8.3: Calculation of Reynolds Stress

The Reynolds stresses such as \(\partial \langle u' w' \rangle \partial z\) 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.

From Experiments

We can calculate Reynolds stresses from direct measurements of \((u', v', w')\) made in the laboratory or ocean. This is accurate, but hard to generalize to other flows. So we seek more general approaches.

By Analogy with Molecular Viscosity

Let’s return to the example in figure \(8.1.1\), which shows flow above a surface in the \(x, y\) plane. Prandtl, in a revolutionary paper published in 1904, stated that turbulent viscous 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 in the ocean, or flow in the mixed layer at the sea surface. See the box Turbulent Boundary Layer Over a Flat Plate.

To calculate flow in a boundary layer, 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 \(\partial/\partial t = \partial/\partial x = \partial/\partial y = 0\), and \((8.2.8)\) can be written: \[2\Omega \ V \sin \varphi + \frac{\partial}{\partial z} \left[\nu \frac{\partial U}{\partial z} - \langle u'w' \rangle\right] = 0 \nonumber \]

We now assume, in analogy with \((8.1.2)\): \[-\rho \langle u'w' \rangle = T_{xz} = \rho A_{z} \frac{\partial U}{\partial z} \nonumber \]

where \(A_{z}\) is an eddy viscosity or eddy diffusivity which replaces the molecular viscosity \(\nu\) in \((8.1.2)\). With this assumption, \[\frac{\partial T_{xz}}{\partial z} = \frac{\partial}{\partial z} \left(A_{z} \frac{\partial U}{\partial z}\right) \approx A_{z} \frac{\partial^{2} U}{\partial z^{2}} \nonumber \]

where I have assumed that \(A_{z}\) is either constant or that it varies more slowly in the \(z\) direction than \(\partial U/\partial z\). Later, I will assume that \(A_{z} \approx z\).

Because eddies can mix heat, salt, or other properties as well as momentum, I will use the term eddy diffusivity. It is more general than eddy viscosity, which is the mixing of momentum.

The \(x\) and \(y\) momentum equations for a homogeneous, steady-state, turbulent boundary layer above or below a horizontal surface are: \[\begin{align} \rho f V + \frac{\partial T_{xz}}{\partial z} = 0 \\ \rho f U - \frac{\partial T_{yz}}{\partial z} = 0 \end{align} \nonumber \]

where \(f = 2 \omega \sin \varphi\) is the Coriolis parameter, and I have dropped the molecular viscosity term because it is much smaller than the turbulent eddy viscosity. Note, \((\PageIndex{5})\) follows from a similar derivation from the \(y\)-component of the momentum equation. We will need \((\PageIndex{4}) - (\PageIndex{5})\) 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 in turbulent boundary layers. However, \(A_{z}\) cannot be obtained from theory. It must be calculated from 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 near a flat plate.

The Turbulent Boundary Layer over a Flat Plate

The revolutionary concept of a boundary layer was invented by Prandtl in 1904 (Anderson, 2005). Later, the concept was applied to flow over a flat plate by G.I. Taylor (1886–1975), L. Prandtl (1875–1953), and T. von Karman (1881–1963) who worked independently on the theory from 1915 to 1935. Their empirical theory, sometimes called the mixing-length theory, works well to predict 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 \((8.1.2)\) has the solution \[U = \frac{T_{x}}{\rho \nu} z \nonumber \]

and the mean velocity varies linearly with distance above the boundary. Usually \((\PageIndex{6})\) is written in dimensionless form: \[\frac{U}{u^{*}} = \frac{u^{*} z}{\nu} \nonumber \]

where \(u^{*2} \equiv T_{x}/\rho\) is the friction velocity.

Further from the boundary, the flow is turbulent, and molecular friction is not important. In this regime, we can use \((\PageIndex{2})\), and \[A_{z} \frac{\partial U}{\partial z} = u^{*2} \nonumber \]

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 \(A_{z} = \kappa z u^{*}\), where \(\kappa\) is a dimensionless constant. With this assumption, the equation for the mean velocity profile becomes \[\kappa z u^{*} \frac{\partial U}{\partial z} = u^{*2} \nonumber \]

Because \(U\) is a function only of \(z\), we can write \(dU = u^{*} / (\kappa z) \ dz\), which has the solution \[U = \frac{u^{*}}{\kappa} \ln \left(\frac{z}{z_{0}}\right) \nonumber \]

where \(z_{0}\) is distance from the boundary at which velocity goes to zero.

For airflow over the sea, \(\kappa = 0.4\) and \(z_{0}\) is given by Charnock’s (1955) relation: \(z_{0} = 0.0156 u^{*2}/g\). The mean velocity in the boundary layer just above the sea surface described in Section 4.3 fits the logarithmic profile of \((\PageIndex{10})\) well, as does the mean velocity in the upper few meters of the sea just below the sea surface. Furthermore, using \((4.6.1)\) in the definition of the friction velocity, then using \((\PageIndex{10})\) gives Charnock’s form of the drag coefficient as a function of wind speed.

Prandtl’s theory based on assumption \((\PageIndex{2})\) 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 (1990: 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 beforehand should be avoided.

Problems with the eddy-viscosity 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 Monin-Obukhov scale \(L\) discussed in Section 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: \[\begin{align*} \langle u'v' \rangle &= \langle u'v' \rangle; \quad \text{but} \\ A_{x} \frac{\partial V}{\partial x} &\neq A_{y} \frac{\partial U}{\partial y} \end{align*} \]

From a Statistical Theory of Turbulence

The Reynolds stresses can be calculated from various theories which relate \(\langle u'u' \rangle\) to higher order correlations of the form \(\langle u'u'u' \rangle\). 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.

1. They can be estimated from measurements of turbulent flows. Measurements in the ocean, however, are difficult. Measurements in the lab, although accurate, cannot reach Reynolds numbers of \(10^{11}\), which are typical of the ocean.
2. The statistical theory of turbulence gives useful insight into the role of turbulence in the ocean, and this is an area of active research.

Table \(\PageIndex{1}\). Some Values for Viscosity.

\(\nu_{water}\)	=	\(10^{-6} \ \text{m}^{2}/\text{s}\)
\(\nu_{tar \ at \ 15^{\circ}\text{C}}\)	=	\(10^{6} \ \text{m}^{2}/\text{s}\)
\(\nu_{glacier / ice}\)	=	\(10^{10} \ \text{m}^{2}/\text{s}\)
\(A_{y, ocean}\)	=	\(10^{4} \ \text{m}^{2}/\text{s}\)
\(A_{z, ocean}\)	=	\(\left(10^{-5} - 10^{-3}\right) \ \text{m}^{2}/\text{s}\)