Restratification by Mixed Layer Eddies

Baylor Fox-Kemper

Collaborators:R. Ferrari, R. Hallberg, G. Flierl, G. Boccaletti and

the CPT-EMiLIe team

AMS 16th Conference on Atmospheric & Oceanic Fluid Dynamics

Santa Fe, NMMonday 6/25/07, 14:00-14:15

The Stratification Permits Two Types of Baroclinic Instability:

Mesoscale and SubMesoscale (Boccaletti et al., 2006)42

!"!#

!"!$

"

%""

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!%""

$"""

&'()*+,-.

/+,0!!.

!"!1

!"!%

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"

%""

!"""

!%""

$"""

&'()*+,-.

34+,0!!.

Figure 2: Buoyancy frequency N2 =!g!z/!0 and vertical shearUz = g!x/ f!0 estimated from the133-130" W SeaSoar section shown in Fig. 1. The vertical gradients are computed across 8 m,

while the horizontal gradients are computed across 10 km. The profiles are extended to the ocean

bottom by matching the SeaSoar estimates in the upper 320 m with estimates based on Levitus

climatology for the rest of the water column. Details of the calculation are given in Appendix A.

The Stratification Permits Two Types of Baroclinic Instability:

Mesoscale and SubMesoscale (Boccaletti et al., 2006)42

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/+,0!!.

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$"""

&'()*+,-.

34+,0!!.

Figure 2: Buoyancy frequency N2 =!g!z/!0 and vertical shearUz = g!x/ f!0 estimated from the133-130" W SeaSoar section shown in Fig. 1. The vertical gradients are computed across 8 m,

while the horizontal gradients are computed across 10 km. The profiles are extended to the ocean

bottom by matching the SeaSoar estimates in the upper 320 m with estimates based on Levitus

climatology for the rest of the water column. Details of the calculation are given in Appendix A.

43

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Figure 3: Stability analysis of the mean shear shown in Fig. 2.The instability is dominated by

two distinct modes: an interior instability with wavelength close to the internal deformation radius

(approx60 km) and a mixed-layer instability (MLI) peaking at wavelength close to the ML defor-

mation radius (! 2 km). The interior instability has a spatial structure (upper left panel) spanningthe whole thermocline depth and represents the mesoscale restratification due to quasigeostrophic

baroclinic instability (Eady, 1949). TheMLI (upper left panel) is confined to the ML and represents

restratification due to ageostrophic instability within the ML (Stone, 1971).

The Stratification Permits Two Types of Baroclinic Instability:

Mesoscale and SubMesoscale (Boccaletti et al., 2006)42

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&'()*+,-.

/+,0!!.

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&'()*+,-.

34+,0!!.

Figure 2: Buoyancy frequency N2 =!g!z/!0 and vertical shearUz = g!x/ f!0 estimated from the133-130" W SeaSoar section shown in Fig. 1. The vertical gradients are computed across 8 m,

while the horizontal gradients are computed across 10 km. The profiles are extended to the ocean

bottom by matching the SeaSoar estimates in the upper 320 m with estimates based on Levitus

climatology for the rest of the water column. Details of the calculation are given in Appendix A.

43

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89:;+01<=+.12-=>!$3

?=@.A,'B.912'!$3

Figure 3: Stability analysis of the mean shear shown in Fig. 2.The instability is dominated by

two distinct modes: an interior instability with wavelength close to the internal deformation radius

(approx60 km) and a mixed-layer instability (MLI) peaking at wavelength close to the ML defor-

mation radius (! 2 km). The interior instability has a spatial structure (upper left panel) spanningthe whole thermocline depth and represents the mesoscale restratification due to quasigeostrophic

baroclinic instability (Eady, 1949). TheMLI (upper left panel) is confined to the ML and represents

restratification due to ageostrophic instability within the ML (Stone, 1971).

43

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!

!"%

!"5

!"4

!"7

89:;+01<=+.12-=>!$3

?=@.A,'B.912'!$3

Figure 3: Stability analysis of the mean shear shown in Fig. 2.The instability is dominated by

two distinct modes: an interior instability with wavelength close to the internal deformation radius

(approx60 km) and a mixed-layer instability (MLI) peaking at wavelength close to the ML defor-

mation radius (! 2 km). The interior instability has a spatial structure (upper left panel) spanningthe whole thermocline depth and represents the mesoscale restratification due to quasigeostrophic

baroclinic instability (Eady, 1949). TheMLI (upper left panel) is confined to the ML and represents

restratification due to ageostrophic instability within the ML (Stone, 1971).

