influence of thermodynamic and mechanic properties of cold pools upon convective triggering (Torri et al. 2015). Similar to this study, those authors also found that mechanical forces dominate over thermodynamic ones in triggering low-level mass flux. Furthermore, they employed a Lagrangian particle dispersion, which allowed them to quantify the influence of cold-pool thermodynamics in reducing particles’ lifting condensation levels. They also introduced a novel algorithm for tracking the lifetimes of cold pools and the residence times of particles within them, providing new insights into the origins of mass flux triggered by cold pools.

Acknowledgments

This work was supported by the U.S. Department of Energy’s Earth System Modeling, an Office of Science, Office of Biological and Environmental Research program under Contract DE-AC02-05CH11231. This research used computing resources of the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation Grant OCI-1053575. N.J. thanks Wolfgang Langhans for discussions and assistance.

APPENDIX A

Comparison of Two Approaches to Effective Buoyancy

Solving (10) is not the only way to obtain the buoyant acceleration. One can use the momentum equation [(4)] and take its divergence to obtain a Poisson equation for , which has as one of its source terms. One can then define a “buoyancy perturbation pressure” as the solution to [modulo ambiguous boundary conditions; Markowski and Richardson (2011, p. 29)], and it can be shown (DJ03) that
ea1
Thus, one can compute by computing B, solving , and then summing. This approach has been taken by other authors (e.g., Xu and Randall 2001; Krueger et al. 1995a). Though the two approaches must yield the same result, they lend themselves to different interpretations. The expression says that Archimedean buoyancy drives buoyant accelerations and gives the environmental response. To analogously interpret as computed via (10), we proceed as follows. We define a “buoyancy pressure” (not to be confused with the buoyancy perturbation pressure ) as the field resulting from zeroing out the wind fields (i.e., ). It follows from (8) that
ea2
and so arises to enforce mass continuity in the face of horizontal hydrostatic pressure forces . Furthermore, (19) implies , and hence the buoyant acceleration can be seen as the vertical acceleration needed to compensate for horizontal motions driven by the hydrostatic pressure gradient, as in the stack (or “chimney”) effect.

We thus have two ways of thinking about the buoyant force. One significant disadvantage of the approach is that it treats B as primary, even though B suffers significant arbitrariness because of its dependence on an arbitrary reference state [as pointed out in section 2b and emphasized by Doswell and Markowski (2004)]. The approach does not suffer this ambiguity and also lends itself to a straightforward derivation of (17).

APPENDIX B

Defining, Interpreting, and Calculating the Inertial Pressure

a. Defining and interpreting

We mentioned in section 2a that can be viewed as the z component of the gradient of an associated pressure, the inertial pressure , defined here as
eb1
Applying this definition to (6) and taking its divergence yields the following Poisson equation for :
eb2
Equation (7) yields Neumann boundary conditions (BCs) at model top and bottom. Applying to both sides of this equation, comparing with (11), and noting that Neumann BCs for imply Dirichlet BCs for show that indeed .

Equation (B2) can be interpreted as enforcing the cancellation of the tendency of mass divergence generated by the inertial pressure with that generated by advection, in order to maintain anelastic continuity. In other words, the field produces whatever force is needed to ensure (anelastic) mass continuity in the face of the divergent tendencies generated by fluid inertia.

To gain further intuition for , we refer to Bradshaw and Koh (1981), who showed that the source term , which we define to be the right-hand side of (B2), can be written (neglecting variations) as
eb3
where e is the strain tensor with components , is the vorticity vector with components , and the norm squared of a vector or matrix denotes the sum of the squares of the components. Thus, strain is a source of positive pressure and vorticity a source of negative pressure.
We can understand this as follows. First consider a 2D velocity field , vanishing at infinity, that, to first order around the origin, is given by the solid-body rotation field (Fig. B1a). If is allowed to evolve solely under its own inertia, then, near the origin,
eq4
In other words, will develop a component pointing radially outward from the origin (Fig. B1b) as a consequence of the familiar centrifugal “force.” This advective tendency will cause a divergence of mass, and since it is the job of to generate a convergence of mass to counteract this, must have a low at the origin. Thus, vorticity must be a source of negative .
Fig. B1.
Fig. B1.

