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\documentclass[11pt,letterpaper,english]{article} |
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\usepackage{amsmath} |
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%\usepackage{graphicx} |
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\usepackage{amssymb} |
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\usepackage{babel} |
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%use epsfig package for figs |
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%MICK - STUFF TO USE HELVETICA FONTS IN TEX |
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%%\usepackage[scaled=0.92]{helvet} |
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\usepackage{helvet} |
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\usepackage[sf]{titlesec} |
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%use lgrind to include code listing |
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%\usepackage{lgrind} |
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\textwidth 465pt |
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\textheight 610pt |
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\begin{document} |
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% SOME MACROS |
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\newcommand{\etal}{{\em et al.}} |
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\newcommand{\ux}{{\underline{x}}} |
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\newcommand{\tdt}{{t}} |
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{\bf {\large A1. Ecosystem Model Parametrization}} |
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The ecosystem model equations are similar that used in Follows et al. (2007). |
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Most significant change is that the grazing term is now includes variable |
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palatibility of phytoplankton and sloppy feeding as treated in Dutkiewicz et al |
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(2005), and the nitrogen limitation term has been slightly modified. For |
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fuller discussions we refer the reader to the Online Supplemental material of |
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Follows et al. (2007). |
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Several nutrients $N_i$ nourish many phytoplankton types $P_j$ which are grazed |
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by several zooplankton types $Z_k$. Mortality of and excretion from plankton, |
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and sloppy feeding by zooplankton contribute to a dissolved organic matter |
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$DOM_i$ pool and a sinking particulate organic matter pool $POM_i$. Subscript |
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$i$ refers to a nutrient/element, $j$ for a specific phytoplankton type, and |
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$k$ for a zooplankton type. |
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\begin{eqnarray} |
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\frac{\partial N_i}{\partial t} & = & |
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-\nabla \cdot (\textbf{u} N_i) +\nabla \cdot (\kappa\nabla N_i)- |
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\sum_j [\mu_j P_j M_{ij}]+S_{N_i} |
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\nonumber \\ |
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\frac{\partial P_j}{\partial t} & = & |
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-\nabla \cdot (\textbf{u} P_j) + \nabla \cdot (\kappa\nabla P_j)+ |
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\mu_j P_j - m_j^P P_j-\sum_k [g_{jk} Z_{k,i=1}] |
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-\frac{\partial(w_j^P P_j)}{\partial z} |
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\nonumber \\ |
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\frac{\partial Z_{ki}}{\partial t} & = & |
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- \nabla \cdot (\textbf{u} Z_{ki}) + \nabla \cdot (\kappa\nabla Z_{ki}) |
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+Z_{ki}\sum_j [\zeta_{jk} g_{jk} M_{ij}] -m_k^Z Z_{ki} |
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\nonumber \\ |
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\frac{\partial POM_i}{\partial t} & = & |
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-\nabla \cdot (\textbf{u} POM_i) + \nabla \cdot (\kappa\nabla POM_i)- |
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r_{PO_i}POM_i-\frac{\partial(w_{POi} POM_i)}{\partial z}+S_{POM_i} |
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\nonumber \\ |
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\frac{\partial DOM_i}{\partial t} & = & |
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-\nabla \cdot (\textbf{u} DOM_i) + \nabla \cdot (\kappa\nabla DOM_i)- |
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r_{DO_i}DOM_i + S_{DOM_i} |
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\nonumber |
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\end{eqnarray} |
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where:\\ |
