Download USERS GUIDE for a modesplit σ-coordinate numerical ocean model

2 THE σ-COORDINATE MODEL
2
Euler forward method is used as a predictor. The motivation for introducing more
implicitness into the model is to avoid noise and growth of numerical errors and
thereby allow simulations with less viscosity and more physical free waves. The
cost is more computer time. If Θ is larger than 0 in both the 2D steps and the
3D steps, the computing time will almost be doubled since the method is based on
computing first a predictor and then a corrector. In version 3.0 there is also an
option for using the 4th order advection scheme presented in [36]. This method
is more accurate than the default TVD-scheme with a superbee limiter. However,
over- and under-shooting near fronts may occur. The method is not monotonic.
In version 4.0, the time stepping of the 3-D steps is changed. The implicit Θmethod is replaced by a predictor-corrector method with the leapfrog method as the
predictor and the Adams-Moulton 2-level method as the corrector. This method is
used in ROMS [37, 36] and it is also analyzed in [22]. The method has a stability
region that includes parts of the imaginary axis and the right half complex plane,
and it is third order accurate. It requires the storage of some variables at three time
levels, but since it is more accurate and stable than previous methods the 3D time
step may be substantially increased in many applications. When the 3D time steps
are increased, the number of 2D time steps per 3D step may be increased to keep
the 2D Courant number at an acceptable level. Also in version 4.0, it is introduced
as an option to use the fourth order method described in [27] for estimation of
internal pressure gradients. In some test cases, the internal pressure gradient errors
are substantially reduced when using higher order methods. In version 4.1, errors
in the weights in the time stepping were corrected.
2
2.1
The σ-coordinate model
The Basic Variables and Equations
The symbols used in the description of the model are given in Appendix A. The
model assumes that the weight of the fluid identically balances the pressure (hydrostatic assumption), and that density differences are neglected unless the differences
are multiplied by gravity (Boussinesq approximation). The following equations are
used to describe the variables as functions of the Cartesian coordinates x, y, z.
The continuity equation is
~ + ∂W = 0,
∇·U
∂z
and the Reynolds momentum equations are
∂U
~ · ∇U + W ∂U − f V = − 1 ∂P + ∂ (KM ∂U ) + Fx ,
+U
∂t
∂z
ρ0 ∂x
∂z
∂z
∂V
~ · ∇V + W ∂V + f U = − 1 ∂P + ∂ (KM ∂V ) + Fy ,
+U
∂t
∂z
ρ0 ∂y
∂z
∂z
(1)
(2)
(3)