Ariane, a Lagrangian analysis application for ocean circulation models
developed by Bruno Blanke and Nicolas Grima
Graphical representation of transfers
In this last part of the course
still concerning quantitative analysis,
we will deal with the graphical representation of transfers.
We have seen that we are able to estimate
the transfers from section to section,
and to schematise them with arbitrary arrows.
But one of Ariane’s strengths
is to propose Lagrangian stream functions.
Let’s go back to the previous simple illustration.
In its trajectory calculations,
Ariane memorises the passage of each particle
between two contiguous grid cells of the model.
To do this, Ariane performs a local addition
of the weights of the particles
as they pass from grid cell to grid cell.
This sum is calculated algebraically.
The weight is summed as a positive quantity
when the displacement is towards the east or the north.
The weight of a particle is summed as a negative quantity
when the displacement is towards the south or the west.
The addition of the weights is done exactly
on the velocity grid points of the C grid.
For example, particle number 1
first heads south
before turning west.
Therefore, we count -T1
on the V grid points
when it moves south
and -T1 on the U grid point
when it reaches section 2.
Particle number 6 first moves south,
so we subtract T6
on the corresponding V grid points.
Then it loops back to the north.
At some grid points,
it's therefore the quantity +T6 that's added.
We note that particle number 6
has made the same mesh crossing,
first from north to south,
then from south to north,
at the same V grid point.
The two transports, +T6 and -T6,
cancel out locally in the sum performed by Ariane.
Here, on average, particle number 6
has not caused any local transfer of volume
from one grid cell to another.
We therefore define on the velocity grid points of the domain,
a transport field,
which is the signature of the transfer of particles
from section 1 to each of the control sections.
Here, to simplify the calculations,
the values of the weights have all been chosen equal to 1.
This transport field is non-divergent by construction.
Indeed, in each grid cell of the domain,
a particle has left the same signature
at its entrance and at its exit.
Because of the algebraic sum of the transports,
this result remains true
after the summation of the passage of all the particles.
We thus have a 2D non-divergent transport field,
which we can plot as a stream function.
The contours of the Lagrangian stream function
give the integrated view,
in time and here vertically,
of the transfers made from section 1.
The spacing of the contours
shows the local intensity of the connection.
The number of contours that intercept each control section
is a measure of the intensity of each transfer.
The summation of the transports
for Lagrangian stream function
can be done from all particles
originating from a geographical section,
as just done,
or from subsets of particles
selected according to specific criteria,
such as their destination.
In this case,
the Lagrangian stream function can be refined,
and show the geometry of very specific connections.
By construction,
the contours of the stream function cannot intersect,
so one must remember
that it is no longer trajectories that are visualised.
What is represented is an overall movement,
integrated in time and, here, on the vertical.
Here is a classic example of the use of these Lagrangian stream functions.
This study by Sabrina Speich and her collaborators
focuses on the origins of the upper branch
of the Atlantic overturning circulation.
The control sections for the Lagrangian experiment
were 20°N latitude in the Atlantic,
the Drake Passage
through the Antarctic Circumpolar Current
south of South America,
the Indonesian Throughflow north of Australia,
and a section between Australia and Antarctica
to isolate the South Indian Ocean from the South Pacific Ocean.
The initial positions of the particles
were chosen at 20°N,
and the trajectories were integrated backwards in time
until the particles intercepted
any of the 4 control sections.
For each of the destinations outside the North Atlantic,
we have here the representation of the Lagrangian stream function.
The intensity of each connection
is read as the number of contours.
The geometry of the transfers is expressed
by the organisation of the contours of the stream function.
In this study, we distinguish
the cold route that feeds the North Atlantic
from the Pacific via the Drake Passage,
and the warm route
that feeds the North Atlantic from the tropical Pacific
via the Indonesian Throughflow.
A third route could be identified,
for intermediate waters,
from the Pacific Ocean south of Australia.
We must end this course with a new warning
about the time characteristics
of the velocities archived on file,
and used by Ariane in the Lagrangian calculations.
This concerns turbulent mixing,
which causes transformations of water masses
along their journeys.
In our Lagrangian calculations,
only the velocity components
explicitly calculated and resolved by the ocean model are used.
These components may include eddy-induced velocities
if the ocean model has activated such a physical scheme.
But turbulent lateral mixing
and turbulent vertical mixing,
which correspond to motions smaller than the grid cells of the ocean model,
are not considered in the trajectory equation.
In fact, the evolution of temperature
or another tracer along a Lagrangian trajectory,
computed with the full, explicit velocity field of the model,
corresponds to the integrated effect of this turbulent mixing.
In other words, the Lagrangian analysis accounts
for the water mass transformations as parameterised in the model
with its numerical schemes for turbulent mixing.
But this is rigorous under the strict condition
that one integrates the trajectories
with all the scales of motion
that the ocean model has explicitly calculated.
However, the archiving of model velocities
is not done at the model time step scale.
Thus, the ocean variability
between the model time step
and the archival scale
is no longer available for Lagrangian calculations.
This truncation of the time spectrum
of model variability
has consequences in the Lagrangian analysis of tracers,
such as temperature and salinity.
The variations of the tracers
on the computed trajectories
are not exactly the variations
that would have been diagnosed
if the trajectories had been computed
with velocities evolving on the time step scale of the model.
The approach remains relevant
if we can verify that the correlations
of velocity and tracer fluctuations
on filtered time scales
that is to say, non-archived,
are negligible compared with the effect of the true physics,
i.e., the turbulent mixing
that led to the evolution of water masses in the model.
To conclude,
here is an illustration of a water mass conversion diagnostic.
These are results that we published
in our seminal paper on Lagrangian stream functions
in 1999.
In that study, we were interested in
the northward transfer of surface and subsurface water masses
in the tropical Atlantic.
A quantitative Lagrangian approach
allowed us to estimate the intensity of this transfer,
evaluated here at 17.4 Sv
for the numerical simulation we chose to analyse.
The Lagrangian stream function
highlights the geographical nature of this transfer,
with an intense flow along the South American continent,
but also an important recirculation in the equatorial region,
and a progressive opening towards the east
of the surface water circulation.
Under the conditions specified above,
it is possible to analyse in a Lagrangian way
the evolution of water mass properties
between 10°S and 10°N.
The water masses are identified here by their temperature,
from the surface for the warmest of them,
to the intermediate Antarctic waters
whose temperature is less than 7°C,
passing through the waters of the thermocline
and subthermocline.
The comparison of particle temperatures
between their entry into the study area at 10°S
and their exit at 10°N
highlights the dominant effect of the equatorial upwelling,
bringing to the surface the thermocline waters
and, to a lesser extent, slightly deeper waters.
The quantification is robust,
as the model archive used for the Lagrangian calculations
included most of the scales of variability
calculated by the ocean model.
Thus, the changes in properties diagnosed
along the Lagrangian trajectories
can be associated
with the mean effect of turbulent mixing,
and thus to a physical conversion of water masses.
This diagnostic would not have been legitimate
with a too degraded archive
of model velocities and tracers.