Mixing From Stresses Wind stresses Bottom stresses Internal stresses Non-stress Instabilities Cooling Double Diffusion Tidal Straining Shear ProductionBuoyancy Production

Mixing vs. Stratification To mix the water column, kinetic energy has to be converted to potential energy. Mixing increases the potential energy of the water column z z2z2 z1z1 Stratification from: estuarine circulation tidal straining heating

Potential energy per unit volume: Potential energy of the water column: But The potential energy per unit area of a mixed water column is: Ψ has units of energy per unit area

The energy difference between a mixed and a stratified water column is: with units of [ Joules/m 2 ] φ is the energy required to mix the water column completely, i.e., the energy required to bring the profile ρ(z) to ρ hat It is called the POTENTIAL ENERGY ANOMALY z z2z2 z1z1 It is a proxy for stratification The greater the φ the more stratified the water column If no energy is required to mix the water column

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 ]

B x and B y are the along-estuary and cross-estuary straining terms A x and A y are the advection terms C x and C y 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 H x and H y are horizontal dispersion; F s and F b are surface and bottom density fluxes De Boer et al (2008, Ocean Modeling, 22, 1)

Burchard and Hofmeister (2008, ECSS, 77, 679) 1-D idealized numerical simulation of tidal straining

Burchard and Hofmeister (2008, ECSS, 77, 679) stratified entire period end of flood

B x and B y are the along-estuary and cross-estuary straining terms A x and A y are the advection terms C x and C y 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 H x and H y are horizontal dispersion; F s and F b are surface and bottom density fluxes De Boer et al (2008, Ocean Modeling, 22, 1)

Mixing Power From Wind 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:

Alternatively, δ is a mixing efficiency coefficient = k s is a drag factor that equals 6.4x10 -5 or ( C d u / W ) Mixing Power From Wind (cont.)

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: But only a fraction of this goes to mixing ε is a mixing efficiency [ 0.002, 0037 ] C b is a bottom drag coefficient =

Tidal Straining assuming the along-estuary density gradient is independent of depth, i.e., Considering advection of mass by ‘u’ only: We need u(z) from tidal currents to determine the power to stratify/destratify from tidal straining

Tidal Straining (cont.) Taking, (Bowden and Fairbairn, 1952, Proc. Roy. Soc. London, A214:371:392.)

Tidal Straining (cont.) The water column will stratify at ebb as is positive, and vice versa

Taking again: Gravitational Circulation and using will tend to stratify the water column

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 P r is the precipitation rate (m/s) α is the thermal expansion coefficient of seawater ~ 1.6x10 -4 °C -1 c p is the specific heat of seawater ~ 4x10 3 J/(kg °C) Q is the cooling/heating rate (Watts/m 2 )

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). The alternative way of representing the riverine influence on stratification is by assuming that increased R enhances Δρ / Δx. This may be parameterized with Caution! Increased R does not necessarily mean increased gradients Assuming that each stratifying/destratifying mechanism can be superimposed separately:

Example: Let’s compare the stratifying tendencies of rain as compared to a low heating rate of 10 W/m 2 α = 1.6x10 -4 °C -1 c p = 4x10 3 J/(kg °C) H = 10 m If the contrast between rain water and sea water is 20 kg/m 3, then A precipitation rate of 1.7 mm per day is comparable to a heating rate of 10 W/m 2 Where can this happen?

Competition between buoyancy from Heating and mixing from Bottom Stress If stratification remains unaltered (or if buoyancy = mixing), For a prescribed Q, the only variables are H and u 0 If Q increases, u 0 needs to increase to keep H/u 0 3 constant If u 0 does not increase then stratification ensues H/u 0 3 is then indicative of regions where mixed waters meet stratified

Simpson-Hunter parameter H/u 0 3 ~ 1.6x10 4 / Q Line where mixed waters are separated from stratified waters. LOG10 (H / U 3 )

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

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

M M 2 -N M 2 -N 2 -S M 2 +N 2 +S Loder and Greenberg, 1996, Cont. Shelf Res., 6(3),

Restrictions of the approach? Dominant Stratifying Power from Heating Dominant Destratifying Power from bottom stresses

Another example: Assume a system with Δρ / Δx of 10 kg/m 3 over 50 km = 2x10 -4, H = 10 m, A z =0.005 m 2 /s 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 1 kg/m 3 over 3 km, H = 20 m, A v =0.001 m 2 /s In order to balance such stratifying power, we need a wind power of: or a current power of:

From Heating/Cooling

From Density Gradient (grav circ)

Examples of successful applications of this approach: Simpson et al. (1990), Estuaries, 13(2), Lund-Hansen et al. (1996), Estuar. Coast. Shelf Sci., 42,