Chapter 8 Contents

The
Influence of Viscosity (8.1)

Turbulence
(8.2)

Calculation
of Reynolds Stress (8.3)

Stability
(8.4)

Mixing
in the Ocean (8.5)

Important
Concepts (8.6)

Throughout most of the interior of the ocean and atmosphere
frictional forces are relatively small, and we can safely assume that the
flow is frictionless. At the boundaries, friction, in the form of
viscosity, becomes important. This thin,viscous layer is called a
boundary layer. Within the layer, the velocity of the flow slows
from values typical of the interior to zero at a solid boundary. If the
boundary is not solid, then the boundary layer is a thin layerof rapidly
changing velocity where by velocity on one side of the boundary changes to
match the velocity on the other side of the boundary. For example, there
is a boundary layer at the bottom of the atmosphere, the planetary
boundary layer we described in Chapter 3. Within the planetary boundary
layer, the velocity goes from many meters per second in the free
atmosphere to tens of centimeters per second at the sea surface. Below the
sea surface, another boundary layer, the Ekman layer described in Chapter
9, matches the flow at the sea surface to the deeper flow inside the
ocean.In this chapter we consider the role of friction in fluid flows, and
the stability of the flows to small changes in velocity or density.

The Influence of Viscosity

In the last chapter we wrote the x-component of the momentum
equation for afluid in the form:

(8.1)

where Fx was a frictional force per unit mass. Now we
can consider the form of this term if it is due to viscosity.

Molecules in a fluid close to a solid boundary can strike the boundary
and transfer momentum to the boundary (Figure 8.1). Molecules further from
the boundarycollide with molecules that have struck the boundary, further
transferring thechange in momentum into the interior of the fluid. This
transfer of momentum is molecular viscosity. Molecules, however,
travel only micrometers between collisions, and the process is very
inefficient for transferring momentum even a few centimeters. Molecular
viscosity is important only within a few millimeters of a boundary.

Molecular viscosity is the ratio of the stress Tx
tangential to the boundaryof a fluid and the shear of the fluid at the
boundary. Therefore, the stress has the form:

(8.2)

where v is the kinematic molecular viscosity. Typical value of
v for waterat 20° C is 10-6 m2/s.

Generalizing (8.2) to three dimensions leads to a stress tensor giving
the nine components of stress at a point in the fluid, including pressure,
which is a normal stress, and shear stresses. A derivation of the stress
tensor is beyond the scope of this book, but you can find the details in
Lamb (1945: §328) or Kundu (1990: p. 93). For an incompressible fluid, the
frictional force per unit mass in (8.1) takes the form:

(8.3)

This form of the frictional term of the momentum equation was first
published by Navier (1785-1836) in 1827.

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Figure 8.1
Molecules colliding with the wall andwith each other transfer momentum
from the fluid to the wall, slowing the fluid velocity.

Figure 8.2 Left: Reynolds apparatus for investigating
the transition to turbulence in pipe flow. Right: Photographs of near-laminar flow
(left) and turbulent flow (right) in a clearpipe much like the one used by
Reynolds. (From Binder 1953).

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Turbulence

If molecular viscosity is important only over distances of a few
millimeters, and if it is not important for most oceanic flows, unless of
course you are a zooplankter trying to swim in the ocean, how then is the
influence of a boundary transferred into the interior of the flow? The
answer is: through turbulence.

Turbulence arises from the
non-linear terms in the momentum equation . The importance of these terms isgiven by a
non-dimensional number, the Reynolds Number, which is the ratio ofthe
non-linear terms to the viscous terms:

(8.4)

where, U is a typical velocity of the flow and L is a
typical length describing the flow. You are free to pick whatever
U,L might be typical ofthe flow. For example L can be either
a typical cross-stream distance, or an along-stream distance. Typical
values in the open ocean are U = 0.1 m/s and L = 1
megameter, so Re = 1011. Because non-linear terms are
important if Re >10 - 1000, they are certainly important in the
ocean. The ocean is turbulent.

The Reynolds number is named after Osborne Reynolds (1842-1912) who
conducted experiments in the late 19th century to understand turbulence.
In one oft-cited experiment (Reynolds 1883), he injected dye into water
flowing at various speedsthrough a tube (Figure 8.2). If the speed was
small, the flow was smooth. This is called laminar flow. At higher
speeds, the flow became irregular and turbulent. The transition occurred
at Re = VD/v 2000, where V is the average speed in
the pipe, and D is the diameter of the pipe.

