# Conduit flow dynamics during Caldera-forming eruptions

## Basic equations

\begin{eqnarray}
&& \rho w \pi (r_2^2 - r_1^2) = Q
\label{eq:mass} \\
&& \rho w \dfrac{d w}{d z} = -\dfrac{d p}{d z} - \rho g - F_\mathrm{w}
\label{eq:mom} \\
&& \dfrac{1}{\rho} = \dfrac{n}{\rho_\mathrm{g}} + \dfrac{1-n}{\rho_\mathrm{lc}}
\label{eq:rho} \\
&& \rho_\mathrm{g} = \dfrac{p}{RT}
\label{eq:rhog} \\
&& \rho_\mathrm{lc} = (1-\beta) \rho_\mathrm{l} + \beta \rho_\mathrm{c}
\label{eq:rholc} \\
&& n = \dfrac{n_0 - n_\mathrm{l} c}{1 - n_\mathrm{l} c}
\label{eq:n} \\
&& c = s p^{m}
\label{eq:c} \\
&& n_\mathrm{l} = \dfrac{(1-\beta) \rho_\mathrm{l}}{\rho_\mathrm{lc}}
\label{eq:nl} \\
&& F_\mathrm{w} = \left\{
\begin{array}{ll}
\dfrac{8 \eta}{(r_2-r_1)^2} w & (\phi < \phi_\mathrm{cr}) \\
\dfrac{\lambda \rho}{4(r_2-r_1)} w^2 & (\phi \ge \phi_\mathrm{cr})
\end{array}
\right.
\label{eq:Fw} \\
&& \eta = \eta_\mathrm{l}(c, T) f_{\beta}(\beta, R_{\beta},\dot{\varepsilon}) f_{\phi}(\phi)
\label{eq:etafunc} \\
&& \log_{10} \eta_\mathrm{l}(c, T) = −3.545+0.833 \ln (100c)+
\dfrac{9601−2368 \ln (100c)}{T −[195.7+32.25 \ln (100c)]}
\label{eq:etal} \\
&& f_{\beta}(\beta, R_{\beta},\dot{\varepsilon}) = \dfrac{1+\left( \dfrac{\beta}{\beta^{*}} \right)^{\delta}}
{\left\{1-(1-\xi) \mathrm{erf} \left[ \dfrac{\sqrt{\pi}}{2(1-\xi)} \dfrac{\beta}{\beta^{*}} \left\{ 1 + \left( \dfrac{\beta}{\beta^{*}} \right)^{\gamma} \right\} \right] \right\}^{B \beta^{*}}}
\label{eq:funcetabeta} \\
&& \hspace{4em} \beta^{*} = \left[ \beta_\mathrm{m} + \Delta \beta \dfrac{(\dot{\varepsilon}/\varepsilon_\mathrm{c})^{n_\beta} - (\varepsilon_\mathrm{c}/\dot{\varepsilon})^{n_\beta}}{(\dot{\varepsilon}/\varepsilon_\mathrm{c})^{n_\beta} + (\varepsilon_\mathrm{c}/\dot{\varepsilon})^{n_\beta}} \right] \cdot R_\beta^{-b_1}
\label{eq:betast} \\
&& \hspace{4em} \delta = \left[ \delta_\mathrm{m} + \Delta \delta \dfrac{(\dot{\varepsilon}/\varepsilon_\mathrm{c})^{n_\beta} - (\varepsilon_\mathrm{c}/\dot{\varepsilon})^{n_\beta}}{(\dot{\varepsilon}/\varepsilon_\mathrm{c})^{n_\beta} + (\varepsilon_\mathrm{c}/\dot{\varepsilon})^{n_\beta}} \right] \cdot e^{-b_2(R_\beta - 1)}
\label{eq:delta} \\
&& \hspace{4em} \xi = \left[ \xi_\mathrm{m} + \Delta \xi \dfrac{(\dot{\varepsilon}/\varepsilon_\mathrm{c})^{n_\beta} - (\varepsilon_\mathrm{c}/\dot{\varepsilon})^{n_\beta}}{(\dot{\varepsilon}/\varepsilon_\mathrm{c})^{n_\beta} + (\varepsilon_\mathrm{c}/\dot{\varepsilon})^{n_\beta}} \right] \cdot R_\beta^{-b_3}
\label{eq:xi} \\
&& \hspace{4em} \gamma = \left[ \gamma_\mathrm{m} + \Delta \gamma \dfrac{(\dot{\varepsilon}/\varepsilon_\mathrm{c})^{n_\beta} - (\varepsilon_\mathrm{c}/\dot{\varepsilon})^{n_\beta}}{(\dot{\varepsilon}/\varepsilon_\mathrm{c})^{n_\beta} + (\varepsilon_\mathrm{c}/\dot{\varepsilon})^{n_\beta}} \right] \cdot R_\beta^{-b_4}
\label{eq:gamma} \\
&& \hspace{4em} \dot{\varepsilon} = \dfrac{w}{r_2 - r_1}
\label{eq:strainrate} \\
&& f_{\phi}(\phi) = \left\{
\begin{array}{ll}
(1-\phi)^{-1} & (\mathrm{Ca} \ll 1) \\
(1-\phi)^{5/3} & (\mathrm{Ca} \gg 1)
\end{array}
\right.
\label{eq:funcetaphi}
\end{eqnarray}

