11

JongJin ParkWoods Hole Oceanographic Institution

Decay Time Scale ofMixed Layer Inertial Motions in the

World Ocean

(Observations from Satellite Tracked Drifters)

Internal Wave Workshop, 3-4 October 2008, Applied Physics Laboratory-University of Washington, Seattle

22

Inertial energy budget in the mixed layer

Mix

ed L

ayer

Inertial kinetic energy( )Global Inertial Kinetic Energy ( EI )

Park et al. [2005]: Mixed layer KE

Alford and Whitmont [2007]: Depth integrated

IE

Previous Studies

Inertial energy flux from wind ( ) Inertial energy flux from wind ( )Alford [2001; 2003]

Watanabe and Hibiya [2001]

Jiang et al. [2005]

~ based on a slab ocean model Plueddemann and Farrar [2007]

windwind

Long-Term Goal :Global inertial energy budget in the oceanic mixed layer

Global distributionof decay time scale

~Deep e IE

Inertial energy efflux out of the mixed layer ~( )Ideep eE

33

What is Inertial Decay Timescale ?

Wind

Mixed Layer,x

y

ufv u

t H

vfu v

t H

Pollard and Millard [1970]’s slab ocean model1 : decay time-scale ( )

Parameterization of decaying inertial motion in the mixed layer

Inertial motion decays exponentially

Q: How is the decay time scale distributed in the

global ocean ?

Dynamics of inertial motion decay

44

Two ways of decaying inertial motion in the mixed layer

- Propagation of inertial-internal wave (Non-Turbulent process) : [Gill, 1984; D’Asaro, 1989; Zervakis and Levine, 1995; Meurs, 1999; etc…]

- Turbulent mixing at the base of the mixed layer (Turbulent process) : [D’Asaro, 1995; Eriksen, 1991; Hebert and Moum, 1994]

• Most of the previous studies focused on the wave propagation as a major decaying process.

• The wave propagation may be primarily responsible for the fast decay of mixed layer inertial energy [Balmforth and Young, 1999; Moehlis and Smith, 2001].

- Buoyancy Frequency - Forcing scale : Gill [1984], D’Asaro [1995]- Wave number change by Beta effect : D’Asaro [1989]- Mixed layer depth : Zervakis and Levine [1995]

- Flow convergence : Weller [1982] - Relative vorticity : Kunze [1985], Balmforth and Young [1999] - Relative vorticity gradient : Van Meurs [1999] - Etc : Advection by background flow Vertical shear of the flow

Without background flow With background flow

Q: Which factor can play more important role to control the global distribution of inertial decay timescale?

Method to estimate inertial amplitude from Satellite Tracked Drifter

Weighted Function Fitting Method

-4 -2 0 2 4 6Z o n a l D ista n ce (k m )

-2

0

2

4

6

8

Mer

idio

nal

Dis

tan

ce (

km

)

P 5m o

P 4m o

P 3m o

P 2m o

P 1m o

P 6m o

P 1m f

P 2m f

P 3m f

P 4m f

P 5m f

P 6m f

m : cycle number

( , )

;

( , )

;

o o ok k k

f f fk k k

P x y

observed

P x y

estimated

rectilinear inertialu u u e

(Park et al., 2004)

1, ( 1, , )k kt t t k N

( )*cos( ) ( )*sin( )

( )*sin( ) ( )*cos( )

fk L k o r o k r o k

fk L k o r o k r o k

x u t x x x f t y y f t

y v t y x x f t y y f t

Inertial Recti-linear 1

exp[ ( )]kt

rtU i ft dt P

Trajectory segment length : > 0.7 * local inertial period Number of fixes : > 5Data latitude : 60oS~60oN except 29o~31o Rectilinear velocity : < 50 cm/s

Data Criteria

r oinertial oU u P P f

Inertial amplitude

Distribution of inertial amplitudes (U) estimated from Satellite tracked drifters (1990~2004)

Global distribution of inertial amplitude (U)

66

Mean Inertial amplitude(2ox2o) 1990~2004

(cm/s)Drifter measurement of U

Inertial energy fluxestimated by a slab model andNCEP wind

77

( , )

1( ) ( ( , ) )( ( , ) ),

( ) 40

k

k

N

k i i k j j ki jk

i j k k i j o

U x t U U x t UN

t t t and x x km

Assumption : Homogeneous amplitude within (Uncorrelated observation error, homogeneity of error, homogeneity of variance)

Freeland et al. [1975]

Separation Time (day)

