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...

  • 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

    2

    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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    3

    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.