| 1 |
\documentclass[11pt,letterpaper,english]{article} |
| 2 |
\usepackage[T1]{fontenc} |
| 3 |
\usepackage[latin1]{inputenc} |
| 4 |
\setlength\parskip{\medskipamount} |
| 5 |
\setlength\parindent{0pt} |
| 6 |
\usepackage{amsmath} |
| 7 |
%\usepackage{graphicx} |
| 8 |
\usepackage{amssymb} |
| 9 |
\usepackage{babel} |
| 10 |
%use epsfig package for figs |
| 11 |
\usepackage{epsfig} |
| 12 |
\makeatother |
| 13 |
|
| 14 |
|
| 15 |
%MICK - STUFF TO USE HELVETICA FONTS IN TEX |
| 16 |
%%\usepackage[scaled=0.92]{helvet} |
| 17 |
\usepackage{helvet} |
| 18 |
\usepackage[sf]{titlesec} |
| 19 |
\renewcommand\familydefault{\sfdefault} |
| 20 |
|
| 21 |
%use lgrind to include code listing |
| 22 |
%\usepackage{lgrind} |
| 23 |
|
| 24 |
%other formatting stuff |
| 25 |
\oddsidemargin 0pt |
| 26 |
\flushbottom |
| 27 |
\parskip 10pt |
| 28 |
\parindent 0pt |
| 29 |
\textwidth 465pt |
| 30 |
\topmargin 10pt |
| 31 |
\textheight 610pt |
| 32 |
\renewcommand{\baselinestretch}{1.0} |
| 33 |
|
| 34 |
|
| 35 |
\begin{document} |
| 36 |
|
| 37 |
% SOME MACROS |
| 38 |
\newcommand{\etal}{{\em et al.}} |
| 39 |
\newcommand{\ux}{{\underline{x}}} |
| 40 |
\newcommand{\tdt}{{t}} |
| 41 |
|
| 42 |
|
| 43 |
{\bf {\large A1. Ecosystem Model Parametrization}} |
| 44 |
|
| 45 |
The ecosystem model equations are similar that used in Follows et al. (2007). |
| 46 |
Most significant change is that the grazing term is now includes variable |
| 47 |
palatibility of phytoplankton and sloppy feeding as treated in Dutkiewicz et al |
| 48 |
(2005), and the nitrogen limitation term has been slightly modified. For |
| 49 |
fuller discussions we refer the reader to the Online Supplemental material of |
| 50 |
Follows et al. (2007). |
| 51 |
|
| 52 |
Several nutrients $N_i$ nourish many phytoplankton types $P_j$ which are grazed |
| 53 |
by several zooplankton types $Z_k$. Mortality of and excretion from plankton, |
| 54 |
and sloppy feeding by zooplankton contribute to a dissolved organic matter |
| 55 |
$DOM_i$ pool and a sinking particulate organic matter pool $POM_i$. Subscript |
| 56 |
$i$ refers to a nutrient/element, $j$ for a specific phytoplankton type, and |
| 57 |
$k$ for a zooplankton type. |
| 58 |
|
| 59 |
\begin{eqnarray} |
| 60 |
\frac{\partial N_i}{\partial t} & = & |
| 61 |
-\nabla \cdot (\textbf{u} N_i) +\nabla \cdot (\kappa\nabla N_i)- |
| 62 |
\sum_j [\mu_j P_j M_{ij}]+S_{N_i} |
| 63 |
\nonumber \\ |
| 64 |
\frac{\partial P_j}{\partial t} & = & |
| 65 |
-\nabla \cdot (\textbf{u} P_j) + \nabla \cdot (\kappa\nabla P_j)+ |
| 66 |
\mu_j P_j - m_j^P P_j-\sum_k [g_{jk} Z_{k,i=1}] |
| 67 |
-\frac{\partial(w_j^P P_j)}{\partial z} |
