OC 530: Double diffusion and conservation of stuff Oct 12, 2009

1. OC 530: Double diffusion and conservation of stuff Oct 12, 2009 • Upcoming instructor absences : Mon Oct 19 and Wed Oct 21, lectures presented by Levi Kilcher • Double diffusion • Begin conservation of stuff

2. The “perpetual” salt fountain Imagine an ocean which is stratified such that warm, salty water overlies relatively cold, fresh water. What would happen if you inserted a tube into the water and drew some water up into the tube? If the tube were thermally conducting, then the water you drew into the tube would warm up, become less dense and continue rising. Eventually, water would bubble out the top! Stommel and Arons did this little experiment in the early ’60’s and dubbed it the perpetual salt fountain.

3. Double Diffusion • Because kT >> ks (1.5x10-7m2/s vs 1x10-9m2/s) something very much like this can occur (salt fingering) • The differing molecular diffusivities of temperature and salinity give rise to other effects depending on the relative layering of fresh and salt water.

4. Double Diffusive effects • Warm salty over colder less salty → salt fingering (common in Atlantic) • Colder less salty over warm salty → diffusive convection sharpens interface between layers (less common, found at high latitudes) • Cold salty over warmer less salty → statically unstable • Warmer less salty over cold salty → statically stable, double diffusion diffuses interface between layers (common in Pacific)

5. Salt fingering http://www.planetwater.ca/research/oceanmixing/saltfingers.html http://www.phys.ocean.dal.ca/programs/doubdiff/labdemos.html Profiles taken near red box Schmitt et al. (2005)

6. Diffusive convection http://www.phys.ocean.dal.ca/programs/doubdiff/labdemos.html Rudels et al. (1996)

7. Conservation of stuff Consider a box of seawater with fixed volume V = δx δy δz, and define the amount of stuff per unit volume as Γ [in SI units: stuff/m3]. Then the total amount of stuff in the box is ΓV. Change of stuff in box over time interval δt: [Γ(t + δt) -Γ(t)] V Amount of stuff transported into box at x over time δt: Fx(x) ∙ δy ∙ δz · δt Amount of stuff transported out of box at (x + δx) over time δt: Fx(x + δx) · δy · δz · δt

8. Conservation of stuff Note: For the moment we are not considering sources or sinks of the stuff within the box (e.g. radioactive decay, uptake or excretion etc.) Then the conservation of Γ in one dimension can be written: dividing through by V δt in the limit as δx, δy, δz, δt approach 0

9. Conservation of stuff with growth and decay If we are considering sources or sinks of the stuff within the box (e.g. radioactive decay, uptake or excretion etc.), just add a term to represent this growth Then the conservation of q in one dimension can be written: dividing through by V δt in the limit as δx, δy, δz, δt approach 0 Where Gq represents the growth or decay rate of stuff in the box

10. Conservation of stuff (3-D) We can generalize the one-dimensional case to three dimensions: Or, in different notation where we have used the vector operator (del) to express the divergence of F (div F)

11. Advective and diffusive fluxes

12. Advective flux: the flux due to some large-scale velocity field where the units are [(stuff / m3 ) · (m/s)] = [stuff / (m2 · s)], and u is a vector = (u, v, w) Example: a steady breeze blowing pollen into your open window

13. Diffusive flux: the flux due to small-scale (turbulent) eddies or, at small enough scale, molecular viscosity (in component form) Sometimes k is identical for different components (e.g., molecular processes) as written above, sometimes k is different for different components (e.g. if k represents eddy processes (often use symbol A here), then AH≠ AZ) Example: small gusts (turbulence) bringing pollen into your open window (Hold off on radiative fluxes for now)

14. Conservation of Mass If we neglect diffusion of matter, and the creation/destruction of matter, we can write the conservation equation as Where instead of q we use ρ. Writing the above in component form we get Under the Boussinesq approximation, we can neglect density, ρ, changes wherever they are not multiplied by gravitational acceleration, g. Hence A approximately equals zero.

15. Conservation of Mass (Volume) Dividing by ρ, the simplest relation expressing conservation of mass can be written: Although this is often termed the conservation of mass, it is more accurately termed the conservation of volume. In throwing away the density variability, we have made the fluid incompressible and thrown out acoustic wave solutions.

16. The material derivative Using we can rewrite (simplify) the conservation of stuff to: or

17. The material derivative where we have defined the material or total derivative The total derivative is the change of stuff as seen by an observer moving with the water, and is the sum of the local and advective change.

18. Conservation of Salt We can also write conservation relations for salt (assuming salt is not deposited or removed from the sea):

19. Conservation of Heat We can also write conservation relations for heat (heat is of course deposited or removed from the sea): where QH is heating or cooling due to phase change, chemical change or viscous dissipation, and is the net heating or cooling due to solar and infrared (wave) radiation.