1. N EW S Lecture 7: Finite-Volume Methods Unlike finite-difference and finite-element methods, the computational domain in the finite-volume methods is divided into many control volumes (CV) and the governing equations are solved in its integral form in individual control volumes. For example: (7.1) Structured grids uDuDuDuD vDvD vDvD 1. Assign the elevation at the center of each rectangular control volume; 2. Define that outflow is positive and inflow is negative; 3. Calculate the net flux

2. Approximation of volume integrals The first order upwind scheme n-1 n n+1 The second order central scheme

3. Consider an arbitrary function like nennw w sw s se e On the east side, for the first order approximation, For the second order approximation, For the fourth order approximation, P Consider an arbitrary function like For the first order approximation For the second order approximation,

4. The fourth order approximation can be obtained by using the bi-quadratic shape funcion: Need 9 coefficients,which can determined by fitting the function to the value of f at 9 locations (nw,w,sw, n, p,s, ne,e,and se). For the cell-centered grids, the value at P point is known,but values at other points must be obtained by interpolation from surrounding cell-centered nodes. Comments; Structured grid finite-volume model is a special type of the finite-difference methods and they always can convert from one to another. So, little efforts need to make to convert a finite- difference model to a finite-volume model under structured grids.

5. 3. Popular unstructured triangular FVM grid in CFD: 1.Cell-centered2.Cell-vertex overlapping3.Cell-vertex median Characteristics of the oceanic motion: ●Free surface----How to calculate accurately the integral form of the surface pressure gradient forcing? ●Steep bottom topography----How to ensure the mass conservation in a two mode model system? ●Open boundary conditions----How to minimize the wave energy reflection at open boundaries?

6. A Grid: All variables ( ,,u,v, , , s..) at nodes Cell vertex median grid Disadvantage: The accuracy of the surface elevation gradient forcing is sensitive to the shape of the control element (due to interpolation) Hard to ensure the mass conservation at open boundaries Advantage: 1) Simple 2) Guarantee the mass conservation for tracers Cell-centered B Grid: All variables ( ,,u,v, , , s..) at centroids Advantage: 1) Simple 2) Better to advection calculation Disadvantage: Hard to guarantee The accuracy of the surface elevation gradient forcing Hard to ensure the mass conservation at open boundaries Hard to ensure the mass conservation for tracer calculation C Grid: , , , s, K m, K h … Advantage:  Combine the best of A and B Grids;  Easy to ensure the mass conservation for tracers  Easy to introduce the mass conservative open boundary conditions FVCOM Cell-vertex-centered

7. FVCOM (Unstructured Grid, Finite-Volume Coastal Ocean Model) Hydrostatic and free surface primitive equation ocean models; Mellor and Yamada turbulence model for vertical mixing and Smagorinsky eddy parameterization for horizontal mixing, recently upgraded to include general ocean turbulence models (GOTM). Sigma transformation in the vertical direction, and unstructured triangular grids in the horizontal plane; Two mode approaches: barotropic 2D mode and baroclinic 3D mode; with second order approximation; Exact form of the no flux boundary conditions on the slope; Flooding/drying processes; Multi-choices of the radiation open boundary conditions; NPZ, multi-species NPZD, sediment suspension model and water quality model.

8. The  -transformation coordinates: The governing equations in the  -transformation coordinate:  = 0;,  = 0; The boundary conditions: at  = 0 at  = -1

9. The 2-D (vertically integrated) equations: Two-mode (external and internal) approach: The 2D equations: solve for  The 3D equations: solve for u, v, w, , s, K m, K h

10. Turbulence Closure Submodels 1. Horizontal diffusion coefficient: A Smagorinsky eddy parameterization method where C : a constant parameter;  u : the area of the individual momentum control element a) for momentum: b) for tracers: where   : the area of the individual tracer control element; : the Prandtl number.

