Waves and Tides
5.3 Buoyancy and Stability
it is natural then to define a local scale height, H,
( z ) ,
which is the height over which the density increases by e:H, ( 2 ) = c 2 ( z ) / g
.
(5.20)Since H, is approximately 225 km in the ocean, many problems having small amplitudes of motion allow the constant density approximation.
Nevertheless, Eq. 5.15, in which we have not made the Boussinesq approxi- mation, has greater generality and will be used in a number of problems re- quiring it.
Waves and Tides 167
is initially at the equilibrium depth,
z,,
, and conceptually assumed to be en- closed in a kind of flaccid balloon that prevents interchange of properties with its surroundings. It now receives an upward impulse and undergoes a vertical upward displacement,l ( z ) ,
from its equilibrium position. When it arrives atz,, +
[, the parcel will have expanded slightly due to decompres- sion and will be a bit more buoyant as a result. If, however, the background density, p o ( z ) , has decreased sufficiently rapidly over the heightC;,
the par- cel will find itself heavier than its surroundings, will halt, and will reverse its direction of motion. As it sinks, its kinetic energy carries it past its equilibri- um position and it will move to a depth of approximately zeq - [, where it is now slightly compressed. Here, if the background water column is suffi- ciently dense, the parcel will be forced upward once again. Assuming the profile dp,/dz permits it over the entire region of excursion, the parcel will oscillate about its equilibrium depth with a characteristic radian frequency, N, variously called the buoyancy frequency, the Brunf- Viiisiilii frequency, or the sfabilify frequency. If the water column is only weakly stratified, how- ever, the parcel might expand or contract sufficiently under pressure change to find itself increasingly lighter than its surroundings upon ascent, or in- creasingly heavier upon descent, and further rise or sink accordingly. In the moist atmosphere, the buoyant rising motion, which is accompanied by re- lease of the latent heat of condensation of the water vapor and attendant warming of the rising parcel of air, often becomes completely unstable at some point, with the air rising t o high altitudes. This process is termed con- vective instabi/ity and is a major source of the vertical uplifting of clouds discussed in Chapter 2. For the sea, analogous sinking motions exist as a result of surface cooling or salinity increases.In the ocean, the background temperature, salinity, and compressibility collectively determine the vertical density profile. In its upper reaches (i.e., in the mixed layer and seasonal thermocline) the temperature and salinity effects are most pronounced and variable, while in the deep regions, the ocean usually ranges between neutrally stratified to very slightly stable. In this re- gion, density and temperature changes are almost entirely due t o adiabatic compression of seawater by the pressure of the overlying fluid (Fig. 4.11).
In order to derive a stability condition for vertical motions, consider again the vertical momentum and continuity equations in the near-steady-state case, allowing only the possibility of small vertical velocities. Under these condi- tions, we may neglect advective, Coriolis, and viscous forces and again ex- pand pressure and density in terms of small perturbations:
Dw
a
Dt az
p- = - - ( Po
+
P ’ ) - ( P o+
p ’ ) g (5.21)while from the continuity equation, one obtains DP - = - p v . u = - p o v * u
Dt
(5.22)
(5.23)
Since the vertical velocity is assumed small, we may neglect the product of
p ’ and
v
.u in Eq. 5.23. For isentropic, isohaline vertical displacements of a fluid column, we obtain from Eq. 4.38,Dp I D p 1 D P O W
Dt - - -
_ _
c2 Dt - - - - c2 Dt ( P g o = - C2 * (5.24) Here we have used the hydrostatic approximation, so that the perturbation pressure is proportional to the actual density times the vertical excursion,4 :
P ’ = P g s . (5.25)
Equation 5.24 states that for buoyancy oscillations, the temporal changes in the density are caused simply by vertical advection of the background pro- file, p o , by the vertical velocity, w. Then the velocity divergence may be written as
1 a P
- -
--(-& +
w ; ).
P (5.26)
The vertical component of the momentum equation (Eq. 4.53) becomes - Pg
P - Dw = - - aP
Dt az (5.27)
We next totally differentiate Eq. 5.27 with respect to time and obtain a second- order differential equation for w:
Waves and Tides 169
J P g 2
-
-
- g - w - T p w ,az C (5.28)
where we have used Eq. 5.24 for the terms on the right. This may be re- arranged to obtain
D2w
Dt2 P az
= N ' w .
Here the radian buoyancy frequency, N , is defined as
(5.29)
(5.30)
The solutions t o Eq. 5.29 are clearly either oscillatory or exponential, ac- cording to whether N i s real, corresponding to buoyancy oscillations, or im- aginary, corresponding to rising/sinking motions. Thus the buoyancy frequency measures the stability of the water column against small vertical perturbations. A stable water column is one that is sharply enough stratified so that (1 / p ) ( d p / a z ) , which is normally negative, will overcome the compres- sive change, g / c 2 , so that
N 2 >
0.An alternative expression for the buoyancy parameter, N2, may be de- rived from purely thermodynamic arguments by writing a generalized small density change, dp, in terms of changes in pressure, temperature, and salini- t y , and comparing it with an isentropic change (dp),
.
The result is (see Gill, loc. cit.),(5.31)
which is useful when the temperature and salinity profiles are known separately.
Examples of actual vertical profiles of salinity, temperature, density anom- aly, and buoyancy frequency are shown in Fig. 5.2. This case is for a tropi- cal mediterranean (inland) sea with very little wind-mixed layer present, and
(a)
-250
E . --750
Q
n
- 1000
-1250
/
Salinity, s(psu1
3 2 33 34 35 3,6
Temperature, T ( C I
-\
\cJ
- 1 5 o o - / L , , I , ,
21 22 23 24 25 26
4
27 -1600- 0 51 Density anomaly, mt ( k g m - 3 ) Brunt-Vaisala frequency, N i 2 r (cycles h - ' Fig. 5.2 (a) Vertical profiles of salinity, temperature, and density anomaly for a trop- ical, landlocked sea under low wind conditions. (b) Profile of buoyancy frequency cal- culated from data of (a). Density is controlled mainly by temperature, which is characteristic of most nonpolar waters. [Adapted from Apel, J. R., eta/., J. Phys. Ocean- ography (1985).]
nearly isohaline conditions existing at depths below approximately 200 m.
The maximum Brunt-VBisBIB frequency occurs where the temperature gra- dient is largest (near 150 m depth) and has a value in excess of 10 cycles h - ' (0.01745 rad s - I ) . Another example, which represents more typical temper- ate Atlantic conditions, is shown in Fig. 5.3. Here Nshows a secondary max- imum near a depth of 750 dbar, probably due to the presence of a differing water mass. The calculated quantity dtl/dp on the graph is the potential tem- perature gradient, and its constancy beneath some 2000 dbar, despite decreasing N, indicates that the buoyancy frequency variations below that depth are due to adiabatic compressibility alone (i.e., g/c*).