\(\newcommand{\p}[1]{\frac{\partial }{\partial #1}}\) \(\newcommand{\pp}[2]{\frac{\partial #1}{\partial #2}}\) \(\newcommand{\dd}[2]{\frac{d #1}{d #2}}\) \(\newcommand{\h}{\frac{1}{2}}\) \(\newcommand{\op}[1]{\operatorname{#1}}\)

8.7.3.5. Spectral Light

Spectral light throughout the water column is computed following [DHJ+15]. The RADTRANS package 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:

\[ \begin{align}\begin{aligned}a_l &= a^{\op{w}}_l + a^{\op{plank}}_l + a^{\op{part}}_{\op{P}l} P_{\op{part}} + a^{\op{CDOM}}_l\\b_l &= b^{\op{w}}_l + b^{\op{plank}}_l + b^{\op{part}}_{\op{P}l} P_{\op{part}}\\b_{\op{b}l} &= \left[ \tilde b^{\op{w}}_{\op{b}} b^{\op{w}}_l + b^{\op{plank}}_{\op{b}l} + b^{\op{part}}_{\op{b}\op{P}l} P_{\op{part}} \right]_{\ge b_{\op{b}}^{\min}}\end{aligned}\end{align} \]

Water IOPs, \(a^{\op{w}}_l\) and \(b^{\op{w}}_l\), are read in from darwin_waterAbsorbFile. Plankton IOPs are computed from individual functional types,

\[ \begin{align}\begin{aligned}a^{\op{plank}}_l &= \sum_j \op{Chl}_j a^{\op{chl}}_{\op{phy}j,l} + \sum_j 12\, c_j a^{\op{C}}_{\op{phy}j,l}\\b^{\op{plank}}_l &= \sum_j 12\, c_j b^{\op{C}}_{\op{phy}j,l}\\b^{\op{plank}}_{\op{b}l} &= \sum_j 12\, c_j b^{\op{C}}_{\op{b}\op{phy}j,l}\end{aligned}\end{align} \]

The spectra are selected based on optical type, grp_aptype, from spectra read in from darwin_phytoAbsorbFile. Usually, phytoplankton absorption spectra are given per amount of Chlorophyll, via \(a^{\op{chl}}_{\op{phy}j,l}\), while bacteria absorption spectra are given in terms of carbon, via \(a^{\op{C}}_{\op{phy}j,l}\). Note that all plankton types can have carbon-specific absorption and scattering, but only phytoplankton can have Chlorophyll-specific absorption. With DARWIN_SCATTER_CHL defined, scattering and backscattering spectra are assumed to be per mg Chl and only available for phytoplankton.

The particulate spectra, \(a^{\op{part}}_{\op{P}l}\), …, are read in from darwin_particleAbsorbFile. \(P_{\op{part}}\) is particulate organic matter in phosphorus units, including a recalcitrant component,

\[P_{\op{part}} = \op{POP} + \frac{1}{120} \op{POC}_{\op{recalc}} \;.\]

Absorption by CDOM is computed from the CDOM tracer and a recalcitrant component,

\[a^{\op{CDOM}}_l = c_{\op{CDOM}} e^{\op{CDOM}}_l (\op{CDOM} + \op{CDOM}_{\op{recalc}}) \;,\]

if DARWIN_ALLOW_CDOM is defined, and estimated from that of water and plankton otherwise,

\[a^{\op{CDOM}}_l = f_{\op{aCDOM}} e^{\op{CDOM}}_l ( a^{\op{w}}_{l_{\op{aCDOM}}} + a^{\op{plank}\prime}_{l_{\op{aCDOM}}} ) \;.\]

Here, the primed quantity does not contain contributions from carbon-specific absorption and \(l_{\op{aCDOM}}\) is the index of the waveband in which \(\lambda_{\op{aCDOM}}\) falls. The spectral dependence in both cases is

\[e^{\op{CDOM}}_l = \mathrm{e}^{-S_{\op{DOM}}(\lambda_l - \lambda_{\op{aCDOM}})}\]

Table 8.32 summarizes the model parameters relevant to spectral light.

