The corresponding diffusive mass flux, relative to the center of mass, is
Jw = pw (uw — u) = pwsm = ps (1 — s)m > 0 (3.39)
Note that the center of mass moves upward with a velocity of u = (1 — s)m, slower than the rate of evaporation because the salt component is stagnant; the diffusive salt flux defined here is the diffusive flux relative to the center of mass. The advective salt flux associated with the center is ups = Jw, which is equal to the rate of diffusive salt flux, but with an opposite sign, i.e., Jw = Js. Thus, the advective salt flux is exactly cancelled by the diffusive saltflux. The balance of water and salt fluxes for this model is shown in Table 3.1. The salt balance in fixed standard Eulerian coordinates is
Since the salt is in an exact balance, there is a salt flux which is opposite to the advective salt flux,
Therefore, the diffusive salt flux defined in the coordinates moving with the center of mass equation (3.40) is slightly (about 3.5%) smaller than the diffusive salt flux defined in the fixed Eulerian coordinates (see Eqn. (3.42)). This difference is due to the fact that the velocity of the center of mass is (1 - s)m, which is slower than the evaporation rate m.
The discussion above applies to regime below a thin layer of pure water. Within a thin layer of pure water, the velocity of center of mass is exactly the same as the velocity of pure water because there is no salt within this layer. As a result, both the salt flux and water flux relative to the center of mass are identically zero. The introduction of an imaginary thin layer of pure water on the top of the ocean has therefore helped us to clarify the dynamical processes associated with evaporation and precipitation, and such a procedure will be detailed in the analysis of entropy balance in the world's oceans at the end of this chapter.
3.4 Balance of mass, energy, and entropy
As a dynamical system, oceanic circulation must satisfy several balance laws, which also act as the most important theoretical tools in the study of oceanic dynamics. In most cases these laws are also called the conservation laws; however, there is an exception for entropy, because it is a peculiar thermodynamic variable which is not conserved and never decreases for any macroscopic thermodynamic systems. Thus, we will adopt the terminology of "balance laws."
The fundamental conservation/balance laws of dynamics and thermodynamics are extensively discussed in various standard textbooks. We here present a concise derivation of these laws in the form most suitable for studies of wind-driven circulation and thermohaline circulation. In this section the standard notation of tensor will be utilized. In particular, the summation convention for tensors will be used. For example, dUj/dxj means ^3=1 d Ui/dXi =d U1/d X1 + d U2/d X2 + d U3/d X3.
3.4.1 Mass conservation dp dp Uj
91 d xj where \Xj\ = (x1,x2,x3) are the coordinates, p is the density, and \Uj\ = (u1, u2, u3) is the velocity.
3.4.2 Momentum conservation dp U< dpUiUj dp dê0 dxij déT
— + = -2siikj - — - p— + —^ - p —, i = 1,2,3. (3.44)
91 d Xj d Xi d Xi d Xj d Xi where eijk is the three-dimensional permutation symbol; the geopotential is separated into two parts:
where 0T is the tidal potential and d0T/dXi = [0, 0, gT (x, y, t)k] is the tidal force, which is treated as a body force specified a priori and remains the same after the ensemble averaging. All forces act as sources of momentum changes. In fluid dynamics, the stress tensor is defined as
where the first term is called the viscosity stress tensor, and for isotropic viscosity is in the form derived by Landau and Lifshitz (1959, p. 48):
In many oceanographic studies, the seawater is treated in terms of a Boussinesq fluid; thus, the divergence terms vanish, and the corresponding viscosity stress is reduced to the following form
3.4.3 Gravitational potential energy conservation
Multiplying the continuity equation (3.4.1) by $ = gz+$T, we obtain the GPE conservation dp$ dpuj$ d$ d$T ,r.
3.4.4 Kinetic energy conservation
From the momentum equations, we can derive the kinetic energy (KE) balance equation by the inner product of the velocity and the momentum equation. Since individual terms of the momentum equation can be interpreted as forces, the rate of doing work is the scalar product of forces times the velocity:
d t d Xj d Xj d Xj d Xj d Xj where K = uiui/2 is the KE per unit mass. Note that the Coriolis force term makes no contribution to the energetics.
A second energy conservation equation can be obtained from the thermodynamic conservation of energy. For a water parcel with volume V and lateral boundary S, the sum of the internal and kinetic energy must change at a rate equal to the sum of the work done by the individual forces plus internal sources, i.e.,
— f P(e + K)dV = — f P(e + K)UjUjdA — f pUjnjdA + f uirijnjdA — f FjnjdA dt Jv Js Js Js J
where fS p(e + K)UjUjdA is the energy flux due to mass exchange through boundaries,
¡SpUjUjdA is due to pressure work on boundaries, fS UiTijUjdA is due to work by frictional stress on boundaries,
JV puj-dV is the work by gravity on the whole water parcel,
JV PUj "t—dV is due to energy input from tides on the whole water parcel, 9 xj fS FjUjdA is due to diffusion of heat and mass through the boundaries, and flux Fi includes heat flux, enthalpy flux due to salt water and freshwater mixing across the boundaries.
Using Eqns. (3.29) and (2.66), this flux can be rewritten as d h
Fi = qi + hsJs,i + hwJw,i = qi + (hs - hw) Js,i = qi + — Js,i (3.50)
where hs and hw are the specific partial enthalpy of salt and pure water, and qi and Js,i are the i-th component of the interfacial heat and salt fluxes.
Since Eqn. (3.49) is valid for any infinitesimal volume, we obtain the differential equation for conservation of kinetic and internal energy:
dp(e + K) dpUj (e + K) dpUj d$o dtyj dUi Tj dFj ^ --1----=---- pUj--pUj--1------ (3.51)
Subtracting Eqn. (3.48) from Eqn. (3.51) leads to the conservation of internal energy:
dpe dpUje d Uj 3Fj
is the dissipation rate per unit volume. The kinetic conservation law can be rewritten as dp K dp UjK dTijUi dp 9(0o + )
Using the continuity equation, the corresponding equation for internal energy (Eqn. (3.52)) is reduced to de d Uj d Fj p— = -P1t1 + pe - -t-1 (3.55)
dt dXj dXj
In summary, now we have the following differential forms of conservation of kinetic energy, internal energy, and GPE
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