## 1. Introduction

A holepunch cloud is the picture-worthy phenomenon of a near-circular patch of clear air puncturing a thin supercooled cloud layer (Schumacher 1940). It is usually accompanied by a fallstreak of ice crystals associated with the center of the clearing, so it is sometimes called a fallstreak hole. Most often initiated by the penetration of a cloud layer by either ascending or descending aircraft, these holes can grow to several kilometers in radius over time periods exceeding an hour. A linear version, called a canal cloud (Fig. 1), can also result from a level flight track within a cloud layer (Poulter 1948; Simon 1966). Long an atmospheric fascination for scientists (Hobbs 1985; Pedgley 2008) and weather enthusiasts alike [eliciting editorial responses from Ludlam (1956) and Scorer (1964)], recent advances have provided new quantitative insights into the holepunch phenomenon. In 2010, two instrumented observations of holepunch cloud events were reported (Westbrook and Davies 2010; Heymsfield et al. 2010) that reveal structural details, such as lidar retrievals of vertical velocities and liquid water content. Subsequently, a full-microphysics simulation based upon these observations has since shown that ice crystal growth can initiate both the dissipation of cloud and the flow dynamics leading to the subsequent expansion of a circular hole (Heymsfield et al. 2011). This article presents a fluid mechanical explanation for the growth of the hole within the cloud layer. In particular, we identify the “wave perturbation propagating outward from a central initiation area,” as observed by Johnson and Holle (1969), to be a gravity wave whose leading-edge front of downward motion evaporates the cloud and enlarges the hole.

### a. Evolution of the holepunch

The numerical simulation of Heymsfield et al. (2011) provides a clear narrative for the holepunch cloud development (observed values upon which this case is based are shown in parentheses). The background environment is a thin (150 m) supercooled cloud layer (−30°C at 7.8-km altitude) bounded above and below by stable, dry air. The dynamics are initiated by a localized (250-m wide) introduction of ice crystal particles throughout the depth of the moist layer. The microphysical response begins with rapid depletion of the liquid cloud by the Wegener–Bergeron–Findeisen (WBF) process, with a net release of the latent heat of fusion, and continuation of ice growth by vapor deposition, with its latent heat of sublimation. The latent heating results in upward motion with compensating downdrafts peripheral to the central ice particle region. As simulated in (Heymsfield et al. 2011), these downdrafts accelerate evaporation of liquid water, and the hole rapidly widens over time (4.4-km diameter after 90 min). While the simulation was clear on this sequencing of events for the formation of the hole, a definitive causal mechanism for the sustained expansion of the hole was not identified. Specifically, factors that determine the speed of the cloud edge remained unaddressed.

Candidate explanations for the growth of the holes are discussed in both Pedgley (2008), Heymsfield et al. (2010), as well as in follow-up comments to the latter (Hindman 2013; Heymsfield 2013). The first category of these involves WBF activity at the cloud edge, where the radial dispersion of ice particles could be driven by turbulent diffusion, weak convection, or wake turbulence. The second category is related to a global circulation (on the scale of the hole) whereby continuous latent heating within clear air (by WBF vapor deposition) strengthens the central updraft, thus inducing subsidence with an increasing outward extent. Both lines of thought require persistent ice microphysical activity, either local to the cloud edge, or at the core. In this article, we present the distinct mechanism of a free-propagating gravity wave that requires no direct microphysical forcing to sustain its motion.

### b. An explanation for the growth of the holepunch

Our theory for the holepunch growth begins by understanding the particular idealization to a marginal cloud layer whose liquid water content is sufficiently low that its dissipation can be captured by small displacement, linear dynamics. Only a two-dimensional cross-sectional version of the holepunch is considered—this corresponds to the canal cloud geometry (Fig. 1). The theory is developed through a sequence of three idealizations for the holepunch dynamics: a full-microphysics simulation of an artificially weak cloud layer, a two-dimensional fluid model with a buoyancy response that switches for saturated and unsaturated moist air, and a further reduced one-dimensional analysis for the outward propagation speed of the cloud edge.

