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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.
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)
Every time you write down and use an equation, you are using a model.
F  ma
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 F=ma 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!
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?
Validation and
Calibration.
Note that the requirement
for an accurate, welltested, 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
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.
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).
s
s
 u( t )
t
x
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.
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.
s
s
 u( t )
t
x
snew  sold
s
 u( t new )
t new  told
x
snew  sold
s 

 t    u t new  
x 

s
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.
u( t )  u0 sin t 
START
Set initial conditions
[horizontal gradient,
start time, initial
salinity sold]
Calculate tidal
current speed u(t)
Set sold = snew
Calculate the new
salinity snew=sold+s
u0 = tidal current amplitude (m s-1)
 = tidal frequency
= 2/(12.42x3600) (s-1)
Save or output data
as required
END
Increment t by
t
Flood tide :
salinity
increases
Ebb tide :
salinity
decreases
Have a look at the program advect1.m if you are know any Matlab
The advection example used a fixed
time step to integrate the advection
equation at the surface of the water
column through time.
Surface
i=N
i=N-1
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:

u( )  usurface  1.15  0.425 2
zi

h
h=Nz
z
i=3
i=2

zi = depth a grid cell i from surface
i=1
Seabed
START
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 i=1 to N for both current
speed and salinity (in the bold
boxes).
Set initial conditions
[horizontal gradient,
start time, initial
salinity profile sold(z)]
Calculate tidal
current speed u(z,t)
Set sold(z) =
snew(z)
Calculate the new
salinity profile
snew(z)=sold(z)+s(z)
Save or output data
as required
END
Increment t by
t
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 reformed
Have a look at the program advect2.m if you are know any Matlab
As a final step in complexity, consider the full advection-diffusion equation:
s
s   s 
 u   K z 
t
x z  z 
  2s 
s
s
 u
 K z  2 
t
x
 z 
The model does this:
Salinity changes by horizontal advection
and vertical turbulent mixing.
To simplify the problem we assume the
vertical turbulent diffusivity is independent
of depth.
s

 si 1  si   si  si 1   
s( z,t )  t    u( z,t )  K z 

2
z



Advection
part
Vertical mixing part
If you are comfortable / keen with Matlab, look at advect_diffus1,2, & 3.m
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!