Post on 15-Dec-2015
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