The presentation starts with a summary of the background environment for a control run showing the holepunch dynamics. It is demonstrated that, within simulations that reproduce the holepunch, the WBF and ice processes that initiate the hole are not the primary driver for its continued expansion. Then, it is shown that the behavior of the holepunch cloud is essentially unaffected by a weakening of the liquid water content of the cloud. From these simulations, we identify the key gravity wave feature to be a warm anomaly within the clear air that is propagated by vertical motions that warm on the outward side and cool on the inward side. The downward motion on the leading edge also evaporates cloud water and expands the hole. These results motivate the development of a two-dimension fluid dynamical model that simplifies the moisture physics: cloudy air is moist neutral, and clear air is weakly, but stably, stratified. For initial conditions and latent heat forcing consistent with the WBF process, this fluid model reproduces a gravity wave front whose cloud-edge dynamics compares well against the full-microphysics simulation. The frontal nature of the wave is a consequence of the fact that cloudy air that is moist neutral does not permit a buoyancy response that supports gravity waves—thus the cloud edge represents an impediment to propagation. Finally, to address the question of what happens when the outward-propagating wave impinges upon this moist neutral barrier, a type of immersed boundary approximation is applied to reduce the fluid model to a one-dimensional analysis of the gravity wave front speed. It is shown that the cloud-edge velocity satisfies a Rankine–Hugoniot condition that quantifies the slowing of the gravity wave front by the neutrality of the cloudy air.

## 2. Simulations of holepunch clouds

### a. Two-dimensional, full-physics experiments

A baseline for understanding the holepunch dynamics is established through a two-dimensional adaptation of the Weather Research and Forecasting (WRF) Model simulations of Heymsfield et al. (2011). Results from our experiments are described here, with details of the model setup and experimental design deferred to the appendix. The initial thermodynamic profiles (Fig. 2) are based upon sonde observations from the original holepunch simulation, but with a refinement that imposes exact moist neutral and moist adiabatic conditions consistent with the WRF thermodynamics (Miglietta and Rotunno 2005). This consistency was found to be important for limiting convective instability leading to dissipation of the cloud layer. In addition, background wind shear is removed so that the flow pattern is driven solely by the holepunch initiation.

The control (CTL) run (Fig. 3) produces the expected holepunch in a 250-m-deep moist layer whose cross-sectional features evolve analogously to the three-dimensional simulations of Heymsfield et al. (2011). Rapid growth of ice by vapor deposition in the supersaturated environment leads to an immediate loss of liquid cloud water by the WBF mechanism. Continued latent heating drives a warm anomaly in the hole center, which in turn drives upward air motion (Fig. 3a). This sets up a circulation in which compensatory downward motion at the hole edges erodes the cloud, and horizontal spreading of the warm anomaly then leads to its eventual splitting (Fig. 3b). This outward dynamics continues and results in a lateral growth of the hole at a near-constant rate of 0.6 ± 0.1 m s^{−1} (Fig. 3c). Conversion of cloud ice to snow produces precipitation beneath the hole (not shown, but is like the fallstreak in Fig. 1) that is consistent with previous observations of holepunch clouds (Westbrook and Davies 2010; Heymsfield et al. 2010). Evolutions for longer time (as illustrated by the LIM run in section 5) indicate a weakening of the central updraft that runs counter to an enlarging global circulation as a simple explanation for the growth of the hole. Moreover, the phasing of the warm anomalies with downward motion on the outward side does support the idea of a gravity wave feature driving the cloud-edge motion.

Next, we eliminate the other ice physics-driven hypothesis (direct WBF erosion) for the cloud-edge motion. The second experiment (LIM) is identical to the CTL run in initialization but limits the WBF microphysics to a center zone (−250 < *x* < 250 m, green shaded) containing the region of initially injected ice. Figure 4a shows this LIM run to be virtually identical at 15 min with the CTL evolution (Fig. 3c)—except for slightly weaker warm anomalies. The similarity of the LIM holepunch beyond the WBF zone to the CTL runs clearly demonstrates that the movement of the cloud edge is not a direct result of local microphysics but again implicates a more dynamical explanation.