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\mbox{} \hspace{.5cm} $\textbf{u}=(u,v,w),$ velocity in physical model, \\ |
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\mbox{} \hspace{.5cm} $\kappa=$Mixing coefficients used in physical model,\\ |
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\mbox{} \hspace{.5cm} $\mu_j=$Growth rate of phytoplankton $j$ (see below),\\ |
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\mbox{} \hspace{.5cm} $M_{ij}=$Matrix of Redfield ratio of element $i$ to P |
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for phytoplankton $j$\\ |
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\mbox{} \hspace{.5cm} $\zeta_{jk}=$ Grazing efficiency of zooplankton $k$ on phytoplankton $j$ (represents sloppy feeding), \\ |
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\mbox{} \hspace{.5cm} $g_{jk}=$Grazing of zooplankton $k$ on phytoplankton $j$ (see below),\\ |
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\mbox{} \hspace{.5cm} $m_j^P=$Mortality/Excretion rate for phytoplankton $j$,\\ |
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\mbox{} \hspace{.5cm} $m_k^Z=$Mortality/Excretion rate for zooplankton $k$,\\ |
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\mbox{} \hspace{.5cm} $w_j^P=$Sinking rate for phytoplankton $j$,\\ |
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\mbox{} \hspace{.5cm} $w_{POi}=$Sinking rate for POM $i$,\\ |
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\mbox{} \hspace{.5cm} $r_{DOM_i}=$Remineralization rate of DOM for element |
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$i$,\\ |
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\mbox{} \hspace{.5cm} $r_{POM_i}=$Remineralization rate of POM for element |
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$i$,\\ |
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\mbox{} \hspace{.5cm} $S_{N_i}=$Additional source or sink for nutrient $i$ |
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(see below),\\ |
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\mbox{} \hspace{.5cm} $S_{DOM_i}=$ Source of DOM $i$, |
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for element $i$ (see below),\\ |
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\mbox{} \hspace{.5cm} $S_{POM_i}=$ Source of POM $i$, |
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for element $i$ (see below),\\ |
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\vspace{.2cm} |
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{\it {\bf A1.1. Phytoplankton growth:}}\\ |
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\[ |
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\mu_j = \mu_{max_{j}} \gamma_j^T \gamma_j^I \gamma_j^N |
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\] |
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where\\ |
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\mbox{} \hspace{.5cm} $\mu_{max_{j}}=$ maximum growth rate of phytoplankton $j$,\\ |
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\mbox{} \hspace{.5cm} $\gamma_j^T=$Modification of growth rate by |
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temperature for phytoplankton $j$,\\ |
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\mbox{} \hspace{.5cm} $\gamma_j^I=$Modification of growth rate by light for |
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phytoplankton $j$,\\ |
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\mbox{} \hspace{.5cm} $\gamma_j^N=$Modification of growth rate by nutrients |
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for phytoplankton $j$.\\ |
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Temperature modification (Fig. \ref{fig-growexp1}a):\\ |
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\[ |
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\gamma_j^T= \frac{1}{\tau_1} (A^T e^{-B(T-T_o)^c} - \tau_2 ) |
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\] |
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where coefficients $\tau_1$ and $\tau_2$ normalize the maximum |
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value, and $A,B,T_o$ and $C$ regulate the form of the temperature |
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modification function. $T$ is the local model ocean temperature. |
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Light modification (Fig. \ref{fig-growexp1}b):\\ |
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\[ |
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\gamma_j^I= \frac{1}{F_o} (1-e^{k_{par} I} ) e^{-k_{inhib} I} |
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\] |
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where $F_{o}$ is a factor controlling the maximum value, $k_{par}$ is the |
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PAR saturation coefficient and $k_{inhib}$ is the PAR inhibition factor. |
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$I$ is the local PAR, that has been attenuated through the water column |
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(including the effects of self-shading). |
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Nutrient limitation is determined by the most limiting nutrient: |
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\[ |
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\gamma_j^N = \min(N_i^{lim}) |
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\] |
