Pressure Gradient

The pressure gradient term in the momentum equation represents the force per unit mass due to horizontal variations in pressure and geopotential. This term is essential for capturing the dynamics of ocean circulation, including both barotropic and baroclinic motions.

Physical Background

In the layered non-Boussinesq momentum equation solved in Omega, the pressure gradient tendency for each edge and layer includes three contributions:

  1. Montgomery potential gradient: The horizontal gradient of the Montgomery potential (\(\alpha p + g z\)), averaged across the top and bottom interfaces of each layer. The Montgomery potential combines the pressure gradient and the geopotential, and its gradient along coordinate surfaces accounts for both the direct pressure force and the effect of tilted layer interfaces that arise when using a general vertical coordinate.

  2. Specific volume correction: A correction term proportional to the gradient of specific volume (inverse density) at each edge. This term ensures that horizontal density variations between the two cells sharing an edge are properly represented in the pressure gradient force.

  3. External geopotential forcing: Contributions from the tidal potential and the self-attraction and loading (SAL) terms. These represent gravitational forcing from astronomical tides and the deformation of the solid Earth and ocean surface in response to the ocean mass distribution. These terms are currently set to zero and will be provided by a future tidal forcing module.

Configuration Options

The pressure gradient method is configured in the input YAML file under the PressureGrad section:

PressureGrad:
   PressureGradType: 'centered'

Available Methods

Centered Difference ('centered' or 'Centered')

  • Computes the pressure gradient using a centered finite-difference approximation of the Montgomery potential gradient and specific volume correction

  • Suitable for global ocean simulations without ice shelf cavities

  • Default and currently the only fully implemented option

High-Order ('HighOrder1')

  • Placeholder for a future high-order pressure gradient method based on volume integral formulations

  • Intended for simulations with ice shelf cavities and steep bathymetry where the centered scheme may be inaccurate

  • Not yet implemented; selecting this option produces zero pressure gradient tendency

Dependencies

The pressure gradient calculation requires the following Omega components to be initialized first:

  • Horizontal Mesh: provides mesh geometry including distances between cell centers and edge connectivity

  • Vertical Coordinate: provides pressure at layer mid-points and interfaces, interface heights (\(z\)), and geopotential

  • Equation of State: provides the specific volume field

  • Ocean State: provides the current layer thicknesses