### b. Idealized experiments

To investigate the gravity wave hypothesis for the hole growth, we introduce two simulations with more idealized background profiles. Both the adiabatic cloud (ADC) and uniform cloud (UNC) runs involve moist neutral layers embedded within a more symmetric (relatively) dry environment. Specifically, the air immediately above and below the moist layer is given constant static stability

However, it is a property of the adiabatic cloud profile that the lifting condensation level for any cloud parcel is the initial cloud bottom. So, clear air appearing within the moist-layer region resulting from resolved-scale vertical motion (rather than microphysics or numerical mixing) must be a (relatively dry) parcel originating from outside the moist layer. The expansion of the holepunch in an ADC layer therefore implies vertical displacements that scale with the moist-layer thickness itself.

This motivates the consideration of an alternative cloud profile: one with uniform liquid water. As discussed in the appendix, such a layer has displacements from the lifting condensation level that are controlled by the amount of liquid water in the cloud [Eq. (A1)]. In addition, this uniform liquid water cloud (UNC) has a moist layer whose vertically integrated liquid water is reduced to one-quarter of the ADC case. We estimate that for most of the UNC initial cloud (Fig. 2), evaporation of all the liquid water in any parcel requires, at most, a downward displacement of roughly 25 m. This is considerably less than the 250-m total depth of the moist layer and, thus, embodies the marginal cloud assumption, whose theory is presented in the next section. The simulated holepunch for the UNC case is shown in Fig. 5 and, despite that the dynamics involved are weaker relative to CTL and ADC (note the differing wind arrow scales), there follows a qualitatively similar narrative for the growth of the hole. The times are chosen to reflect development stages that roughly parallel those of CTL (Fig. 3). Note that the gray shading of the liquid water mixing ratio *t* = 17 min and beyond).

In all of the simulations, the outward expansion of the cloud edge is trailed by a localized warm anomaly with a circulation whose phase strongly suggests a propagating gravity wave (Cushman-Roisin 1994). However, a question arises regarding the interplay between this gravity wave feature and the moist neutral cloud, where the absence of buoyancy effects precludes wave propagation. This is further investigated through the development of theoretical models presented in the next two sections.

## 3. Fluid dynamics for a moist neutral layer

### a. Equations for gravity wave motions

*x*and

*z*levels above and below—all sufficiently remote to minimize spurious wave effects. It then remains to present the fourth and remaining equation for

### b. Background profiles for dry and moist air

### c. The dimensionless model

The dimensions of the independent variables (*x*, *z*, and *t*) are based on a horizontal scale *L* and the dry Brunt–Väisälä time scale *u*, *w*) scale on *η*) on *ψ* on *b* scales on

### d. Initiation of the holepunch

Without the explicit modeling of ice, the WBF initialization process cannot be represented directly within this fluid mechanical framework. Nonetheless, in this section we design an initial condensation level

The microphysical response to the introduction of ice particles has two phases. It starts with the rapid depletion of cloud liquid (with its net release of latent heat of fusion) by the WBF process and follows with the deposition of vapor to ice (with its release of latent heat of sublimation). The simulations of section 2 confirm this two-stage sequence: the WBF liquid-to-ice conversion completes within the first 2 min, while much of the vapor-to-ice conversion occurs over the next 4 min. With our fluid model, we address the liquid conversion by the initial condition and the vapor conversion by the prescribed time-dependent buoyancy forcing [Eq. (5)].

