On Non-Gaussian SST VariabilityOn Non-Gaussian SST Variabilityin the Gulf Stream andin the Gulf Stream and otherother

Strong CurrentsStrong CurrentsPhilip Sura

Florida State UniversityDepartment of Meteorology

Collaborator:Prashant SardeshmukhNOAA/CIRESBoulder, CO

Introduction Since the very early days of physical oceanography the Gulf Stream system plays a central role in the dynamical description of the general circulation of the ocean.

Here we will study the physics of non-Gaussian SST variability in the Gulf Stream and other strong currents in a recently developed stochastic framework.

Measures of Non-Gaussianity:Skewness and Kurtosis

skew !

"To

3

# 3

kurt !

"To

4

# 4

$ 3

Skewness and kurtosis are non-dimensional measuresdescribing the shape of a probability density function (PDF)

The shape of a PDF can provide information about thedynamics of the underlying system!

SST anomaly skewness and kurtosisin the Gulf Stream System

Daily AVHRR SSTs blended with in situ data, 1985-2005

Probability Density Functionsin the Gulf Stream System

PDFs from full year daily Reynolds data

PDFs Follow a Power-Lawin the Gulf Stream System

p(x) ! x"#

Modeling SST AnomaliesThe heat budget equation for SST To is:

!To

!t= " vo i#To +

Q

$Ch "

we

h(To " To

b) + %#2

To & F

A B C D

Advection through ocean currents (A) Surface heat flux (B) Vertical entrainment (C) Horizontal mixing (D)

! "To

!t=

!F

!To

"To

+! "F

!To

"To

+ "F

A Taylor expansion of the total heat flux F yields:

Modeling SST Anomalieswith Additive Noise

! "To

!t=

!F

!To

"To + "F

! "To

!t= #$eff "To + F '

Neglecting the anomalous heat flux derivative, Modeling the anomalous heat flux F’ as white-noise yields:

Pros:Pros: The red-noise spectrum is consistent with observations.

Cons:Cons: The PDF is strictly Gaussian and not consistent with observations.

Frankignoul and Hasselmann (1977)

! "To

!t=

!F

!To

"To

+! "F

!To

"To

+ "F !>

Modeling SST Anomalieswith Linear Multiplicative Noise

Including the anomalous heat flux derivative, Modeling the anomalous heat flux and the anomalous heat flux derivative as white noise yields:

Pros:Pros: The red-noise spectrum is consistent with observations. Skewness and kurtosis are consistent with observation.

! "To

!t=

!F

!To

"To +! "F

!To

"To + "F + "R

! "To

!t= #$eff "To #% "F "To + "F + "R

Equation for the Moments:Skewness and Kurtosis

0 =ddx

!eff

xp"

#$

%

&'+

1

2

d2

dx2( )F

2+( )R

2+*2( )F

2x2 + 2*( )F2x,

-./01p

"

#$

%

&'

The Fokker-Planck equation (FPE) for the stationary PDF p(x) of SST anomalies is:

!eff

" n"12

#$ %F&'

()

2*

+

,,

-

.

//< xn

> = " (n"1) # $ %F2

< xn"1> +

n"12

$ %F2

+$ %R2&

'() < xn"2

>

Moments < xn > are obtained by multiplying the FPE by xn and integrating by parts:

< x2> = ! "F

2+! "R

2#$%

&'(

/ 2)eff

* +! "F#$%

&'(2,

-

.

.

/

0

11,

< x3> = * 2+! "F

2< x2

> / )eff

* +! "F#$%

&'(2,

-

.

.

/

0

11,

< x4> = *3+! "F

2< x3

> +(3/ 2) ! "F2

+! "R2#

$%&'(< x2

>,

-.

/

01 / )

eff* (3/ 2) +! "F

#$%

&'(2,

-

.

.

/

0

11 .

In particular, the second, third, and fourth moments are:

Equation for the Moments:Skewness and Kurtosis

kurt = A skew2+ B

A =

32!"

eff+ #$ %F

&'

()

2&

'**

(

)++

!"eff

+ 32#$ %F&'

()

2&

'**

(

)++

,A- 3

2

B =

3 !"eff

+ 12

#$ %F&'

()

2&

'**

(

)++

!"eff

+ 32

#$ %F&'

()

2&

'**

(

)++

! 3

,B-0

kurt!3

2skew2

Skewness and Kurtosis - SST AnomaliesDaily AVHRR SSTs blended with in situ data, 1985-2005

Dataset from Reynolds et al. (2006)

Skewness and Kurtosis - SST Anomalies

kurt !3

2skew

2

The observed skewness-kurtosis link is consistentwith our univariate linear

model with linearmultiplicative noise.

