Quantifying the organic carbon pump

Jorn BruggemanTheoretical BiologyVrije Universiteit, AmsterdamPhD March 2004 – 2009

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

The biological carbon pump

Ocean top layer: CO2

consumed by phytoplankton Phytoplankton biomass

enters food web Biomass coagulates, sinks,

enters deep Carbon from atmosphere,

accumulates in deep water

CO2 (aq) biomass

POC

CO2 (g)

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

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

Biological model: mixotroph

maintenance

N-reserve

detritus

growth activebiomass

light

nutrient

CO2

DOC

death

C-reserve

death

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

CO2

nutrient = constant

biotaturbulence

carbon transport

Sample results: 10 year cycle

biomass

turbulence

nutrient

temperature

Integration algorithms

(bio)chemical criteria:– Positive– Conservative– Order of accuracy

Even if model meets requirements, integration results may not

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

2 2 2 6 12 6CO H O O6 6 6 1C H Or

Systems of transformations

Integration operates on (components of) ODEs Transformation fluxes distributed over multiple

ODEs:2

2

2

6 12 6

6

6

6

CO

H O

O

C H O

dcr

dtdc

rdtdc

rdt

dcr

dt

2 2 2 6 12 66 6 6CO H O O C H Or

Forward Euler, Runge-Kutta

1 ,

i

n n n ni i i

c

c c t f t

c

Non-positive Conservative (stoichiometric ratios preserved) Order: 1, 2, 4 etc.

2

2

2

6 12 6

6

6

6

CO

H O

O

C H O

c t r

c t r

c t r

c t r

2 2 2 6 12 66 6 6CO H O O C H Or

Backward Euler, Gear

1 11 ,n ni i i

n nc c t f t c

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

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)– Pij = rate of conversion from j to i

– Dij = rate of conversion from i to j

– Substrate fluxes in D, product fluxes in P

Modified Patankar: integration

1 1

1

1 1

I In ni

n n

i ij ijj j

j in nj i

c cc c t P D

c c

Flux-specific multiplication factors cn+1/cn

Represent ratio: (substrate after) : (substrate before) Multiple substrates in transformation: multiple,

different cn+1/cn factors Then: stoichiometric ratios not preserved!

Modified Patankar: example and conclusion

2

2 2

2

2

2 2

2

11

11

6

6

nCOn n

CO CO nCO

nH On n

H O H O nH O

cc c t r

c

cc c t r

c

Positive Conservative for single-substrate transformations only! Order 1, 2 (higher possible) Requires solving linear system of equations

2 2

2 2

1 1n nCO H O

n nCO H O

c c

c c

2 2 2 6 12 66 6 6CO H O O C H Or

11 , with

: ( , ) 0 for all {1,..., }

njn n n n

i i i nj J j

n ni

cpc c t f t p

c

J i f t i I

c

c

Extended Modified Patankar 1

Non-linear system of equations Positivity requirement fixes domain of product term p:

0

1

min,

nin nj J

i

p

p

cp

t f t

c

Extended Modified Patankar 2

11 ,1 with

,1

n n nni ji

n n nj Ji i j

n nj

nj J j

t f t ccp p

c c c

t f tp p

c

c

c

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

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

Test case

Nitrogen + carbon phytoplankton

16106

1 0.1

rN C P

C Nr

C N

Test case: Modified Patankar

Modified Patankar first order scheme

0

20

40

60

80

100

120

0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8 2,0

time (days)

con

cen

trat

ion

carbon MP1

carbon reference

nitrogen MP1

nitrogen reference

phytoplankton MP1

phytoplankton reference

Test case: Extended Modified Patankar

Extended Modified Patankar first order scheme

0

20

40

60

80

100

120

0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8 2,0

time (days)

con

cen

trat

ion

carbon EMP1

carbon reference

nitrogen EMP1

nitrogen reference

phytoplankton EMP1

phytoplankton reference

Test case: conservation?

Nitrogen conservation

98

98,5

99

99,5

100

100,5

0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8 2,0

time (days)

tota

l n

itro

gen

MP1

EMP1

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