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a4f2018aab Oliv* .. include:: ../defs.hrst
                 
                 .. _Spectral:
                 
                 Spectral Light
                 ^^^^^^^^^^^^^^
                 
                 Spectral light throughout the water column is computed following
                 :cite:`dutkiewicz:2015`.  The :ref:`sub_phys_pkg_radtrans` has to be enabled
                 and will attenuate light using intrinsic optical properties provided by the
                 darwin package.  They are computed from concentrations of plankton, particles
                 and CDOM:
                 
                 .. math::
                 
                     a_l &= a^{\op{w}}_l + a^{\op{plank}}_l
                          + a^{\op{part}}_{\op{P}l} P_{\op{part}} + a^{\op{CDOM}}_l
                 
                     b_l &= b^{\op{w}}_l + b^{\op{plank}}_l
                          + b^{\op{part}}_{\op{P}l} P_{\op{part}}
                 
                     b_{\op{b}l} &= \left[ \tilde b^{\op{w}}_{\op{b}} b^{\op{w}}_l
                                         + b^{\op{plank}}_{\op{b}l}
                                         + b^{\op{part}}_{\op{b}\op{P}l} P_{\op{part}}
                                    \right]_{\ge b_{\op{b}}^{\min}}
                 
                 Water IOPs, :math:`a^{\op{w}}_l` and :math:`b^{\op{w}}_l`, are read in from
                 :varlink:`darwin_waterAbsorbFile`.  Plankton IOPs are computed from
                 individual functional types,
                 
                 .. math::
                 
                    a^{\op{plank}}_l &= \sum_j \op{Chl}_j a^{\op{chl}}_{\op{phy}j,l}
                                      + \sum_j 12\, c_j a^{\op{C}}_{\op{phy}j,l}
                 
                    b^{\op{plank}}_l &= \sum_j 12\, c_j b^{\op{C}}_{\op{phy}j,l}
                 
                    b^{\op{plank}}_{\op{b}l} &= \sum_j 12\, c_j b^{\op{C}}_{\op{b}\op{phy}j,l}
                 
faa67d1773 Oliv* The spectra are selected based on optical type, :varlink:`grp_aptype`, from
                 spectra read in from :varlink:`darwin_phytoAbsorbFile`.  Usually, phytoplankton
a4f2018aab Oliv* absorption spectra are given per amount of Chlorophyll, via
                 :math:`a^{\op{chl}}_{\op{phy}j,l}`, while bacteria absorption spectra are given
                 in terms of carbon, via :math:`a^{\op{C}}_{\op{phy}j,l}`.  Note that all
                 plankton types can have carbon-specific absorption and scattering, but only
                 phytoplankton can have Chlorophyll-specific absorption.  With
                 :varlink:`DARWIN_SCATTER_CHL` defined, scattering and backscattering spectra
                 are assumed to be per mg Chl and only available for phytoplankton.
                 
                 The particulate spectra, :math:`a^{\op{part}}_{\op{P}l}`, ..., are read in from
                 :varlink:`darwin_particleAbsorbFile`.  :math:`P_{\op{part}}` is particulate
                 organic matter in phosphorus units, including a recalcitrant component,
                 
                 .. math::
                 
                    P_{\op{part}} = \op{POP} + \frac{1}{120} \op{POC}_{\op{recalc}}
                    \;.
                 
                 Absorption by CDOM is computed from the CDOM tracer and a recalcitrant
                 component,
                 
                 .. math::
                 
                    a^{\op{CDOM}}_l = c_{\op{CDOM}} e^{\op{CDOM}}_l (\op{CDOM}
                                    + \op{CDOM}_{\op{recalc}})
                    \;,
                 
                 if :varlink:`DARWIN_ALLOW_CDOM` is defined, and estimated from that of water
                 and plankton otherwise,
                 
                 .. math::
                 
                    a^{\op{CDOM}}_l = f_{\op{aCDOM}} e^{\op{CDOM}}_l
                        ( a^{\op{w}}_{l_{\op{aCDOM}}} + a^{\op{plank}\prime}_{l_{\op{aCDOM}}} )
                    \;.
                 
                 Here, the primed quantity does not contain contributions from carbon-specific
                 absorption and :math:`l_{\op{aCDOM}}` is the index of the waveband in which
                 :math:`\lambda_{\op{aCDOM}}` falls.  The spectral dependence in both cases is
                 
                 .. math::
                 
                    e^{\op{CDOM}}_l = \mathrm{e}^{-S_{\op{DOM}}(\lambda_l - \lambda_{\op{aCDOM}})}
                 
                 :numref:`tab_phys_pkgs_darwin_spectral` summarizes the model parameters
                 relevant to spectral light.
                 
