Regional Coastal Ocean
Modeling
Patrick Marchesiello
Brest, 2005
Patrick Marchesiello IRD 2005
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The Coastal Ocean:
A Challenging Environment
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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
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Numerical Modeling
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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
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Coastal Model Inventory
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POM
ROMS
MARS3D
SYMPHONIE
GHERM
HAMSOM
QUODDY
MOG3D
SEOM
Finite-Difference Models
Finite-Elements Models
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Hydrodynamics
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Primitive
Equations:
Hydrostatic,
Incompressible,
Boussinesq
Momentum
Tracer
Hydrostatic
Similar transport equations
for other tracers: passive or
actives
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Continuity
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Vertical Coordinate System
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Bottom following
coordinate (sigma):
best representation of
bottom dynamics:
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but subject to pressure
gradient errors on steep
bathymetry
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GENERALIZED
-COORDINATE
Stretching &
condensing of
vertical resolution
(a)
(b)
(c)
(d)
Ts=0, Tb=0
Ts=8, Tb=0
Ts=8, Tb=1
Ts=5, Tb=0.4
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Horizontal Coordinate System
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Orthogonal curvilinear
coordinates
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Primitive
Equations in
Curvilinear
Coordinate
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Simplified Equations
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2D barotropic
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2D vertical
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Tidal problems
Upwelling
1D vertical
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Turbulent mixing problems (with
boundary layer parameterization)
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Barotropic Equations
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Vertical Problems:
Parameterization of Surface and
Bottom Boundary Layers
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Boundary Layer
Parameterization
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Boundary layers are
characterized by strong
turbulent mixing
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Turbulent Mixing depends on:
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Surface/bottom forcing:
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Wind / bottom-shear stress stirring
Stable/unstable buoyancy forcing
Reynolds term:
w’T’
Interior conditions:
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Current shear instability
Stratification
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Surface and Bottom Forcing
Wind stress
Heat Flux
Salt Flux
Bottom stress
Drag Coefficient CD:
γ1=3.10-4 m/s Linear
γ2=2.5 10-3 Quadratic
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Boundary Layer Parameterization
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All mixed layer schemes are based on
one-dimentional « column physics »
Boundary layer parameterizations are
based either on:
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Turbulent closure (Mellor-Yamada, TKE)
K profile (KPP)
Note: Hydrostatic stability may require
large vertical diffusivities:
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implicit numerical methods are best suited.
convective adjustment methods (infinite
diffusivity) for explicit methods
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Application: Tidal Fronts
ROMS
Simulation in
the Iroise
Sea (Front
d’Ouessant)
Simpson-Hunter
criterium for tidal
fronts position
1.5 <
<2
H. Muller, 2004
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Bottom Shear Stress – Wave effect
 c   /ln(za /z0 ) u2z z
2
 w  12 f w ub2 ;
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a
f w  1.39(ub /z0 )0.52
Waves enhance bottom shear stress (Soulsby 1995):
3.2

