## 1. Introduction

[2] The 2010 Eyjafjallajökull and 2011 Puyehue-Cordón Caulle eruptions demonstrated how even moderate volcanic eruptions can inject hazardous amounts of ash into the atmosphere that disrupt the air-travel infrastructure worldwide. Real-time forecasting of cloud spreading and atmospheric ash concentration based on volcanic ash transport and dispersion models (VATDMs) is, therefore, essential to reduce impact [*Guffanti et al.*, 2010; *Bonadonna et al.*, 2011a; *Langmann et al.*, 2012]. Currently, ash concentration in the atmosphere forecasted by VATDMs can only be constrained within a factor of ten [*Mastin et al.*, 2009]. This is mainly due to the large uncertainty associated with the operationally used relations between plume height and mass flow rate, which omit first order effects related to the variability of the eruption source and the atmosphere. The mass flow rates estimated from the characterization of tephra deposits are based on similar expressions [*Wilson et al.*, 1980; *Wilson and Walker*, 1987; *Sparks et al.*, 1997] and consequently suffer from similar uncertainties. Here we address this issue by developing an expression that more flexibly incorporates variable atmospheric and source conditions.

[3] Analytical and one-dimensional models of volcanic plumes are based on the general theory developed for turbulent gravitational convection [*Morton et al.*, 1956]. Under the assumptions that (i) the vertical velocity and buoyancy profile are self-similar at all heights, (ii) the rate of entrainment is proportional to the characteristic velocity of the plume at every height, and (iii) the largest local density variations in the plume are small in comparison with the density of the atmosphere at the source, an analytical expression can be derived for the maximum height*H* reached by a purely buoyant plume (i.e., zero initial mass and momentum flow rate) released from a maintained source into a calm and dry atmosphere [see *Morton et al.*, 1956, equation (10)]:

with being the buoyancy flow rate at the source, and *α*the radial entrainment coefficient under the assumption of a top-hat velocity and buoyancy profile, determined through observations and experiments [*Morton et al.*, 1956; *Carazzo et al.*, 2008; *Devenish et al.*, 2010] (auxiliary material). *Morton et al.* [1956, equation (10)] uses the radial entrainment coefficient of a Gaussian profile, which is that of a top-hat profile divided by 2^{1/2}. We use *α* = 0.1 unless stated otherwise (see auxiliary material). *N* is the buoyancy frequency and quantifies the density stratification of the fluid in which the source is released (for a standard atmosphere *N* = 1.065 × 10^{−2} s^{−1}). Note that *Morton et al.* [1956] use the parameter *G* defined in equation (24) in their paper, which is identical too *N*^{2}. *z*_{1}is the maximum non-dimensional height of*Morton et al.* [1956]and was determined by numerical integration of the non-dimensional governing equations. It has a value of 2.8 [*Morton et al.*, 1956, Table 1]. In the case of a purely buoyant plume rising in a cross flow with a uniform velocity *v* much larger than the characteristic plume rise velocity, an analogous expression to equation (1) can be obtained [see *Hewett et al.*, 1971, equation (A.35); *Briggs*, 1972, equation (2)]:

with *β* being the wind entrainment coefficient, which is determined from observations and experiments [*Briggs*, 1972; *Devenish et al.*, 2010]. We use a value of *β* = 0.5 unless stated otherwise (auxiliary material). Furthermore, a moist atmosphere can significantly increase plume height, through the production of latent heat released by the formation of water droplets [*Morton*, 1957]. The addition of the latent heat term in the conservation of energy, however, does not allow for a general scaling expression [*Morton*, 1957], and a numerical integration of the governing equations is necessary to obtain a relationship between buoyancy flow rate and maximum height.

[4] These findings have been applied to volcanic plumes [*Settle*, 1978; *Wilson et al.*, 1978, 1980; *Woods*, 1988, 1993; *Glaze et al.*, 1997; *Sparks et al.*, 1997; *Hort and Gardner*, 2000; *Bursik*, 2001; *Mastin*, 2007], which differ from purely buoyant plumes in that (i) they consist of a multiphase mixture of particles and gas [*Woods*, 1988; *Hort and Gardner*, 2000], (ii) they are characterized by a non-zero mass and momentum flow rate [*Morton*, 1959; *Woods*, 1988], and (iii) their initial density is much higher than the surroundings [*Wilson et al.*, 1980; *Woods*, 1988]. This is highly important for the initial jet phase and leads to plume collapse if mixing with the surrounding air is insufficient [*Wilson et al.*, 1980; *Woods*, 1988]. If we consider that the particles and the gas are in thermal and mechanical equilibrium, which is reasonable for particles <1 cm and <100 *μ*m respectively [*Woods*, 1988; *Hort and Gardner*, 2000; *Stroberg et al.*, 2010], and the plume becomes buoyant beyond the jet phase, equation (1) remains robust in describing plume height of volcanic plumes in a dry and calm atmosphere [*Woods*, 1988]. However, volcanic plumes are often released in atmospheres with variable wind and humidity conditions, which are shown to have significant effects [*Woods*, 1993; *Glaze et al.*, 1997; *Bursik*, 2001; *Mastin*, 2007].