MesoscaleEddies

The Stratification Permits Two Types of Baroclinic Instability:

Mesoscale and SubMesoscale (Boccaletti et al., 2006)42

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!"!$

"

%""

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&'()*+,-.

/+,0!!.

!"!1

!"!%

!"!2

"

%""

!"""

!%""

$"""

&'()*+,-.

34+,0!!.

Figure 2: Buoyancy frequency N2 =!g!z/!0 and vertical shearUz = g!x/ f!0 estimated from the133-130" W SeaSoar section shown in Fig. 1. The vertical gradients are computed across 8 m,

while the horizontal gradients are computed across 10 km. The profiles are extended to the ocean

bottom by matching the SeaSoar estimates in the upper 320 m with estimates based on Levitus

climatology for the rest of the water column. Details of the calculation are given in Appendix A.

43

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!"5

!"4

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89:;+01<=+.12-=>!$3

?=@.A,'B.912'!$3

Figure 3: Stability analysis of the mean shear shown in Fig. 2.The instability is dominated by

two distinct modes: an interior instability with wavelength close to the internal deformation radius

(approx60 km) and a mixed-layer instability (MLI) peaking at wavelength close to the ML defor-

mation radius (! 2 km). The interior instability has a spatial structure (upper left panel) spanningthe whole thermocline depth and represents the mesoscale restratification due to quasigeostrophic

baroclinic instability (Eady, 1949). TheMLI (upper left panel) is confined to the ML and represents

restratification due to ageostrophic instability within the ML (Stone, 1971).

43

! !"# $

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%!!!

&'()*+,-.

/.(+012'3

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!

!"%

!"5

!"4

!"7

89:;+01<=+.12-=>!$3

?=@.A,'B.912'!$3

Figure 3: Stability analysis of the mean shear shown in Fig. 2.The instability is dominated by

two distinct modes: an interior instability with wavelength close to the internal deformation radius

(approx60 km) and a mixed-layer instability (MLI) peaking at wavelength close to the ML defor-

mation radius (! 2 km). The interior instability has a spatial structure (upper left panel) spanningthe whole thermocline depth and represents the mesoscale restratification due to quasigeostrophic

baroclinic instability (Eady, 1949). TheMLI (upper left panel) is confined to the ML and represents

restratification due to ageostrophic instability within the ML (Stone, 1971).

43

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&'()*+,-.

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#!!

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%!!!

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/.(+012'3

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$!!5

$!!6

$!!%

!

!"%

!"5

!"4

!"7

89:;+01<=+.12-=>!$3

?=@.A,'B.912'!$3

Figure 3: Stability analysis of the mean shear shown in Fig. 2.The instability is dominated by

two distinct modes: an interior instability with wavelength close to the internal deformation radius

(approx60 km) and a mixed-layer instability (MLI) peaking at wavelength close to the ML defor-

mation radius (! 2 km). The interior instability has a spatial structure (upper left panel) spanningthe whole thermocline depth and represents the mesoscale restratification due to quasigeostrophic

baroclinic instability (Eady, 1949). TheMLI (upper left panel) is confined to the ML and represents

restratification due to ageostrophic instability within the ML (Stone, 1971).

MesoscaleEddies

SubMesoscaleMixed Layer Eddies

The Stratification Permits Two Types of Baroclinic Instability:

Mesoscale and SubMesoscale (Boccaletti et al., 2006)42

!"!#

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"

%""

!"""

!%""

$"""

&'()*+,-.

/+,0!!.

!"!1

!"!%

!"!2

"

%""

!"""

!%""

$"""

&'()*+,-.

34+,0!!.

Figure 2: Buoyancy frequency N2 =!g!z/!0 and vertical shearUz = g!x/ f!0 estimated from the133-130" W SeaSoar section shown in Fig. 1. The vertical gradients are computed across 8 m,

while the horizontal gradients are computed across 10 km. The profiles are extended to the ocean

bottom by matching the SeaSoar estimates in the upper 320 m with estimates based on Levitus

climatology for the rest of the water column. Details of the calculation are given in Appendix A.

43

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#!!

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%!!!

&'()*+,-.

/.(+012'3

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#!!

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/.(+012'3

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!

