Table 1 List of material properties and values of parameters

\(C_{\mathrm {p}}\)

specific heat of bulk mixture

-

J/kg/K

\(C_{\mathrm {pa}}\)

specific heat of air

1004

J/kg/K

\(C_{\mathrm {pv}}\)

specific heat of volcanic gas (water vapor)

1810

J/kg/K

\(C_{\mathrm {ps}}\)

specific heat of pyroclast

1100

J/kg/K

D

power-law exponent for the TDSD at the vent (Girault et al. 2014)

3

-

\(\frac{dQ_\phi }{ds}\)

emission mass along a plume per unit time and length (SMD=-\(\frac{dQ_\phi }{ds}\))

-

kg/sm

ds

step size of integral

1

m

g

acceleration of gravity (\(\varvec{g}=(0,0,g)\))

9.80665

m/s\(^2\)

H

plume height

-

m

h

thickness of the downwind gravity current

-

m

\(\dot{M}\)

mass eruption rate

-

kg/s

N

Brunt-Väisälä frequency

-

1/s

n

gas mass fraction

-

-

\(n_{\mathrm {a}}\)

air mass fraction

-

-

\(n_{\mathrm {v}}\)

volcanic gas mass fraction (water vapor)

-

-

\(n_{{\mathrm {v}}0}\)

volcanic gas mass fraction (water vapor) at the vent

0.05

-

p

pressure in a plume

-

hPa

\(Q_\phi\)

mass flux for grainsize \(\phi\)

-

kg/s

\(R_{\mathrm {g}}\)

gas constant for bulk mixture

-

J/kg/K

\(R_{\mathrm {a}}\)

gas constant for air

287

J/kg/K

\(R_{\mathrm {v}}\)

gas constant of volcanic gas (water vapor)

462

J/kg/K

r

characteristics radius of eruption column

-

m

s

local coordinate for the vent location along the centerline of the plume

-

m

T

temperature of bulk mixture

-

K

\(T_0\)

temperature of bulk mixture at the vent

1000

K

\(T_{\mathrm {a}}\)

temperature of ambient air

-

K

\(U_{\mathrm {e}}\)

entrainment velocity

-

m/s

L

width of the downwind gravity current

-

m

\(V_0\)

volumetric flux at NBL in the eruption column

-

m\(^3\)/s

\(V_1\)

volumetric flux for the downwind gravity current

-

m\(^3\)/s

\(v_\phi\)

terminal velocity of pyrocrast with grainsize \(\phi\)

-

m/s

\(v_0\)

velocity of bulk mixture at the vent (Mach 1; \(v_0=\sqrt{n R_{\mathrm {g}}T}\))

152

m/s

\(\varvec{v}\)

velocity of bulk mixture (\(v=|\varvec{v}|\))

-

m/s

\(\varvec{w}=(w_x, w_y, 0)\)

horizontal ambient wind velocity (\(w=|\varvec{w}|\))

-

m/s

\(w_{\text {nbl}}\)

ambient wind speed at the NBL

-

m/s

x, y

horizontal distance from the vent

-

m

z

altitude

-

m

\(\alpha\)

radial entrainment coefficient

0.06

-

\(\beta\)

wind entrainment coefficient

0.2

-

\(\phi\)

grainsize of pyroclasts

\(-6,-5,\cdot \cdot \cdot , +6\)

\(\phi\)-scale

\(\lambda\)

spreading parameter for the downwind gravity current (Bursik et al. 1992)

0.83

-

\(\mu\)

ratio of volumetric flux (\(\mu =V_1/V_0\))

-

-

\(\rho\)

density of bulk mixture

-

kg/m\(^3\)

\(\rho _{\text {p}}\)

density of pyroclasts

1000

kg/m\(^3\)

\(\rho _{\text {a}}\)

density of ambient air

-

kg/m\(^3\)

\(\theta\)

local angle of the plume (vending of plume)

-

radian