patrick marchesiello ird 20051 regional coastal ocean modeling patrick marchesiello brest, 2005

Patrick Marchesiello IRD 2005 1

Regional Coastal Ocean Modeling

Patrick Marchesiello

Brest, 2005


The Coastal Ocean: A Challenging Environment Geometrical constraints: irregular coastlines and highly

variable bathymetry Forcing is internal (intrinsic), lateral and superficial: tides,

winds, buoyancy Broad range of space/time scales of coastal structures

and dynamics: fronts, intense currents, coastal trapped waves, (sub)mesoscale variability, turbulent mixing in surface and bottom boundary layers

Heterogeneity of regional and local characteristics: eastern/western boundary systems; regions can be dominated by tides, opened/closed to deep ocean

Complexe Physical-biogeochemical interactions


Numerical Modeling

Require highly optimized models of significant dynamical complexity

In the past: simplified models due to limited computer resources

In recent years: based on fully nonlinear stratified Primitive Equations


Coastal Model Inventory

POM ROMS MARS3D SYMPHONIE GHERM HAMSOM QUODDY MOG3D SEOM

Finite-Difference Models

Finite-Elements Models


Hydrodynamics


Primitive Equations:Hydrostatic, Incompressible,Boussinesq

Similar transport equations for other tracers: passive or actives

Hydrostatic

Continuity

Tracer

Momentum


Vertical Coordinate System

Bottom following coordinate (sigma): best representation of bottom dynamics:

but subject to pressure gradient errors on steep bathymetry


GENERALIZED

-COORDINATE

Stretching & condensing of vertical resolution

(a) Ts=0, Tb=0(b) Ts=8, Tb=0(c) Ts=8, Tb=1(d) Ts=5, Tb=0.4


Horizontal Coordinate System Orthogonal curvilinear

coordinates


Primitive Equations in Curvilinear Coordinate


Simplified Equations

2D barotropic Tidal problems

2D vertical Upwelling

1D vertical Turbulent mixing problems (with

boundary layer parameterization)


Barotropic Equations


Vertical Problems:Parameterization of Surface and Bottom Boundary Layers


Boundary Layer Parameterization

Boundary layers are characterized by strong turbulent mixing

Turbulent Mixing depends on: Surface/bottom forcing:

Wind / bottom-shear stress stirring Stable/unstable buoyancy forcing

Interior conditions: Current shear instability Stratification

w’T’

Reynolds term:

K theory


Surface and Bottom Forcing

Wind stress

Heat FluxSalt Flux

Bottom stress

Drag Coefficient CD:γ1=3.10-4 m/s Linearγ2=2.5 10-3 Quadratic


Boundary Layer Parameterization All mixed layer schemes are based on

one-dimentional « column physics » Boundary layer parameterizations are

based either on: Turbulent closure (Mellor-Yamada, TKE) K profile (KPP)

Note: Hydrostatic stability may require large vertical diffusivities: implicit numerical methods are best suited. convective adjustment methods (infinite

diffusivity) for explicit methods


Application: Tidal Fronts

ROMS Simulation in the Iroise Sea (Front d’Ouessant)

Simpson-Hunter Simpson-Hunter criterium for tidal criterium for tidal

fronts positionfronts position

1.5 < < 2

H. Muller, 2004


Bottom Shear Stress – Wave effect

c /ln(za /z0) 2uzza

2

w 12fwub

2; fw 1.39(ub /z0 ) 0.52

Waves enhance bottom shear stress (Soulsby 1995):

cw c 11.2w

c w

3.2


Numerical Discretization


A Discrete Ocean


Structured / Unstructured GridsFinite Differences / Elements

Structured grids: the grid cells have the same number of sides

Unstructured grids: the domain is tiled using more general geometrical shapes (triangles, …) pieced together to optimally fit details of the geometry Good for tidal modeling, engineering applications Problems: geostrophic balance accuracy, wave scattering

by non-uniform grids, conservation and positivity properties, …


Finite Difference (Grid Point) Method If we know:

The ocean state at time t (u,v,w,T,S, …) Boundary conditions (surface, bottom, lateral

sides) We can compute the ocean state at t+dt

using numerical approximations of Primitive Equations


Horizontal and Vertical Grids


Consistent Schemes: Taylor series expansion, truncation errors We need to find an consistent approximation

for the equations derivatives Taylor series expansion of f at point x:

