Estimates of Mixing
Jody M. Klymak Jonathan D. Nash
May 14, 2007
Abstract
Vertical fluxes of tracers and momentum in the ocean are dominated by tur-
bulent transports. Since it is impossible to measure the complete set of motions
associated with turbulent processes, estimates of mixing rely on sets of assump-
tions. In this article, we describe the most common techniques used for estimating
ocean mixing, which may be based on (1) large- or (2) micro-scale observations.
Large-scale estimates may involve the purposeful release of a man-made tracer
and the subsequent measurement of its release over time. Alternatively, transport
budgets can be used to determine mixing if a closed system can be defined.
Microscale estimates generally rely on direct measurement of the turbulent mo-
tions and/or the consequences of these motions. Direct eddy correlations rely on
quantifying the turbulent stirring. In contrast, the Osborn-Cox method quantifies
the rate of irreversible molecular mixing associated with turbulent structures. The
most commonly used microscale estimate (the Osborn method) assumes an energy
balance in which turbulent energy production is balanced by turbulent buoyancy
flux and viscous dissipation. Several variants on these techniques are also dis-
cussed, including mixing estimates from density overturns (Thorpe-scale analyses)
and finescale shear based parameterizations (Gregg-Henyey/Polzin).
Introduction
Mixing in the ocean redistributes tracers, driving physical and biogeochemical dy-
namics. The mixing of the active tracers, temperature and salinity, changes the
density of seawater, creating pressure gradients that drive mean currents. Circula-
tions from small estuaries all the way up to the global overturning circulation are
ultimately controlled by the mixing of buoyant fresh or warm water. Momentum
is similarly diffused by turbulent mixing, which acts to transmit forces through the
ocean surface and boundaries into the interior. The mixing of passive scalars like
nutrients, carbon dioxide, and oxygen, is important to understanding biological
cycles in the ocean. Phytoplankton rely on vertical mixing processes to transport
recycled nutrients into the sunlit near-surface waters. The mixing of carbon diox-
ide ultimately affects its storage in the ocean and removal from the atmosphere.
In the interior of the ocean, most of the mixing takes place when internal waves
break, driving dense water over light. Breaking is the result of instabilities of the
internal wave flow due to focusing of wave energy. A numerical simulation of
a shear instability (Smyth et al., 2001) illustrates the anatomy of a mixing event
(figure 1). In this example, a Kelvin-Helmholtz billow is triggered from an initially
1

uniform shear-flow. As the flow evolves, a wave-like instability grows and rolls up
into two vortices (a) that pair to create a single breaking event that is initially
mostly two-dimensional (b). Further instabilities ensue, creating a fully turbulent
and three-dimensional flow field (c).
Breaking events like this are important because molecular diffusivity on large-
scale gradients (tens of meters) is very ineffective at mixing. Mixing is ultimately
accomplished by molecular processes via Fickian diffusion, i.e., the irreversible
flux of property C is proportional to its three-dimensional gradient and the molec-
ular diffusion coefficient κC:
fC =−κC∇C. (1)
For temperature, a thermodynamic tracer, κT ≈ 10−7 m2 s−1 and for salinity and
other tracers κS ≈ 10−9 m2 s−1. At large scales, representing the non-turbulent
flow, gradients are small and the molecular flux is slow. However, the stirring
driven by the breaking of finescale (order 10–1 m) waves, creates gradients at the
microscale (order 1 cm–1 mm). The microscale gradients can be very large, as can
be seen visually in figure 1b and c, and the molecular flux becomes significant.
1972 VOLUME 31J O U R N A L O F P H Y S I C A L O C E A N O G R A P H Y
FIG. 1. Cross sections of instantaneous flow fields from simulation 1. Temperature is represented in nondimensional form as !/!o in (a)–(d): (a) t " 565 s, (b) t " 1414 s, (c) t " 4242 s, and (d) t " 6222 s. The arrow in (d) corresponds to Fig. 2. The turbulent kinetic energy
dissipation rate # is shown at (e) t " 4242 s and (f ) t " 6222 s. The scalar variance dissipation rate $ is also shown at (g) t " 4242 s and(h) t " 6222 s.
was set to 0.0099 s%1. Most of our conclusions are dem-
onstrated using only the first three simulations shown
in Table 1.
Cross sections of the evolving flow fields in simu-
lation 1 are shown in Fig. 1. Figures 1a–d show scaled
temperature cross sections taken at four instants during
the simulation. Figures 1e and 1f show the kinetic en-
ergy dissipation rate # at the latter two instants, and
Figs. 1g and 1h show the scalar variance dissipation rate
at the same two instants. [The dissipation rates # and $
are defined explicitly in section 2c(3).] At t " 565 s(Fig. 1a), a pair of primary KH billows has grown and
Figure 1: A numerical simulation of turbulent mixing (Smyth et al., 2001). The event
is triggered by a shear instability between an upper layer of warm water (red) moving
to the right and a layer of cold water (blue) moving to the left. The initial pair of
vortices (a) pair to create a single large breaking event (b). This becomes fully turbulent
and three dimensional (c) at which point there is large irreversible diffusion of the
temperature. Diffusion continues until a large volume of mixed fluid results (d).
In this example, and in the ocean in general, turbulence acts to stir the fluid,
greatly increasing the flux due to mixing, FC. It is very useful to parametrize the
turbulent flux in terms of gradients of the mean fields C. We do this by defining a
turbulent diffusivity KC so that
FC ≈−KC∇C. (2)
2