  w  

 cw   c 1 1.2

 c   w  

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Numerical Discretization
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A Discrete Ocean
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Structured / Unstructured Grids
Finite Differences / Elements
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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
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Good for tidal modeling, engineering applications
Problems: geostrophic balance accuracy, wave scattering
by non-uniform grids, conservation and positivity
properties, …
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Finite Difference (Grid Point) Method
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If we know:
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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
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Horizontal and Vertical Grids
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Consistent Schemes:
Taylor series expansion, truncation errors
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We need to find an consistent approximation
for the equations derivatives
Taylor series expansion of f at point x:
Truncation
error
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Exemple: Advection Equation
Dx
Dt
grid space
time step
Dt
Dx
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Order of Accuracy
First order
Downstream
Upstream
2nd order
Centered
4th order
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Numerical properties:
stability, dispersion/diffusion
Leapfrog / Centered
Tin+1 = Ti n-1 - C (Ti+1n - Ti-1n) ; C = u0 dt / dx
Conditionally stable: CFL condition C <
1 but dispersive (computational modes)
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Euler / Centered
Tin+1 = Ti n - C (Ti+1n - Ti-1n)
Unconditionally unstable
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Upstream
Tin+1 = Ti n - C (Tin - Ti-1n) , C > 0
Tin+1 = Ti n - C (Ti+1n - Tin) , C < 0
Conditionally stable,
not dispersive but diffusive
(monotone linear scheme)
Advection equation:
should be non-dispersive:the
phase speed ω/k and group
speed δω/δk are equal and
constant (uo)
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Patrick Marchesiello IRD 2005
2nd order approx to the
modified equation:
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Numerical Properties
A numerical scheme can be:
• Dispersive: ripples, overshoot and
extrema (centered)
• Diffusive (upstream)
• Unstable (Euler/centered)
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Weakly Dispersive, Weakly Diffusive Schemes
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Using high order upstream schemes:
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Using a right combination of a centered scheme
and a diffusive upstream scheme
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3rd order upstream biased
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
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Upstream
Centered
2nd order
flux limited
3rd order
flux limited
Durran, 2004
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Accuracy
Numerical dispersion
2nd order
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
4th order
2nd order
double resolution
Spectral method
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Sensitivity to the Methods: Example
OPA - 0.25 deg
ROMS – 0.25 deg
C. Blanc
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C. Blanc
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Properties of Horizontal
Grids
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Arakawa Staggered Grids
Linear 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
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Horizontal Arakawa grids
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B grid is prefered at coarse resolution:
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C grid is prefered at fine resolution:
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Superior for poorly resolved inertia-gravity waves.
Good for Rossby waves: collocation of velocity points.
Bad for gravity waves: computational checkboard mode.
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)
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Arakawa-C Grid
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Vertical
Staggered
Grid
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Numerical Round-off Errors
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Round-off Errors
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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.
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Time Stepping
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Time Stepping: Standard
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Leapfrog: φin+1 = φi n-1 + 2 Δt F(φin)
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computational mode amplifies when applied to
nonlinear equations (Burger, PE)
Leapfrog + Asselin-Robert filter:
φin+1 = φfi n-1 + 2 Δt F(φin)
φfi n = φi n + 0.5 α (φin+1 - 2 φin + φfin-1)
 reduction of accuracy to 1rst order depending on
α (usually 0.1)
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Time Stepping: Performance
C = 0.5
C = 0.2
Kantha and Clayson (2000) after Durran (1991)
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Time Stepping: New Standards
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Multi-time level schemes:
Multi-time level
scheme
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Adams-Bashforth 3rd order (AB3)
Adams-Moulton 3rd order (AM3)
Multi-stage Predictor/Corrector scheme
Multi-stage
scheme
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Increase of robustness and stability range
LF-Trapezoidal, LF-AM3, Forward-Backward
Runge-Kutta 4: best but expensive
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Barotropic Dynamics
and Time Splitting
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Time step restrictions
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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
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Barotropic Dynamics
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The fastest mode (barotropic) imposes a
short time step
3 methods for releasing the time-step
constraint:
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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)
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Rigid-lid Streamfunction Method
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Advantage: fast mode is properly filtered
Disadvantages:
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Preclude direct incorporation of tidal processes, storm
surges, surface gravity waves.
Elliptic problem to solve:
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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
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Implicit Free Surface Method
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Numerical damping to supress barotropic
waves
Disadvantanges:
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Not really adapted to tidal processes unless Δt is
reduced, then optimality is lost
Involves an elliptic problem
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matrix inversion
Bad parallelization performances
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Time Splitting
Explicit free surface method
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Barotropic Dynamics:Time Splitting
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Direct integration of barotropic equations, only few
assumptions; competitive with previous methods at
high resolution (avoid penalty on elliptic solver);
good parallelization performances
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Disadvantages: potential instability issues involving
difficulty of cleanly separating fast and slow modes
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Solution:
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time averaging over the barotropic sub-cycle
finer mode coupling
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Time Splitting: Averaging
Averaging
weights
ROMS
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Time Splitting: Coupling terms
Coupling terms: advection (dispersion) + baroclinic PGF
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Internal
mode
Flow
Diagram
of POM
External
mode
Forcing terms
of external mode
Replace barotropic part
in internal mode
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Vertical Diffusion
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Vertical
Diffusion
Semiimplicit
CrankNicholson
scheme
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Pressure Gradient Force
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PGF Problem
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Truncation errors are made
from calculating the baroclinic
pressure gradients across
sharp topographic changes
such as the continental slope
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Difference between 2 large
terms
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Errors can appear in the
unforced flat stratification
experiment
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Reducing PGF Truncation Errors
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Smoothing the topography using a
nonlinear filter and a criterium:
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Using a density formulation
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Using high order schemes to reduce
the truncation error (4th order,
McCalpin, 1994)
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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
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r = Δh / h < 0.2
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Equation of State
Full UNESCO EOS:
30% of total CPU!
Jackett &
McDougall,
1995: 10%
of CPU
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Linearization
(ROMS):
reduces PGF
errors
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Smoothing methods
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r = Δh / h is the slope of the logarithm of h
One method (ROMS) consists of smoothing ln(h)
until r < rmax
Res: 1 km
r < 0.25
Res: 5 km
r < 0.25
Senegal Bathymetry Profil
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Smoothing method and resolution
Standard Deviation [m]
Bathymetry Smoothing Error off Senegal
Convergence at
~ 4 km resolution
Grid Resolution [deg]
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Errors in Bathymetry data compilations
Gebco1 compilation
Etopo2: Satellite observations
Shelf errors
(noise)
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Wetting and Drying Schemes
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Wetting and Drying: Principles
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Application:
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Principles:
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mask/unmask
drying/wetting
areas at every time
step
Criterium based on
a minimum depth
Requirements
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Intertidal zone
Storm surges
Conservation
properties
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