As Reynolds number increases above some critical value, the flow
becomes more and more turbulent. Note that flow pattern is a function of
Reynold's number. All flows with the same geometry and the same Reynolds
number have the same flow pattern. Thus flow around all circular
cylinders, whether 1 mm or 1 m in diameter, look the same as the flow at
the top of Figure 8.3 if the Reynolds number is 20. Furthermore, the
boundary layer is confined to a very thin layerclose to the cylinder, in a
layer too thin to show in the figure.

Figure 8.3
Flow past a circular cylinderas a function of Reynolds number between one
and a million (From Richardson1961). The appropriate flows are: A-a
toothpick moving at 1 mm/s; B-finger moving at 2 cm/s; F-hand out a car
window at 60 mph. All flow at the same Reynolds number has the same
streamlines. Flow past a 10 cm diameter cylinder at1 cm/s looks the same
as 10 cm/s flow past a cylinder 1 cm in diameter because inboth cases Re =
1000.

Turbulent Stresses: The Reynolds Stress
Those who studied fluid mechanics in the early 20th century
hypothesized that parcels of fluid in a turbulent flow played the same
role in transferring momentum within the flow that molecules played in
laminar flow. The work led to the idea of turbulent stresses.

To see how these stresses might arise, consider the momentum equation
for a flow with mean and a turbulent components of flow:

(8.5)

where the mean value U is calculated from a time or space
average:

(8.6)

The non-linear terms in the momentum equation can be written:

(8.7)

The second equation follows from the first because both and , which follow from the definition of
U: .

Using (8.7), the continuity equation splits into two equations:

(8.8a)

(8.8b)

And the x-component of the momentum equation becomes:

(8.9)

where 2Wv'sin
j has been dropped because it is small. Thus the turbulent
frictional term is:

(8.10)

The terms r<u' u'>, r<u' v'>, and r<u' w'> transfer momentum in the
x, y, and z directions. For example, the term r<u' w'> gives thedownward transport of
eastward momentum across a horizontal plane. Because they transfer
momentum, and because they were first derived by Osborne Reynolds, theyare
called Reynolds Stresses.

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Calculation of Reynolds
Stress:

The friction terms such as are virtual stresses (cf. Goldstein, 1965: 69
& 80). We now assume that they play the same role as the viscous terms
in the equation of motion. The problem becomes one of obtaining values or
functional form for the Reynolds stress.Several approaches are used.

By Analogy with Molecular Viscosity Let's return to the simple
example shown in figure 8.1, which shows a boundary layer above a flat
plate in the x, y plane. Now let's assume that the flow
above the plate is turbulent. This is a very common type of boundary layer
flow,and it a type of flow that we will describe various times in later
chapters. It can be wind flow above the sea surface or flow at the bottom
boundary layer in theocean or flow in the mixed layer at the sea
surface.

For flow above a boundary, we assume that flow is constant
in the x, y direction, that the statistical properties of
the flow vary only in the z direction, and that the mean flow is
steady. Therefore . The stress term is:

(8.11)

and

(8.12)

We now assume, in analogy with (8.2)

(8.13a)

(8.13b)

where Az is the eddy viscosity which replaces
the molecular viscosity v in (8.2). Then

(8.14a)

(8.14b)

assuming Az is either constant or that it varies more
slowly in the z direction than . Thus, we will assume later that
Azz.

With these assumptions, the x and y momentum equations
are:

(8.15a)

(8.15b)

where f = 2wsinj is the Coriolis parameter.

The assumption that
Az varies with distance from the boundary works well for
describing the flow over flat plates where U is a function of
distance z from the plate, and W, the mean velocity
perpendicular to the plate is zero (See the box Turbulent
Boundary Layer Over a Flat Plate).This is the classical approach
first described in 1925 by Prandtl, who introduced the concept of a
boundary layer, and by others. Here Az is determined by
an empirical fit to data collected in wind tunnels or measured in the
surface boundary layer at sea. See Hinze (1975, §5-2 and §7-5) and
Goldstein (1965: §80) for more on the theory of turbulence flow neara flat
plate.Assumption (8.13) and the classical theory works well only where
friction is much larger than the Coriolis force. This is true for air flow
within tens of meters of the sea surface and for water flow within a few
meters of the surface. The application of the technique to other flows in
the ocean is less clear. For example, the flow in the mixed layer at
depths below about ten meters is less well described by the classical
turbulent theory. Tennekes and Lumley (1970: 57) write:

Mixing-length and eddy viscosity models should be used only to
generate analytical expressions for the Reynolds stress and
mean-velocity profile if those are desired for curve fitting purposes in
turbulent flows characterized by a single length scale and a single
velocity scale. The use of mixing-length theory in turbulent flows whose
scaling laws are not known before hand should be avoided.

