THE NEAR-SURFACE LAYER OF THE OCEAN
The upper ocean boundary layer is turbulent, yet it reveals features of
organization. From this point of view, coherent structures in the ocean can
be interpreted as a form of self-organization. Self-organization is a
fundamentally nonlinear process; the mathematical description of the
organized structures is hence complicated.
A distinction between three-dimensional and two-dimensional
hydrodynamic processes is related to the effect of vortex stretching. Vortex
stretching pulls matter towards the rotation axis of the vortex; owing to
compares this effect with the pirouette of a figure skater, who can increase
her rotation rate by bringing both arms against her body. The absence of
vortex stretching in two-dimensional flow leads to a spectral flux of the
kinetic energy from small to large spatial scales. This property directly
follows from the equation for the vorticity
u
Z u
G
G of the flow, where u
G
is the velocity vector in the Eulerian co-ordinate system.
In a two-dimensional flow field u
G = (u, v, 0), as a result, the vorticity
vector
0,0,
Z
Z
G
is always directed perpendicular to the flow direction.
For inviscid flows the vorticity vector Z
G is a conserved quantity. Two other
conservation laws that follow from the inviscid vorticity equation are:
and
b const
W const
(5.1)
where b is the kinetic energy and W is the enstrophy (defined in Section 5.1).
The inverse energy cascade in the wavenumber domain, a fundamental
property of two-dimensional flows, is illustrated below by considering the
conserved quantities b and V in spectral form (Van Heijst, 1993). The kinetic
energy and enstrophy are represented as follows:
0
~
( ; )
b
e k t dk const
f
³
(5.2)
2
0
~
;
W
k e k t dk const
f
³
(5.3)
where
;
e k t is the wavenumber spectrum of kinetic energy at time t.
Suppose that the energy spectrum at some initial moment t = 0 has a
peak around a wavenumber k 0 (see schematic diagram in Figure 5-1). Due to
nonlinear interactions in the flow, the kinetic energy peak will broaden with
time. According to (5.2), the total kinetic energy (i.e., the area underneath
the curve of e) should remain constant. In order to satisfy (5.2) the spectrum
broadening has to be associated with a decrease of the spectral peak value.
At the same time, according to (5.3), the spectral distribution of enstrophy,
288
conservation of angular momentum it will rotate faster. Van Heijst (1993).
The upper ocean boundary layer is turbulent, yet it reveals features of
organization. From this point of view, coherent structures in the ocean can
be interpreted as a form of self-organization. Self-organization is a
fundamentally nonlinear process; the mathematical description of the
organized structures is hence complicated.
A distinction between three-dimensional and two-dimensional
hydrodynamic processes is related to the effect of vortex stretching. Vortex
stretching pulls matter towards the rotation axis of the vortex; owing to
compares this effect with the pirouette of a figure skater, who can increase
her rotation rate by bringing both arms against her body. The absence of
vortex stretching in two-dimensional flow leads to a spectral flux of the
kinetic energy from small to large spatial scales. This property directly
follows from the equation for the vorticity
u
Z u
G
G of the flow, where u
G
is the velocity vector in the Eulerian co-ordinate system.
In a two-dimensional flow field u
G = (u, v, 0), as a result, the vorticity
vector
0,0,
Z
Z
G
is always directed perpendicular to the flow direction.
For inviscid flows the vorticity vector Z
G is a conserved quantity. Two other
conservation laws that follow from the inviscid vorticity equation are:
and
b const
W const
(5.1)
where b is the kinetic energy and W is the enstrophy (defined in Section 5.1).
The inverse energy cascade in the wavenumber domain, a fundamental
property of two-dimensional flows, is illustrated below by considering the
conserved quantities b and V in spectral form (Van Heijst, 1993). The kinetic
energy and enstrophy are represented as follows:
0
~
( ; )
b
e k t dk const
f
³
(5.2)
2
0
~
;
W
k e k t dk const
f
³
(5.3)
where
;
e k t is the wavenumber spectrum of kinetic energy at time t.
Suppose that the energy spectrum at some initial moment t = 0 has a
peak around a wavenumber k 0 (see schematic diagram in Figure 5-1). Due to
nonlinear interactions in the flow, the kinetic energy peak will broaden with
time. According to (5.2), the total kinetic energy (i.e., the area underneath
the curve of e) should remain constant. In order to satisfy (5.2) the spectrum
broadening has to be associated with a decrease of the spectral peak value.
At the same time, according to (5.3), the spectral distribution of enstrophy,
288
conservation of angular momentum it will rotate faster. Van Heijst (1993).