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light-attenuation.rst

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Light attenuation formulation

Starting point is equation 6 of :cite:`Morel-2001`

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}

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

\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 :eq:`chl_tot_def` into equation :eq:`ze_chl_tot` and solving for Z_e yields

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}

The euphotic zone depth is defined to be the depth where PAR is 1% of its surface value :cite:`Morel-1988`.

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

Z_e = - \log 0.01 / \overline{K} = \log 100 / \overline{K}.

Substituting equation :eq:`Kbar_def` into equation :eq:`ze_chl_avg` and solving for \overline{K} yields

\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}

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 :eq:`Kbar_chl_avg` are 1/m.

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

.. only:: html

  .. rubric:: References


.. bibliography:: ../references.bib
  :filter: docname in docnames