We analyze frontal dynamics of dilute powder snow avalanches sustained by rapid blow-out behind the front. Such material injection arises as a weakly cohesive snow cover is fluidized by the very pore pressure gradient that the particle cloud induces within the snowpack. We model cloud fluid mechanics as a potential flow consisting of a traveling source of denser fluid thrust into a uniform airflow. Stability analysis of a mass balance involving snow cover and powder cloud yields relations among scouring depth, frontal height, speed, mixed-mean density, and impact pressure when the frontal region achieves a stable growth rate. We compare predictions with field measurements, show that powder clouds cannot reach steady frontal speed on a uniform snowpack with constant cloud width and derive a criterion for cloud ignition. Because static pressure is continuous across the mean air-cloud interface and deviatoric stresses are negligible, frontal acceleration is insensitive to local slope, but instead arises from a deficit of flow-induced suction in the wake. We calculate how far a powder cloud travels until its frontal mixed-mean density becomes stable, and show how topographic spread can hasten its collapse.
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 Gravity currents are common in geophysics [Simpson, 1997]. They include lahars [Naranjo et al., 1986], snow avalanches [Hopfinger, 1983], pyroclastic clouds [Cole et al.1998], lava flows [Huppert, 1982], deep sea turbidity currents [Heezen and Ewing, 1952], and haboob sand storms [Lawson, 1971]. A crucial distinction is whether the density difference between particle-laden “cloud” and ambient fluid is sustained by material scoured from the base, or whether such material is fed from the tail. Most studies of gravity currents focus on the latter [Simpson and Britter, 1978; Britter and Linden, 1980], for example by analyzing the canonical lock exchange problem [Shin et al., 2004; Birman et al., 2007], in which a denser fluid is released upstream into a lighter one at rest.
 In contrast, powder snow avalanches absorb massive amounts of material at the front [Louge et al., 2011], producing frontal speeds up to U=60 m/s and heights up to H′=50 m [Turnbull and McElwaine, 2007]. Although direct density measurements have yet to be published, observations suggest that the fast-moving frontal region of a powder avalanche is stratified, with a bottom layer of estimated density ∼30 kg/m3 and a more dilute suspension above with ∼3 kg/m3[Issler, 2003]. Both regions are lighter than the density ρc∼100–300 kg/m3 of the snowpack that feeds them. The relatively dilute, short frontal region may also be followed by a denser slide traveling well behind [Issler, 2003; Sovilla et al., 2008a]. As recent frequency-modulated continuous-wave (FMCW) radar measurements showed [Sovilla et al., 2006, 2010], powder clouds over weakly cohesive snowpacks are sustained by intense particle eruption confined to a narrow region just behind the front. Such material input contrasts with more gradual entrainment rate from surface shear [Carroll et al., 2012], and it greatly exceeds the particle sedimentation rate that occurs over much longer time scales [Blanchette et al., 2005].
 Static pressure measurements carried out by McElwaine and Turnbull  and Turnbull and McElwaine  at the Vallée de la Sionne (Switzerland) confirmed that powder clouds behave as an “air wave” described by Grigorian et al. , who recorded velocity and pressure in avalanches near Mount Elbrus in the Caucasus. The air wave first raises static pressure as flow stagnates in the reference frame moving with the front. Then, a significant depression is observed shortly behind passage of the front [Nishimura et al., 1995; McElwaine and Turnbull, 2005; Turnbull and McElwaine, 2010]. Roche et al.  observed a similar depression in laboratory releases of gas-solid suspensions traveling over a horizontal surface.
 In this context, Louge et al.  suggested that powder snow avalanches behave as “eruption currents,” which grow continuously by extracting material from the underlying sediment in a synergistic process. In such currents, the very static pressure that the flow imposes at the surface creates pressure gradients fluidizing the porous sediment by defeating its modest cohesion. The fluidization acts as a narrow traveling source of material thrust into ambient fluid. Therefore, although eruption currents possess a frontal flow field resembling gravity currents [McElwaine, 2005; Ancey et al., 2007], their treatment must account for the source explicitly.
 In this paper, focusing exclusive attention on the short frontal region of the avalanche, our objective is to elucidate mechanisms governing frontal dynamics of dilute powder clouds, which expand by drawing material from a relatively light snow cover. We neither expect these mechanisms to prevail in denser slides, nor imply that a dilute cloud cannot co-exist with the latter. To keep its analysis in closed-form, our theory poses four simplifying assumptions:
 The frontal region is defined as the relatively short section of the cloud where its interface with ambient air has favorable pressure gradient.
 Recognizing that powder clouds possess very large Reynolds number , we model the frontal region as an inviscid Eulerian fluid without deviatoric stresses. Thus, inspired by the success of McElwaine and Turnbull  and Turnbull and McElwaine , we follow Carroll et al.  in treating this region in the rest frame of the moving avalanche as a Rankine half-body (RHB) [Rankine, 1864, 1871], consisting of a moving point source of denser fluid injected into a uniform air stream.
 Using large-eddy numerical simulations (LES) and water flume experiments, Carroll et al. showed that little ambient fluid is entrained into the frontal region through the mean interface (albeit much more into the tail). Therefore, our frontal analysis ignores air entrainment.
