We consider a volcanic plume as a multiphase mixture of volatiles, suspended
particles (tephra) and entrained ambient air. For simplicity, water (in
vapour, liquid or ice phase) is assumed the only volatile species, being
either of magmatic origin or incorporated through the ingestion of moist
ambient air. Erupted tephra particles can form by magma fragmentation or by
erosion of the volcanic conduit, and can vary notably in size, shape and
density. For historical reasons, field volcanologists describe the continuous
spectrum of particle sizes in terms of the dimensionless Φ-scale
:
d(Φ)=d*2-Φ=d*e-Φlog2,
where d is the particle size and d*=10-3m is a reference
length (i.e. 2-Φ is the direction-averaged particle size expressed
in mm). The vast majority of modelling strategies, discretize the
continuous particle grain size distribution (GSD) by grouping particles in
n different Φ-bins, each with an associated particle mass fraction
(the models based on moments e.g. are the exception).
Because particle size exerts a primary control on sedimentation,
Φ-classes are often identified with terminal settling velocity classes
although, strictly, a particle settling velocity class is defined not only by
particle size but also by its density and shape. We propose a model for
volcanic plumes as a multiphase homogeneous mixture of water (in any phase),
entrained air, and n particle classes, including a parameterization for the
air entrainment coefficients and a wet aggregation model. Because the
governing equations based upon the BPT are not adequate above NBL, we also
propose a new semi-empirical model to describe such a region.
Governing equations
The steady-state cross-section-averaged governing equations for axisymmetric
plume motion in a turbulent wind (see Fig. ) are the
following (for the meaning of the used symbols see
Tables and ):
dM^ds=2πrρaue+∑i=1ndM^ids,
dP^ds=πr2ρa-ρ^gsinθ+uacosθ2πrρaue+u^∑i=1ndM^ids,
P^dθds=πr2ρa-ρ^gcosθ-uasinθ2πrρaue,
dE^ds=2πrρaue(1-wa)caTa+wahwa(Ta)+gz+12ue2+cpT^∑i=1ndM^ids,
dM^ads=2πrρaue(1-wa),
dM^wds=2πrρauewa,
dM^ids=-χusiru^1+fueusidr/ds-1M^i+Ai+-Ai-,
dxds=cosθcosΦa,
dyds=cosθsinΦa,
dzds=sinθ,
where M^=πr2ρ^u^ is the total mass flow rate,
P^=M^u^ is the total axial (stream-wise) momentum flow
rate, θ is the plume bent over angle with respect to the horizontal
(i.e. θ=90∘ for a plume raising vertically), E^=M^(H^+gz+12u^2) is the total energy flow rate,
H^ is the enthalpy flow rate of the mixture,
T^=T^(H^) is the mixture temperature,
M^a is the mass flow rate of dry air, M^w=M^x^w is the mass flow rate of volatiles (including water vapour,
liquid and ice), hwa is the enthalpy per unit mass of the water
in the atmosphere, M^i=M^x^pfi is the mass
flow rate of particles of class i(i=1:n), x and y are the horizontal
coordinates, z is height and s is the distance along the plume axis (see
Tables and for the definition of
all symbols and variables appearing in the manuscript).
The equations above derive from conservation principles assuming axial
(stream-wise) symmetry and considering bulk quantities integrated over a
plume cross section using a top-hat profile in which a generic quantity
ϕ has a constant value ϕ^(s) at a given plume cross section
and vanishes outside (here we refer to section-averaged quantities as bulk
quantities, denoted by a hat). We have derived these equations by combining
formulations from different previous plume models
in order to include in a single model effects from plume bending by wind,
particle fallout and re-entrainment at plume margins, transport of volatiles
(water) accounting also for ingestion of ambient moisture, phase changes
(water vapour condensation and deposition) and particle aggregation.
Equation () expresses the conservation of total mass, accounting
in the right-hand side (rhs) for the mass of air entrained through the plume
margins and the loss/gain of mass by particle fallout/re-entrainment.