Cor

rela

tion

e-folding (δ) = 4.9 (4.1- 6.1)(95% confidence interval)

North Pacific(Winter)

UI(tj)

UI(t

i)

Lag - 1dayCorrcoef. = 0.84

UI(tj)

UI(t

i)

Lag - 5day

Corrcoef. = 0.44

Estimating decay time scale of inertial amplitude (U)

o

88

0 500 1000 1500 20000

20

40

60

Time

Am

pli

tud

e

0 10 20 30 400

0.2

0.4

0.6

0.8

1

Time lag

Co

rrela

tio

n

Concept of estimating decay time scale

0 50 100 150 2000

5

10

15

20

Time

Am

pli

tud

e

0 10 20 30 40-0.5

0

0.5

1

Time lag

Co

rrela

tio

n

( ) /

1

( ) ( ) i

Nt t

i ii

U t t t Fe

0 50 100 150 2000

5

10

15

20

Time

Am

pli

tud

e

0 10 20 30 40-0.2

0

0.2

0.4

0.6

0.8

Time lag

Co

rrela

tio

n

/e

0 500 1000 1500 20000

20

40

60

Time

Am

pli

tud

e

0 10 20 30 40-0.2

0

0.2

0.4

0.6

0.8

Time lag

Co

rrela

tio

n/e

0 500 1000 1500 20000

20

40

60

Time

Am

pli

tud

e

0 10 20 30 40-0.2

0

0.2

0.4

0.6

0.8

Time lag

Co

rrela

tio

n

0 50 100 150 2000

5

10

15

20

Time

Am

pli

tud

e

0 10 20 30 40-0.2

0

0.2

0.4

0.6

0.8

Time lag

Co

rrela

tio

n

2

( '( ) '( ))( )

E U t U tR

Preset DecayFunction

Auto-Correlation

RandomPair Sampling

Inertial amplitudes from a short-term trajectory segment

Independent datasetfor 15 years

Temporal correlation function in the basin average sense

Utilizing the whole data in a certain area by the pair-sampling method

Separation Time (day)C

orre

lati

on

NA (50~60)

= 4.8 ( 4.2 - 5.5)

0 5 10 15 200

0.2

0.4

0.6

0.8

0

5

10

15

20

25

Separation Time (day)

Cor

rela

tion

NA (20~30)

= 4.0 ( 3.4 - 4.5)

0 5 10 15 200

0.2

0.4

0.6

0.8

0

5

10

15

20

25

Separation Time (day)

Cor

rela

tion

NP (20~30)

= 3.7 ( 3.0 - 4.4)

0 5 10 15 200

0.2

0.4

0.6

0.8

0

5

10

15

20

25

Separation Time (day)

Cor

rela

tion

NP (50~60)

=12.9 ( 9.2 -16.6)

0 5 10 15 200

0.2

0.4

0.6

0.8

0

5

10

15

20

25

Examples of Correlation Function (Bootstrap resampling)

99

(%)

(%)(%)

(%)Temporal correlation function of inertial amplitudes from the Drifter Observation

δ=12.9 (9.2-16.6)North Pacific (50oN~60oN)

δ= 3.7 (3.0-4.4)North Pacific (20oN~30oN)

δ= 4.0 (3.4-4.5)North Atlantic (20oN~30oN)

δ= 4.8 (4.2-5.5)North Atlantic (50oN~60oN)

• Exponential shape

• Basin wide difference

• Meridional difference

1010

Decay time scale of inertial amplitude (U)

LOW MID HIGHLOW MID HIGH

North AtlanticNorth Pacific

Winter (D-A)

Summer (J-O)

95% confidenceinterval

Low = 15N~30N, Mid = 30N~45N, High = 45N-60N

★★ Winter

Summer

PreviousMoored Obs. North Pacific : Slow decay in high latitude

North Pacific : Slow decay in summer North Atlantic : No significant meridional

distribution

E-folding timescale of observed correlation function

★

★

★★★ ★

★

★ ★

Meridional distribution of decay time scale

-60 -40 -20 0 20 40 600

5

10

15

20

Latitude

Dec

ayti

mes

cale

(d

ay)

NANPSA+IOSP

Drifter Observation ( )ObsA

-60 -40 -20 0 20 40 600

5

10

15

20

Latitude

Dec

ayti

mes

cale

(d

ay)

NANPSA+IOSP

North Atlantic (60W~0)

South Pacific (150E~80W)

North Pacific (140E~100W)

South Atlantic + Indian Ocean (80W~150E)

Decay time scale increases with latitude

Decay time scale hardly varies from 20o to 40o and rapidly increases with latitude higher than 45o

No significant meridional variation in the North Atlantic

Q: How is the decay time scale distributed in

the global ocean ?