Problems with the eddy-viscosity approach:

Except in boundary layers a few meters thick, geophysical flows may
be influenced by several characteristic scales. For example, in the
atmospheric boundary layer above the sea, at least three scales may be
important: i) the height above the sea z, ii) the Monin-Obukhov
scale L discussed in §4.3, and iii) the typical velocity U
divided by the Coriolis parameter U/f.

The velocities u',\,w' are a property of the fluid, while
Az is a property of the flow;

Eddy viscosity terms are not symmetric:

The Turbulent Boundary Layer Over a
Flat Plate

The theory for the mean velocity distribution in a
turbulent boundary layer over a flat plate was worked out
independently by G.I.Taylor (1886-1975), L. Prandtl
(1875-1953), and T. von Karman (1818-1963) from 1915 to 1935.
Their empirical theory, sometimes called the mixing-length
theory predicts well the mean velocity profile close to
the boundary. Of interest to us, it predicts the mean flow of
air above the sea. Here's asimplified version of the theory
applied to a smooth surface.

We begin by assuming that the mean flow in the boundary
layer is steady and that it varies only in the z
direction. Within a few millimeters of the boundary, friction
is important and (8.2) has the solution

(8.16)

and the mean velocity varies linearly with distance above
the boundary. Usually(8.16) is written in dimensionless
form:

(8.17)

where u*2 Tx/r is the friction velocity.

Further from the boundary, the flow is turbulent, and
molecular friction is not important. In this regime, we can
use (8.13), and

(8.18)

Prandtl and Taylor assumed that large eddies are more
effective in mixing momentum than small eddies, and therefore
Az ought to vary with distance from the
wall. Karman assumed that it had the particular functional
form Az =kzu*, where k is a dimensionless constant. With this
assumption, the equation for the mean velocity profile
becomes

(8.19)

Because U is a function only of z, we can
write dU =u*/(kz)dz, which has the
solution

(8.20)

where z0 is distance from the boundary at
which velocity goes to zero.

For airflow over the sea, k = 0.4
and z0 is given by Charnock's (1955)
relation z0 =
0.0156u*2/g. The mean velocity in the
atmospheric boundary layer just above the sea surface
described in § 4.3 fits well the logarithmic profile of
(8.20), as does the mean velocity in the upper few meters of
the sea just below the sea surface. Furthermore, using (4.1)
in the definition of the friction velocity, then using (8.20)
gives Charnock's form ofthe drag coefficient as a function of
wind speed in Figure 4.6

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From a Statistical Theory of Turbulence The Reynolds
stresses can be calculated from various theories which relate <
u'u'> to higher order correlations of the form
<u'u'u'>. The problem then becomes: How
to calculate the higher order terms? This is the closure problem in
turbulence. There is no general solution, but the approach leads to useful
understanding of some forms of turbulence such as isotropic turbulence
downstream of a grid in a wind tunnel (Batchelor 1967). - Isotropic
turbulence is turbulence who statistical properties are independent of
direction.

The approach can be modified somewhat for flow in the
ocean. In the idealized case of high Reynolds flow, we can calculate the
statistical properties of a flow in thermodynamic equilibrium. Because the
actual flow in the ocean is far from equilibrium, we assume it will evolve
towards equilibrium. Holloway(1986) provides a good review of this
approach, showing how it can be used to derive the influence of turbulence
on mixing and heat transports. One interesting result of the work is that
zonal mixing ought to be larger than meridional mixing.

Summary The turbulent eddy viscosities Ax,
Ay, and Az cannot be calculated
accurately for most oceanic flows.

They can be estimated from measurements of turbulent flows.
Measurements in the ocean, however, are difficult; and measurements in
the lab, although accurate, cannot reach Reynolds numbers of
1011 typical of the ocean.

The statistical theory of turbulence gives useful insight into the
role of turbulence in the ocean, and this is an area of active research.

Some Values for Viscosity

vwater

=10-6
m2/s

vtar at 15 °C

=106 m2/s

vglacier ice

=1010 m2/s

Ay

=104
m2/s

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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.