11. 0 z l n  The surface and bottom boundary conditions for temperature are: The absorption of downward irradiance is included in the temperature (heat) equation in the form of The surface and bottom boundary conditions for salinity are: The kinematic and heat and salt flux conditions on the solid boundary:

12. The 2-D finite-volume discrete approach: 1. The surface elevation: u,vu,v u,vu,v u,vu,v u,vu,v u,vu,v u,vu,v  x determined by a least-square method based on velocity values at the four cell centered points The second-order accuracy upwind scheme (Kobayashi, 1999) to calculate the momentum and volume fluxes 2. Velocity: y

13. Time integration : The modified fourth-order Runge-Kutta time-stepping scheme. This is a multi- stage time-stepping approach with second-order accuracy. ; and ; k =1,2,3,4 The area of the tracer element (TCE) The area of individual triangle (momentum element) (MCE)

14. The 3-D Internal Mode: Two step: 1.Use either explicit scheme or second-order accuracy Runge-Kutta time stepping scheme to calculate “immediate” velocities by solving the momentum equations 2. Use the implicit scheme to solve the equation including vertical diffusion terms The implicit method used to calculate the vertical diffusion is the same as POM/ECOM-si

15. The calculation of the tracer advection terms: u,vu,v u,vu,v u,vu,v u,vu,v u,vu,v u,vu,v s s s s ss s a) The second-order upwind scheme: b) The first-order central scheme ?? (The very early version of the FVCOM code uses this approximation) (The current version of the FVCOM code uses this approximation) s s s c) The working-on scheme: Keep the upwind scheme as (a) but using the least square to determine the tracer distribution in the big area with inclusion of the central point. This could allow a highly order accuracy scheme.

16. Transport Consistency of External and Internal Modes: tEtE tItI Internal mode: To conserve the volume on the ith TCE, the vertically-integrated form of the above equation must satisfy Since the sea level is calculated through the vertically integrated continuity equation over I split external time steps, then the above condition requires

17. Mass Conservative Open Boundary Conditions:   ,,  FAFA FBFB FCFC  Case 1: The sea level at the open boundary is specified (tidal elevation) Step 1: Step 2: Determine the tracer flux by Step 3: Apply a radiation condition for the perturbation of the tracer such as Case 2: The sea level at the open boundary is unknown Step 1: Apply a radiation condition to determine the sea level at the open boundary Step 2: Repeat steps 1-3 listed in Case 1

18. ,  u,v ,  u,v Recommended shapes of triangular grids at open boundaries: Advantage: Avoid the numerical error due to interpolation Note: For a region in which the topography varies abruptly at the open boundary, a sponge layer is always recommended in these area after the above treatment.

19. Methods to Add the Discharge from the Coast or a River : Method 1: The TCE Method---Inject the water into the tracer control element (TCE) u,vu,v SoSo SS SS ii jj u,vu,v  k u,vu,v is the water volume transport into a TCE with an area of and a depth of D The vertically averaged velocity caused by Q equals to is the angle of the coastline relative to the x direction. The contributions of Q to the x and y vertically-integrated momentum equation in the MCE with an area of or are given as For the external mode: For the internal mode: is the percentage of Q in the kth sigma layers which satisfy the condition of

20. Methods to Calculate the Salinity at the Freshwater Discharge Node Point 1) Specified: The value of the salinity is specified by users. 2) Calculated: The value of the salinity at the freshwater discharge node point is calculated using the salinity equation as follows: is the velocity component normal to the boundary of the TCE in the kth sigma layer is the salinity at nodes of triangles connecting to the coastal node of the TCE is the horizontal and vertical diffusion terms in the salinity equation is the salinity contained in the water volume of Q

21. Method 2: The MCE Method ---Inject the water through the boundary edge The vertically averaged velocity caused by Q: The x and y components of the velocity in the kth sigma layer in the MCE are given as The contributions to the internal x and y momentum equations of the MCE in the kth sigma layer are equal to The transport inject to the TCEs that include the river edge: And then, the sea level at the node point is calculated by

22. Criterions for horizontal resolution or time step: Considering a case with no vertical and horizontal diffusion, the vertical integral form of the salinity equation with a point freshwater discharge can be written as Let us start with a simple first-order time integration scheme: Including the continuity equation, yield: For a freshwater discharge case, For mass conservation, or

23. m 3 /s, D ~ 10 m, and For example:, L ~ 100 m

24. 0 z H  hBhB , S u, v The wet/dry point treatment: Criterion for node points: Criterion for triangular cells: Mass conservative!

