116
RAINER BLECK
transitions from surface cooling to warming. Not only do fluctuations in
vertical grid spacing spawn fluctuations in the magnitude of the truncation error in the finite difference equations, but the large ˙
s values in (1)
resulting from rapid coordinate movement are likely to unduly disperse
water mass properties in the vertical. These effects are of particular
concern if the mixed layer depth responds, as it does in nature, to the
24-hr cycle in solar radiation.
If suppression of excessive vertical migration of coordinate surfaces
during the daily or annual heating-cooling cycle is deemed important,
the optimal strategy is to “park” coordinate layers near the surface at
those times (night or winter, respectively) when their target density does
not exist. When surface warming makes the target density reappear, it
will reappear at the sea surface. A coordinate layer lying in waiting near
the surface can reattach itself to the target density with minimal vertical
displacement. Reattachment will also take place sooner if intervening
fixed-depth layers, associated with lighter target densities yet to appear,
are kept as thin as possible.
False numerical dispersion of water mass properties caused by truncation errors in the finite-difference advection operators is a major concern
in ocean modeling. The problem is particularly acute in the z direction
where undulating vertical velocities associated with gravity wave trains
can have a noticeable dispersive effect; in fact, elimination of vertical
dispersion by gravity waves is often mentioned as one of the points in
favor of using a material coordinate system. The dispersive effect of
gravity waves is illustrated in Fig. 1.
While HYCOM’s coordinate surfaces are for the most part material
and thus remain unaffected by the dispersion problem just mentioned,
the enforcement of minimum layer thickness constraints in the upper
part of the ocean does open the door to vertical dispersion. This is of
particular concern if the vertical regridding process displaces coordinate
surfaces over large distances, because a large first term in (1) is likely to
produce an ˙
s term of similar magnitude. Dispersion in this case likely
will be larger than in a fixed-grid model whose ˙
s is given by the righthand side of (1) and thus is bounded by dynamic constraints.
In HYCOM’s original grid generator (see Bleck, 2002), the vertical
“remapping” of prognostic variables following the “regridding” by the
grid generator is formulated as a donor cell process: the amount of a variable X transferred from one grid cell to the next due to interface movement is computed under the assumption that property X is distributed
uniformly within each grid cell. The donor cell scheme is known to be
quite diffusive (as illustrated, for example, in Fig. 1), and upper-ocean
vertical dispersion in HYCOM therefore has been a persistent concern.
RAINER BLECK
transitions from surface cooling to warming. Not only do fluctuations in
vertical grid spacing spawn fluctuations in the magnitude of the truncation error in the finite difference equations, but the large ˙
s values in (1)
resulting from rapid coordinate movement are likely to unduly disperse
water mass properties in the vertical. These effects are of particular
concern if the mixed layer depth responds, as it does in nature, to the
24-hr cycle in solar radiation.
If suppression of excessive vertical migration of coordinate surfaces
during the daily or annual heating-cooling cycle is deemed important,
the optimal strategy is to “park” coordinate layers near the surface at
those times (night or winter, respectively) when their target density does
not exist. When surface warming makes the target density reappear, it
will reappear at the sea surface. A coordinate layer lying in waiting near
the surface can reattach itself to the target density with minimal vertical
displacement. Reattachment will also take place sooner if intervening
fixed-depth layers, associated with lighter target densities yet to appear,
are kept as thin as possible.
False numerical dispersion of water mass properties caused by truncation errors in the finite-difference advection operators is a major concern
in ocean modeling. The problem is particularly acute in the z direction
where undulating vertical velocities associated with gravity wave trains
can have a noticeable dispersive effect; in fact, elimination of vertical
dispersion by gravity waves is often mentioned as one of the points in
favor of using a material coordinate system. The dispersive effect of
gravity waves is illustrated in Fig. 1.
While HYCOM’s coordinate surfaces are for the most part material
and thus remain unaffected by the dispersion problem just mentioned,
the enforcement of minimum layer thickness constraints in the upper
part of the ocean does open the door to vertical dispersion. This is of
particular concern if the vertical regridding process displaces coordinate
surfaces over large distances, because a large first term in (1) is likely to
produce an ˙
s term of similar magnitude. Dispersion in this case likely
will be larger than in a fixed-grid model whose ˙
s is given by the righthand side of (1) and thus is bounded by dynamic constraints.
In HYCOM’s original grid generator (see Bleck, 2002), the vertical
“remapping” of prognostic variables following the “regridding” by the
grid generator is formulated as a donor cell process: the amount of a variable X transferred from one grid cell to the next due to interface movement is computed under the assumption that property X is distributed
uniformly within each grid cell. The donor cell scheme is known to be
quite diffusive (as illustrated, for example, in Fig. 1), and upper-ocean
vertical dispersion in HYCOM therefore has been a persistent concern.