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where typically |
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$N_i^{lim}=\frac{N_i}{N_i+\kappa_{N_{ij}}}$ |
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(Fig. \ref{fig-growexp1}c) and $\kappa_{N_{ij}}$ is the half saturation constant of nutrient $i$ for phytoplankton $j$. |
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When we include the nitrogen as a potential limiting nutrient (EXP2) we |
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modify $N_i^{lim}$ to take into account the uptake inhibition caused by ammonium: |
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\begin{align*} |
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N_N^{lim} &= \frac{NO_2}{NO_2+\kappa_{IN}} e^{-\psi NH_4} |
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+\frac{NH_4}{NH_4 + \kappa_{NH4}} && \text{(nsource=1)} \\ |
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N_N^{lim} &= \frac{NH_4}{NH_4 + \kappa_{NH4}} && \text{(nsource=2)} \\ |
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N_N^{lim} &= \frac{NO_3 + NO_2}{NO_3+NO_2+\kappa_{IN}} e^{-\psi NH_4} |
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+\frac{NH_4}{NH_4 + \kappa_{NH4}} && \text{(nsource=3)} |
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\end{align*} |
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where $\psi$ reflects the inhibition and $\kappa_{IN}$ and $ \kappa_{NH4}$ |
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are the half saturation constant of $IN=NO_3+NO_2$ and $NH_4$ respectively. |
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\vspace{.2cm} |
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{\it {\bf A1.2. Zooplankton grazing:}}\\ |
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\[ |
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g_{jk} =g_{max_{jk}} \frac{\eta_{jk} P_j}{A_k} \frac{A_k}{A_k+\kappa^P_k} |
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\] |
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where\\ |
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\mbox{} \hspace{.5cm} $g_{max_{jk}}=$ Maximum grazing rate of zooplankton $k$ on |
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phytoplankton $j$,\\ |
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\mbox{} \hspace{.5cm} $\eta_{jk}=$ Palatibility of plankton $j$ to zooplankton $k$,\\ |
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\mbox{} \hspace{.5cm} $A_k=$ Palatibility (for zooplankton $k$) weighted total phytoplankton concentration,\\ |
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\mbox{} \hspace{1.1cm} $=\sum_j [\eta_{jk} P_j$] \\ |
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\mbox{} \hspace{.5cm} $\kappa^P_k=$Half-saturation constant for grazing of zooplankton $k$,\\ |
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\vspace{.2cm} |
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{\it {\bf A1.3. Inorganic nutrient Source/Sink terms:}}\\ |
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$S_{N_i}$ depends on the specific nutrient, and includes the remineralization |
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of organic matter, external sources and other non-biological transformations: |
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\begin{eqnarray} |
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S_{PO4} & = & r_{DOP} DOP + r_{POP} POP \nonumber \\ |
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S_{Si} & = & r_{POSi} POSi \nonumber \\ |
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S_{FeT} & = & r_{DOFe} DOFe + r_{POFe} POFe -c_{scav} Fe' + \alpha F_{atmos} \nonumber \\ |
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S_{NO3} & = & \zeta_{NO3} NO_2 \nonumber \\ |
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S_{NO2} & = & \zeta_{NO2} NH4 - \zeta_{NO3} NO_2 \nonumber \\ |
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S_{NH4} & = & r_{DON} DON + r_{PON} PON - \zeta_{NO2} NH_4 \nonumber |
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\end{eqnarray} |
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where:\\ |
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\mbox{} \hspace{.5cm} $r_{DOM_i}=$Remineralization rate of DOM for element |
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$i$, here P, Fe, N,\\ |
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\mbox{} \hspace{.5cm} $r_{POM_i}=$Remineralization rate of POM for element |
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$i$, here P, Si, Fe, N,\\ |
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\mbox{} \hspace{.5cm} $c_{scav}=$scavenging rate for free iron,\\ |
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\mbox{} \hspace{.5cm} $Fe'=$free iron, modelled as in Parekh et al (2004), \\ |
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\mbox{} \hspace{.5cm} $alpha=$solubility of iron dust in ocean water, \\ |
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\mbox{} \hspace{.5cm} $F_{atmos}=$atmospheric deposition of iron dust on surface of model ocean,\\ |
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\mbox{} \hspace{.5cm} $\zeta_{NO3}=\zeta_{NO3}^0(1-I/I_0)_+=$oxidation rate of NO$_2$ to NO$_3$,\\ |
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\mbox{} \hspace{.5cm} $\zeta_{NO2}=\zeta_{NO2}^0(1-I/I_0)_+=$oxidation rate of NH$_4$ to NO$_2$ (is photoinhibited),\\ |