Table 8.32 Spectral light parameters

Param	Symbol	Default	Units	Description
darwin_bbmin	\(b_{\op{b}}^{\min}\)	0.0002	1/m	minimum backscattering ratio
darwin_bbw	\(\tilde b_{\op{b}}^{\op{w}}\)	0.5		backscattering ratio of water
darwin_RPOC	\(\op{POC}_{\op{recalc}}\)	0.0	mmol C/m3	recalcitrant POC concentration
darwin_rCDOM	\(\op{CDOM}_{\op{recalc}}\)	0.0	mmol P/m3	recalcitrant CDOM concentration
		0.0	mmol C/m3	if #define DARWIN_CDOM_UNITS_CARBON
CDOMcoeff	\(c_{\op{CDOM}}\)	100.0	m2 / mmol P	P-specific absorption coefficient of CDOM at \(\lambda_{\op{CDOM}}\)
		100/120	m2 / mmol C	if #define DARWIN_CDOM_UNITS_CARBON
darwin_lambda_aCDOM	\(\lambda_{\op{aCDOM}}\)	450.0	nm	reference wavelength for CDOM absorption spectra
darwin_Sdom	\(S_{\op{DOM}}\)	0.014	1/nm	coefficient for CDOM absorption spectra
darwin_aCDOM_fac	\(f_{\op{aCDOM}}\)	0.2		factor for computing aCDOM from water+Chlorophyll absorption
darwin_part_size_P	\(q^{\op{part}}_{\op{P}}\)	1E-15	mmol P / particle	conversion factor for particle absorption and scattering spectra

Table 8.33 Spectral light traits

Trait	Param	Symbol	Units	Description
aphy_chl	via grp_aptype	\(a^{\op{chl}}_{\op{phy}j,l}\)	m2 (mg Chl)–1	phytoplankton Chl-specific absorption coefficient
aphy_chl_ps	via grp_aptype	\(a^{\op{chl}}_{\op{ps}j,l}\)	m2 (mg Chl)–1	part of aphy_chl that is used in photosynthesis
aphy_mgC	via grp_aptype	\(a^{\op{C}}_{\op{phy}j,l}\)	m2 (mg C)–1	plankton carbon-specific absorption coefficient
bphy_mgC	via grp_aptype	\(b^{\op{C}}_{\op{phy}j,l}\)	m2 (mg C)–1	carbon-specific total scattering coefficient
bbphy_mgC	via grp_aptype	\(b^{\op{C}}_{\op{b}\op{phy}j,l}\)	m2 (mg C)–1	carbon-specific backscattering coefficient

8.7.3.5.1. 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 Table 8.34. 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 section used for each type is selected by grp_aptype.

Table 8.34 Format of data lines in optical spectra files

File	Format	variables
darwin_waterAbsorbFile	(I5,F15,F10)	\(\lambda_l\), \(a^{\op{w}}_l\), \(b^{\op{w}}_l\)
darwin_particleAbsorbFile	(I4,F15,F15,F15)	\(\lambda_l\), \(a^{\op{part}}_{l}\), \(b^{\op{part}}_{l}\), \(b^{\op{part}}_{\op{b}l}\)
darwin_phytoAbsorbFile	(I4,F10,F10,F10,F20,F10)	\(\lambda_l\), \(a^{\op{chl}}_{\op{phy}l}\), \(a^{\op{chl}}_{\op{ps}l}\), \(b^{\op{C}}_{\op{phy}l}\), \(b^{\op{C}}_{\op{b}\op{phy}l}\), \(a^{\op{C}}_{\op{phy}l}\)
		first line in sec: *, \(d^{\op{a}}\), *, \(d^{\op{b}}\), *, \(d^{\op{aC}}\)

Particle spectra are read in units of m²/particle and converted to m²/mmol P using a fixed conversion factor,

\[ \begin{align}\begin{aligned}a^{\op{part}}_{\op{P}l} &= a^{\op{part}}_{l}/q^{\op{part}}_{\op{P}}\\b^{\op{part}}_{\op{P}l} &= b^{\op{part}}_{l}/q^{\op{part}}_{\op{P}}\\b^{\op{part}}_{\op{b}\op{P}l} &= b^{\op{part}}_{\op{b}l}/q^{\op{part}}_{\op{P}}\end{aligned}\end{align} \]

8.7.3.5.2. Allometric scaling of absorption and scattering spectra

If darwin_allomSpectra is set to .TRUE., read-in absorption and scattering spectra for each optical type \(i\) (grp_aptype) are scaled according to size before being assigned to a specific model plankton type \(j\) following [DCJ+20]. Reference sizes for absorption and scattering are read in as effective spherical diameters, \(d^{\op{a}}_i\), \(d^{\op{aC}}_i\), \(d^{\op{b}}_i\), and converted to volumes, \(V^{\op{a}}_i\), \(V^{\op{aC}}_i\), \(V^{\op{b}}_i\) via \(V=\frac{\pi}{6}d^3\).

