unit 3 – part i: mass conservation; water, salt & heat budgets; fluxes

SundermeyerMAR 555 Fall, 2009

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Unit 3 – Part I:Mass Conservation; water, salt

& heat budgets; fluxes

Introductory Physical Oceanography (MAR 555) - Fall 2009

Miles A. Sundermeyer

Assigned Reading: OC 6.2 and IPO Chapter 7


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Key Concepts:

1. Preamable: term balances (a.k.a., shortcut math)2. Property budgets:

Sources + Sinks = Net Change

3. Mass continuity / conservation equation

4. Divergence & gradient – physical interpretation

5. Intuitive notion of conservation equation,

e.g., , S, v (a.k.a., momentum – stay tuned …)

0

uz

w

y

v

x

u 0

uz

w

y

v

x

u


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Term balancesExample: driving from UMass Boston -> SMAST

Goo

gle

Map

s (w

ww

.goo

gle.

com

)

Dtotal = DUMB-I95split + DI95-Rt24-I195 + DI195-SMAST

highway drive vs. city drive

Ttotal = TUMB-I95split + TI95-Rt24-I195 + TI195-SMAST


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Global Heat BudgetsOceanic Radiation Balance

From Lecture notes for Intro PO by H. Brydenhttp://www.ocean.washington.edu/people/faculty/luanne/classes/pcc586/papers/bryden.pdf


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(Property) BudgetsExample: Bank Checking/Debit Account

Direct Deposit


Net Flux in or out change in $ value / unit time)

Accrued Interest

Other Deposits

Checks Written

Debits

Other Withdrawals$


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Property BudgetsExample: Mass Conservation

u u + u)


Flux = mass / (unit area · unit time) = mass/vol. · length/time= v

analogous to heat flux = Energy/(unit area · unit time)

u ≠ 0 indicates flux divergence


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Property Budgets (cont’d)Example: Mass Conservation - Flux Divergence


Flux divergence = wz

vy

ux

v

u + u) – udx

u u + u)


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Property Budgets (cont’d)Example: Mass conservation

Assumes:• = constant• no sources or sinks• incompressible

wz

vy

ux

v

0

uz

w

y

v

x

u

continuity equation

u u + u)


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Property Budgets (cont’d)Example: Mass conservation (cont’d)

yzu in flow Mass

u ( + )u + u)

z

xy

yzuu out flow Mass

See also Section 7.7 of Stewart’s Book

Flux Mass D-3 zyx

z

w

y

v

x

u

Balances zyxt

01

z

w

y

v

x

u

Dt

D


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Property Budgets (cont’d)Example: North Atlantic Meridional Circulation

From Lecture notes for Intro PO by H. Brydenhttp://www.ocean.washington.edu/people/faculty/luanne/classes/pcc586/papers/bryden.pdf

N. Atlantic Ocean Transports @ 25 oN:• Gulf Stream 30 Sv• Northward Ekman Transport 4 Sv• Southward Interior Geostrophic Flow 34 Sv

North

East

Gulf StreamInterior Geostrophic Flow

Ekman Transport

(1 Sv = 106 m2s-1)


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Property Budgets (cont’d)Examples: Theoretical flows

Horiz. Non-divergent; Hv=0:u=x, v=-y

Curlz = v = const:u=-y, v=x

y

u

x

v

y

v

x

u


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Property Budgets (cont’d)Examples: Theoretical flows

Constant oblique:u=Uo, v=Uo

... other whacky possibilities ...u=y2, v=x2

y

u

x

v

y

u,v

x

u,v


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General Conservation EquationsExample: Salt conservation

Swz

Svy

Suxt

S

0

uz

w

y

v

x

u

z

Sw

z

wS

y

Sv

y

vS

x

Su

x

uS

since

0

z

Sw

y

Sv

x

Su

t

S

mass conservation equation for salt

0)(

Sut

Sor equivalently:


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OceanSalt

Pond32 PSU20 PSU

x

y

xS

ut

S

Dt

DS

advectiveflux

gradient

timerate of change

General Conservation EquationsExample: Salt conservation (cont’d)


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General Conservation EquationsExample: mass conservation (alternate justification)

wz

vy

uxt

z

wz

w

yv

y

v

xu

x

u

(1024 kg/m3 · U/L) vs. (U · [1/100 ·1024 kg/m3]/L)

0

uz

w

y

v

x

u

continuity equation


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General Conservation EquationsExample: Weather

z

Tw

y

Tv

x

Tu

t

T??? ... rhs?

