The Inertial Boundary Layer
47
the fluid in the direction of flow. Were it otherwise, the sublayer would tend to
stall next to the wall and separate from the boundary, and the structure
hypothesized in Fig. 2.8.1 would break down.
It is important to note that most of the streamlines in the western
boundary layer do not go through the sublayer in this picture. They remain in
the outer inertial layer. This means that for fluid elements on those streamlines
dissipation still remains unimportant. However, as we have already noted,
dissipation must be important on every streamline. It therefore appears unlikely that in the limit of large boundary layer Reynolds number the structure
of the motion can remain split between an inertial region and a dissipative
region throughout the passage of fluid through the boundary layer. The resolution of this perplexing difficulty forms the crux of the remaining discussion
of this chapter.
To confront this question we must go beyond the heuristic discussion of
this section and consider a detailed analysis of the boundary layer dynamics for
arbitrary Reynolds number. First, however, we describe an additional important feature of the purely inertial theory.
The Fofonoff Mode
The purely inertial model contains a free, nonlinear mode of motion first
discovered by Fofonoff (1954). Fofonoff looked for a solution of (2.2.9) in the
absence of both forcing and dissipation. The mode is therefore a free, unforced
mode of motion for steady flow. It must be nonlinear, i.e., relative vorticity
must be important in the solution; otherwise the total vorticity would be simply
f3y. For inertial motion, isolines of total vorticity and streamlines coincide. If
the total vorticity were just {3y, the isolines of total vorticity would be latitude
circles and intersect the eastern boundary where t/J = 0. There can be no flow
emanating from the eastern boundary so that the only way lines of constant y
and lines of constant t/J could coincide is if t/1 were 0 everywhere. This would
imply no motion.
In the purely inertial limit (2.2.9) reduces to:
(2.8.11)
which is the generalization of (2.8.1) to flow in the entire basin. In analogy with
(2.8.2), a first integral of (2.8.11) is just the statement that the total vorticity is
constant on streamlines, that is:
(2.8.12)
Whereas the function Q( t/1) could be deduced from the structure of the interior
field for the boundary layer problem, here Q(t/1) is completely arbitrary.
Fofonoff arbitrarily chose the linear form:
47
the fluid in the direction of flow. Were it otherwise, the sublayer would tend to
stall next to the wall and separate from the boundary, and the structure
hypothesized in Fig. 2.8.1 would break down.
It is important to note that most of the streamlines in the western
boundary layer do not go through the sublayer in this picture. They remain in
the outer inertial layer. This means that for fluid elements on those streamlines
dissipation still remains unimportant. However, as we have already noted,
dissipation must be important on every streamline. It therefore appears unlikely that in the limit of large boundary layer Reynolds number the structure
of the motion can remain split between an inertial region and a dissipative
region throughout the passage of fluid through the boundary layer. The resolution of this perplexing difficulty forms the crux of the remaining discussion
of this chapter.
To confront this question we must go beyond the heuristic discussion of
this section and consider a detailed analysis of the boundary layer dynamics for
arbitrary Reynolds number. First, however, we describe an additional important feature of the purely inertial theory.
The Fofonoff Mode
The purely inertial model contains a free, nonlinear mode of motion first
discovered by Fofonoff (1954). Fofonoff looked for a solution of (2.2.9) in the
absence of both forcing and dissipation. The mode is therefore a free, unforced
mode of motion for steady flow. It must be nonlinear, i.e., relative vorticity
must be important in the solution; otherwise the total vorticity would be simply
f3y. For inertial motion, isolines of total vorticity and streamlines coincide. If
the total vorticity were just {3y, the isolines of total vorticity would be latitude
circles and intersect the eastern boundary where t/J = 0. There can be no flow
emanating from the eastern boundary so that the only way lines of constant y
and lines of constant t/J could coincide is if t/1 were 0 everywhere. This would
imply no motion.
In the purely inertial limit (2.2.9) reduces to:
(2.8.11)
which is the generalization of (2.8.1) to flow in the entire basin. In analogy with
(2.8.2), a first integral of (2.8.11) is just the statement that the total vorticity is
constant on streamlines, that is:
(2.8.12)
Whereas the function Q( t/1) could be deduced from the structure of the interior
field for the boundary layer problem, here Q(t/1) is completely arbitrary.
Fofonoff arbitrarily chose the linear form: