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Coordinate System
Origin Subtraction
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)
Continuity in
1-dir
mass in massouttime time
=
1 1 2 2V Vt t
ρ ρ=
1 1 2 2A u A uρ = ρ
V = volume
V (m3/s) = A (m2) • u(m/s)t
A1u1 =A2u2
• 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?)
U1 U2
X1 X2
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
2-D Continuity
If u1=5m/s and u2=2m/s, what is du/dx?
All sides of the box are10 m in length.
What is dv/dy?What are some ways to get dv/dy = .3 s-1?
du dv0
dx dy+ =
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3-D Continuitydu dv dw
0dx dy dz
+ + =
If du/dx + dv/dydoes not add up to zero, what has to happen?
Divergences/ Convergences
Use of Continuity Equation
• du + dv = - dwdx dy dz
ud - ub + va - vc = - (W2 - W1)
dx dy (-Z2 - -Z1)
Always choose Z1to be 0 m since w = 0 there
Continuity Equation
• 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.
DiffusionDiffusion 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•s
cp = 1 cal/gm•°C
• Example:Diffusion of Heat = - cp kq dT/dnkq = 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
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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.
X
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.
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