Equation () and () express the conservation of
axial (stream-wise) and radial momentum, respectively, accounting on the rhs
for contributions from buoyancy (first term), entrainment of air, and
particle fallout/re-entrainment. Note that generally the buoyancy term,
acting only along the vertical direction z, represents a sink of momentum
in the basal gas-thrust jet region (where ρ^>ρa) and
a source of momentum where the plume is positively buoyant (ρ^<ρa). Equation () expresses the conservation of
energy, accounting on the rhs for gain of energy (enthalpy, potential and
kinetic) by ambient air entrainment (first term), loss/gain by particle
fallout/re-entrainment (second term), and gain of energy by conversion of
water vapour into liquid (condensation) or into ice (deposition).
Equation (), () and ()
express, respectively, the conservation of mass of dry air, water (vapour,
liquid and ice) and solid particles. The latter set of equations, one for
each particle class, account on the rhs for particle re-entrainment (first
term), particle fallout (second term) and particle aggregation. Here we have
included to terms (Ai+ and Ai-) that account for the creation of
mass from smaller particles aggregating into particle class i and for the
destruction of mass resulting from particles of class i contributing to the
formation of larger-size aggregates. Finally, Eq. () to
() determine the 3-D plume trajectory as a function of the length
parameter s. All these equations constitute a set of 9+n first order
ordinary differential equations in s for 9+n unknowns: M^,
P^, θ, E^, Ma^, Mw^,
M^i (for each particle class), x, y and z. Note that, using the
definitions of M^-P^-E^, the equations can also be
expressed in terms of u^-r-T^ given the bulk density.
Assuming an homogeneous mixture, the bulk density ρ^ of the mixture is
1ρ^=x^pρp+x^lρl+x^sρs+(1-x^p-x^l-x^s)ρg,
where x^p, x^l and x^s
are, respectively, the mass fractions of particles, liquid water and ice,
ρp is the class-weighted average density of particles
(pyroclasts), ρl and ρs are liquid water and
ice densities, and ρg is the gas phase (i.e. dry air plus water
vapour) density. Under the assumption of mechanical equilibrium (i.e.
assuming the same bulk velocity u^ for all phases and components) it
holds that
x^p=∑M^iM^=∑M^i∑M^i+M^w+M^a.
The enthalpy flow rate of the mixture is a non-decreasing function of the
temperature T^ given by
H^=M^[xacaT^+xpcpT^+xvhv(T^)+xlhl(T^)+xshs(T^)],
where hv, hl and hs are, respectively,
the enthalpy per unit mass of water vapour, liquid and ice:
hs(T^)=csT^,
hl(T^)=hl0+cl(T^-T0),
hv(T^)=hv0+cv(T^-T0),
where cs=2108 J K-1 kg-1 is the specific heat of
ice, T0 is a reference temperature,
hl0=3.337×105 J kg-1 is the enthalpy of the liquid
water at the reference temperature,
cl=4187 J K-1 kg-1 is the specific heat of
liquid water, hv0=2.501×106 J kg-1 is the enthalpy
of vapour water at the reference temperature and
cv=1996 J K-1 kg-1 is the specific heat of
vapour water. For convenience, the reference temperature T0 is taken equal
to the temperature of triple point of the water (T0=273.15 K). The energy
and the enthalpy flow rate are related by
E^=H^+M^(gz+12u2).
For the integration of Eq. () and for evaluating the aggregation
rate terms in Eq. (), the temperature T^ and the mass
fractions of ice (xs), liquid water (xl) and vapour
(xv) need to be evaluated. These quantities are obtained by the
direct inversion of Eq. (), with the use of
Eqs. () and () and by assuming that the pressure
inside the plume P is equal to the atmospheric pressure at the same
altitude (z).
