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1d947889e4 Oliv* .. include:: ../defs.hrst
                 
                 .. _Iron:
                 
                 Iron chemistry
                 ^^^^^^^^^^^^^^
                 
                 The tracer representing inorganic iron in darwin is total dissolved inorganic
                 iron, FeT.  Its source terms are
                 
                 .. math::
                 
9f79ba70a9 Oliv*    S_{\op{Fe}} &= \delta_{k,1} \frac{\alpha_{\op{Fe}}}{\Delta r_{\mathrm{F}}h_{\mathrm{C}}} F_{\op{Fe}}
                              + \delta_{\op{bottom}} \delta_{|r|\le d_{\op{sed}}}
                                \frac{1}{\Delta r_{\mathrm{F}}h_{\mathrm{C}}} F_{\op{Fe}}^{{\text{sed}}}
                                \\
                              &+ \delta_{\op{bottom}} \delta_{|r|\ge d_{\op{vents}}}
                                \frac{1}{\Delta r_{\mathrm{F}}h_{\mathrm{C}}} F_{\op{Fe}}^{{\text{vents}}}
                              - r_{{\text{scav}}}\op{Fe}'
1d947889e4 Oliv* 
02be68c5dc Oliv* where :math:`\delta_{\op{bottom}}` selects the last ocean grid cell above the sea floor.
                 
                 
1d947889e4 Oliv* Dust deposition
                 '''''''''''''''
                 
                 The first term is iron input at the surface from dust deposition.
                 The rate of dust deposition is read in from :varlink:`ironfile`.
                 :math:`\alpha_{\op{Fe}}` (:varlink:`alpfe`) is the solubility of iron dust;
                 set it to 1 if the deposition rate in :varlink:`ironfile` is already of
                 soluble iron.  :varlink:`darwin_inscal_iron` can be used to convert units on the fly.
                 
                 
                 Sedimentation
                 '''''''''''''
                 
                 The second term represents iron input from sediments at the ocean floor.
02be68c5dc Oliv* It only occurs above :varlink:`depthfesed`.  The flux is either a fixed number,
                 :varlink:`fesedflux` or, if :varlink:`DARWIN_IRON_SED_SOURCE_VARIABLE` is defined,
1d947889e4 Oliv* 
                 .. math::
                 
02be68c5dc Oliv*    F_{\op{Fe}}^{{\text{sed}}}=
                        \left[
                        F_{\op{Fe}}^{\op{sed pcm}} w^{\mathrm{C}}_{{{\text{sink}}}} \op{POC}
                        - F_{\op{Fe}}^{\op{sed min}}
                        \right]_0
                 
                 For backwards compatibility, the variable sediment flux can be expressed in
                 terms of POP if :varlink:`DARWIN_IRON_SED_SOURCE_POP` is defined:
                 
                 .. math::
                 
                    F_{\op{Fe}}^{{\text{sed}}}=
                        F_{\op{Fe}}^{\op{sed pcm}} w^{\mathrm{P}}_{{{\text{sink}}}}
                        R^{{\mathrm{C}}:{\mathrm{P}}}_{{\text{sed}}}  \op{POP}^{\op{up}}
1d947889e4 Oliv* 
02be68c5dc Oliv* where :math:`\op{POP}^{\op{up}}` is POP in the second-lowest wet grid cell
                 of the water column.
1d947889e4 Oliv* 
                 
9f79ba70a9 Oliv* Hydrothermal vents
                 ''''''''''''''''''
                 
                 The third term represents iron input from hydrothermal vents.  To enable it,
                 define :varlink:`DARWIN_ALLOW_HYDROTHERMAL_VENTS`.  This iron source is only
                 active below :varlink:`depthFeVent`.  The flux is proportional to the
                 Helium-3 flux :math:`F_{^3\text{He}}^{{\text{vents}}}` given in
                 :varlink:`ventHe3file` in units of mmol :sup:`3`\ He m\ :sup:`-2` s\ :sup:`-1`,
                 
                 .. math::
                 
                    F_{\op{Fe}}^{\text{vents}} = \alpha_{\op{Fe}}^{\op{vents}}
                        R^{\op{Fe:^3He}}_{\op{vents}} F_{^3\text{He}}^{{\text{vents}}}
                    \;.
                 
                 Here, :math:`\alpha_{\op{Fe}}^{\op{vents}}` (:varlink:`solFeVent`) is the
                 solubility of iron from vents and :math:`R^{\op{Fe:^3He}}_{\op{vents}}`
                 (:varlink:`R_FeHe3_vent`) is the iron to Helium-3 ratio of the vents.
                 
