Marine Benthic Boundary Layers
Shear Stress

Physical and Geological Perspective

Sediment transport in the marine environment is driven by fluid motion. Some important transport mechanisms are depicted in Figure 1 in the context of the continental shelf:

• Wind-driven alongshore (driving upwelling and downwelling) and across-shore
• Surface gravity waves
• Tidal currents (not depicted in figure)
• Internal waves
• Infragravity oscillations
• Buoyant Plumes (with both positive and negative buoyancy depending on sediment concentration)
• Wave driven surf zone processes (e.g., waves breaking, rip currents)

Different regimes exist across the continental shelf:

Figure 34-2

Context for Study of Boundary Layers

A boundary layer exhibits a velocity profile connecting a background oceanographic flow and a no-slip, zero velocity condition at a solid boundary.

Figure 34-3

Transfer of momentum toward the boundary imposes a shear stress; when it is sufficient (i.e., exceeds a critical value), it can cause transport of sediment either as bedload or suspended load.

We begin by considering how a moving fluid interacts with a solid boundary and the nature of the velocity profile.

Shear Stress

Stress, denoted t, is a force applied to a plane surface per unit area. Dimensionally:

Eq 34-1

Consider a volume element. Each of the three surfaces can have three forces acting on it, so there are nine components in all. They are denoted where the i subscript describes the surface (e.g., z would indicate the surface perpendicular to the z-axis) and the j subscript describes the direction of the force (e.g., x would indicate the force acting in the direction of the x-axis).

Figure 34-4

Eq 34-2

Of the remaining six components, we are interested in the two acting in the x and y directions on the plane perpendicular to the z-axis :

Eq 34-3

At the seabed these stresses are denoted and .

For simplicity we will consider flow in the x-direction that is steady () and uniform () so henceforth we denote as and the stress at the boundary as .

Consider the case of laminar flow (no turbulence) between a fixed surface and a plate of area A, separated by a height h, and moving at a steady velocity u (such that the applied force is opposed equally by the drag of the solid surface).

Figure 34-5

Eq 34-4:

For laminar flow (of a Newtonian fluid, see below), the velocity profile would be linear:

Eq 34-5:

The gradient of velocity which appears in this equation is called the shear. Since the stress is the force per unit area, we use Eq 34-4 and 34-5 to write:

Eq 34-6:

The proportionality constant µ is the absolute viscosity and is a measure of the resistance to deformation of a fluid. Dimensionally the stress has units of force per unit area (g cm/s)/(cm) and the velocity gradient units of (s) so the viscosity has units of g/(cm s). One g/(cm s) is one poise. In seawater the viscosity depends on temperature and salinity:

An alternative way of thinking about the stress due to shear is as a transfer of momentum in a direction perpendicular to the surface on which the stress is acting. Multiplying the numerator and denominator by r :

Eq 34-7

The stress is the product of the gradient of fluid momentum and a constant n, the kinematic viscosity.

The discussion to this point pertains to a laminar flow and is for a Newtonian fluid. We say a fluid is Newtonian when the viscosity as defined by the proportionality in Eq 34-6 is a constant.

Figure 34-6