                 
                 .. csv-table:: Spectral light parameters
                    :delim: &
                    :widths: auto
                    :class: longtable
                    :header: Param, Symbol, Default, Units, Description
                    :name: tab_phys_pkgs_darwin_spectral
                 
faa67d1773 Oliv*    :varlink:`darwin_bbmin`        & :math:`b_{\op{b}}^{\min}`          & 0.0002  & 1/m                  & minimum backscattering ratio
                    :varlink:`darwin_bbw`          & :math:`\tilde b_{\op{b}}^{\op{w}}` & 0.5     &                      & backscattering ratio of water
                    :varlink:`darwin_RPOC`         & :math:`\op{POC}_{\op{recalc}}`     & 0.0     & mmol C/m\ :sup:`3`   & recalcitrant POC concentration
                    :varlink:`darwin_rCDOM`        & :math:`\op{CDOM}_{\op{recalc}}`    & 0.0     & mmol P/m\ :sup:`3`   & recalcitrant CDOM concentration
                                                   &                                    & 0.0     & mmol C/m\ :sup:`3`   & - if #define DARWIN_CDOM_UNITS_CARBON
                    :varlink:`CDOMcoeff`           & :math:`c_{\op{CDOM}}`              & 100.0   & m\ :sup:`2` / mmol P & P-specific absorption coefficient of CDOM at :math:`\lambda_{\op{CDOM}}`
                                                   &                                    & 100/120 & m\ :sup:`2` / mmol C & - if #define DARWIN_CDOM_UNITS_CARBON
                    :varlink:`darwin_lambda_aCDOM` & :math:`\lambda_{\op{aCDOM}}`       & 450.0   & nm                   & reference wavelength for CDOM absorption spectra
                    :varlink:`darwin_Sdom`         & :math:`S_{\op{DOM}}`               & 0.014   & 1/nm                 & coefficient for CDOM absorption spectra
                    :varlink:`darwin_aCDOM_fac`    & :math:`f_{\op{aCDOM}}`             & 0.2     &                      & factor for computing aCDOM from water+Chlorophyll absorption
                    :varlink:`darwin_part_size_P`  & :math:`q^{\op{part}}_{\op{P}}`     & 1E-15   & mmol P / particle    & conversion factor for particle absorption and scattering spectra
a4f2018aab Oliv* 
                 .. csv-table:: Spectral light traits
                    :delim: &
faa67d1773 Oliv*    :widths: 16,16,12,18,38
a4f2018aab Oliv*    :class: longtable
faa67d1773 Oliv*    :header: Trait, Param, Symbol, Units, Description
a4f2018aab Oliv* 
faa67d1773 Oliv*    :varlink:`aphy_chl`    & via :varlink:`grp_aptype` & :math:`a^{\op{chl}}_{\op{phy}j,l}`     & m\ :sup:`2` (mg Chl)\ :sup:`--1` & phytoplankton Chl-specific absorption coefficient
                    :varlink:`aphy_chl_ps` & via :varlink:`grp_aptype` & :math:`a^{\op{chl}}_{\op{ps}j,l}`      & m\ :sup:`2` (mg Chl)\ :sup:`--1` & part of :varlink:`aphy_chl` that is used in photosynthesis
                    :varlink:`aphy_mgC`    & via :varlink:`grp_aptype` & :math:`a^{\op{C}}_{\op{phy}j,l}`       & m\ :sup:`2` (mg C)\ :sup:`--1`   & plankton carbon-specific absorption coefficient
                    :varlink:`bphy_mgC`    & via :varlink:`grp_aptype` & :math:`b^{\op{C}}_{\op{phy}j,l}`       & m\ :sup:`2` (mg C)\ :sup:`--1`   & carbon-specific total scattering coefficient
                    :varlink:`bbphy_mgC`   & via :varlink:`grp_aptype` & :math:`b^{\op{C}}_{\op{b}\op{phy}j,l}` & m\ :sup:`2` (mg C)\ :sup:`--1`   & carbon-specific backscattering coefficient
a4f2018aab Oliv* 
                 
                 Format of optical spectra files
                 '''''''''''''''''''''''''''''''
                 
                 The spectra files have 6 header lines which will be ignored.  The format of the
                 data lines for each file is given in :numref:`tab_phys_pkg_darwin_spectra`.
                 The plankton spectra file contains multiple sections for the different optical
                 types.  Each starts with one line with reference sizes (ESD in microns; same
                 format, first column ignored), followed by a line for each waveband.  The
faa67d1773 Oliv* section used for each type is selected by :varlink:`grp_aptype`.
a4f2018aab Oliv* 
                 .. csv-table:: Format of data lines in optical spectra files
                    :delim: &
                    :widths: auto
                    :class: longtable
                    :header: File, Format, variables
                    :name: tab_phys_pkg_darwin_spectra
                 
