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1d947889e4 Oliv*0001 .. include:: ../defs.hrst
0002
0003 Model equations
0004 ^^^^^^^^^^^^^^^
0005
faa67d1773 Oliv*0006 The basic model equations are (omitting transport terms handled by the ptracers
3c930300af Oliv*0007 package, :ref:`para_phys_pkg_darwin_sink`, :ref:`AirSea` and terms correcting
0008 conservation with the linear free surface formulation discussed in
0009 :numref:`para_phys_pkg_darwin_cons`):
1d947889e4 Oliv*0010
0011 .. math::
0012
ab3dad0141 Oliv*0013 \partial_t\op{DIC} &= \sum_j \bigl( (-U^{\mathrm{DIC}}_j + R^{\mathrm{C}}_j)
81872c4188 Oliv*0014 \cdot (1 + R^{\text{PIC:POC}}_j)
ab3dad0141 Oliv*0015 + R^{\mathrm{DIC}}_j
81872c4188 Oliv*0016 \bigr)
ab3dad0141 Oliv*0017 + R_{\mathrm{DOC}} + [R_{\mathrm{POC}}]
3c930300af Oliv*0018 + D_{\mathrm{PIC}} \\
ab3dad0141 Oliv*0019 \partial_t\op{PO}_4 &= \sum_j \bigl( -U^{\mathrm{PO4}}_j + R^{\mathrm{P}}_j + R^{\mathrm{PO4}}_j \bigr)
0020 + R_{\mathrm{DOP}} + [R_{\mathrm{POP}}] \\
0021 \partial_t\op{NH}_4 &= \sum_j \bigl( -U^{\mathrm{NH4}}_j + R^{\mathrm{N,NH4}}_j + R^{\mathrm{NH4}}_j \bigr)
0022 + R_{\mathrm{DON}} + [R_{\mathrm{PON}}] - P_{\mathrm{NO2}} - D_{\mathrm{NH4}} \\
0023 \partial_t\op{NO}_2 &= \sum_j \bigl( -U^{\mathrm{NO2}}_j + R^{\mathrm{N,NO2}}_j + R^{\mathrm{NO2}}_j \bigr)
0024 + P_{\mathrm{NO2}} - P_{\mathrm{NO3}} \\
0025 \partial_t\op{NO}_3 &= \sum_j \bigl( -U^{\mathrm{NO3}}_j + R^{\mathrm{N,NO3}}_j + R^{\mathrm{NO3}}_j \bigr)
0026 + P_{\mathrm{NO3}} - D_{\mathrm{NO3}} \\
0027 \partial_t\op{FeT} &= \sum_j \bigl( -U^{\mathrm{Fe}}_j + R^{\mathrm{Fe}}_j + R^{\mathrm{FeT}}_j \bigr)
0028 + R_{\mathrm{DOFe}} + [R_{\mathrm{POFe}}] + S_{\mathrm{Fe}} \\
0029 \partial_t\op{SiO}_2 &= \sum_j \bigl( -U^{\mathrm{SiO2}}_j + R^{\mathrm{Si}}_j \bigr)
0030 + R_{\mathrm{POSi}} \\
0031 \partial_t c_j &= U^{\mathrm{DIC}}_j - M_j - R^{\mathrm{C}}_j - G_j + g_j^{\mathrm{C}} \\
0032 \partial_t p_j &= U^{\mathrm{PO4}}_j - M_j Q^{\mathrm{P}}_j - G_j Q^{\mathrm{P}}_j + g_j^{\mathrm{P}} \qquad\text{(with P quota)} \\
0033 \partial_t n_j &= U^{{\mathrm{N}}}_j \;\;\; - M_j Q^{\mathrm{N}}_j - G_j Q^{\mathrm{N}}_j + g_j^{\mathrm{N}} \qquad\text{(with N quota)} \\
0034 \partial_t\op{fe}_j &= U^{\mathrm{Fe}}_j \;\; - M_j Q^{\mathrm{Fe}}_j - G_j Q^{\mathrm{Fe}}_j + g_j^{\mathrm{Fe}} \qquad\text{(with Fe quota)} \\
0035 \partial_t\op{si}_j &= U^{\mathrm{SiO2}}_j - M_j Q^{\mathrm{Si}}_j - G_j Q^{\mathrm{Si}}_j \qquad\text{(with Si quota)} \\
0036 \partial_t\op{Chl}_j &= S^{\mathrm{Chl}}_j \; - M_j Q^{\mathrm{Chl}}_j - G_j Q^{\mathrm{Chl}}_j \qquad\text{(with Chl quota)} \\
0037 \partial_t\op{DOC} &= \sum_j M_j^{\mathrm{DOM}} \;\;\;\;\; + g^{\mathrm{DOC}} + \sum_j \left( H^{\mathrm{POC}}_j - U^{\mathrm{DOC}}_j \right) - R_{\mathrm{DOC}} - S_{\mathrm{CDOM}} R^{{\mathrm{C}}:{\mathrm{P}}}_{\mathrm{CDOM}} \\
0038 \partial_t\op{DOP} &= \sum_j M_j^{\mathrm{DOM}} Q_j^{{\mathrm{P}}} + g^{\mathrm{DOP}} + \sum_j \left( H^{\mathrm{POP}}_j - U^{\mathrm{DOP}}_j \right) - R_{\mathrm{DOP}} - S_{\mathrm{CDOM}} \\