The model uses a pseudo-gas assumption considering that the mixture of air
and water vapour behaves as an ideal gas:
P=Pv+Pa;Pv=nvP;Pa=naP,
nv=xv/mvxv/mv+xa/ma;na=xa/maxv/mv+xa/ma,
where Pv and Pa are, respectively, the partial
pressures of the water vapour and of the air in the plume, nv and
na are the molar fractions of vapour and air in the gas phase
(nv+na=1) and mv=0.018 kg/mole and
ma=0.029 kg/mole are the molar weights of vapour and air.
Following and , we consider that, if the
air-water mixture becomes saturated in water vapour, condensation or
deposition occurs and the plume remains just saturated. This assumption
implies that the partial pressure of water vapour Pv equals the
saturation pressure of vapour over liquid (el) or over ice
(es) at the bulk temperature, and the saturation pressures over
liquid and ice are given (in hPa) by
el=6.112exp17.67T^-273.16T^-29.65loges=-9.097(273.16T^-1)-3.566log(273.16T^),+0.876(1-T^273.16)+log(6.1071).
Equation () holds for T^≥Tf and Eq. () is
valid for T^≤Tf, where Tf is the temperature of the triple
point of the water (here set at Pf=611.2 Pa, Tf=273.16 K). Therefore,
if T^>Tf and Pv<el, the plume is
undersaturated and there is no water vapour condensation (i.e.
x^v=x^w and
x^l=x^s=0). In contrast, if Pv≥el, the vapour in excess is immediately converted into liquid and
(P-el)nv=elnax^s=0x^l=x^w-x^v.
The vapour and air mass fractions xv and xa are
evaluated by combining Eq. () and (). On the other
hand, if T^≤Tf and Pv<es the plume is
undersaturated and there is no water vapour deposition. In contrast, if
Pv≥es, the vapour in excess is immediately
converted into ice and
(P-es)nv=esnax^l=0x^s=x^w-x^v.
Again, the vapour and air mass fractions xv and xa
are evaluated by combining Eq. () and ().
For the particle re-entrainment parameter f we adopt the fit proposed by
using data for plumes not affected by wind:
f=0.431+0.78usPo1/4Fo1/26-1,
where Po=ro2u^o2 and Fo=ro2u^oH^o are the
specific momentum and thermal fluxes at the vent (s=0), and H^o
is the enthalpy per unit mass of the mixture at the vent. This expression may
overestimate re-entrainment for bent over plumes . Finally,
particle terminal settling velocity usi is parameterized as
usi=4g(ρpi-ρ^)di3Cdρ^,
where di is the class particle diameter and Cd is a drag
coefficient that depends on the Reynolds number Re = diusiρ^/μ^. Several empirical fits exist for
drag coefficients of spherical and non-spherical particles
e.g.. In
particular, gave a fit valid over a wide range of particle
sizes and shapes covering the spectrum of volcanic particles considered in
volcanic column models (lapilli and ash):
Cd=24ReK11+0.1118ReK1K20.6567+0.4305K21+3305ReK1K2,
where K1 and K2 are two shape factors depending on particle sphericity,
Ψ, and particle orientation. Given that the Cd depends on
Re (i.e. on us), Eq. is solved iteratively
using a bisection algorithm. Given a closure equation for the turbulent air
entrainment velocity ue, and an aggregation model (defining the
mass aggregation coefficients Ai+ and Ai-), Eq. ()
to () can be integrated along the plume axis from the inlet
(volcanic vent) up to the neutral buoyancy level. Inflow (boundary)
conditions are required at the vent (s=0) for, e.g., total mass flow rate
M^o, bent over angle θo=90∘, temperature T^o,
exit velocity u^o, fraction of water x^wo, null air
mass flow rate M^a=0, vent coordinates (xo,yo and
zo), and mass flow rate for each particle class M^io. The latter
is obtained from the total mass flow rate at inflow given the particle grain
size distribution at the vent:
M^io=fioM^o(1-x^wo),
where fio is the mass fraction of class i at the vent.
