Ariane, a Lagrangian analysis application for ocean circulation models
developed by Bruno Blanke and Nicolas Grima
Equations and history of ocean modelling
This third part of the course
will briefly discuss the equations
and above all, the main historical stages
of numerical ocean modelling.
In order to introduce the trajectory calculations performed by Ariane,
it is necessary to look back at
the equations describing ocean circulation,
and how a model is able to solve them numerically.
The classical system of equations,
used by ocean general circulation models,
is the so-called primitive equations system.
These are the Navier-Stokes equations
adapted to the specificities of the ocean.
Thus,
it is assumed that the horizontal scale of motion
and the depth of the ocean
are small compared with the Earth's radius.
The hydrostatic hypothesis is made
with a simple integral relation
linking the pressure to the density of the ocean.
The hypothesis of an incompressible flow is made,
which results in a non-divergence relation
for the 3D velocity field.
We also use
the Boussinesq approximation,
where the density
is chosen equal to a constant rau0,
except in the calculation of the pressure.
Finally,
we assume the existence
of an equation of state
linking density to
temperature, salinity and pressure.
Here is a short history
of ocean modelling,
and, first, that of the atmospheric modelling,
which preceded it.
In the 19th century,
the equations of meteorology
(hydro-thermodynamics)
were known,
and scientists tried to solve them.
In 1904,
the Norwegian physicist Vilhelm Bjerknes
formalised these efforts
and concluded that it was possible
to effectively solve these equations,
provided the following two conditions were met:
- sufficiently precise knowledge of the state of the atmosphere at a given initial time;
- sufficient knowledge of the laws governing the transition from one state of the atmosphere to another.
Thus,
starting from a perfectly defined state
(deduced from observations),
he associated the forecast of meteorological weather
to the resolution of a system of partial differential equations.
Bjerknes proposed a system of seven equations
whose solution leads,
in principle,
to the prediction of large-scale atmospheric movements.
In 1910,
an English scientist
named Lewis Fry Richardson
presented a 50-page manuscript
in which he discussed
the iterative solution
of reference differential equations,
such as the Laplace equation
or the biharmonic equation.
He clearly distinguished between
solving elliptic problems
and solving hyperbolic problems
and treated boundary conditions in a rigorous manner.
In 1922,
Richardson used finite difference numerical methods
in an attempt at numerical prediction.
This was the first numerical weather forecasting system...
The techniques used
(space meshing, finite differences)
are the same as those that would be adopted
for the first successful atmospheric models.
His method was based
on a simplified version of the Bjerknes equations,
in order to reduce
the volume of calculations to
an acceptable level for 'manual' resolution,
in the manner of a 'forecasting factory',
with a one-man band, organising and supervising all the calculations.
Nevertheless, the work appeared so colossal
that Richardson himself
did not think of associating it
with a real operational numerical forecasting experiment.
The first efforts to solve such a problem
(for a six-hour forecast)
took six weeks
and were a failure.
The reasons for this failure
were only understood later,
and some of them are related
to the stability of the numerical schemes used.
Indeed, in 1928,
Richard Courant,
Kurt Otto Friedrichs
and Hans Lewy
showed that for an integration
performed in finite differences,
the spatial and temporal increments
must verify an appropriate stability criterion,
the CFL criterion.
In the late 1930s and early 1940s,
Carl-Gustaf Rossby, a Swedish meteorologist,
and his colleagues at MIT
successfully produced a system of simplified equations,
that were also robust and rigorous,
to describe the large-scale motions of the atmosphere.
This system is based on the so-called vorticity equation,
and, in particular, allows us
to calculate the evolution of planetary waves.
In 1945,
the first numerical computers appeared,
including the ENIAC
Electronic Numerical Integrator and Computer
in an American military base in Philadelphia,
the Aberdeen Proving Ground.
In 1950,
on a computer linked to the Princeton Project,
a phenomenal machine for the time,
Jule Gregory Charney,
Ragnar Fjørtoft
and John von Neumann
made the first numerical weather forecast
on a computer,
using the vorticity equation for the 500 mb level.
This was a 24-hour forecast,
made at the time,
in computing time,
over 24 hours.
The funding came from the Weather Bureau,
the Navy,
and the US Air Force.
This first forecast covered North America
using a horizontal grid
of 270 points,
spaced about 700 km apart,
and the model's time step was 3 hours.
Although the results of this forecast
were not perfect,
they helped to justify
further efforts in the same direction.
The first operational forecasts
were made from 1954,
in a joint unit,
the Joint Numerical Weather Prediction Unit
of the Weather Bureau,
the Air Weather Service,
the US Air Force
and the US Navy,
using a barotropic atmosphere model.