!"%

!"5

!"4

!"7

89:;+01<=+.12-=>!$3

?=@.A,'B.912'!$3

Figure 3: Stability analysis of the mean shear shown in Fig. 2.The instability is dominated by

two distinct modes: an interior instability with wavelength close to the internal deformation radius

(approx60 km) and a mixed-layer instability (MLI) peaking at wavelength close to the ML defor-

mation radius (! 2 km). The interior instability has a spatial structure (upper left panel) spanningthe whole thermocline depth and represents the mesoscale restratification due to quasigeostrophic

baroclinic instability (Eady, 1949). TheMLI (upper left panel) is confined to the ML and represents

restratification due to ageostrophic instability within the ML (Stone, 1971).

43

! !"# $

!

#!!

$!!!

$#!!

%!!!

&'()*+,-.

/.(+012'3

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!

#!!

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$!!5

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!

!"%

!"5

!"4

!"7

89:;+01<=+.12-=>!$3

?=@.A,'B.912'!$3

Figure 3: Stability analysis of the mean shear shown in Fig. 2.The instability is dominated by

two distinct modes: an interior instability with wavelength close to the internal deformation radius

(approx60 km) and a mixed-layer instability (MLI) peaking at wavelength close to the ML defor-

mation radius (! 2 km). The interior instability has a spatial structure (upper left panel) spanningthe whole thermocline depth and represents the mesoscale restratification due to quasigeostrophic

baroclinic instability (Eady, 1949). TheMLI (upper left panel) is confined to the ML and represents

restratification due to ageostrophic instability within the ML (Stone, 1971).

43

! !"# $

!

#!!

$!!!

$#!!

%!!!

&'()*+,-.

/.(+012'3

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#!!

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/.(+012'3

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$!!5

$!!6

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!

!"%

!"5

!"4

!"7

89:;+01<=+.12-=>!$3

?=@.A,'B.912'!$3

Figure 3: Stability analysis of the mean shear shown in Fig. 2.The instability is dominated by

two distinct modes: an interior instability with wavelength close to the internal deformation radius

(approx60 km) and a mixed-layer instability (MLI) peaking at wavelength close to the ML defor-

mation radius (! 2 km). The interior instability has a spatial structure (upper left panel) spanningthe whole thermocline depth and represents the mesoscale restratification due to quasigeostrophic

baroclinic instability (Eady, 1949). TheMLI (upper left panel) is confined to the ML and represents

restratification due to ageostrophic instability within the ML (Stone, 1971).

MesoscaleEddies

SubMesoscaleMixed Layer EddiesO(100km)

1 month

O(1km)1 day

Mesoscale and SubMesoscale

areCoupled Together:

ML Fronts are formed by Mesoscale Straining.

Submesoscale eddies remove PE from those

fronts.

Prototype: Mixed Layer Front Overturning

Simple Spindown Plus, Diurnal Cycleand KPP

Note: initial geostrophic adjustment overwhelmed by eddy restratification

Parameterization of Finite Amp. Eddies: Ingredients

0 2 4 6 8 10 12 14 160

50

100

150

200

250N

2/f

2

time (days)

unbalanced, Ri0=0

unbalanced, Ri0=1

balanced, Ri0=1

hi!res unbal, Ri0=0

bal, Ri0=0

Linear Solution <w’b’> for vert. structure.

Parameterization of Finite Amp. Eddies: Ingredients

FiniteAmplitude

0 2 4 6 8 10 12 14 160

50

100

150

200

250N

2/f

2

time (days)

unbalanced, Ri0=0

unbalanced, Ri0=1

balanced, Ri0=1

hi!res unbal, Ri0=0

bal, Ri0=0

Linear Solution <w’b’> for vert. structure.

Parameterization of Finite Amp. Eddies: Ingredients

FiniteAmplitude

0 5 10 15 20 25 30

10!8

10!6

10!4

10!2

time (days)

kin

etic e

nerg

y (

m2/s

2)

basin!avg. pert. KE

linear predict. pert. KE.

initial mean KE2: 1/2(M

2 H/f)

2

avg. pert. v2 in front

0 2 4 6 8 10 12 14 160

50

100

150

200

250N

2/f

2

time (days)

unbalanced, Ri0=0

unbalanced, Ri0=1

balanced, Ri0=1

hi!res unbal, Ri0=0

bal, Ri0=0

Linear Solution <w’b’> for vert. structure.