Entrainment coefficients
Turbulent entrainment of ambient air plays a key role on the dynamics of jets
and buoyant plumes. In the basal region of volcanic columns, the rate of
entrainment dictates whether the volcanic jet enters into a collapse regime
by exhaustion of momentum before the mixture becomes positively buoyant, or
whether it evolves into a convective regime reaching much higher altitudes.
Early laboratory experiments e.g. already indicated
that the velocity of entrainment of ambient air is proportional to velocity
differences parallel and normal to the plume axis (see inset in
Fig. ):
ue=αs|u^-uacosθ|+αv|uasinθ|,
where αs and αv are dimensionless
coefficients that control the entrainment along the stream-wise (shear) and
cross-flow (vortex) directions, respectively. Note that, in absence of wind
(i.e. ua=0), the equation above reduces to ue=αsu^ and the classical expression for entrainment
velocity of is recovered. In contrast, under a wind field,
both along-plume (proportional to the relative velocity differences
parallel to the plume) and cross-flow (proportional to the wind normal
component) contributions appear. However it is worth noting that
Eq. () has not a solid theoretical justification and is used on
an empirical basis. A vast literature exists regarding the experimental
e.g. and numerical e.g.
determination of entrainment coefficients for jets and buoyant plumes. Based
on these results, most 1-D integral plume models available in literature
consider (i) same constant entrainment coefficients along the plume,
(ii) piecewise constant values at the different regions or, (iii) piecewise
constant values corrected by a factor ρ^/ρa
. Typical values for the entrainment coefficients derived
from experiments are of the order of αs≈ 0.07–0.1
for the jet region, αs≈ 0.1–0.17 for the buoyant
region and αv≈ 0.3–1.0
e.g.. However, more recent experimental
and sensitivity analysis numerical studies
concluded that piecewise constant functions are valid
only as a first approach, implying that 1-D integral models assuming constant
entrainment coefficients do not always provide satisfactory results. This has
also been corroborated by 3-D numerical simulations of volcanic plumes
, which indicate that 1-D integral models overestimate the
effects of wind on turbulent mixing efficiency (i.e. the value of
αv) and, consequently, underestimate plume heights under
strong wind fields. For example, recent 3-D numerical simulation results for
small-scale eruptions under strong wind fields suggest lower values of
αv, in the range 0.1–0.3 . For this
reason, besides the option of constant entrainment coefficients, FPLUME
allows for considering also a parameterization of αs and
αv based on the local Richardson number. In particular, we
use the empirical parameterization of and
that describes
αs for jets and plumes as a function of the local Richardson
number as
αs=0.0675+1-1A(zs)Ri+r21A(zs)dAdz,
where A(zs) is an entrainment function depending on the
dimensionless length zs=z/2ro (ro is the vent radius) and
Ri=g(ρa-ρ^)r/ρau^2 is the
Richardson number. Beside the local Richardson number, the entrainment
coefficient αs depends on plume orientation
e.g., therefore we modify Eq. as:
αs=0.0675+1-1A(zs)Risinθ+r21A(zs)dAdz.
Moreover, in order to use a compact analytical expression and extend it to
values of zs≤10 we fitted the experimental data of
considering the following empirical function:
A(zs)=cozs2+c1zs2+c2,
1A(zs)dAdz=12r02(c2-c1)zszs2+c1zs2+c2,
and in order to extrapolate to low zs we multiply
A(zs) for the following function h(zs) that affects
the behaviour only for small values of zs:
h(zs)=11-c4exp-5(zs/10-1),
where ci are dimensionless fitting constants. Best-fit results and
entrainment functions resulting from fitting
Eq. ()–() are shown in Table
and Fig. , respectively. However, the veracity of the empirical
parameterization in Eq. () was not observed by
in their experiments nor has it been seen in Direct Numerical Simulation (DNS) or Large Eddy Simulations (LES)
simulations of buoyant plumes e.g.. Finally, for the
vortex entrainment coefficient αv, we adopt a
parameterization proposed by based on a few laboratory
experiments:
αv=0.342|Ri|u¯au^o-0.125,
where u^o is the mixture velocity at the vent and
u¯a is the average wind velocity. For illustrative purposes,
Fig. shows the entrainment coefficients
αs and αv predicted by
Eqs. () and () for weak and strong plume
cases under a prescribed wind profile. It is important stressing that air
entrainment rates play a first-order role on eruptive plume dynamics and a
simple description in terms of entrainment coefficients, both assuming them
as empirical constants or describing them like in Eq. (),
represents an over-simplification of the real physics characterizing the
processes. A better quantification of entrainment rates is one of the current
main challenges of the volcanological community seeand references
therein.