What makes the time scale so different in space?

1212

Understanding spatial variation of observed decay time scale

20

( , )( ) 0,

( , ) 2 2t

A iA f A i y A

x y

[Young and Ben Jelloul, 1997; Balmforth and Young, 1999]

2 2( ) .o z zA f N A

[Local change]

[Wave advection]

[Wave dispersion]

[Wave refraction]

Propagation equation of Near-Inertial Wave

( , )~ 0

( , )

A

x y

~ 0

( 100 )Y Y km / 2@North Pacific/ 2@North Atlantic

No zonal variation of and A:

Small relative vorticity :

20

0

( 0)

( )m

m

N N H z

N N z H

Linear density profile with mixed layer (Hm)

1

Assumptions

exp( ) ,ou iv if t A

20 0

2t

iA f A i y A

[Moehlis and Smith, 2001]

1313

* * * *0/ , / , 1 / , ,mA A A y y Y z z H t t * *

0/ , o oN N N l Yl

2 21/30

0

( ) ,mH NY

f

Non-dimensionalize

2 2 21/30

0

( )mH N

f

Initial condition

0

0

oil ym

m

Ue H zu iv

z H

* 1 1/300 0 2 2 2

0

( ) ( ) ( )ModelA

m

fl l Y

H N

Simplified Analytical Model

t*

l* 0

0 1 2 3 4 5-4

-2

0

2

4

0

0.2

0.4

0.6

0.8

1

(Discussed with Stefan L. Smith at Scripps)

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

t*

A*

l0 = -2

l0 = -1

l0 = 0

U*

Solution for amplitude evolution in the mixed layer

* * * *3( )* * * / 6 *3/ 2

* * * 3 *3 1/30

1( )2 3

{( ) }

oiy t l iML ML

o

iU u iv e e erfc

where t l l

Initial Scale, MLD, and N from QuikSCAT and Argo floats (2000~2007)

180oW 120oW 60oW 0o 60oE 120oE 180oW 60oS

45oS

30oS

15oS

0o

15oN

30oN

45oN

60oN

0

100

200

300

400

λU ~Meridional scale of correlation (R)

(km)Forcing scale (λU)With 72 hours high pass filtered QuikSCAT wind (Uw)

1( , ) ( , ) ( , )w w

x x x

R x x U x t U x x t dt

180oW 120

oW 60

oW 0

o 60

oE 120

oE 180

oW

60oS

45oS

30oS

15oS

0o

15oN

30oN

45oN

60oN

0

100

200

300

400

MLD (Hm)

( , ) ( , ),

0.8 o

T T S T S

where T C

Density based method of Kara et al. [2000]

(m)

Nmax (N)

180oW 120

oW 60

oW 0

o 60

oE 120

oE 180

oW

60oS

45oS

30oS

15oS

0o

15oN

30oN

45oN

60oN

0

0.01

0.02

0.03

0.04(s-1)

1/300 2 2 2

0

( ) ( )ModelA

m

fl Y

H N

180oW 120

oW 60

oW 0

o 60

oE 120

oE 180

oW

60oS

45oS

30oS

15oS

0o

15oN

30oN

45oN

60oN

0

5

10

15

20

180oW 120

oW 60

oW 0

o 60

oE 120

oE 180

oW

60oS

45oS

30oS

15oS

0o

15oN

30oN

45oN

60oN

0

1

2

3

4

5

180oW 120

oW 60

oW 0

o 60

oE 120

oE 180

oW

60oS

45oS

30oS

15oS

0o

15oN

30oN

45oN

60oN

0

5

10

15

20

Decay timescale based on simplified analytical model

(day)

95% ConfidenceLevel estimated byBootstrap method

Decay timescalesimulated by theoretical model

(day)

ModelA 1/30

0 2 2 20

( ) ( )m

fl Y

H N

-60 -40 -20 0 20 40 600

5

10

15

20

Latitude

Dec

ayti

mes

cale

(d

ay)

NANPSA+IOSP

Comparison of observation and analytical model

-60 -40 -20 0 20 40 600

5

10

15

20

Latitude

Dec

ayti

mes

cale

(da

y)

NANPSASPIO

Theoretical Model

Drifter Observation ( )ObsA

( )ModelA

Q: Which factor can play an important role to control the global distribution of inertial decay timescale?

