Lecture-1: FVCOM-An unstructured grid Finite-Volume Community Ocean Model

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1 Lecture-1: FVCOM-An unstructured grid Finite-Volume Community Ocean Model Changsheng Chen School for Marine Science and Technology University of Massachusetts-Dartmouth New Bedford, MA Website: Chen-FVCOM Chile

2 Critical Issues in Coastal Ocean Modeling Irregular geometry Mass Conservation? Steep topography Intertidal wetlands

3 Critical Issues for Global Ocean: Basin-shelf interaction Coastal process Basin scale u Multi-scale dynamics: Basin-shelf interaction, convection via advection, etc. u Resolving irregular coastal geometries connected to the North Atlantic Ocean and Pacific Ocean

calculating the variables by a net flux through control volumes An example: ζ dxdy t = ( ud) ( vd) [ x + y

4 Possible solutions: 1. Finite-volume algorithms: Configuring the computational domains with individual control volumes and calculating the variables by a net flux through control volumes An example: ζ dxdy t = ( ud) ( vd) [ x + y ]dxdy = S v n Dds 2. Unstructured grids: Configuring the computational domains with unstructured grids to provide the accurate fitting of the irregular coastal geometry. An example: Chen-FVCOM Chile

5 FVCOM: Unstructured-grid, Finite-Volume Coastal Ocean Model (Chen, C. R. H. Liu and R. C. Beardsley, JAOT, 2003) All variables are computed in the integral form of the equations, which provides a better representation of the conservative laws of mass, momentum and heat in the coastal region with complex geometry. F u,v u,v F u,v F u,v F u,v u,v F The numerical computational domain consists of non-overlapping unstructured cells. F F F : ", T, S,!, H, K m, K h, etc yʹ Combines the best from the finiteelement method for the geometric flexibility and finite difference method for the simplest discrete computation. u,v u,v u,v u,v Both current and tracer remain the second-order accuracy.

6 For example: The Continuity Equation:! I : u, v, : H,!, ", D, s, #, q 2, q 2 l, A m, K h ( ud) #( vd) #$ #!! dxdy = " [ ]dxdy t!! + = " # # x # y! S v n Dds

7 FVCOM Wet/Dry Treatment Technology-Coastal Inundation ζ u, v

8 In the Cartesian coordinates, Spherical Coordinate at the Arctic! = ;! dy = s dx 0 0 A line integral is closed s In the spherical coordinates, North Pole " s r cos $ d#! 0 A line integral is not always closed &# ' $$ r cos" d! d" % r& " ( 2? $ s ( ) d!

9 Spherical Coordinate FVCOM u,vcosϕ F: Scalar variables such as ζ, T, S, K m, K h,..and vertical velocity ω. u,vcosϕ F u,vcosϕ : The node of triangles where scalar variable or vertical velocity is calculated u,vcosϕ u,vcosϕ u,vcosϕ : The centroid of a triangle where the horizontal velocity is calculated. Example: Continuity equation! $ 1! ud! v cos" D + ( + ) = 0! t r cos"!#!" T T u,vcosϕ u,vcosϕ T T u,vcosϕ T "" #!& r! t $ ' $ t 2 cos$ d% d$ + "" r = " [! ( Du ) d& " #! # 1! ud! v cos$ D ( + ) r r cos$!%! $ ( Dv cos& ) d% ] 2 cos$ d% d$ = 0 u,vcosϕ T u,vcosϕ u,vcosϕ T The gradient of the water temperature (or salinity) is determined by the Green s function through the integration over the larger volume (with boundaries linked to nodes).

10 Treatment of North Pole Node calculated using the polar stereographic projection coordinate. Centroid calculated using the polar stereographic projection coordinate. Node calculated directly in the spherical coordinate system. Centroid calculated directly in the spherical coordinate system. Conversion formulae between (u p, v p ) and (u s, v s ): & u $ % v p p #! " = & $ % ' sin ( cos( ' cos( #& u! $ ' sin ( "% v s s #! " u v p p = h = h x y dx dt dy dt h h x y 1+ sin! = 1+ sin 2! = 2

11 Nudging/OI Assimilation Ensemble/Reduced Kalman Filters North Pole Nested System Modules of FVCOM Forcings: Tides (Equilibrium+ O.B.) Winds, Heat flux, Precipitation/ Evaporation River discharges, Groundwater O.B. fluxes 3-D Wet/Dry Treatment General Ocean Turbulence Model (GOTM) 3-D Sediment Model Model Field Sampling Adjoint Assimilation Surface Wave Model Multiple Nesting Non-hydrostatic FVCOM-Main Code Cartesian/Spherical Coordinates MPI Parallel NetCDF Output ViSiT Monitoring GUI Post-process Tools Generalized Biological Model Water Quality Models Multi OB Radiations Lagrangian-IBM Ice model Key: Existing Modules Under Development Solver: Mode-split or semi-implicit; 2-D and 3-D, Types: Research version (FVCOM-v2.7) and Forecast version (FVCOM-v3.1).