0039 \partial_t\op{DON} &= \sum_j M_j^{\mathrm{DOM}} Q_j^{{\mathrm{N}}} + g^{\mathrm{DON}} + \sum_j \left( H^{\mathrm{PON}}_j - U^{\mathrm{DON}}_j \right) - R_{\mathrm{DON}} - S_{\mathrm{CDOM}} R^{{\mathrm{N}}:{\mathrm{P}}}_{\mathrm{CDOM}} \\
0040 \partial_t\op{DOFe} &= \sum_j M_j^{\mathrm{DOM}} Q_j^{\mathrm{Fe}} + g^{\mathrm{DOFe}} + \sum_j \left( H^{\mathrm{POFe}}_j- U^{\mathrm{DOFe}}_j\right) - R_{\mathrm{DOFe}}- S_{\mathrm{CDOM}} R^{{\mathrm{Fe}}:{\mathrm{P}}}_{\mathrm{CDOM}} \\
3c930300af Oliv*0041 \partial_t\op{PIC} &= \sum_j M_j R_j^{\text{PIC:POC}} + g^{\mathrm{PIC}} - D_{\mathrm{PIC}} \\
ab3dad0141 Oliv*0042 \partial_t\op{POC} &= \sum_j M_j^{\mathrm{POM}} \;\;\;\;\; + g^{\mathrm{POC}} - \sum_j U^{\mathrm{POC}}_j - R_{\mathrm{POC}} \\
0043 \partial_t\op{POP} &= \sum_j M_j^{\mathrm{POM}} Q_j^{{\mathrm{P}}} + g^{\mathrm{POP}} - \sum_j U^{\mathrm{POP}}_j - R_{\mathrm{POP}} \\
0044 \partial_t\op{PON} &= \sum_j M_j^{\mathrm{POM}} Q_j^{{\mathrm{N}}} + g^{\mathrm{PON}} - \sum_j U^{\mathrm{PON}}_j - R_{\mathrm{PON}} \\
0045 \partial_t\op{POFe} &= \sum_j M_j^{\mathrm{POM}} Q_j^{\mathrm{Fe}} + g^{\mathrm{POFe}} - \sum_j U^{\mathrm{POFe}}_j - R_{\mathrm{POFe}} \\
0046 \partial_t\op{POSi} &= \sum_j M_j Q_j^{\mathrm{Si}} \;\;\;\;\; + g^{\mathrm{POSi}} - R_{\mathrm{POSi}} \\
0047 \partial_t\op{ALK} &= -\biggl( P_{\mathrm{NO3}} - \sum_j U^{\mathrm{NO3}}_j \biggr)
3c930300af Oliv*0048 - 2\biggl( \sum_j U^{\mathrm{DIC}}_j R^{{\text{PIC:POC}}}_j - D_{\mathrm{PIC}} \biggr)
ab3dad0141 Oliv*0049 + D_{\mathrm{NO3}} \\
faa67d1773 Oliv*0050 \partial_t{\mathrm{O}}_2 &= R_{\mathrm{O}_2:\mathrm{P}} \biggl(
ab3dad0141 Oliv*0051 \sum_j U^{\mathrm{PO4}}_j
0052 - R_{\mathrm{DOP}} - [R_{\mathrm{POP}}]
faa67d1773 Oliv*0053 \biggr)
0054 \\
ab3dad0141 Oliv*0055 \partial_t\op{CDOM} &= S_{\mathrm{CDOM}} \qquad\text{(with CDOM tracer)} \\
faa67d1773 Oliv*0056
1d947889e4 Oliv*0057
ab3dad0141 Oliv*0058 The quotas are defined as :math:`Q^{\mathrm{P}}_j=p_j/c_j`, etc. The following
0059 abbreviated source terms are described in sections below:
1d947889e4 Oliv*0060
faa67d1773 Oliv*0061 - :math:`U`: uptake, see :ref:`Uptake` and :ref:`Bacteria`,
8916dab93b Oliv*0062 also :ref:`Growth`, :ref:`CarbonStore`, :ref:`Light` and :ref:`Spectral`
81872c4188 Oliv*0063 - :math:`M`: :ref:`Mortality`
51e33025af Oliv*0064 - :math:`R^{\mathrm{C}}`, :math:`R^{\mathrm{P}}`, ...: :ref:`Respiration`
0065 - :math:`H`, :math:`R^{\mathrm{DIC}}`, ...: bacterial hydrolysis and remineralization, see :ref:`Bacteria`
faa67d1773 Oliv*0066 - :math:`R`, :math:`P`: parameterized :ref:`Remineralization`
3c930300af Oliv*0067 - :math:`D_{\mathrm{NO3}}`: denitrification, see :ref:`Denitrification`
0068 - :math:`D_{\mathrm{PIC}}`: dissolution of PIC, see :ref:`CarbonChemistry`
faa67d1773 Oliv*0069 - :math:`g`, :math:`G`: grazing gains and losses, see :ref:`Grazing`
ab3dad0141 Oliv*0070 - :math:`S^{\mathrm{Chl}}`: synthesis, see :ref:`Synthesis`
0071 - :math:`S_{\mathrm{Fe}}`: iron sources, see :ref:`Iron`
0072 - :math:`S_{\mathrm{CDOM}}`: see :ref:`CDOM`.