                    :varlink:`darwin_waterAbsorbFile`    & (I5,F15,F10)             & :math:`\lambda_l`, :math:`a^{\op{w}}_l`, :math:`b^{\op{w}}_l`
faa67d1773 Oliv*    :varlink:`darwin_particleAbsorbFile` & (I4,F15,F15,F15)         & :math:`\lambda_l`, :math:`a^{\op{part}}_{l}`, :math:`b^{\op{part}}_{l}`, :math:`b^{\op{part}}_{\op{b}l}`
a4f2018aab Oliv*    :varlink:`darwin_phytoAbsorbFile`    & (I4,F10,F10,F10,F20,F10) & :math:`\lambda_l`, :math:`a^{\op{chl}}_{\op{phy}l}`, :math:`a^{\op{chl}}_{\op{ps}l}`, :math:`b^{\op{C}}_{\op{phy}l}`, :math:`b^{\op{C}}_{\op{b}\op{phy}l}`, :math:`a^{\op{C}}_{\op{phy}l}`
                                                         &                          & first line in sec: \*, :math:`d^{\op{a}}`, \*, :math:`d^{\op{b}}`, \*, :math:`d^{\op{aC}}`
                 
faa67d1773 Oliv* Particle spectra are read in units of m\ :sup:`2`/particle and converted to
                 m\ :sup:`2`/mmol P using a fixed conversion factor,
                 
                 .. math::
                 
                    a^{\op{part}}_{\op{P}l}       &= a^{\op{part}}_{l}/q^{\op{part}}_{\op{P}}
                 
                    b^{\op{part}}_{\op{P}l}       &= b^{\op{part}}_{l}/q^{\op{part}}_{\op{P}}
                 
                    b^{\op{part}}_{\op{b}\op{P}l} &= b^{\op{part}}_{\op{b}l}/q^{\op{part}}_{\op{P}}
                 
a4f2018aab Oliv* 
                 .. _allomSpectra:
                 
                 Allometric scaling of absorption and scattering spectra
                 '''''''''''''''''''''''''''''''''''''''''''''''''''''''
                 
                 If :varlink:`darwin_allomSpectra` is set to .TRUE., read-in absorption and
faa67d1773 Oliv* scattering spectra for each optical type :math:`i` (grp_aptype) are
                 scaled according to size before being assigned to a specific model plankton
                 type :math:`j` following :cite:`dutkiewicz:2020`.  Reference sizes for
                 absorption and scattering are read in as effective spherical diameters,
                 :math:`d^{\op{a}}_i`, :math:`d^{\op{aC}}_i`, :math:`d^{\op{b}}_i`, and
                 converted to volumes, :math:`V^{\op{a}}_i`, :math:`V^{\op{aC}}_i`,
                 :math:`V^{\op{b}}_i` via :math:`V=\frac{\pi}{6}d^3`.
a4f2018aab Oliv* 
                 Absorption
                 ++++++++++
                 
                 Read-in absorption spectra, :math:`a^{\op{meas}}_i`, are scaled in terms of volume,
                 
                 .. math::
                 
faa67d1773 Oliv*       a^{\op{chl}}_{\op{phy}j,l} &= a^{\op{meas}}_{i l} \cdot
                                       (V_j/V^{\op{a}}_i)^{s^{\op{a}}}
a4f2018aab Oliv*       \;,
                 
faa67d1773 Oliv*       a^{\op{chl}}_{\op{ps}j,l} &= a^{\op{ps\ meas}}_{i l} \cdot
                                                 (V_j/V^{\op{a}}_i)^{s^{\op{a}}}
a4f2018aab Oliv*       \;.
                 
                 Carbon-specific absorption is scaled similarly but with a different reference size,
                 
                 .. math::
                 
faa67d1773 Oliv*       a^{\op{C}}_{\op{phy}j,l} = a^{\op{C\,meas}}_{i l} \cdot
                                               (V_j/V^{\op{aC}}_i)^{s^{\op{a}}}
a4f2018aab Oliv*       \;.
                 
                 Total scattering
                 ++++++++++++++++
                 
                 Total scattering coefficients are converted from carbon to cell-density specific
                 using the relation between volume and carbon content of :cite:`montagnes:1994`,
                 
                 .. math::
                 
                    Q^{\op{C}} = a^{\op{C}}_{\op{cell}} V^{b^{\op{C}}_{\op{cell}}}
                    \;.
                 
