Chapter 15 - Numerical Models
Many of the models we have described so far have output, such as current velocity
or surface topography, constrained by oceanic observations of the variables
they calculate. Such models are called assimilation
models. In this section,
we will consider how data can be assimilated into numerical models.
Let's begin with a primitive-equation, eddy-admitting
numerical model to calculate the position of the Gulf
Stream. Let's assume that the model is driven with real-time surface winds
from the ECMWF weather model. Using the model, we can calculate the position
of the current and also the sea-surface topography associated with the current.
We find that the position of the Gulf Stream wiggles offshore of Cape Hatteras
due to instabilities, and the position calculated by the model is just one
of many possible positions for the same wind forcing. Which position is correct,
that is, what is the position of the current today? We know, from satellite
altimetry, the position of the current at a few points a few days ago. Can
we use this information to calculate the current's position today? How do we
assimilate this information into the model?
Many different approaches are being explored (Malanotte-Rizzoli, 1996). Roger
Daley (1991) gives a complete description of how data are used with atmospheric
models. Andrew Bennet (1992) and Carl Wunsch (1996) describe oceanic applications.
Assimilation of data into models is not easy.
- Data assimilation is an inverse problem:
A finite number of observations are used to estimate a continuous field -
a function, which has an infinite number of points. The calculated fields,
the solution to the inverse problem, are completely under-determined. There
are many fields that fit the observations and the model precisely, and the
solutions are not unique. In our example, the position of the Gulf Stream
is a function. We may not need an infinite number of values to specify the
position of the stream if we assume the position is somewhat smooth in space.
But we certainly need hundreds of values along the stream's axis. Yet, we
have only a few satellite points to constrain the position of the Stream.
To learn more about inverse problems and their solution, read Parker (1994)
who gives a very good introduction based on geophysical examples.
- Ocean dynamics are non-linear, while most methods for calculating solutions
to inverse problems depend on linear approximations. For example the position
of the Gulf Stream is a very nonlinear function of the forcing by wind and
heat fluxes over the North Atlantic.
- Both the model and the data are incomplete and both have errors. For example,
we have altimeter measurements only along the tracks shown in Figure
and the measurements have errors of ± 4cm.
- Most data available for assimilation into data comes from the surface,
such as AVHRR and altimeter data. Surface data obviously constrain the surface
geostrophic velocity, and surface velocity is related to deeper velocities.
The trick is to couple the surface observations to deeper currents.
While various techniques are used to constrain numerical models in oceanography,
perhaps the most practical are techniques borrowed from meteorology.
Most major ocean currents have dynamics which are
significantly nonlinear. This precludes the ready development of inverse
methods. .. Accordingly, most attempts to combine ocean models and measurements
have followed the practice in operational meteorology: measurements are
used to prepare initial conditions for the model, which is then integrated
forward in time until further measurements are available. The model is
thereupon reinitialized. Such a strategy may be described as sequential.
From Bennet (1992).
Let's see how Professor Allan Robinson and colleagues at Harvard University
used sequential estimation techniques to forecast the position of the Gulf
Harvard Open-Ocean Model
This is an eddy-admitting,
quasi-geostropic model of the Gulf Stream east of Cape Hatteras (Robinson et
al. 1989). It has six levels in the vertical, 15 km resolution, and
one-hour time steps. It uses a simple filter to smooth high-frequency variability
and to damp grid-scale variability.
By quasi-geostrophic we mean that the flow
field is close to geostrophic balance. The equations of motion include the
acceleration terms D /Dt ,
where D /Dt is
the substantial derivative and t is time. The
flow can be stratified, but there is no change in density due to heat fluxes
or vertical mixing. Thus the quasi-geostrophic equations are simpler than the
primitive equations, and they can be integrated much faster. Cushman-Roisin
(1994: 204) gives a good description of the development of quasi-geostrophic
equations of motion.
The model reproduces the important features of the Gulf Stream and it's extension,
including meanders, cold-and warm-core rings, the interaction of rings with
the stream, and baroclinic instability. Because the model was designed to forecast
the dynamics of the Gulf Stream, it must be constrained by oceanic measurements:
- Data provide the initial conditions for the model. Satellite measurements
of sea-surface temperature from the AVHRR and topography from an altimeter
are used to determine the location of features in the region. Expendable
bathythermograph, AXBT measurements of subsurface temperature, and historical
measurements of internal density are also used. The features are represented
by simple analytic functions in the model.
- The data are introduced into the
numerical model, which interpolates and smoothes the data to produce the
best estimate of the initial fields of density and velocity. The resulting
fields are called an analysis.
- The model is
integrated forward for one week, when new data are available, to produce
- Finally, the new data are introduced into the model as in the
first step above, and the processes is repeated.
The model has been used for making successful, one-week forecasts of the Gulf
Stream and region (Figure 15.4).
|| Figure 15.4 Output from the Harvard Open-Ocean
Model: A the initial state of the model,
the analysis, and B Data used to produce
the analysis for 2 March 1988. C The forecast
for 9 March 1988. D The analysis for 9 March.
Although the Gulf Stream changed substantially in one week, the model forecasts
the changes well. From Robinson et al.
Much more advanced models with much higher resolution are now being used to make
global forecasts of ocean currents up to one month in advance in support of the
Global Ocean Data Assimilation Experiment GODAE that started in 2003. The goal
of GODAE is produce routine oceanic forecasts similar to todays weather forecasts.
Navy Layered Ocean Model
An example of a GODAE model is the global US
Navy Layered Ocean Model. It is
a primitive equation model with 1/32° resolution in the horizontal and
seven layers in the vertical. It assimilates altimeter data from Jason, Geosat
Follow-on GFO, and ERS-2 satellites and sea-surface temperature from AVHRR
on NOAA satellites. The model is forced with winds and heat fluxes for up to
five days in the future using output from the Navy Operational Global Atmospheric
Prediction System. Beyond five days, seasonal mean winds and fluxes are used.
The model is run daily and produces forecasts for up to one month in the future.
The model has useful skill out to about 20 days.
Ocean surface currents calculated by the US Navy Layered Ocean Model for the
Gulf Stream region on 6 November 2007.
A group of French laboratories and agencies operates a similar operational
forecasting system, Mercator, based on assimilation of altimeter measurements
of sea-surface height, satellite measurements of sea-surface temperature, and
internal density fields in the ocean, and currents at 1000 m from thousands
of Argo floats. Their model has 1/15° resolution in the Atlantic and 2° globally.