 Like Carroll et al. , we preserve the mathematical framework of potential flow theory by ignoring stratification or local density fluctuations. However, we account for the higher “mixed-mean” fluid density ρ′ of the frontal region using the well-stirred reactor ansatz of Chemical Engineering [Levenspiel, 1999]. Although density uniformity is our least realistic assumption, it allows closed-form predictions, from which insight on frontal dynamics may arise.
 We begin the paper by summarizing the results of [Carroll et al., 2012] for eruption currents. We then show how the interaction between porous snowpack and powder cloud regulates frontal mass balance and leads to a stable state relating mixed-mean density, speed, and height of the frontal region. After calculating forces exerted on this region, we analyze its peculiar dynamics, and deduce the role of topography on acceleration, densification, and collapse. We close by comparing predictions with available data. To limit the size of this article, we archive calculations as supporting information available online.
2 Eruption Current
 This section summarizes potential-flow results that Carroll et al.  derived for eruption currents, a class of gravity currents driven by narrow fluid injection shortly behind the front (Figure 1). Details are available in the supporting information (Appendices A: “Cloud velocity and static pressure fields” and B: “Size of the frontal region”) published online. Carroll et al.  showed that, below the mean interface dividing denser source and lighter ambient fluid, the greater density swells the RHB uniformly by a factor 1/δwith
where ζ represents source density ρ′ relative to ambient fluid ρ,
and Ri is a bulk Richardson number based on asymptotic RHB height h′ and speed U′,
In this paper, primes denote conditions within the cloud and g is gravitational acceleration. Carroll et al. also found that, in the rest frame of the source, the velocity field below the interface is uniformly reduced by δ. In particular, they calculated
such that far fluid below the interface moves away from the source at velocity , whereas ambient fluid does so at , where is the unit vector pointing in the direction of cloud motion and U is the absolute frontal speed. Finally, Carroll et al.  provided a more complicated expression for velocity above the interface, quoting it as an infinite series. They carried out these calculations on horizontal surfaces, for which symmetries yield results in closed form. In the supporting information, Appendix C (“Quasi-steady approximation”) justifies using the same approach on inclined slopes.
 Because the tail of powder snow avalanches incorporates much ambient air [Ancey, 2004], potential theory is invalid there, as it is in the wake of the cylinder in cross-flow [Prandtl, 1904]. However, for the frontal region, Carroll et al.  tested the potential flow theory with large-eddy numerical simulations (LES) and experiments in a water flume. As in powder snow avalanches, they observed instantaneous interface oscillations, which they attributed to the form of the velocity profile at the source. Despite these oscillations, they showed that the relative swelling of the mean interface conformed to equation (1). Inspired by Ancey  and Beghin et al. , Carroll et al.  exploited the LES to measure ambient fluid entrainment into the frontal region. Although, as Bartelt and Buser  and Louge et al.  discussed, a small rate of ambient air mass must cross into the frontal region to bring the mass of air separately into balance, Carroll et al.  showed that the source mass flow rate greatly exceeds , even with vigorous interface oscillations. Therefore, although air entrainment at the ambient interface contributes to dilution of the frontal region, and dominates volume growth in the tail, it hardly affects the frontal dynamics of powder snow clouds.
 In this relatively simple view, the frontal region of a powder snow avalanche consists of a dense particle-laden fluid emerging from a source traveling at the frontal speed U. The source interacts with ambient air behaving, in the rest frame of the source, as a uniform flow moving toward the source with speed −U far upstream. The resulting RHB then induces a synergistic static pressure field that regulates the source strength γ. Such convenient mathematical framework makes it possible to calculate the velocity and static pressure fields on and below the mean interface exactly. In turn, this yields an expression for static pressure at the surface of the snowpack, which is used as boundary condition to calculate pore pressure gradients within, and to deduce how deeply such gradients can fluidize the snow cover by defeating its cohesion and friction [Louge et al., 2011].
 All lengths in this potential flow scale with the distance b′=γ/(2πU′) between the stagnation point and the source (Figure 1). In particular, the asymptotic flow height is
Comparisons of RHB predictions with measurements at the Vallée de la Sionne [McElwaine and Turnbull, 2005; Louge et al., 2011; Carroll et al., 2012] indicate that static pressure returns to its upstream value for x≲−b′, so steep pressure gradients driving snow expulsion are mainly confined within −b′≲x<b′. Coincidentally, as shown in the supporting information (Appendix B), the plane at x≃−b′ roughly intersects the location x=xf<0 where the static pressure gradient along the interface becomes adverse. Therefore, it is convenient to define the frontal region as the domain below the interface that is bounded by the stagnation point and the exit plane at x=xf. In this definition, the frontal region lies within a distance roughly ±b′ on either side of the source, where surface pressure gradients are steepest.
