Introduction Monday 21 st 9-10am Turbulence theory Monday 21 st 11-12am Measuring turbulence Monday 28 th 9-10am Modelling turbulence Monday 28 th 11-12am . Marine Turbulence and Mixing. 2. Turbulence theory. Dr. Matthew Palmer [email protected]
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u = U + u’
Consider the momentum flux (unit area) in the x direction of a particle through area A at the velocity u, or U+u’.
Momentum flux =
(mass(m-3) x velocity) x flux through A
The mean force per unit area, or stress, in the x direction by flow u is,
and in the y and z direction,
but , so
In the ocean, vertical gradients are much stronger than horizontal so we may disregard horizontal components of stress
The 9 cartesian Reynolds stress tensor
So we may now only consider txz and tyz or r u’w’ and r v’w’
Reynolds decomposition is also applicable to scalar quantities (temperature, salinity, density, nutrients etc.)
s = S + s’
s‘ = 0
But remember, turbulence is random, chaotic, broad spectrum and so difficult to measure. So we often need to parameterise the process of generating turbulence.
Consider how turbulent eddies interact.
Reynolds stress transfers energy from the horizontal to vertical and vortical turbulent motion.
txz = r u’w’
tyz = r v’w’
Due to its diffusive nature we may approximate these stresses to act in a similar way to molecular diffusivity but on larger scales (Bousinesq).
We may then parameterise the generation of turbulence by gradients in the mean flow.
Nz is the vertical eddy viscosity coefficient (m2s-1)
Remember that we may consider the turbulent properties of scalar quantities such as density or temperature in the same was as momentum.
s = S + s’
s‘ = 0
We may also approximate eddy diffusivity from molecular processes.
Kz is the vertical eddy diffusivity coefficient (m2s-1)
Which gives us the advection-diffusion equation for a scalar quantity:
Which represents the local rate of change of the scalar s in terms of horizontal advection by the mean flow U and vertical diffusion by the turbulent fluctuations within the water column.
We now have methods for quantifying complex turbulent processes by representing the statistical properties of turbulent flow and by considering them to behave similar to molecular processes,
within the equations of motion
and within the turbulent kinetic energy (E) budget,
Where P = production (+ve)
B = buoyant production (+ve or –ve)
e = dissipation (-ve)
In the absence of convection, there is TKE production is in balance with work against buoyancy and TKE dissipation.
But there is a limiting factor!
Above a certain ratio of B/P turbulence will be suppressed.
So we can define an expression to calculate the eddy diffusivity coefficient Kz.
Where G ~0.2 (Osborn, 1980)
If velocity changes with depth in a stable, stratified flow, then the flow may become unstable providing that the shear flow is large enough.
The limiting condition for the development of turbulence is that the energy supply from the mean flow is greater than the energy demands of dissipation and mixing.
Turbulence is hungry!
Assuming that NzKz, then we may consider the relative stability as a competition between destabilising shear flow and stabilising buoyancy. The gradient Richardson number,
Where Rig ≤ ¼ somewhere within the flow instability may exist and disturbances may grow (Miles (1961) and Howard (1961) )
ADCPs are able to sample u fast enough to measure u’w’ and v’w’ and so we can directly calculate Reynolds stresses, eddy viscosity and TKE production……