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. 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?

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 Validation and Calibration.

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. 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). 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. Horizontal salinity gradient Tidal current Change of salinity through time

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.

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 0 = tidal current amplitude (m s -1 )  = tidal frequency = 2  /(12.42x3600) (s -1 ) Set initial conditions [horizontal gradient, start time, initial salinity s old ] Calculate tidal current speed u(t) Calculate the new salinity s new =s old +  s Save or output data as required END START Increment t by  t Set s old = s new

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. 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: i=1 i=2 i=3 i=N-1 i=N Seabed Surface zz h=N  z z i = depth a grid cell i from surface

Set initial conditions [horizontal gradient, start time, initial salinity profile s old (z)] Calculate tidal current speed u(z,t) Calculate the new salinity profile s new (z)=s old (z)+  s(z) Save or output data as required END START Increment t by  t Set s old (z) = s new (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 i=1 to N for both current speed and salinity (in the bold boxes).

Have a look at the program advect2.m if you are know any Matlab 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

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. Advection part Vertical mixing part The model does this: ss 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!