Surface ocean physical processes (Thorpe, 1995) How to model all this ?
Basic physical principles: conservation of volume (incompressibility) conservation of mass (water, salt, …) conservation of momentum (velocity) conservation of angular momentum conservation of total energy (mechanical & thermodynamic) plus material laws for water (viscosity, …) gives Dynamic equations for momentum, heat, salt, …
These dynamic equation are valid on all scales, and all these scale are relevant. Problem: in numerical models, we cannot resolve from millimeter to kilometer. Therefore, equations are statistically separated into mean (expected) and fluctuating part (Reynolds decomposition):
The Reynolds decomposition allows to derive dynamic equations for mean-flow quantaties, but … … new (unknown) terms are introduced, the turbulent fluxes: Reynolds stresses Turbulent heat flux Turbulent salt flux Eddy viscosity Eddy diffusivity
The TKE equation Reynolds stresses cause loss of kinetic energy from mean flow, which is a source of turbulent kinetic energy (TKE, k). TKE may is produced as large eddy sizes and is dissipated into heat by small eddy sizes at rate (dissipation rate). At stable stratification, TKE is converted into potential energy (vertical mixing, depening of mixed layer). Unstable stratification converts potential energy into TKE (convective mixing).
How to calculate the eddy viscosity / eddy diffusivity ? Turbulent macro length scale The well-known k- model uses dynamic equations for the TKE and its dissipation rate. There are however many other models in use …
Tree of turbulence closure models (extention of Haidvogel & Beckmann, 1999) Bulk modelsDifferential models Kraus-Turner type models KPP Empirical models Statistical models Ri number depending models Flow depending models Algebraic stress models Full Reynolds stress models Treatment of TKE and length scale Treatment of algebraic stresses Non-equilibrium models Quasi-equilibrium models One-equation models Zero-equation models Two-equation models MY k- k- Generic length scale Mixing length formulations Blackadar-type length scale Gaspar et al. (1990) type models
Convenient approximations: Hydrostatic approximation (vertical velocity dynamically irrelevant) One-dimensional approximation (horizontal homogeneity, far away from coasts and fronts)
The dynamic equations for momentum, temperature, etc., are PDEs (Partial Differential Equations), and therefore need initial and boundary conditions. Initial conditions are either from observations, idealised, or simply set to dummy values (because they may be forgotten after a while). Surface boundary conditions for physical properties come from atmospheric conditions:
Simulated short-wave radiation profile in water, I(z) Surface radiationAttenuation lengths Bio-shadingWeighting Short-wave radiation in water The local heating depends on the vertical gradient of I(z).