Regional Coastal Ocean Modeling

1
Regional Coastal Ocean Modeling

• Patrick Marchesiello
• Brest, 2005

2
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

3
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

4
Coastal Model Inventory

• POM
• ROMS
• MARS3D
• SYMPHONIE
• GHERM
• HAMSOM
• QUODDY
• MOG3D
• SEOM

Finite-Difference Models
Finite-Elements Models
5
(No Transcript)
6
Hydrodynamics
7
Primitive EquationsHydrostatic,
Incompressible,Boussinesq
Momentum
Tracer
Hydrostatic
Similar transport equations for other tracers
passive or actives
Continuity
8
Vertical Coordinate System

• Bottom following coordinate (sigma) best
  representation of bottom dynamics
• but subject to pressure gradient errors on steep
  bathymetry

9
GENERALIZED ?-COORDINATE Stretching condensing
of vertical resolution

• Ts0, Tb0
• Ts8, Tb0
• Ts8, Tb1
• Ts5, Tb0.4

10
Horizontal Coordinate System

• Orthogonal curvilinear coordinates

11
Primitive Equations in Curvilinear Coordinate
12
Simplified Equations

• 2D barotropic
• Tidal problems
• 2D vertical
• Upwelling
• 1D vertical
• Turbulent mixing problems (with boundary layer
  parameterization)

13
Barotropic Equations
14
Vertical ProblemsParameterization of Surface
and Bottom Boundary Layers
15
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

Reynolds term
K theory
16
Surface and Bottom Forcing
Wind stress Heat Flux Salt Flux
17
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

18
Application Tidal Fronts
ROMS Simulation in the Iroise Sea (Front
dOuessant)
H. Muller, 2004
19
Bottom Shear Stress Wave effect

• Waves enhance bottom shear stress (Soulsby 1995)

20
Numerical Discretization
21
A Discrete Ocean
22
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,

23
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 tdt using
  numerical approximations of Primitive Equations

24
Horizontal and Vertical Grids
25
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
26
Exemple Advection Equation
Dx grid space Dt time step
Dt
Dx
27
Order of Accuracy
First order
Downstream
Upstream
2nd order
Centered
4th order
28
Numerical properties stability,
dispersion/diffusion
Advection equation

• Leapfrog / Centered
• Tin1 Ti n-1 - C (Ti1n - Ti-1n) C u0 dt
  / dx
• Conditionally stable CFL condition C lt 1 but
  dispersive (computational modes)
• Euler / Centered
• Tin1 Ti n - C (Ti1n - Ti-1n)
• Unconditionally unstable
• Upstream
• Tin1 Ti n - C (Tin - Ti-1n) , C gt 0
• Tin1 Ti n - C (Ti1n - Tin) , C lt 0
• Conditionally stable,
• not dispersive but diffusive
• (monotone linear scheme)

should be non-dispersivethe phase speed ?/k and
group speed d?/dk are equal and constant (uo)
2nd order approx to the modified equation
29
Numerical Properties

• A numerical scheme can be
• Dispersive ripples, overshoot and extrema
  (centered)
• Diffusive (upstream)
• Unstable (Euler/centered)

30
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

31
Upstream
Centered
2nd order flux limited
3rd order flux limited
Durran, 2004
32
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
33
Sensitivity to the Methods Example
ROMS 0.25 deg
OPA - 0.25 deg
C. Blanc
C. Blanc
34
Properties of Horizontal Grids
35
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

36
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)

37
Arakawa-C Grid
38
Vertical Staggered Grid
39
Numerical Round-off Errors
40
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.
• 1byte8bits, ex10100100
• Simple precision machine (32-bit)
• 1 word4 bytes, 6 significant digits
• Double precision machine (64-bit)
• 1 word8 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.

41
Time Stepping
42
Time Stepping Standard

• Leapfrog fin1 fi n-1 2 ?t F(fin)
• computational mode amplifies when applied to
  nonlinear equations (Burger, PE)
• Leapfrog Asselin-Robert filter
• fin1 ffi n-1 2 ?t F(fin)
• ffi n fi n 0.5 a (fin1 - 2 fin ffin-1)
• reduction of accuracy to 1rst order depending on
  a (usually 0.1)

43
Time Stepping Performance
C 0.5
C 0.2
Kantha and Clayson (2000) after Durran (1991)
44
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
45
Barotropic Dynamicsand Time Splitting
46
Time step restrictions

• The Courant-Friedrichs-Levy CFL stability
  condition on the barotropic (external) fast mode
  limits the time step
• ?text lt ?x / Cext where Cext vgH Uemax
• ex H 4000 m, Cext 200 m/s (700 km/h)
• ?x 1 km, ?text lt 5 s
• Baroclinic (internal) slow mode
• Cin 2 m/s Uimax (internal gravity wave phase
  speed max advective velocity)
• ?x 1 km, ?text lt 8 mn
• ?tin / ?text 60-100 !
• Additional diffusion and rotational conditions
• ?tin lt ?x2 / 2 Ah and ?tin lt 1 / f

47
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)

48
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

49
Implicit Free Surface Method

• Numerical damping to supress barotropic waves
• Disadvantanges
• Not really adapted to tidal processes unless ?t
  is reduced, then optimality is lost
• Involves an elliptic problem
• matrix inversion
• Bad parallelization performances

50
Time SplittingExplicit free surface method
51
Barotropic DynamicsTime 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

52
Time Splitting Averaging
Averaging weights
ROMS
53
Time Splitting Coupling terms
Coupling terms advection (dispersion)
baroclinic PGF
54
Internal mode
Flow Diagram of POM
External mode
Forcing terms of external mode
Replace barotropic part in internal mode
55
Vertical Diffusion
56
Vertical Diffusion
Semi-implicit Crank-Nicholson scheme
57
Pressure Gradient Force
58
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

59
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 (? ? - ?a) before
  computing pressure gradient
• Rewritting Equation of State reduce passive
  compressibility effects on pressure gradient

r ?h / h lt 0.2
60
Equation of State
Full UNESCO EOS 30 of total CPU!
Jackett McDougall, 1995 10 of CPU
Linearization (ROMS) reduces PGF errors
61
Smoothing methods

• r ?h / h is the slope of the logarithm of h
• One method (ROMS) consists of smoothing ln(h)
  until r lt rmax

Res 1 km r lt 0.25
Res 5 km r lt 0.25
Senegal Bathymetry Profil
62
Smoothing method and resolution
Bathymetry Smoothing Error off Senegal
Convergence at 4 km resolution
Standard Deviation m
Grid Resolution deg
63
Errors in Bathymetry data compilations
Etopo2 Satellite observations
Gebco1 compilation
Shelf errors (noise)
64
Wetting and Drying Schemes
65
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