| 68 |
\nonumber \\ |
| 69 |
\frac{\partial Z_{ki}}{\partial t} & = & |
| 70 |
- \nabla \cdot (\textbf{u} Z_{ki}) + \nabla \cdot (\kappa\nabla Z_{ki}) |
| 71 |
+Z_{ki}\sum_j [\zeta_{jk} g_{jk} M_{ij}] -m_k^Z Z_{ki} |
| 72 |
\nonumber \\ |
| 73 |
\frac{\partial POM_i}{\partial t} & = & |
| 74 |
-\nabla \cdot (\textbf{u} POM_i) + \nabla \cdot (\kappa\nabla POM_i)- |
| 75 |
r_{PO_i}POM_i-\frac{\partial(w_{POi} POM_i)}{\partial z}+S_{POM_i} |
| 76 |
\nonumber \\ |
| 77 |
\frac{\partial DOM_i}{\partial t} & = & |
| 78 |
-\nabla \cdot (\textbf{u} DOM_i) + \nabla \cdot (\kappa\nabla DOM_i)- |
| 79 |
r_{DO_i}DOM_i + S_{DOM_i} |
| 80 |
\nonumber |
| 81 |
\end{eqnarray} |
| 82 |
|
| 83 |
where:\\ |
| 84 |
\mbox{} \hspace{.5cm} $\textbf{u}=(u,v,w),$ velocity in physical model, \\ |
| 85 |
\mbox{} \hspace{.5cm} $\kappa=$Mixing coefficients used in physical model,\\ |
| 86 |
\mbox{} \hspace{.5cm} $\mu_j=$Growth rate of phytoplankton $j$ (see below),\\ |
| 87 |
\mbox{} \hspace{.5cm} $M_{ij}=$Matrix of Redfield ratio of element $i$ to P |
| 88 |
for phytoplankton $j$\\ |
| 89 |
\mbox{} \hspace{.5cm} $\zeta_{jk}=$ Grazing efficiency of zooplankton $k$ on phytoplankton $j$ (represents sloppy feeding), \\ |
| 90 |
\mbox{} \hspace{.5cm} $g_{jk}=$Grazing of zooplankton $k$ on phytoplankton $j$ (see below),\\ |
| 91 |
\mbox{} \hspace{.5cm} $m_j^P=$Mortality/Excretion rate for phytoplankton $j$,\\ |
| 92 |
\mbox{} \hspace{.5cm} $m_k^Z=$Mortality/Excretion rate for zooplankton $k$,\\ |
| 93 |
\mbox{} \hspace{.5cm} $w_j^P=$Sinking rate for phytoplankton $j$,\\ |
| 94 |
\mbox{} \hspace{.5cm} $w_{POi}=$Sinking rate for POM $i$,\\ |
| 95 |
\mbox{} \hspace{.5cm} $r_{DOM_i}=$Remineralization rate of DOM for element |
| 96 |
$i$,\\ |
| 97 |
\mbox{} \hspace{.5cm} $r_{POM_i}=$Remineralization rate of POM for element |
| 98 |
$i$,\\ |
| 99 |
\mbox{} \hspace{.5cm} $S_{N_i}=$Additional source or sink for nutrient $i$ |
| 100 |
(see below),\\ |
| 101 |
\mbox{} \hspace{.5cm} $S_{DOM_i}=$ Source of DOM $i$, |
| 102 |
for element $i$ (see below),\\ |
| 103 |
\mbox{} \hspace{.5cm} $S_{POM_i}=$ Source of POM $i$, |
| 104 |
for element $i$ (see below),\\ |
| 105 |
|
| 106 |
|
| 107 |
|
| 108 |
\vspace{.2cm} |
| 109 |
|
| 110 |
{\it {\bf A1.1. Phytoplankton growth:}}\\ |
| 111 |
\[ |
| 112 |
\mu_j = \mu_{max_{j}} \gamma_j^T \gamma_j^I \gamma_j^N |
| 113 |
\] |
| 114 |
where\\ |
| 115 |
\mbox{} \hspace{.5cm} $\mu_{max_{j}}=$ maximum growth rate of phytoplankton $j$,\\ |
| 116 |
\mbox{} \hspace{.5cm} $\gamma_j^T=$Modification of growth rate by |
| 117 |
temperature for phytoplankton $j$,\\ |
| 118 |
\mbox{} \hspace{.5cm} $\gamma_j^I=$Modification of growth rate by light for |
