Tidal flux and resonance in the Gulf of Tonkin

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Description of the Regional Ocean Modeling System (ROMS)

ROMS (Shchepetkin and McWilliams, 2005) is an evolutionary descendent from the S-Coordinate Rutgers University Model (SCRUM: Song and Haidvogel, 1994), which started as a joint venture between UCLA and Rutgers University oceanographers. From this partnership, two branches of ROMS have emerged. A third version branched from UCLA: ROMS_AGRIF. This version is maintained jointly by IRD (P. Marchesiello, P. Penven and G. Cambon) and INRIA (L.
Debreu), which are French institutes working on environmental sciences and applied mathematics. The main originality of this version is its 2 way nesting capability but is otherwise close to the UCLA version regarding the numerical kernel. The model evolution still benefit from frequent exchanges with UCLA (J.
McWilliams, A. Shchepetkin, F. Lemarié, J. Molemaker and others) and with a growing body of developers and users of the model. ROMS is briefly presented here with references for further details on numerical schemes and physical parameterizations. Additional information can be found in a series of papers and documents available on the ROMS_AGRIF web site: http://www.romsagrif.org (in the documentation section).
ROMS solves the primitive equations in an Earth-centered rotating environment, based on the Boussinesq approximation and hydrostatic vertical momentum balance. It is discretized in coastline- and terrain-following curvilinear coordinates using high-order numerical methods. ROMS is a split-explicit, free-surface ocean  model, where short time steps are used to advance the surface elevation and barotropic momentum, with a much larger time step used for temperature, salinity, and baroclinic momentum. The model has a 2-way, time-averaging procedure for the barotropic mode, which satisfies the 3D continuity equation. The specially designed 3rd-order predictor-corrector time step algorithm allows a substantial increase in the permissible time-step size and a reduction of small-scale   numerical dispersion and diffusion. The complete time stepping algorithm is  described in Shchepetkin and McWilliams (2005). Associated with the 3rd-order time stepping, a 3rd-order, upstream-biased advection scheme allows the generation of steep gradients, enhancing the effective resolution of the solution for a given grid size (Shchepetkin and McWilliams, 1998). Because of the implicit diffusion in the upstream-biased advection scheme, explicit lateral viscosity is not needed in ROMS, except in sponge layers near the open boundaries where it increases smoothly close to the lateral open boundaries. For tracers, a 3rd-order upstream-biased advection scheme is also implemented for tracers but the diffusion part of this scheme is rotated along isopycnal surfaces to avoid spurious diapycnal mixing and loss of water masses (Marchesiello et al., 2009; Lemarié et al., 2012). A non-local, K-profile planetary (KPP) boundary layer scheme (Large et al., 1994) parameterizes the unresolved physical vertical subgrid-scale processes at the surface, bottom and interior of the ocean. If a lateral boundary faces the open ocean, an active, implicit, upstream biased, radiation condition connects the model solution to the surroundings (Marchesiello et al., 2001). ROMS also include an accurate pressure gradient algorithm (Shchepetkin and McWilliams, 2003) and a variety of features including air-sea bulk formulae, surface and benthic boundary layers. The model was therefore developed to simulate both coastal and oceanic regions and their interactions. Its computational methods allow for realistic, long-term integrations in a fine-mesh regional domain. ROMSTOOLS (Penven et al., 2008) is a collection of global data sets and a series of Matlab programs collected in an integrated toolbox, developed for generating the grid, the surface forcing, initial conditions, open boundary conditions and tides for climatologically and inter-annual ROMS ocean simulation.

