## 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.

*ρ*is the system density (including the weight of hydrometeors). Up to a factor of

We will show in the next section that

A central feature of the definition (1) is that *B* as well as the environmental response to the accelerations produced by *B*. We will see that *B*, to the degree that *B* cannot always be considered a first approximation for *B*, as advocated by Doswell and Markowski (2004).

We begin by using the definitions (1) and (2) to derive diagnostic Poisson equations for

## 2. Buoyant and inertial accelerations

### a. Vertical force decomposition

*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

*local*hydrostatic pressure field and

*z*component of (6) is simply

*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

Note that the determination of

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

### b. Contrasting Archimedean and effective buoyancies

Before describing our experiments and their results, let us get a feel for how *ρ*, so that buoyant accelerations tend to be strongest at local extrema of density (or, more generally, regions of *B*). This means that

Since (10) is a Poisson equation, effective buoyancy is nonlocal: that is, localized extrema of density give rise to accelerations everywhere, even where *B*. For isolated regions of significant buoyancy, we thus expect that

*x*–

*z*cross sections of

*B*and

*n*= 2 and 4. Here,

*B*with respect to the horizontal average of

*ρ*rather than

*B*(Doswell and Markowski 2004).

Perhaps the most striking feature (for both values of *n*) of Fig. 1 is the degree to which

Finally, for *B* field, where the maximum is found at *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 *B*; we will see even more dramatic examples of this in the next section.

## 3. RCE simulations

With a preliminary understanding of

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

For a first diagnosis of *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

## 4. RCE results

Plan views of the vertical velocity *w* at *B*, *B* field, and incipient convection at the cold-pool gust fronts is evident in the *w* field. Comparison of *B*, again requiring a severely stretched color bar; in this circumstance, *B* is not even a first approximation for *B* is a result of the extreme aspect ratio of the cold pools as well as their proximity to the ground, where an *B* can be used as a proxy for

Plot of *B*, *w*, *B* field and that the vertical acceleration there appears strongly dominated by *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

Plot of *B*, *w*, *B* field and that the vertical acceleration there appears strongly dominated by *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

Plot of *B*, *w*, *B* field and that the vertical acceleration there appears strongly dominated by *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 *y*–*z* 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 *w* > 1 m s^{−1}) just above at around *B*, *θ*, *q*_{υ}, and *θ*_{e} fields at (*y*, *z*) ≈ (38 km, 150 m), and the gust front and plume indeed exhibit anomalously high

Plot of *υ*, *w*, *B*, *θ*, *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*, *q*_{υ}, and *B* but is nonetheless triggered primarily by *B* fields.

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

Plot of *υ*, *w*, *B*, *θ*, *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*, *q*_{υ}, and *B* but is nonetheless triggered primarily by *B* fields.

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

Plot of *υ*, *w*, *B*, *θ*, *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*, *q*_{υ}, and *B* but is nonetheless triggered primarily by *B* fields.

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

To quantitatively test the hypothesis that the *w* of the nascent plume at ^{−1}, is equal to *h* is the height of the *w*.

*δx*,

*δy*, and

*z*, against

*C*

_{b}and

*C*

_{i}are negligible and ignored henceforth. We use

*w*because (by the work-energy theorem) a linear relationship with the forces is expected only for

*z*are shown in Fig. 4. [The units and order of magnitude of the coefficients are given by

Regression coefficients *r*_{i} and *r*_{b} (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

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

Regression coefficients *r*_{i} and *r*_{b} (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

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

Regression coefficients *r*_{i} and *r*_{b} (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

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

*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

*i*= 1 or 2 in (15) yields a factor of

*j*. Also,

*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

*H*.

^{1}Here

*H*is a characteristic height for the negatively buoyant downdraft that spawned the cold pool, and

*H*to

*h*as a determining factor in the dominance of

Cartoon depicting the dominance of *H* to *h*, as in (17).

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

Cartoon depicting the dominance of *H* to *h*, as in (17).

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

Cartoon depicting the dominance 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

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 *q*_{υ} 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 *q*_{υ} profile via relative humidity (RH) by interpolating

We run this simulation for 2 days, saving 3-hourly snapshots. For each snapshot, we calculate

Time series of *z* but with

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

Time series of *z* but with

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

Time series of *z* but with

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

Finally, our result that the inertial acceleration

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

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

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

*B*, solving

We thus have two ways of thinking about the buoyant force. One significant disadvantage of the *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

## APPENDIX B

### Defining, Interpreting, and Calculating the Inertial Pressure

#### a. Defining and interpreting

*z*component of the gradient of an associated pressure, the inertial pressure

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

**e**is the strain tensor with components

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

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

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

*x*axis and diverges along the

*y*axis and thus has nonzero strain at the origin. If

#### b. Calculating

*z*derivative of this is just the right-hand side of (14).] We solve (B4) by first Fourier transforming from

**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.

As an aside, we should note here that numerically, the summed source term

*A*. We evaluate

As a final note, (B9) and the Dirichlet BCs on *w* show that

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