Absorption

Read-in absorption spectra, \(a^{\op{meas}}_i\), are scaled in terms of volume,

\[ \begin{align}\begin{aligned}a^{\op{chl}}_{\op{phy}j,l} &= a^{\op{meas}}_{i l} \cdot (V_j/V^{\op{a}}_i)^{s^{\op{a}}} \;,\\a^{\op{chl}}_{\op{ps}j,l} &= a^{\op{ps\ meas}}_{i l} \cdot (V_j/V^{\op{a}}_i)^{s^{\op{a}}} \;.\end{aligned}\end{align} \]

Carbon-specific absorption is scaled similarly but with a different reference size,

\[a^{\op{C}}_{\op{phy}j,l} = a^{\op{C\,meas}}_{i l} \cdot (V_j/V^{\op{aC}}_i)^{s^{\op{a}}} \;.\]

Total scattering

Total scattering coefficients are converted from carbon to cell-density specific using the relation between volume and carbon content of [MBHT94],

\[Q^{\op{C}} = a^{\op{C}}_{\op{cell}} V^{b^{\op{C}}_{\op{cell}}} \;.\]

The cell-density-specific coefficients are then scaled in terms of diameter and converted back to carbon specific,

\[b^{\op{C}}_{\op{phy}j,l} Q^{\op{C}}_j = b^{\op{meas}}_{i l} Q^{\op{C\,b}}_i \cdot \left( d_j/d^{\op{b}}_i \right)^{s^{\op{b}}_{i l}} \;.\]

There are 2 slopes for small and large measured cell sizes:

\[\begin{split}s^{\op{b}}_{i l} = \begin{cases} s^{\op{bl}}_l & \text{if } d^{\op{b}}_i \ge 10^{\ell^{\op{b}}_l} \\ s^{\op{bs}}_l & \text{else.} \end{cases}\end{split}\]

Backscattering

We scale the non-spectral mean backscattering ratio using the reference diameter for total scattering,

\[\tilde b_{\op{b}j} = \tilde b_{\op{b}i}^{\op{meas}} \cdot \left( d_j/d^{\op{b}}_i \right)^{s^{\op{bbb}}} \;,\]

where

\[\tilde b_{\op{b}i}^{\op{meas}} = \frac{\sum_l b^{\op{meas}}_{\op{b}i l} \Delta\lambda_l} {\sum_l b^{\op{meas}}_{i l} \Delta\lambda_l}\]

and compute spectral backscattering from total scattering,

\[b^{\op{C}}_{\op{b}\op{phy}j,l} = b^{\op{C}}_{\op{phy}j,l} \tilde b_{\op{b}j} \;.\]

Table 8.35 Allometric scaling parameters

Param	Symbol	Default	Units	Description
darwin_allomSpectra		.FALSE.		enable/disable allometric scaling of plankton absorption and scattering spectra
darwin_aCarCell	\(a^{\op{C}}_{\op{cell}}\)	0.109E-9	mg C/cell	coefficient coefficient for scaling plankton spectra
darwin_bCarCell	\(b^{\op{C}}_{\op{cell}}\)	0.991		coefficient coefficient for scaling plankton spectra
darwin_absorpSlope	\(s^{\op{a}}\)	-0.075		slope for scaled absorption spectra
darwin_bbbSlope	\(s^{\op{bbb}}\)	-1.458		slope for scaled backscattering ratio spectra
darwin_scatSwitchSizeLog	\(\ell^{\op{b}}_l\)	0	log(μm)	log of size for switching slopes
darwin_scatSlopeSmall	\(s^{\op{bs}}_l\)	1.5		slope for small plankton
darwin_scatSlopeLarge	\(s^{\op{bl}}_l\)	1.5		slope for large plankton

8.7.3.5.3. Photosynthetically Active Radation

Radtrans provides spectral radiances in W m^-2 at vertical grid cell boundaries, \(E_0^{\op{F}}\). These are converted to photosynthetically available radiation,

\[I^{\op{F}}_l = 10^{-3} \frac{\lambda_l}{N_{\op{A}}h c} E^{\op{F}}_{0\,l}\]

where \(h=6.6256\cdot 10^{-34}\), \(c=2.998\cdot 10^8\) and \(N_{\op{A}}=6.023\cdot 10^{23}\), and the pre-factor is for converting \(\lambda\) from nm to m and the result from Ein to µEin. PAR at the grid-cell center is computed as a geometric mean,

\[I_l(r^{\op{C}}_k) = \sqrt{ I^{\op{F}}_l(r^{\op{F}}_k) I^{\op{F}}_l(r^{\op{F}}_{k+1})} \;.\]