Surface layer

The surface layer is the layer in a fluid where the scale of turbulent eddy is limited by the eddies' proximity to an interface. The objects highlighted in white above are turbulent eddies whose size is constrained by the proximity of the center of each eddy to the surface.

The surface layer is the layer of a turbulent fluid most affected by interaction with a solid surface or the surface separating a gas and a liquid where the characteristics of the turbulence depend on distance from the interface. Surface layers are characterized by large normal gradients of tangential velocity and large concentration gradients of any substances (temperature, moisture, sediments et cetera) transported to or from the interface.

The term boundary layer is used in meteorology and in physical oceanography. The atmospheric surface layer is the lowest part of the atmospheric boundary layer (typically the bottom 10% where the log wind profile is valid). The ocean has two surface layers: the benthic, found immediately above the sea floor and the marine surface layer, at the air-sea interface.

Mathematical Formulation

A simple model of the surface layer can be derived by first examining the turbulent momentum flux through a surface. Using Reynolds Decomposition to express the horizontal flow in the $x$ direction as the sum of a slowly varying component,${\overline {u}}$, and a turbulent component,$\ u'$,:

$u={\overline {u}}+u'$

and the vertical flow, $\ w$, in an analogous fashion:

$w={\overline {w}}+w'$

we can express the flux of turbulent momentum through a surface, $\ u_{*}$ as the time averaged magnitude of vertical turbulent transport of horizontal turbulent momentum, $\ u'w'$:

$u_{*}^{2}=\left|{\overline {(u'w')_{s}}}\right|$.

If the flow is homogeneous within the region, we can set the product of the vertical gradient of the mean horizontal flow and the eddy viscosity coefficient $\ K_{m}$ equal to $\ u_{*}$:

$\ K_{m}{\frac {\partial {\overline {u}}}{\partial z}}=u_{*}^{2}$,

where $\ K_{m}$ is defined in terms of Prandtl's mixing length hypothesis:

$\ K_{m}={\overline {\xi '^{2}}}\left|{\frac {\partial {\overline {u}}}{\partial z}}\right|$

where $\ \xi '$ is the mixing length.

We can then express $\ u_{*}$ as:

${\frac {\partial {\overline {u}}}{\partial z}}={\frac {u_{*}}{\overline {\xi '}}}$.

Assumptions about the mixing length

From the figure above, we can see that the size of a turbulent eddy near the surface is constrained by its proximity to the surface; turbulent eddies centered near the surface cannot be as large as those centered further from the surface. From this consideration, it is reasonable to assume that the mixing length, $\ \xi '$ is proportional to the eddy's depth in the surface:

$\ \xi '=kz$,

where $\ z$ is the depth and $\ k$ is known as the von Kármán constant. Thus the gradient can be integrated to solve for $\ {\overline {u}}$:

${\overline {u}}={\frac {u_{*}}{k}}\ln {\frac {z}{z_{o}}}$.

So we see that the mean flow in the surface layer has a logarithmic relationship with depth.

The Surface layer in oceanography

The surface layer is studied in oceanography, as both the wind stress and action of surface waves can cause turbulent mixing necessary for the formation of a surface layer.

Discrepancies with traditional theory

The logarithmic flow profile has long been observed in the ocean, but recent, highly sensitive measurements reveal a sublayer within the surface layer in which turbulent eddies are enhanced by the action of surface waves. It is becoming clear that the surface layer of the ocean is only poorly modeled as being up against the "wall" of the air-sea interaction. Observations of turbulence in Lake Ontario reveal under wave-breaking conditions the traditional theory significantly underestimates the production of turbulent kinetic energy within the surface layer.

Diurnal cycle

After nighttime convection over the ocean, the turbulent surface layer is found to completely decay and restratify. The decay is caused by the decrease in solar insolation, divergence of turbulent flux and relaxation of lateral gradients.

See also

• Boundary layer
• Mixed layer
• Density
• Salinity