Weibull Distribution for the Global Surface Current Speeds Obtained

from Satellite Altimetry

Peter C. Chu Naval Postgraduate School, Monterey,

CA93943, USA pcchu@nps.edu,

http://faculty.nps.edu/pcchutel: 831-656-3688, fax: 831-656-3686

Reference

• Chu, P. C., 2008: Probability distribution function of the upper equatorial Pacific current speeds. Geophysical Research Letters, doi:10.1029/2008GL033669

Thermohaline Circulation

Poleward Heat Transport Global Climate Change

• Nonlinear dependence on the current speed (w) and temperature (T)

• Space or time average flux not generally equal to the flux of the averaged filed

• Urgent needs to know the probability distribution function (PDF) of w and T

wT w T≠

Ocean Surface Velocity

Satellite Altimeters (JASON-1, GFO, ENVISAT)

Scatterometer (QSCAT)

Ocean Surface Current Analyses –Realtime (OSCAR) Data

(1) Ocean Surface currents data available for whole world’ oceans at www.oscar.noaa.gov

(2) Ocean Currents are computed from Sea Surface Height (SSH) data which is derived from satellite based altimeters JASON-1, GFO, Envisat and wind data which is derived from QUICKSCAT satellite

(3) Data continuously available every 5 days

Stochastic Dynamics for the Ocean Surface Currents

Lentz, 1992

What does the oceanic surface boundary layer look like?Q, surface heat flux

“slab-like” layer velocity 90° to right of wind (in northern hemisphere), well-mixed

u*, shear velocity

Vertically Averaged Horizontal Velocity (u, v) within the Mixed Layer

-- Slab Model --

2

1u

u Ku

t h h

∂= Λ −

∂

2

1v

v K

t h hv∂

= Λ −∂

, yxu E v EfV fU

ττ

ρ ρΛ ≡ + Λ ≡ − +

h mixed layer depth

K eddy viscosity

,x yτ τ( ) Surface windstress

(1)

(2)

Ekman Transport (UE, VE)

0

( , ) ( , ) ,E E g ghU V u u v v dz

−= − −∫

,u v( ) Vertically Varying Velocity

(ug, vg) geostrophic velocity

(ug, vg) = 0 Eqs(1) (2) Wind-forced Slab model

Ensemble Mean and Stochastic Fluctuations of the Forcing

• (1) Ensemble Mean Ekman Transport is determined by the surface wind stress

• (2) Fluctuations ( strength)

0, 0u vΛ = Λ =

1 2( ) ( ) , ( ) ( )v vu ut W t h t W t hΛ = Λ + Σ Λ = Λ + Σ

1 2 1 2( ) ( ) ( )i j ijW t W t t tδ δ= −

Σ(3)

Stochastic Dynamic System

Eq(3) Eqs(1) (2)

12( )

u Ku W t

t h

∂= − + Σ

∂

22( )

v Kv W t

t h

∂= − + Σ

∂

(4)

(5)

Fokker-Planck Equation forPDF of (u, v)

2 2 2

2 2 2 2( ) ( )

2

p p p K Ku p v p

t u v u h v h

∂ Σ ∂ ∂ ∂ ∂= + + +

∂ ∂ ∂ ∂ ∂

⎛ ⎞⎛ ⎞ ⎡ ⎤ ⎡ ⎤⎜ ⎟⎜ ⎟ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎝ ⎠⎝ ⎠

Polar Coordinate cos , sinu w v wϕ ϕ= =

w Current Speed

(6)

For Constant K PDF of w the Rayleigh Distribution

(Special Case of the Weibull Distribution)

2

2

2( ) exp ,

w w hp w a

a a K

Σ= − ≡

⎡ ⎤⎛ ⎞⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

For Non-Constant K Weibull Distribution

1 2

( ) expbb w w

p wa a a

−

= −⎡ ⎤⎛ ⎞ ⎛ ⎞

⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦

1mean( ) 1w a

b= Γ +⎛ ⎞

⎜ ⎟⎝ ⎠

1 / 2

22 1std( ) 1 1w a

b b= Γ + − Γ +⎡ ⎛ ⎞ ⎛ ⎞⎤

⎜ ⎟ ⎜ ⎟⎢ ⎥⎣ ⎝ ⎠ ⎝ ⎠⎦

Γ Gamma Function

Characteristics of the WeibullDistribution

1.086mean( ) mean( )

.std( ) (1 1/ )

, w wb a

w b=Γ +

⎡ ⎤⎢ ⎥⎣ ⎦

3

3 / 2

2

3 1 2 11 3 1 1 2 1

skew( )2 1

1 1

b b b bw

b b

Γ + − Γ + Γ + + Γ +=

Γ + − Γ +

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

⎡ ⎛ ⎞ ⎛ ⎞⎤⎜ ⎟ ⎜ ⎟⎢ ⎥⎣ ⎝ ⎠ ⎝ ⎠⎦

2 4

2 2

2 2

4 1 3 1 2 11 4 1 1 6 1 1 3 1

kurt( ) 32 1 2 1

1 1 1 1

b b b b b bw

b b b b

Γ + − Γ + Γ + Γ + Γ + − Γ += −

Γ + − Γ + Γ + − Γ +

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠+⎡ ⎛ ⎞ ⎛ ⎞⎤ ⎡ ⎛ ⎞ ⎛ ⎞⎤⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥⎣ ⎝ ⎠ ⎝ ⎠⎦ ⎣ ⎝ ⎠ ⎝ ⎠⎦

Kurt Skew

• Since

Skew = f1(b), Kurt = f2(b)

Kurt = F(Skew)

Mean

Standard Deviation

Skewness

Kurtosis

Statistical Characteristics of Global Surface Current Speed

Weibull Parameters (a, b)

60oS

30oS

0

30oN

60oN

Latit

ude

60oE 120oE 180 120oW 60oW 0

60oS

30oS

0

30oN

60oN

Latit

ude

Longitude

Kernel Density Estimates of Joint PDFs of skewness and ‘b’ (DJF)

Kernel Density Estimates of Joint PDFs of skewness and ‘b’ (JJA)

Kernel Density Estimates of Joint PDFsof kurtosis and skewness (DJF)

Kernel Density Estimates of Joint PDFsof kurtosis and skewness (JJA)

Conclusions

• The Weibull distribution provides a reasonable empirical approximation to the PDF of the surface current speeds (w), which presents the possibility of improving the representation of the horizontal fluxes that are at the heart of the coupled physical–biogeochemical dynamics of the marine system and climate system.