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Collaborator
There was a problem hiding this comment. would it be better to make it a part of the web-publishable doxygen html? |
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| \documentclass{article} | ||
| \usepackage{amsmath} | ||
| \usepackage[margin=2cm]{geometry} | ||
| \usepackage{longtable} | ||
| \usepackage{comment} | ||
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| \begin{document} | ||
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| \section{Symbol list} | ||
| \label{sec:symbol-list} | ||
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| \newcommand{\rr}{\raggedright} | ||
| \newcommand{\tn}{\tabularnewline\hline} | ||
| \renewcommand{\arraystretch}{1.5} | ||
| \begin{longtable}{|l|p{5.5cm}|l|l|p{4.5cm}|} | ||
| \hline \textbf{Symbol} & \textbf{Meaning} & \textbf{Type} & \textbf{Units} & \textbf{Reference} \tn | ||
| \hline \endhead | ||
| \(i\) & species index. & index & discrete & \tn | ||
| \( A_j \) & The total surface area of the INP. & function & $m^2$ & \tn | ||
| \( A_j^\text{(i)} \) & Surface area of \( i \)th species. & function & $m^2$ & \tn | ||
| \( a_{\text{w}} \) & Water activity of the droplet in which the INP is immersed. & function & 1 & eq.~\ref{eq:4}\tn | ||
| \( a_\text{w}^\text{ice} \) & Water activity in equilibrium with ice. & function & 1 & eq.~\ref{eq:5} \tn | ||
| \(a_\text{INAS}\) & INAS parameter for singular scheme. & constant & $K^{-1}$ & Niemand et al., 2012 \tn | ||
| \(b_\text{INAS}\) & INAS parameter for singular scheme. & constant & 1 & Niemand et al., 2012 \tn | ||
| \( \Delta t \) & Time interval. & function & $s$ & \tn | ||
| \( e_\text{s} \) & Saturated vapor pressure with respect to water. & function & Pa & eq.~\ref{eq:2} \tn | ||
| \( e_\text{s}^\text{ice} \) & Saturated vapor pressure with respect to ice. & function & Pa & eq.~\ref{eq:3} \tn | ||
| \( H \) & Relative humidity with respect to water. & dynamics & 1 & \tn | ||
| \( J_{\text{het}} \) & A constant heterogeneous freezing rate for all particles (for the constant nucleation rate scheme only) & constant & $m^{-2}s^{-1}$ & \tn | ||
| \( J_{\text{het}}^{(i)} \) & Heterogeneous freezing rate coefficients of the \( i \)th species. & function & $m^{-2}s^{-1}$ & eq.~\ref{eq:6} \tn | ||
| \( m_i, c_i \) & ABIFM parameters of \( i^\text{th} \) species. & constant & 1 & Knopf et al., 2013 \tn | ||
| \(n_0\) & An unit constant (=1 $m^{-2}$). & constant & $m^{-2}$ & \tn | ||
| \( N_\text{s} \) & The total number of species covering the surface of the INP. & constant & 1 & \tn | ||
| \( P_{\text{frz,}j} (\Delta t)\) & The freezing probability of $j^\text{th}$ particle within a time interval of $\Delta t$. & function & 1 & eq.~\ref{eq:1},eq.~\ref{eq:7} \tn | ||
| \(p_j\) & A random number uniformly distributed in [0,1] regarding the sampling of $j^\text{th}$ particle. & function & 1 & eq.~\ref{eq:8} \tn | ||
| \(T\) & Current environmental temperature & prescribed & $K$ & \tn | ||
| \(T_0\) & Water freezing temperature & constant & K & \verb+env_state%temp+ \tn | ||
| \(T_{\text{m,}j}\) & The freezing temperature of the $j^\text{th}$ particle. & function & K & eq.~\ref{eq:9} \tn | ||
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| \end{longtable} | ||
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| \begin{comment} | ||
| \( J_{\text{het}}^{(i)} \) & Heterogeneous freezing rate coefficients of the \( i \)th species. \\ | ||
| \( A_i \) & Surface area of \( i \)th species. \\ | ||
| \( A \) & The total surface area of the INP. \\ | ||
| \( \Delta t \) & Time interval. \\ | ||
| \( a_{\text{w}} \) & Water activity of the droplet in which the INP is immersed. \\ | ||
| \( a_\text{w}^\text{ice} \) & Water activity in equilibrium with ice. \\ | ||
| \( RH \) & Relative humidity with respect to water. \\ | ||
| \( e_\text{s} \) & Saturated vapor pressure with respect to water. \\ | ||
| \( e_\text{s}^\text{ice} \) & Saturated vapor pressure with respect to ice. \\ | ||
