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, w. Mass per volume per time (kg/(m 2 s). , v. , u. Flux of mass out (kg/s) =. Flux of mass in (kg/s) =. Net Flux of mass in ‘ x ’ =. Net Flux of mass in ‘ y ’ =. Net Flux of mass in ‘ z ’ =. which is the same as:. or.
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, w Mass per volume per time (kg/(m2 s) , v , u Flux of mass out (kg/s) = Flux of mass in (kg/s) = Net Flux of mass in ‘x’ = Net Flux of mass in ‘y’ = Net Flux of mass in ‘z’ =
which is the same as: or 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:
Boussinesq approximation But 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.) = 1.1510-5
NablaorDeloperator Special case of variations in the horizontal only:
w w z u u x w w z u u x
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?
Continuity Equation in Bulk Form: Continuity assuming: steady state one dimensional motion flow integrated over a closed surface
, w Advective flux of salt , v Diffusive flux of salt , u Flux of salt into dydz = Flux of salt out of dydz = Net Flux of Salt in ‘x’ = Conservation of Salt
Net Salt Flux per unit volume = 0 Salt Conservation Net Flux of Salt in ‘y’ = Net Flux of Salt in ‘z’ =
Advection-Diffusion Equation Conservation of Salt: At steady state and assuming constant diffusivities: Further assuming motion in one direction and integrated over the volume considered, the statement of CONSERVATION OF SALT may be given as: VinSin = VoutSout
Continuity Equation in Bulk Form: Sb S0 Salt Conservation Equation in Bulk Form: VbSb =V0S0
Example of Conservation Principles What is the volume inflow and outflow at the Chesapeake Bay entrance if the mean river discharge is 2,200 m3/s, and the outflowing salinity is 24 and the inflowing salinity is 30? Conservation of Mass: Vout = Vin + R Conservation of Salt: Vout Sout = VinSin
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:
Earth doesn’t gain or lose heat, but the ocean exchanges it with the atmosphere Qheat flux through air-water interface (Watts/m2) rhoa density of air (1.2 kg/m3) Cp specific heat of moist air [~1000 joules/(kg oC)] Qs sensible heat or direct thermal transfer Ql latent heat flux or evaporation QB long wave radiation from the ocean Qi short-wave radiation or incident solar radiation Conservation of Heat:
Heating and Cooling Processes 64 emitted by clouds, water vapor and CO2 (long-wave radiation) 1370 W/m2 To Space 100 6 to space 19 30 reflected from clouds, by water and land, and backscattered by air (shortwave radiation) Absorbed by Atmosphere 51 15 Ql QB Qi 21 long-wave radiation Qs 23 Evaporation 7 Sensible 51 absorbed in ocean
Advection-Diffusion for a Non-conservative Tracer C could be chlorophyll_a dissolved oxygen nutrient species pollutant planktonic species sediment concentration
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?
