# T

Free energy (Helmholtz free energy, available work)

f = g - P£3 = i - P (yiT - yo) e(ao+o.5aiT+yiP)T-fiS-yoP + C (T, S) (2.io3)

_ 2Y1T - Y0 — (Y1T - Y0) (a0 + a1T + Y1P) T — P (Y1T — Y0)2 e(a0+0.5a1T+Y1P)T-pS-Y0P — 2 p0 (Y1T - Y0)2

Chemical potential p = ^ =--P-^o+o^T+y^T-ps-yoP + dC (T,S)

Entropy n = = - (Y1T - Y0) (ao + aiT + Y1P) - Y1 e(ao+0.5aiT+YiP)T-fiS-yoP

Other thermodynamic functions can be obtained similarly.

For simplicity, we list the thermodynamic functions for the case when the equation of state is a simple linear function of temperature, salinity, and pressure. Omitting the arbitrary function C(T, S) introduced in Eqn. (2.99), the Gibbs function takes the form eaT-pS-Y P

where a, P, y and po are all consistent. From the Gibbs function, we can derive the following list of thermodynamic variables which are consistent with each other.

d T PoY

d P poY

d s PoY

d T PoY

Note that under the assumption of a linear equation of state, all basic thermodynamic functions have a factor which is an exponential function of temperature, salinity, and pressure. In particular, entropy increases with temperature exponentially. As a result, the Gibbs function (the free enthalpy) declines with the increase in temperature, although both the internal energy and enthalpy increase with temperature.

### 2.6 Scaling and different approximations

The partial differential equations describing oceanic motions constitute a very complicated equation system. These equations can describe thousands of kinds of phenomena having time scales from seconds to thousands of years, and length scales from millimeters to thousands of kilometers. Thus, trying to find the "complete solution" or "general solution" of such a system is unrealistic. In order to understand the real-world oceanic motions by means of such an equation system, we have to limit ourselves to certain scales, thereby simplifying the general equation system.

The basic principle of scaling is that we want to focus our study on some specific phenomenon which has certain time and space scales. Therefore, we will use these scales to estimate and compare the magnitude of all terms in the equations. We keep only the "important terms" and discard the smaller terms, assuming that they are not as important for studying the phenomenon in which we are really interested. It is crucial to emphasize that scale analysis simplifies equations and helps to highlight the physics important for our study; however, the result of scaling is implicitly defined in the basic assumptions of the scaling. Different scales will certainly lead to quite different equations, which describe different dynamical processes.

Note that scaling is an art for dealing with complicated systems, so it is not a foolproof method. There are indeed many cases in which some terms that are much smaller than other terms may play crucial roles in the overall balance of vorticity and energy, so they cannot be discarded. For example, terms associated with small-scale turbulence dissipation/mixing often have magnitudes much smaller than other terms; however, the interaction between large and small scales is one of the major players that ultimately regulates the global-scale wind-driven circulation and thermohaline circulation. Since we will derive equations in the local Cartesian coordinates, the following notations are used in this section: dx = r cos 0 dk, dy = rd0, and dz = dr.

2.6.1 Hydrostatic approximation

Scaling of the vertical momentum equation

For basin-scale motions in the oceans the magnitude of terms in the vertical momentum equation can be estimated as follows

Magnitude 10'