Large explosive volcanic eruptions can generate ash clouds from rising plumes that spread in the atmosphere around a Neutral Buoyancy Level (NBL). These ash clouds spread as inertial intrusions and are advected by atmospheric winds. For low mass flow rates, tephra transport is mainly dictated by wind advection, because ash cloud spreading due to gravity current effects is negligible (passive transport). For large mass flow rates, gravity-driven transport at the NBL can be the dominant transport mechanism. Conditions under which the passive transport assumption is valid have not yet been critically studied. We analyze the conditions when gravity-driven transport is dominant in terms of the cloud Richardson number. Moreover, we couple an analytical model that describes cloud spreading as a gravity current with an advection-diffusion model. This coupled model is used to simulate the evolution of the volcanic cloud during the climatic phase of the 1991 Pinatubo eruption.
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 Atmospheric transport of tephra released during explosive volcanic eruptions is significantly affected by the interaction of the volcanic plume and the atmospheric wind field. Weak volcanic plumes develop in the troposphere and follow bent-over trajectories as a result of the wind advection [e.g., Carey and Sparks, 1986; Bonadonna and Phillips, 2003]. Strong plumes typically rise above the tropopause, developing a vertical eruption column that, on reaching the Neutral Buoyancy Level (NBL), spreads laterally as a turbulent gravity current. Under these conditions, the vertical plume rises above the NBL due to its excess momentum and, once its vertical velocity approaches zero, subsides feeding the current that spreads as a lateral intrusion [e.g., Woods and Kienle, 1994; Sparks et al., 1997]. The larger and higher the intensity of the eruption, the more this transport mechanism dominates over passive wind advection at distances from tens to several hundreds of kilometers from the source [Baines and Sparks, 2005].
 Most Tephra Transport and Dispersal Models (TTDM) [Folch, 2012] build on the assumption of passive transport, i.e., they assume that the dispersion and sedimentation of tephra particles in the atmosphere are governed by wind advection, atmospheric turbulent diffusion, and settling of particles by gravity (e.g., Advection-Diffusion-Sedimentation (ADS) models) [e.g., Armienti et al., 1988; Folch et al., 2009]. Conditions under which this passive transport assumption is valid have not been critically examined previously, although it has been shown that ADS models capture the features of past deposits and reproduce the correct order of magnitude of observations, such as accumulated tephra thickness, grain size distribution, and concentration in the atmosphere. Good performance of passive TTDMs (see Folch  for details) is in apparent contradiction to gravity current models for volcanic plumes, especially for high-intensity eruptions and for proximal transport and proximal depositional facies. One of the reasons even simple analytical ADS models can model many tephra deposits is that these models use an effective diffusion coefficient (commonly calibrated from past deposits) that is larger than the actual atmospheric turbulent diffusion, although the actual spreading due to gravity-driven transport follows a different time evolution and should be localized around the source. In this way, even simpler ADS models are able to mimic the effective increase of the cloud area at the NBL observed in large-magnitude eruptions.
 Here we describe the conditions when gravity-driven transport is dominant and when it is negligible in terms of the cloud Richardson number, using an analytical model describing the radial growth of the cloud. Then we couple this analytical gravity-driven model with the FALL3D [Costa et al., 2006; Folch et al., 2009] ADS model in order to evaluate the relative importance of gravity current effects. Finally, the coupled model is used to simulate the evolution of the volcanic cloud during the climatic phase of the 1991 Pinatubo eruption, for which satellite observations are available and previous studies can help to constrain the key eruption source parameters [e.g., Koyaguchi, 1996; Holasek et al., 1996a; Suzuki and Koyaguchi, 2009].
2 Density-Driven Cloud Model
 In the initial phase of a Plinian eruption, volcanic clouds spread as a gravity current around the NBL [e.g., Woods and Kienle, 1994; Sparks et al., 1997]. The radius of the umbrella cloud, R, can be written as a function of time as [e.g., Woods and Kienle, 1994; Sparks et al., 1997]
where t is time, λ is an empirical constant, N is the frequency of Brunt-Väisälä due to the ambient stratification of the atmosphere, and q is the volumetric flow rate into the umbrella region. The value of λ was first estimated to be in the range λ=0.1−0.6 from laboratory experiments and satellite observations [e.g., Holasek et al., 1996a; Holasek et al., 1996b] and then constrained to λ≈0.2 from Direct Numerical Simulations (DNS) [e.g., Suzuki and Koyaguchi, 2009]. The volumetric flow rate into the umbrella region can be estimated as a function of the efficiency of air entrainment, k, and the mass eruption rate as [Morton et al., 1956; Suzuki and Koyaguchi, 2009]
where, from the results of Suzuki and Koyaguchi 
The relationship (1) derives from the conservation equation
with the assumption that the velocity of the leading edge of the spreading current, ub, scales with the average cloud thickness, h, as
Combining (1) and (5), one obtains the radial velocity of the umbrella spreading as a function of time:
As the cloud entrains air, it becomes more dilute and, at a certain distance, atmospheric turbulence and wind advection transport mechanisms dominate cloud transport. In order to estimate the radial distance at which this critical transition between density-driven and passive transport occurs, we compare the umbrella front velocity ub with the mean wind velocity uw at the NBL estimating the Richardson number:
Atmospheric studies [Zilitinkevich et al., 2008] have shown that the interval 0.25<Ri<1 separates two different turbulent regimes (associated with strong and weak mixing, respectively) rather than the turbulent and laminar regimes usually assumed for low and high Ri numbers. That does not support the existence of a critical Richardson number above which turbulent mixing is inhibited. Actually, experimental and observational data indicate that turbulence survives for Ri≫1 [Galperin et al., 2007]. For example, in the free atmosphere, where Ri typically varies from 1 to 100, significant turbulence has been observed at all levels [Lawrence et al., 2004]. Similar observations are valid for the deep ocean [Galperin et al., 2007] and for dispersion of dense CO2 clouds, where Cortis and Oldenburg  found that the threshold between passive and density-dominated regimes appears around values of Ri≃0.25 (note that their Richardson number corresponds to the square root of the one used here), much larger than the value suggested by Britter and McQuaid . Based on these observations, we consider that when Ri>1, the transport is mainly density driven, whereas for Ri<0.25, transport is substantially passive.
