Introduc)ontoOceanNumericalModeling#2-Discre)za)on
regional model SST Global model SSH
GildasCambon,IRD/LOPS,France [email protected]
Oceanmodelingprinciple
2
Ifweknow:• Theoceanstateat)met:u,v,w,T,S,…• Boundarycondi)ons:surface,bo-om,lateralsides
Wecancomputetheoceanstateat7met+dtbyresolvingnumericallytheprimi7veequa7ons:numericalmodeling
Oceanmodelingprinciple
Theoceanisdividedintoboxes:Discre)za)on
Exampleofafinitedifferencegrid
Discre)za)on
4
StructuredgridsThegridcellshavethesamenumberofsides.UnstructuredgridsThedomainis)ledusingmoregeneralgeometricalshapes(triangles,…)piecedtogethertoop)mallyfitdetailsofthegeometry.
ü Goodfor)dalmodeling,engineeringapplica)ons.ü Problems:geostrophicbalanceaccuracy,wavescaVeringbynon-uniformgrids,conserva)onandposi)vityproper)es,…
ROMS
Horizontaldiscre)za)on
5
Linearshallowwaterequa7on:
n Astaggereddifferenceis4)mesmoreaccuratethannon-staggeredandimprovesthedispersionrela)onbecauseofreduceduseofaveragingoperators
Horizontaldiscre)za)on
6
§ Bgridispreferedatcoarseresolu)on,whenCoriolisisimportant:
§ Superiorforpoorlyresolvediner)a-gravitywaves.§ GoodforRossbywaves:colloca)onofvelocitypoints.§ Badforgravitywaves:computa)onalcheckboardmode.
§ Cgridispreferedatfineresolu)on,whenCoriolisislessimportant:
§ Superiorforgravitywaves.§ Goodforwellresolvediner)a-gravitywaves.§ Badforpoorlyresolvedwaves:Rossbywaves(computa)onalcheckboardmode)andiner)a-gravitywavesduetoaveragingtheCoriolisforce.
§ Combina)onscanalsobeused(A+C)
ROMS
Horizontaldiscre)za)on
7
ROMS:ArakawaC-grid
Horizontalcurvilineargrid
8
• Discre)zedincoastline-andterrain-followingcurvilinearcoordinate• ArakawaC-grid
m, n: scale factors relating the differential distances to the physical arc lengths
Ver)caldiscre)za)on
9
Zcoordinate:NEMO
sigma(&stretched)coordinate:ROMS
11
Horizontal curvilinear grid • This is a possible grid: However in practice variations in dx and
dy should be minimized to minimize errors and optimize computation time.
Prefer rotated rectangular grids + use land/sea mask
LandSeaMask12
Land/sea Mask Variables within the masked region are set to zero by multiplying by the mask for either the u, v or rho points :
CBA
D E
G H I J
K L
M N
F
– u points
– v points
– ⇢ points
– points
Figure 3: Masked region within the domain
3.2 Masking of land areas
ROMS has the ability to work with interior land areas, although the computations occur overthe entire model domain. One grid cell is shown in Fig. 1 while several cells are shown in Fig.3, including two land cells. The process of defining which areas are to be masked is external toROMS and is usually accomplished in Matlab; this section describes how the masking a�ects thecomputation of the various terms in the equations of motion.
3.2.1 Velocity
At the end of every time step, the values of many variables within the masked region are set to zeroby multiplying by the mask for either the u, v or ⇢ points. This is appropriate for the v points Eand L in Fig. 3, since the flow in and out of the land should be zero. It is likewise appropriate forthe u point at I, but is not necessarily correct for point G. The only term in the u equation thatrequires the u value at point G is the horizontal viscosity, which has a term of the form @
@⌘⌫@u@⌘ .
Since point G is used in this term by both points A and M, it is not su�cient to replace its valuewith that of the image point for A. Instead, the term @u
@⌘ is computed and the values at points Dand K are replaced with the values appropriate for either free-slip or no-slip boundary conditions.Likewise, the term @
@⇠⌫@v@⇠ in the v equation must be corrected at the mask boundaries.
This is accomplished by having a fourth mask array defined at the points, in which the valuesare set to be no-slip in metrics. For no-slip boundaries, we count on the values inside the land(point G) having been zeroed out. For point D, the image point at G should contain minus thevalue of u at point A. The desired value of @u
@⌘ is therefore 2uA while instead we have simply uA.In order to achieve the correct result, we multiply by a mask which contains the value 2 at pointD. It also contains a 2 at point K so that @u
@⌘ there will acquire the desired value of �2uM. Thecorner point F is set to have a value of 1.
3.2.2 Temperature, salinity and surface elevation
The handling of masks by the temperature, salinity and surface elevation equations is similar tothat in the momentum equations, and is in fact simpler. Values of T , S and ⇣ inside the land
10
Ver)caldiscre)za)onVertical discretization
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Surface mixed layer
Interior
Bottom
Ver)caldiscre)za)onVertical discretization
z-coordinates. In this
coordinate system, the
vertical coordinate is depth,
or "z".
sigma (σ )-coordinates.
In this type of model, the
vertical coordinate
follows the bathymetry.
isopycnal coordinates/
layered models. These
models use the potential
density referenced to a
given pressure as the
vertical coordinate.
Vertical Grid Types
Naval Postgraduate SchoolROMS
Ver)caldiscre)za)onVertical discretization
z-coordinates provide fine
resolution needed to
represent turbulence, but
intersect bathymetry and
may cause unrealistic
vertical velocities.
sigma (σ )-coordinates
are most appropriate for
continental shelf and
coastal regions, but have
difficulty handling sharp
topographic changes
and can give rise to
unrealistic flows.
isopycnal coordinates/
layered models
preserve water mass
characteristics through
centuries of integration,
but have limited
applicability in coastal
regions and surface and
bottom boundary layers.
Vertical Grid Types
Naval Postgraduate School
ROMS
Ver)caldiscre)za)onVertical discretization • The representation of tracer advection and diffusion along inclined density surfaces in the ocean interior is
cumbersome.
• Representation and parameterization of the BBL is unnatural.
• Representation of bottom topography is difficult.
• The representation of advection and diffusion along inclined density surfaces in the ocean interior is even more cumbersome
• Diffculty accurately representing the horizontal pressure gradient
• Representing the effects of a realistic (non-linear) equation of state is cumbersome.
• A coordinate is an inappropriate framework for representing the surface mixed layer or BBL since these boundary layers are mostly unstratified.
z-coordinates provide fine
resolution needed to
represent turbulence, but
intersect bathymetry and
may cause unrealistic
vertical velocities.
sigma (σ )-coordinates
are most appropriate for
continental shelf and
coastal regions, but have
difficulty handling sharp
topographic changes
and can give rise to
unrealistic flows.
isopycnal coordinates/
layered models
preserve water mass
characteristics through
centuries of integration,
but have limited
applicability in coastal
regions and surface and
bottom boundary layers.
Vertical Grid Types
Naval Postgraduate School
ROMS
Ver)caldiscre)za)onVertical grid : σ generalized coordinate • 50 vertical levels
θ=7, b=2, hc=300 m
• 80 vertical levels
θ=6, b=4, hc=300 m