Introduc)ontoOceanNumericalModeling#2-Discre)za)on

regional model SST Global model SSH

GildasCambon,IRD/LOPS,France gildas.cambon@ird.fr

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

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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