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\mbox{} \hspace{.5cm} $I_0=$critical light level below which oxidation occurs,\\ |
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The remineralization timescale $r_{DOi}$ and $r_{POi}$ parameterizes the break |
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down of organic matter to an inorganic form through the microbial loop. |
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{\it {\bf A1.3.1 Fe chemistry:}}\\ |
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\begin{eqnarray} |
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Fe' & = & FeT - FeL \nonumber \\ |
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FeL & = & L_{tot} - |
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\frac{ L_{stab} (L_{tot} - FeT) - 1 |
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+\sqrt{(1 - L_{stab} (L_{tot} - FeT))^2 + 4 L_{stab} L_{tot}}} |
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{2 L_{stab}} \nonumber |
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\end{eqnarray} |
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($Fe'$ may be constrained to be less than $Fe'_{max}$ while preserving $FeT$). |
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{\it {\bf A1.4 DOM and POM Source terms:}}\\ |
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$S_{DOM_i}$ and $S_{POM_i}$ are the sources of dissolved and particulate |
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organic detritus arising from mortality, excretion and sloppy feeding of the |
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plantkon. We simply define that a fixed fraction $\lambda_m$ of the the |
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mortality/excretion term and the non-consumed grazed phytoplankton |
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($\lambda_g$) go into the dissolved pool and the remainder into the particulate |
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pool. |
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\begin{eqnarray} |
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S_{DOM_i} & = & \sum_{j} [\lambda_{mp_{ij}} m^p_j P_j M_{ij}] |
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+ \sum_{k} [\lambda_{mz_{ik}} m^z_k Z_{ik}] |
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+ \sum_{k} \sum_{j} [\lambda_{g_{ijk}} (1-\zeta_{jk}) |
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g_{ij} M_{ij} Z_k ] |
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\nonumber \\ |
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S_{POM_i} & = & \sum_{j} [(1-\lambda_{m_{ij}}) m^p_j P_j M_{ij}] |
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+ \sum_{k} [(1-\lambda_{mz_{ik}}) m^z_k Z_{ik}] |
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+ \sum_{k} \sum_{j} [(1-\lambda_{g_{ijk}}) (1-\zeta_{jk}) |
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g_{ij} M_{ij} Z_k ] |
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\nonumber |
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\end{eqnarray} |
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\newcommand{\pcm}[1]{P^C_{m#1}} |
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\newcommand{\pcmax}[1]{P^C_{\textrm{MAX}#1}} |
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\newcommand{\pcarbon}{P^C} |
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\newcommand{\chltoc}{\theta} |
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\newcommand{\chltocmax}{\theta^{\textrm{max}}} |
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\newcommand{\chltocmin}{\theta^{\textrm{min}}} |
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\newcommand{\alphachl}{\alpha^{\textrm{Chl}}} |
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\newcommand{\mQyield}{\mathit{mQ}^{\textrm{yield}}} |
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\newcommand{\RPC}{R^{PC}} |
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\newcommand{\phychl}{\mathit{Chl}} |
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\newcommand{\aphychlave}{A^{\mathrm{phy}}_{\mathrm{Chl,ave}}} |
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{\it {\bf A1.4 Geider light limitation model:}}\\ |
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The phytoplankton growth rate is given by the carbon-specific photosynthesis rate |
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(rate of carbon synthesized per carbon present), |
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\[ |
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\mu_j = \pcarbon_j |
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\] |
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The carbon-specific photosynthesis rate |
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\[ |
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\pcarbon_j = \pcm{,j} \begin{cases} |
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1 - e^{-\alphachl_j I \chltoc_j/\pcm{,j}} & \text{if }I>0.1 \\ |
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0 & \text{otherwise} |
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\end{cases} |
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\] |
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depends on the carbon-specific, light-saturated photosynthesis rate |
