Light attenuation models

Nearly all BGC models require some model of the attenuation of PAR through the water. Usually this depends on the concentration of chlorophyll in the water (in phytoplankton), and may depend on the concentration of coloured dissolved organic matter or particulates.

We have two models implemented, a two band model by Karleskind et al. (2011), and a more generic "multi band" model which can have the PAR split into arbitary many wavelength bands, but default to the widely used three band model by Morel (1988). As the light level is diagnostic of the phytoplankton concentration these models are implemented with the light level as various auxiliary fields which are updated within the biogeochemical model.

Models requiring light attenuation models will set these up automatically, for example LOBSTER sets light_attenuation_model = TwoBandPhotosyntheticallyActiveRadiation(). You may choose others. Additionally, you can pass the surface PAR as a function of horizontal position and time. The default for LOBSTER is (x, y, t) -> 100*max(0.0, cos(t*π/(12hours))).

The two band and prescribed attenuation models (below) share a common internal implementation of exponential light attenuation, which also allows them to optionally track $PAR$ at cell faces as well as cell centres — see Recording PAR at cell faces below.

The multi band model

The surface intensity is split into multiple bands (usually with equal weight, but users may specify custom weights), and the attenuation of each band (i) is computed from the radiative transfer equation:

\[\frac{\partial PAR^i}{\partial z} = PAR^i (k^w(i) + \chi(i)Chl^{e(i)}),\]

where $Chl$ is the concentration of chlorophyll, $k^w(i)$ is the band specific water attenuation coefficient, $\chi(i)$ the chlorophyll attenuation coefficient, and $e(i)$ the chlorophyll exponent.

The water concentration of chlorophyll is returned by a function chlorophyll with arguments biogeochemistry and model. For the LOBSTER model this returns a constant ratio of the phytoplankton concentration, but may be different for other models.

The two band model

Light attenuation is calculated by integrating attenuation (from the surface). The $PAR$ is considered as two components attenuated at different rates. At depth $z$ the total $PAR$ is given by:

\[PAR = \frac{PAR_0}{2} \left[\exp\left(k_rz + \chi_r\int_{z=0}^z Chl_r dz\right) + \exp\left(k_bz + \chi_b\int_{z=0}^z Chl_b dz\right)\right],\]

where $PAR_0$ is the surface value, $k_r$ and $k_b$ are the red and blue attenuation coefficients of water, $\chi_r$ and $\chi_b$ are the red and blue chlorophyll attenuation coefficients, and $Chl_r$ and $Chl_b$ are the red and blue chlorophyll pigment concentrations. Total chlorophyll, $Chl$, is obtained from chlorophyll(model.biogeochemistry, model).

The red and blue pigment concentrations are then found as $Chl_r = \left(\frac{Chl}{r_\text{pig}}\right)^{e_r}$ and $Chl_b = \left(\frac{Chl}{r_\text{pig}}\right)^{e_b}$.

Parameter variable names

Recording PAR at cell faces

By default, the light attenuation models above return $PAR$ sampled at the centre of each grid cell, found by integrating the attenuation downward from the surface. TwoBandPhotosyntheticallyActiveRadiation and PrescribedAttenuationPAR can additionally be given a field on which to record the attenuation at cell faces (interfaces), by passing the interface_field keyword argument, for example:

using OceanBioME
using Oceananigans.Fields: ZFaceField

interface_field = ZFaceField(grid)
light_attenuation = TwoBandPhotosyntheticallyActiveRadiation(grid, surface_PAR; interface_field)

When interface_field is provided, attenuation is instead computed directly between successive interfaces (faces), and the value stored at each cell centre is the exponentially-weighted average of $PAR$ over that cell, consistent with the tracked face values, rather than a pointwise sample taken at the centre. This is probably a more accurate representation of the light available to a cell than the default pointwise sampling. If $K$ is the local attenuation coefficient and $PAR_{k+1}$ is the value at the shallower face of a cell of thickness $\Delta z$, the cell-averaged value used is

\[\overline{PAR} = PAR_{k+1} \frac{1 - e^{-K \Delta z}}{K \Delta z}.\]

For TwoBandPhotosyntheticallyActiveRadiation, the face values are also exposed as an additional PAR_interface auxiliary field. This can be useful on coarse grids, or wherever a flux-consistent estimate of the light available over a whole cell is preferred to a single point sample.

Prescribed attenuation model

PrescribedAttenuationPAR is a simpler model that integrates a prescribed attenuation coefficient downward from the surface, rather than computing attenuation from the chlorophyll concentration. At depth $z$ the PAR is:

\[PAR(z) = PAR_0 \exp\!\left(-\int_0^z K\, dz'\right),\]

where $K$ is the attenuation coefficient (units: 1/m). By default $K$ is a constant, but it may also be a function of position and time.

This model is used by default in ImplicitBiology and is useful when no chlorophyll-based feedback on light is desired. It is set up as:

using OceanBioME

light_attenuation = PrescribedAttenuationPAR(grid, surface_PAR; attenuation = 0.1)

where surface_PAR may be a constant or a function f(x, y, t), and attenuation may be a constant or a function f(x, y, z, t). PrescribedAttenuationPAR also accepts the interface_field keyword described in Recording PAR at cell faces.