 From (7), we can estimate the critical time scales characterizing density-driven (tb) and passive (tp) transport processes:
Note that the classic passive transport assumption of ADS models for describing tephra dispersal should not be considered valid for timescales t<tb, whereas for tb<t<tp, both transport mechanisms are relevant. However, as discussed previously, some ADS models appear to work well even for timescales t<tp because they use an effective diffusion coefficient calibrated from observations rather than the actual atmospheric turbulent diffusion.
 Considering equations (4) and (5) and solving the ordinary differential equation for ub(R) yields the following relationship between the front velocity and the radius of the cloud:
For practical purposes, this radial velocity field is considered only at distances less than Rp=R(tp) and for heights between H−h/2 and H+h/2, where H is the NBL height and h the umbrella thickness. Within this region, the variation of the velocity field with distance r is given by
consistent with the equations above. Our strategy, therefore, consists of using equations (10) and (11) to compute a time-dependent radial velocity field centered above the vent in the umbrella region. This radial velocity is added to the wind field furnished by the Numerical Weather Prediction Model to estimate the combined windfield for the ADS model (FALL3D in our case). By doing this, ADS models can run with an “effective” velocity field accounting for contributions from both passive and density-driven mechanisms. Note that, depending on the balance between wind intensity and volumetric flow rate at the NBL, the added radial velocity field can change the wind advection significantly. The box model given by equations (10) and (11) provides a reasonable approximation about the position of the flow front. Because it is based on the assumption of uniform cloud thickness, the velocity field inside the cloud predicted by equation (11), although consistent with equations (4) and (5), should be viewed as a crude approximation of the real velocity field. Moreover, because the radial velocity (11) diverges as r→0, numerical calculations are truncated assuming a minimum radius , being Δx and Δy the horizontal computational grid sizes. The method of superposition of the radial flow of density current and the ambient wind field is suitable for large steady eruption columns for which the wind velocity near the vent is typically much smaller than the radial density current velocity and the movement of the source position can be assumed negligible.
3 Application to 1991 Pinatubo Eruption
 Mount Pinatubo erupted early in the afternoon of 15 June 1991, generating a Plinian column of 37–39 km high [Holasek et al., 1996a]. Satellite images revealed a giant disk-shaped umbrella cloud that expanded radially for about 5 h. The cloud extended up to about 140 km in radius covering an area of about 60,000 km2 by 14:40 Philippine Local Time (PLT = UTC + 8). One hour later, at 15:40 PLT, the cloud radius further expanded up to ∼200 km, covering an area larger than 120,000 km2 [Koyaguchi, 1996; Holasek et al., 1996a]. At 19:40 PLT, the cloud reached a stagnation point upwind but continued to grow downwind [Koyaguchi, 1996; Holasek et al., 1996a]. Here we simulate the climatic phase of the eruption from 13:40 to 23:40 PLT [Holasek et al., 1996a] using version 7.0 of the FALL3D code (http://bsccase02.bsc.es/projects/fall3d/). For computational reasons, we divided the climatic phase in three intervals named C1, C2, and C3, as follows: C1 from 13:40 to 16:10 PLT with an average column height of about 37 km above the vent level (a.v.l), C2 from 16:10 to 20:40 PLT with an average column height of about 31.5 km a.v.l, and C3 from 20:40 to 23:40 PLT with an average column height of about 26 km a.v.l. [Holasek et al., 1996a].