ViSiT Software developed by Lawrence Livermore National Laboratory http://www.llnl.

12 ViSiT Software developed by Lawrence Livermore National Laboratory Open source; Parallel visualization; Interactive simulation support Multiple platform support (LINUX, UNIX, PC, MAC) FVCOM Plug-in for VISIT FVCOM NETCDF files Visualization and animation of 3D vector and scalar fields Database linking to NETCDF formatted particle tracking output Example of density isosurfaces from a high resolution FVCOM GOM model

FVCOM Users 38 countries(including HK/Taiwan) >1000 users or institutions Australia, Bangladesh, Brazil, Canada, Chile, China, Colombia, Denmark, Egypt, France, Germany, Hong Kong, Hungary India,

13 FVCOM Users 38 countries(including HK/Taiwan) >1000 users or institutions Australia, Bangladesh, Brazil, Canada, Chile, China, Colombia, Denmark, Egypt, France, Germany, Hong Kong, Hungary India, Indonesia, Iran, Israel, Italy, Jamaica, Japan, Korea, Kuwait, Malaysia, Mexico, Netherlands, New Zealand, Norway Peru, Saudi Arabia, Singapore, Sweden, Taiwan, Thailand, Turkey, U.S., UK, Venezuela, Vietnam

14 1. Advection scheme; Hydrostatic FVCOM Validation Experiments 2. Wind-induced oscillation (POM, ECOM-si); 3. Wind-induced waves over the slope bottom topography (POM, ECOM-si); 4. Tidal Resonances in a semi-enclosed channel and a sector (POM, ECOM-si); 5. Freshwater discharge plume (POM, ECOM-si); 6. Bottom boundary layer over a step bottom slope (POM; ECOM-si) 7. Equatorial Rossby soliton (ROMs) 8. Wind-induced lateral boundary (ROMs) 9. Super-critical current (ROMs) Non-Hydrostatic FVCOM Validation Experiments 1. Standing surface short gravity waves; 2. Lock exchanges; 3. Solitary waves in both homogeneous and layer fluids.

15 1. Advection Scheme!F!t +C!F!x = 0 # % F(x,0) = $ % & 5,!2 " x " 2 0, otherwise and C = 1 Δt = 0.05, Δx=0.1 Upwind finitedifference stream Central finitedifference stream FVCOM finite-volume flux scheme

16 Δt = 0.005, Δx=0.01 Upwind finitedifference stream Central finitedifference stream FVCOM finitevolume flux scheme

17 Wind-induced oscillation Linear, non-dimensional equations: where u ζ v = λ t r v ζ + u = λ t r θ ζ λ ( ru) v + [ + ] = 0 t r r θ " = gd ;# = $ % ˆ $ ; ˆ $ = & rcos' o ;& r o f " 4 o = g& r 3 o f 4 d=75 m Wind Wind is suddenly imposed at initial r o =67.5 km and u r=1 = 0; (u,v,") r=0 # finite; u t =0 = v t =0 = 0;" t =0 = $ ˆ % (r,&) Solution:!(r,!,t) =! o! [A (r)cos! + a A (r)cos(!!" t)] 4 o k k k u(r,!,t) =! o! [( A (r) " o!1)sin!!# b 3 r k F k (r)sin(!!" k t)] " # k=1 v(r,!,t) =! o! [(da (r) " o!1)cos!!# b 3 dr k G k (r)cos(!!" k t)] k=1 k=1 Reference: Csanady ( 1968) Birchfield (1969)

18 Unstructured (FVCOM) Structured (POM) Chen- FVCOM

19 Water elevation Alongshore transport Chen-FVCOM Chile

20 Chen-FVCOM Chile

21 Chen-FVCOM Chile

22 Radial mode: k=1, 2: gravity waves, k=3: topographic wave Birchfield and Hickie (1977) JPO Analytical FVCOM (5 km) POM (2.5 km) 1 h 1 d Chen-FVCOM Chile

