Hi ecosystem modellers,

In the ecosystem models, I noticed that light is attenuated twice for each vertical cell. Once to the approximate centre of the cell, and then again to the bottom. The light at the centre of the cell is then used for calculating photosynthetic processes.

As light decays exponentially with depth, the light halfway through the cell (the median light for the cell) will be less than the average in the cell. The average light is given by:

1/d \int_0^d I exp(-k z} dz = I(1-exp(-kd))/kd,

[I = light at the top of the layer,k=attenuation rate;d=layer thickness]

while the median value is I exp(-kd/2).

To illustrate the difference, run the following matlab commands:

**Code:**

kd = 0.1:0.1:100;

Iav = (1-exp(-kd))./kd;

I2 = exp(-kd/2);

semilogx(kd,I2./Iav);line([0 109999],[1 1]);

xlabel('kd');

ylabel('exp(-kd/2) / ((1-exp(-kd))./(kd))');

set(gca,'xlim',[0.1 100],'ylim',[0 1.1]);grid on

or go to:

http://www.maths.unsw.edu.au/~mbaird/li ... lation.pdfFor small kd the two forms are the same, but as kd increases, the halfway calculation underestimates the light compared to the average calculation. In a sigma co-ordinate model with increasing layer thickness with depth, this would be most pronounced in a deep chlorophyll maximum in deep waters. For a layer 30 m thick, with an attenuation rate of 0.3 m-1, the halfway calculation would give a calculated light for the cell of 10% of the average calculation.

The median form also has the undesirable effect of producing a different vertically resolved light field (for photosynthesis) depending on the vertical resolution of the grid. The average

formulation does not have this problem. The median forms merges to the vertical profile of the average form for high resolution

(because kd becomes small), suggesting the average form is better.

To illustrate the difference, I have run the upwelling case with fennel.h editted by:

**Code:**

diff fennel_orig.h fennel.h

293c293,294

<

---

> ! added Itop,ExpAtt for new calculation of light for photosynthesis

> real(r8) :: Itop,ExpAtt

546c547,548

< Att=EXP(-0.5_r8*(AttSW(ng)+ &

---

> ! Edit to base photosytnesis on average light in cell

> Att=(AttSW(ng)+ &

549,550c551,554

< & (z_w(i,j,k)-z_w(i,j,k-1)))

< PAR=PAR*Att

---

> & (z_w(i,j,k)-z_w(i,j,k-1))

> ExpAtt = EXP(-Att)

> Itop=PAR

> PAR = PAR*(1-ExpAtt)/Att

652c656,657

< PAR=PAR*Att

---

> ! Correct for above change in use of variable PAR.

> PAR = Itop*ExpAtt

This includes two more real variables, and two extra lines of codes. There is only 1 call to the EXP function, as per the original.

As the effect is more pronounced in opaque waters, I have run the model with the upwelling case for water column attenuation due to water increased to 0.25 and 0.5. These are modest increases for a coastal system. The other configuration that will show

differences are open ocean cases with thick layers.

You can find the output at:

http://www.maths.unsw.edu.au/~mbaird/up ... kw0p50.pdf and

http://www.maths.unsw.edu.au/~mbaird/up ... kw0p25.pdfThe images show primary production for the average form (left col),

median (original) form (centre col) and the percent difference (right col) for days 2.5, 5.5 and 8.5.

The average form always increases PP if all other things are equal (i.e. NO3, T, P). For the kw=0.25, the deeper (thicker layers) centre columns increased by 5-15 %, while this increases to 15-25 % for kw=0.5. It is not long before feedbacks (i.e. changes in P biomass changing) result in greater changes in PP. At the edges, where layer thickness are smaller, the difference is less than 5%.

In any case, while this shows it working in the simulations, lightformulation.pdf above tells the story best, because it is free of biological feedbacks. The difference will certainly be greatest for thick layers in opaque waters.

I understand there is discussion about merging the vertical attenuation of light for the purposes of water column heating and photosynthesis. This might be a good time to consider the calculation of light for photosynthesis for which I would be happy to work on with anyone interested.

Yours sincerely,

Mark Baird

m.baird@unsw.edu.au