Chapter 8  Equations of Motion With Viscosity
8.4 Stability We saw in the last section that fluid flow with a sufficiently large Reynolds number is turbulent. This is one form of instability. Many other types of instability occur in the in the ocean. Here, let's consider three of the more important ones:
Static Stability and the Stability Frequency
Consider a parcel of water that is displaced vertically and adiabatically in a stratified fluid (Figure 8.4). The buoyancy force F acting on the displaced parcel is the difference between its weight gV_{g} ρ_{}' and the weight of the surrounding water gV_{g }ρ_{2 }, where V is the volume of the parcel:
The acceleration of the displaced parcel is:
but
Using (8.21) and (8.22) in (8.20), ignoring terms proportional to δz^{ 2}, we obtain:
where E ≡ a / ( g dz ) is the stability of the water column. This can be written in terms of the measured temperature and salinity t(z), S(z) in the water column (McDougall, 1987; Sverdrup, Johnson, and Fleming, 1942: 416; or Gill, 1982: 50):
where
and where α is the thermal expansion coefficient, β is the saline contraction coefficient, and Γ is the adiabatic lapse rate, the change of temperature with pressure as the water parcel moves without exchanging heat with it's surroundings. p is pressure, t is temperature in celsius, ρ_{} is density, and S is salinity. In the upper kilometer of the ocean stability is large, and the first term in (8.23) is much larger than the second. The first term is proportional to the rate of change of density of the water column; the second term is proportional to the compressibility of sea water, which is very small. Neglecting the second term, we can write the stability equation:
The approximation used to derive (8.26) is valid for E > 50 × 10^{8}/m. Below about a kilometer in the ocean, the change in density with depth is so small that we must consider the small change in density of the parcel due to changes in pressure as it is moved vertically, and (8.24) must be used. Stability is defined such that
In the upper kilometer of the ocean, z < 1,000m, E = (501000)×10^{8}/m, and in deep trenches where z > 7,000m, E = 1×10^{8}/m. The influence of stability is usually expressed by a stability frequency N:
The stability frequency is often called the BruntVaisala frequency or the stratification frequency. The frequency quantifies the importance of stability, and it is a fundamental variable in the dynamics of stratified flow. In simplest terms, the frequency can be interpreted as the vertical frequency excited by a vertical displacement of a fluid parcel. Thus, it is the maximum frequency of internal waves in the ocean. Typical values of N are a few cycles per hour (Figure 8.5).
Dynamic Stability and Richardson's Number
This is an example of dynamic instability in which a stable fluid is made unstable by velocity shear. Another example of dynamic instability, the KelvinHelmholtz instability, occurs when the density contrast in a sheared flow is much less than at the sea surface, such as in the thermocline or at the top of a stable, atmospheric boundary layer (Figure 8.6).
The relative importance of static stability and dynamic instability is expressed by the Richardson Number:
where the numerator is the strength of the static stability, and the denominator is the strength of the velocity shear.
Note that a small Richardson number is not the only criterion for instability. The Reynolds number must be large and the Richardson number must be less than 0.25 for turbulence. These criteria are met in some oceanic flows. The turbulence mixes fluid in the vertical, leading to a vertical eddy viscosity and eddy diffusivity. Because the ocean tends to be strongly stratified and currents tend to be weak, turbulent mixing is intermittent and rare. Measurements of density as a function of depth rarely show more dense fluid over less dense fluid as seen in the breaking waves in Figure 8.6 (Moum and Caldwell 1985). Double Diffusion and Salt Fingers
Consider two thin layers a few meters thick separated by a sharp interface (Figure 8.7). If the upper layer is warm and salty, and if the lower is colder and less salty than the upper layer, the interface becomes unstable even if the upper layer is less dense than the lower.
Here's what happens. Heat diffuses across the interface faster than salt, leading to a thin, cold, salty layer between the two initial layers. The cold salty layer is more dense than the cold, lesssalty layer below, and the water in the layer sinks. Because the layer is thin, the fluid sinks in fingers 15cm in diameter and 10s of centimeters long, not much different in size and shape from our fingers. This is salt fingering. Because two constituents diffuse across the interface, the process is called double diffusion. There are four variations on this theme. Two variables taken two at a time leads to four possible combinations:
Double diffusion mixes ocean water, and it cannot be ignored. Merryfield et al., (1999), using a numerical model of the ocean circulation that included double diffusion, found that doublediffusive mixing changed the regional distributions of temperature and salinity although it had little influence on largescale circulation of the ocean.


Department of Oceanography, Texas A&M University Robert H. Stewart, stewart@ocean.tamu.edu All contents copyright © 2005 Robert H. Stewart, All rights reserved Updated on February 4, 2009 