| \( m_i, c_i \) & ABIFM parameters of \( i^\text{th} \) species. \\ | ||
| \( N_\text{s} \) & The total number of species covering the surface of the INP. \\ | ||
| \hline | ||
| \end{comment} | ||
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| \section{Equations} | ||
| \label{sec:equations} | ||
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| \subsection{Time-dependent scheme with a constant nucleation rate} | ||
| \begin{equation} | ||
| P_{\text{frz,}j}(\Delta t) = 1 - \exp\left(J_{\text{het}} \cdot A_j \cdot \Delta t \right) \label{eq:1} | ||
| \end{equation} | ||
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| \subsection{Time-dependent ABIFM scheme} | ||
| \begin{equation} | ||
| \begin{split} | ||
| e_\text{s}(T) &= \exp \left\{ 54.842763 - \frac{6763.22}{T} - 4.21 * \ln{T} + 0.000367T \right.\\ | ||
| &+ \left. \tanh \left[{0.0415 \cdot (T - 218.8) \cdot (53.878 - \frac{1331.22}{T} - 9.44523 \ln{T} + 0.014025T)}\right] \right\} \label{eq:2} | ||
| \end{split} | ||
| \end{equation} | ||
| \begin{equation} | ||
| e_\text{s}^\text{ice} (T) = \exp \left( 9.550426 - \frac{5723.265}{T} + 3.53068 \cdot \ln{T} - 0.00728332T\right). \label{eq:3} | ||
| \end{equation} | ||
| \begin{align} | ||
| a_{\text{w}} &= 1 \label{eq:4} \\ | ||
| a_\text{w}^\text{ice} &= \frac{e_\text{s}^\text{ice}(T)}{e_\text{s}(T)} \label{eq:5} \\ | ||
| \log_{10}J_{\text{het}}^{(i)} &= m_{\text{i}} \cdot (a_{\text{w}} - a_\text{w}^\text{ice}) + c_i \label{eq:6} \\ | ||
| P_{\text{frz,}j}(\Delta t) &= 1 - \exp\left[ -\sum_{i=1}^{N_{\text{s}}} (A_j^\text{(i)} \cdot J_{\text{het}}^{(i)}) \cdot \Delta t \right] \label{eq:7} | ||
| \end{align} | ||
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| \subsection{Singular scheme} | ||
| \begin{equation} | ||
| p_j = random() \in [0,1] \label{eq:8} | ||
| \end{equation} | ||
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| \begin{equation} | ||
| T_{\text{m,}j} = T_0 + \frac{1}{a_\text{INAS}} \left[\ln{\left( \frac{\ln{(1-p_j)} + \exp{\left(-A_j n_0 \exp{\left(-a_\text{INAS} T_0 + b_\text{INAS}\right)}\right)}}{-A_j n_0} \right)} - b_\text{INAS}\right] \label{eq:9} | ||
| \end{equation} | ||
| \end{document} |
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| Original file line number | Diff line number | Diff line change |
|---|---|---|
| @@ -1,21 +1,21 @@ | ||
| # dens (kg/m^3) ions in soln (1) molec wght (kg/mole) kappa (1) abifm_m (1) abifm_c (1) | ||
| SO4 1800 0 96d-3 0.65 0 0 | ||
| NO3 1800 0 62d-3 0.65 0 0 | ||
| Cl 2200 0 35.5d-3 1.28 0 0 | ||
| NH4 1800 0 18d-3 0.65 0 0 | ||
| MSA 1800 0 95d-3 0.53 0 0 | ||
| ARO1 1400 0 150d-3 0.1 0 0 | ||
| ARO2 1400 0 150d-3 0.1 0 0 | ||
| ALK1 1400 0 140d-3 0.1 0 0 | ||
| OLE1 1400 0 140d-3 0.1 0 0 | ||
| API1 1400 0 184d-3 0.1 0 0 | ||
| API2 1400 0 184d-3 0.1 0 0 | ||
| LIM1 1400 0 200d-3 0.1 0 0 | ||
| LIM2 1400 0 200d-3 0.1 0 0 | ||
| CO3 2600 0 60d-3 0.53 0 0 | ||
| Na 2200 0 23d-3 1.28 0 0 | ||
| Ca 2600 0 40d-3 0.53 0 0 | ||
| OIN 2600 0 1d-3 0.1 0 0 | ||
| OC 1000 0 1d-3 0.001 0 0 | ||
| BC 1800 0 1d-3 0 0 0 | ||
| H2O 1000 0 18d-3 0 0 0 |
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| Original file line number | Diff line number | Diff line change |
|---|---|---|
| @@ -1,21 +1,21 @@ | ||
| # dens (kg/m^3) ions in soln (1) molec wght (kg/mole) kappa (1) abifm_m (1) abifm_c (1) | ||
| SO4 1800 0 96d-3 0.65 0 0 | ||
| NO3 1800 0 62d-3 0.65 0 0 | ||
| Cl 2200 0 35.5d-3 1.28 0 0 | ||
| NH4 1800 0 18d-3 0.65 0 0 | ||
| MSA 1800 0 95d-3 0.53 0 0 | ||
| ARO1 1400 0 150d-3 0.1 0 0 | ||
| ARO2 1400 0 150d-3 0.1 0 0 | ||
| ALK1 1400 0 140d-3 0.1 0 0 | ||
| OLE1 1400 0 140d-3 0.1 0 0 | ||
| API1 1400 0 184d-3 0.1 0 0 | ||
| API2 1400 0 184d-3 0.1 0 0 | ||
| LIM1 1400 0 200d-3 0.1 0 0 | ||
| LIM2 1400 0 200d-3 0.1 0 0 | ||
| CO3 2600 0 60d-3 0.53 0 0 | ||
| Na 2200 0 23d-3 1.28 0 0 | ||
| Ca 2600 0 40d-3 0.53 0 0 | ||
| OIN 2600 0 1d-3 0.1 0 0 | ||
| OC 1000 0 1d-3 0.001 0 0 | ||
| BC 1800 0 1d-3 0 0 0 | ||
| H2O 1000 0 18d-3 0 0 0 |
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