266
8 Transport in the Oceans and Coastal Zone
variances (Okubo, 1980):
Kx
1 da;
2 dt
1 da 2
Ky
--y
2 dt
(8.34)
1 da 2
Kz
z
2 dt
in which a;, a~, a; are the variances of particle distances in the x, y and z
direction, respectively.
Although simple in theory, this approach to quantify K x, Ky and K z values
can be difficult in practice, as it requires an experimental determination of
variances of particle distances in all directions.
The differences in observed values of coefficients of horizontal turbulent diffusion indicate that coefficients K x and K yare related to the scale of a given
diffusion phenomenon. In the cloud of particles of dimension [, there are active turbulent eddies of dimension [ (and smaller). Eddies larger than [ simply
transport the cloud as a one body, not participating in its diffusion. Okubo
(1980) and Ozmidov (1986) provided experimental evidence for the relationship
between coefficient Kx (Ky) and the scale, [, in the form:
K ~ k f (±) [4/3,
(8.35)
in which [ is the scale of the turbulent flow (for example, eddies), k is a constant
coefficient, and the function f (l / L) is a certain universal function in which L
is the characteristic scale of the main turbulent flow. For example, for water
motion with a scale of 10 6 m, the coefficient of horizontal turbulent diffusion
can be of the order of 10 4 m 2 /s, while for motion with a scale of 10 2 m, it can be
of the order of 10- 1 m 2 /s. For an in-depth discussion on scaling and Eq. (8.35),
the reader should consult Csanady (1973), Okubo (1980) or Ozmidov (1986).
The relationship (8.35) applies only to horizontal diffusivity. For the diffusion of particles in the vertical direction, water stratification has to be taken
into account. In an ocean with stable stratification, any vertical turbulent fluctuations of particles will be suppressed as more energy is needed to transfer
more dense water to a higher level, or vice versa. The suppressing action of
stable stratification may preclude the generation of turbulence and turbulent
diffusion. The criteria for turbulent hydrodynamic instability and appearance
of turbulence is known as the Richardson number Ri which, for stationary and
horizontally uniform flow with stable stratification, takes the form:
(8.36)
8 Transport in the Oceans and Coastal Zone
variances (Okubo, 1980):
Kx
1 da;
2 dt
1 da 2
Ky
--y
2 dt
(8.34)
1 da 2
Kz
z
2 dt
in which a;, a~, a; are the variances of particle distances in the x, y and z
direction, respectively.
Although simple in theory, this approach to quantify K x, Ky and K z values
can be difficult in practice, as it requires an experimental determination of
variances of particle distances in all directions.
The differences in observed values of coefficients of horizontal turbulent diffusion indicate that coefficients K x and K yare related to the scale of a given
diffusion phenomenon. In the cloud of particles of dimension [, there are active turbulent eddies of dimension [ (and smaller). Eddies larger than [ simply
transport the cloud as a one body, not participating in its diffusion. Okubo
(1980) and Ozmidov (1986) provided experimental evidence for the relationship
between coefficient Kx (Ky) and the scale, [, in the form:
K ~ k f (±) [4/3,
(8.35)
in which [ is the scale of the turbulent flow (for example, eddies), k is a constant
coefficient, and the function f (l / L) is a certain universal function in which L
is the characteristic scale of the main turbulent flow. For example, for water
motion with a scale of 10 6 m, the coefficient of horizontal turbulent diffusion
can be of the order of 10 4 m 2 /s, while for motion with a scale of 10 2 m, it can be
of the order of 10- 1 m 2 /s. For an in-depth discussion on scaling and Eq. (8.35),
the reader should consult Csanady (1973), Okubo (1980) or Ozmidov (1986).
The relationship (8.35) applies only to horizontal diffusivity. For the diffusion of particles in the vertical direction, water stratification has to be taken
into account. In an ocean with stable stratification, any vertical turbulent fluctuations of particles will be suppressed as more energy is needed to transfer
more dense water to a higher level, or vice versa. The suppressing action of
stable stratification may preclude the generation of turbulence and turbulent
diffusion. The criteria for turbulent hydrodynamic instability and appearance
of turbulence is known as the Richardson number Ri which, for stationary and
horizontally uniform flow with stable stratification, takes the form:
(8.36)