Open boundary conditions

Open boundary conditions are more problematic because they constitute an artificial procedure destined to confine the computational domain within the region of interest. It saves computational resources but must be used with care to limit the propagation of boundary condition errors to the interior domain (Marchesiello et al., 2001; Blayo and Debreu, 2005). In ROMS, a combined set of passive (radiative) and active (relaxation to forcing data) open boundary conditions are provided for u, v, T, S and ! , following the method of Marchesiello et al. (2001).
Coastal modeling requires well-behaved, long-term solutions for configurations with open boundaries. A numerical boundary scheme should allow the inner solutions to radiate through the boundary without reflection and information from the surrounding ocean to enter the model. The active open boundary scheme implemented in ROMS estimates the two dimensional horizontal phase velocities in the vicinity of the boundary. For each model variable ! , the normal (cx) and tangential (cy) phase velocities are: c x = ! « ! « t « !  $ % & ‘ ( 2.

Sigma-coordinate errors: The pressure gradient and diffusion terms

The major advantage of sigma coordinate models is the transformation of the surface and sea bottom to coordinate surfaces. Unfortunately, this is also the source of their major disadvantage: the well-known pressure gradient error that is most acute on steep continental slopes.
In sigma-coordinate, the x-component of the pressure gradient force is determined by: !p !x z=cst = !p !x ! =cst  » ! h !p !! !h !x (2.27).
The first term of the right hand side involves the variation of pressure along a constant ! -surface and the second is the hydrostatic correction. Near steep topography, these 2 terms are large, comparable in magnitude and tend to cancel each other. A small error in computing either term can result in a relatively large error in the resulting horizontal pressure gradient. Beckman and Haidvogel (1993) show that this error is proportional to the slope number: the ratio of topographic variation and topographic depth. They found empirically that this ratio must be lower than a certain threshold (|h(i+1)-h(i)/(h(i+1)+h(i)|) < 0.2; i is the x grid index) to yield acceptable errors (erroneous currents lower than about 1 cm/s).
Another source of pressure-gradient error is associated with vertical resolution and is called hydrostatic inconsistency (Haney, 1991). Consistency requires a limit to vertical resolution (given a certain horizontal resolution) but this limit depends on the pressure gradient scheme (Barnier et al., 1998).
The scheme implemented in ROMS is the formulation proposed by Shchepetkin and McWilliams (2003), based on high order density profile reconstruction and rewriting of the equation of state to reduce spurious water compressibility effects on the pressure gradient. This formulation has been designed to minimize truncation errors and is one of the most successful among the large list of schemes proposed in the literature.

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Lagrangian modeling: Ariane

ARIANE (Blanke et al., 1997) is developed in the “Laboratoire de Physique des Oceans” (LPO) in Brest by Bruno Blanke and Nicolas Grima. It is a code that temporally integrates the velocity field to compute trajectories. The package contains the FORTRAN 95 code and Matlab tools for graphic outputs and streamfunction calculation. ARIANE computes trajectories from the 3-dimensional Eulerian velocity field. For that, the velocity field must be interpolated at each particle location along their trajectory. The algorithm is detailed in Blanke and Arhan (1999). It takes advantage of the C-grid used to discretize ROMS equations and provides trajectories for a given velocity field through the computation of 3- dimensional streamlines. The 3 components of the velocity field are known over the six faces of each cell. The non-divergence of the velocity field ensures continuous trajectories within the cell.
ARIANE can be used in two different modes, qualitative and quantitative:
• The qualitative mode uses individual particles to display realistic trajectories. This mode is useful to simulate the trajectory of individual Lagrangian drifters or buoys and assesses the connectivity between various locations. The maximum duration of the particle drift corresponds to the duration of stored data. To initialize the particles in the domain, ARIANE  uses five parameters, three spatial ones, a temporal one and a last parameter read but not used in qualitative mode (it is present for consistency with the quantitative mode inputs). The spatial parameters are given in number of cell grid, i.e. we do not indicate longitude, latitude and depth but the corresponding grid cell number.
• The quantitative mode uses thousands or more particles to compute a Lagrangian streamfunction and determine Lagrangian transports. Th streamfunction is computed on the horizontal plan of a non-divergent flow, diagnosed by the movement of particles each associated with a weight (volume of water in m3). In this mode, particles are initialized on a section defined by longitude, latitude and depth limits and intercepted by the same section or different ones. The ensemble of sections must form a closed domain to ensure that a quantitative diagnostic is possible.

