Title: Simple coupled physicalbiogeochemical models of marine ecosystems
1Simple coupled physical-biogeochemical models of
marine ecosystems
- Formulating quantitative mathematical models of
conceptual ecosystems
2Why 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 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 evolution over time
of the ecosystem state
3What 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 - The uptake of nutrients during the bloom and loss
of particulates below the euphotic zone gives us
quantitative information on the corresponding net
primary production and carbon export quantities
we cannot easily observe directly
4Reality 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 CN ratio
- Limit state variables (one or two limiting
nutrients) - Discrete spatial points and time intervals
- Average behavior based on ad hoc assumptions
- Must parameterize unknowns
5The 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.
6State variables and Processes
- NPZD model characterized by its state
variables - State variables are concentrations (in a common
currency) that depend on space and time
7Processes
- Biology
- Growth
- Death
- Photosynthesis
- Grazing
- Bacterial regeneration of nutrients
- Physics
- Mixing
- Transport
- Light
- Air-sea interaction (winds, heat fluxes,
precipitation)
8State variables and Processes
- Can use Redfield ratio to give e.g. carbon
biomass from nitrogen equivalent - Carbon-chlorophyll ratio
- Where is the physics?
9Examples 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 clases of zooplankton
- Detritus
- Predation (predators and their behavior)
- Pigments and bio-optical properties
- Photo-adaptation, self-shading
- 3 spatial dimensions in the physical environment,
diurnal cycle of atmospheric forcing, tides
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13Examples 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
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15Mathematical formulation
- Mass conservation
- Mass M (kilograms) of e.g. carbon or nitrogen in
the system - Concentration Cn (kilograms m-3) of state
variable n (mass per unit volume V)
16Mathematical 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
17Some calculus
Slope of a continuous function of x is
Baron Gottfried Wilhelm von Leibniz 1646-1716
18Example f distance x time df/dx speed
Which comes from
19State variables Nutrient and PhytoplanktonProce
ss Photosynthetic production of organic matter
Large N Small N
Michaelis and Menten (1913)
µmax is maximum growth ratekn is
half-saturation concentration f(kn)0.5
20Representative results from 32Si kinetic
experiments measuring the rate of Si uptake as a
function of the silicic acid concentration
(ambientadded). Four of the 26
multi-concentration experiments are shown,
representing the main kinetic responses observed
in this study (Southern Ocean). Nelson et al.
2001 Deep-Sea Research Volume 48, Issues 19-20 ,
2001, Pages 3973-3995
21Uptake expressions
22State variables Nutrient and PhytoplanktonProce
ss Photosynthetic production of organic matter
The nitrogen consumed by the phytoplankton must
be lost from the Nutrients state variable
23- Suppose there are ample nutrients so N is not
limiting then f(N) 1 - Growth of P will be exponential
24- 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
25- 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
26Can 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.
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28There are many possible parameterizations for
processes e.g. Zooplankton grazing
Zooplankton grazing rates might be parameterized
as proportional to Z i.e. g constant or if
P is small the grazing rate might be less because
the Z have to find them or catch them first
Ivlev (1945) function Grazing parameter Iv
29Light
Irradiance I Initial slope of the P-I curve a
30Coupling to physical processes
- Advection-diffusion-equation
turbulent mixing
Biological dynamics
advection
C is the concentration of any biological state
variable
31I0
spring
summer
fall
winter
32Simple 1-dimensional vertical model of mixed
layer and N-P-Z type 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.
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35I0
spring
summer
fall
winter
bloom
36I0
spring
summer
fall
winter
bloom
secondary bloom
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