 By analyzing the solid mechanics of a homogeneous, porous snowpack, Louge et al. showed that an avalanche can fluidize the latter if its cohesion is less than the difference between stagnation pressure at the front and the significant depression observed shortly behind it [McElwaine and Turnbull, 2005]. Postulating Mohr-Coulomb failure, they calculated the depth h′ of snowpack fluidization,
which Sovilla et al.  also called “plowing” depth. In this expression, the parameter
represents conflict between mechanical properties of the snowpack resisting fluidization and the strength of surface pressure gradients scaling as (1/2)ρ′U′2/b′. In equation (7), ρc is snowpack density and μe is an effective internal friction incorporating snowpack cohesion [Louge et al., 2011]. For convenience, Louge et al.  recast their infinite-series solution for h′/b′ versus as a simpler equation (6), from which they established the constants
Note that these exact constants would change if the snowpack no longer possessed a uniform permeability, for example, if it was substantially stratified. However, a similar framework could be used to calculate them numerically.
Louge et al.  also showed that snowpacks of excessive friction and/or density could not sustain fluidization right up to the free surface. Accordingly, they derived the necessary condition for sustainable fluidization of a homogeneous snow cover
where κ is an exact constant ≃47.9 or, using equation (7),
Inequality (11) sets upper limits for cohesion, density, and internal friction allowing an avalanche to blossom into a powder cloud of relatively small mixed-mean density.
 Whereas air entrainment driven by large-scale vortices at the air-cloud interface further dilutes the suspension and increases the apparent height of its tail, the frontal motion of powder snow avalanches is dominated instead by rapid material ejection, which is driven by static pressure gradients that are steepest in the narrow region from stagnation to source. Therefore, this paper focuses on mass and momentum balances on the frontal region. Doing so avoids complications arising in the tail, including expansion by air entrainment and sedimentation.
 This model of frontal dynamics departs significantly from earlier work. Previous theories of powder snow avalanches such as those of Rastello and Hopfinger  and Fukushima and Parker  were inspired by the analysis of a powder avalanche as a cloud [Kulikovskiy and Sveshnikova, 1977] and the turbidity current analysis of Parker et al. , who modeled erosion beyond a critical Shields number in a classical open-channel approach to bed load transport. Beghin and Olagne , Ancey , and Turnbull et al. closed their integral models more simply, assuming that the avalanche entrains the full depth of loose snowpack, but without modeling the erosion mechanism explicitly. Although our theory only applies to cases where frontal snow eruption dominates particle entrainment at the base, its framework makes it possible to link erosion rates and dynamics of the frontal region directly.
 The first step is to analyze how the eruption rate regulates the cloud velocity field and, conversely, how the resulting static pressure gradients control snowpack fluidization and therefore eruption rate. In the next section, we show that this feedback mechanism is unstable when the cloud ignites at low density and height, but that it can reach stable growth thereafter.
3 Regulation of Frontal Mass Balance
 We derive frontal stability by inspecting the mass balance between snow ejected at the source and cloud material leaving the exit plane at x=xf (Figure 1). Once the frontal region of a powder cloud establishes a steep pressure gradient between stagnation and source, equation (6) determines the depth h′ of snowpack that it fluidizes. In terms of Richardson number,
However, the avalanche-induced depression only draws part of the fluidized material into the cloud. Although the flow casting this material is not known, the failure surface on which snowpack fluidization first takes place is likely vertical, as it is where an imposed pore pressure gradient can most easily defeat gravity [Louge et al., 2011]. If so, defining λ as the fraction of the fluidized depth that is “scoured,” i.e., actually entrained into the cloud, the mass flow rate of material emerging from the source is, in the rest frame of the front barreling down the slope of inclination α at speed U,
where the term in parentheses represents the vertical surface through which material is ejected across the width W of the frontal region (Figure 1), and 0<λ<1. Ignoring sedimentation and interface mixing, the mass flow rate leaving the frontal region is, from equation (5),
 As shown in the supporting information (Appendix C), because the mean particle residence time through the frontal region is small relative to the characteristic time for cloud growth, we can assume that the mixed-mean density ρ′ is established instantaneously. Similarly, the control volume may be considered fixed to write the frontal mass balance. At such quasi steady-state, inlet and exit mass flow rates are equal, , or equivalently, using equations (1)–(4),
 Avalanche operating conditions (Ri, ζ, λ) are fixed by satisfying equations (13) and (16). However, these conditions are not always stable. As shown in the supporting information (Appendix D: “Stability”), one can invoke an argument that is analogous to mass balance stability of well-stirred reactors [Strehlow, 1984]. That analysis indicates that solutions can be plotted in a phase diagram of ζ versus Ri at different values of λ (Figure 2). The plane (Ri,ζ) is divided in four quadrants determined by a1 alone: at fixed ζ, solutions have stable Ri in the two quadrants where
similarly, at fixed Ri, solutions have a stable ζin the two quadrants where
In Figure 2, intersection of the four quadrants is a node henceforth denoted by the subscript n. The inequality a1<1/2 guarantees the existence of a quadrant of stable solutions.