| 119 |
phytoplankton $j$,\\ |
| 120 |
\mbox{} \hspace{.5cm} $\gamma_j^N=$Modification of growth rate by nutrients |
| 121 |
for phytoplankton $j$.\\ |
| 122 |
|
| 123 |
Temperature modification (Fig. \ref{fig-growexp1}a):\\ |
| 124 |
\[ |
| 125 |
\gamma_j^T= \frac{1}{\tau_1} (A^T e^{-B(T-T_o)^c} - \tau_2 ) |
| 126 |
\] |
| 127 |
where coefficients $\tau_1$ and $\tau_2$ normalize the maximum |
| 128 |
value, and $A,B,T_o$ and $C$ regulate the form of the temperature |
| 129 |
modification function. $T$ is the local model ocean temperature. |
| 130 |
|
| 131 |
Light modification (Fig. \ref{fig-growexp1}b):\\ |
| 132 |
\[ |
| 133 |
\gamma_j^I= \frac{1}{F_o} (1-e^{k_{par} I} ) e^{-k_{inhib} I} |
| 134 |
\] |
| 135 |
where $F_{o}$ is a factor controlling the maximum value, $k_{par}$ is the |
| 136 |
PAR saturation coefficient and $k_{inhib}$ is the PAR inhibition factor. |
| 137 |
$I$ is the local PAR, that has been attenuated through the water column |
| 138 |
(including the effects of self-shading). |
| 139 |
|
| 140 |
Nutrient limitation is determined by the most limiting nutrient: |
| 141 |
\[ |
| 142 |
\gamma_j^N = \min(N_i^{lim}) |
| 143 |
\] |
| 144 |
where typically |
| 145 |
$N_i^{lim}=\frac{N_i}{N_i+\kappa_{N_{ij}}}$ |
| 146 |
(Fig. \ref{fig-growexp1}c) and $\kappa_{N_{ij}}$ is the half saturation constant of nutrient $i$ for phytoplankton $j$. |
| 147 |
|
| 148 |
When we include the nitrogen as a potential limiting nutrient (EXP2) we |
| 149 |
modify $N_i^{lim}$ to take into account the uptake inhibition caused by ammonium: |
| 150 |
\begin{align*} |
| 151 |
N_N^{lim} &= \frac{NO_2}{NO_2+\kappa_{IN}} e^{-\psi NH_4} |
| 152 |
+\frac{NH_4}{NH_4 + \kappa_{NH4}} && \text{(nsource=1)} \\ |
| 153 |
N_N^{lim} &= \frac{NH_4}{NH_4 + \kappa_{NH4}} && \text{(nsource=2)} \\ |
| 154 |
N_N^{lim} &= \frac{NO_3 + NO_2}{NO_3+NO_2+\kappa_{IN}} e^{-\psi NH_4} |
| 155 |
+\frac{NH_4}{NH_4 + \kappa_{NH4}} && \text{(nsource=3)} |
| 156 |
\end{align*} |
| 157 |
where $\psi$ reflects the inhibition and $\kappa_{IN}$ and $ \kappa_{NH4}$ |
| 158 |
are the half saturation constant of $IN=NO_3+NO_2$ and $NH_4$ respectively. |
| 159 |
|
| 160 |
\vspace{.2cm} |
| 161 |
|
| 162 |
{\it {\bf A1.2. Zooplankton grazing:}}\\ |
| 163 |
\[ |
| 164 |
g_{jk} =g_{max_{jk}} \frac{\eta_{jk} P_j}{A_k} \frac{A_k}{A_k+\kappa^P_k} |
| 165 |
\] |
| 166 |
where\\ |
| 167 |
\mbox{} \hspace{.5cm} $g_{max_{jk}}=$ Maximum grazing rate of zooplankton $k$ on |
| 168 |
phytoplankton $j$,\\ |
| 169 |
\mbox{} \hspace{.5cm} $\eta_{jk}=$ Palatibility of plankton $j$ to zooplankton $k$,\\ |
| 170 |