Harmonic tidal analysis: Detidor

Detidor is a post-processing tool developed at LEGOS for treatment of tidal signals in time series of model and observational data. One of the primary uses of Detidor calculations is to separate out the tidal from the non-tidal components by performing a harmonic tidal analysis. Harmonic analysis is most useful for the analysis and prediction of tidal elevations and currents. The use of this technique for tides appears to have originated with Lord Kelvin (1824-1907) around 1867. A discussion of tidal harmonic analysis can be found in Admiralty manual of Tides (Doodson and Warburf, 1941) and Godin (1972). The harmonic method of analysis treats every tidal record as consisting of a sum of harmonic constituents of known frequency. At time t, the harmonic expression of the ocean tidal height at location can be written as: ! (« ,#, t) = H k k ! (« ,# )cos[$k (t)+ %k « G(« ,# )] (2.28) where , are the unknown amplitude and Greenwich phase lag of tide k at location , and and are the amplitude and phase corrections. To avoid the singularities of the amplitude at the amphidromes, it is more common to give equation 2.28 the following form with cosine and sine functions: « (#,$, t) = [C k k % (#,$)cos(&k (t) + ‘k ) + S k (#,$)sin(&k (t) + ‘k )].

Table of contents :

Chapter 1: Description of the Gulf of Tonkin
1.1 Geography
1.2 Tidal forcing
1.3 Meteorological conditions
1.3.1 Winds
1.3.2 Precipitation
1.3.3 Surface heat flux
1.4 Ocean Climatology
1.4.1 Sea Surface Temperature patterns
1.4.2 Subtidal circulation
Chapter 2: Methods
2.1 Description of the Regional Ocean Modeling System (ROMS)
2.1.1 Equations of continuity and momentum balance
2.1.2 Vertical boundary conditions.
2.1.3 Terrain-following coordinate systems
2.1.4 Horizontal curvilinear coordinates
2.1.5 Open boundary conditions
2.1.6 Time-stepping
2.1.7 Sigma-coordinate errors: The pressure gradient and diffusion terms .
2.1.8 Turbulent closure
2.1.9 Nesting
2.2 Lagrangian modeling: Ariane
2.3 Data for model verification
2.3.1 Tide gauges
2.3.2 Altimetry data
2.4 Harmonic tidal analysis: Detidor
2.5 Tidal energy budget: COMODO-energy
Chapter 3: Model validation and sensitivity analysis in the Gulf of Tonkin .
3.1 Previous modeling work
3.2 Model setup
3.2.1 Grid generation
3.2.2 Surface fluxes
3.2.3 Initial and boundary conditions
3.3 Model validation
3.3.1 Tidal gauges
3.3.2 Satellite altimetry
3.4 Model sensitivity
3.4.1 Sensitivity to bottom stress formulation
3.4.2 Sensitivity to the horizontal resolution
3.4.3 Comparison of two- and three-dimensional model solutions
3.4.4 Sensitivity to bathymetry
3.5 Conclusion
Appendix
A.1 Comparison of altimeter and tide gauge measurements
A.2 Relative errors
A.3 OTIS forcing
Chapter 4: Tidal flux and resonance in the Gulf of Tonkin
4.1 Tidal energy flux
4.2 Tidal resonance
4.2.1 The rectangular bay model
4.2.2 Numerical simulations
Chapter 5: Residual transports
5.1 Tide-induced residual current and transport
5.1.1 Eulerian residuals
5.1.2 Tide-induced Lagrangian residual current
5.2 Subtidal residual flow
5.2.1 Wind-driven circulation
5.2.2 Density circulation
5.3 Connectivity to Ha-Long bay
5.4 Conclusion
Chapter 6: Heat budget
6.1 Coastal cooling in winter
6.2 Frontogenesis in spring-summer
Conclusion
References 

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