Coordinate Systems

ADCIRC can operate in different coordinate systems depending on the application requirements and domain scale. This section describes the coordinate transformations and implementations in the model.

Spherical Coordinates

For large domains where the Earth’s curvature is significant, ADCIRC uses a spherical coordinate system:

  • Longitude (\(\lambda\)): Angular distance east or west from the Prime Meridian

  • Latitude (\(\phi\)): Angular distance north or south from the equator

  • Vertical coordinate (\(z\)): Distance above or below the reference geoid

The governing equations presented in the previous sections are formulated in spherical coordinates for global and regional applications.

Cartesian Coordinates

For smaller domains where Earth’s curvature effects are negligible, ADCIRC can operate in a Cartesian coordinate system:

  • \(x\): Eastward distance

  • \(y\): Northward distance

  • \(z\): Vertical distance from reference level

The continuity equation in Cartesian coordinates is:

\[\frac{\partial \zeta}{\partial t} + \frac{\partial (UH)}{\partial x} + \frac{\partial (VH)}{\partial y} = 0\]

The momentum equations in Cartesian coordinates are:

\[\frac{\partial U}{\partial t} + \frac{\partial}{\partial x} \left( \frac{U^2}{H} \right) + \frac{\partial}{\partial y} \left( \frac{UV}{H} \right) - fV + g \frac{\partial \zeta}{\partial x} + \frac{\tau_{sx}}{\rho_0 H} - \frac{\tau_{bx}}{\rho_0 H} - \frac{1}{\rho_0 H} \frac{\partial p_s}{\partial x} + \frac{\partial}{\partial x} \left[ \frac{2 N_x}{H} \frac{\partial U}{\partial x} \right] + \frac{\partial}{\partial y} \left[ \frac{N_y}{H} \left( \frac{\partial U}{\partial y} + \frac{\partial V}{\partial x} \right) \right] = 0\]
\[\frac{\partial V}{\partial t} + \frac{\partial}{\partial x} \left( \frac{UV}{H} \right) + \frac{\partial}{\partial y} \left( \frac{V^2}{H} \right) + fU + g \frac{\partial \zeta}{\partial y} + \frac{\tau_{sy}}{\rho_0 H} - \frac{\tau_{by}}{\rho_0 H} - \frac{1}{\rho_0 H} \frac{\partial p_s}{\partial y} + \frac{\partial}{\partial x} \left[ \frac{N_x}{H} \left( \frac{\partial U}{\partial y} + \frac{\partial V}{\partial x} \right) \right] + \frac{\partial}{\partial y} \left[ \frac{2 N_y}{H} \frac{\partial V}{\partial y} \right] = 0\]

Coordinate Transformations

For regional applications, ADCIRC often employs a conformal map projection to transform between spherical and projected Cartesian coordinates. The most commonly used projections are:

  1. Mercator Projection: * Preserves angles (conformal) * Distorts areas, especially at high latitudes * Transformation equations:

    \[x = R \lambda\]
    \[y = R \ln\left[\tan\left(\frac{\pi}{4} + \frac{\phi}{2}\right)\right]\]

  2. Lambert Conformal Conic: * Preserves angles * Minimizes distortion in mid-latitudes * Commonly used for regional modeling in mid-latitudes

  3. Stereographic Projection: * Conformal projection * Useful for polar regions * Minimal distortion near the projection center

CPP Coordinate System

For some specialized applications, ADCIRC uses a Cartesian Coordinates with Polar Projection (CPP) system. This hybrid approach:

  • Maintains the simplicity of Cartesian equations

  • Accounts for Earth’s curvature through carefully designed projections

  • Applies correction factors to the Coriolis and other terms

The CPP coordinate system is defined by:

\[x = (R + h) \cos \phi \sin(\lambda - \lambda_0)\]
\[y = (R + h) [\sin\phi\cos\phi_0 - \cos\phi\sin\phi_0\cos(\lambda - \lambda_0)]\]

where \((\lambda_0, \phi_0)\) are the coordinates of the projection origin.

Vertical Coordinate Systems

ADCIRC employs several vertical coordinate systems:

  1. Sigma Coordinates: * Terrain-following coordinates that map the water column to a uniform layer * \(\sigma = \frac{z - \zeta}{\zeta + h}\) ranges from 0 (surface) to -1 (bottom) * Advantages include natural handling of bathymetry

  2. Z-level Coordinates: * Fixed vertical levels * Typically used in deeper waters * Provides better representation of stratification

  3. Hybrid Systems: * Combination of sigma and z-level approaches * Optimizes advantages of both systems