Conservation Equations for Mass and Salinity The Continuity Equation

Hydrodynamic Equations of the Sea

3.10 Conservation Equations for Mass and Salinity The Continuity Equation

reasons for choosing anisotropic eddy diffusion coefficients follow from two of the properties of the sea discussed earlier: (1) its essential shallowness when compared with the horizontal scales of many of its motions, and (2) the strong vertical stability that the ocean generally possesses. Both of these charac- teristics make mixing and diffusion in the vertical much smaller than in the horizontal.

TABLE 3.2 Typical Values of Eddy Diffusion Coefficients

K , = A , / p = 3 x lo-’ to 2 x m’ s - ’

Kh = A,/p = 10’ to lo5 m’ s - ’

In summary, the internal body forces in the ocean are modeled by pres- sure and eddy viscosity, viz:

(3.65) where the second term is given by Eq. 3.61.

3.10 Conservation Equations for Mass and Salinity

Hydrodynamic Equations of the Sea

a

ax - ( P U ) dx dy dz

101

Similar expressions for the other two pairs of faces give an overall net con- vergence (negative divergence) into the parcel of

V . p u d3x

.

(3.66)

This must obviously equal the rate at which mass is accumulating within the fixed volume, which is just the local rate of density change, ap/at, times d3x. Equating these rates, we obtain the continuity equation, one of the ba- sic equations of fluid dynamics:

J P

-

+

^{v . p u = 0}

at (3.67)

An alternative form of the continuity equation can be derived using the convective derivative notation (Eq. 3.8). By expanding Eq. 3.67, one obtains

a P DP

-

+

^{u * v p}

+

^{p v * u = -}

+

^{p v * u = 0 ,}

at Dt (3.68)

which ascribes the rate of change of density following the fluid motion to the velocity divergence.

If the fluid is incompressible, the density of a fluid element remains con- stant and DplDt = 0; then the continuity equation requires that through any closed surface the inward flux be equal to the outward flux. That equa- tion reduces to

(3.69)

In the ocean, the density of seawater varies slowly in space and time, hav- ing a typical surface value in temperate zones of p = 1.026 x lo3 kg m - 3 , depending on salinity and temperature. Seawater is slightly compressible, with an isothermal compressibility, ap = ( l / p ) ( a p / a p ) , = 4.3 x lo-'' Pa-'

= 43 x bar-' at 2000 m depth and 4°C (see Chapter 4). This leads to an average fractional increase in density with depth, assuming only hydrostatic pressure, of

!

^{( $ ) T}

(g)

^{= a,pg = 4.3 x lo-(' m - '}

^.

^(3.70)

P

Thus seawater increases its density due to the weight of overlying fluid at a rate of approximately 4.3 parts per million (ppm) per meter. The recipro- cal of this quantity is called the scale height, H, , and has a value of approx- imately 230 km, which is the depth at which the ocean would have increased its density by a factor of e ⁼ 2.718 because of isothermal compressibility alone. Since the average depth of the sea is approximately 5 km, this means that the density of the ocean varies only slightly, even in the deepest por- tions of the sea. However, it does not mean that this variation is negligible;

if seawater were truly incompressible, the surface of the ocean would be per- haps 30 m higher than it is. Futhermore, the compressibility must be taken into account in temperature and salinity measurements made in the deep ocean. However, for many calculations, especially those involving accelera- tions, the density can be assumed constant and the resultant equations con- siderably simplified; this forms the basis of the Boussinesq approximation, to be discussed in Chapter 5 .

Returning to Eq. 3.67: For acoustic waves, the continuity equation as shown must be used. However, for slower fluid motions, even those with time scales characteristic of surface gravity waves, say, the incompressible version (Eq. 3.69) may generally be used. This forms the basis of one meth- od of estimating the very slow vertical velocities of flow in the sea. If mea- surements can be made of the divergence of the horizontal flow, V,, 'u, over some area as a function of depth, these can be integrated to arrive at an esti- mate of the vertical flow speed, w, between any two depths,

zI

and z 2 :

(3.71)

Thus upwelling ( w

>

0) or downwelling ( w

<

0 ) speeds can be derived from an array of current meter moorings measuring horizontal currents alone, sim- ply by assuming divergence-free flow. Such speeds are typically of order

to m s - I and are essentially impossible to determine by other

means.