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\[ |
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\pcm{,j}=\pcmax{j} \gamma^N_j \gamma^T_j |
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\] |
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and the Chl $a$ to carbon ratio |
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\[ |
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\chltoc_j = \left[ \frac{\chltocmax_j} |
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{1 + \chltocmax_j \alphachl_j I / (2 \pcm{,j})} |
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\right]^{\chltocmax_j}_{\chltocmin_j} |
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\] |
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The chlorophyll concentration is |
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\[ |
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\phychl_j=P_j \RPC_j \chltoc_j |
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\] |
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The light limitation factor can be diagnosed |
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\[ |
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\gamma^I_j=\pcarbon_j/\pcm{,j} |
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\] |
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\[ |
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\alphachl_j = \mQyield_j \aphychlave |
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\] |
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Parameters:\\ |
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\begin{tabular}{@{\qquad}r@{}l} |
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$\pcmax{j} ={}$& Maximum C-spec.\ photosynthesis rate at reference temperature of phytoplankton $j$\\ |
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$\chltocmax_j ={}$& Maximum Chl a to C ratio if phytoplankton $j$\\ |
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$\RPC_j ={}$& Carbon to phosphorus (!) ratio of phytoplankton $j$\\ |
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$\alphachl_j ={}$& Chl a-specific initial slope of the photosynthesis-light curve\\ |
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$\mQyield_j ={}$& slope of the photosynthesis-light curve per absorption\\ |
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$\aphychlave ={}$& absorption ($m^{-1}$) per mg Chl a |
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\end{tabular} |
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\newcommand{\Ptot}{P_{\mathrm{tot}}} |
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{\it {\bf A2 Diagnostics:}}\\ |
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Total phytoplankton biomass: |
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\[ |
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\Ptot = \sum_j P_j |
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\] |
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\begin{tabular}{llll} |
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name & definition && units \\ |
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\hline |
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\texttt{PhyTot } & $\Ptot$ && $\mu\mathrm{M\,P}$ \\ |
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\texttt{PhyGrp1 } & Total biomass of small phytoplankton with $\texttt{nsrc}=1$ && $\mu\mathrm{M\,P}$ \\ |
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\texttt{PhyGrp2 } & Total biomass of small phytoplankton with $\texttt{nsrc}=2$ && $\mu\mathrm{M\,P}$ \\ |
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\texttt{PhyGrp3 } & Total biomass of small phytoplankton with $\texttt{nsrc}=3$ && $\mu\mathrm{M\,P}$ \\ |
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\texttt{PhyGrp4 } & Total biomass of large non-diatoms && $\mu\mathrm{M\,P}$ \\ |
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\texttt{PhyGrp5 } & Total biomass of diatoms && $\mu\mathrm{M\,P}$ \\ |
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\texttt{PP } & Primary production && $\mu\mathrm{M\,P}\, \mathrm{s}^{-1}$ \\ |
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\texttt{Nfix } & Nitrogen fixation && $\mu\mathrm{M\,N}\, \mathrm{s}^{-1}$ \\ |
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\texttt{PAR } & Photosynthetically active radiation && $\mu\mathrm{Ein}\, \mathrm{m}^{-2}\,\mathrm{s}^{-1}$ \\ |
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\texttt{Rstar01 } & $R^*_{\mathrm{PO4}}$ of Phytoplankton species \#1, \dots && $\mu\mathrm{M\,P}$ \\ |
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\texttt{Diver1 } & Number of species with $P_j > 10^{-8}\,\mu\mathrm{M\,P}$ & where $\Ptot>10^{-12}$ \\ |
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\texttt{Diver2 } & Number of species with $P_j > 0.1\%\, \Ptot$ & where $\Ptot>10^{-12}$ \\ |
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\texttt{Diver3 } & Number of species that constitute 99.9\% of $\Ptot$ & where $\Ptot>10^{-12}$ \\ |
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\texttt{Diver4 } & Number of species with $P_j > 10^{-5} \cdot \max\limits_j P_j$ & where $\Ptot>10^{-12}$ \\ |
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\end{tabular} |
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\end{document} |
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