Patrick Marchesiello IRD Regional Coastal Ocean Modeling Patrick Marchesiello Brest, 2005

Patrick Marchesiello IRD 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

Patrick Marchesiello IRD 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

Patrick Marchesiello IRD Coastal Model Inventory POM ROMS MARS3D SYMPHONIE GHERM HAMSOM QUODDY MOG3D SEOM Finite-Difference Models Finite-Elements Models

Patrick Marchesiello IRD

6 Hydrodynamics

Patrick Marchesiello IRD Primitive Equations: Hydrostatic, Incompressible, Boussinesq Similar transport equations for other tracers: passive or actives Hydrostatic Continuity Tracer Momentum

Patrick Marchesiello IRD Vertical Coordinate System Bottom following coordinate (sigma): best representation of bottom dynamics: but subject to pressure gradient errors on steep bathymetry

Patrick Marchesiello IRD 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

Patrick Marchesiello IRD Horizontal Coordinate System Orthogonal curvilinear coordinates

Patrick Marchesiello IRD Primitive Equations in Curvilinear Coordinate

Patrick Marchesiello IRD Simplified Equations 2D barotropic  Tidal problems 2D vertical  Upwelling 1D vertical  Turbulent mixing problems (with boundary layer parameterization)

Patrick Marchesiello IRD Barotropic Equations

Patrick Marchesiello IRD Vertical Problems: Parameterization of Surface and Bottom Boundary Layers

Patrick Marchesiello IRD 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

Patrick Marchesiello IRD Surface and Bottom Forcing Wind stress Heat Flux Salt Flux Bottom stress Drag Coefficient C D : γ 1 = m/s Linear γ 2 = Quadratic

Patrick Marchesiello IRD 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

Patrick Marchesiello IRD 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

Patrick Marchesiello IRD Bottom Shear Stress – Wave effect Waves enhance bottom shear stress (Soulsby 1995):

Patrick Marchesiello IRD Numerical Discretization

Patrick Marchesiello IRD A Discrete Ocean

Patrick Marchesiello IRD Structured / Unstructured Grids Finite 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, …

Patrick Marchesiello IRD 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

Patrick Marchesiello IRD Horizontal and Vertical Grids

Patrick Marchesiello IRD 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

Patrick Marchesiello IRD Exemple: Advection Equation xx tt  x grid space  t time step

Patrick Marchesiello IRD Order of Accuracy First order 2 nd order 4 th order Downstream Upstream Centered

Patrick Marchesiello IRD Numerical properties: stability, dispersion/diffusion Leapfrog / Centered T i n+1 = T i n-1 - C (T i+1 n - T i-1 n ) ; C = u 0 dt / dx Conditionally stable: CFL condition C < 1 but dispersive (computational modes) Euler / Centered T i n+1 = T i n - C (T i+1 n - T i-1 n ) Unconditionally unstable Upstream T i n+1 = T i n - C (T i n - T i-1 n ), C > 0 T i n+1 = T i n - C (T i+1 n - T i n ), C < 0 Conditionally 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 (u o )

Patrick Marchesiello IRD Numerical Properties A numerical scheme can be: Dispersive: ripples, overshoot and extrema (centered) Diffusive (upstream) Unstable (Euler/centered)

Patrick Marchesiello IRD 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

Patrick Marchesiello IRD Upstream Centered 2nd order flux limited 3rd order flux limited Durran, 2004

Patrick Marchesiello IRD Accuracy 2 nd order 4 th order 2 nd order double resolution Spectral method Numerical dispersion High order accurate methods: optimal choice (lower cost for a given accuracy) for general ocean circulation models is 3 RD OR 4 TH ORDER accurate methods (Sanderson, 1998) With special care to: dispersion / diffusion monotonicity and positivity Combination of methods

Patrick Marchesiello IRD OPA deg ROMS – 0.25 deg C. Blanc Sensitivity to the Methods: Example

Patrick Marchesiello IRD Properties of Horizontal Grids

Patrick Marchesiello IRD 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

Patrick Marchesiello IRD 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)

Patrick Marchesiello IRD Arakawa-C Grid

Patrick Marchesiello IRD Vertical Staggered Grid

Patrick Marchesiello IRD Numerical Round-off Errors

Patrick Marchesiello IRD 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: 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.

Patrick Marchesiello IRD Time Stepping

Patrick Marchesiello IRD Time Stepping: Standard Leapfrog: φ i n+1 = φ i n Δt F(φ i n )  computational mode amplifies when applied to nonlinear equations (Burger, PE) Leapfrog + Asselin-Robert filter: φ i n+1 = φf i n Δt F(φ i n ) φf i n = φ i n α (φ i n φ i n + φf i n-1 )  reduction of accuracy to 1rst order depending on α (usually 0.1)

Patrick Marchesiello IRD Kantha and Clayson (2000) after Durran (1991) Time Stepping: Performance C = 0.5C = 0.2

Patrick Marchesiello IRD 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

Patrick Marchesiello IRD Barotropic Dynamics and Time Splitting

Patrick Marchesiello IRD Time step restrictions The Courant-Friedrichs-Levy CFL stability condition on the barotropic (external) fast mode limits the time step: Δt ext < Δx / C ext where C ext = √gH + Ue max ex: H =4000 m, C ext = 200 m/s (700 km/h) Δx = 1 km, Δt ext < 5 s Baroclinic (internal) slow mode: C in ~ 2 m/s + Ui max (internal gravity wave phase speed + max advective velocity) Δx = 1 km, Δt ext < 8 mn Δt in / Δt ext ~ ! Additional diffusion and rotational conditions: Δt in < Δx 2 / 2 Ah and Δt in < 1 / f

Patrick Marchesiello IRD 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)

Patrick Marchesiello IRD 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

Patrick Marchesiello IRD 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

Patrick Marchesiello IRD Time Splitting Explicit free surface method

Patrick Marchesiello IRD 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

Patrick Marchesiello IRD Time Splitting: Averaging ROMS Averaging weights

Patrick Marchesiello IRD Time Splitting: Coupling terms Coupling terms: advection (dispersion) + baroclinic PGF

Patrick Marchesiello IRD Flow Diagram of POM External mode Internal mode Forcing terms of external mode Replace barotropic part in internal mode

Patrick Marchesiello IRD Vertical Diffusion

Patrick Marchesiello IRD Vertical Diffusion Semi- implicit Crank- Nicholson scheme

Patrick Marchesiello IRD Pressure Gradient Force

Patrick Marchesiello IRD 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

Patrick Marchesiello IRD 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 - ρ a ) before computing pressure gradient Rewritting Equation of State: reduce passive compressibility effects on pressure gradient r = Δh / h < 0.2

Patrick Marchesiello IRD Equation of State Jackett & McDougall, 1995: 10% of CPU Full UNESCO EOS: 30% of total CPU! Linearization (ROMS): reduces PGF errors

Patrick Marchesiello IRD Smoothing methods r = Δh / h is the slope of the logarithm of h One method (ROMS) consists of smoothing ln(h) until r < r max Res: 5 km r < 0.25 Res: 1 km r < 0.25 Senegal Bathymetry Profil

Patrick Marchesiello IRD Smoothing method and resolution Grid Resolution [deg] Bathymetry Smoothing Error off Senegal Convergence at ~ 4 km resolution Standard Deviation [m]

Patrick Marchesiello IRD Errors in Bathymetry data compilations Shelf errors (noise) Etopo2: Satellite observationsGebco1 compilation

Patrick Marchesiello IRD Wetting and Drying Schemes

Patrick Marchesiello IRD 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