1 / 33

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

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. Key Concepts: Preamable: term balances (a.k.a., shortcut math) Property budgets:

Download Presentation

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

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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 …)

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

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

  20. Random Walks – 2D

  21. Random Walks – 2D

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

  23. 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 )

  24. 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 =

  25. 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

  26. 45 40 35 30 25 20 15 10 5 0 Diffusive Flux / Flux Divergence (cont’d) Examples of diffusive flux and flux divergence

  27. 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

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

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

  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: Laboratory Studies Lab experiments by Grant Stuart, SMAST

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

More Related