Static Stability and the Stratification Frequency Consider a
parcel of water that is displaced vertically in a stratified fluid (Figure
8.4). The buoyancy force F acting on the displaced parcel is the
difference between its weight Vgr' and the
weight of the surrounding water Vgr2, where V is the volume of the
parcel:

F = Vg(r2-r')

The acceleration of the displaced parcel is:

(8.21)

but

(8.22)

(8.23)

Using (8.22) and (8.23) in (8.21), ignoring terms proportional to dz2, we obtain:

We define stability E -a/g for dz = 1:

(8.24)

In the upper kilometer of the ocean stability is large, and the first
term in (8.24) 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

(8.25)

Recalling that r(S,T,0) - 1000 = st, we can also write:

(8.26)

The approximation used to derive (8.25) and (8.26) is valid for
E > 50x10-8/m.
Deeper 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. At these depths, a more accurate form
of (8.17) is:

(8.27)

where G is the adiabatic temperature gradient.
See Sverdrup, Johnson, and Fleming (1942: 416) or Gill (1982: 50) for a
more complete derivation.

Stability is defined such that

E > 0

Stable

E = 0

Neutral Stability

E < 0

Unstable

In the upper kilometer of the ocean, z < 1,000 m, E =
(100-1000) x 10-8/m, and in
deep trenches where z > 7,000 m, E = 1 x 10-8/m.

The influence of stability is usually expressed by a stability
frequency N:

(8.28)

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 r, and (8.18) 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).

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

(8.29)

where the numerator is the strength of the static stability, and the
denominator is the strength of the velocity shear.

Ri > 0.25

Stable

Ri <0.25

Velocity Shear Enhances
Turbulence

Note that a small Richarson 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).

Figure 8.6
Billow clouds showing a Kelvin-Helmholtz instability at the top of a
stable atmospheric boundary layer near Denver, Colorado (From Drazin and
Reid1981). Note that the billows become large enough that more dense air
overlies less dense air, and the billows collapse into turbulence.

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 (Figure8.7). 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, less-salty layer below,
and the water in the layer sinks. Because the layer is thin, the fluid
sinks in fingers acentimeter or so in diameter 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 three other variations on this theme. Two variables taken two
at a timeleads to four possible combinations:

Warm salty over colder less salty: Leads to salt fingers. Found in
central waters of sub-tropical gyres, western tropical North Atlantic,
and the North-east Atlantic beneath the outflow from the Mediterranean
Sea.

Colder less salty over warm salty: Double diffusion sharpens the
interface between the layers, leading to a stair-step increase of
density with 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.
Eventually step discontinuities of density are created that are
separated by layers of constant density. This process is called
diffusive convection.

Cold salty over warmer less salty: Always statically unstable.

Warmer less salty over cold salty: Always stable and double
diffusion diffuses the interface between the two layers.

Examples of double diffusion are common in the ocean. We saw in Chapter
7 that warm, salty, water flows into the Atlantic through the Strait of
Gibraltar. The water is more dense than the colder but fresher waters on
the surface of the North Atlantic. The Mediterranean Sea water sinks to a
depth of a about a kilometer, where it has the same density as the
surrounding water. It then spreads into the North Atlantic as a warm,
salty intrusion between colder, less-salty water. Double diffusion leads
to density steps at the top of the intrusion and salt fingering below the
intrusion.

Salt fingering may seem to be unimportant because salt fingers are so
small. Yet the opposite is true. Unlike turbulent mixing, the process
occurs continuously, and the net effect over many yaers is large. Salt
fingering and double diffusion may be the most inportant processes
producing vertical mixing in large regions of the ocean.

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.

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Mixing in the Ocean

Instability in the ocean leads to mixing. Because the ocean has stable
stratification and any 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.

In the ocean, mixing by turbulent eddies is far more important than
mixing by molecular diffusion (Munk 1966). The equation for vertical
mixing by eddies of a tracer q such as salt or
temperature is:

(8.30)

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

Average Vertical Mixing Average mixing rates in the ocean have
been calculated for many years from the distribution of mean properties in
the ocean. Munk (1966) considered the important case of the thermocline.
He noted that the thermocline is constant in time. Measurements of
temperature as a function of depth made decades apart showed the same
structure (Figure 8.8). For a steady-state thermocline with no sources or
sinks of heat, (8.30) reduces to:

(8.31)

where T is temperature as a function of depth in the
thermocline.The steady-state thermocline requires that the downward mixing
of heat by turbulence be balanced by an upward transport of heat by a mean
vertical current W.

The equation has the solution:

T T0exp( z/H)

(8.32)

where H=Kz/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. An exponential function fit through
the observations of T(z) gives H,from which
Kz can be calculated if W is known.

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 x
10-4 years-1. The length and time scales gave

W =1.2 cm/day and

<Kz> = 1.3 x 10-4 m2/s

Average Vertical Eddy Diffusivity

(8.33)

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.

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 (bullet), and by the
Thompson in 1985 (square).
Data from Atlas of Ocean Sections produced by Swift, Rhines, and Schlitzer.

