Reynolds Decomposition
Av* du/dz
it is the mixing of momentum
much like
Kv dS/dz
Is the mixing of salt
1)Draw boxes showing high salt in one box and low salt in lower box
2) Think about the net transport of salt that would occur if you mixed between those two
boxes
3) The same is true for momentum flux – fast moving fluid from one box mix I
into slow moving box will increase the velocity.
4) Mixing actually happen via fluctuations in the fields: Reynolds Decompostion
u=<u>+u’
w=<w> + w’
s=<s>+s’
where <s’>=<w’>=<u’>=0
So the vertical transport of salt would be
wS +<w’s’>
if there is a correlation between upward vertical velocity and salinity then the product of
the fluctuating terms will produce a net flux
This will happen if there is a mean gradient,. i.e. upward vertical fluctuations will tend to
transport salty water—while downward vertical fluctuations will tend to transport fresh
waters. Thus the turbulent motions will tend to transport salt upwards.
The same is true for momentum. Consider the flow in a frictional boundary layer where
there is more momentum aloft. Downward vertical fluctuations transport high momentum
fluid downward while upward vertical fluctuations will tend to bring low momentum
fluid upwards. Consequently there is a tendency for the vertical fluctuations to transport
momentum downward.
Where does it go?
A: The momentum is transferred to the bottom where it does work ,i.e. sediment transport
What is the impact of stratification?
A: Stratification will suppress the vertical fluctuations thus reducing the vertical eddy
viscosity and the vertical transport of momentum
Trick question: Do you get more turbulent salt flux in a water column with salinity
stratification or one with no salinity stratification
A: The one with salinity stratification. Because while turbulence is suppressed and the
mixing coefficient will be lower—the salt flux is the eddy diffusivity times the vertical
salt gradient. If there no salinity stratification then there is no salinity gradient and thus
no vertical salt flux.
However this does not mean that you get bigger salt flux with bigger stratification—
because if the eddy diffusivity drops more quickly than the salt gradient increases then
the vertical salt flux would be reduced.
The eddy diffusivity (we think) depends on the state of the flow in terms of the
stratification and vertical shear.
(Av, Kv)= some function of the Richardson Number.
Ekman
Ekman transport happens when the momentum balance is between the Coriolis force and
the vertical mixing of momentum.
 fv  Av
fu  Av
 2u 1  x

z
 z
 2 v 1  y

z  z
The vertical scale of the Ekman motion can be estimated with a scaling analysis.
fv is of magnitude (we also call this of order) o(fv)
Av d2u/dz2 is of order o(Av V/de2)
Since they are equal
fV= Avv/de2
from which we can determine the scale of depth to be
de~sqrt(Av/f)
Basically it is the depth that the vertical mixing can penetrate in an inertial period.
A more detailed approach would yield
de= (2 Av/f)1/2
For Av=.01 Ekman depth is ~40 meters.
This is the depth of the wind driven layer.
It’s very small compared to the depth of the ocean.
Intuitively one would think that the stronger the wind the deeper should be the winddriven layer. But the above equation doesn’t explicitly indicate this. Why?
Because the eddy viscosity is proportional to the wind speed! So as wind speed increases
eddy viscosity increases and the Ekman depth increases.
Draw slaps of fluid along with free-body diagrams to show intuitive example of the
Ekman spiral.
Show Dye picture
An exact solution can be written as
ue=Vo cos(De)z) exp (De)z
ve=Vo sin(De)z) exp (De)z
Where
Vo =(sqrt(2) (Deee f)
A subtle yet real important effect of the Ekman Transport on the ocean is that it produces
pressure gradients. This occurs both at the basin scale due to the curl of the wind i.e
westerlies in the temperate latitudes and easterlies in the tropics drives the subtropical
gyre.
It also is responsible for coastal upwelling and Ekman Pumping, and the Bottom
Ekman Layer—but we’ll discuss these in more detail later.
Geostrophic flow
In both the atmosphere and the ocean a the first order balance is often between the
pressure gradient and the corilios force—particularly away from the
1 P
 fv
 x
1 P

 fu
 y

and is called geostrophic flow. It is the first order momentum balance that occurs in much
of the atmosphere and the ocean. It is why winds flow clockwise around a high (in the
northern hemisphere) and counter-clockwise around a low.