_Advection.dox

/*! \page Tracer_Advection Tracer Advection

MOM6 implements a generalised tracer advection scheme, which is a

combination of the modified flux advection scheme \cite easter1993 with

reconstructed tracer distributions. The tracer distributions may be

piecewise linear (PLM) or piecewise parabolic (PPM), which may itself

use either the \cite colella1984 (CW84) or \cite huynh1997 (H3) reconstruction.

\section Flux_advection Flux advection

The modified flux advection scheme preserves the tracer mixing ratio in

a cell across directional splitting by accounting for changes in mass

changes. Fluxes are applied to alternate directions in turn, restricting

the applied flux so as not to evacuate all mass out of a cell. Because

of this, we need to know the stencil used during the calculation of the

reconstruction. Every iteration of the splitting algorithm, cells at the

edge of a processor's data domain are invalidated. When this invalidation

region extends below the halo, a group pass is required to refresh the

halo. A larger stencil (such as for the CW84 reconstruction) therefore

introduces more frequent updates, and may impact performance.

\section Tracer_reconstruction Tracer reconstruction

While MOM6 only carries the mean tracer concentration in a cell,

a higher order reconstruction is computed for the purpose of

advection. Reconstructions are also modified to ensure that monotonicity

is preserved (i.e. spurious minima or maxima cannot be introduced).

The piecewise linear (PLM) reconstruction uses the monotonic modified

van Leer scheme \cite lin1994. One might think to use the average

of the one-sided differences of mean tracer concentration within a cell

to calculate the slope of the linear reconstruction, however this method

guarantees neither monotonicity, nor positive definiteness. Instead, the

method is locally limited to the minimum of this average slope and each

of the one-sided slopes, i.e. \f[\Delta \Phi_i = \min\left\{\left|[\Delta

\Phi_i]_\text{avg}\right|, 2\left(\Phi_i - \Phi_i^\text{min}\right),

2\left(\Phi_i^\text{max} - \Phi_i\right)\right\}\f] (where

\f$\Phi_i^\text{min}\f$ is the minimum in the 3-point stencil).

In a PPM scheme (\ref PPM), for a cell with mean tracer concentration \f$\Phi_i\f$,

the values at the left and right interfaces, \f$\Phi_{L,i}\f$

and \f$\Phi_{R,i}\f$ must be estimated. First, an interpolation is

used to calculate \f$\Phi_{i-1/2}\f$ and \f$\Phi_{i+1/2}\f$. These

values are then modified to preserve monotonicity in each cell, which

introduces discontinuities between cell edges (e.g. \f$\Phi_{R,i}\f$

and \f$\Phi_{L,i+1}\f$).

The reconstruction \f$\Phi_i(\xi)\f$ then satisfies three properties:

- total amount of tracer is conserved, \f$\int_{\xi_{i-1/2}}^{\xi_{i+1/2}} \Phi_i(\xi') \,\mathrm d\xi' = \Phi_i\f$

- left interface value matches, \f$\Phi(\xi_{i-1/2}) = \Phi_{L,i}\f$

- right interface value matches, \f$\Phi(\xi_{i+1/2}) = \Phi_{R,i}\f$

There are two methods of reconstruction for a piecewise parabolic

(PPM) profile. They differ in the estimate of interface values

\f$\Phi_{i+1/2}\f$ prior to monotonicity limiting. The CW84

scheme makes use of the limited slope \f$\Delta\Phi_i\f$

from PLM, above. This has the effect of requiring a larger stencil

for each reconstruction. On the other hand, the H3 scheme

reduces the requirement of this stencil, by only examining the tracer

concentrations in adjacent cells, at the same time reducing order of

accuracy of the reconstruction.

*/