 All the needed model parameters, given by Holasek et al. [1996a] and Suzuki and Koyaguchi , are summarized in Table 1. For volcanological input parameters, FALL3D requires specification of the source term, i.e., the vertical distribution of Mass Flow Rate (MFR) for each particle class, the column height, and the total grain size distribution (TGSD). The source term was estimated by means of the eruption column model implemented in FALL3D [Costa et al., 2006; Folch et al., 2009], based on the Buoyant Plume Theory (BPT). The BPT plume model embedded in FALL3D 7.0 is similar to that of Bursik , but the radial entrainment coefficient is calculated similar to Carazzo et al.  and the crosswind entrainment coefficient is estimated on the basis of the local Richardson number [Folch et al., 2012]. This model allowed us to compute the MFR and the vertical distribution of mass along the column as function of the column height and mixture properties at the vent (exit temperature, velocity, water content, and TGSD). In order to account for ash aggregation processes [Costa et al., 2010; Folch et al., 2010], a simple aggregation model similar to that of Cornell et al.  was used [Costa et al., 2012]. Concerning meteorological data, we used the European Centre for Medium-Range Weather Forecasts (ECMWF) ERA-Interim reanalysis at 0.25° resolution and 37 pressure levels (top at 1 hPa). Unfortunately, reanalysis data were inconsistent with the stratospheric wind intensity and direction suggested by cloud satellite images, giving a greater westward advection than observed in the satellite imagery [Holasek et al., 1996a; Koyaguchi, 1996]. This discrepancy, already pointed out by Fero et al.  also for the National Centers for Environmental Prediction reanalysis data set, can be attributed to the lack of regional meteorological observations during the pass of a typhoon [Guo et al., 2004] and to the fact that no real-time atmospheric soundings were possible around Mount Pinatubo during the eruption [Fero et al., 2009]. In order to match satellite observations, we rotated the original ERA-Interim reanalysis wind field 30° anticlockwise around the vent and halved the wind intensity.
Table 1. Parameters Used for the Simulation of the Volcanic Plume Evolution of the Climatic Phase of 15 June 1991 Pinatubo Eruption
Assumed as Bi-Gaussian Distribution similar to TGSD of other Plinian eruptions [Folch et al., 2009; Costa et al., 2012]; reported values refer to the two means and variances of the distribution.
MFR and mixture exit parameters at the vent were fixed in order to reproduce the average column heights estimated from Figure 1 of Holasek et al. [1996a] using the BPT model coupled with the wind field [Bursik, 2001; Folch et al., 2012].
Computed in accord to equation (3). For comparison, Suzuki and Koyaguchi  estimated q in the range 0.5–1.5×1011m3/s.
Original ERA-Interim wind fields were rotated 30° anticlockwise around the vent and their intensities halved.
 Figure 1 compares observations of the time evolution of the plume derived from Geostationary Meteorological Satellite (GMS) thermal infrared (IR) satellite images [Holasek et al., 1996a] with the simulation results, with and without considering the gravity current model. Clearly, the simulation considering the gravity current model at the NBL reproduces eruption cloud features much better than that using atmospheric turbulent diffusion only, showing a dominant effect of gravity current spreading both in cross and downwind directions. Assuming that an integral column load of 1 g/m2 represents the satellite detection threshold, the coupled model is able to reproduce the volcanic cloud areas with an average error of ∼15%, the crosswind axes with an error of ∼5%, the downwind axes (that are more affected by the wind field) with an error of ∼20%, and the cloud-spreading velocity with an error of ∼5%. Using the values estimated by Suzuki and Koyaguchi  for the 1991 Pinatubo eruption (see Table 1) and considering an average wind velocity of uw≈15 m/s, from equations (8) and (9), we obtain tb≈5 h (corresponding to a radius of ∼450 km), which is in good agreement with the observed time that the cloud traveled upwind [Holasek et al., 1996a; Koyaguchi, 1996]. The passive dispersion approximation can be reasonably applied for times larger than tb, but it is fully valid only for times larger than tp≈40 h (corresponding to a radius of ∼1800 km). This implies that for long-lasting eruptions having an intensity similar to or larger than that of Pinatubo 1991, density-driven transport dominates for most of the duration of the transport, whereas for smaller eruptions, the mechanism is relevant only in the very initial phase of the eruption (see Figure 2).
 We analyzed the conditions under which the classical passive transport assumption is valid for describing tephra dispersal. An analytical model describing the spreading of the cloud as a gravity current was coupled with an advection-diffusion-sedimentation model. Conditions when cloud spreading due to gravity current effects are dominant, or conversely are negligible, were identified on the basis of the cloud Richardson number. The coupled model satisfactorily reproduced the observed evolution of the volcanic cloud during the climatic phase of the 1991 Pinatubo eruption.
 This work has benefited from funding provided by the Italian Presidenza del Consiglio dei Ministri - Dipartimento della Protezione Civile (DPC), agreement INGV-DPC 2012-2013. This paper does not necessarily represent DPC official opinion and policies. A.F. acknowledges funding by the Spanish project ATMOST (CGL2009-10244). ERA-Interim reanalysis data were provided by European Centre for Medium-Range Weather Forecasts (ECMWF). We are grateful to reviewers of the paper, T. Koyaguchi and J. Telling, for useful comments that improved the paper. We also thank S. Self, C. Connor, and C. Bonadonna for helpful suggestions on an early version of the paper.
 The Editor thanks Takehiro Koyaguchi and an anonymous reviewer for assistance in evaluating this manuscript.