8.3 Concentration of Matter for Molecular and Thrbulent Diffusion
267
in which Pw is the water density, u and v are mean flow velocities in the x and
y directions, and N is the Brunt-Viiisiila frequency (see Eq. 6.10).
Hydrodynamic instability and turbulence can appear when (Monin and Ozmidov, 1985):
1
Ri < -.
4
(8.37)
If the Richardson number is known, the coefficient of turbulent viscosity A z
can be approximated as (Peters et at., 1988):
(8.38)
There is a convenient criterion to distinguish laminar and turbulent motion of
ocean water, known as the Cox number C x:
(8.39)
in which, for example, f) = T (temperature) or f) = S (salt). For Cx » 1, the
motion is turbulent and for C x < 1 the motion is laminar.
We note that turbulence can also be generated in wind-induced currents due
to a strong shearing velocity. However, in this case the stability criterion is
expressed as a function of the Reynolds number.
8.3.3 Shear Flow Dispersion
Brief Orientation. Hitherto in the discussion of mixing phenomena, we neglected the presence of any boundaries, such as the sea surface or sea bottom,
channel or pipe walls. What is the effect of these boundaries on the spreading
of contaminants? It is a well known fact that spreading in rivers and estuaries in the direction of flow is primarily caused by the velocity profile across
the river. As this mechanism of spreading is associated with shear flows, it is
known as shear effect.
If we consider two molecules being carried in a river flow, one well above
the bottom and one near the bottom, the rate of separation caused by the
difference in advection velocity will exceed that caused by molecular motion.
At each instant of time, the velocity of any single molecule is essentially that of
the stream velocity at a given cross-sectional position plus the velocity of the
random movement across the cross-section due to molecular diffusion. When
we adopt a coordinate system moving at the mean velocity, these random steps
(with respect to this coordinate system) will have the same probability of being
either backward or forward.
After a sufficiently long time, the motion becomes similar to the random
walk, discussed in Sect. 8.2.2, and we should expect that Fick's equation (8.22)
267
in which Pw is the water density, u and v are mean flow velocities in the x and
y directions, and N is the Brunt-Viiisiila frequency (see Eq. 6.10).
Hydrodynamic instability and turbulence can appear when (Monin and Ozmidov, 1985):
1
Ri < -.
4
(8.37)
If the Richardson number is known, the coefficient of turbulent viscosity A z
can be approximated as (Peters et at., 1988):
(8.38)
There is a convenient criterion to distinguish laminar and turbulent motion of
ocean water, known as the Cox number C x:
(8.39)
in which, for example, f) = T (temperature) or f) = S (salt). For Cx » 1, the
motion is turbulent and for C x < 1 the motion is laminar.
We note that turbulence can also be generated in wind-induced currents due
to a strong shearing velocity. However, in this case the stability criterion is
expressed as a function of the Reynolds number.
8.3.3 Shear Flow Dispersion
Brief Orientation. Hitherto in the discussion of mixing phenomena, we neglected the presence of any boundaries, such as the sea surface or sea bottom,
channel or pipe walls. What is the effect of these boundaries on the spreading
of contaminants? It is a well known fact that spreading in rivers and estuaries in the direction of flow is primarily caused by the velocity profile across
the river. As this mechanism of spreading is associated with shear flows, it is
known as shear effect.
If we consider two molecules being carried in a river flow, one well above
the bottom and one near the bottom, the rate of separation caused by the
difference in advection velocity will exceed that caused by molecular motion.
At each instant of time, the velocity of any single molecule is essentially that of
the stream velocity at a given cross-sectional position plus the velocity of the
random movement across the cross-section due to molecular diffusion. When
we adopt a coordinate system moving at the mean velocity, these random steps
(with respect to this coordinate system) will have the same probability of being
either backward or forward.
After a sufficiently long time, the motion becomes similar to the random
walk, discussed in Sect. 8.2.2, and we should expect that Fick's equation (8.22)