Modelling 1:

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Modelling 1 Basic Introduction.

• What constitutes a model?
• Why do we use models?
• Calibration and validation.
• The basic concept of numerical integration.

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What constitutes a model?
A model is some simplified description of a real
system that can be used to understand or make
predictions/hindcasts of the system
behaviour. Most of the examples we will look at
are numerical models of equations that describe
the system, running on computers. However, real
hydraulic models are often used for particular
applications (e.g. coastal engineering, ship
design). Useful web pages Princeton Ocean Model
Proudman Lab Coastal Ocean Model GOTM 1-D
turbulence model HR-Wallingford (commercial
modelling) ABP Research (commercial modelling)
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Every time you write down and use an equation,
you are using a model.
Newtons 2nd law force mass x
acceleration. Or, acceleration force / mass,
the basis for much of our modelling of the
dynamics of the ocean (i.e. the Equation of
Motion). The model of Fma works well in some
cases, and breaks down when the conditions do not
satisfy the requirements of the model. All models
make assumptions, simplifications, and
compromises. Many models can work well in some
areas, but poorly in others (i.e a model is not
always transferable between different
regions). No model is perfect!
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Why do we use models?

• Investigating processes. Oceanography is an
  observational science - we can rarely manipulate
  the real environment to conduct an experiment.
  But we can manipulate a model environment.
• Making predictions or hindcasts. How will the
  climate change in the future? How did the climate
  behave in the past? How will an approaching storm
  affect the local coast? How will a new
  installation in a port affect the environment?

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Validation and Calibration.
Note that the requirement for an accurate,
well-tested, reliable model is less important for
the process investigation, but critically
important in the case of operational
oceanographic modelling.
Climate modelling at the SOC Proudman Lab
operational shelf model Met Office operational
modelling Tampa Bay modelling and observation
system Proudman Lab coastal observatory
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All models require Calibration and
Validation. Typically - use one dataset to
calibrate the model, and then validate the model
by running it in comparison with another
independent dataset. Operational models require
continuous calibration data assimilation from
observation networks.
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The basic concept of numerical integration.
Horizontal salinity gradient
We will concentrate on the basics of finite
difference modelling. Consider the equation that
describes the advection of a property (e.g.
salinity along an estuarine horizontal salinity
gradient).
Tidal current
Change of salinity through time
i.e. at one position within the estuary you
observe the salinity change through time. The
change in the salinity is caused by the
horizontal tidal current moving water past your
boat, bringing with it higher salinity water from
the sea, or lower salinity water from nearer the
river.
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Remember that the use of the ?s refers to
infinitesimal changes. The basic concept of the
finite difference modelling method is to
approximate these infinitesimal changes as finite
changes, i.e.
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START
So, if you know the horizontal salinity gradient
(from observations), and you can describe the
tidal change of the current speed, you can get
the model to calculate how the salinity changes
through time.
Set initial conditions horizontal gradient,
start time, initial salinity sold
Calculate tidal current speed u(t)
Set sold snew
Calculate the new salinity snewsold?s
Increment t by ?t
u0 tidal current amplitude (m s-1) ? tidal
frequency 2?/(12.42x3600) (s-1)
Save or output data as required
END
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Flood tide salinity increases
Ebb tide salinity decreases
Have a look at the program advect1.m if you are
know any Matlab
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The advection example used a fixed time step to
integrate the advection equation at the surface
of the water column through time. Now we will
consider modelling the whole water column. This
requires us splitting the water column up into a
series of evenly-spaced grid cells. Bed friction
slows down the tidal current, and so near the
seabed the salinity will not vary as much. We can
extend the model to include the effect of a real
vertical velocity profile by including a simple
depth-variability in velocity
Surface
iN
iN-1
hN??z
i3
?z
i2
i1
Seabed
zi depth a grid cell i from surface
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START
Set initial conditions horizontal gradient,
start time, initial salinity profile sold(z)
The new model is very similar to the original
model, except now we need to calculate vertical
profiles of current speed and salinity. This
involves loops in the model between i1 to N for
both current speed and salinity (in the bold
boxes).
Calculate tidal current speed u(z,t)
Set sold(z) snew(z)
Calculate the new salinity profile
snew(z)sold(z)?s(z)
Increment t by ?t
Save or output data as required
END
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1. Ebb tide surface salinity decreases faster
than bottom salinity
2. End of ebb tide maximum salinity
stratification
3. Flood tide stratification decreases
4. End of flood tide mixed profile re-formed
Have a look at the program advect2.m if you are
know any Matlab
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As a final step in complexity, consider the full
advection-diffusion equation
Salinity changes by horizontal advection and
vertical turbulent mixing.
To simplify the problem we assume the vertical
turbulent diffusivity is independent of depth.
The model does this
?s
Advection part
Vertical mixing part
If you are comfortable / keen with Matlab, look
at advect_diffus1,2, 3.m
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Main points to be aware of

• A numerical model calculates a time series by
  incrementing the parameter over a finite time
  step, knowing the forces that influence that
  parameter.
• Vertical profiles are calculated on a model grid
  with a specified grid cell size.
• The model works by changing the infinitesimal
  calculus of the differential equations into
  simple sums over a finite difference.
• As long as you can parameterise a process in
  terms of the forces that influence it, you can
  model it!