23 5 days FVCOM (5 km) POM (2.5 km) ζ: Elevation V : Currents Chen-FVCOM Chile

24 Chen- FVCOM

25 Chen-FVCOM Chile

26 Structured (POM) Unstructured (FVCOM) Chen-FVCOM Chile

27 Structured (POM) Unstructured (FVCOM) Chen- FVCOM

Tidal Resonance in A Semi-closed Channel = 0 + r g t V r η = 0 + θ η θ r g t V Consider a 2-D linear, non-rotated initial problem such as 0 0 0 = + + θ η θ r H V r r H rv t r { The solution: ] 2) / (

28 Tidal Resonance in A Semi-closed Channel = 0 + r g t V r η = 0 + θ η θ r g t V Consider a 2-D linear, non-rotated initial problem such as = + + θ η θ r H V r r H rv t r { The solution: ] 2) / ( cos[ )] ( ) ( [ ), ( α α θ π ω ω θ η γ γ + + = m gh r Y c gh r J c r m m ) ( ) ( ) ( ) ( ) /[ ( ' 0 1 ' ' 1 gh L Y gh L J gh L Y gh L J gh L Y A c m m m m m ω ω ω ω ω γ γ γ γ γ = ) ( ) ( ) ( ) ( ) /[ ( ' 0 1 ' ' 2 gh L Y gh L J gh L Y gh L J gh L J A c m m m m m ω ω ω ω ω γ γ γ γ γ = α π γ / m = m where Chen-FVCOM Chile

29 1. Normal condition (non-resonance) 2. Near-resonance condition Chen-FVCOM Chile

30 Chen-FVCOM Chile A Normal Case

31 Chen-FVCOM Chile A Near-resonance Case

32 Chen-FVCOM Chile Near-resonance, 2 km (FVCOM), 1 km (POM&ECOM-si)

33 Chen-FVCOM Chile Near-resonance, 2km, Curvilinear

34 0.5-4 km 4 km 2 km Chen-FVCOM Chile

35 Chen-FVCOM Chile Slope topography fitting

36 Equatorial Rossby Soliton L = 48 Grid A D = 24 C=0. 4 Grid B Periodic boundary condition 1. Nonlinear shallow water equation in equatorial β-plane 2. Inviscid flow 3. Asymptotic solutions available to zero and first orders (Boyd 1980,1985) Grid C Chen-FVCOM Chile

37 Δx (ND) FVCOM (2 nd ) ROMS (4 th ) SEOM (7-9 th ) Grid A 0.5 h n /h t C n /C t h n /h t C n /C t h n /h t C n /C t Grid B Grid C h n : Computed peak of the sea surface elevation at 120 units h t : Analytical peak of the sea surface elevation at 120 units C n : Computed average speed C t : Analytical averaged speed. Comments: 1. Analytical solution only represents the zero and 1st modes, while the numerical solution contains a complete set of higher order modes. This is not surprised to see numerical models can not exactly reach the analytical solutions. 2. FVCOM shows a fast convergence with increase of horizontal resolution.

38 Chen- FVCOM

39 Hydraulic Jump No gradient condition Analytical solution: Maximum sea level:! max = 0.5 m Minimum sea level:! = 0 min m (m) u v = 8.57 cm / s = 0! = 0 Mean sea level: Mean velocity: Mean Froude #:! mean u = Fr = 0.5 m u = gd m/s = o Shock angle: 0! = 30 Thickness:! = 0 m φ = 8.95 o (m) Mean deviation: dy = 0 m Characteristics: Barotropic shallow water equations No rotation considered, i.e. f =0, β = 0 Steady analytical solutions for u, ζ and the jump angle relative to the x axis. Chen-FVCOM Chile

The case with no horizontal diffusion: FVCOM quickly reaches steady status. x = 0.5 m, t = 0.002 x = 0.25 m, t = 0.001 x = 0.125 m, t = 0.0005 Model grids t ζ max ζ min ζ mean ū F r α δ dy True 0.

