1. Quantifying the organic carbon pump Jorn Bruggeman Theoretical Biology Vrije Universiteit, Amsterdam PhD March 2004 – 2009

2. Contents The project – Organic carbon pump – General aims – Biota modeling – Physics modeling New integration algorithm – Criteria – Mass and energy conservation – Existing algorithms – Extended modified Patankar Plans

3. The biological carbon pump Ocean top layer: CO 2 consumed by phytoplankton Phytoplankton biomass enters food web Biomass coagulates, sinks, enters deep Carbon from atmosphere, accumulates in deep water CO 2 (aq)biomass POC CO 2 (g)

4. The project Title – “Understanding the ‘organic carbon pump’ in meso-scale ocean flows” 3 PhDs – Physical oceanography, biology, numerical mathematics Aim: – quantitative prediction of global organic carbon pump from 3D models My role: – biota modeling, 1D water column

5. Biota modeling Dynamic Energy Budget theory (Kooijman 2000) Based on individual, extended to populations Defines generic kinetics for: – food uptake – food buffering – compound conversion – reproduction, growth Integrates existing approaches: – Michaelis-Menten functional response – Droop quota – Marr-Pirt maintenance – Von Bertalanffy growth – Body size scaling relationships

6. Biological model: mixotroph maintenance N-reserve detritus growth active biomass light nutrient CO 2 DOC death C-reserve death

7. Physics modeling GOTM water column Open ocean test cases (less influence of horizontal advection) weather: light air temperature air pressure relative humidity wind speed nutrient = 0 CO 2 nutrient = constant biotaturbulence carbon transport

8. Sample results: 10 year cycle biomass turbulence nutrient temperature

9. Integration algorithms (bio)chemical criteria: – Positive – Conservative – Order of accuracy Even if model meets requirements, integration results may not

10. Mass and energy conservation Model building block: transformation Conservation – for any element, sums on left and right must be equal Property of conservation – is independent of r – does depend on stoichiometric coefficients Complete conservation requires preservation of stoichiometric ratios

11. Systems of transformations Integration operates on (components of) ODEs Transformation fluxes distributed over multiple ODEs:

12. Forward Euler, Runge-Kutta Non-positive Conservative (stoichiometric ratios preserved) Order: 1, 2, 4 etc.

13. Backward Euler, Gear Positive for order 1 Conservative (stoichiometric ratios preserved) Generalization to higher order eliminates positivity Slow! – requires numerical approximation of partial derivatives – requires solving linear system of equations

14. Modified Patankar: concepts Burchard, Deleersnijder, Meister (2003) – “A high-order conservative Patankar-type discretisation for stiff systems of production-destruction equations” Approach – Transformation fluxes in production, destruction matrices (P, D) – P ij = rate of conversion from j to i – D ij = rate of conversion from i to j – Substrate fluxes in D, product fluxes in P

15. Modified Patankar: integration Flux-specific multiplication factors c n+1 /c n Represent ratio: (substrate after) : (substrate before) Multiple substrates in transformation: multiple, different c n+1 /c n factors Then: stoichiometric ratios not preserved!

16. Modified Patankar: example and conclusion Positive Conservative for single-substrate transformations only! Order 1, 2 (higher possible) Requires solving linear system of equations

17. Extended Modified Patankar 1 Non-linear system of equations Positivity requirement fixes domain of product term p:

18. Extended Modified Patankar 2 Polynomial for p: positive at left bound of p, negative at right bound Derivative of polynomial is negative within p domain: only one valid p Bisection technique is guaranteed to find p

19. Extended Modified Patankar 3 Positive Conservative (stoichiometric ratios preserved) Order 1, 2 (higher should be possible) ±20 bisection iterations (evaluations of polynomial) – Always cheaper than Backward Euler, Modified Patankar

20. Test case Nitrogen + carbon  phytoplankton

21. Test case: Modified Patankar

22. Test case: Extended Modified Patankar

23. Test case: conservation?

24. Plans Publish Extended Modified Patankar Short term – Modeling ecosystems – Aggregation into functional groups – Modeling coagulation (marine snow) Extension to complete ocean and world; longer timescales with surfacing of deep water – GOTM in MOM/POM/…, GETM? – Integration with meso-scale eddy results