*x*defines a localized disturbance that is continuous at the hole edges. The variance parameter

### e. Gravity wave front

The numerical solutions to the nondimensionalized equations of section 3a [Eq.s. (2), (3), and (5)] are found using the method described in Muraki and Rotunno (2013, their section 3), but in this application all physical-diffusion and time-filter coefficients are set to zero. The dimensionless domain for the numerical solution is

Despite the simplified representations [Eqs. (8) and (9)] for the WBF microphysics, Fig. 6 shows a holepunch dynamics that reflects well the key features of the UNC simulated sequence. The earliest time^{1} displayed (

## 4. A theory for the velocity of the cloud edge

Simulation using the fluid model of section 3 suggests that the expansion of the holepunch occurs by an outgoing gravity wave—propagating warm anomalies whose leading-edge front of downward motion clears the cloud. A similar gravity wave structure associated with a moving cloud-edge boundary arose in Muraki and Rotunno (2013) within the context of moist neutral, saturated airflow past a topographic ridge. In that situation, the appearance of clear air by sinking motion in the lee of the ridge generated both upstream- and downstream-propagating gravity waves. Specifically, the upstream-propagating feature was a front of subsidence which desaturated the incoming airstream. The influence of the moist neutral cloud was in determining the velocity of the (upstream) disturbance. It was slower than the natural gravity wave speed and was determined by jump conditions local to the front. We demonstrate here that, within the buoyancy model of the previous section, the outward propagation of the holepunch cloud edge is similarly controlled.

### a. Rankine–Hugoniot formulas for the cloud-edge velocity

*z*in the moist layer, cloud edges are moving points

*x*derivative of the vorticity equation [Eq. (11)] following the expectation that

^{2}Here, we will use confirmation of this R–H speed condition [Eq. (12)] as a demonstration that the cloud edge is a propagating front.

Previous analysis of conservation laws in Muraki and Rotunno (2013) was simplified by the additional mathematical property that the system of equations was also hyperbolic. This permitted the use of the method of characteristics and the theory of shocks (Lax 1973). Here, however, the streamfunction inversion [Eq. (3)], by virtue of its elliptic PDE nature, breaks this hyperbolic property. It is usually the case that this loss of hyperbolicity also leads to the nonexistence (by smoothing or dispersion) of propagating discontinuities; but with discontinuous behavior built into the buoyancy equation [Eq. (5)] via the stratification [Eq. (6)], the cloud-edge fronts within our fluid model are a novel case of a free boundary (Ockendon 2003).

The computational confirmation of the R–H condition [Eq. (12)] raises two challenges: achieving sufficient numerical accuracy at the discontinuities and capturing the two-dimensional front structure. As a first approximation, we choose to bypass these complications by a reduction of our fluid model to a thin-layer analog that generates a one-dimensional front structure.

### b. An immersed layer approach

The approximation strategy represents the moist region by a thin layer whose properties derive only from the values at level

### c. The IL holepunch computation

*x*derivatives. A key advantage of the IL simplification, where the moist switch [Eq. (5)] is only applied at the cloud level

*z*derivative and lends additional robustness to the spectral computation. For spectral accuracy, the discontinuous buoyancy derivative [Eq. (18)] is calculated with the switch [Eq. (19)] applied after pseudospectral differentiation.

### d. The IL holepunch dynamics

Buoyancy and flow velocity from computing the IL approximation are shown as Fig. 7 for *t* = 5, 10, and 15. The physical parameters used are identical to those of the 2D simulation of Fig. 6. The spectral collocation gives a resolution of 1/512 in both *x* and *z* directions, with adaptive time stepping. Intervals of saturated moist buoyancy (cloud) are indicated by a thick dark line along

*t*derivative of the zero cloud water condition [Eq. (7)] at

^{−1}, which compares very well with the two-dimensional fluid model. The slight oscillations in the front speed are attributed to waves propagating within the clear-air layer. The upshot of this 1D analysis is that the cloud edge propagates as a frontal discontinuity, and its propagation speed is determined by local jump conditions.