… other variables?

www.weather.com


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Summary of Key Concepts:

1. Preamable: term balances (a.k.a., shortcut math)2. Property budgets:


3. Mass continuity / conservation equation

4. Divergence & gradient – physical interpretation

5. Intuitive notion of conservation equation,

e.g., , S, v (a.k.a., momentum – stay tuned …)

0

uz

w

y

v

x

u 0

uz

w

y

v

x

u


18


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Unit 3 – Part II:Diffusion; Diffusive fluxes;

Advection/Diffusion Eq.

Introductory Physical Oceanography (MAR 555) - Fall 2009

Miles A. Sundermeyer


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Key Concepts:1. Random walks

2. Diffusion:

1. molecular diffusivity

2. eddy diffusivity

3. Diffusive flux / flux divergence

4. Advection/Diffusion Equations

5. Viscosity

SSut

S 2)(


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Random Walks – 2D


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Random Walks – 2D


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Molecular Diffusivity:

2

2

2

2

2

22

z

S

y

S

x

SS zyx

Molecular diffusivity:

Or, more generally:

z

S

zy

S

yx

S

x zyx


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Molecular Diffusivity (cont’d)Molecular diffusivities for ocean properties:

• Heat: = 1.5 x 10-7 m2 s-1

• Salt: = 1.5 x 10-9 m2 s-1

• Velocity/Momentum: = 1 x 10-6 m2 s-1

(Note: for velocity, we call it “viscosity” rather than “diffusivity”; oceanic values vary from 1.8 x 10-6 m2 s-1 at 0 oC to 0.9 x 10-6 m2 s-1 at 25 oC )

TTut

T 2)(

SSut

S 2)(

uuut

u 2)(


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Diffusive Flux / Flux Divergence

Consider 1-D diffusion equation:

x

S

xt

S

Recall from advective fluxes:


Flux divergence = wz

vy

ux

v

L

S

T

L

L

21


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Diffusive Flux / Flux Divergence (cont’d)Examples of flux and flux divergence

Consider 1-D diffusion equation:

x

S

xt

S

Consider 3 scenarios:

S

x

0

0

2

2

x

S

x

S

constS S

x

02

2

x

S

constx

S

axS S

x

constx

S

bxx

S

bxS

2

2

2


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Diffusive Flux / Flux Divergence (cont’d)Examples of diffusive flux and flux divergence

x

S

xt

S

15

20

10

25

35

30

40

0

5

45

02

2

x

S02

2

x

S


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Eddy Diffusivity / Viscosity:

2

2

2

2

2

22

z

SA

y

SA

x

SASA zyx

Eddy diffusivity:

Or, more generally:

z

SA

zy

SA

yx

SA

x zyx

Note: • Eddy diffusivities can be (and generally are) different in

different coordinate directions• Eddy diffusivities can vary spatially and temporally• Eddy diffusivities can be scale dependent


292

2

2

2

2

22

z

SA

y

SA

x

SASA zyx

Eddy Diffusivity / Viscosity (cont’d):Example: Eddy stirring

From W. Young lecture notes: WHOI 1999

Summer GFD Program, after Welander (1955)


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Eddy Diffusivity / Viscosity (cont’d)Example: Numerical Modeling Studies

Numerical experiments by Sundermeyer and Lelong (J. Phys. Oceanogr., 2005)


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Eddy Diffusivity / Viscosity (cont’d)Example: Numerical Modeling Studies

Numerical experiments by Sundermeyer and Lelong (J. Phys. Oceanogr., 2005)


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Eddy Diffusivity / Viscosity (cont’d)Example: Laboratory Studies

Lab experiments by Grant Stuart, SMAST


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Key Concepts:1. Random walks

2. Diffusion:

1. molecular diffusivity

2. eddy diffusivity

3. Diffusive flux / flux divergence

4. Advection/Diffusion Equations

5. Viscosity

SSut

S 2)(

unit 3 – part i: mass conservation; water, salt & heat budgets; fluxes

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