Physics Overview

GOTM models a one-dimensional water column — the vertical dimension only. Horizontal gradients of velocity, temperature, and salinity can be prescribed as external forcing, but are not computed. This is appropriate for open-ocean and lake sites where horizontal homogeneity is a reasonable assumption over the simulation period.

Governing Equations

The state vector of a GOTM column consists of:

  • Horizontal velocity components \(U(z,t)\) and \(V(z,t)\)

  • Potential temperature \(\theta(z,t)\)

  • Salinity \(S(z,t)\)

  • Turbulent kinetic energy \(k(z,t)\) and a second turbulence quantity (\(\varepsilon\), \(\omega\), or a length-scale \(l\))

Each quantity is governed by a 1D transport equation of the generic form

\[\frac{\partial \phi}{\partial t} = \frac{\partial}{\partial z}\left(\nu_\phi \frac{\partial \phi}{\partial z}\right) + S_\phi\]

where \(\nu_\phi\) is the appropriate eddy diffusivity and \(S_\phi\) collects all remaining source/sink terms (Coriolis, pressure gradients, radiation, etc.).

Vertical Grid

GOTM uses a staggered finite-difference grid with \(N_\mathrm{lev}\) layers:

  • Scalar quantities (\(\theta\), \(S\), \(k\), \(\varepsilon\)) are located at layer centres (indices \(1, \dots, N_\mathrm{lev}\)).

  • Fluxes and diffusivities are located at layer interfaces (indices \(0, \dots, N_\mathrm{lev}\)).

  • Layer thicknesses \(h_i\) are uniform by default but can be configured.

Staggered vertical grid

Figure 1 — Staggered vertical grid. Filled circles mark cell interfaces (indices \(i = 0, \dots, N\)) where turbulent quantities (\(k\), \(\varepsilon\), \(\nu_t\), \(\kappa_t\)) are stored. Open squares mark cell centres (indices \(i = 1, \dots, N\)) where mean-flow quantities (\(U\), \(V\), \(\theta\), \(S\)) are stored. Layer thickness \(h_i\) connects adjacent interfaces. (After GOTM manual Fig. 1, p.25.)

Solution Sequence

Each timestep follows this sequence:

  1. Coriolis rotation — rotate \((U,V)\) by angle \(f\Delta t\) (Section 3.2.4).

  2. External pressure gradient — add the depth-uniform barotropic pressure gradient (Section 3.2.7).

  3. Internal pressure gradient — add the baroclinic pressure gradient from the density field (Section 3.2.8).

  4. U/V momentum equations — advance \(U\) and \(V\) with implicit vertical diffusion (Sections 3.2.5–3.2.6).

  5. Ice thermodynamics — update ice cover, thickness, albedo, and transmissivity; diagnose the ocean–ice heat flux that modifies the temperature upper boundary condition (see Ice Thermodynamics).

  6. Temperature equation — advance \(\theta\) including short-wave radiation extinction with the ice-modified albedo and transmissivity (Section 3.2.10).

  7. Salinity equation — advance \(S\) (Section 3.2.11).

  8. Equation of state — update density \(\rho\) and buoyancy frequency \(N^2\).

  9. Shear frequency — compute \(M^2\) from the updated velocity field (Section 3.2.13).

  10. Turbulence closure — advance \(k\) and the second turbulence quantity; update stability functions and diffusivities (Chapter 4).

All section references are to the GOTM manual (Umlauf, Burchard & Bolding).

When FABM biogeochemistry is enabled, the physics loop above runs for a chunk of timesteps (default: ~1 day), storing snapshots of \(T\), \(S\), \(\rho\), \(h\), \(\nu_h\), and radiation. The biogeochemical engine then steps through those snapshots at the same \(\Delta t\). See Biogeochemistry and FABM Coupling for the full description of the coupled loop.

See Also