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SIMPLE COUPLED PHYSICAL-BIOGEOCHEMICAL MODELS OF MARINE ECOSYSTEMS. Why use mathematical models?. Conceptual models often characterize an ecosystem as a set of “ boxes ” linked by processes Processes e.g. photosynthesis, growth, grazing, and mortality link elements of the …

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why use mathematical models
Why use mathematical models?
  • Conceptual models often characterize an ecosystem as a set of “boxes” linked by processes
  • Processes e.g. photosynthesis, growth, grazing, and mortality link elements of the …
  • State (“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

SUMBER: marine.rutgers.edu/.../2013-09-13-Coupled-physical-b...‎Rutgers University

what can we learn
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 model 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

SUMBER: marine.rutgers.edu/.../2013-09-13-Coupled-physical-b...‎Rutgers University

reality model
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

express in terms of biomass and 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

SUMBER: marine.rutgers.edu/.../2013-09-13-Coupled-physical-b...‎Rutgers University

the steps in constructing a model
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.

SUMBER: marine.rutgers.edu/.../2013-09-13-Coupled-physical-b...‎Rutgers University

state variables and processes
State variables and Processes

“NPZD”: model named for and characterized by its state variables

State variables are concentrations (in a common “currency”) that depend on space and time

Processes link the state variable boxes

SUMBER: marine.rutgers.edu/.../2013-09-13-Coupled-physical-b...‎Rutgers University

processes
Processes
  • Biological:
    • Growth
    • Death
    • Photosynthesis
    • Grazing
    • Bacterial regeneration of nutrients
  • Physical:
    • Mixing
    • Transport (by currents from tides, winds …)
    • Light
    • Air-sea interaction (winds, heat fluxes, precipitation)

SUMBER: marine.rutgers.edu/.../2013-09-13-Coupled-physical-b...‎Rutgers University

state variables and processes1
State variables and Processes

Can use Redfield ratio to give e.g. carbon biomass from nitrogen equivalent

Carbon-chlorophyll ratio

Where is the physics?

SUMBER: marine.rutgers.edu/.../2013-09-13-Coupled-physical-b...‎Rutgers University

examples of conceptual ecosystems that have been modeled
Examples of conceptual ecosystems that have been modeled
  • 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, tides

SUMBER: marine.rutgers.edu/.../2013-09-13-Coupled-physical-b...‎Rutgers University

slide10

particulate silicon

Silicic acid – important limiting nutrient in N. Pacific

gelatinous zooplankton, euphausids, krill

copepods

ciliates

Fig. 1 – Schematic view of the NEMURO lower trophic level ecosystem model. Solid black arrows indicate nitrogen flows and dashed blue arrows indicate silicon. Dotted black arrows represent the exchange or sinking of the materials between the modeled box below the mixed layer depth.

Kishi, M., M. Kashiwai, and others, (2007), NEMURO - a lower trophic level model for the North Pacific marine ecosystem, Ecological Modelling, 202(1-2), 12-25.

slide11

Soetaert K, Middelburg JJ, Herman PMJ, Buis K. 2000. On the coupling of benthic and pelagic bio-geochemical models. Earth-Sci. Rev. 51:173-201

slide12

Phytoplankton concentration absorbs light

Att(x,z) = AttSW + AttChl*Chlorophyll(x,z,t)

Schematic of ROMS “Bio_Fennel”ecosystem model

SUMBER: marine.rutgers.edu/.../2013-09-13-Coupled-physical-b...‎Rutgers University

examples of conceptual ecosystems that have been modeled1
Examples of conceptual ecosystems that have been modeled
  • In simpler models, elements of the state and processes can be combined if time and space scales justify this
    • e.g. bacterial regeneration can be treated as a flux from zooplankton mortality directly to nutrients
  • A very simple model might be just: N – P – Z
    • Nutrients
    • Phytoplankton
    • Zooplankton… all expressed in terms of equivalent nitrogen concentration

