Chapter 8 - Equations of Motion With Viscosity

8.5 Mixing in the Ocean

Turbulence in the ocean leads to mixing. Because the ocean has stable stratification, vertical displacement must work against the buoyancy force. Vertical mixing requires more energy than horizontal mixing. The larger the stability frequency the greater the work required for vertical mixing. As a result, horizontal mixing along surfaces of constant density is much larger than vertical mixing across surfaces of constant density. The latter, however, usually called diapycnal mixing, is very important because it changes the vertical structure of the ocean, and it controls to a large extent the rate at which deep water eventually reaches the surface in mid and low latitudes.

The equation for vertical mixing depends on many processes. See Garrett (2006) for a good overview of the subject. Here I consider some simple flows. A simple equation for vertical mixing by eddies of a tracer Θ such as salt or temperature is:

(8.29)

where K_z is the vertical eddy diffusivity, W is a mean vertical velocity, and S is a source term.

Figure 8.8 Potential temperature measured as a function of depth (pressure) near 24.7°N, 161.4°W in the central North Pacific by the Yaquina in 1966 (·), and by the Thompson in 1985 ( ). Data from Atlas of Ocean Sections produced by Swift, Rhines, and Schlitzer. Data from Atlas of Ocean Sections produced by Swift, Rhines, and Schlitzer.

Average Vertical Mixing
Walter Munk (1966) used a very simple observation to calculate vertical mixing in the ocean. He observed that the ocean has a thermocline almost everywhere, and the deeper part of the thermocline does not change even over decades (Figure 8.8). This was a remarkable observation because we expect downward mixing would continuously deepen the thermocline. But it doesn't. Therefore, a steady-state thermocline requires that the downward mixing of heat by turbulence must be balanced by an upward transport of heat by a mean vertical current W. This follows from, (8.29) for steady state with no sources or sinks:

(8.30)

where T is temperature as a function of depth in the thermocline.

The equation has the solution:

where H = K_z /W is the scale depth of the thermocline, and T₀ is the temperature near the top of the thermocline. Observations of the shape of the deep thermocline are indeed very close to a exponential function. Munk used an exponential function fit through the observations of T (z) to get H.

Munk calculated W from the observed vertical distribution of ¹⁴C, a radioactive isotope of carbon, to obtain a vertical time scale. In this case, S = -1.24 × 10^-4 years^-1. The length and time scales gave W = 1.2 cm/day and

where the brackets denote average eddy diffusivity in the thermocline.

Munk also used W to calculate the average vertical flux of water through the thermocline in the Pacific, and the flux agreed well with the rate of formation of bottom water assuming that bottom water upwells almost everywhere at a constant rate in the Pacific. Globally, his theory requires upward mixing of 25 to 30 Sverdrups of water, where one Sverdrup is 10⁶ cubic meters per second.

Munk's observations have another important consequence. Because the thermocline exists almost everywhere, there must be upward currents almost every-where. We will come back to this when we study Stommel's theory for the deep circulation in §13.2.

Measured Vertical Mixing
Direct observations of vertical mixing required the development of techniques for measuring:

i) the fine structure of turbulence, including probes able to measure temperature and salinity with a spatial resolution of a few centimeters (Gregg 1991), and
ii) the distribution of tracers such as sulphur hexafluoride (SF₆) which can be easily detected at concentrations as small as one gram in a cubic kilometer of seawater.

Direct measurements of open-ocean turbulence and the diffusion of SF₆ yield an eddy diffusivity:

For example, Ledwell, Watson, and Law (1991) injected 139kg of SF₆ in the Atlantic near 26°N, 29°W 1200 km west of the Canary Islands at a depth of 310m. They then measured the concentration for five months as it mixed over hundreds of kilometers to obtain a diapycnal eddy diffusivity of K_z = 1.1 ± 0.2 × 10^-5 m²/s.

These and other open-ocean experiments indicate that turbulent mixing is driven by breaking internal waves and shear instability at boundaries. Furthermore, mixing by turbulence seems to be more important than double diffusion (Gregg 1987).