Truncation error


Exemple: Advection Equation

x

t

x grid space

t time step


Order of Accuracy

First order

2nd order

4th order

Downstream

Upstream

Centered


Numerical properties: stability, dispersion/diffusion

Leapfrog / CenteredTi

n+1 = Ti n-1 - C (Ti+1n - Ti-1

n) ; C = u0 dt / dxConditionally stable: CFL condition C < 1 but dispersive (computational modes)

Euler / CenteredTi

n+1 = Ti n - C (Ti+1n - Ti-1

n)Unconditionally unstable

UpstreamTi

n+1 = Ti n - C (Tin - Ti-1

n) , C > 0Ti

n+1 = Ti n - C (Ti+1n - Ti

n) , C < 0Conditionally stable, not dispersive but diffusive

(monotone linear scheme)

Advection equation:

2nd order approx to the modified equation:

should be non-dispersive:the phase speed ω/k and group speed δω/δk are equal and constant (uo)


Numerical Properties

A numerical scheme can be:

• Dispersive: ripples, overshoot and extrema (centered)

• Diffusive (upstream)

• Unstable (Euler/centered)


Weakly Dispersive, Weakly Diffusive Schemes Using high order upstream schemes:

3rd order upstream biased Using a right combination of a centered scheme

and a diffusive upstream scheme TVD, FCT, QUICK, MPDATA, UTOPIA, PPM

Using flux limiters to build nolinear monotone schemes and guarantee positivity and monotonicity for tracers and avoid false extrema (FCT, TVD)

Note: order of accuracy does not reduce dispersion of shorter waves


Upstream

Centered

2nd order flux limited

3rd order flux limited

Durran, 2004


Accuracy

2nd order

4th order

2nd orderdouble resolution

Spectral method

Numerical dispersion

High order accurate methods: optimal choice (lower cost for a given accuracy) for general ocean circulation models is 3RD OR 4TH ORDER accurate methods (Sanderson, 1998)

With special care to:• dispersion / diffusion• monotonicity and positivity• Combination of methods


OPA - 0.25 deg ROMS – 0.25 deg

C. Blanc C. Blanc

Sensitivity to the Methods: Example


Properties of Horizontal Grids


Arakawa Staggered GridsLinear shallow water equation:

A staggered difference is 4 times more accurate than non-staggered and improves the dispersion relation because of reduced use of averaging operators


Horizontal Arakawa grids B grid is prefered at coarse resolution:

Superior for poorly resolved inertia-gravity waves. Good for Rossby waves: collocation of velocity points. Bad for gravity waves: computational checkboard mode.

C grid is prefered at fine resolution: Superior for gravity waves. Good for well resolved inertia-gravity waves. Bad for poorly resolved waves: Rossby waves

(computational checkboard mode) and inertia-gravity waves due to averaging the Coriolis force.

Combinations can also be used (A + C)


Arakawa-C Grid


Vertical Staggered Grid


Numerical Round-off Errors


Round-off Errors Round-off errors result from inability of computers to represent a

floating point number to infinite precision. Round-off errors tend to accumulate but little control on the

magnitude of cumulative errors is possible. 1byte=8bits, ex:10100100 Simple precision machine (32-bit):

1 word=4 bytes, 6 significant digits Double precision machine (64-bit):

1 word=8 bytes, 15 significant digits Accuracy depends on word length and fractions assigned to

mantissa and exponent. Double precision is possible on a machine of any given basic

precision (using software instructions), but penalty is: slowdown in computation.


Time Stepping


Time Stepping: Standard

Leapfrog: φin+1 = φi n-1 + 2 Δt F(φi

n) computational mode amplifies when applied to

nonlinear equations (Burger, PE)

Leapfrog + Asselin-Robert filter:φi

n+1 = φfi n-1 + 2 Δt F(φin)

φfi n = φi n + 0.5 α (φin+1 - 2 φi

n + φfin-1)

reduction of accuracy to 1rst order depending on α (usually 0.1)

Patrick Marchesiello IRD 2005 43Kantha and Clayson (2000) after Durran (1991)

Time Stepping: Performance

C = 0.5 C = 0.2


Time Stepping: New Standards Multi-time level schemes:

Adams-Bashforth 3rd order (AB3) Adams-Moulton 3rd order (AM3)

Multi-stage Predictor/Corrector scheme

Increase of robustness and stability range LF-Trapezoidal, LF-AM3, Forward-Backward

