Ariane, a Lagrangian analysis application for ocean circulation models
developed by Bruno Blanke and Nicolas Grima
Grids and grid cells
In this part of the course,
we will briefly discuss
the notion of grids and their cells for ocean models,
and how the fluid mechanics equations are discretised.
Most ocean models
solve the so-called primitive equations,
after discretising them on a geographical grid.
As an example,
we will work here with the barotropic model,
which is a simplification of
the primitive equations system.
For the barotropic model,
we have three equations
with three unknowns.
The three unknowns are
the two components U and V
of the horizontal velocity,
and the geopotential PHI.
KSI and K are intermediate variables
that are the relative vorticity of the flow
and its kinetic energy, respectively.
It is usual to designate the grids
on which ocean models are discretised
by the so-called Arakawa classification,
which differentiates
the possible arrangements of the computational variables.
We speak of an A grid,
a B grid,
a C grid, etc.
Arakawa was an American scientist of Japanese descent.
He died in March 2021.
He was a pioneer in atmospheric modelling,
and the grid classification named after him
is but one of the many important results we owe him.
For the A grid, for example
all quantities useful for solving the barotropic model
are chosen at the same location,
at the same grid point.
Some operations, such as the addition
or multiplication of two variables,
are done naturally and simply,
because these variables are available
at the same location on the grid,
and because the result is estimated by default
at the expected location.
Other operations,
however,
require information to be obtained
from further west,
east,
north,
or south.
For example,
a correct finite difference estimate
of the zonal derivative of the meridional velocity
requires at least
one point east
and one point west
of where this derivative is needed.
For this purpose, we can use
a centred finite difference scheme,
using the zonal indices
i+1 and i-1,
for an estimate of the result
at the point with zonal index i.
In this representation,
the horizontal bar above a variable,
with the index following it,
represents the averaging operator
needed to estimate the information
at the correct location,
by a centred scheme (a half-sum).
In another grid,
the B grid
according to the Arakawa classification
the U and V components of the velocity
are not located at the same location as the geopotential.
The method of calculating the different terms of the equations
is therefore different.
We focus on the C grid,
since this is the grid used for
the NEMO and ROMS models.
For this grid,
the three variables U, V and PHI
are located at three different grid points.
The relative vorticity is also expressed on another grid point.
In this particular case,
we see that some calculations are very simple:
there is no need to calculate an average
to estimate the vorticity at the chosen point.
The zonal derivative of the meridional velocity
and the meridional derivative of the zonal velocity,
by simple centred finite differences,
are naturally positioned
on the KSI points.
In the same way,
the gradient term
in the equations for U and V
is deduced by the simple finite difference
of the quantity PHI + K
in the zonal direction or the meridional direction,
with, in each case,
the centred result of the finite difference
exactly on the expected grid point.
The trade-off, of course,
is a more complicated calculation for other terms.
Kinetic energy, for example,
requires that the squares of the two velocity components
be averaged,
in the zonal direction for U,
and in the meridional direction for V.
The detailed study of the numerical properties of the different grids
is a science in itself.
It consists of numerical analysis.
We simply note here
that some general properties have been demonstrated.
The greater the precision of the advection terms,
the better the representation
of the transport terms in the model,
mainly at small scales.
The greater the accuracy of the geostrophic adaptation terms,
the better the representation of the group velocity of gravity waves,
with minimal numerical dispersion for these waves.
The choice of the most relevant grid
for a given study framework
also requires attention
to the numerical stability criteria of the schemes used
for the discretisation of the equations.
Finally,
the accuracy of the calculations
and also their cost
depend on
the ensemble of averaging and half-sum operations
that must be performed to estimate
the additions, multiplications or derivatives
of the different variables involved in the equations to be resolved.
For the rest of the course,
here is what we need to visualise for an elementary grid cell
in a C grid.
An elementary grid cell
is a simple cube of unit 1
for indices i, j and k,
in the 3 directions.
In this cube,
we know the components of the velocity field,
transversally to each of the 6 faces:
2 faces for U,
2 faces for V
and 2 faces for W.
The dimensions of the mesh being known,
it is natural to convert the velocities into transports,
simply by multiplying them individually
by the area of the corresponding faces.
The non-divergence property of the flow,
at the scale of each grid cell of the model,
is then simply expressed by a conservation relation.
Algebraically,
and at any time,
the total transport leaving an elementary grid cell
must be balanced
by an equivalent transport entering the same grid cell.