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1d947889e4 Oliv* .. include:: ../defs.hrst
                 
                 .. _Grazing:
                 
                 Grazing
                 ^^^^^^^
                 
                 Grazing loss of plankton :math:`j`:
                 
                 .. math:: G_j = \sum_{z\in\op{pred}} G_{j,z}
                 
                 where
                 
                 .. math::
                 
                    G_{j,z} = g^{\max}_z
                           \frac{({p}_{j,z} {c}_j)^s}{A_z}
                           \frac{p_z^h}{p_z^h + {k^{{{\text{graz}}}}_z}^h}
faa67d1773 Oliv*           (1 - {\mathrm{e}}^{-i_{{{\text{graz}}}} p_z})^{e_{\op{inhib}}}
                           f^{{{\text{graz}}}}_z(T)^{e^{\op{graz}}_j}
1d947889e4 Oliv*           {c}_z
                 
                 with
                 
faa67d1773 Oliv* .. math:: A_z = \biggl[ \sum_j ({p}_{j,z} {c}_j)^s \biggr]_{\ge c^{\min}_{\op{graz}}}
1d947889e4 Oliv* 
faa67d1773 Oliv* .. math:: p_z = \biggl[ \sum_j {p}_{j,z} {c}_j - c^{\min}_{\op{graz}} \biggr]_{\ge 0}
1d947889e4 Oliv* 
                 :math:`s` is 1 for non-switching and 2 for switching grazers
                 (#define :varlink:`DARWIN_GRAZING_SWITCH`). The exponent :math:`h` defaults to
                 1.
                 
                 **Note:** For non-switching grazers (:math:`s=1`), Ben has an additional
                 factor
                 
                 .. math:: \frac{S^{[j]}_z}{S^{{{\text{phy}}}}_z + S^{{\text{zoo}}}_z}
                 
                 in :math:`G_{j,z}` where
                 
                 .. math::
                 
                      S^{{{\text{phy}}}}_z &= \sum_{j\in{{\text{phy}}}} {p}_{j,z} {c}_j
                 
                      S^{{\text{zoo}}}_z &= \sum_{j\in{{\text{zoo}}}} {p}_{j,z} {c}_j
                 
                 and :math:`S^{[j]}_z` is the sum for the class plankton :math:`j`
                 belongs to. **This is not implemented yet!**
                 
                 Gains from grazing:
                 
                 .. math::
                 
                        g^{{\mathrm{C}}}_z &= \sum_j G_{j,z} a_{j,z} {{\text{reg}}}^{Q{\mathrm{C}}}_z
                 
                        g^{{\mathrm{P}}}_z &= \sum_j G_{j,z} a_{j,z} {{\text{reg}}}^{Q{\mathrm{P}}}_z Q^{\mathrm{P}}_j
                        \qquad\text{if }\op{DARWIN\_ALLOW\_PQUOTA}
                 
                        &\dots
                 
                        g^{\op{DOC}} &= \sum_{j,z} G_{j,z} (1 - a_{j,z} {{\text{reg}}}^{Q{\mathrm{C}}}_z) (1 - f^{\text{exp graz}}_{j,z})
                 
                        g^{\op{DOP}} &= \sum_{j,z} \begin{cases}
                          G_{j,z} (1 - a_{j,z} {{\text{reg}}}^{Q{\mathrm{P}}}_z) (1 - f^{\text{exp graz}}_{j,z}) Q^{\mathrm{P}}_j
                          &\text{if }\op{DARWIN\_ALLOW\_PQUOTA}
                 
                          G_{j,z} (R^{{\mathrm{P}}:{\mathrm{C}}}_j - a_{j,z} R^{{\mathrm{P}}:{\mathrm{C}}}_z) (1 - f^{\text{exp graz}}_{j,z})
                          &\text{else}
                        \end{cases}
                 
                        &\dots
                 
                        g^{\op{POC}} &= \sum_{j,z} G_{j,z} (1 - a_{j,z} {{\text{reg}}}^{Q{\mathrm{C}}}_z) f^{\text{exp graz}}_{j,z}
                 
                        g^{\op{POP}} &= \sum_{j,z} \begin{cases}
                          G_{j,z} (1 - a_{j,z} {{\text{reg}}}^{Q{\mathrm{P}}}_z) f^{\text{exp graz}}_{j,z} Q^{\mathrm{P}}_j
                          &\text{if }\op{DARWIN\_ALLOW\_PQUOTA}
                 
                          G_{j,z} (R^{{\mathrm{P}}:{\mathrm{C}}}_j - a_{j,z} R^{{\mathrm{P}}:{\mathrm{C}}}_z) f^{\text{exp graz}}_{j,z}
                          &\text{else}
                          \end{cases}
                 