Measured Vertical Mixing Direct observations of 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 (Gregg1991), and ii) the
distribution of tracers such as sulpher hexafluoride(SF6) with
concentrations of 10-15 mole (Ledwell, Watson, and Law
1993).

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

Kx 1 x 10-5 m2/s

Open-Ocean Vertical Eddy Diffusivity

(8.34)

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

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
hunderd kilometers for geostrophic eddies discussed in Chapter 10.

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

(8.35)

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

Equation (8.35) implies Kx ~ 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 theNorth Atlantic. They report

Kx = PL

(8.36)

10 km < L < 1500 km

P = 0.01 ± 0.005 m/s

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

The horizontal eddy diffusivity (8.36) 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 floats, and
McWilliams (1976) who used observations of currents, calculated

Kx 8 x 102 m2/s

Geostrophic Horizontal Eddy Diffusivity

(8.37)

Using (8.36) and the measured Kx 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

Kx 1 - 3 m2/s

Open-Ocean Horizontal Eddy Diffusivity

(8.38)

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.

Processes Causing Turbulent Mixing The large discrepancy between
the mean eddy diffusivity for vertical mixing and the observed values in
the deep ocean led to further experiments to find resolve the difference.
Two recent experiments are especially interesting.

Polzin et al. (1997) measured the vertical structure of temperature
in the Brazil Basin in the South Atlantic. They found
Kz > 10-3 m2/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 diffusivities along the flank of the
seamount. Summing the influence over all Pacific seamounts, they found,
however, that the mixing near seamounts does not account for Munk's
basin-wide average.

Comments

The open-ocean experiments noted above 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). Over rough bathymetric features

Kz 10-3 m2/s

Rough Bottom Vertical Eddy Diffusivity

(8.39)

The results imply that
mixing occurs mostly at oceanic boundaries: along continental slopes,
above seamounts and mid-ocean ridges, at fronts, and in the mixed layer
at the sea surface. 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 1985). One particular case is particularly note
worthy. 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.

The observations of mixing in the ocean imply that numerical models
of the oceanic circulation should use mixing schemes that have different
eddy diffusivities 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.

The observed mixing in the open ocean away from bundaries is too
small to account for the mixing calculated by Munk. Recent work reported
at theWorld Ocean Circulation Experiment Conference on Circulation and
Climate 1998 and by Munk and Wunsch (1998) indicate thet the dilemma may
be resolved severalways:

First, separate studies by Gargett, Salmon, and Marotzke show that
we must separate the concept of deep convection from that of the
meridional overturning circulation (see chapter 13). Deep convection
may mix properties not mass, and the mass of upwelled water required
by Munk may be overestimated, and the vertical mixing needed to
balance the upwelling may be smaller than he calculated.

Second, mixing probably takes place along boundaries or in the
source regions for thermocline waters (Gnadadesikan, 1999). 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 at a depth of 1100 m. Thus water at 1200 m and at 1100
m may reach their position along entirely different paths.

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Important Concepts

Friction in the ocean is important only over distances of a few
millimeters. For most flows, friction can be ignored.

The ocean is turbulent for all flows whose typical dimension exceeds
a few centimeters, yet the theory for turbulent flow in the ocean is
poorly understood.

The influence of turbulence is a function of the Reynolds number of
the flow. Flows with the same geometry and Reynolds number have the same
streamlines.

Oceanographers assume that turbulence influences flows over
distances greater than a few centimeters in the same way that molecular
viscosity influences flow over much smaller distances.

The influence of turbulence leads to Reynolds stress terms in the
momentum equation.

The influence of static stability in the ocean is expressed as
afrequency, the stability frequency. The larger the frequency, the more
stable the water column.

The influence of shear stability is expressed through the Richardson
number. The greater the velocity shear, and the weaker the static
stability, the more likely the flow will become turbulent.

Molecular diffusion of heat is much faster than the diffusion of
salt. This leads to a double-diffusion instability which modifies the
density distribution in the water column in many regions of the ocean.

Instability in the ocean leads to mixing. Mixing across surfaces of
constant density is much smaller than mixing along such surfaces.

Calculations of the average eddy diffusivity in the interior of the
oceanis much smaller than measured diffusivity.

Measurements of eddy diffusivity indicate water is mixed vertically
near oceanic boundaries such as above seamounts and mid-ocean ridges.
This may explain the small measured values of open-ocean diffusivity.

chapter
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Department of Oceanography, Texas A&M
University
Robert H. Stewart, [email-masked]
All contents copyright © 1996, 1998 Robert H. Stewart, All
rights reserved
Updated

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