Ariane, a Lagrangian analysis application for ocean circulation models
developed by Bruno Blanke and Nicolas Grima
Eulerian and Lagrangian views
In this second part of the course,
we discuss the Eulerian view of velocities,
which is usually opposed to the Lagrangian view.
To do this,
we'll start by looking at the atmosphere.
This allows an easier presentation
that's also compatible with an oceanic equivalent.
A windsock,
near an airfield or a viaduct,
gives us local information
about the wind direction and speed,
thanks to the direction of the cloth tube and
its angle.
Regular readings of these two parameters,
horizontal direction and horizontal speed,
allow us to locally define a time series
of the horizontal wind speed.
This is an Eulerian knowledge of the wind speed,
and therefore of the atmospheric flow.
It is defined at a precise point and at consecutive times.
The oceanic equivalent would be a current meter,
mounted on a mooring line.
Thanks to a magnetic and mechanical system,
its orientation,
given by a magnetised compass,
and the rotational speed of an internal propeller
allow us to measure the direction
and speed of the horizontal current,
at the precise spot where the current meter is immersed.
The vertical component of the velocity
is much more difficult to measure directly,
as it is several orders of magnitude smaller
than the horizontal velocity component.
But, under specific conditions,
it can also be determined by mechanical propeller systems.
For the ocean, nowadays,
acoustic Doppler current profilers
(ADCP)
are more commonly used to measure current velocities
over a given depth range,
by measuring the Doppler effect
of sound waves backscattered by particles in the water column.
The vision angle of the sound waves is
usually about 30 degrees away from the vertical.
Measuring the Doppler effect
gives access to the horizontal component of the velocity,
that of the particles in suspension,
which is projected on that view.
By using several beams simultaneously,
we can determine unambiguously,
the direction and intensity of the horizontal current.
Careful processing of the transmit and receive pulses
provides information on
the velocity distributed over an entire thickness of water.
In both cases,
by ADCP or by mechanical current meter,
it is the local speed and the local direction of the current
that are estimated at regular intervals,
in order to constitute a time series of the Eulerian speed.
At regular intervals, at the same point.
Now, let us imagine that
we can precisely follow
the successive positions of a dead leaf carried by the wind
or the positions of a small polystyrene ball in water
without buoyancy,
so without any intrinsic speed.
The dead leaf
and the polystyrene ball
are comparable to a point particle.
This particle follows a trajectory,
more or less complex,
depending on the complexity of the velocity field
in which it is immersed.
Between two successive positions,
every second for example,
we can evaluate the straight-line displacement
of the particle.
We therefore have successive velocity vectors,
over the entire length of the trajectory.
These velocity vectors are Lagrangian velocities.
The velocities are now calculated
along a trajectory,
rather than at a fixed point as before.
The Lagrangian velocity at a particular point of a trajectory
obviously corresponds to
the Eulerian velocity
that can be measured at this point at this precise time.
The notion of Lagrangian velocity
refers to the sequence of velocities
that can be constructed along a trajectory in the fluid.
The Eulerian velocity
refers to the measurement of the velocity of this same fluid
at a precise point,
a bit like a speed camera
that flashes different vehicles,
always at the same place.
Many global, basin or regional estimates of ocean velocity
are Eulerian velocities.
They can be velocities calculated by an ocean model.
In this case,
on a given mesh
i.e., on a set of fixed points,
the model calculates at regular intervals
the values of the three components of velocity in the ocean.
The horizontal components of the velocity can also be estimated from observations.
For example, on a large scale,
the horizontal velocity of surface currents is
in the first place,
the sum of a component related to the slope of the sea surface
and a component directly induced by the action of wind on the ocean,
the Ekman circulation.
The geostrophic component,
related to the slope of the sea level,
can be deduced from satellite measurements
of the ocean surface topography.
The wind component
can be deduced from meteorological analyses.
Whether modelled or deduced from topography and wind measurements,
the velocity field here is
an Eulerian velocity field,
available on an a priori regular mesh
and at regular time intervals.
However, we should note the multiplication of Lagrangian measurement instruments.
Scientists can use, for example,
surface buoys,
anchored at a depth of about ten metres,
thanks to a floating anchor.
They can also use Argo-type profilers,
drifting freely at a depth of 1000 metres,
between two vertical profiles made through the water column.
In both cases,
knowing the successive positions of the floats,
communicated to land by regular satellite transmissions
(when the antenna of the float is emerged),
allows the building of atlases of ocean velocities
this time from Lagrangian estimates.
Let us finish with
the notion of spatial and temporal resolution
of velocity fields.
This spatial and temporal resolution
is indeed crucial
for the accuracy of trajectory calculations.
The ocean contains movements at all scales of time and space.
Their representation in a velocity file,
hence by a kind of atlas,
is necessarily truncated.
Ocean movements
of a finer scale than the spatial grid
or of a faster frequency than the chosen sampling period
are not represented in this atlas.
Let us consider for example a sea
subject to significant tidal effects,
such as the semidiurnal tide off the coast of Brittany.
A simple daily velocity field
will not account
for tidal movements
because the trajectories that could be computed from this field
will not reproduce the usual tidal ellipses
made possible by the rotation of velocity vectors
on scales of a few hours.
As noted earlier,
the information about the ocean dynamics
is contained in the velocity fields provided by the user.
The Lagrangian computations
that will be performed based on these velocity fields
will not reconstruct
spatial and temporal scales
missing from the available Eulerian field.
While it is usual to archive the fields of an ocean model
on its native grid,
a temporal subsampling
is unfortunately needed
to limit the size of the output files.