                 
1d947889e4 Oliv* Scavenging
                 ''''''''''
                 
9f79ba70a9 Oliv* The fourth term represents losses due to particle scavenging.
1d947889e4 Oliv* The scavenging rate for free iron is
                 
                 .. math::
                 
                    r_{{\text{scav}}}= \begin{cases}
                        \op{scav}
4ee77fca61 Oliv*        & \text{for fixed-rate scavenging,} \\
                        \tau_{{\text{scav}}}I_{{\text{scav}}}\op{POM}^{e_{{\text{scav}}}}
                        & \text{for particle-based scavenging.}
1d947889e4 Oliv*        \end{cases}
                 
4ee77fca61 Oliv* To select particle-based scavenging following Parekh et al. (2005)
                 :cite:`parekh:2005`, define :varlink:`DARWIN_PART_SCAV`.  POM is the
                 concentration of particulate organic matter in units of mg/L.  It is
28ac947a3e Oliv* parameterized in terms of POC, PSi and PIC,
1d947889e4 Oliv* 
4ee77fca61 Oliv* .. math::
                 
                    \op{POM} = w^{\text{scav}}_{\text{POC}} \op{POC}
28ac947a3e Oliv*             + w^{\text{scav}}_{\text{PSi}} \op{PSi}
4ee77fca61 Oliv*             + w^{\text{scav}}_{\text{PIC}} \op{PIC}
28ac947a3e Oliv*             + \op{rPOM}
4ee77fca61 Oliv*    \;.
                 
28ac947a3e Oliv* By default, only POC contributes, see
4ee77fca61 Oliv* :numref:`tab_phys_pkgs_darwin_iron_params`.
                 
28ac947a3e Oliv* .. admonition:: deprecated formulation
eea283df88 Oliv* 
                    The old (and deprecated) formulation of scavenging in terms of POP can be
                    recovered by defining :varlink:`DARWIN_PART_SCAV_POP`, in which case
                    :math:`\op{POM}` is replaced by
                    :math:`\op{POP}\!/R^{\op{POP}:\op{POC}}_{{\text{scav}}}`.  Parameter names
                    and defaults are different in this case, see
28ac947a3e Oliv*    :numref:`tab_phys_pkgs_darwin_scav_pop`.  The new formulation with default
                    parameters closely approximates the old formulation with its default
                    parameters when POC:POP is 120:1.
be00149aee Oliv* 
4ee77fca61 Oliv* The concentration of free iron, Fe', is determined following Parekh et al.
                 (2004) :cite:`parekh:2004` and Dutkiewicz et al. (2005)
                 :cite:`dutkiewicz:2005`.  Free dissolved iron is assumed to be in equilibrium
                 with dissolved iron bound to ligands, FeL,
1d947889e4 Oliv* 
                 .. math:: \op{Fe}' + L' \rightleftharpoons \op{FeL}
                 
                 At equilibrium,
                 
                 .. math:: \frac{[\op{FeL}]}{[\op{Fe}'][L']} = {\beta_{{\text{stab}}}}\;,
                 
                 Using
                 :math:`\op{FeL}+\op{Fe}'={\op{FeT}}` and :math:`\op{FeL}+L'={L_{{\mathrm{T}}}}`,
                 the solution is obtained as
                 
                 .. math::
                 
                      L' &= \frac{ {\beta_{{\text{stab}}}}({L_{{\mathrm{T}}}}- {\op{FeT}}) - 1
                                 +\sqrt{(1 - {\beta_{{\text{stab}}}}({L_{{\mathrm{T}}}}- {\op{FeT}}))^2 + 4 {\beta_{{\text{stab}}}}{L_{{\mathrm{T}}}}}}
                              {2 {\beta_{{\text{stab}}}}}
                 
                      \op{FeL} &= {L_{{\mathrm{T}}}}- L'
                 
                      \op{Fe}' &= \op{FeT} - \op{FeL}\;.
                 
                 If :varlink:`DARWIN_MINFE` is defined, Fe' will be
                 constrained to be no more than Fe':sub:`max`, and FeT adjusted accordingly,
                 assuming that excess free iron is scavenged away. This is done before
                 and after each biogeochemical subtimestep.
                 
