Light attenuation formulation

Starting point is equation 6 of [MM01]

(1)\[\begin{split}Z_e = \begin{cases} 912.5 \left[ \mathrm{Chl_{tot}} \right]^{-0.839} & 10 < Z_e < 102 \\ 426.3 \left[ \mathrm{Chl_{tot}} \right]^{-0.547} & 102 < Z_e < 180 \end{cases}\end{split}\]

In this approximation, \(\left[ \mathrm{Chl_{tot}} \right]\) is \(\left[ \mathrm{Chl}(z) \right]\) integrated from the surface to \(Z_e\), the depth of the euphotic zone.

We will convert this formula to one based on \(\left[ \mathrm{Chl}(z) \right]\) averaged over the euphotic zone

(2)\[\left[ \mathrm{Chl_{tot}} \right] = Z_e \overline{\left[ \mathrm{Chl} \right]}\]

The crossover point \(Z_e=102\) corresponds to \(\left[ \mathrm{Chl_{tot}} \right]=13.65\), which is equivalent to \(\overline{\left[ \mathrm{Chl} \right]}=0.1338\).

Substituting equation (2) into equation (1) and solving for \(Z_e\) yields

(3)\[\begin{split}Z_e = \begin{cases} 40.710 \overline{\left[ \mathrm{Chl} \right]}^{-0.4562} & \overline{\left[ \mathrm{Chl} \right]} > 0.1338 \\ 50.105 \overline{\left[ \mathrm{Chl} \right]}^{-0.3536} & \overline{\left[ \mathrm{Chl} \right]} < 0.1338 \end{cases}\end{split}\]

The euphotic zone depth is defined to be the depth where PAR is 1% of its surface value [Mor88].

We denote the attenuation coefficient of PAR as \(K\), and its effective average over the euphotic zone as \(\overline{K}\).

So we have

\[0.01 = e^{-Z_e \overline{K}}.\]

Solving for \(Z_e\) yields

(4)\[Z_e = - \log 0.01 / \overline{K} = \log 100 / \overline{K}.\]

Substituting equation (4) into equation (3) and solving for \(\overline{K}\) yields

(5)\[\begin{split}\overline{K} = \begin{cases} 0.1131 \overline{\left[ \mathrm{Chl} \right]}^{0.4562} & \overline{\left[ \mathrm{Chl} \right]} > 0.1338 \\ 0.0919 \overline{\left[ \mathrm{Chl} \right]}^{0.3536} & \overline{\left[ \mathrm{Chl} \right]} < 0.1338 \end{cases}\end{split}\]

In the model implementation, this equation relating \(\overline{K}\) to \(\overline{\left[ \mathrm{Chl} \right]}\) is applied to each model layer.

The crossover point was recomputed to be where the curves cross, yielding \(\overline{\left[ \mathrm{Chl} \right]}=0.13224\).

The units of \(K\) in equation (5) are 1/m.

Model units are cm, so the model implementation includes multiplication by 0.01.

References

[Mor88]

André Morel. Optical modeling of the upper ocean in relation to its biogenous matter content (case I waters). J. Geophys. Res., 93(C9):10749–10768, 1988. doi:10.1029/jc093ic09p10749.

[MM01]

André Morel and Stéphane Maritorena. Bio-optical properties of oceanic waters: a reappraisal. J. Geophys. Res., 106(C4):7163–7180, Apr 2001. doi:10.1029/2000jc000319.