Simple coupled physical-biogeochemical models of marine ecosystems

1. mathematical ^ Simple coupled physical-biogeochemical models of marine ecosystems Formulating quantitative mathematical models of conceptual ecosystems MS320: John Wilkin

2. Why use mathematical models? • Conceptual models often characterize an ecosystem as a set of “boxes” linked by processes • Processese.g. photosynthesis, growth, grazing, and mortality link elements of the … • State variables (“the boxes”) e.g. nutrient concentration, phytoplankton abundance, biomass, dissolved gases, of an ecosystem • In the lab, field, or mesocosm, we can observe some of the complexity of an ecosystem and quantify these processes • With quantitative rules for linking the boxes, we can attempt to simulate the changes over time of the ecosystem state

3. What can we learn? • Suppose a model can simulate the spring bloom chlorophyll concentration observed by satellite using: observed light, a climatology of winter nutrients, ocean temperature and mixed layer depth … • … then the modeled rates of uptake of nutrients during the bloom and loss of particulates below the euphotic zone give us quantitative information on net primary production and carbon export – quantities we cannot easily observe directly

4. Individual plants and animals Many influences from nutrients and trace elements Continuous functions of space and time Varying behavior, choice, chance Unknown or incompletely understood interactions Lump similar individuals into groups Might express some biomass as C:N ratio Small number of state variables (one or two limiting nutrients) Discrete spatial points and time intervals Average behavior based on ad hoc assumptions Must parameterize unknowns Reality Model

5. The steps in constructing a model • Identify the scientific problem(e.g. seasonal cycle of nutrients and plankton in mid-latitudes; short-term blooms associated with coastal upwelling events; human-induced eutrophication and water quality; global climate change) • Determine relevant variables and processes that need to be considered • Develop mathematical formulation • Numerical implementation, provide forcing, parameters, etc.

6. State variables and Processes “NPZD”: model named for and characterized by its state variables Statevariables are concentrations (in a common “currency”) that depend on space and time Processes link the state variable boxes

7. Processes • Biological: • Growth • Death / mortality • Photosynthesis • Grazing • Bacterial regeneration of nutrients • Excretion of dissolved matter • Egestion of undigested food • Physical: • Mixing • Transport (by currents from tides, winds …) • Light • Air-sea interaction (winds, heat fluxes, precipitation)

8. State variables and Processes Can use Redfield ratio to give e.g. carbon biomass from nitrogen equivalent Carbon-chlorophyll ratio might be assumed Where is the physics?

9. A model of a food web might be relatively complex: • Several nutrients • Different size/species classes of phytoplankton • Different size/species classes of zooplankton • Detritus (multiple size classes) • Predation (predators and their behavior) • Multiple trophic levels • Pigments and bio-optical properties • Photo-adaptation, self-shading • 3 spatial dimensions in the physical environment, diurnal cycle of atmospheric forcing and light; tides

10. A very simple model N – P – Z • Nutrients • Phytoplankton • Zooplankton… all expressed in terms of equivalent nitrogen concentration • Several elements of the state and multiple processes are combined • e.g. the action of bacterial regeneration is treated as a flux from zooplankton mortality directly to nutrients with no explicit bacteria in the state

11. To run this yourself download the java program NPZ Visualizer from the class web site ROMS fennel.h(carbon off, oxygen off, chl not shown) Banas, N. S., E. J. Lessard, R. M. Kudela, P. MacCready, T. D. Peterson, B. M. Hickey, and E. Frame (2009), Planktonic growth and grazing in the Columbia River plume region: A biophysical model study, J. Geophys. Res., 114, C00B06, doi:10.1029/2008JC004993.

12. Phytoplankton concentration absorbs light Att(x,z) = AttSW + AttChl*Chlorophyll(x,z,t) Schematic of ROMS “Fennel”ecosystem model

13. Mathematical formulation e.g. inputs of nutrients from rivers or sediments e.g. burial in sediments e.g. nutrient uptake by phytoplankton The key to model building is finding appropriate formulations for transfers, and not omitting important state variables

14. Some calculus Slope of a continuous function of x is Baron Gottfried Wilhelm von Leibniz 1646-1716

15. For example:State variables: Nutrient and PhytoplanktonProcess: Photosynthetic production of organic matter Large N Small N Michaelis and Menten (1913) vmax is maximum growth rate (units are time-1) kn is “half-saturation” concentration; at N=kn f(kn)=0.5

16. State variables: Nutrient and PhytoplanktonProcess: Photosynthetic production of organic matter The nitrogen consumed by the phytoplankton for growth must be lost from the Nutrients state variable The total inventory of nitrogen is conserved

17. Suppose there are ample nutrients so N is not limiting: then f(N) = 1 • Growth of P will be exponential

18. Suppose the plankton concentration held constant, and nutrients again are not limiting: f(N) = 1 • N will decrease linearly with time as it is consumed to grow P

19. Suppose the plankton concentration held constant, but nutrients become limiting: then f(N) = N/kn • N will exponentially decay to zero until it is exhausted

20. Can the right-hand-side of the P equation be negative? Can the right-hand-side of the N equation be positive? … So we need other processes to complete our model.

22. Rate of vertical mixing from below depends on vertical gradient in nutrients and turbulent mixing coefficient N ~~~~~~~~~~~ No

23. Coupling to physical processes Advection-diffusion-equation: physics turbulent mixing Biological dynamics advection C is the concentration of any biological state variable

24. I0 spring summer fall winter

25. Simple 1-dimensional vertical model of mixed layer and N-P ecosystem • Windows program and inputs files are at: http://marine.rutgers.edu/dmcs/ms320/Phyto1d/ • Run the program called Phyto_1d.exe using the default inputfiles • Sharples, J., Investigating theseasonal vertical structure of phytoplankton in shelf seas, Marine Models Online, vol 1, 1999, 3-38.

26. wind stress drives mixing at surface z PAR sea surface ~~~~~~~~~~~~~~~~ n n-1 n-2 PAR Biological equations are solved in each grid element depending on the local N, P and Z and available light. PAR Vertical turbulent mixes causes N, P and Z to exchange between the grid element, and P and Z can sink. tidal currents drive mixing at the bottom

27. wind stress drives mixing at surface z PAR sea surface ~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~ O O O n n-1 n-2 O O O PAR PAR ChlDIN tidal currents drive mixing at the bottom

28. The rate of change of phytoplankton biomass is described by the equation vertical sinking at velocity ws grazing mortality phytoplankton growth vertical turbulentmixing of phytoplankton The first term on the right describes the vertical turbulent transport of biomass.The second term represents growth of phytoplankton, with μ the specific instantaneous growth rate (s-1). Growth can be either light-limited or nutrient-limited, so the growth rate is taken as the lesser of: (19a) describes nutrient-determined growth. μmaxis the max growth rate, kQ the subsistence cell nutrient quota. (19b) describes light-determined growth, driven by mean photosynthetically-available radiation (PAR) in a depth element (I W/m2) of the model. qchl is cell chlorophyll:carbon ratio, α is the max quantum yield, and rB is the respiration rate.

29. Run Change settings …. Physicsc.dat: stronger PAR attenuation eliminates mid-depth chl-max Phyto1d.dat: greater respiration rate delaysbloom until photosynthesis rate is greater

30. I0 spring summer fall winter bloom secondary bloom

31. I0 spring summer fall winter bloom secondary bloom