### Refereneces
-- https://www.mohno-dispenser.jp/compass/compass16.html
-- http://crystallization.eng.niigata-u.ac.jp/%E5%86%86%E7%AE%A1%E5%86%85%E6%B5%81%E5%8B%95.pdf

## Variables and parameters

| Name | Description | Value |
|:------------------|:-----|:-----|
| $\rho$ | Density (kg m$^{-3}$) | - |
| $w$ | Velocity (m s$^{-1}$) | - |
| $r_1$ | Inner conduit radius (m) | - |
| $r_2$ | Outer conduit radius (m) | - |
| $Q$ | Discharge rate (kg s$^{-1}$) | - |
| $p$ | Pressure (Pa) | - |
| $z$ | Vertical coordinate (m) | - |
| $c$ | H$_2$O concentration | - |
| $n$ | Mass fraction of gas | - |
| $\phi$ | Gas volume fraction | - |
| $\rho_\mathrm{g}$ | Gas density (kg m$^{-3}$) | - |
| $n_\mathrm{l}$ | Liquid volume fraction with respect to liquid+crystal | - |
| $\rho_\mathrm{lc}$| Liquid-crystal density (kg m$^{-3}$) | - |
| $\eta$ | Magma viscosity (Pa s) | - |
| $R_\beta$ | Aspect ratio of crystal | - |
| $\dot{\varepsilon}$| Strain rate (s$^{-1}$) | - |
| $n_0$ | Initial H$_2$O content (wt \%) | $3.9$ |
| $T$ | Temperature (K) | $1086$ |
| $\beta$ | Crystal volume fraction | $0.51$ |
| $\phi_\mathrm{cr}$| Critical $\phi$ for fragmentation | $0.51$ |
| $L$| Conduit length (m) | $10000$ |
| $p_\mathrm{ch}$| Chamber pressure (MPa) | $245.1$ |
| $\rho_\mathrm{l}$ | Liquid density (kg m$^{-3}$) | $2500$ |
| $\rho_\mathrm{c}$ | Crystal density (kg m$^{-3}$) | $2500$ |
| $R$ | Gas constant (J K$^{−1}$ kg$^{−1}$ mol$^{−1}$) | 462 |
| $s$ | Solubility constant (Pa$^{−0.5}$) | $4.11 \times 10^{-6}$ |
| $m$ | Solubility constant | $0.5$ |
| $\lambda$ | Friction coefficient | $0.03$ |
| $B$ | Coefficient for $f_\beta$ | $2.5$ |
| $n_\beta$ | Coefficient for $f_\beta$ | $0.3$ |
| $\varepsilon_\mathrm{c}$ | Coefficient for $f_\beta$ | $4.07 \times 10^{-4}$ |
| $\beta_\mathrm{m}$ | Coefficient for $f_\beta$ | $0.62$ |
| $\Delta \beta$ | Coefficient for $f_\beta$ | $0.09$ |
| $\delta_\mathrm{m}$ | Coefficient for $f_\beta$ | $7.73$ |
| $\Delta \delta$ | Coefficient for $f_\beta$ | $5.73$ |
| $\xi_\mathrm{m}$ | Coefficient for $f_\beta$ | $4.75 \times 10^{-4}$ |
| $\Delta \xi$ | Coefficient for $f_\beta$ | $4.56 \times 10^{-4}$ |
| $\gamma_\mathrm{m}$ | Coefficient for $f_\beta$ | $5.92$ |
| $\Delta \gamma$ | Coefficient for $f_\beta$ | $5.11$ |
| $b_1$ | Coefficient for $f_\beta$ | $0.538$ |
| $b_2$ | Coefficient for $f_\beta$ | $0.092$ |
| $b_3$ | Coefficient for $f_\beta$ | $4.568$ |
| $b_4$ | Coefficient for $f_\beta$ | $0.718$ |

## Results

- Fig. 1: $r_2 = 5000$ (m), $R_\beta = 1.87$

- Fig. 2a: $r_2 = 5000$ (m), $R_\beta = 1$

- Fig. 2b: $r_2 = 5000$ (m), $R_\beta = 1.5$

- Fig. 2c: $r_2 = 5000$ (m), $R_\beta = 2$

- Fig. 3a: $r_2 = 1000$ (m), $R_\beta = 1.87$

- Fig. 3b: $r_2 = 10000$ (m), $R_\beta = 1.87$