Weibull Distribution for the Global Surface Current Speeds...
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Transcript of Weibull Distribution for the Global Surface Current Speeds...
Weibull Distribution for the Global Surface Current Speeds Obtained
from Satellite Altimetry
Peter C. Chu Naval Postgraduate School, Monterey,
CA93943, USA [email protected],
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.