From: A physics-based source model for real-time tephra-dispersal forecasting for weak eruption plumes
Symbol | Definition | Value | Unit |
---|---|---|---|
\(C_{\mathrm {p}}\) | 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 |