Parameterization of Finite Amp. Eddies: Ingredients

FiniteAmplitude

0 5 10 15 20 25 30

10!8

10!6

10!4

10!2

time (days)

kin

etic e

nerg

y (

m2/s

2)

basin!avg. pert. KE

linear predict. pert. KE.

initial mean KE2: 1/2(M

2 H/f)

2

avg. pert. v2 in front

Eddy Velocity Saturates

0 2 4 6 8 10 12 14 160

50

100

150

200

250N

2/f

2

time (days)

unbalanced, Ri0=0

unbalanced, Ri0=1

balanced, Ri0=1

hi!res unbal, Ri0=0

bal, Ri0=0

Linear Solution <w’b’> for vert. structure.

Parameterization of Finite Amp. Eddies: Ingredients

FiniteAmplitude

0 5 10 15 20 25 30

10!8

10!6

10!4

10!2

time (days)

kin

etic e

nerg

y (

m2/s

2)

basin!avg. pert. KE

linear predict. pert. KE.

initial mean KE2: 1/2(M

2 H/f)

2

avg. pert. v2 in front

Eddy Velocity Saturates

Near Mean KE

0 2 4 6 8 10 12 14 160

50

100

150

200

250N

2/f

2

time (days)

unbalanced, Ri0=0

unbalanced, Ri0=1

balanced, Ri0=1

hi!res unbal, Ri0=0

bal, Ri0=0

Linear Solution <w’b’> for vert. structure.

Parameterization of Finite Amp. Eddies: Ingredients

Vert. Excursions(b’ /N )

scale with H

FiniteAmplitude

0 5 10 150

0.2

0.4

0.6

0.8

1

time (days)

!/H

0 5 10 15 20 25 30

10!8

10!6

10!4

10!2

time (days)

kin

etic e

nerg

y (

m2/s

2)

basin!avg. pert. KE

linear predict. pert. KE.

initial mean KE2: 1/2(M

2 H/f)

2

avg. pert. v2 in front

Eddy Velocity Saturates

Near Mean KE

rms

2

0 2 4 6 8 10 12 14 160

50

100

150

200

250N

2/f

2

time (days)

unbalanced, Ri0=0

unbalanced, Ri0=1

balanced, Ri0=1

hi!res unbal, Ri0=0

bal, Ri0=0

Linear Solution <w’b’> for vert. structure.

Parameterization of Finite Amp. Eddies: Ingredients

Vert. Excursions(b’ /N )

scale with H

FiniteAmplitude

0 5 10 150

0.2

0.4

0.6

0.8

1

time (days)

!/H

0 5 10 15 20 25 30

10!8

10!6

10!4

10!2

time (days)

kin

etic e

nerg

y (

m2/s

2)

basin!avg. pert. KE

linear predict. pert. KE.

initial mean KE2: 1/2(M

2 H/f)

2

avg. pert. v2 in front

Eddy Velocity Saturates

Near Mean KE

rms

2

0 2 4 6 8 10 12 14 160

50

100

150

200

250N

2/f

2

time (days)

unbalanced, Ri0=0

unbalanced, Ri0=1

balanced, Ri0=1

hi!res unbal, Ri0=0

bal, Ri0=0

Eddy Fluxes are at nearly 1/2 the mean

isopycnal slope

Linear Solution <w’b’> for vert. structure.

The Parameterization:! =

CeH2µ(z)

|f |!b " z

w!b! =CeH

2µ(z)

|f ||!b|2

u!

Hb! = !

CeH2µ(z) !b

!z

|f |"H b

µ(z) =

!

1 !

"

2z

H+ 1

#2$ !

1 +5

21

"

2z

H+ 1

#2$

The horizontal fluxes are downgradient:

Vertical fluxes always upward to restratify:

Adjustments for coarse resolution and f->0 are known

It works for Prototype Sims:

Circles: Balanced Initial Cond.Squares: Unbalanced Initial Cond.

>2 orders ofmagnitude!

Red: No Diurnal Blue: With Diurnal

10!3

10!2

10!1

100

101

10!3

10!2

10!1

100

101

Ce H

2 M

2 |f|!1

!d

10!3

10!2

10!1

100

101

10!3

10!2

10!1

100

101

Ce H

2 M

2 |f|!1

!d

29Ja

nuar

y20

07FO

X-K

EM

PER

,FER

RA

RI,

and

HA

LLB

ER

G7

boun

dary

-nor

mal

velo

citi

esva

nish

.

!b

!t

+!

·ub+!

·u! b

!=

D,

(6)

dPE

dt=

d dt"

zbxyz

="

wbx

yz.

