Chapter 8  Equations of Motion With
Viscosity
8.3 Calculation of Reynolds Stress
The Reynolds stresses such as ∂< u'
w' >/∂z are
called virtual stresses
(cf. Goldstein, 1965: §69 & §81) because
we assume that they play the same role as the viscous terms in the equation of
motion. To proceed further,
we need 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. Prandtl,
in a revolutionary paper published in 1904, stated that turbulent effects are
only important in a very thin layer close to the surface, the boundary layer.
Prandtl's invention of the boundary layer allows us to describe very accurately
turbulent flow of wind above the sea surface, or flow at the bottom boundary
layer of the ocean. See the box below.
To calculate flow in a boundary leyer, 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
∂ /∂ t =
∂ /∂x =
∂ /∂y =
0, and (8.9) can be written:
We now assume, in analogy with (8.2)
where A_{z} is an eddy viscosity
which replaces the molecular viscosity in (8.2). With this assumption,
where we have assumed that A_{z} is
either constant or that it
varies more slowly in the z direction than
∂U/∂z.
Later, we will assume that A_{z}∼ z.
The x and y momentum
equations for a homogeneous, steadystate, turbulent boundary layer above or
below a horizontal surface are:
where f = 2ω sin
φ is the Coriolis parameter, and we have dropped the molecular viscosity
term because it is much smaller than the turbulent eddy viscosity.
Note, (8.14b) follows from a similar derivation from the ycomponent of the
momentum equation. We will need (8.14) when we describe flow near the surface.
The assumption that an eddy viscosity A_{z} can be used to relate
the Reynolds stress to the mean flow works well for describing the flow near a horizontal surface where
U is a function of distance z from the
surface, and W, the mean velocity perpendicular to the surface is zero
(See the box Turbulent Boundary Layer Over a Flat Plate). This is the
approach first described in 1925 by Prandtl, who introduced the concept of a boundary layer, and by others.
Please notice that a value for A_{z} cannot be obtained from theory. Instead, it must be calculated from data
collected in wind tunnels or measured in the surface
boundary layer at sea. See Hinze (1975, §52 and §75) and Goldstein (1965: §80)
for more on the theory
of turbulence flow near a flat plate.
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
(18861975), L. Prandtl (18751953), and T. von Karman
(18181963) from 1915 to 1935. Their empirical theory,
sometimes called the mixinglength
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 a simplified 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
and the mean velocity varies linearly with distance above
the boundary. Usually (8.15) is written in dimensionless
form:
where u^{*2} ≡ T_{x }/ρ
is the friction velocity.
Further from
the boundary, the flow is turbulent, and molecular friction
is not important. In this regime, we
can use (8.12), and
Prandtl and Taylor assumed that large
eddies are more
effective in mixing momentum than small eddies, and therefore
A_{z} ought to vary with distance from
the wall. Karman assumed that it had the particular functional
form A_{z} = kzu^{*},
where k is a dimensionless
constant. With this assumption, the equation for the
mean velocity profile becomes
Because U is a function only of
z, we can
write dU = u^{*}/(kz) dz,
which has the solution
where z_{0} is distance from the boundary
at which velocity goes to zero.
For airflow over the sea, k = 0.4 and
z_{0} is given by Charnock's
(1955) relation z_{0} =
0.0156 u^{*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 (8.19), 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 (8.19) gives Charnock's
form of the drag coefficient as a function of wind speed
in Figure 4.6

Prandtl's theory based on assumption (8.12) 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:
Mixinglength and eddy viscosity models should be used only to generate analytical expressions for the
Reynolds stress and meanvelocity
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 mixinglength theory in turbulent flows whose scaling laws are not
known beforehand should be avoided.
Problems with the eddyviscosity approach:
1. 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 MoninObukhov scale L discussed
in §4.3, and
iii) the typical velocity U divided by the
Coriolis parameter
U / f.
2. The velocities u', w' are a property
of the fluid, while A_{z} is a
property of the flow;
3. Eddy viscosity terms are not symmetric:
< u' v' > = < v'
u' >;
but

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 with statistical
properties
that 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 A_{x},
A_{y}, and A_{z} 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 10^{11} 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
ν_{water} = 10^{6}m^{2}/s 
ν_{tar} at
15°C =
10^{6}m^{2}/s 
ν_{glacier ice} = 10^{10}m^{2}/s 
A_{y} = 10^{4}m^{2}/s 
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