As a next step, we have developed a general method to calculate the entropy increase rate for a non-steady synthetic system (Shimokawa and Ozawa). The entropy increase rate for the synthetic system is composed of a term of the non-linear system and that of the surrounding system connected by the system (Fig.4):

This expression can be rewritten into a different form by using the heat balance equation

  where F the diabatic heat flux (e.g. heat conduction), and Φ the dissipation function, representing the rate of dissipation of kinetic energy by viscosity. It is interesting to note that the thermodynamic concept (_syn=Max.) is now identical to Busse's maximum dissipation hypothesis (Φ=Max.) suggested for turbulent Couette flows under the isothermal condition (grad T 0). Thus, Busse's suggestion on maximum dissipation and Malkus-Howard's suggestion on maximum heat transport are consistent with the thermodynamic concept (2) presented here.
  An attempt has been made to evaluate the entropy increase rates using an ocean circulation model. In the case of the ocean system, the entropy increase rate by the salt transport is taken into account (Shimokawa and Ozawa) as,

where ν 2 the dissociation effect, k the Boltzmann constant, C the number density of salt, and F_s the surface salt flux.