## 5. In closing

The analysis presented here determines that, in the case of a marginal cloud, linear wave theory identifies a gravity wave front of subsidence as the mechanism for the sustained expansion of a holepunch cloud. However, the full-physics WRF simulations show that the holepunch is possible over a range of cloud profiles encompassing small-displacement (UNC) to finite-displacement (ADC) dynamics. To illustrate the importance of the frontal nature of the disturbance to the opening of the holepunch even in the case of an adiabatic cloud, we contrast with a final WRF run in which the front cannot form. The WAV run is identical to the LIM run except that the latent heating of non-WBF cloud evaporation is suppressed (Fig. 9), so the entire WAV moist layer acts as a uniform, weakly stratified fluid for both clear and cloudy air. As a result, the WAV run produces the expected wave train of outward-propagating warm cells (Fig. 9b). The outcome of this difference is apparent in the detailed cloud structure where the coherent front of subsidence (LIM) produces more clearing by permanent downward displacement than a laterally dispersing wave train (WAV). This supports the idea that the frontal mechanism identified within the specialized marginal cloud case is a general dynamical feature of the holepunch.

In this study, an alternative explanation for the continued growth of the holepunch by a gravity wave front is presented. The mechanism identified here is different from existing hypotheses that rely upon continuous ice processes and is encapsulated in the cartoon in Fig. 10. The initial ice injection creates a localized updraft, but the outward propagation of the warm anomalies proceeds without the requirement of direct forcing by further WBF processes. An important consequence of the slowing of the front is a persistent subsidence that enhances the clearing of the cloud.

## Acknowledgments

Support for DJM provided by NSERC RGPIN-238928. DJM also thanks the MMM Division of NCAR for their intellectual hospitality during the course of this work.

## APPENDIX

### WRF Model Description and Experimental Design

This study employs the Weather Research and Forecasting (WRF) Model, version 3.5.1 (Skamarock et al. 2008). WRF is a nonhydrostatic, compressible atmospheric model. The governing equations are solved using a time-split integration with third-order Runge–Kutta scheme. Horizontal and vertical advection are calculated using fifth- and third-order discretization schemes, respectively, with modifications to ensure monotonicity (Wang et al. 2009). The model setup is two dimensional with periodic lateral boundaries and a domain 30 km wide and 20 km deep. The upper boundary is a rigid lid with a Rayleigh damper and damping coefficient of 0.003 s^{−1} applied to the top 5 km. The horizontal grid spacing is 50 m and vertical grid spacing approximately 50 m, with a slight stretching from model bottom to top. Diffusion is implicit through the use of odd-order advection schemes. The model time step is 1 s, with substepping applied for acoustic modes. Microphysical processes are treated using the two-moment bulk scheme of Morrison et al. (2005, 2009). Other physical processes such as radiation are neglected for simplicity.

A set of simulations with differing initial conditions was performed to explore the behavior of the holepunch dynamics simulated by WRF—an overview is given in Table A1. These experiments were only slightly modified from those used for the simulations of Heymsfield et al. (2011), and the initial thermodynamic profiles used are shown in Fig. 2. The difference in the cloud layers is an initialization that imposes moist neutrality in a manner consistent with the WRF thermodynamics [Heymsfield et al. (2011) assumed only quasi–moist neutral conditions]. The condition for a zero Brunt–Väisälä frequency (Emanuel 1994), along with the WRF equations for the first law of thermodynamics and hydrostatic balance, were discretized and integrated upward from cloud base [Miglietta and Rotunno (2005), see their section 2]. At each vertical level of the discretized equations, the moist thermodynamics and hydrostatic equations were solved by iteration to produce consistent values of potential temperature, saturation vapor pressure, and air density—ensuring initial

Summary of the WRF simulations.

Ice initiation of the holepunch was done by injecting a concentration of ice crystals 10 s into the simulation over a region five grid points (250 m) wide throughout the depth of the cloud layer. The number mixing ratio of injected particles was 500 kg^{−1} ^{−1}). Note that in all simulations the only conversion process from liquid water to ice is the WBF mechanism; riming is neglected for simplicity.

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