(7)

a.M

agni

tude

ofth

eO

vert

urni

ng

The

first

step

info

rmin

gth

epa

ram

eter

izat

ion

isde

-te

rmin

ing

the

mag

nitu

desc

alin

gof

the

vert

ical

and

hori

zont

aled

dybu

oyan

cyflu

xes.

MLE

sre

sult

from

baro

clin

icin

stab

ility

,whi

chre

-le

ases

PE

from

the

mea

nflo

w.

Con

side

rth

eP

Eex

trac

tion

byex

chan

geof

fluid

parc

els

over

deco

r-re

lation

dist

ance

s!

yan

d!

zin

ati

me

!t.

!P

E!

t#

"!

z! !

yM2

+!

zN2"

!t

,

We

may

estim

ate

the

extr

acti

onra

teby

assu

min

g

1.T

here

leva

ntti

mes

cale

!t

isth

eti

me

itta

kes

for

aned

dyto

trav

erse

the

deco

rrel

atio

nle

ngth

(!t#

|!y/

U|=

|!z/

W|).

2.T

heho

rizo

ntal

eddy

velo

city

scal

esas

the

mea

nth

erm

alw

ind

U#

# # M2H

/f# # (

see

Fig

.5).

3.T

heve

rtic

alde

corr

elat

ion

leng

thsc

ales

wit

hth

eM

Lde

pth

!z#

H(s

eeFig

.6).

4.Flu

idex

chan

geoc

curs

alon

ga

slop

ebe

low

(i.e

.,P

Eex

trac

ting

)an

dpr

opor

tion

alto

the

mea

nis

opyc

nals

lope

(# # !yM

2# # =

C!

zN2

whe

reC

>1,

see

Fig

.7).

Thu

s,

!P

E!

t#

"C

+1

C

M4H

2

|f|

.(8

)

The

MLE

flux

scal

ings

may

befo

und

byno

ting

that

our

sim

ulat

ions

,th

eM

LEve

rtic

alflu

xdo

mi-

nate

sth

em

ean

in(7

).T

hus,

w! b

!xyz#

C"

1C

M4H

2

|f|

.(9

)

Ass

umpt

ion

4al

soim

plie

s

v!b!

xyz

="

Cw

! b!x

yzN

2

M2

#"

(C"

1)M

2N

2H

2

|f|

.(10

)

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inFig

.8,th

eve

rtic

alflu

xis

upw

ard

and

v!b!

isdo

wn

the

mea

nbu

oyan

cygr

adie

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b.The

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rtur

ning

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amfu

nctio

n

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ram

eter

izat

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coul

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rect

lyfr

om(1

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),bu

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over

turn

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stre

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nction

aids

impl

emen

tation

ina

num

eric

alm

odel

.T

heed

dybu

oyan

cyflu

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may

bebr

oken

into

ask

ewflu

xge

nera

ted

bya

stre

amfu

ncti

on(v

! sb!$

""

b z,w

! sb!$

"b y

)an

dth

ere

mai

ning

resi

dual

flux.

Usi

nga

stre

amfu

ncti

onin

(6)

yiel

ds,

!·u

! b!

="

! !y

! "b z

" +! !z

! "b y

"(1

1)

+!(v

! b!"

v! sb!

)!y

+!(w

! b!"

w! sb!

)!z

,

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ere

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nish

,the

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est

ream

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ould

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ique

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What does it look like?Parameterization (2d, 10km grid) Submesoscale-Resolving (3d,500m grid)

N2

Changes To Mixing Layer Depth inEddy-Resolving Southern Ocean ModelBulk Mixed Layer New Mixed Layer Model

Bulk Mixed Layer New Mixed Layer Model Changes To Mixing Layer Depth in

Eddy-Resolving Southern Ocean Model

Improves Restratification after Deep ConvectionNote: param. reproduces Haine&Marshall (98) and Jones&Marshall (93,97)

Equator (f->0) and coarse resolution (up to 1 deg) are manageable

Contoured: 5-yr mean mixing layer depth (m) in HIM. Shaded: change (m) with parameterization

Conclusion:Submesoscale features, and mixed layer eddies in particular, exhibit large vertical fluxes of buoyancy that are presently ignored in climate models.

A parameterization of mixed layer eddy fluxes as an overturning streamfunction is proposed. The magnitude comes from extraction of potential energy, and the vertical structure resembles the linear Eady solution.

Many observations are consistent, and model biases are reduced. Biogeochemical effects are likely, as vertical fluxes and mixed layer depth are changed.

In HIM, soon to be in MITgcm and CCSM.

3 Papers so far... Just ask me for them.