Review of conservation equations State, Mass and Momentum

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Presentation transcript:

Review of conservation equations State, Mass and Momentum

Equation of State (EOS-80) Determines water density from T, S, and P A through N are polynomials T is temperature in oC S salinity P pressure in bars K secant bulk modulus – measure of compresibility

Check values: A B C D T0 999.842594 8.24493E-1 -5.72466E-3 4.8314E-4 F G 19652.21 54.6746 7.944E-2 148.4206 -0.603459 1.6483E-2 -2.327105 1.09987E-2 -5.3009E-4 1.360477E-2 -6.1670E-5 -5.155288E-5 H I J 3.239908 2.2838E-3 1.91075E-4 1.43713E-3 -1.0981E-5 1.16092E-4 -1.6078E-6 -5.77905E-7 M N 8.50935E-5 -9.9348E-7 -6.12293E-6 2.0816E-8 5.2787E-8 9.1697E-10 Check values:

90 % of Ocean Water Greater influence of salinity on density Mean T & S for World Ocean Greater influence of salinity on density

Effects of Temperature and Salinity on Density Thermal Expansion Saline Contraction x 10-4 oC-1 x 10-4 S-1 Density changes by 0.2 kg/m3 for a T change of 1oC, and by 0.8 kg/m3 for a S change of 1.

Flux of mass out (kg/s) = , w , v Mass per area per time (kg/(m2 s)) 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’ =

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

Boussinesq approximation 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 But

z x Continuity Equation in Bulk Form:

Conservation of Salt: Conservation of Heat: Equation of State:

z Sb S0 x Continuity Equation in Bulk Form: Salt Conservation Equation in Bulk Form: VbSb = V0S0

Conservation of Momentum (Equations of Motion)

Pressure gradient + friction + tides + gravity + Coriolis Pressure gradient: Barotropic and Baroclinic Friction: Surface, bottom, internal Tides: Boundary condition Gravity: Only in the vertical Coriolis: Only in the horizontal REMEMBER, these are FORCES PER UNIT MASS

Pressure gradient + friction + tides + gravity + Coriolis Pressure gradient: Barotropic and Baroclinic Friction: Surface, bottom, internal Tides: Boundary condition Gravity: Only in the vertical Coriolis: Only in the horizontal REMEMBER, these are FORCES PER UNIT MASS

Hydrostatic Pressure Total Pressure Pressure gradient force per unit mass Barometric Barotropic Baroclinic Note that even if the density is constant with depth, the horizontal pressure gradient increases with depth if there is a horizontal density gradient

Pressure gradient + friction + tides + gravity + Coriolis Pressure gradient: Barotropic and Baroclinic Friction: Surface, bottom, internal Tides: Boundary condition Gravity: Only in the vertical Coriolis: Only in the horizontal REMEMBER, these are FORCES PER UNIT MASS

Friction @ surface: @ bottom: @ interior:

Pressure gradient + friction + tides + gravity + Coriolis Pressure gradient: Barotropic and Baroclinic Friction: Surface, bottom, internal Tides: Boundary condition Gravity: Only in the vertical Coriolis: Only in the horizontal REMEMBER, these are FORCES PER UNIT MASS

Gravity [0, 0, g] = [0, 0, 9.81] Coriolis [-fv, fu, 0]