2 2 2 2 2 2

ˆ ˆ ˆ ˆlog( / )ˆ ˆ ˆlog( / ); (1/ 3) log( /( ) ); (1/ 3) log( /( ) )

cA A A

c c co o m mf f N H N H

Control Factors for decay time scale

( )c : Basin-averaged value of the North Pacific

1/3 2 2 1/300 02

( ) ( ) ( )ModelA m

fl Y H N

Forcing Scale

ˆA

Beta Effect

Bouyancy Effect

Decay Time Scale

10 20 30 40 50 60-1

-0.5

0

0.5

1

Latitude

Non

dim

ensi

onal

Fac

tor

10 20 30 40 50 60-1

-0.5

0

0.5

1

Latitude

Non

dim

ensi

onal

Fac

tor

-60-50-40-30-20-10-1

-0.5

0

0.5

1

Latitude

Non

dim

ensi

onal

Fac

tor

-60-50-40-30-20-10-1

-0.5

0

0.5

1

Latitude

Non

dim

ensi

onal

Fac

tor

North Pacific North Atlantic

South Pacific South Atlantic

Why are the meridional structures of the buoyancy effect so different?

2 2

log( / )

2 log( / )

2 log( / )

m

cm

c

cm m m

b N H

b b b N H

N N N

H H H

b

mH

N

2 2

2 1

5.9

2.2 10

~ 110

c

c

cm

b m s

N s

H m

Buoyancy structure

180oW 120

oW 60

oW 0

o 60

oE 120

oE 180

oW

60oS

45oS

30oS

15oS

0o

15oN

30oN

45oN

60oN

-3

-2

-1

0

1

2

3

180oW 120

oW 60

oW 0

o 60

oE 120

oE 180

oW

60oS

45oS

30oS

15oS

0o

15oN

30oN

45oN

60oN

-3

-2

-1

0

1

2

3

180oW 120

oW 60

oW 0

o 60

oE 120

oE 180

oW

60oS

45oS

30oS

15oS

0o

15oN

30oN

45oN

60oN

-3

-2

-1

0

1

2

3

• N and Hm seem to be canceled out in terms of spatial distribution.

• Shallow Hm in the high latitude of the North Pacific is responsible for the longer decay time scale.

• Weaker stratification in the Southern Ocean makes the time scale longer.

• In the North Atlantic, deep mixed layer and yet strong buoyancy may be the major cause of the shortest decay time scale in the high latitudes.

longer δ

longer δ

longer δ

1919

Understanding Dynamics

From Kunze [1985]’s dispersion relation2 2 2

2

( ) 1[ ( )]

2 2o o

N k l U Vf y l k

fm m z z

Group velocity of inertial-internal wave ignoring vertical shear of low frequency background current

2 2

3o

gz

N lC

m fm

N2 fo

2 2

3gz

N lC

fm

or

0k assuming

0 0( ) ( 0)l t l t consider l [D’Asaro, 1989]

exp( ( )) exp( ( ))I oZ U i ft ly U if t iy l t

l gzC

Stratification and Local inertial frequency

Beta Effect and Forcing Scale

2020

Understanding Dynamics : MLD

2 2

3o

gz

N lC

m fm

With a continuously varying density structure, a perturbation is separated into several modes (normal modes). Large MLD induces lower modes to have larger energy [Zervakis and Levine, 1995]

mH

m gzC

[Zervakis and Levine, 1995]

Deep MLD

Shallow MLDLowMode

HighMode

Mixed Layer Depth

2121

Summary & Conclusion

Global distribution of inertial decay timescale from the drifter observation : Increasing with latitudes in all the basins except in the North Atlantic

The analytical model with beta dispersion dynamics reproduces global distribution of the decay timescale fairly comparable to the observation.

Dephasing process by beta effect is primarily responsible for the meridional variation of the decay timescale in the North Pacific and the Southern Ocean.

In the North Atlantic, buoyancy effect seems to compensate the beta effect which leads to a lack of meridional variation.

Temporal correlation function

Theoretical solution Shape of exponential function

AcceptableRayleigh damping

The decay time scale distribution shown in this study suggested that the mixed layer inertial energy budget may have basin-dependency.

2222

Special thanks : Ray Schmitt, Young-Oh Kwon, Chris Garrett, Stefan Smith, Kurt Polzin, Tom Farrar, Julie Deshayes

2323

Special thanks : Ray Schmitt, Young-Oh Kwon, Chris Garrett, Stefan Smith, Kurt Polzin, Tom Farrar, Julie Deshayes