25. , Critical issues: How to ensure the mass conservation of the individual TCE in which the moving boundary is crossed? During the I split external mode time steps, due to drying of nodes or triangles, In this case, the external and internal mode adjustment can not guarantee that  reaches zero at  =-1. An additional adjustment of  is added in the  equation Note: This adjustment works in general but fails in the case where  =  -D min <10 -1 if I split =1

26. The key parameters controlling the mass conservation: QS. How could we estimate these constraints? Conduct a sensitivity study using the model with an simple geometric river with inclusion of intertidal zones. Criterion to define the wet/dry points; Time step used for numerical integration; Horizontal and vertical resolutions of model grids; Amplitudes of surface elevation; Bathymetry.

27. Wind Stress, Heat Flux, Precipitation/Evaporation and Tidal Forcing a) Wind stress: 1) Calculated directly in the model (time and spatial dependent or uniform) 2) Input directly from the output of MM5 or other meteorological models b) Heat flux: 1) Specified by users (net flux and short-wave radiation) 2) Input directly from the output of MM5 or other meteorological models c) Precipitation/Evaporation: The code has never been used! d) Tidal forcing: the mean elevation relative to a water level at rest the amplitude of the ith tidal constituent the frequency of the ith tidal constituent the phase of the ith tidal constituent the total number of tidal constituent Six tidal constituents are 1)S 2 tide (period = 12 hours); 2)M 2 tide (period = 12.42 hours); 3)N 2 tide (period = 12.66 hours); 4)K 1 tide (period = 23.94 hours); 5)P 1 tide (period = 24.06 hours), 6)O 1 tide (period = 25.82 hours).

28. The Data Assimilation Methods 1)Nudging method: Directly adopted from the weather forecast model (MM5) Simple, computational efficiency and practical to the forecast application 2) The OI method: This method will be added into the Fortran 90 code this summer Simple, statistical meaningful, computational efficiency and practical to the forecast application, but requires covariance data. 3) The adjoint method: This method will be added by collaborating with Carl Wunsch at MIT Better to sensitive studies, but requires huge computational times for iteration. 1.Sea surface SST 2.Current mooring data at fixed locations 3.Hydrographic data in the 3-D domain

29. The 3-D Lagrangian Particle Tracking FVCOM has two online (run with the code) and offline Lagrangian particle tracking programs by solving a nonlinear system of ordinary differential equations (ODE) as follows This equation is solved by using 4 th order explicit Runge-Kutta (ERK) multistep methods 1.An individual-based zooplankton and fish larvae model has been developed on this Lagrangian frame. 2.Vertical and horizontal random walk processes have been included (Chen et al., 2004)

30. Mortality N Uptake PZ Grazing Egestion Figure 8.1: Schematic of the food web loop of the NPZ model The Nutrient-Phytoplankton-Zooplankton Model:

31. Figure 8.2: Schematic of the lower trophic level food web model for a phosphorus-controlled ecosystem system. The NPZDB Model

32. Figure 8.3: Schematic of the flow chart of the multi-species NPZD model. The NPZD Model

33. The Water Quality Model

34. Schematic of the water quality model in the sediment layer.

35. The On-going Improvement 1. Add GOTM modules into the current FVCOM version 2 to provide multiple choices for vertical mixing ; 2. Add the OI data assimilation method (which we had for the old version) to the current FVCOM version 2 3. Add the high-order advection scheme for tracer equations 4. Test the off-line biological models The future Improvement 1.Convert FVCOM to the non-hydrostatic version 2.Add adjoint data assimilation to FVCOM for both online and offline 3.Convert SWAN to unstructured grids and then couple into FVCOM The Final Goal Build a atmospheric-ocean coupled community model system.

