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### Transcript of 5) continuity Continuity •Conservation of Mass •Mass In = Mass Out •“per unit...

• 9/24/19

1

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 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

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 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 + =

• 9/24/19

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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?

Divergences/ Convergences

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

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.

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•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

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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.