_Discrete_Coriolis.dox

/*! \page Discrete_Coriolis Discrete Coriolis Term

\section Coriolis Coriolis Term

In general, the discrete equations are written as simple difference equations

based on the Arakawa C-grid as described in section \ref horizontal_grids.

One of the more interesting exceptions is the Coriolis term. It is computed in the

form shown in \eqref{eq:h-horz-momentum,h-equations,momentum}, or:

\f[

    \frac{( f + \zeta )}{h} \, \hat{\mathbf{z}} \times h \, \mathbf{u}

\f]

This term needs to be evaluated at \f$u\f$ points for the \f$v\f$ equation and

vice versa, plus we need to keep the thickness, \f$h\f$, positive definite.

MOM6 contains a number of options for how to compute this term.

\li SADOURNY75_ENERGY Sadourny \cite sadourny1975 figured out how to

conserve energy or enstrophy but not both. This option is energy conserving.

The term in the \f$u\f$ equation becomes:

\f[

    \frac{1}{4 dx} \left( q_{i,j} (vh_{i+1,j} + vh_{i,j}) +

    q_{i,j-1} (vh_{i+1,j-1} + vh_{i,j-1}) \right)

\f]

where \f$q = \frac{f + \zeta}{h}\f$ and \f$h\f$ is an area-weighted

average of the four thicknesses surrounding the \f$q\f$ point, such that

it is guaranteed to be positive definite.

There is a variant on this scheme with the CORIOLIS_EN_DIS option. If true,

two estimates of the thickness fluxes \f$vh\f$ are used to estimate the Coriolis

term, and the one that dissipates energy relative to the other one

is used.

\li SADOURNY75_ENSTRO Also from \cite sadourny1975, this option is enstrophy

conserving.

\f[

    \frac{1}{8 dx} ( q_{i,j} + q_{i,j-1} ) ((vh_{i+1,j} + vh_{i,j}) +

    (vh_{i+1,j-1} + vh_{i,j-1}) )

\f]

\li ARAKAWA_LAMB81 From \cite arakawa1981 is a scheme which is both

energy and enstrophy conserving. Its weaknesses are a large stencil and differing

thickness stencils in the numerator and denominator.

This scheme and several others (with differing values of \f$a,

b, c, d\f$ and \f$ep\f$) are implemented as:

\f{eqnarray}{

    \frac{1}{dx} (a_{i,j} vh_{i+1,j} &+ b_{i,j} vh_{i,j} +

    d_{i,j} vh_{i+1,j-1} + c_{i,j} vh_{i,j-1} \\

    &+ ep_{i,j}*uh_{i-1,j} -

    ep_{i+1,j}*uh_{i+1,j}) \label{eq:Coriolis_abcd}

\f}

with

\f{eqnarray}{

   a_{i,j} &= \frac{1}{24} (2.0*(q_{i+1,j} + q_{i,j-1}) + (q_{i,j} + q_{i+1,j-1})) \\

   b_{i,j} &= \frac{1}{24} ((q_{i,j} + q_{i-1,j-1}) + 2.0*(q_{i-1,j} + q_{i,j-1})) \\

   c_{i,j} &= \frac{1}{24} (2.0*(q_{i,j} + q_{i-1,j-1}) + (q_{i-1,j} + q_{i,j-1})) \\

   d_{i,j} &= \frac{1}{24} ((q_{i+1,j} + q_{i,j-1}) + 2.0*(q_{i,j} + q_{i+1,j-1})) \\

   ep_{i,j} &= \frac{1}{24}((q_{i,j} - q_{i-1,j-1}) + (q_{i-1,j} - q_{i,j-1}))

\f}

\li ARAKAWA_HSU90 From \cite arakawa1990 is a scheme which always conserves

energy and conserves enstrophy in the limit of non-divergent flow. This one

has a larger stencil than Sadourny's energy scheme, but it's much better behaved

in terms of handling vanishing layers than Arakawa and Lamb.

This scheme is implemented with:

\f[

    \frac{1}{dx} (a_{i,j} vh_{i+1,j} + b_{i,j} vh_{i,j} +

    d_{i,j} vh_{i+1,j-1} + c_{i,j} vh_{i,j-1})

\f]

and

\f{eqnarray}{

   a_{i,j} &= \frac{1}{12} (q_{i,j} + (q_{i+1,j} + q_{i,j-1})) \\

   b_{i,j} &= \frac{1}{12} (q_{i,j} + (q_{i-1,j} + q_{i,j-1})) \\

   c_{i,j} &= \frac{1}{12} (q_{i,j} + (q_{i-1,j-1} + q_{i,j-1})) \\

   d_{i,j} &= \frac{1}{12} (q_{i,j} + (q_{i+1,j-1} + q_{i,j-1}))

\f}

\li ARAKAWA_LAMB_BLEND This is a blending of Arakawa and Lamb, Arakawa and Hsu,

and the Sadourny Energy scheme. There are weights CORIOLIS_BLEND_WT_LIN and

CORIOLIS_BLEND_F_EFF_MAX to control this scheme. The equation is the same as for

Arakawa and Lamb \eqref{eq:Coriolis_abcd}, but the values of \f$a, b, c, d\f$ and

\f$ep\f$ differ when the pure Arakawa and Lamb scheme breaks down due to thickness

variations.

\li ROBUST_ENSTRO An enstrophy-conserving scheme which is robust to vanishing

layers.

Some of these options also support the BOUND_CORIOLIS flag. If true,

the Coriolis terms in the \f$u\f$ equation are bounded by the four estimates of

\f$\frac{(f+\zeta)}{h}vh\f$ from the four neighboring \f$v\f$ points, and

similarly in the \f$v\f$ equation.  This option would have no effect

on the SADOURNY75_ENERGY scheme if it were possible to use centered

difference thickness fluxes.

Note, if BOUND_CORIOLIS is on, it will also turn on the

BOUND_CORIOLIS_BIHARM option by default. This option uses a viscosity

that increases with the square of the velocity shears, so that the

resulting viscous drag is of comparable magnitude to the Coriolis

terms when the velocity differences between adjacent grid points is

0.5*BOUND_CORIOLIS_VEL.

\subsection Coriolis_BC Wall boundary conditions

Two sets of boundary conditions have been coded in the

definition of relative vorticity.  These are written as:

NOSLIP defined (in spherical coordinates):

\f{eqnarray}{

  \mbox{relvort} &= dv/dx \mbox{ (east $\&$ west)}, \mbox{ with } v = 0. \\

  \mbox{relvort} &= -\sec(\phi) * d(u \cos(\phi))/dy \mbox{ (north $\&$

  south)}, \mbox{ with } u = 0.

\f}

Free slip (NOSLIP not defined):

\f[

  \mbox{relvort} = 0 \mbox{ (all boundaries)}

\f]

with \f$\phi\f$ defined as latitude.  The free slip boundary

condition is much more natural on a C-grid.

*/