 Once ignited, the frontal region begins with relatively low height and small mixed-mean density or, equivalently, at both ζand Ri low enough to start in the unstable lower-left quadrant of Figure 2. Because these conditions are unstable, frontal growth progressively raises Ri and densifies the frontal region to greater ζ. Once the frontal region achieves either Ri=Rin or ζ=ζn, it likely remains pegged at the value of the stable parameter while raising the other toward stability, eventually reaching the stable node at (Rin,ζn). At these first stable operating conditions, equation (1) implies that the factor δ is
while the scoured fraction of fluidized snowpack stays at
in which the constant χ0 combines snowpack properties, air density and inclination,
 Although nascent clouds follow an unknown path in the (Ri,ζ) diagram, they must accelerate at values of Ri that are consistent with equation (11). Therefore, if (Rin,ζn) represents the first stable state, it can only be reached while the snowpack surface is fluidized or, from equations (7)–(12), as long as
Any smaller κ0 would permit the frontal region of a nascent cloud to reach a stable ζ=ζn but, as its Richardson rises toward stability, it would force the frontal region to stroll into the excluded domain where Ri>κ0ζ/(1−ζ), at which point the snowpack surface would de-fluidize, thus quenching the avalanche. Condition ((21)) may therefore be interpreted as an “ignition limit” for powder clouds. In terms of snowpack density and effective friction, it is
For example, with ρ=1.2 kg/m3 and μe=0.45, this condition implies that snowpacks with ρc>400 kg/m3 cannot ignite into an eruption current.
 When the frontal region reaches Ri=Rin, it grows steadily while accelerating down slope, since its size and speed are linked. In that case, combining equations (2)–(4), (17), (18) and (19),
This result resembles what von Kármán  and Benjamin  obtained in gravity currents. Such quasi-steady growth may eventually be stalled by lack of a weakly cohesive snowpack of sufficient depth. Upon combining equations (5), (16)–(18), (20) and ((24)), the scouring depth of fluidized material actually entrained into the cloud is
Because in general λn≪1, the depth of fluidized snow cover is much larger than what the source rapidly ejects into the frontal region. However, the weakening that accompanies such fluidization can promote deeper mobilization of the snowpack behind the frontal region, and may cause additional material entrainment in a secondary slide.
4 Mixed-Mean Density
 Our principal, and perhaps weakest, simplifying assumption is to ignore density stratification in the frontal region. An analysis in closed-form is intractable without it. Although density must be larger near the source than higher in the cloud, vigorous mixing makes failure of this assumption less likely. As the next section will show, mean frontal dynamics are well captured by the theory, as they are in the analysis of well-stirred reactors, where the same mixture assumption is invoked [Levenspiel, 1999].
 In this section, we focus on predictions for mixed-mean density. The form of equation ((24)) suggests that the latter can be estimated from records of cloud height h′ and front speed U. Together, these measurable quantities form a Froude number
which, after combining with equations (2) and (3), relates Ri and ζ,
At the node (Rin,ζn), this Froude number is
Because ζ grows with Fr, a large Froude number implies a high ρ′. If Ri≫1 (for example, when Ri=Rin≃5.3), equation (27) reduces to ζ≃(1+Fr−2)−1 or, equivalently,
thus making it possible to estimate mixed-mean density from records of front speed and height in sufficiently tall powder clouds. If the first stable conditions at (Rin,ζn) persist, equations (2) and (18) imply that the mixed-mean density
(i.e., ) is lower in powder clouds than in dense slab avalanches.
 However, after the first stable state (Rin,ζn) is reached, it is yet possible for the frontal region to develop higher stable Ri or ζ in the upper right quadrant of the phase diagram (Figure 2). (As we will discuss in section 8, such brief incursion occurred in “avalanche 509”). Thus, it is possible that a powder avalanche, once ignited on a cover satisfying criterion ((23)), could acquire sufficient momentum to persist on a subsequently denser, more frictional and/or cohesive snowpack, as long as the snow cover is fluidized up to the free surface, as prescribed by equation (11). This may be what happened after avalanche 509 reached its stable state: as it rejoined a path likely hardened by a similar event two days earlier, it continued unabated. Another example may be the May 1983 powder avalanche in the Chamonix Valley, which featured a wet snowpack on part of its path [Ancey, 2006].
5 Frontal Dynamics
 In this section, inspired by the work of Ancey  and Turnbull et al. , we establish the dynamic equation for streamwise momentum of the frontal region in the inertial frame of the mountain, and we derive an approximate mass balance. To do so, we assume that the cloud of width W is engaged in two-dimensional flow with negligible transverse velocity. Because the frontal region is enclosed within an accelerating control volume V(t) of surface S(t) bounded by the snowpack, the interface and the exit plane at xf (Figure 1), the transport equation for -momentum is
The first term accounts for acceleration of the frontal region of mass relative to the inertial mountain frame. For consistency, the last term involves the velocity u′ relative to the accelerating coordinate system attached to the traveling source [White, 2003]. It represents a reaction thrust to net momentum lost through S, where is the projection on the outward unit vector normal to the boundary of the relative velocity urel between fluid and boundary. Because u′ and urel are invariant in a Galilean transformation, so is the thrust. The second term on the left is the rate of change of total momentum of the frontal region relative to the source. Finally, includes the gravity volume force on V and surface forces on S.
 An advantage of potential flow theory is that velocity and pressure fields can be calculated exactly. However, while this approach conserves mass, it is subject to d'Alembert's paradox, whereby the sum of all steady forces vanishes [d'Alembert, 1752]. Although sections 2-4, which are rooted upon mass balance, are unaffected by the paradox, an analysis of cloud frontal dynamics must first resolve it, since it clearly contradicts observations that the front accelerates.
