Unit 3 – Part I: Mass Conservation; water, salt & heat budgets; fluxes

1. Unit 3 – Part I:Mass Conservation; water, salt & heat budgets; fluxes Introductory Physical Oceanography (MAR 555) - Fall 2009 Miles A. Sundermeyer Assigned Reading: OC 6.2 and IPO Chapter 7

2. Key Concepts: • Preamable: term balances (a.k.a., shortcut math) • Property budgets: • Sources + Sinks = Net Change • Mass continuity / conservation equation • Divergence & gradient – physical interpretation • Intuitive notion of conservation equation, • e.g., r, S, rv (a.k.a., momentum – stay tuned …)

3. Term balances Example: driving from UMass Boston -> SMAST Google Maps (www.google.com) Dtotal = DUMB-I95split + DI95-Rt24-I195 + DI195-SMAST highway drive vs. city drive Ttotal = TUMB-I95split + TI95-Rt24-I195 + TI195-SMAST

4. Global Heat Budgets Oceanic Radiation Balance From Lecture notes for Intro PO by H. Bryden http://www.ocean.washington.edu/people/faculty/luanne/classes/pcc586/papers/bryden.pdf

5. (Property) Budgets Example: Bank Checking/Debit Account Net Flux in or out  change in $ value / unit time) Direct Deposit Checks Written Accrued Interest Debits $ Other Withdrawals Other Deposits Sources + Sinks = Net Change

6. Property Budgets Example: Mass Conservation analogous to heat flux = Energy/(unit area · unit time) Flux = mass / (unit area · unit time) = mass/vol. · length/time = rv du≠ 0 indicates flux divergence ru r(u + du) Sources + Sinks = Net Change

7. ru r(u + du) Property Budgets (cont’d) Example: Mass Conservation - Flux Divergence Flux = mass / (unit area · unit time) = mass/vol. · length/time = rv Flux divergence = r(u + du) – ru dx

8. ru r(u + du) Property Budgets (cont’d) Example: Mass conservation • Assumes: • r = constant • no sources or sinks • incompressible continuity equation

9. Property Budgets (cont’d) Example: Mass conservation (cont’d) r, u (r + dr), (u + du) dz See also Section 7.7 of Stewart’s Book dy dx

10. North East Property Budgets (cont’d) Example: North Atlantic Meridional Circulation Ekman Transport Gulf Stream Interior Geostrophic Flow From Lecture notes for Intro PO by H. Bryden http://www.ocean.washington.edu/people/faculty/luanne/classes/pcc586/papers/bryden.pdf • N. Atlantic Ocean Transports @ 25 oN: • Gulf Stream  30 Sv • Northward Ekman Transport  4 Sv • Southward Interior Geostrophic Flow  34 Sv (1 Sv = 106 m2s-1)

11. Property Budgets (cont’d) Examples: Theoretical flows Horiz. Non-divergent; Hv=0: u=gx, v=-gy Curlz = v = const: u=-gy, v=gx y y v u u v x x

12. Property Budgets (cont’d) Examples: Theoretical flows Constant oblique: u=Uo, v=Uo ... other whacky possibilities ... u=ay2, v=ax2 y y u,v u u,v v x x

13. General Conservation Equations Example: Salt conservation since or equivalently: mass conservation equation for salt

14. y x General Conservation Equations Example: Salt conservation (cont’d) Ocean Salt Pond 32 PSU 20 PSU gradient time rate of change advective flux

15. General Conservation Equations Example: mass conservation (alternate justification) (1024 kg/m3· U/L) vs. (U · [1/100 ·1024 kg/m3]/L) continuity equation

16. General Conservation Equations Example: Weather www.weather.com ??? ... rhs? … other variables?

17. Summary of Key Concepts: • Preamable: term balances (a.k.a., shortcut math) • Property budgets: • Sources + Sinks = Net Change • Mass continuity / conservation equation • Divergence & gradient – physical interpretation • Intuitive notion of conservation equation, • e.g., r, S, rv (a.k.a., momentum – stay tuned …)

19. Unit 3 – Part II:Diffusion; Diffusive fluxes; Advection/Diffusion Eq. Introductory Physical Oceanography (MAR 555) - Fall 2009 Miles A. Sundermeyer

20. Key Concepts: • Random walks • Diffusion: • molecular diffusivity • eddy diffusivity • Diffusive flux / flux divergence • Advection/Diffusion Equations • Viscosity

21. Random Walks – 2D

22. Random Walks – 2D

23. Molecular Diffusivity: Molecular diffusivity: Or, more generally:

24. Molecular Diffusivity (cont’d) • Molecular diffusivities for ocean properties: • Heat: k = 1.5 x 10-7 m2 s-1 • Salt: k = 1.5 x 10-9 m2 s-1 • Velocity/Momentum: n = 1 x 10-6 m2 s-1 • (Note: for velocity, we call it “viscosity” rather than “diffusivity”; oceanic values vary from 1.8 x 10-6m2 s-1 at 0 oC to 0.9 x 10-6 • m2 s-1 at 25 oC )

25. Diffusive Flux / Flux Divergence Consider 1-D diffusion equation: Recall from advective fluxes: Flux = mass / (unit area · unit time) = mass/vol. · length/time = rv Flux divergence =

26. Diffusive Flux / Flux Divergence (cont’d) Examples of flux and flux divergence Consider 1-D diffusion equation: Consider 3 scenarios: S S S x x x

27. 45 40 35 30 25 20 15 10 5 0 Diffusive Flux / Flux Divergence (cont’d) Examples of diffusive flux and flux divergence

28. Eddy Diffusivity / Viscosity: Eddy diffusivity: Or, more generally: • Note: • Eddy diffusivities can be (and generally are) different in different coordinate directions • Eddy diffusivities can vary spatially and temporally • Eddy diffusivities can be scale dependent

29. Eddy Diffusivity / Viscosity (cont’d): Example: Eddy stirring From W. Young lecture notes: WHOI 1999 Summer GFD Program, after Welander (1955)

30. Eddy Diffusivity / Viscosity (cont’d) Example: Numerical Modeling Studies Numerical experiments by Sundermeyer and Lelong (J. Phys. Oceanogr., 2005)

31. Eddy Diffusivity / Viscosity (cont’d) Example: Numerical Modeling Studies Numerical experiments by Sundermeyer and Lelong (J. Phys. Oceanogr., 2005)

32. Eddy Diffusivity / Viscosity (cont’d) Example: Laboratory Studies Lab experiments by Grant Stuart, SMAST

33. Key Concepts: • Random walks • Diffusion: • molecular diffusivity • eddy diffusivity • Diffusive flux / flux divergence • Advection/Diffusion Equations • Viscosity