1d947889e4 Oliv*0073
faa67d1773 Oliv*0074 With :varlink:`DARWIN_ALLOW_CDOM`, all particulate remineralization terms
0075 (in square brackets [...]) except Si are absent.
ab3dad0141 Oliv*0076 Without :varlink:`DARWIN_ALLOW_CDOM`, :math:`f_{\mathrm{CDOM}}=0` and there is no CDOM tracer.
3c930300af Oliv*0077 The Alk and O\ :sub:`2` tracers are only present with :varlink:`DARWIN_ALLOW_CARBON`.
1d947889e4 Oliv*0078
a4f2018aab Oliv*0079
0080 .. csv-table:: General parameters
0081 :delim: &
faa67d1773 Oliv*0082 :widths: 15,17,14,11,18,25
a4f2018aab Oliv*0083 :class: longtable
0084 :header: Trait, Param, Symbol, Default, Units, Description
0085
ab3dad0141 Oliv*0086 :varlink:`R_PICPOC` & :varlink:`a_R_PICPOC` & :math:`R^{\text{PIC:POC}}_j` & 0.8 & mmol PIC / mmol POC & inorganic-organic carbon ratio
0087 & :varlink:`R_OP` & :math:`R_{\mathrm{O}_2:\mathrm{P}}` & 170 & mmol O\ :sub:`2` / mmol P & O\ :sub:`2`:P ratio for respiration and consumption
a4f2018aab Oliv*0088
ab3dad0141 Oliv*0089 For CDOM elemental ratios, see :ref:`CDOM`.
3b34245348 Oliv*0090
0091 Flow diagrams for the various chemical elements:
0092
0093 .. figure:: figs/darwin_P_noCDOM_nograz.*
0094 :width: 100%
0095 :align: center
0096 :alt: Flow of phosphorus schematic
0097 :name: figdarwinP
0098
0099 The flow of phosphorus without CDOM (for grazing see :numref:`figdarwinPgraz`)
0100
0101 .. figure:: figs/darwin_P_CDOM_nograz.*
0102 :width: 100%
0103 :align: center
0104 :alt: Flow of phosphorus schematic
0105 :name: figdarwinPCDOM
0106
1b4f0c760e Oliv*0107 The flow of phosphorus with CDOM.
3b34245348 Oliv*0108
0109 .. figure:: figs/darwin_P_graz.*
0110 :width: 75%
0111 :align: center
0112 :alt: Flow of phosphorus schematic
0113 :name: figdarwinPgraz
0114
0115 The flow of phosphorus due to grazing.
0116
0117 .. figure:: figs/darwin_N_noCDOM_nograz.*
0118 :width: 100%
0119 :align: center
0120 :alt: Flow of nitrogen schematic
0121 :name: figdarwinN
0122
0123 The flow of nitrogen (no CDOM, grazing suppressed).
0124
0125 .. figure:: figs/darwin_Fe_noCDOM_nograz.*
0126 :width: 100%
0127 :align: center
0128 :alt: Flow of iron schematic
0129 :name: figdarwinFe
0130
0131 The flow of iron (no CDOM, grazing suppressed).
0132
0133 .. figure:: figs/darwin_Si_noCDOM_nograz.*
0134 :width: 80%
0135 :align: center
0136 :alt: Flow of silica schematic
0137 :name: figdarwinSi
0138
0139 The flow of silica (no CDOM, grazing suppressed).
0140
0141 .. figure:: figs/darwin_ALK.*
0142 :width: 50%
0143 :align: center
0144 :alt: Flow of alkalinity schematic
0145 :name: figdarwinALK
0146
0147 Sources and sinks of alkalinity.
0148
0149 .. figure:: figs/darwin_O2.*
0150 :width: 50%
0151 :align: center
0152 :alt: Flow of oxygen schematic
0153 :name: figdarwinO2
0154
0155 Sources and sinks of oxygen.