                 .. csv-table:: Iron parameters set in DARWIN_PARAMS
                    :delim: &
                    :widths: auto
                    :header: Parameter, Symbol, Default, Units, Description
4ee77fca61 Oliv*    :name: tab_phys_pkgs_darwin_iron_params
1d947889e4 Oliv* 
                    :varlink:`alpfe`         & :math:`\alpha_{\op{Fe}}`             & 0.04        &                        & solubility of Fe dust
9f79ba70a9 Oliv*    :varlink:`depthfesed`    & :math:`d_{\op{sed}}`                 & -1.0        & m                      & depth above which to add sediment source
                    :varlink:`fesedflux`     & fesedflux                            & 1E-3 / day  & mmol Fe /m\ :sup:`2`/s & fixed iron flux from sediment
                    :varlink:`fesedflux_pcm` & :math:`F_{\op{Fe}}^{\op{sed pcm}}`   & 0.68E-3     & mmol Fe / mmol C       & iron input per POC sinking into bottom for :varlink:`DARWIN_IRON_SED_SOURCE_VARIABLE`
                    :varlink:`fesedflux_min` & :math:`F_{\op{Fe}}^{\op{sed min}}`   & 0.5E-3 / day& mmol Fe /s             & minimum iron input rate subtracted from fesedflux_pcm*wc_sink*POC
1d947889e4 Oliv*    :varlink:`R_CP_fesed`    & :math:`R^{\op{C:P}}_{\op{sed}}`      & 106         & mmol C / mmol P        & POC:POP conversion for :varlink:`DARWIN_IRON_SED_SOURCE_VARIABLE`
9f79ba70a9 Oliv*    :varlink:`depthFeVent`   & :math:`d_{\op{vents}}`               & 750         & m                      & depth below which iron from hydrothermal vents is added
                    :varlink:`solFeVent`     & :math:`\alpha_{\op{Fe}}^{\op{vents}}` & 0.002      &                        & solubility of iron from hydrothermal vents
                    :varlink:`R_FeHe3_vent`  & :math:`R^{\op{Fe:^3He}}_{\op{vents}}` & 4.5E8      & mol Fe / mol :sup:`3`\ He & Fe:\ :sup:`3`\ He ratio for hydrothermal vents
1d947889e4 Oliv*    :varlink:`scav`          & scav                                 & 0.4/year    & 1/s                    & fixed iron scavenging rate
4ee77fca61 Oliv*    :varlink:`scav_tau`      & :math:`\tau_{\op{scav}}`             & 0.2         &                        & factor for converting Th scavenging rates to iron ones
                    :varlink:`scav_inter`    & :math:`I_{\op{scav}}`                & 0.079 / day & L\ :sup:`e` mg\ :sup:`-e` s\ :sup:`-1` & intercept of scavenging power law (e=e\ :sub:`scav`)
1d947889e4 Oliv*    :varlink:`scav_exp`      & :math:`e_{\op{scav}}`                & 0.58        &                        & exponent of scavenging power law
28ac947a3e Oliv*    :varlink:`scav_POC_wgt`  & :math:`w^{\op{scav}}_{\op{POC}}`     & 0.12728     & g/mmol |nbsp| C        & weight POC contributes to POM
                    :varlink:`scav_PSi_wgt`  & :math:`w^{\op{scav}}_{\op{PSi}}`     & 0.0         & g/mmol |nbsp| Si       & weight PSi contributes to POM
                    :varlink:`scav_PIC_wgt`  & :math:`w^{\op{scav}}_{\op{PIC}}`     & 0.0         & g/mmol |nbsp| C        & weight PIC contributes to POM
                    :varlink:`scav_rPOM`     & :math:`\op{rPOM}`                    & 0.0         & g/m\ :sup:`3`          & concentration of non-labile POM for scavenging
9f79ba70a9 Oliv*    :varlink:`ligand_tot`    & :math:`L_{\op{T}}`                   & 1E-3        & mmol/m\ :sup:`3`       & total ligand concentration
4ee77fca61 Oliv*    :varlink:`ligand_stab`   & :math:`\beta_{\op{stab}}`            & 0.2E6       & m\ :sup:`3`/mmol       & ligand stability rate ratio
9f79ba70a9 Oliv*    :varlink:`freefemax`     & :math:`\op{Fe}'_{\op{max}}`          & 0.4E-3      & mmol/m\ :sup:`3`       & max concentration of free iron
4ee77fca61 Oliv* 
                 
                 .. csv-table:: Iron parameters for :varlink:`DARWIN_PART_SCAV_POP`
                    :delim: &
                    :widths: auto
                    :header: Parameter, Symbol, Default, Units, Description
                    :name: tab_phys_pkgs_darwin_scav_pop
                 
                    :varlink:`scav_rat`      & :math:`\tau_{\op{scav}}`             & 0.005 / day & 1/s                       & rate factor
                    :varlink:`scav_inter`    & :math:`I_{\op{scav}}`                & 0.079       & L\ :sup:`e` mg\ :sup:`-e` & intercept of scavenging power law (e=e\ :sub:`scav`)
                    :varlink:`scav_exp`      & :math:`e_{\op{scav}}`                & 0.58        &                           & exponent of scavenging power law
                    :varlink:`scav_R_POPPOC` & :math:`R^{\op{POP:POC}}_{\op{scav}}` & 1.1321E-4   & mmol |nbsp| P / g         & POP\:POC ratio

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