D t p

Equation (3.33b) can be rewritten as

Using Eqns. (3.34), (3.36), and the definition of Js, this gives rise to a formula for the diffusive salt flux at the sea surface

However, it is important to notice that this expression holds only below the sea surface. Above the sea surface, there is no salt and salt flux is identically zero.

Our discussion here excludes the water-sea ice interface. Since sea ice formation and melting involves the movement of both pure water and salt through the water-ice interface, the corresponding boundary conditions are more complicated. The detailed discussion of the boundary condition applied to the water-ice interface can be found in an article by Huang and Jin (2007).

3.3.3 A one-dimensional model with evaporation

To gain a better understanding of the boundary condition for salinity, we examine a one-dimensional model (Fig. 3.6). The basic idea can best be illustrated by adding on an imaginary, infinitely thin, layer of freshwater with S = 0 on the top.

Suppose the ocean is subject to evaporation of m. In a one-dimensional model, mass conservation requires the balance of pure water, i.e., the vertical flux of water should be continuous. Thus, over the whole depth of the water column, freshwater, with density z = 0

Js i Jw

Fig. 3.6 A one-dimensional balance of transport in the upper ocean, including an imaginary infinitely thin layer of pure water on the top. All fluxes are defined in the coordinates moving with the center of mass.

Table 3.1. Balance of water and salt flux in a steady one-dimensional model

Diffusive flux relative to center of mass Advective flux Sum

Sum 0 p (1 — s)m p (1 — s)m pw = p(1 — s), must move upward with the same vertical velocity uw = m; however, the other component, salt (density ps = ps), is stagnant, i.e., us = 0 (Fig. 3.6). The velocity of the center of mass is pwuw + psus pw ,, . ,0.

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