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:

i) static stability associated with change of density with depth,
ii) dynamic stability associated with velocity shear, and
iii) double-diffusion associated with salinity and temperature gradients in the ocean.

Static Stability and the Stability Frequency
Consider first static stability. If more dense water lies above less dense water, the fluid is unstable. The more dense water will sink beneath the less dense. Conversely, if less dense water lies above more dense water, the interface between the two is stable. But how stable? We might guess that the larger the density contrast across the interface, the more stable the interface. This is an example of static stability. Static stability is important in any stratified flow where density increases with depth; and we need some criterion for determining the importance of the stability.

Figure 8.4 Sketch for calculating static stability and stratification 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 ρ₂ , where V is the volume of the parcel:

The acceleration of the displaced parcel is:

(8.20)

but

	(8.21)
	(8.22)

Using (8.21) and (8.22) in (8.20), ignoring terms proportional to δz ², we obtain:

(8.23)

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):

(8.24)

where

(8.25)

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:

(8.26)

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 = (50-1000)×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 Brunt-Vaisala 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).

Figure 8.5. Observed stratification frequency in the Pacific. Left: Stability of the deep thermocline east of the Kuroshio. Right: Stability of a shallow thermocline typical of the tropics. Note the change of scales.

Dynamic Stability and Richardson's Number
If velocity changes with depth in a stable, stratified flow, then the flow may become unstable if the change in velocity with depth, the current shear , is large enough. The simplest example is wind blowing over the ocean. In this case, stability is very large across the sea surface. We might say it is infinite because there is a step discontinuity in ρ, and (8.27) is infinite. Yet, wind blowing on the ocean creates waves, and if the wind is strong enough, the surface becomes unstable and the waves break.

This is an example of dynamic instability in which a stable fluid is made unstable by velocity shear. Another example of dynamic instability, the Kelvin-Helmholtz 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).

Figure 8.6 Billow clouds showing a Kelvin-Helmholtz instability at the top of a stable atmospheric boundary layer. Some billows can become large enough that more dense air overlies less dense air, and then the billows collapse into turbulence. Photography copyright Brooks Martner, NOAA Environmental Technology Laboratory.

The relative importance of static stability and dynamic instability is expressed by the Richardson Number:

(8.28)

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
In some regions of the ocean, less dense water overlies more dense water, yet the water column is unstable even if there are no currents. The instability occurs because the molecular diffusion of heat is about 100 times faster than the molecular diffusion of salt. The instability was first discovered by Melvin Stern in 1960 who quickly realized its importance in oceanography.

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.

Figure 8.7 Left: Initial distribution of density in the vertical. Right: After some time, the diffusion of heat leads to a thin unstable layer between the two initially stable layers. The thin unstable layer sinks into the lower layer as salty fingers. The vertical scale in the figures is a few centimeters.

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, less-salty layer below, and the water in the layer sinks. Because the layer is thin, the fluid sinks in fingers 1-5cm 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:

1. Warm salty over colder less salty. This process is called salt fingering It occurs in central waters of sub-tropical gyres, western tropical North Atlantic, and the North-east Atlantic beneath the out flow from the Mediterranean Sea. Salt fingering eventually leads to density increasing with depth in a series of steps. Layers of constant-density are separated by thin layers with large changes in density, and the profile of density as a function of depth looks like stair steps. Schmitt et al., (1987) observed 5-30m thick steps in the western, tropical North Atlantic that were coherent over 200-400km and that lasted for at least eight months. Kerr (2002) reports a recent experiment by Raymond Schmitt, James Leswell, John Toole, and Kurt Polzin showed salt fingering o Barbados mixed water 10 times faster than turbulence.
2. Colder less salty over warm salty. This process is called diffusive convection. It is much less common than salt fingering, and it us mostly found at high latitudes. Diffusive convection also leads to a stair step of density as a function of depth. Here's what happens in this case. Double diffusion leads to a thin, warm, less-salty layer at the base of the upper, colder, less-salty layer. The thin layer of water rises and mixes with water in the upper layer. A similar processes occurs in the lower layer where a colder, salty layer forms at the interface. As a result of the convection in the upper and lower layers, the interface is sharpened; and any small gradients of density in either layer are reduced. Neal et al. (1969) observed 2-10m thick layers in the sea beneath the Arctic ice.
3. Cold salty over warmer less salty. Always statically unstable.
4. Warmer less salty over cold salty. Always stable and double diffusion diffuses the interface between the two layers.

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 double-diffusive mixing changed the regional distributions of temperature and salinity although it had little influence on large-scale circulation of the ocean.

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