The large discrepancy between Munk's calculation of the the mean eddy diffusivity for vertical mixing and the observed values in the open ocean has been resolved by recent studies that show:

Polzin et al., (1997) measured the vertical structure of temperature in the Brazil Basin in the South Atlantic. They found K_z > 10^-3m²/s close to the bottom when the water flowed over the western flank of the mid-Atlantic ridge at the eastern edge of the basin. Kunze and Toole (1997) calculated enhanced eddy diffusivity as large as K = 10^-3 m²/s above Fieberling Guyot in the Northwest Pacific and smaller diffusivity along the flank of the seamount. And Garbato et al (2004) calculated even stronger mixing in the Scotia Sea where the Antarctic Circumpolar Current flows between Antarctica and South America.

The results of these and other experiments show that mixing occurs mostly by breaking internal waves and by shear at oceanic boundaries: along continental slopes, above seamounts and mid-ocean ridges, at fronts, and in the mixed layer at the sea surface. To a large extent, the mixing is driven by deep-ocean tidal currents, which become turbulent when they flow past obstacles on the sea floor, including seamounts and mid-ocean ridges (Jayne et al, 2004).

Because water is mixed along boundaries or in other regions (Gnadadesikan, 1999), we must take care in interpreting temperature profiles such as that in Figure 8.8. For example, water at 1200 m in the central north Atlantic could move horizontally to the Gulf Stream, where it mixes with water from 1000 m. The mixed water may then move horizontally back into the central north Atlantic ata depth of 1100 m. Thus the parcel of water at 1200 m and 1100 m at some location may reach their position along entirely different paths.

Measured Horizontal Mixing
Eddies mix fluid in the horizontal, and large eddies mix more fluid than small eddies. Eddies range in size from a few meters due to turbulence in the thermocline up to several hundred kilometers for geostrophic eddies discussed in Chapter 10.

In general, mixing depends on Reynolds number R (Tennekes 1990: p.11)

(8.35)

where γ is the molecular diffusivity of heat. Furthermore, horizontal eddy diffusivity is ten thousand to ten million times larger than the average vertical eddy diffusivity.

Equation (8.35) implies K_x ~ UL. This functional form agrees well with Joseph and Sender's (1958) analysis, as reported in (Bowden 1962) of spreading of radioactive tracers, optical turbidity, and Mediterranean Sea water in the North Atlantic. They report

where L is the distance from the source, and U is a constant.

The horizontal eddy diffusivity (8.35) also agrees well with more recent reports of horizontal diffusivity. Work by Holloway (1986) who used satellite altimeter observations of geostrophic currents, Freeland et al., (1975) who tracked SOFAR underwater drifters, McWilliams (1976) and Ledwell et al., (1998) who used observations of currents and tracers to find

Using (8.36) and the measured K_x implies eddies with typical scales of 80 km, a value near the size of geostrophic eddies responsible for the mixing.

Ledwell, Watson, and Law (1991) also measured a horizontal eddy diffusivity. They found

over scales of meters due to turbulence in the thermocline probably driven by breaking internal waves. This value, when used in (8.36) implies typical lengths of 100 m for the small eddies responsible for mixing in this experiment.

Comments on horizontal mixing

1. Horizontal eddy diffusivity is 10⁵ — 10⁸ times larger than vertical eddy diffusivity.
2. Water in the interior of the ocean seems to move along sloping surfaces of constant density with little local mixing until it reaches a boundary where it is mixed vertically. The mixed water then moves back into the open ocean again along surfaces of constant density (Gregg 1987).
   One particular case is particularly noteworthy. When water mixed downward through the base of the mixed layer flows out into the thermocline along surfaces of constant density, the mixing leads to the ventilated thermocline model of oceanic density distributions.
3. The observations of mixing in the ocean imply that numerical models of the oceanic circulation should use mixing schemes that have different eddy diffusivity parallel and perpendicular to surfaces of constant density, not parallel and perpendicular to level surfaces of constant z as we used above. Horizontal mixing along surfaces of constant z leads to mixing across layers of constant density because layers of constant density are inclined to the horizontal by about 10^-3 radians (see §10.7 and Figure 10.13).
   Studies by Danabasoglu, McWilliams, and Gent (1994) show that numerical models using isopycnal and diapycnal mixing leads to much more realistic simulations of the oceanic circulation.
4. Mixing is horizontal and two dimensional for horizontal scales greater than N H/(2f) where H is the water depth, N is the stability frequency, and f is the Coriois parameter (Dritschel, Juarez, and Ambaum (1999).

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