Mixing
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Mixing. Shear Production. Buoyancy Production. z 1. To mix the water column, kinetic energy has to be converted to potential energy. Mixing increases the potential energy of the water column. z 2. z. Mixing vs. Stratification. Potential energy of the water column:.

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Mixing

Mixing

Shear Production

Buoyancy Production


Mixing

z1

To mix the water column, kinetic energy has to be converted to potential energy.

Mixing increases the potential energy of the water column

z2

z

Mixing vs. Stratification


Mixing

Potential energy of the water column:

Potential energy per unit volume:

But

The potential energy per unit area of a mixed

water column is:

Ψ has units ofenergy per unit area


Mixing

z1

z2

If

z

no energy is required to mix the water column

The energy difference between a mixed and a stratified water column is:

with units of [ Joules/m2 ]

φ is the energy required to mix the water column completely, i.e., the energy required to bring the profile ρ(z) to ρhat

It is called the POTENTIAL ENERGY ANOMALY

It is a proxy for stratification

The greater the φ the more stratified the water column


Mixing

We’re really interested in determining whether the water column remains stratified or mixes as a result of the forcings acting on the water column.

For that we need to study

[ Watts per squared meter ]


Mixing

De Boer et al (2008, Ocean Modeling, 22, 1)

Bx and By are the along-estuary and cross-estuary straining terms

Ax and Ay are the advection terms

Cx and Cy interaction of density and flow deviations in the vertical

C’x and C’y correlation between vertical shear and density variations in the vertical; depth-averaged counterparts of C

E is vertical mixing and D is vertical advection

Hx and Hy are horizontal dispersion;

Fs and Fb are surface and bottom density fluxes


Mixing

1-D idealized numerical simulation of tidal straining

Burchard and Hofmeister (2008, ECSS, 77, 679)


Mixing

destratified @

end of flood

stratified entire period

Burchard and Hofmeister (2008, ECSS, 77, 679)


Mixing

De Boer et al (2008, Ocean Modeling, 22, 1)

Bx and By are the along-estuary and cross-estuary straining terms

Ax and Ay are the advection terms

Cx and Cy interaction of density and flow deviations in the vertical

C’x and C’y correlation between vertical shear and density variations in the vertical; depth-averaged counterparts of C

E is vertical mixing and D is vertical advection

Hx and Hy are horizontal dispersion;

Fs and Fb are surface and bottom density fluxes


Mixing

The power/unit area generated by the wind at a height of 10 m is given by:

But the power/unit area generated by the wind stress on the sea surface is:

W* is the wind shear velocity at the surface and equals:

Mixing Power From Wind


Mixing

Alternatively,

δ is a mixing efficiency coefficient = 0.023

ksis a drag factor that equals 6.4x10-5 or( Cd u / W )

Mixing Power From Wind (cont.)


Mixing

But only a fraction of this goes to mixing

Mixing Power From Tidal Currents

Can also be expressed in terms of bottom stress.

The power/unit area produced by tidal flow interacting with the bottom is:

ε is a mixing efficiency [ 0.002, 0037 ]

Cb is a bottom drag coefficient = 0.0025


Mixing

Considering advection of mass by ‘u’ only:

assuming the along-estuary density gradient is independent of depth, i.e.,

Tidal Straining

We need u(z) from tidal currents to determine the power to stratify/destratify from tidal straining


Mixing

Taking,

(Bowden and Fairbairn, 1952,

Proc. Roy. Soc. London, A214:371:392.)

1.15

0.72

Tidal Straining (cont.)


Mixing

The water column will stratify at ebb as is positive, and vice versa

Tidal Straining (cont.)


Mixing

Taking again:

and using

Gravitational Circulation

will tend to stratify the water column


Mixing

α is the thermal expansion coefficient of seawater ~ 1.6x10-4 °C-1

cp is the specific heat of seawater ~ 4x103 J/(kg °C)

Q is the cooling/heating rate (Watts/m2)

Heating/Cooling

In addition to buoyancy from heating, it may come from precipitation (rain)

Δρ is the density contrast between fresh water and sea water

Pris the precipitation rate (m/s)


Mixing

The alternative way of representing the riverine influence on stratification is by assuming that increased R enhances Δρ / Δx.

This may be parameterized with

Assuming that each stratifying/destratifying mechanism can be superimposed separately:

In estuaries, however, the main input of freshwater buoyancy is from river discharge.

There is no simple way of dealing with feshwater input as specified by the discharge

rate R because R is not distributed uniformly over a prescribed area (as is the case for

wind, bottom stress, rain, heat).

Caution! Increased R does not necessarily mean increased gradients


Mixing

α = 1.6x10-4 °C-1

cp = 4x103 J/(kg °C)

H = 10 m

If the contrast between rain water and sea water is 20 kg/m3, then

Example: Let’s compare the stratifying tendencies of rain as compared to a low heating rate of 10 W/m2

A precipitation rate of 1.7 mm per day is comparable to a heating rate of 10 W/m2

Where can this happen?


Mixing

If stratification remains unaltered (or if buoyancy = mixing),

For a prescribed Q, the only variables are H and u0

Competition between buoyancy from Heating and mixing from Bottom Stress

If Q increases, u0 needs to increase to keep H/u03constant

If u0 does not increase then stratification ensues

H/u03is then indicative of regions where mixed waters meet stratified


Mixing

Simpson-Hunter parameter

H/u03 ~ 1.6x104/ Q

Line where mixed waters are separated from stratified waters.

LOG10 (H / U3)


Mixing

Bowman and Esaias, 1981, JGR, 86(C5), 4260.


Mixing

Loder and Greenberg, 1996, Cont. Shelf Res., 6(3), 397-414.


Mixing

Loder and Greenberg, 1996,

Cont. Shelf Res., 6(3), 397-414.

M2 -----------------

M2-N2 ------------

M2-N2-S2--------

M2+N2+S2------


Mixing

Restrictions of the approach?

Dominant Stratifying Power from Heating

Dominant Destratifying Power from bottom stresses


Mixing

In order to balance that stratifying power, we need a wind power of:

or a current power of:

Another example:

Assume a system with Δρ / Δx of 10 kg/m3 over 50 km = 2x10-4, H = 10 m, Az =0.005 m2/s


Mixing

In order to balance such stratifying power, we need a wind power of:

or a current power of:

Another example:

Assume a system with Δρ / Δx of 1 kg/m3 over 3 km, H = 20 m, Av =0.001 m2/s


Mixing

From Heating/Cooling


Mixing

From Density Gradient (grav circ)


Mixing

Examples of successful applications of this approach:

Simpson et al. (1990), Estuaries, 13(2), 125-132.

Lund-Hansen et al. (1996), Estuar. Coast. Shelf Sci., 42, 45-54.


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