 In the supporting information (Appendix E: “Force and momentum integrals”), we calculate unsteady terms and forces in equation (31) as predicted by the potential theory. Combining forces into a term proportional to volume (aVb′2W) of the frontal region, we write equation (31) as
where W is width of the frontal region, and the constants aV, aM and aμare calculated (see supporting information). For an object of uniform density ρ′, the dimensionless acceleration Γ∗ combines steady integrals for weight, hydrostatic buoyancy, pressure drag, and reaction to momentum output. Its introduction in equation ((32)) allows us to contrast the role of these forces in powder clouds to that in similar flows. In d'Alembert's paradox, Γ∗=0.
 To resolve the latter, it would be tempting to regard the frontal region as a solid of density ρ′ subject to gravity and buoyancy with resultant along the slope or, equivalently, Γ∗=(ρ′−ρ)/ρ′=ζ. However, the frontal region cannot sustain a static pressure jump across its outer surface. Instead, continuity p=p′ on the interface [von Kármán, 1940; Carroll et al., 2012] and a corollary of Gauss' divergence theorem imply that the net upward buoyancy force exerted by hydrostatic pressure pz on the outer surface S of the frontal region cancels exactly its downward weight , even if ρ′ varies. Therefore, if indeed the dilute, turbulent frontal region has negligible deviatoric stresses, its momentum balance only involves forces that are strictly related to the flow itself, but not to gravity or slope. These forces include the surface integral of flow-induced pressure pu (Appendix A of the supporting information) and reactions to the rate of momentum output (Appendix E of the supporting information).
 This observation has two principal consequences. First, it suggests that frontal acceleration of a powder cloud is much less than what a buoyant solid body of the same density would experience. As shown in the supporting information (Appendix G: “Longitudinal acceleration”), Γ∗ can be estimated from available data of frontal height, speed, width, and slope [Sovilla et al., 2006; Vallet et al., 2004]. Although evaluating Γ∗ from field records relies on differentiation, and therefore exhibits noise, the resulting values of Γ∗ are clearly <ζ, thus dismissing Fweight + buoyancy mentioned earlier. One might argue that such small frontal acceleration is due to shear stresses at the cloud base. However, it is shown in the supporting information (Appendix E) that turbulent shear forces at the base of the dilute cloud are several orders of magnitude smaller than streamwise cloud weight, and thus are unlikely to contribute significantly to the force balance.
 The second consequence is that frontal acceleration is independent of slope. Because static pressure on the cloud's boundary is continuous and basal shear stress is negligible, the frontal region experiences a net streamwise force that is entirely generated by the flow. To leading order, such net force is independent of sinα, just as air drag on a land vehicle is unrelated to road inclination. This surprising prediction, which might explain why powder clouds can survive for a time on moderate or even negative slopes, is borne by frontal speed data for inclined gravity currents of light particles or snow suspensions in air [Turnbull and McElwaine, 2008], and in a water tank [Britter and Linden, 1980].
 Observations in avalanche 509 [Sovilla et al., 2006] also confirm that slope is not a leading order effect in powder clouds. For this event, slope rose sharply from α≃27.3° to α≃37.9° at a distance x†≃950 m on the map, remaining steep for at least ±90 m around that point, while barely eliciting any discernible change in acceleration. Conversely, a sharp decrease in slope from α≃39.7° at x†≃370 m down to α≃31.2° at x†≃500 m was accompanied by a noticeable increase in speed. (Section 6 attributes this counterintuitive correlation to a narrowing width).
 However, note that flows with significant deviatoric stress, e.g., collision-driven fully developed granular slides, would not have a frontal speed independent of slope [Louge, 2003]. In such denser flows, hydrostatic streamwise pressure gradients are insignificant compared to shear stress gradients, and therefore no longer balance weight along . Thus, if an avalanche featured a dense region with significant deviatoric stress, its frontal speed should instead grow with slope.
 As shown in the supporting information (Appendix F: “Resolution of d'Alembert's paradox”), d'Alembert's paradox can be resolved by analogy with the cylinder in cross-flow. In this case, frontal acceleration is attributed to a deficit of suction at the exit plane. From this calculation, equation ((32)) becomes
where αeff≃33° is a constant effective inclination calculated in the supporting information (Appendix F). Eliminating ρ′ and U′ using equations (2) and (4), and invoking the quasi-steady approximation (b′≃constant), the equation governing frontal dynamics is
6 Channel Width
 The stability analysis in section 3 ignored variations in front speed, cloud density, cloud height, or channel width along the frontal region. If instead any of these quantities varied from source to exit plane, the mass balance would no longer reduce to , but instead becomes
In particular, if the powder cloud spreaded, its front would fluidize more material than is ejected at the exit plane. To capture such effect, we write that source and exit mass flow rates are related by a Taylor expansion of equation (15) truncated to first order along x,
Consistent with quasi-steady growth (see supporting information, Appendix C), we assume in equation (36) that b′ varies more slowly along x than other quantities. Then, substituting xf and aM from the supporting information [(equations (B2) and (E3), respectively] eliminating ρ′ and U′ using equations (2) and (4), and transforming the spatial derivative to a Lagrangian formulation using
To illustrate the role of channel width, we first consider a frontal region maintaining stable growth at (Rin,ζn) as it travels along a channel of variable width W. Substituting δn for δ and ζn for ζ, expanding derivatives in equation (34), and using equation (38), we obtain
To compare with frontal speed data reported in terms of projected distance x† on a map, we transform equation (39) to the Eulerian formulation with distance x along the avalanche path using equation (37), and find
We then project x† onto x by integrating topographical slope data α(x†),
invert the resulting function to convert known slope and width to α(x) and W(x), and substitute the latter in equation (41). Integration yields frontal velocity U(x), which we convert to U(x†) for comparison with data. We will present such comparison in section 8.
