1. Patrick Marchesiello IRD 20051 Regional Coastal Ocean Modeling Patrick Marchesiello Brest, 2005

2. Patrick Marchesiello IRD 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. Patrick Marchesiello IRD 2005 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. Patrick Marchesiello IRD 2005 4 Coastal Model Inventory POM ROMS MARS3D SYMPHONIE GHERM HAMSOM QUODDY MOG3D SEOM Finite-Difference Models Finite-Elements Models

5. Patrick Marchesiello IRD 2005 5

6. 6 Hydrodynamics

7. Patrick Marchesiello IRD 2005 7 Primitive Equations: Hydrostatic, Incompressible, Boussinesq Similar transport equations for other tracers: passive or actives Hydrostatic Continuity Tracer Momentum

8. Patrick Marchesiello IRD 2005 8 Vertical Coordinate System Bottom following coordinate (sigma): best representation of bottom dynamics: but subject to pressure gradient errors on steep bathymetry

9. Patrick Marchesiello IRD 2005 9 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

10. Patrick Marchesiello IRD 2005 10 Horizontal Coordinate System Orthogonal curvilinear coordinates

11. Patrick Marchesiello IRD 2005 11 Primitive Equations in Curvilinear Coordinate

12. Patrick Marchesiello IRD 2005 12 Simplified Equations 2D barotropic  Tidal problems 2D vertical  Upwelling 1D vertical  Turbulent mixing problems (with boundary layer parameterization)

13. Patrick Marchesiello IRD 2005 13 Barotropic Equations

14. Patrick Marchesiello IRD 200514 Vertical Problems: Parameterization of Surface and Bottom Boundary Layers

15. Patrick Marchesiello IRD 2005 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 w’T’ Reynolds term: K theory

16. Patrick Marchesiello IRD 2005 16 Surface and Bottom Forcing Wind stress Heat Flux Salt Flux Bottom stress Drag Coefficient C D : γ 1 =3.10-4 m/s Linear γ 2 =2.5 10-3 Quadratic

17. Patrick Marchesiello IRD 2005 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. Patrick Marchesiello IRD 2005 18 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

19. Patrick Marchesiello IRD 2005 19 Bottom Shear Stress – Wave effect Waves enhance bottom shear stress (Soulsby 1995):

20. Patrick Marchesiello IRD 200520 Numerical Discretization

21. Patrick Marchesiello IRD 2005 21 A Discrete Ocean

22. Patrick Marchesiello IRD 2005 22 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, …

23. Patrick Marchesiello IRD 2005 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 t+dt using numerical approximations of Primitive Equations

24. Patrick Marchesiello IRD 2005 24 Horizontal and Vertical Grids

25. Patrick Marchesiello IRD 2005 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. Patrick Marchesiello IRD 2005 26 Exemple: Advection Equation xx tt  x grid space  t time step

27. Patrick Marchesiello IRD 2005 27 Order of Accuracy First order 2 nd order 4 th order Downstream Upstream Centered

28. Patrick Marchesiello IRD 2005 28 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 )

29. Patrick Marchesiello IRD 2005 29 Numerical Properties A numerical scheme can be: Dispersive: ripples, overshoot and extrema (centered) Diffusive (upstream) Unstable (Euler/centered)

30. Patrick Marchesiello IRD 2005 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. Patrick Marchesiello IRD 2005 31 Upstream Centered 2nd order flux limited 3rd order flux limited Durran, 2004

32. Patrick Marchesiello IRD 2005 32 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

33. Patrick Marchesiello IRD 2005 33 OPA - 0.25 deg ROMS – 0.25 deg C. Blanc Sensitivity to the Methods: Example

34. Patrick Marchesiello IRD 200534 Properties of Horizontal Grids

35. Patrick Marchesiello IRD 2005 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. Patrick Marchesiello IRD 2005 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. Patrick Marchesiello IRD 2005 37 Arakawa-C Grid

38. Patrick Marchesiello IRD 2005 38 Vertical Staggered Grid

39. Patrick Marchesiello IRD 200539 Numerical Round-off Errors

40. Patrick Marchesiello IRD 2005 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. 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.

41. Patrick Marchesiello IRD 200541 Time Stepping

42. Patrick Marchesiello IRD 2005 42 Time Stepping: Standard Leapfrog: φ i n+1 = φ i n-1 + 2 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-1 + 2 Δt F(φ i n ) φf i n = φ i n + 0.5 α (φ i n+1 - 2 φ i n + φf i n-1 )  reduction of accuracy to 1rst order depending on α (usually 0.1)

43. Patrick Marchesiello IRD 2005 43 Kantha and Clayson (2000) after Durran (1991) Time Stepping: Performance C = 0.5C = 0.2

44. Patrick Marchesiello IRD 2005 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. Patrick Marchesiello IRD 200545 Barotropic Dynamics and Time Splitting

46. Patrick Marchesiello IRD 2005 46 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 ~ 60-100 ! Additional diffusion and rotational conditions: Δt in < Δx 2 / 2 Ah and Δt in < 1 / f

47. Patrick Marchesiello IRD 2005 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. Patrick Marchesiello IRD 2005 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. Patrick Marchesiello IRD 2005 49 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

50. Patrick Marchesiello IRD 2005 50 Time Splitting Explicit free surface method

51. Patrick Marchesiello IRD 2005 51 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

52. Patrick Marchesiello IRD 2005 52 Time Splitting: Averaging ROMS Averaging weights

53. Patrick Marchesiello IRD 2005 53 Time Splitting: Coupling terms Coupling terms: advection (dispersion) + baroclinic PGF

54. Patrick Marchesiello IRD 2005 54 Flow Diagram of POM External mode Internal mode Forcing terms of external mode Replace barotropic part in internal mode

55. Patrick Marchesiello IRD 200555 Vertical Diffusion

56. Patrick Marchesiello IRD 2005 56 Vertical Diffusion Semi- implicit Crank- Nicholson scheme

57. Patrick Marchesiello IRD 200557 Pressure Gradient Force

58. Patrick Marchesiello IRD 2005 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. Patrick Marchesiello IRD 2005 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 (ρ’= ρ - ρ a ) before computing pressure gradient Rewritting Equation of State: reduce passive compressibility effects on pressure gradient r = Δh / h < 0.2

60. Patrick Marchesiello IRD 2005 60 Equation of State Jackett & McDougall, 1995: 10% of CPU Full UNESCO EOS: 30% of total CPU! Linearization (ROMS): reduces PGF errors

61. Patrick Marchesiello IRD 2005 61 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

62. Patrick Marchesiello IRD 2005 62 Smoothing method and resolution Grid Resolution [deg] Bathymetry Smoothing Error off Senegal Convergence at ~ 4 km resolution Standard Deviation [m]

63. Patrick Marchesiello IRD 2005 63 Errors in Bathymetry data compilations Shelf errors (noise) Etopo2: Satellite observationsGebco1 compilation

64. Patrick Marchesiello IRD 200564 Wetting and Drying Schemes

65. Patrick Marchesiello IRD 2005 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