During the 1955-1965 period,
the weather forecasting centres
used the same numerical methods,
but with increasingly sophisticated atmospheric models:
more vertical levels,
use of primitive equations,
more realistic physics.
It was at this time
that many laboratories dedicated to atmospheric research,
and later to climate research, were created:
the General Circulation Research Section,
GCRS,
Maryland,
which would become in 1968 the GFDL
at Princeton University,
with Joseph Smagorinsky and Syukuro Manabe, among others;
the Department of Meteorology at UCLA,
with Akio Arakawa and Yale Mintz;
the Lawrence Livermore Laboratory;
and the National Center for Atmospheric Research.
Ten years later,
the atmospheric modelling groups developed,
with innovative modelling techniques
and new research topics.
The 1965-75 decade saw,
in particular, the appearance
of the first coupled ocean-atmosphere models,
initially with a very simple component for the ocean,
e.g., a simple mixed layer.
The first publications implementing
a coupling with an ocean general circulation model
date from 1969,
for an idealised ocean geometry,
and 1975,
for a realistic configuration.
And since then?
The numerical and physical developments
are still going on,
and always take advantage
of the growing potential of computing centres,
such as the Jean Zay supercomputer in France,
on which many atmosphere and climate models are integrated.
Very fine scale regional models,
non-hydrostatic models,
data assimilation techniques,
ensemble forecasts, etc.
have now spread among most laboratories
and meteorological centres
involved in atmospheric studies.
The quality of weather forecasts
benefits from all these improvements.
The numerical ocean
has been somewhat left out.
However, let’s underline the importance of the oceans,
for life and for mankind.
The oceans cover 70% of the surface of the globe,
and contain 97% of the water on Earth.
They are an essential component
of our life and our environment.
Through complex exchanges with the atmosphere,
and their capacity to store enormous quantities of energy,
they play a major role in the climate.
Let’s remember that the top two and a half metres of the ocean
contain as much heat
as the entire air column,
nearly 40 km,
above them.
Nevertheless,
the study of ocean dynamics is a rather young science.
Numerical developments
to understand ocean dynamics
have been rather late.
This delay
has two main origins:
On the one hand, the ocean
has not been studied for as long as the atmosphere
for obvious reasons of mankind's interest in its immediate environment,
the air.
Thus,
the observational bases necessary
for a better validation of the modelled processes
are still insufficient,
due to the complexity of ocean observation.
On the other hand, the ocean
has its own characteristics,
which tend to make it more complicated
to represent numerically than the atmosphere:
complex lateral boundaries,
smaller spatial and temporal scales for motions
due to the 1000 ratio
between the density of water and air,
heterogeneity of forcings...
The detailed structure of currents,
movements of water masses,
and the distribution of physical
and chemical properties in the ocean
are far from being fully understood.
The physical processes
that create currents and determine oceanic properties
are indeed numerous and complex.
They are mainly due to
the action of the wind on the sea surface,
and to the circulation induced
by the heterogeneous spatial distribution
of temperature and salinity,
the thermohaline circulation.
Thus, there has been a kind of transfer of skills
between the study of the atmosphere
and the study of the ocean.
The understanding and numerical simulation of the ocean
have, of course, common points with meteorology
and weather forecasting.
This is why many techniques developed for meteorology
have been progressively applied to ocean problems.
In particular,
for the last 50 years,
increasingly powerful computers
have been used
to solve the Navier-Stokes equations
applied to the ocean.
Here are two important dates to remember:
1960:
the first ocean circulation model was created
by Kirk Bryan and Michael Cox, at Princeton.
1990:
coupled ocean-atmosphere (and sometimes ice) models
were developed
in most ocean research institutes.
And finally,
here is a list of some numerical ocean models,
explicitly listed
in a recent review article on the challenges and prospects
of ocean circulation models.
We can say a few words about certain models.
MOM,
for Modular Ocean Model:
this is the GFDL model,
the first numerical ocean model.
POP,
for Princeton Ocean Model:
a model with sigma coordinates
(not bathymetric),
and with a free surface.
HYCOM,
for Hybrid Coordinate Model:
an immediate descendant of MICOM
(Miami Isopycnic Coordinate Ocean Model),
written in isopycnal coordinates.
ROMS,
for Regional Ocean Model:
a distant descendant of SPEM
(Semi-spectral Primitive Equation Model),
which included spectral coordinates on the vertical.
NEMO/OPA,
for Nucleus for European Modelling of the Ocean / Océan Parallélisé:
the general circulation model
initially developed in France,
and now on a European scale.
CROCO,
for Coastal and Regional Coastal Community model:
a model that benefits
from the convergence of HYCOM, ROMS [and MARS3D].