 Equation (39) suggests that width rate of change can be represented by a length scale
which has positive or negative values whether the avalanche spreads or narrows, respectively. As equation (38) shows, cloud spread causes acceleration to weaken, as extra momentum must be supplied to uptake new material into the frontal region. Conversely, any narrowing enhances acceleration. However, if a powder cloud grows at a fixed width, its front keeps accelerating at a constant rate aA/(1+δnaM/aV) g sinαeff≃0.074g. Therefore, an avalanching cloud on uniform snow cover cannot reach a steady frontal speed at constant width, accelerating instead ad infinitum. Conversely, we predict that, if mixed-mean cloud density were stable and snowpack properties remained invariant, a steady frontal speed could only occur if the cloud widened at a rate that balances the right side of equation (38).
7 Densification and Collapse
 In this section, we examine how far the nascent cloud must travel before its frontal density achieves the first stable value , and we consider final collapse of the cloud. In the framework of Figure 2, frontal densification brings conditions from near the origin (Ri≃0, ζ≃0), where the frontal region has finite speed, small height, and density near air, to small finite values of (Ri, ζ) just beyond the unstable solutions marked by open symbols in the supporting information [Figures (1) and (2) of Appendix D], thus forcing Ri and ζto increase in tandem toward the stable quadrant, until they eventually reach stable values at (Rin,ζn).
 During this evolution, there are three unknown variables (avalanche density ρ′, speed U and height h′), but only two available equations (34) and ((37)), so the problem is not closed. To do so, we conjecture that a nascent cloud starting at low Ri and ζfirst densifies at the highest possible Ri consistent with fluidization of the snowpack surface (i.e., Ri=κ0ζ/(1−ζ), see inequality (11). This initial phase is marked by the letter A in Figure 2. It ends when the Richardson number reaches its stable value Rin. Then, the frontal region progressively densifies at constant Ri=Rin along “path B” by increasing ζ from
until ζ reaches its stable value ζn. On path B, equations (1), (3), (17) and (28) yield
 To estimate how far an avalanche must travel before reaching the stable density , we first consider a frontal region of constant width, densifying along path B initially at ρ′=ρ/(1−ζ0) and U=U0. In this case, the mass conservation equation (38) becomes, after rearrangement,
with dimensionless solution
where is an exact constant, and
Equation (48) predicts that, on a uniform snowpack, an incipient avalanche of constant width densifies on path B as long as its front speed grows. Conversely, it thins down as speed decreases.
 Using equation (45), the momentum balance (34) becomes, after expanding its left derivative in Eulerian form with equation (37), combining with mass balance (47), and solving for ∂ζ/∂x,
where U∗2 is given in equation (47) and distance is made dimensionless using
Integrating equation (50) numerically from ζκ at the bottom of path B to ζn, we find that densification at constant width is slow. For a typical snow cover (ρ=1.2 kg/m3, ρc=200 kg/m3, μe=0.45) such that κ0≃2, the dimensionless densification distance is x∗≃515/ sinαeff. If a powder cloud began forming at U0=10 m/s and proceeded at constant width, it would reach its first stable density at x≃9600 m. Thus, if the width of an avalanche stayed constant, its frontal region would only achieve a relatively small density by the time it reached the bottom of a valley.
 Instead, densification is greatly hastened in a narrowing avalanche. To show this, we rewrite equations (34) and (38) on path B in dimensionless form using equations (49) and (51),
where . We solve these two ODEs simultaneously using a fourth-order Runge-Kutta method implemented as ODE45 in MATLAB, subject to U∗=1 and ζ=ζκ at x∗=0. Results indicate that the distance xdens travelled by the front on path B until mixed-mean density reaches its first stable value at ρ′=ρ/(1−ζn) is much shorter than if the width of the frontal region stayed constant. For example, in avalanche 509 with U0=10 m/s, we calculate that the first stable density is reached at x≃580 m (i.e., at a projected distance x†≃450 m).
 Once the frontal region has densified to the first stable value ρ′=ρ/(1−ζn), its speed is governed by equation (40). If it stayed at that condition but encountered a region where topography widens the avalanche rapidly (e.g., ℓ≃constant>0), then speed should decrease as soon as ∂U2/∂x<0 or, equivalently, when
Although avalanche deceleration does not necessarily imply ineluctable collapse, equation (48) suggests that reductions in front speed lead to a decreasing density. Thus, criterion (54) is a necessary, but not sufficient condition for eventual collapse on a uniform snowpack. However, it implies that powder clouds are more prone to collapse at large speeds. If it is satisfied, speed and density decline in concert, as equation (48) suggests, unless width promptly narrows again to reenergize the frontal region. In the collapse phase, we conjecture that ζ decreases on path B at the stable Ri=Rin until ζreaches ζκ, at which point the free surface of the snowpack defluidizes, further hastening the cloud's demise. Beside rapid frontal spread, collapse is exacerbated by changes in snowpack density or friction that would cause the avalanche to violate equation (11). In practice, variations in snow cover with altitude can therefore dominate collapse.