36. Model Validation 1.Advection Scheme 2.Wind-induced oscillation; 3.Wind-induced waves over the slope bottom topography; 4.Tidal Resonances in semi-enclosed channel; 5.Freshwater discharge plume; 6.Bottom boundary layer over a step bottom slope 7.Equatorial Rossby soliton 8.Flooding/drying process 9.Dye Experiments

37. 1. Advection Scheme and C = 1 Upwind finite- difference stream Central finite- difference stream FVCOM finite-volume flux scheme

38. Upwind finite- difference stream Central finite- difference stream FVCOM finite- volume flux scheme

39. r o =67.5 km d=75 m Wind Linear, non-dimensional equations: Wind-induced oscillation where and Solution: Wind is suddenly imposed at initial Reference: Csanady ( 1968) Birchfield (1969)

40. FVCOM POM

41. Water elevation Alongshore transport

44. Birchfield and Hickie (1977) JPO Radial mode: k=1, 2: gravity waves, k=3: topographic wave AnalyticalFVCOM (5 km)POM (2.5 km) 1 h 1 d

45. FVCOM (5 km)POM (2.5 km) 5 days  : Elevation : Current

48. Tidal Resonance in A Rectangular Channel HoHo L L1L1 B.W O.B. B x z Solution: 1) Normal case:2) Near-resonance case:

50. Normal case: FVCOM POM ECOM-si Analytical Computed

51. Near-resonance case FVCOM POM ECOM-si

52. Y Z 400 100 m 10 0 X 400 300 m Freshwater discharge Case 1: Continental shelfCase 2: Circular lake shelf

53. Sea LevelSalinity FVCOM POM FVCOM POM

54. Central difference scheme Upward difference scheme

56. 4.2 2.0 0.9  x (km)

58. Background mixing coefficient K m = 10 -4 m 2 /s The Bottom Boundary Layer Bottom boundary condition : Slope:

59. FVCOMPOM10 days Interval: 0.1 Interval: 0.2

60. FVCOMPOM 10 days 5 days

61. Interval: 0.05 U (cm/s) W (cm/s) Vertical velocity is one order of magnitude smaller!

62. Equatorial Rossby Soliton  x=0.125 Non-dimensional grid size (scaled by the wave length) Zero-order analytical solution Model-calculated full solution

63. 10 m 3 km2.3 km 30 km A simple estuarine model used to test the model sensitivity FVCOM: Two time steps: T E : External mode; T I : Internal mode

65.  = 4  10 -4  = 7  10 -4  = 9  10 -4 Figure 3.10: The model-predicted relationship of the ratio of the internal to external mode time steps () with the tidal forcing amplitude () and the bathymetric slope of the inter-tidal zone (). In these experiments, = 4.14 sec, kb = 6, = 5 cm.

66.  = 4  10 -4  = 7  10 -4  = 9  10 -4 Figure 3.11: The model-derived relationship of Isplit with D min for the three cases with = 4.0  10 -4, 7.0  10 -4 and 9.0  10 -4. In the three cases, = 4.14 sec, kb = 6, and = 1.5 m. kb=11 kb=6 Figure 3.12: The model-derived relationship of Isplit with D min for the two cases with kb = 6 and 11, respectively. In these two cases, = 4.14 sec, = 4.0  10 -4, and = 1.5 m.

67.  L = 600 m  L = 300 m  t E =2.07 sec  t E =4.14 sec Figure 3.13: The model-derived relationship of with (upper panel) and (lower panel). In the upper panel case, = 4.14 sec, = 7.0  10 -4, kb =6, and = 1.5 m. In the lower panel case, = 300 m, = 7.0  10 -4, kb =6, and = 1.5 m.

68. The 1999 Dye Experiment (carried by Bob Houghton)

69. 4km grid2km grid ` At 16: 33 GMT, 22 May, 1999

70. 0.25 km grid0.5 km grid At 16: 33 GMT, 22 May, 1999

76. 2 1: 3:59 GMT, 23 May 2: 11: 43 GMT, 23 May 3: 15:22 GMT, 23 May 4: 19:24 GMT, 23 May 5: 21:34 GMT, 23 May 6: 00:53 GMT, 25 May Observed Simulated Assimilation with T and S 1 3 4 5 6

77. Trajectory of a near- bottom particle Trajectory of a near- surface particle Trajectory of the center of the vertically averaged model dye concentration. Simulated 0.5 kmAssimilation 0.5 km

81. 0 1 2 3 4 5 Lapse times (days)

82. End here! More model validations will be given by Dr.Wang in the next lectures