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Lateral Open Boundary Conditions

This section describes how SURF processes lateral open boundary condition data to enable accurate and consistent downscaling from coarse-resolution parent models to high-resolution child models. The process involves two main steps:

  • Boundary Data Extraction: Extracting required ocean variables along the open boundaries of the child domain from the remapped parent model fields.

  • Correction: Adjusting the boundary velocity fields using an integral constraint to ensure volume transport consistency between the parent and child models.

The following subsections detail these stages.

Boundary Data Extraction

After remapping data to the 3D high-resolution grid, SURF extracts the open boundary conditions. These conditions are structured as 1D arrays (or 2D arrays if depth is considered) along the child domain boundary. This procedure generates the following output files:

  • coordinates.bdy.nc: A NetCDF file containing the coordinates for the open boundary points along the child model’s T, U, and V staggered grids.

  • Ocean Variable Boundary Files: A set of NetCDF files containing the ocean variables extracted at the open boundary points. These include:

    • On the T-grid: Temperature, Salinity, and Sea Surface Height (SSH)
    • On the U-grid: Barotropic and baroclinic zonal velocity components
    • On the V-grid: Barotropic and baroclinic meridional velocity components

💡 A Note on Velocity Fields

Parent ocean models — such as those provided by CMEMS — usually provide only the total velocity field, which represents the combined effects of both the barotropic (depth-averaged) and baroclinic (depth-varying) components.

To ensure consistency and accuracy in the child model, SURF performs a velocity decomposition on the interpolated velocity fields:

  • Barotropic Component: The vertically averaged flow, representing the large-scale transport and tidal motion.

  • Baroclinic Component: The residual, depth-dependent structure of the flow, capturing internal dynamics and density-driven stratification.

Integral Constraint at the Open Boundary

The downscaling process in SURF is designed so that the volume transport across the open boundary of the downscaled model matches that across the corresponding section of the parent model. By ensuring that the interpolation process preserves the net transport across the child model's lateral open boundary, SURF avoids any artificial modifications to the total transport across the downscaled model boundary.

At the eastern and western boundaries, the U-points (zonal velocities) are imposed using the following condition:

\[ \int_{y_2}^{y_1} \int_{-H_{child}}^{\eta_{child}} U_{child} \, dz\, dy = \int_{y_2}^{y_1} \int_{-H_{parent}}^{\eta_{parent}} U_{parent} \, dz\, dy \]

Here, \(y_1, y_2\) represent the limits of the open boundary section, \(\eta_{child}, H_{child}\) are the surface elevation and bathymetry at the boundary in the child model, while \(\eta_{parent}, H_{parent}\) represent the surface elevation and bathymetry in the parent model.
The terms \(U_{parent}, U_{child}\) denote the total zonal velocities in the parent and child models, respectively, normal to the western/eastern boundaries.

The corrected velocity component normal to the boundary, \(U_{child}\), is calculated according to the method described in N. Pinardi et al. (2003):

\[ U_{child} (x,y,z,t) = U_{interp} - U_{correction} \]

where \(U_{interp}\) is the \(U_{parent}\) interpolated on the child open boundary points, and the velocity correction is given by:

\[ U_{correction} = \frac{M_{interp} - M_{parent}}{S} \]

In this equation:

  • \(M_{interp} = \int_{y_2}^{y_1} \int_{-H_{child}}^{\eta_{child}} U_{interp} \, dz \, dy\) is the volume transport of the interpolated zonal velocity across the open boundary in the child model.
  • \(M_{parent} = \int_{y_2}^{y_1} \int_{-H_{parent}}^{\eta_{parent}} U_{parent} \, dz \, dy\) is the volume transport of the parent model across the corresponding boundary section.
  • \(S = \int_{y_2}^{y_1} \int_{-H_{child}}^{\eta_{child}} dz \, dy\) is the area of the boundary section in the child model.

These conditions are similarly applied for the meridional velocity component (\(V\)-points) at the northern and southern boundaries.