 “Avalanche 509” of 7 February 2003 is a rare powder cloud with published data on frontal velocity U, slope αand width W versus distance [Sovilla et al., 2006], total cloud volume and frontal height h′ versus time [Vallet et al., 2004; Turnbull and McElwaine, 2007], and estimates of scouring depth λh′[Sovilla et al., 2006]. Nishimura et al.  also published comprehensive data for a powder avalanche in Ryggfonn, Norway. In this section, we test predictions against data for avalanche 509. However, because static pressure measurements are unavailable, we do not compare p′ with the same data set. Carroll et al.  already showed that the model captures the time-history of p′ that McElwaine and Turnbull  recorded for avalanche 629 on 19 January 2004. Because of the practical interest that “impact pressure” pI elicits [Sovilla et al., 2008b], Appendix H (“Impact pressure”, supporting information) also predicts the latter near the front and shows why p′≪pI.
 We begin by testing frontal speed predictions of equation (41), which assumes for simplicity that the frontal region has instantly reached its first stable state at (Rin,ζn). Powder clouds rarely proceed on a transversely uniform incline. Topography often channels them instead. If so, predicting avalanche spread by solving a lateral momentum balance would not be instructive. Thus, rather than attempting to predict W, we adopt values recorded by Sovilla et al. . We use smoothing splines to fit them in terms of projected distance x†, integrate equation (42) to convert x† to distance x along the avalanche path, and evaluate d lnW/dxnumerically. We then integrate equation (41) subject to the typical speed U0=10 m/s that is commonly observed as a dense slide degenerates into a nascent powder cloud [Ancey, 1998].
 Figure 3 illustrates predictions for frontal speed and height. Near the pylon where Turnbull and McElwaine  and Sovilla et al. [2008a] carried out observations, local topography widens the channel available to the avalanche, thus producing a nearly steady speed. In the range 1050<x†<1650 m, Sovilla et al.  reported the average frontal speed m/s. Equation (41) predicts m/s. As the dashed line indicates, ignoring variations in width would underestimate speed in the early runout, when acceleration rises as width decreases, and it would preclude the apparent steady speed that is observed. Figure 3 also shows that the added mass inertial force of equation (E5) (in the supporting information) accounts for no more than 10% of predicted front speed, and thus its exact form matters little.
 Because the cloud reaches 73%of asymptotic tail height h′ at the exit plane, and because interface oscillations exaggerate mean cloud height in videogrammetry, it is reasonable to confuse h′ with height of the frontal region at x=xf when comparing the latter with data. For avalanche 509, Turnbull and McElwaine  reported H′≃20 m at the pylon. From equation ((23)), we predict H′≃29 m. The inset of Figure 3 also compares predictions with average cloud height that Vallet et al.  calculated from videogrammetry as the ratio of total avalanche volume and surface footprint area. Agreement is reasonable given uncertainties in interpreting such data [Turnbull and McElwaine, 2007]. Recently, Louge et al.  exploited the present model to interpret the volume growth that Vallet et al.  had recorded.
Sovilla et al.  reported an average “entrainment depth” ≃13 cm, which they calculated at an average snowpack density by subtracting the total mass of snow deposited from that released over the active avalanche footprint. They also estimated a scouring depth of ≃30 cm from FMCW radar records. At U=57 m/s, equation (25) predicts a scouring depth for a snowpack of density ρc=200 kg/m3 (or for ρc=100 kg/m3). However, the depth of fluidization is much larger, e.g., with μe=0.4. This suggests that passage of the frontal region weakens the snowpack deeply, even though only a fraction λn≃7% of that depth is actually drawn into the cloud. If the latter developed greater stable density, then λ would increase.
 As Figure 4 illustrates, it is likely that the frontal region left the first stable state at (Rin,ζn) to explore the stable quadrant of the (Ri,ζ) diagram with greater mixed-mean density. To show this, the inset in Figure 4 plots Froude number in equation (26) calculated from the data of Sovilla et al.  and Vallet et al. . Fr first increased after ignition, suggesting progressive densification of the nascent frontal region, according to equation ((28)). It leveled off at the first stable value for 510< x†< 740 m. Fr then rose above Frn in the range 740< x†< 1430 m, before returning to Frneventually.
 As expected, the mixed-mean density predicted by equation (30) at (Rin,ζn) is intermediate between estimates for the “saltation layer” (∼30 kg/m3) and the more dilute suspension above (∼3 kg/m3) [Issler, 2003]. Figure 4 shows that avalanche 509 reached a mixed-mean density as high as 10.4 kg/m3. Recently, Sovilla  carried out density measurements at the Vallée de la Sionne with a capacitance technique described by Louge et al. , and used in denser avalanches [Dent et al., 1998] and snowpacks [Louge et al., 1998]. Her preliminary results indicate that, as the front passes, time-averaged density is greater near the snowpack than the mixed-mean value predicted by equation (30), and that the instantaneous ρ′ oscillates. Sovilla  also recorded smaller densities from a similar sensor higher on the pylon, suggesting that the flow is stratified.