\mbox{} \hspace{.5cm} $A_k=$ Palatibility (for zooplankton $k$) weighted total phytoplankton concentration,\\ |
| 171 |
\mbox{} \hspace{1.1cm} $=\sum_j [\eta_{jk} P_j$] \\ |
| 172 |
\mbox{} \hspace{.5cm} $\kappa^P_k=$Half-saturation constant for grazing of zooplankton $k$,\\ |
| 173 |
|
| 174 |
|
| 175 |
\vspace{.2cm} |
| 176 |
|
| 177 |
{\it {\bf A1.3. Inorganic nutrient Source/Sink terms:}}\\ |
| 178 |
$S_{N_i}$ depends on the specific nutrient, and includes the remineralization |
| 179 |
of organic matter, external sources and other non-biological transformations: |
| 180 |
\begin{eqnarray} |
| 181 |
S_{PO4} & = & r_{DOP} DOP + r_{POP} POP \nonumber \\ |
| 182 |
S_{Si} & = & r_{POSi} POSi \nonumber \\ |
| 183 |
S_{FeT} & = & r_{DOFe} DOFe + r_{POFe} POFe -c_{scav} Fe' + \alpha F_{atmos} \nonumber \\ |
| 184 |
S_{NO3} & = & \zeta_{NO3} NO_2 \nonumber \\ |
| 185 |
S_{NO2} & = & \zeta_{NO2} NH4 - \zeta_{NO3} NO_2 \nonumber \\ |
| 186 |
S_{NH4} & = & r_{DON} DON + r_{PON} PON - \zeta_{NO2} NH_4 \nonumber |
| 187 |
\end{eqnarray} |
| 188 |
|
| 189 |
where:\\ |
| 190 |
\mbox{} \hspace{.5cm} $r_{DOM_i}=$Remineralization rate of DOM for element |
| 191 |
$i$, here P, Fe, N,\\ |
| 192 |
\mbox{} \hspace{.5cm} $r_{POM_i}=$Remineralization rate of POM for element |
| 193 |
$i$, here P, Si, Fe, N,\\ |
| 194 |
\mbox{} \hspace{.5cm} $c_{scav}=$scavenging rate for free iron,\\ |
| 195 |
\mbox{} \hspace{.5cm} $Fe'=$free iron, modelled as in Parekh et al (2004), \\ |
| 196 |
\mbox{} \hspace{.5cm} $alpha=$solubility of iron dust in ocean water, \\ |
| 197 |
\mbox{} \hspace{.5cm} $F_{atmos}=$atmospheric deposition of iron dust on surface of model ocean,\\ |
| 198 |
\mbox{} \hspace{.5cm} $\zeta_{NO3}=\zeta_{NO3}^0(1-I/I_0)_+=$oxidation rate of NO$_2$ to NO$_3$,\\ |
| 199 |
\mbox{} \hspace{.5cm} $\zeta_{NO2}=\zeta_{NO2}^0(1-I/I_0)_+=$oxidation rate of NH$_4$ to NO$_2$ (is photoinhibited),\\ |
| 200 |
\mbox{} \hspace{.5cm} $I_0=$critical light level below which oxidation occurs,\\ |
| 201 |
|
| 202 |
The remineralization timescale $r_{DOi}$ and $r_{POi}$ parameterizes the break |
| 203 |
down of organic matter to an inorganic form through the microbial loop. |
| 204 |
|
| 205 |
|
| 206 |
{\it {\bf A1.3.1 Fe chemistry:}}\\ |
| 207 |
\begin{eqnarray} |
| 208 |
Fe' & = & FeT - FeL \nonumber \\ |
| 209 |
FeL & = & L_{tot} - |
| 210 |
\frac{ L_{stab} (L_{tot} - FeT) - 1 |
| 211 |
+\sqrt{(1 - L_{stab} (L_{tot} - FeT))^2 + 4 L_{stab} L_{tot}}} |
| 212 |
{2 L_{stab}} \nonumber |
| 213 |
\end{eqnarray} |
| 214 |
($Fe'$ may be constrained to be less than $Fe'_{max}$ while preserving $FeT$). |
| 215 |
|
| 216 |
|
| 217 |
{\it {\bf A1.4 DOM and POM Source terms:}}\\ |