Runge-Kutta 4: best but expensive

Multi-time level scheme

Multi-stage scheme


Barotropic Dynamicsand Time Splitting


Time step restrictions The Courant-Friedrichs-Levy CFL stability condition on the

barotropic (external) fast mode limits the time step:

Δtext < Δx / Cext where Cext = √gH + Uemax

ex: H =4000 m, Cext = 200 m/s (700 km/h)

Δx = 1 km, Δtext < 5 s Baroclinic (internal) slow mode:

Cin ~ 2 m/s + Uimax (internal gravity wave phase speed + max advective velocity)

Δx = 1 km, Δtext < 8 mn

Δtin / Δtext ~ 60-100 ! Additional diffusion and rotational conditions:

Δtin < Δx2 / 2 Ah and Δtin < 1 / f


Barotropic Dynamics The fastest mode (barotropic) imposes a

short time step 3 methods for releasing the time-step

constraint: Rigid-lid approximation Implicit time-stepping Explicit time-spitting of barotropic and baroclinic

modes Note: depth-averaged flow is an

approximation of the fast mode (exactly true only for gravity waves in a flat bottom ocean)


Rigid-lid Streamfunction Method Advantage: fast mode is properly filtered Disadvantages:

Preclude direct incorporation of tidal processes, storm surges, surface gravity waves.

Elliptic problem to solve: convergence is difficult with complexe geometry; numerical

instabilities near regions of steep slope (smoothing required) Matrix inversion (expensive for large matrices); Bad

scaling properties on parallel machines Fresh water input difficult Distorts dispersion relation for Rossby waves


Implicit Free Surface Method

Numerical damping to supress barotropic waves

Disadvantanges: Not really adapted to tidal processes unless Δt is

reduced, then optimality is lost Involves an elliptic problem

matrix inversion Bad parallelization performances


Time SplittingExplicit free surface method


Barotropic Dynamics:Time Splitting Direct integration of barotropic equations, only few

assumptions; competitive with previous methods at high resolution (avoid penalty on elliptic solver); good parallelization performances

Disadvantages: potential instability issues involving difficulty of cleanly separating fast and slow modes

Solution: time averaging over the barotropic sub-cycle finer mode coupling


Time Splitting: Averaging

ROMS

Averaging weights


Time Splitting: Coupling terms

Coupling terms: advection (dispersion) + baroclinic PGF


Flow Diagram of POM External

mode

Internal mode

Forcing termsof external mode

Replace barotropic partin internal mode


Vertical Diffusion


Vertical Diffusion

Semi-implicitCrank-Nicholson scheme


Pressure Gradient Force


PGF Problem

Truncation errors are made from calculating the baroclinic pressure gradients across sharp topographic changes such as the continental slope

Difference between 2 large terms

Errors can appear in the unforced flat stratification experiment


Reducing PGF Truncation Errors

Smoothing the topography using a nonlinear filter and a criterium:

Using a density formulation

Using high order schemes to reduce the truncation error (4th order, McCalpin, 1994)

Gary, 1973: substracting a reference horizontal averaged value from density (ρ’= ρ - ρa) before computing pressure gradient

Rewritting Equation of State: reduce passive compressibility effects on pressure gradient

r = Δh / h < 0.2


Equation of State

Jackett & McDougall, 1995: 10% of CPU

Full UNESCO EOS:30% of total CPU!

Linearization (ROMS): reduces PGF errors


Smoothing methods

r = Δh / h is the slope of the logarithm of h One method (ROMS) consists of smoothing ln(h)

until r < rmax

Res: 5 kmr < 0.25

Res: 1 kmr < 0.25

Senegal Bathymetry Profil


Smoothing method and resolution

Grid Resolution [deg]

Bathymetry Smoothing Error off Senegal

Convergence at ~ 4 km resolution

Sta

ndar

d D

evia

tion

[m]


Errors in Bathymetry data compilations

Shelf errors(noise)

Etopo2: Satellite observationsGebco1 compilation


Wetting and Drying Schemes


Wetting and Drying: Principles Application:

Intertidal zone Storm surges

Principles: mask/unmask

drying/wetting areas at every time step

Criterium based on a minimum depth

Requirements Conservation

properties

patrick marchesiello ird 20051 regional coastal ocean modeling patrick marchesiello brest, 2005

Documents

patrick marchesiello

boundary layers slide

theory slide

hydrodynamics slide

quadratic slide

curvilinear coordinate

steep bathymetry slide

vertical problems