Mathematical description
57
Figure 3.4. Buoyancy frequency (N) along the WOCE A16 section as plotted using the ODV
software and the Gouretski and Koltermann (2004) gridded ocean data atlas.
3.1.4. Mixing of momentum, heat and salt
.
With ν =1 0 −6 m 2 s −1 (Table 1.3) being the kinematic viscosity of water and
U , L characteristic velocity and length scales of the flow, the Reynolds number
Re = UL/ν is the ratio of inertial and viscous accelerations in a liquid. For basin
scale flows in the ocean, with U =1 0 −1 ms −1 and L =1 0 6 m, a typical value
of the Reynolds number is Re =10 11 . This is far above typical critical values of
Re marking transitions from laminar to turbulent flow and hence the ocean flow is
highly turbulent. Molecular viscous effects can be neglected on those scales with
respect to inertial effects.
In turbulent flows there are rapid fluctuations that result in a fast redistribution
of momentum, heat and salt. Because of their apparent random nature these fluctuations have the character of diffusion, but one that is much larger than diffusion
due to molecular processes. In one of the simplest representations of these mixing
processes, called first-order closure, the mixing is represented as being down gradient diffusion with turbulent mixing coefficients (also called ‘eddy viscosities’
and ‘eddy diffusivities’). Contrary to the molecular coefficients, these coefficients
are not a material property but are flow dependent.
In this first-order closure theory, the turbulent flux Ψ of a scalar quantity (for
example, heat) ψ is proportional to the gradient ∇ψ, with mixing coefficient K,
i.e.,
Ψ = −K ∇ ψ.
(3.16)
Below, we will indicate the mixing coefficients for momentum with A,a n dt a k e
those of heat and salt the same and indicate them with K. It should be stressed that
the first-order closure theory is certainly incorrect over large parts of the ocean.
57
Figure 3.4. Buoyancy frequency (N) along the WOCE A16 section as plotted using the ODV
software and the Gouretski and Koltermann (2004) gridded ocean data atlas.
3.1.4. Mixing of momentum, heat and salt
.
With ν =1 0 −6 m 2 s −1 (Table 1.3) being the kinematic viscosity of water and
U , L characteristic velocity and length scales of the flow, the Reynolds number
Re = UL/ν is the ratio of inertial and viscous accelerations in a liquid. For basin
scale flows in the ocean, with U =1 0 −1 ms −1 and L =1 0 6 m, a typical value
of the Reynolds number is Re =10 11 . This is far above typical critical values of
Re marking transitions from laminar to turbulent flow and hence the ocean flow is
highly turbulent. Molecular viscous effects can be neglected on those scales with
respect to inertial effects.
In turbulent flows there are rapid fluctuations that result in a fast redistribution
of momentum, heat and salt. Because of their apparent random nature these fluctuations have the character of diffusion, but one that is much larger than diffusion
due to molecular processes. In one of the simplest representations of these mixing
processes, called first-order closure, the mixing is represented as being down gradient diffusion with turbulent mixing coefficients (also called ‘eddy viscosities’
and ‘eddy diffusivities’). Contrary to the molecular coefficients, these coefficients
are not a material property but are flow dependent.
In this first-order closure theory, the turbulent flux Ψ of a scalar quantity (for
example, heat) ψ is proportional to the gradient ∇ψ, with mixing coefficient K,
i.e.,
Ψ = −K ∇ ψ.
(3.16)
Below, we will indicate the mixing coefficients for momentum with A,a n dt a k e
those of heat and salt the same and indicate them with K. It should be stressed that
the first-order closure theory is certainly incorrect over large parts of the ocean.