Phosphorus uptake in Fennel model's implicit scheme

[dullman]

I have been examining the implicit scheme for phosphorus in the Fennel model and it seems to me that there
is no constraint that ensures that the ratio of phosphorus uptake/nitrogen uptake is equal to the model's specified P/N ratio
(R_P2N). If this constraint is not enforced then, it seems to me that phosphorus will not necessarily be conserved (summing over
phytoplankton and dissolved phosphate). I suggest that the factor cff6 (used in the PO4 update: Bio(i,k,iPO4_)=Bio(i,k,iPO4_)/(1+cff6) )
should be calculated from the previously computed cff4 and cff5 as shown below.

the NO3, NH4, and PO4 update equations are:
Bio(i,k,iNO3_)=Bio(i,k,iNO3_)/(1+cff4)
Bio(i,k,iNH4_)=Bio(i,k,iNH4_)/(1+cff5)
Bio(i,k,iPO4_)=Bio(i,k,iPO4_)/(1+cff6)

the uptakes of NO3 and NH4 are:
NO3_uptake=Bio(i,k,iNO3_)*cff4/(1+cff4)
NH4_uptake=Bio(i,k,iNH4_)*cff5/(1+cff5)

the total nitrogen uptake (TNU) is given by: TNU=NO3_uptake+NH4_uptake

the phosphorus uptake should be equal to R_P2N*TNU:
Bio(i,k,iPO4_)*cff6/(1+cff6)=R_P2N*TNU.

solving for cff6 gives:
cff6=R_P2N*TNU/(Bio(i,k,iPO4_) - R_P2N*TNU),
where TNU=Bio(i,k,iNO3_)*cff4/(1+cff4) + Bio(i,k,iNH4_)*cff5/(1+cff5)

Does this make sense? Am I incorrect in my conclusion that the cff6 calculation in the code does not ensure the correct P/N ratio
in the nutrient uptakes?

thanks,
dave

[arnaudlaurent]

The formulations of nutrient uptake with the implicit scheme are detailed in the attached file below. You can see from the derivation of PO4_uptake that with the formulation Bio(i,k,iPO4_)=Bio(i,k,iPO4_)/(1+cff6), the phosphorus uptake is indeed equal to R_P2N*TNU.

[dullman]

hi arnaud,
thanks for your reply. although i follow your derivation of the implicit scheme, i still think that there is a small
deviation from the specified P/N ratio in the actual nutrient uptake. as shown in the attached file, which assumes only
one nitrogenous nutrient (NO3), i arrive at this conclusion by evaluating the uptakes of nitrogen and phosphorus according
to the discretization scheme used. it seems that the uptake ratio (P uptake/N uptake) is exactly the specified P/N ratio only when the ratio P_old/N_old
equals the specified P/N ratio. this discrepancy goes to zero as the timestep goes to zero, implying that it is simply due to finite
difference error. however, for realistic timesteps and dissolved P and N far from the specified P/N ratio, it looks like the discrepancy
may be significant, especially for long model runs where nutrient conservation is important. i note that the scheme i proposed in my initial post
ensures that P and N are taken up at precisely the specified P/N ratio.
dave