 Equations (52) and ((53)) capture the role of an evolving mixed-mean density on frontal dynamics. To compare with data, we assume that the frontal region conforms to path B and remains at the stable node upon reaching it. As Figure 5 shows, the frontal speed predicted by equations (52) and (53) is slightly larger than in the constant-density calculation in Figure 3, indicating that frontal density plays a relatively minor role early in the runout. These equations also predict that the frontal region achieved its first stable mixed-mean density at x†≃450 m, in agreement with the location where the Froude number first reached Frn (Figure 4).
 When criterion ((53)) signals the onset of collapse, integration is restarted with current (U,ζ) for best accuracy. As Figure 5 shows, collapse may be exacerbated by a rapid spread of the cloud. Substituting data from Sovilla et al.  in equation ((54)) indicates that the cloud should collapse beyond x†>1780 m. Although the cloud widened relatively slowly at that location, it is instructive to calculate what would happen if the characteristic length ℓ in equation ((43)) was comparable to the value ℓ≃+400 m reported farther downslope around x≃1800 m. As Figure 5 shows, frontal speed would have fallen more sharply. Thus, it is possible for the spread of a powder cloud to hasten its collapse [Nishimura et al., 1995], beyond what an increase in snowpack density or friction might alone accomplish.
 This paper discussed frontal dynamics of powder snow avalanches behaving as an “eruption current”, in which the very static pressure field caused by rapid snowpack blow-out induces pore pressure gradients sufficient to fluidize the snow cover [Louge et al., 2011].
 By analyzing stability of the balance regulating frontal mass flow rates, we derived frontal height, speed, mixed-mean density, and acceleration of powder clouds with stable growth rate (equations (24), (30), and (39)), and we calculated the corresponding fluidization and scouring depths of a uniform snow cover (equations (7) and (25)). We derived an “ignition” criterion limiting density, internal friction and cohesion of uniform snowpacks able to erupt as a nascent powder cloud (equation (23). In the limit of large bulk Richardson number, we showed how a Froude number Fr measured in experiments (equation (26)) could be used to estimate mixed-mean density in the frontal region (equation (29)). We then compared our predictions with recorded values of Fr, and showed that a typical frontal region first densifies, then remains at or near its first stable state (Figure 4).
 Having defined the frontal region as the domain below the mean air-cloud interface with favorable pressure gradient, we showed how continuity of static pressure and negligible deviatoric stress imply that slope does not substantially affect frontal acceleration. To analyze frontal dynamics, we then suggested a resolution of d'Alembert's paradox arising in the potential flow theory by analogy to Prandtl's treatment for the cylinder in cross-flow. By inspecting the large-eddy-simulations of Carroll et al., we attributed frontal acceleration to a deficit in suction pressure on the exit plane of the frontal region. By analyzing available field data, we also showed in the supporting information (Appendix G) that frontal acceleration is smaller than what a similar buoyant solid object would experience.
 In section 8, we tested predictions for frontal speed with constant or variable mixed-mean density (Figures 3 and 5, respectively) against the only data record from the Vallée de la Sionne, for which mountain slope, avalanche width, frontal speed, scouring depth, and cloud height were all reported [Sovilla et al., 2006; Vallet et al., 2004]. Our numerical integration reproduced initial acceleration, and attributed the subsequent speed plateau to cloud spread. The model also captured cloud height and snowpack scouring depth reasonably well.
 We calculated that the distance required for achieving a stable frontal density would be long if avalanche width stayed constant. However, by integrating numerically the governing equations (52) and (53) with variable frontal density, we reproduced the densification distance recorded in the field (Figure 4), and showed that narrowing the avalanche reduces the densification distance considerably. We also quantified how streamwise variations in avalanche width affect acceleration (equation (39)) and promote collapse (equation (54)).
 This work sharply contrasted the dynamics of “eruption” and “gravity” currents. In the latter, ambient fluid entrainment dominates momentum loss. In eruption currents, the frontal region spends momentum to accelerate material rapidly acquired from the snow cover. This analysis remained tractable by ignoring powder cloud stratification. Although the predicted stable mixed-mean density in equation (30) was consistent with estimates [Grigorian et al., 1982; Issler, 2003], it should be compared against density time-histories recently carried out at the Vallée de la Sionne, so the significance of stratification can be established. Finally, our resolution of d'Alembert's paradox should be tested against future numerical simulations or experiments.
 Acknowledgment is made to the Donors of the American Chemical Society Petroleum Research Fund for support of this research. The authors are grateful to Edwin A. Cowen, Peter J. Diamessis, Paolo Luzzatto-Fegiz, Mason Peck, Olivier Roche, and Charles H. K. Williamson for stimulating discussions, to Siping Wang and Nicolas Gautier for carrying out numerical simulations, and to Betty Sovilla and Christophe Ancey for their suggestions and careful reading of the manuscript.