Entrainment functions A(zs) for jets and
plumes depending on the dimensionless height zs=z/2ro.
Functions have been obtained by fitting experimental data (points) from
(for zs>10) and multiplying by a correction
function () to extend the functions to zs<10
verifying function continuity and convergence to values of A=1.11 for jets
and A=1.31 for plumes when zs→0.
![]()
Constants defining the entrainment
functions for jets and plumes following the formulation introduced by
(see Eq. to ) obtained after
fitting experimental data reported in . For Kaminski-R we
considered all data including that of , whereas for
Kaminski-C, as suggested by , data from
was excluded.
Kaminski-R
Kaminski-C
jets
plumes
jets
plumes
c0
1.92
1.61717
1.92
1.55
c1
3737.26
478.374
3737.26
329.0
c2
4825.98
738.348
4825.98
504.5
c3=2(c2-c1)
2177.44
519.948
1883.81
351.0
c4
0.00235
-0.00145
0.00235
-0.00145
Modelling of the umbrella region
The umbrella region is defined as the upper region of the plume, from about
the NBL to the top of the column. This region can be dominated by fountaining
processes of the eruptive mixture that reaches the top of the column,
dissipating the excess of momentum at the NBL, and then collapsing as a
gravity current e.g.. The 1-D BPT should not be
extended to this region because it assumes that the mixture still entrains
air with the same mechanisms as below NBL and, moreover, predicts that the
radius goes to infinity towards the top of the column. For these reasons, we
describe the umbrella region adopting a simple semi-empirical approximation.
In the umbrella region (from the NBL to the top of the column), we neglect
air entrainment and assume that the mixture is homogeneous, i.e. the
content of air, water vapour, liquid water, ice and total mass of particles
do not vary with z. Pressure P(z) is considered equal to the atmospheric
pressure Pa(z) evaluated at the same level, whereas temperature
decreases with z due to the adiabatic cooling:
P(z)=Pa(z)anddTdP=1c^ρ^.
As a consequence, the density of the mixture varies accordingly. The total
height of the volcanic plume Ht, above the vent, is approximated as
e.g.
Ht=cH(Hb+8ro),
where cH is a dimensionless parameter (typically cH=1.32),
Hb is the height of the neutral buoyancy level (above the vent)
and ro the radius at the vent. Between Hb and Ht, the
coordinates x and y of the position of the plume centre and the plume
radius r are parameterized as a function of the elevation z, with
Hb≤z≤Ht. The position of the plume centre is assumed
to vary linearly with the same slope at the NBL, whereas the effective plume
radius is assumed to decrease as a Gaussian function:
x=xb+(z-Hb)dxdzz=zb,y=yb+(z-Hb)dydzz=zb,r=rbe-(z-Hb)2/2σH2,
where xb, yb, rb are, respectively, the coordinates x and y of
the centre of the plume and the plume radius at the NBL, and
σH=Ht-Hb.
Finally, assuming that the kinetic energy of the mixture is converted to
potential energy, the vertical velocity is approximated to decrease as the
square root of the distance from the NBL:
uz=uzbHt-zHt-Hb,
where uzb is the vertical velocity of the plume at the NBL.Although the proposed empirical parameterization of the region above the NBL
is qualitatively consistent with the trends predicted by 3-D numerical models
, a more rigorous description requires further research.