Method 2, Entropy Increase Rate

For the calculation of entropy increase rate for a system, it is necessary to take account of the heat exchange between the system and the outside of it. In other words, if the exchange of heat with the outside of the system is substantial, the system should be considered to be a dissipative system. Very recently, Shimokawa and Ozawa (2000) derived a formula of the entropy increase rate for the ocean, taking account of the heat exchanges with the outside of the system, i.e., the atmosphere. Their formula is relevant to the present model setting, and can be expressed as follows,

,                           (2)

where is the entropy increase rate, is the specific heat of water (3900 J kg^-1 K^-1), is the water density, the temperature, and the heat input per a unit volume in the mixed layer. The integral indicates the volume integration, and the subscript, (), of the integral symbol indicates the integration over the whole model domain (mixed layer). We will examine whether the gradient of the entropy increase rate is consistent or not with the maximization hypothesis of entropy increase rate in association with the 2-D and 3-D transition in a manner shown in Fig. 1. The entropy increase rates are the averages from 80 to 144 hours.