THE NEAR-SURFACE LAYER OF THE OCEAN
the modal cascade in one of the quadratic quantities is towards small
wavenumbers.
An example of such a hydrodynamic system is the two-dimensional
incompressible fluid. It conserves the total energy,
2
1
2
D
b t
u dV
U ³
G
, as
well as a second quadratic quantity, the total squared vorticity (enstrophy),
2
1
2
D
W t
u dV
u
³
G
, where
,
u x t
G G
is the Eulerian velocity field
defined in a three-dimensional domain D (bounded or unbounded). The
enstrophy is a measure for the total vorticity of the flow.
When finite viscous dissipation is introduced, the energy spectrum for
two-dimensional hydrodynamic turbulence cascades to smaller
wavenumbers. On the other hand, the enstrophy spectrum cascades to higher
wavenumbers (normal cascade) and dissipates due to viscosity.
Consequently, energy accumulates at the longest wavenumber permitted in
the system. From an energy point of view, such a state is considered
organized.
The process of self-organization in two-dimensional turbulence is
discussed in detail in Section 5.2. On a horizontal scale l exceeding the
boundary layer thickness h, the vertical component of motion in the upper
ocean boundary layer is effectively suppressed due to stratification, and the
boundary layer processes are effectively two-dimensional. Mesoscale eddies
and spirals, which are often seen in satellite images of the ocean surface,
have been linked to two-dimensional turbulence. The baroclinic Rossby
radius, L R, determines the maximum horizontal scale of these coherent
structures.
Another example of a nonlinear dissipative system with tendency to selforganization is the helical structure (Moffatt and Tsinober, 1992; Branover
et al., 1999). The helicity of a fluid flow is defined as the integrated scalar
product of the velocity field and the vorticity field in the following way:
D
t
u
u dV
+
u
³
G
G
. On intuitive level the vortex that has a nonzero axial
component of velocity is characterized by nonzero helicity. Langmuir
circulations in the ocean and tornados and tropical cyclones in the
atmosphere are examples of the helical structure.
Organized structures are a characteristic feature of the atmospheric
boundary layer as well. For instance, atmospheric convection produces
clouds. The clouds modulate radiative fluxes and are a source of freshwater
flux at the air-sea interface and thus induce spatial patterns in the sea surface
temperature and salinity. This effect is especially strong in the tropical warm
pools, because of deep atmospheric convection accompanied by heavy
rainfalls. The “images” of the atmospheric processes in the upper ocean
turbulent boundary layer evolve according to the laws of two-dimensional
286
the modal cascade in one of the quadratic quantities is towards small
wavenumbers.
An example of such a hydrodynamic system is the two-dimensional
incompressible fluid. It conserves the total energy,
2
1
2
D
b t
u dV
U ³
G
, as
well as a second quadratic quantity, the total squared vorticity (enstrophy),
2
1
2
D
W t
u dV
u
³
G
, where
,
u x t
G G
is the Eulerian velocity field
defined in a three-dimensional domain D (bounded or unbounded). The
enstrophy is a measure for the total vorticity of the flow.
When finite viscous dissipation is introduced, the energy spectrum for
two-dimensional hydrodynamic turbulence cascades to smaller
wavenumbers. On the other hand, the enstrophy spectrum cascades to higher
wavenumbers (normal cascade) and dissipates due to viscosity.
Consequently, energy accumulates at the longest wavenumber permitted in
the system. From an energy point of view, such a state is considered
organized.
The process of self-organization in two-dimensional turbulence is
discussed in detail in Section 5.2. On a horizontal scale l exceeding the
boundary layer thickness h, the vertical component of motion in the upper
ocean boundary layer is effectively suppressed due to stratification, and the
boundary layer processes are effectively two-dimensional. Mesoscale eddies
and spirals, which are often seen in satellite images of the ocean surface,
have been linked to two-dimensional turbulence. The baroclinic Rossby
radius, L R, determines the maximum horizontal scale of these coherent
structures.
Another example of a nonlinear dissipative system with tendency to selforganization is the helical structure (Moffatt and Tsinober, 1992; Branover
et al., 1999). The helicity of a fluid flow is defined as the integrated scalar
product of the velocity field and the vorticity field in the following way:
D
t
u
u dV
+
u
³
G
G
. On intuitive level the vortex that has a nonzero axial
component of velocity is characterized by nonzero helicity. Langmuir
circulations in the ocean and tornados and tropical cyclones in the
atmosphere are examples of the helical structure.
Organized structures are a characteristic feature of the atmospheric
boundary layer as well. For instance, atmospheric convection produces
clouds. The clouds modulate radiative fluxes and are a source of freshwater
flux at the air-sea interface and thus induce spatial patterns in the sea surface
temperature and salinity. This effect is especially strong in the tropical warm
pools, because of deep atmospheric convection accompanied by heavy
rainfalls. The “images” of the atmospheric processes in the upper ocean
turbulent boundary layer evolve according to the laws of two-dimensional
286