Hydrodynamic Equations of the Sea

The Salinity Equation

103

The continuity equation is an example of a class of conservation equa- tions for the fluxes of various properties, y(x, f), which, unlike mass, may have sources and sinks. Thus in near-surface waters, oceanic salinity (y = s) may increase or decrease due to riverine input, evaporation, rainfall, or freez- ing and thawing. An equation for the rate of change of salinity may then be written:

DS as

Dt at

_ - - -

+

^{u .}

vs

= Si" -

so,,

, (3.72)

where the right-hand side represents the sources and sinks of salinity. In the Mediterranean Sea, for example, the evaporation excess has important con- sequences for the circulation of both surface and subsurface waters (cf. Chap- ter 2), and has led to the accumulation of massive quantities of salt on the floor of that Sea over geological times, when the Strait of Gibraltar was closed.

An explicit form for the balance of constitutive quantities such as salini- ty, heat, humidity, or energy may be derived from conservation arguments, and from these, equations describing advective and diffusive processes may be obtained. As a generic example, let J be the advective flux per unit area of some property such as salinity or density, and C be the quantity of that property per unit volume, or concentration. The advective flux and concen- tration are clearly related by

J = C u . (3.73)

Now the flux through some elemental area, d A , in the fluid is

J.A

dA , (3.74)

where A is the unit normal to the area d A . Since C can change within the volume, the advective flux must also change with a loss rate, or divergence per unit volume, of

aJ, aJ, aJ,

ax ay

az

V - J = ^~

+

^~

+

^{- ,} (3.75)

as obtained by a series expansion, as previously (see Fig. 3.13, for example).

Then the rate of change of concentration due to advection alone is

(3.76)

In addition to a convergence due to advection, the concentration can change due to diffusive processes, both molecular and eddy-like. The diffusive flux, Jd, is usually found to be proportional to the gradient of the concentration, VC, in the lowest approximation, with the constant of proportionality be- ing the diffusion coefficient. As with the diffusion of momentum (cf. Sec- tion 3.9) this quantity may be anisotropic, so that the diffusive flux, which may not be in the same direction as the concentration gradient, may be written

where the negative sign indicates that diffusion proceeds from regions of high concentration to regions of low concentration. For the ocean, we take the diffusion tensor, K , to be diagonal but differing in the horizontal and verti- cal, as was the case with the eddy viscosity:

(3.78)

Here ^K, is an isotropic molecular diffusion coefficient, as before. The rate at which the diffusive flux changes within the elemental volume is similar in functional form to the advective flux, so that the accumulation is propor- tional to the convergence of the total flux, J

+

Jd. The rate equation for the concentration is thus

To apply this to salinity, lets be the fraction, by mass, of dissolved salts, multiplied by 1000; this unit is termed the practical salinity unit (psu), and replaces the former units of 0700, ppt, or parts per thousand. Then the con- centration of salt per unit mass,

C,,

is

c,

^{= ps} (3.80)

and the conservation equation for salinity becomes

Hydrodynamic Equations of the Sea 105

Here the diffsivity of salt in seawater, ^K,, depends on the thermodynamic state of the fluid, i.e., temperature, pressure, etc. For isotropic molecular diffusion alone, ^K, = 1.5 x m2 s - ' at T = 25"C, and at normal oceanic salinities. As with eddy viscosity, the eddy diffusion of salt is much larger than molecular diffusion, of course, but less is known about numeri- cal values for salinity eddy diffusion than for momentum. However, esti- mates from the Mediterranean salt tongue of Figs. 2.24 and 2.25 give values of

K , ~ = 3 x 103 m 2 s - ' and

K,, = 5 x m 2 s - '

,

(3.82) which are somewhat smaller than the momentum diffusivities. The source/

sink terms represent either surface or volumetric rates for evaporation, E, minus precipitation, P, and freezing, F, minus thawing, 8 , minus R , the in- put of fresh water from riverine sources. This equation, together with anal- ogous ones for momentum and heat, can be used to describe thermohaline circulations, salt-fingering and other density-driven motions.