Ariane, a Lagrangian analysis application for ocean circulation models
developed by Bruno Blanke and Nicolas Grima
Quantitative approach
In this seventh part of the course,
we will introduce the ideas behind the quantitative analyses
possible with Ariane.
The strength of a quantitative approach
is the use of thousands
or more often millions of trajectories
to estimate volume transfers.
The objective is to obtain robust statistics
on the pathways of water masses,
on their transformations along their journey,
on the ages, transit times, etc.
The strategy was proposed by Kristofer Döös,
based on our Lagrangian algorithm,
then adapted, formalised and developed by ourselves.
The Eulerian transport of a water mass is expressed
in terms of infinitesimal transport elements
whose sum gives the total transport considered.
In this case, 1 element = 1 particle = 1 initial position.
The initialisation of particles is performed by Ariane
at the scale of the model grid.
We do not work on geographical coordinates
but on the indices I, J and K of the grid,
knowing that the relations of passage
between the model grid and the geographical space are available.
As long as the trajectory scheme preserves the non-divergence of the flow
down to the motion of an individual particle,
the infinitesimal transport assigned to each particle
will be preserved along the motion.
The initial positioning of the particles
in the vein of current that we wish to analyse
in a quantitative way
is the subject of a meticulous approach in Ariane.
Our expertise leads us to group the initial particles
where the transport is the greatest.
Each particle is assigned a fraction of the transport of the grid cell where it is initialised,
with the simple accounting rule
that the total transport,
in a chosen direction and on the initial section considered,
is explained by the sum of the infinitesimal transports
assigned to the particles.
The decomposition of the model grid cells into individual particles
is performed for each time record available for the velocity field.
In addition to being distributed in the two directions covered by the initial section,
the particles are also distributed in time.
This staggering in 3 directions,
2 spatial directions and 1 temporal direction,
is performed automatically by Ariane
satisfying the following relation:
the Eulerian transport Tm of a mesh M of the model
is explained by M particles of the same weight:
Tr=Tm/M³
where M is the smallest integer checking:
Tr ≤ Trmax.
The parameter Trmax is provided by the user.
It is the maximum weight that one chooses to assign to a particle.
For an arbitrary large value of Trmax,
Ariane simply positions one particle per model mesh
at the exact position where the velocity grid point is defined
and in the middle of the period covered by the velocity time sample.
For smaller values of Trmax,
Ariane may have to subdivide each space-time grid cell
into as many sub-cells as necessary,
to ensure that the individual weight given to a particle
does not exceed Trmax.
Then, trajectory calculations can be performed.
The volume transfer from the initial section to another section
will be evaluated by summing the elementary transport associated with the particles,
making the desired connection.
We will illustrate here how these transfers between sections are calculated.
In the last part of the course,
we will introduce the notion of Lagrangian stream function, specific to Ariane.
For convenience,
the following diagrams focus on a purely horizontal flow,
and, moreover, for stationary conditions,
i.e., without temporal evolution.
The integration domain chosen for the trajectories
is included between three control sections,
numbered from 1 to 3,
and a continent,
with a coastline where the transverse velocities are virtually zero.
The initial section is section 1.
The objective is to calculate the volume transfer between section 1
and any of the three sections bordering the domain.
To do this, we use the automatic initialisation of the particles by Ariane on section 1.
As mentioned before,
each particle is assigned an infinitesimal transport.
The sum of the transports of all the particles
gives the total flow entering the domain through section 1,
with a southward motion.
Ariane is able to calculate the trajectories of all the particles,
here they number 6.
Some particles, number 1 and number 3, escape through section 2.
Others escape by section 3: these are the particle numbers 2, 4 and 5.
And one particle, number 6, leaves the study area through the initial section.
The volume transfer from section 1 to each of the other sections
is simply the sum of the transports
carried by the particles that make the connection.
From section 1 to section 3,
the transfer is T2 + T4 + T5.
The first study we published
with this quantitative estimate of volume transfers
concerned the structuring of the equatorial undercurrent in the Pacific,
here represented from observations
in a north-south cross section
and as a function of depth.
At different longitudes,
150°E, the date line, 150°W, 120°W and 90°W,
particles are distributed in latitudes and depth,
over the fraction of the meridional cross section that includes the equatorial undercurrent
and where the velocity is directed eastward.
For each 30° longitude domain we are able to evaluate,
with the previous method,
the volume transfers between the different sections that define the domain.
We thus have a step-by-step estimate of the eastward transfer
of water from the equatorial undercurrent along the equator,
but also towards latitudes further north or south,
or even towards the base of the study domain set here at 500 m depth.
The estimate can be made in direct mode,
with trajectory calculations on the standard model calendar
from an initial section to different control sections.
But it can also be done backwards, starting from a final section
and integrating backwards in time the trajectories to deduce their origins.
It can be verified that the two estimates of the same transfer,
one direct and the other backwards, converge to an equivalent value,
if a sufficient number of particles are used.
This verification provides evidence of the robustness of the transfer estimates.
For this study,
it was shown that the equatorial undercurrent transport
decreases from west to east.
This is simple Eulerian information,
directly available in the model output.
The quantitative Lagrangian approach
allows us to estimate a direct connection of 19 Sv
from 150°E to 120°W,
but also to estimate the volume exchanges
between the undercurrent and the extra-equatorial latitudes:
a loss of transport by Ekman divergence
partially compensated
by an increase of transport by geostrophic convergence.