40 The case with no horizontal diffusion: FVCOM quickly reaches steady status. x = 0.5 m, t = x = 0.25 m, t = x = m, t = Model grids t ζ max ζ min ζ mean ū F r α δ dy True FVCOM 80 X X X ROMs Reach an oscillatory solution without horizontal diffusion. Chen-FVCOM Chile

41 Over shocking can be reduced by introducing a slope limiter method (Hubbard, J. Comput. Phys., 1999). Original code Modified code with limiter Chen-FVCOM Chile

42 3-Dimensional Wind-Driven Flow in an Elongated, Rotating Basin 2L z Winant (J. Phys. Oceano. 2004) y x Governing equations: $ & )! #!! fk "! v + w! ( v B.Cs: r $ r ' s! vz = # & K! r " v = 0 m z = 0 = ' g& * + K w = 0 w = 0 m at z = 0 at z = % h 2! % v 2 % z Length: 2L; width: 2B, and bathymetry: h = h0{ "[ X ( x / L)(1! ( y / B) where X(x) is a function in the form of X! $ 1, x) = # x & 1+ % x 1& [ ],!" % x ( 2 )]} x is a constant specified as 0.3% of the total length of the basin. 2 x < 1& % x x ' 1& % x Steady status analytical solution for this linear equation system is given as: u + iv = w = "! sinh[!( z + h)] " x + i" # 2! cosh(! h)! v y dz y cosh(! z) [1 # ] cosh(! h) where α 2 = 2iδ -2 and δ = (2E) 1/2 (E: Ekman number). " L! x! L, " B! y! B

43 Chen-FVCOM Chile Analytic FVCOM

44 ROMs U V W Analytic Be aware that ROMs underestimates u and overestimates w (color bar scales are different for analytical and ROMs solutions). This figure is scanned from Winant s working note.

45 A non-hydrostatic problem: Lock-exchange flow Heavy fluid Light fluid 0.1m 0.8m Model setup: 2 Initially g ' = gδρ / ρ0 = 0.01m / s Vertical 100 sigma layers Horizontal nodes, dx = 0.002(m) no bottom friction and viscosity/diffusivity

46 Chen-FVCOM Chile

47 Is the total energy conservative? Potential Energy (P E ) = L/2! "L/2 H! 0!gz dx dz Kinetic Energy (K E )= L/2! "L/2 H! 0 1 2!V (u2 + v 2 )dx dz Under an inviscid condition, the total energy is conservative at an order of Blue line: PE Red line: KE

FVCOM-NH (DNS results from Härtel et al., 2000) ROMS s results presented by Kanarska et al.

48 Comparison of FVCOM-NH with a high-order direct numerical simulation (DNS) method, with constant horizontal and vertical eddy viscosity and tracer diffusivity m 2 /s: FVCOM-NH (DNS results from Härtel et al., 2000) ROMS s results presented by Kanarska et al. ( 2007) SUNTRAN s results presented by Fringer et al. (2006) with a first-order finite-volume scheme.

49 Internal solitary waves breaking on a linear varying slope ρ 1 ρ 2 h H

50 Photography (Michallet and Ivey,1999) FVCOM-NH

51 Photography (Michallet and Ivey,1999) FVCOM-NH

52 Chen-FVCOM Chile

53 Under shear instability condition 53

54 Under the vortex condition 54

55 55

56 References Chen, C. H. Liu, R. C. Beardsley, An unstructured, finite-volume, three-dimensional, primitive equation ocean model: application to coastal ocean and estuaries. Journal of Atmospheric and Oceanic Technology, 20, Chen, C, R. C. Beardsley and G. Cowles, An unstructured grid, finite-volume coastal ocean model (FVCOM) system. Special Issue entitled Advance in Computational Oceanography, Oceanography, 19(1), Chen, C. H. Huang, R. C. Beardsley, H. Liu, Q. Xu and G. Cowles, A finite-volume numerical approach for coastal ocean circulation studies: comparisons with finite difference models. Journal of Geophysical. Research. 112, C03018, doi: /2006jc Huang, H, C. Chen, G. W. Cowles, C. D. Winant, R. C. Beardsley, K. S. Hedstrom and D. B Haidvogel, FVCOM validation experiments: comparisons with ROMS for three idealized barotropic test problems. J. Geophys. Res., 113, C07042, doi: /2007JC Lai, Z., C. Chen, G. Cowles, and R. C. Beardsley, A Non-Hydrostatic Version of FVCOM, Part I: Validation Experiments, J. Geophys. Res.-Oceans, 115, doi: /2009jc

An Unstructured Grid, Finite-Volume Coastal Ocean Model (FVCOM), Validations and Applications

An Unstructured Grid, Finite-Volume Coastal Ocean Model (FVCOM), Validations and Applications Bob Beardsley, Changsheng Chen, and Geoff Cowles http://fvcom.smast.umassd.edu Data Assimilation in Support