**1.3 "Missing mixing" problem**

148 Thermodynamics – Interaction Studies – Solids, Liquids and Gases

Sustained abyssal circulation is a manifestation of conversion of potential energy to kinetic energy within the system. Production of potential energy is mainly the result of diapycnal mixing in the ocean interior, geothermal heating through the ocean floor, and the meridional

Diapycnal mixing results from turbulent diffusion by wind and tides. The most reasonable mechanism to transfer energy from the surface to the deeper layer is regarded as breaking and wave–wave interaction of internal waves generated by wind and tides (e.g., Muller & Briscoe, 2000). The wind and tidal dissipation quantities have been estimated respectively as about 1 TW (Wunsch, 1998) and 1 TW (Egbert & Ray, 2000). Using these estimates and *R*f = 0.15 (Osborn, 1980) as the flux Richardson number, γ= *R*f/(1-*R*f)=0.18 as the ratio of potential energy to available energy, and *S*=3.6 × 1014 m2 as the total surface area of the ocean, the production of potential energy caused by diapycnal mixing has been estimated as about 1.0

Geothermal heating through the ocean floor causes a temperature increase and a thermal expansion in seawater, and generates potential energy. Production of potential energy caused by geothermal heating has been estimated as about 0.11 (Gade & Gustafsson, 2004) -

Precipitation (evaporation) is a flux of mass to (from) the sea surface and consequently a flux of potential energy. On average, the warm (cold) tropics with high (low) sea level are regions of evaporation (precipitation). These therefore tend to reduce the potential energy. The value integrated for the entire ocean shows a net loss of potential energy. Loss of potential energy attributable to precipitation, evaporation, and runoff has been estimated as less than 0.02 (Gade & Gustafsson, 2004) – 0.03 (Huang, 1998) × 10-3 W m-2. These

distribution of precipitation, evaporation, and runoff (e.g., Gade & Gustafsson, 2004).

Fig. 1. Illustration of oceanic general circulation (Broecker, 1987).

**1.2 Energy sources of abyssal circulation** 

× 10-3 W m-2 (=2TW/(3.6 × 1014 m2) × 0.18).

0.14 (Huang, 1999) × 10-3 W m-2.

contributions can be negligible.

Munk (1966) estimated that the magnitude of diapycnal mixing to drive and maintain abyssal circulation is about K≈10-4 m2 s-1. He reached that figure by fitting of vertical profiles of tracers with one-dimensional vertical balance equation of advection and diffusion as

$$K\frac{\text{d}^2T}{\text{d}z^2} = w\frac{\text{d}T}{\text{d}z} \,\text{'} \tag{1}$$

where *K* is a diapycnal mixing coefficient, *T* denotes a tracer variable such as temperature, salinity and radioactive tracers, *z* signifies a vertical coordinate, and *w* represents the upwelling velocity. The estimated value has been regarded as reasonable because the total upwelling of deep water estimated using the above *K* is consistent with the total sinking of deep water estimated by observations in the sinking area.

However, some direct observations of turbulence (Gregg, 1989) and dye diffusion (Ledwell et al., 1993) in the deep ocean indicate a diapycnal mixing of only *K*≈10-5 m2 s-1. Moreover, this is consistent with mixing estimated from the energy cascade in an internal wave spectrum (called "background") (McComas & Mullar, 1981). This difference of *K* is designated as the "missing mixing" problem.

On the other hand, recent observations of turbulence show larger diapycnal mixing of *K*≥10-4 m2 s-1 (Ledwell et al., 2000; Polizin et al., 1997), although such observations are limited to areas near places with large topographic changes such as seamounts (called "hot spots"), where internal waves are strongly generated as sources of diapycnal mixing. Munk & Wunsch (1998) reported that the value averaged over the entire ocean including "background" and "hot spots" can be about *K*≈10-4 m2 s-1, which remains controversial.