                        &\dots
                 
faa67d1773 Oliv*        g^{\op{POSi}} &= \sum_{j,z} \begin{cases}
                          G_{j,z} Q^{\op{Si}}_j &\text{if }\op{DARWIN\_ALLOW\_SIQUOTA}
1d947889e4 Oliv* 
faa67d1773 Oliv*          G_{j,z} R^{{\op{Si}}:{\mathrm{C}}}_j &\text{else}
                          \end{cases}
                 
                        g^{\op{PIC}} &= \sum_{j} G_{j} R^{{\text{PIC:POC}}}_j
1d947889e4 Oliv* 
                 where
                 
                 .. math::
                 
                        {{\text{reg}}}^{Q{\mathrm{P}}}_z &= \left( \left[ \frac{Q^{{\mathrm{P}}\max}_z - Q^{{\mathrm{P}}}_z}
                                                    {Q^{{\mathrm{P}}\max}_z - Q^{{\mathrm{P}}\min}_z}
                                        \right]_0^1 \right)^{h_{\op{G}}}
                 
                        &\dots
                 
                        {{\text{reg}}}^{Q{\mathrm{C}}}_z &= \left( \min\left\{
                              \frac{Q^{{\mathrm{P}}}_z - Q^{{\mathrm{P}}\min}_z}{Q^{{\mathrm{P}}\max}_z - Q^{{\mathrm{P}}\min}_z},
                              \frac{Q^{{\mathrm{N}}}_z - Q^{{\mathrm{N}}\min}_z}{Q^{{\mathrm{N}}\max}_z - Q^{{\mathrm{N}}\min}_z},
                              \frac{Q^{\op{Fe}}_z - Q^{\op{Fe}\min}_z}{Q^{\op{Fe}\max}_z - Q^{\op{Fe}\min}_z}
                              \right\}_0^1 \right)^{h_{\op{G}}}
                 
                        & \qquad\text{(only quota elements)}
                 
                 and :math:`h_{\op{G}}` is the Hill number for grazing (:varlink:`hillnumGraz`,
                 default 1).
                 
faa67d1773 Oliv* 
                 Implementation
                 ''''''''''''''
                 
                 In order to reduce the number of double (predator-prey) sums as much as
                 possible while still maintaining some code readability, the above sums are
                 computed in :filelink:`~pkg/darwin/darwin_plankton.F` via :math:`G_j`,
                 :math:`g^{\mathrm{C}}_z` and the following auxiliary sums:
                 
                 .. math::
                 
                    G^{\exp}_j &= \sum_z G_{j,z} f^{\text{exp graz}}_{j,z}
                    \;,
                 
                    g^{\mathrm{C}\exp}_z &= \sum_j G_{j,z} a_{j,z}
                               {{\text{reg}}}^{Q{\mathrm{C}}}_z f^{\text{exp graz}}_{j,z}
                    \;,
                 
                 and for quotas elements additionally :math:`g^{\mathrm{P}}_z`, ..., and
                 
                 .. math::
                 
                    g^{\mathrm{P}\exp}_z &= \sum_j G_{j,z} a_{j,z}
                        {{\text{reg}}}^{Q{\mathrm{P}}}_z Q^{\mathrm{P}}_j
                        f^{\text{exp graz}}_{j,z}
                 
                    &\ldots
                 
                 The remaining terms are then computed as
                 
                 .. math::
                 
                    g^{\op{POC}} &= \sum_j G^{\exp}_j - \sum_z g^{\mathrm{C}\exp}_z
                 
                    g^{\op{DOC}} &= g^{\op{OC}} - g^{\op{POC}}
                 
                 where
                 
                 .. math::
                 
                    g^{\op{OC}} = \sum_j G_j - \sum_z g^{\mathrm{C}}_z
                    \;.
                 
                 For other non-quota elements:
                 
                 .. math::
                 
                    g^{\op{POP}} &= \sum_j G^{\exp}_j R^{\mathrm{P}:\mathrm{C}}_j
                                  - \sum_z g^{\mathrm{C}\exp}_z R^{\mathrm{P}:\mathrm{C}}_z
                 
                    g^{\op{DOP}} &= g^{\op{OP}} - g^{\op{POP}}
                 
                 where
                 
                 .. math::
                 
                    g^{\op{OP}} = \sum_j G_j R^{\mathrm{P}:\mathrm{C}}_j
                                 - \sum_z g^{\mathrm{C}}_z R^{\mathrm{P}:\mathrm{C}}_z
                    \;.
                 