SUMBER: marine.rutgers.edu/.../2013-09-13-Coupled-physical-b...‎Rutgers University

slide14

ROMS fennel.h(carbon off, oxygen off, chl not shown)

SUMBER: marine.rutgers.edu/.../2013-09-13-Coupled-physical-b...‎Rutgers University

mathematical formulation
Mathematical formulation
  • Mass conservation
    • Mass M (kilograms) of e.g. carbon or nitrogen in the system
  • Concentration Cn (kg m-3) of state variable n is mass per unit volume V
  • Source for one state variable will be a sink for another

SUMBER: marine.rutgers.edu/.../2013-09-13-Coupled-physical-b...‎Rutgers University

mathematical formulation1
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

SUMBER: marine.rutgers.edu/.../2013-09-13-Coupled-physical-b...‎Rutgers University

some calculus
Some calculus

Slope of a continuous function of x is

SUMBER: marine.rutgers.edu/.../2013-09-13-Coupled-physical-b...‎Rutgers University

SUMBER: marine.rutgers.edu/.../2013-09-13-Coupled-physical-b...‎Rutgers University

slide18
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

SUMBER: marine.rutgers.edu/.../2013-09-13-Coupled-physical-b...‎Rutgers University

average pp saturates at high par
Average PP saturates at high PAR

PPmax

0.5 PPmax

From Heidi’s lectures

SUMBER: marine.rutgers.edu/.../2013-09-13-Coupled-physical-b...‎Rutgers University

SUMBER: marine.rutgers.edu/.../2013-09-13-Coupled-physical-b...‎Rutgers University

state variables nutrient and phytoplankton process photosynthetic production of organic matter
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

SUMBER: marine.rutgers.edu/.../2013-09-13-Coupled-physical-b...‎Rutgers University

slide21
Suppose there are ample nutrients so N is not limiting: then f(N) = 1
  • Growth of P will be exponential

SUMBER: marine.rutgers.edu/.../2013-09-13-Coupled-physical-b...‎Rutgers University

slide22

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

SUMBER: marine.rutgers.edu/.../2013-09-13-Coupled-physical-b...‎Rutgers University

slide23

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

SUMBER: marine.rutgers.edu/.../2013-09-13-Coupled-physical-b...‎Rutgers University

slide24

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.

SUMBER: marine.rutgers.edu/.../2013-09-13-Coupled-physical-b...‎Rutgers University

slide25

SUMBER: marine.rutgers.edu/.../2013-09-13-Coupled-physical-b...‎Rutgers University

slide26

SUMBER: marine.rutgers.edu/.../2013-09-13-Coupled-physical-b...‎Rutgers University

coupling to physical processes
Coupling to physical processes

Advection-diffusion-equation:

physics

turbulent

mixing

Biological dynamics

advection

C is the concentration of any biological state variable

SUMBER: marine.rutgers.edu/.../2013-09-13-Coupled-physical-b...‎Rutgers University

slide28

I0

spring

summer

fall

winter

SUMBER: marine.rutgers.edu/.../2013-09-13-Coupled-physical-b...‎Rutgers University

simple 1 dimensional vertical model of mixed layer and n p ecosystem
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 input files
  • Sharples, J., Investigating theseasonal vertical structure of phytoplankton in shelf seas, Marine Models Online, vol 1, 1999, 3-38.

SUMBER: marine.rutgers.edu/.../2013-09-13-Coupled-physical-b...‎Rutgers University

slide30

I0

spring

summer

fall

winter

bloom

SUMBER: marine.rutgers.edu/.../2013-09-13-Coupled-physical-b...‎Rutgers University

slide31

I0

spring

summer

fall

winter

bloom

secondary bloom

SUMBER: marine.rutgers.edu/.../2013-09-13-Coupled-physical-b...‎Rutgers University

slide32

I0

spring

summer

fall

winter

bloom

secondary bloom

SUMBER: marine.rutgers.edu/.../2013-09-13-Coupled-physical-b...‎Rutgers University