                 The cell-density-specific coefficients are then scaled in terms of diameter and
                 converted back to carbon specific,
                 
                 .. math::
                 
faa67d1773 Oliv*    b^{\op{C}}_{\op{phy}j,l} Q^{\op{C}}_j = b^{\op{meas}}_{i l} Q^{\op{C\,b}}_i \cdot
                                     \left( d_j/d^{\op{b}}_i \right)^{s^{\op{b}}_{i l}}
a4f2018aab Oliv*    \;.
                 
                 There are 2 slopes for small and large measured cell sizes:
                 
                 .. math::
                 
faa67d1773 Oliv*    s^{\op{b}}_{i l} = \begin{cases}
                         s^{\op{bl}}_l
                             & \text{if } d^{\op{b}}_i \ge 10^{\ell^{\op{b}}_l} \\
                         s^{\op{bs}}_l
a4f2018aab Oliv*             & \text{else.}
                      \end{cases}
                 
                 Backscattering
                 ++++++++++++++
                 
                 We scale the non-spectral mean backscattering ratio using the reference
                 diameter for total scattering,
                 
                 .. math::
                    \tilde b_{\op{b}j} = \tilde b_{\op{b}i}^{\op{meas}} \cdot
                                         \left( d_j/d^{\op{b}}_i \right)^{s^{\op{bbb}}}
                    \;,
                 
                 where
                 
                 .. math::
                 
                    \tilde b_{\op{b}i}^{\op{meas}} =
faa67d1773 Oliv*      \frac{\sum_l b^{\op{meas}}_{\op{b}i l} \Delta\lambda_l}
                           {\sum_l b^{\op{meas}}_{i l} \Delta\lambda_l}
a4f2018aab Oliv* 
                 and compute spectral backscattering from total scattering,
                 
                 .. math::
                 
faa67d1773 Oliv*    b^{\op{C}}_{\op{b}\op{phy}j,l} = b^{\op{C}}_{\op{phy}j,l} \tilde b_{\op{b}j}
a4f2018aab Oliv*      \;.
                 
                 .. csv-table:: Allometric scaling parameters
                    :delim: &
                    :widths: auto
                    :class: longtable
                    :header: Param, Symbol, Default, Units, Description
                 
faa67d1773 Oliv*    :varlink:`darwin_allomSpectra`      &                                & .FALSE.  &           & enable/disable allometric scaling of plankton absorption and scattering spectra
                    :varlink:`darwin_aCarCell`          & :math:`a^{\op{C}}_{\op{cell}}` & 0.109E-9 & mg C/cell & coefficient coefficient for scaling plankton spectra
                    :varlink:`darwin_bCarCell`          & :math:`b^{\op{C}}_{\op{cell}}` & 0.991    &           & coefficient coefficient for scaling plankton spectra
                    :varlink:`darwin_absorpSlope`       & :math:`s^{\op{a}}`             & -0.075   &           & slope for scaled absorption spectra
                    :varlink:`darwin_bbbSlope`          & :math:`s^{\op{bbb}}`           & -1.458   &           & slope for scaled backscattering ratio spectra
                
** Warning **
Wide character in print at /usr/local/share/lxr/source line 1030, <$git> line 261.


    :varlink:`darwin_scatSwitchSizeLog` & :math:`\ell^{\op{b}}_l`        & 0        & log(μm)   & log of size for switching slopes
                    :varlink:`darwin_scatSlopeSmall`    & :math:`s^{\op{bs}}_l`          & 1.5      &           & slope for small plankton
                    :varlink:`darwin_scatSlopeLarge`    & :math:`s^{\op{bl}}_l`          & 1.5      &           & slope for large plankton
a4f2018aab Oliv* 
                 
                 Photosynthetically Active Radation
                 ''''''''''''''''''''''''''''''''''
                 
                 Radtrans provides spectral radiances in W m\ :sup:`-2` at vertical grid cell
                 boundaries, :math:`E_0^{\op{F}}`.  These are converted to photosynthetically
                 available radiation,
                 
                 .. math::
                 
                    I^{\op{F}}_l = 10^{-3} \frac{\lambda_l}{N_{\op{A}}h c}
                                          E^{\op{F}}_{0\,l}
                 
                 where :math:`h=6.6256\cdot 10^{-34}`, :math:`c=2.998\cdot 10^8` and
                 :math:`N_{\op{A}}=6.023\cdot 10^{23}`, and the pre-factor is for converting
                 :math:`\lambda` from nm to m and the result from Ein to µEin.  PAR at the
                 grid-cell center is computed as a geometric mean,
                 
                 .. math::
                 
                    I_l(r^{\op{C}}_k) = \sqrt{
                        I^{\op{F}}_l(r^{\op{F}}_k)
                        I^{\op{F}}_l(r^{\op{F}}_{k+1})}
                    \;.
                 

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