1. Flux of mass in (kg/s) = Flux of mass out (kg/s) = Net Flux of mass in ‘x’ = Net Flux of mass in ‘y’ = Net Flux of mass in ‘z’ =, u, w, v Mass per volume per time (kg/(m 2 s) Conservation of Mass

2. The change of mass per unit time going through the volume element is: And the change of mass per unit time per unit volume is: which is the same as: or

3. This is the Continuity Equation or Equation of Conservation of Mass How valid is the Boussinesq approximation in the OCEAN? How would you determine that? 1 sigma-t throughout one day = 1 / (24*3600.) = 1.15  10 -5 Boussinesq approximation But

4. Nabla or Del operator Special case of variations in the horizontal only:

5. x z uuww x z uu ww

6. Example: What is the vertical velocity in an upwelling region 30 m deep, where the flow accelerates southward by 0.10 m/s in 10 km, and westward by 0.20 m/s in 10 km?

7. Continuity Equation in Bulk Form: Continuity assuming: steady state one dimensional motion flow integrated over a closed surface

8. Flux of salt into dydz = Net Flux of Salt in ‘x’ =, u, w, v Advective flux of salt Conservation of Salt Diffusive flux of salt Flux of salt out of dydz =

9. Net Flux of Salt in ‘y’ = Net Flux of Salt in ‘z’ = Net Salt Flux per unit volume = 0 Salt Conservation

10. Conservation of Salt: At steady state and assuming constant diffusivities: Advection-Diffusion Equation Further assuming motion in one direction and integrated over the volume considered, the statement of CONSERVATION OF SALT may be given as: V in S in = V out S out

11. Continuity Equation in Bulk Form: SbSb S0S0 Salt Conservation Equation in Bulk Form: V b S b = V 0 S 0

12. Example of Conservation Principles (Knudsen Relations or Basin Equations) What is the volume inflow and outflow at the Chesapeake Bay entrance if the mean river discharge is 2,200 m 3 /s, and the outflowing salinity is 24 and the inflowing salinity is 30? Conservation of Mass: V out = V in + R Conservation of Salt: V out S out = V in S in

13. Residence Time Time it takes to flush entire volume of system (this is one definition). May be determined from volume of basin and volume of water that enters the basin per unit time. From the above example:

14. Conservation of Heat: Earth doesn’t gain or lose heat, but the ocean exchanges it with the atmosphere Q heat flux through air-water interface (Watts/m 2 ) rho a density of air (1.2 kg/m 3 ) C p specific heat of moist air [~1000 joules/(kg o C)] Q s sensible heat or direct thermal transfer Q l latent heat flux or evaporation Q B long wave radiation from the ocean Q i short-wave radiation or incident solar radiation

15. Heating and Cooling Processes 100 30 reflected from clouds, by water and land, and backscattered by air (shortwave radiation) 19 51 51 absorbed in ocean To Space 21 long-wave radiation 6 to space 23 Evaporation 7 Sensible Absorbed by Atmosphere 15 64 emitted by clouds, water vapor and CO 2 (long-wave radiation) QiQi QlQl QsQs QBQB 1370 W/m 2 T s sea surface temperature T a air temperature W 10 wind speed at a height of 10 m q s saturation specific humidity q specific humidity of air eta c cloud cover (%) Q io net downward flux of solar radiation α s surface albedo

16. From Kallberg et al 2005ERA-40 atlas. ERA-40 Project Report Series No. 19. © 2005 Robert H. Stewart Short Wave Radiation

17. From Kallberg et al 2005ERA-40 atlas. ERA-40 Project Report Series No. 19. © 2005 Robert H. Stewart Long Wave Radiation

18. From Kallberg et al 2005ERA-40 atlas. ERA-40 Project Report Series No. 19. © 2005 Robert H. Stewart Latent Heat Flux

19. From Kallberg et al 2005ERA-40 atlas. ERA-40 Project Report Series No. 19. © 2005 Robert H. Stewart Sensible Heat Flux

20. Annual mean net heat flux From Kallberg et al 2005ERA-40 atlas. ERA-40 Project Report Series No. 19. © 2005 Robert H. Stewart

21. Advection-Diffusion for a Non-conservative Tracer C could be chlorophyll_a dissolved oxygen nutrient species pollutant planktonic species sediment concentration

22. At oceanic station S1 the [DIN] is 200 µg/l and at station S2, 5 km to the E, it is 300 µg/l. If biochemical processes add 100 µg/l to the entire area each day and mixing/diffusion can be neglected, how much would you expect [DIN] to change in time if the region is swept by a 0.5 m/s eastward current?