Idealized vector fields and their advective tendencies. (a) A pure rotational flow and (b) its divergent advective tendency. (c) A pure (irrotational) strain flow and (d) its convergent advective tendency. The inertial pressure balances these tendencies by generating a low (high) at the origin in the case of pure rotation (strain).

Citation: Journal of the Atmospheric Sciences 72, 8; 10.1175/JAS-D-14-0349.1

Next, consider another field that vanishes at infinity but is given to first order at the origin by the irrotational field (Fig. B1c). This field converges along the x axis and diverges along the y axis and thus has nonzero strain at the origin. If is allowed to evolve solely under its own inertia, then, near the origin,
eq5
In other words, will develop a component pointing radially inward toward the origin (Fig. B1d). This advective tendency will cause a convergence of mass, and since it is the job of to generate a divergence of mass to counteract this, must have a high at the origin. Thus, strain must be a source of positive .

b. Calculating

When calculating , it is computationally expedient to rewrite yet again to obtain the Poisson equation
eb4
[The z derivative of this is just the right-hand side of (14).] We solve (B4) by first Fourier transforming from to in the horizontal, periodic dimensions, which yields a set of algebraic equations for each k which are coupled only in z. This system can be written in terms of a k-dependent tridiagonal matrix, which is (in general) easily inverted, whereupon we Fourier transform back and are done.
One issue that deserves further comment is that of the boundary conditions at model top () and model bottom (). As noted above, obeys the Neumann BCs
eb5
The rub is that these double Neumann BCs do not uniquely specify but only determine it up to a constant; thus, these BCs are degenerate. This manifests computationally in a noninvertible tridiagonal matrix for —that is, the constant Fourier component (denoting horizontal averages with an overbar). We can thus keep our Neumann BCs for as well as for at model bottom and then specify our constant and remove the degeneracy by imposing the Dirichlet BC at model top. This, however, seems potentially inconsistent with (B5), which implies as well. Fortunately, the Poisson equation carries an integral constraint that saves the day. Integrating (B2) over the entire domain, noting that the boundary of the domain consists of just the model top and bottom, and applying the divergence theorem yields
eb6
The fact that at and implies that the right-hand side of (B6) is zero. At the same time, one can recognize the left-hand side as times a constant. This yields the constraint
eb7
Thus, the Neumann BC at model bottom plus the constraint (B6) implies that the Neumann BC holds at model top as well, in addition to our Dirichlet BC there.

As an aside, we should note here that numerically, the summed source term on the right-hand side of (B4) must be quite carefully computed in order for the constraint (B7) to be obeyed. In particular, on an Arakawa C grid (Arakawa and Lamb 1977) the terms with different live at different points of the grid cell, but for a given each factor and lives on the same point of the grid cell. To compute , one must interpolate each term to a common point before summing, but it is imperative to only perform this interpolation after multiplying and together. We found that interpolating before multiplying yielded a field that did not obey (B7).

Returning to analytics, we observe that one can, in fact, go beyond the constraint in (B7) and obtain an explicit expression for . We begin with the anelastic equation of motion
eb8
and take a horizontal average over our domain with area A. We evaluate by applying a horizontal average to (10), which yields . This, along with the BCs , implies . Meanwhile, mass continuity implies . Applying these results to the horizontal average of (B8) yields
eq6
Integrating down from model top then shows that
eb9
We can interpret this equation as follows. The quantity is simply the domain-averaged flux of vertical momentum . The convergence of this flux is a force—namely, the rate at which inertial motions generate . Continuity dictates that , however, and so (B9) just says that provides the force necessary to ensure this, in consonance with our discussion in the last section.

As a final note, (B9) and the Dirichlet BCs on w show that obeys both Neumann and Dirichlet BCs, and so either (or a mix) may be used in practice. Also, the relation in (B9) serves as a useful diagnostic constraint against which one may check their calculation of .

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1

More specifically, if the initial slug has a radius , then is roughly constant until the gust front reaches , at which point the cold pools in our simulation have transitioned to warm pools.

2

As computed via , or by adding in the buoyancy perturbation pressure force , as per (A1) below.

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