                 For quota elements:
                 
                 .. math::
                 
                    g^{\op{POP}} &= \sum_j G^{\exp}_j Q^{\mathrm{P}}_j
                                  - \sum_z g^{\mathrm{P}\exp}_z
                 
                    g^{\op{DOP}} &= g^{\op{OP}} - g^{\op{POP}}
                 
                 where
                 
                 .. math::
                 
                    g^{\op{OP}} = \sum_j G_j Q^{\mathrm{P}}_j
                                 - \sum_z g^{\mathrm{P}}_z
                    \;.
                 
                 
                 Runtime Parameters
                 ''''''''''''''''''
                 
                 Grazing parameters are given in :numref:`tab_phys_pkg_darwin_grazing_params`.
                 
                 .. csv-table:: Grazing parameters
                    :delim: &
                    :widths: 15,19,10,20,13,23
                    :class: longtable
                    :header: Trait, Param, Sym, Default, Units, Description
                    :name: tab_phys_pkg_darwin_grazing_params
                 
                    :varlink:`grazemax`  & :varlink:`a_grazemax`     & :math:`g^{\op{max}}_z`       & 21.9/day·V\ :sup:`-0.16` & s\ :sup:`-1`         & maximum grazing rate
                    :varlink:`kgrazesat` & :varlink:`a_kgrazesat`    & :math:`k^{\op{graz}}_z`      & 1.0                      & mmol C m\ :sup:`-3`  & grazing half-saturation concentration
                    :varlink:`tempGraz`  & :varlink:`grp_tempGraz`   & :math:`e^{\op{graz}}_j`      & 1                        &                      & 1: grazing is temperature dependent, 0: turn dependence off
                                         & :varlink:`inhib_graz`     & :math:`i_{\op{graz}}`        & 1.0                      & m\ :sup:`3` / mmol C & inverse decay scale for grazing inhibition
                                         & :varlink:`inhib_graz_exp` & :math:`e_{\op{inhib}}`       & 0.0                      &                      & exponent for grazing inhibition (0 to turn off inhibition)
                                         & :varlink:`hillnumGraz`    & :math:`h_{\op{G}}`           & 1.0                      &                      & exponent for limiting quota uptake in grazing
                                         & :varlink:`hollexp`        & :math:`h`                    & 1.0                      &                      & grazing exponential 1= "Holling 2", 2= "Holling 3"
                                         & :varlink:`phygrazmin`     & :math:`c^{\min}_{\op{graz}}` & 120E-10                  & mmol C m\ :sup:`-3`  & minimum total prey conc for grazing to occur
                 
                 See :numref:`tab_phys_pkg_darwin_uptake` for stochiometry and quota-related parameters.
                 
                 
                 .. csv-table:: Trait matrices for grazing; indices (prey, pred); unitless
                    :delim: &
                    :widths: auto
                    :class: longtable
                    :header: Trait, Param, Symbol, Default, Description
                 
                    :varlink:`palat`              & see below                         & :math:`p_{j,z}`                     & 0   & palatability matrix
                    :varlink:`asseff`             & :varlink:`grp_ass_eff`            & :math:`a_{j,z}`                     & 0.7 & assimilation efficiency matrix
                    :varlink:`ExportFracPreyPred` & :varlink:`grp_ExportFracPreyPred` & :math:`f^{\op{exp}\op{graz}}_{j,z}` & 0.5 & fraction of unassimilated prey becoming particulate organic matter
                 
                 If :varlink:`DARWIN_ALLOMETRIC_PALAT` is defined, palatabilities are set
                 allometrically,
                 
                 .. math::
                 
                    p_{j,z} = \left[ \frac{1}{2\sigma_{\op{pp}}}
                                     \exp\left\{
                                       -\frac{(\ln(V_z/V_j/r_{\op{opt}}))^2}{2\sigma_{\op{pp}}^2}
                                     \right\}
                              \right]_{\ge p_{\min}}
                 
                 :varlink:`grp_pred` and :varlink:`grp_prey` should be set to select which
                 plankton groups can graze or be grazed.
                 
                 
                 .. csv-table:: Allometric palatability trait parameters (unitless)
                    :delim: &
                    :widths: 20,15,15,50
                    :class: longtable
                    :header: Param, Symbol, Default, Description
                 
                    :varlink:`a <a_ppOpt>`,\ :varlink:`b_ppOpt` & :math:`r_{\op{opt}}`     & 1024·V\ :sup:`0` & optimum predator-prey ratio
                    :varlink:`a_ppSig`     & :math:`\sigma_{\op{pp}}` & 1                & width of predator-prey curve
                    :varlink:`palat_min`   & :math:`p_{\min}`         & 0                & min non-zero palatability, smaller :varlink:`palat` are set to 0 (was 1D-4 in quota)
                    :varlink:`grp_pred`    &                          & 0                & 1: can graze, 0: not
                    :varlink:`grp_prey`    &                          & 1                & 1: can be grazed, 0: not
                 

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