| 218 |
$S_{DOM_i}$ and $S_{POM_i}$ are the sources of dissolved and particulate |
| 219 |
organic detritus arising from mortality, excretion and sloppy feeding of the |
| 220 |
plantkon. We simply define that a fixed fraction $\lambda_m$ of the the |
| 221 |
mortality/excretion term and the non-consumed grazed phytoplankton |
| 222 |
($\lambda_g$) go into the dissolved pool and the remainder into the particulate |
| 223 |
pool. |
| 224 |
\begin{eqnarray} |
| 225 |
S_{DOM_i} & = & \sum_{j} [\lambda_{mp_{ij}} m^p_j P_j M_{ij}] |
| 226 |
+ \sum_{k} [\lambda_{mz_{ik}} m^z_k Z_{ik}] |
| 227 |
+ \sum_{k} \sum_{j} [\lambda_{g_{ijk}} (1-\zeta_{jk}) |
| 228 |
g_{ij} M_{ij} Z_k ] |
| 229 |
\nonumber \\ |
| 230 |
S_{POM_i} & = & \sum_{j} [(1-\lambda_{m_{ij}}) m^p_j P_j M_{ij}] |
| 231 |
+ \sum_{k} [(1-\lambda_{mz_{ik}}) m^z_k Z_{ik}] |
| 232 |
+ \sum_{k} \sum_{j} [(1-\lambda_{g_{ijk}}) (1-\zeta_{jk}) |
| 233 |
g_{ij} M_{ij} Z_k ] |
| 234 |
\nonumber |
| 235 |
\end{eqnarray} |
| 236 |
|
| 237 |
|
| 238 |
\newcommand{\pcm}[1]{P^C_{m#1}} |
| 239 |
\newcommand{\pcmax}[1]{P^C_{\textrm{MAX}#1}} |
| 240 |
\newcommand{\pcarbon}{P^C} |
| 241 |
\newcommand{\chltoc}{\theta} |
| 242 |
\newcommand{\chltocmax}{\theta^{\textrm{max}}} |
| 243 |
\newcommand{\chltocmin}{\theta^{\textrm{min}}} |
| 244 |
\newcommand{\alphachl}{\alpha^{\textrm{Chl}}} |
| 245 |
\newcommand{\mQyield}{\mathit{mQ}^{\textrm{yield}}} |
| 246 |
\newcommand{\RPC}{R^{PC}} |
| 247 |
\newcommand{\phychl}{\mathit{Chl}} |
| 248 |
\newcommand{\aphychlave}{A^{\mathrm{phy}}_{\mathrm{Chl,ave}}} |
| 249 |
|
| 250 |
{\it {\bf A1.4 Geider light limitation model:}}\\ |
| 251 |
The phytoplankton growth rate is given by the carbon-specific photosynthesis rate |
| 252 |
(rate of carbon synthesized per carbon present), |
| 253 |
\[ |
| 254 |
\mu_j = \pcarbon_j |
| 255 |
\] |
| 256 |
The carbon-specific photosynthesis rate |
| 257 |
\[ |
| 258 |
\pcarbon_j = \pcm{,j} \begin{cases} |
| 259 |
1 - e^{-\alphachl_j I \chltoc_j/\pcm{,j}} & \text{if }I>0.1 \\ |
| 260 |
0 & \text{otherwise} |
| 261 |
\end{cases} |
| 262 |
\] |
| 263 |
depends on the carbon-specific, light-saturated photosynthesis rate |
| 264 |
\[ |
| 265 |
\pcm{,j}=\pcmax{j} \gamma^N_j \gamma^T_j |
| 266 |
\] |
| 267 |
and the Chl $a$ to carbon ratio |
| 268 |
\[ |
| 269 |
\chltoc_j = \left[ \frac{\chltocmax_j} |
| 270 |
{1 + \chltocmax_j \alphachl_j I / (2 \pcm{,j})} |
| 271 |
\right]^{\chltocmax_j}_{\chltocmin_j} |
| 272 |
\] |
| 273 |
|
| 274 |
The chlorophyll concentration is |
| 275 |
\[ |
| 276 |
\phychl_j=P_j \RPC_j \chltoc_j |
| 277 |
\] |
| 278 |
|
| 279 |
The light limitation factor can be diagnosed |