Power-Law - SST Anomalies

0 =ddx

!eff

xp"

#$

%

&'+

1

2

d2

dx2( )F

2+( )R

2+*2( )F

2x2 + 2*( )F2x,

-./01p

"

#$

%

&'

The Fokker-Planck equation (FPE) for the stationary PDF p(x) of SST anomalies is:

0 = !eff

xp +1

2

d

dx"

2#

$F2

x2( ) p

For large x the FPE becomes:

The observed power-law is consistent with our univariatelinear model with linear multiplicative noise.

p(x) ! x"#

# = 2$eff

%2& 'F

2+1

(

)*+

,-

Modeling Non-Gaussian SST Anomalies! "To

!t= #$eff "To #% "F "To + "F + "R

The so far unspecified parameter φ determines theskewness of SST variability.

The skewness is positive if the additive and multiplicative noises are positively correlated ( φ < 0 ).

negative if the additive and multiplicative noises are negatively correlated ( φ > 0 ).

This univariate linear model with linear multiplicative noisecaptures the observed non-Gaussianity very well.

Modeling Non-Gaussian SST AnomaliesThe heat budget equation for SST To is:

!To

!t= " vo i#To +

Q

$Ch "

we

h(To " To

b) + %#2

To & F

A B C D

Advection through ocean currents (A) Surface heat flux (B) Vertical entrainment (C) Horizontal mixing (D)

! "To

!t=

!F

!To

"To

+! "F

!To

"To

+ "F

A Taylor expansion of the total heat flux F yields:

Modeling Non-Gaussian SST Anomalies:The Surface Heat Flux

!To

!t= "(T

a# T

o) U $ F

Q

!FQ

!To

= #" U = #"( U + U % )

%FQ

= "(Ta# T

o) U %

(Ta# T

o) < 0

! "To

!t=!FQ

!To

"To + "FQ

#

! "To

!t= $%( U + U " ) "To + %(Ta $ To ) U "

Parameterizing U " as white-noise we get

! "To

!t= $&eff "To + "F "To + "F + "R

' ( $1

!

The skew induced by the surface heat flux is always positivebecause the additive and multiplicative noises are

positively correlated ( φ < 0 ).That’s because the ocean is warmer than the atmosphere.

Modeling Non-Gaussian SST Anomalies:Advection Through Ocean Currents

!To

!t= " vo i#To $ FA

!FA

!To

= "!!To

u!To

!x+ v

!To

!y

%&'

()*

= "!u

!x+!v

!y

%&'

()*

= "# +vo

FA+ = "vo i# +To " +vo i#T o " +vo i# +To + +vo i# +To

Neglecting second order anomaly products we get

FA+ , "vo i# +To " +vo i#T o = " +vo i#T o + slow

Neglecting slow components we get

FA+ , " +vo i#T o = "#( +voT o ) + T o# +vo

The oceanic velocityfield is split into

slow geostrophic fast ageostrophic

components.

Modeling Non-Gaussian SST Anomalies:Advection Through Ocean Currents

! "To

!t=!FA

!To

"To + "FA

#

! "To

!t= $ "To % "vo $ %( "voT o ) + T o% "v

Next we parameterize the rapidly varying divergence of

the ageostrophic velocity field as white-noise, and the remaining fast term

as a noise residual "R . If we also include a damping term we get

! "To

!t= $&eff "To $ "F "To + "F + "R

' ( +1

The skew induced advection through ocean currents is alwaysnegative because the additive and multiplicative noises are

negatively correlated ( φ > 0 ).

Summary and ConclusionsSummary and Conclusions It is the divergence of the rapidly varying ageostrophic velocity field that most likely causes the negative skewness in strong currents.

Through temperature advection the velocity divergence introduces negatively correlated additive and multiplicative noise terms in the evolution equation of SST anomalies.

The required damping of SST anomalies is provided by the heat flux through the sea surface.

A combination of SST advection and heat flux forcing could result in negative or positive skewness. It is the relative strength of each process responsible for the net skewness. In very strong currents such as the Gulf Stream the SST advection is strong enough to significantly affect the mixed-layer heat budget, resulting in the pronounced SST skewness observed along-stream strong currents.