5) continuity Continuity •Conservation of Mass •Mass In = Mass Out •“per unit...
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Transcript of 5) continuity Continuity •Conservation of Mass •Mass In = Mass Out •“per unit...
Continuity •Conservation of Mass •Mass In = Mass Out
• “per unit volume” means M / Vol = r. It is the water's density that represents its mass •Fluid conserves mass by adjusting its flow (its velocity)
mass in massout time time
1 1 2 2V V t t
ρ ρ =
1 1 2 2A u A uρ = ρ
V = volume
V (m3/s) = A (m2) • u(m/s) t
• To conserve mass the east - west velocity adjusts (changes) in the east-west direction:
• Mathematically this is du/dx • du = u2 - u1 • dx x2 - x1
What is du/dx here?
(How would you calculate it?)
What if the flow was North - South
• Mathematically the change in velocity over change is distance is:
• dv/dy • dv = v2 - v1 • dy y2 - y1
If u1=5m/s and u2=2m/s, what is du/dx?
All sides of the box are 10 m in length.
What is dv/dy? What are some ways to get dv/dy = .3 s-1?
du dv 0
dx dy + =
3-D Continuity du dv dw
0 dx dy dz
+ + =
If du/dx + dv/dy does not add up to zero, what has to happen?
Use of Continuity Equation
• du + dv = - dw dx dy dz
ud - ub + va - vc = - (W2 - W1) dx dy (-Z2 - -Z1)
Always choose Z1 to be 0 m since w = 0 there
• The continuity equation can be used to describe the movement of any conservative property by advection.
• Advection – movement due to the flow of the ocean
• There is another process that can transport properties such as heat and salt besides advection.
Diffusion Diffusion of Salt Diffusion of Heat
= - ks dS/dn = - cp kq dT/dn
Molecular diffusivity (ks) molecular conductivity (kq) ks = 2 x 10-5 gm/cm•s kq = 1 x 10-3 gm/cm•s
cp = 1 cal/gm•°C
• Example: Diffusion of Heat = - cp kq dT/dn kq = 1 x 10-3 gm/cm•s cp = 1 cal/gm•°C
• Measurements in Med. Sea found the diffusion of heat in the sea to be 8 W/m2. What is the temperature gradient in the sea? 8 W/m2 = 2 x 10-4 cal/cm2•s
•2 x 10-4 cal/cm2•s = -(1 cal/gm•°C)(1 x 10-3 gm/cm•s ) dT/dn
• dT/dn = - .2°C/cm
• This is - 20,000°C/km
Diffusion • In the real ocean properties such as momentum,
salt and heat will not be transferred only by molecular processes.
• Actions such as winds, waves breaking, currents etc. produces a turbulent ocean filled with eddies.
• The movement of properties by turbulent processes (eddies) is HUGE compared to molecular processes.
• This turbulent internal friction of a fluid is known as Eddy Viscosity or Eddy Diffusion.
Diffusion of Salt Diffusion of Heat (gm/cm2•s) (cal/cm2•s)
= - ks dS/dn = - cp kq dT/dn
Molecular diffusivity (ks) molecular conductivity (kq) ks = 2 x 10-5 gm/cm•s kq = 1 x 10-3 gm/cm•sX
Replace Molecular Terms with Eddy Viscosity Coefficients
Ah = 107 gm/cm s Av = 10 gm/cm s
Diffusion of Salt (east - west direction)
= - Ah dS/dn
Summary: Conservation of Mass
• The ocean conserves mass by adjusting its flow. This is represented by the 3-D Continuity Equation.
• Wherever there is a convergence or divergence in the horizontal flow, there will be vertical flow.
• The transfer of ocean properties such as momentum, salt & heat occurs by turbulent eddy processes not by molecular processes. This turbulent transfer is known as eddy viscosity.
• Eddy viscosity if represented by coefficients(Ah, Az) in the diffusion equations.