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Solution of the Vertical Diffusion Equation
The solution procedure in HYCOM follows the procedure used by Large et al. (1994) and is used for all three Reynolds stress vertical mixing models (KPP, GISS, MY). Decomposing model variables into mean (denoted by an overbar) and turbulent (denoted by a prime) components, the vertical diffusion equations to be solved for potential temperature, salinity, and vector momentum are
Boundary layer diffusivities and viscosity parameterized as follows:
where the γ terms represent nonlocal fluxes. For example, the KPP model includes nonlocal terms for θ and S, but not for momentum. The following solution procedure is valid for any mixing model in HYCOM that calculates the diffusivity/viscosity profiles at model interfaces, whether or not nonlocal terms are parameterized.
The following ma trix problems are formulated and solved:
where superscripts t,t+1 denote model times, and M is the vector of a momentum component, either u or v. The matrices A are tri-diagonal coefficient matrices, while the vectors HT and HS represent the nonlocal flux terms. Given Κ model layers with nonzero thickness, where an individual layer k of thickness δpk is bounded above and below by interfaces located at pressures pk and pk+1, the matrix AS is determined as follows:
with
where pk+0.5 represents the central pressure depth of model layer k. The nonlocal flux arrays are calculated using
The solution is then found by inverting the tri-diagonal matrix A. The matrix problems are formulated and solved in the same manner for potential temperature and momentum components.
Large, W. G., J. C. Mc Williams, and S. C. Doney, 1994: Oceanic vertical mixing: a review and a model with a nonlocal boundary layer parameterization. Rev. Geophys. 32, 363-403.