| 280 |
\[ |
| 281 |
\gamma^I_j=\pcarbon_j/\pcm{,j} |
| 282 |
\] |
| 283 |
|
| 284 |
\[ |
| 285 |
\alphachl_j = \mQyield_j \aphychlave |
| 286 |
\] |
| 287 |
|
| 288 |
Parameters:\\ |
| 289 |
\begin{tabular}{@{\qquad}r@{}l} |
| 290 |
$\pcmax{j} ={}$& Maximum C-spec.\ photosynthesis rate at reference temperature of phytoplankton $j$\\ |
| 291 |
$\chltocmax_j ={}$& Maximum Chl a to C ratio if phytoplankton $j$\\ |
| 292 |
$\RPC_j ={}$& Carbon to phosphorus (!) ratio of phytoplankton $j$\\ |
| 293 |
$\alphachl_j ={}$& Chl a-specific initial slope of the photosynthesis-light curve\\ |
| 294 |
$\mQyield_j ={}$& slope of the photosynthesis-light curve per absorption\\ |
| 295 |
$\aphychlave ={}$& absorption ($m^{-1}$) per mg Chl a |
| 296 |
\end{tabular} |
| 297 |
|
| 298 |
|
| 299 |
|
| 300 |
\newcommand{\Ptot}{P_{\mathrm{tot}}} |
| 301 |
|
| 302 |
{\it {\bf A2 Diagnostics:}}\\ |
| 303 |
Total phytoplankton biomass: |
| 304 |
\[ |
| 305 |
\Ptot = \sum_j P_j |
| 306 |
\] |
| 307 |
|
| 308 |
\begin{tabular}{llll} |
| 309 |
name & definition && units \\ |
| 310 |
\hline |
| 311 |
\texttt{PhyTot } & $\Ptot$ && $\mu\mathrm{M\,P}$ \\ |
| 312 |
\texttt{PhyGrp1 } & Total biomass of small phytoplankton with $\texttt{nsrc}=1$ && $\mu\mathrm{M\,P}$ \\ |
| 313 |
\texttt{PhyGrp2 } & Total biomass of small phytoplankton with $\texttt{nsrc}=2$ && $\mu\mathrm{M\,P}$ \\ |
| 314 |
\texttt{PhyGrp3 } & Total biomass of small phytoplankton with $\texttt{nsrc}=3$ && $\mu\mathrm{M\,P}$ \\ |
| 315 |
\texttt{PhyGrp4 } & Total biomass of large non-diatoms && $\mu\mathrm{M\,P}$ \\ |
| 316 |
\texttt{PhyGrp5 } & Total biomass of diatoms && $\mu\mathrm{M\,P}$ \\ |
| 317 |
\texttt{PP } & Primary production && $\mu\mathrm{M\,P}\, \mathrm{s}^{-1}$ \\ |
| 318 |
\texttt{Nfix } & Nitrogen fixation && $\mu\mathrm{M\,N}\, \mathrm{s}^{-1}$ \\ |
| 319 |
\texttt{PAR } & Photosynthetically active radiation && $\mu\mathrm{Ein}\, \mathrm{m}^{-2}\,\mathrm{s}^{-1}$ \\ |
| 320 |
\texttt{Rstar01 } & $R^*_{\mathrm{PO4}}$ of Phytoplankton species \#1, \dots && $\mu\mathrm{M\,P}$ \\ |
| 321 |
\texttt{Diver1 } & Number of species with $P_j > 10^{-8}\,\mu\mathrm{M\,P}$ & where $\Ptot>10^{-12}$ \\ |
| 322 |
\texttt{Diver2 } & Number of species with $P_j > 0.1\%\, \Ptot$ & where $\Ptot>10^{-12}$ \\ |
| 323 |
\texttt{Diver3 } & Number of species that constitute 99.9\% of $\Ptot$ & where $\Ptot>10^{-12}$ \\ |
| 324 |
\texttt{Diver4 } & Number of species with $P_j > 10^{-5} \cdot \max\limits_j P_j$ & where $\Ptot>10^{-12}$ \\ |
| 325 |
\end{tabular} |
| 326 |
|
| 327 |
|
| 328 |
\end{document} |
Parent Directory
| 
Revision Log
| 
Revision Graph