2. The OSCAR Data. 3. Stochastic Differential Equations for Surface Currents

Transcription

1 Proceedings of t Conference on Satellite Meteorology and Oceanograpy, Poenix, - January 9 JP.8 Weibull Distribution for Global Surface Current Speeds Obtained from Satellite Altimetry Peter C. Cu ) Naval Ocean Analysis and Prediction Laboratory Department of Oceanograpy Naval Postgraduate Scool Monterey, California. Introduction. Te OSCAR Data Te world oceans contribute significantly to te global redistribution of eat necessary to maintain te eart s termal equilibrium. Surface layer orizontal fluxes of momentum, eat, water mass and cemical constituents are typically nonlinear in te speed (Galanis et al. ; Lozano et al. 99), so te space or time average flux is not generally equal to te flux tat would be diagnosed from te averaged current speed. In fact, te average flux will generally depend on iger-order moments of te current speed, suc as te standard deviation, skewness, and kurtosis. From bot diagnostic and modeling perspectives, tere is a need for parameterizations of te probability distribution function (PDF) of te current speed w (called w- PDF ere). Recent study on te equatorial Pacific (Cu 8) sowed tat te w-pdf satisfies te twoparameter Weibull distribution in te upper layer ( m) after analyzing te ourly Acoustic Doppler Current Profiler (ADCP) data (99-7) at all te six stations along during te Tropical Atmospere Ocean (TAO) project. Question arises: Can suc a result (e.g., te Weibull distribution for te equatorial Pacific surface current speeds) be extended to global oceans? To answer tis question, we use te - day Ocean Surface Currents Analyses Realtime (OSCAR) data to construct te observational w-pdf for te global ocean surface circulation. Special caracteristics of te statistical parameters suc as mean, standard deviation, skewness, and kurtosis will also be identified. Te OSCAR data are available for te world oceans from o N to o S on o o grid to a broad-based user community via a web-based interactive data selection interface on a time base wit exactly 7 steps per year (about day interval) starting from October 99 (Fig. ). Te velocity is automatically computed from gridded fields of surface topograpy and wind derived on te base of te Ekman dynamics from satellite altimeter (JASON-, GFO, ENVISAT) and scatterometer (QSCAT) vector wind data. See website: ttp:// for detailed information. Fig.. Global ocean surface current vectors (-day mean and anomaly) derived from satellite altimeter and scatterometer data. Te data can be downloaded from te website: ttp:// Stocastic Differential Equations for Surface Currents ) Corresponding autor address: Peter C. Cu, Naval Postgraduate Scool, Monterey, CA 99, pccu@nps.edu Let (x, y) be te orizontal coordinates and z be te vertical coordinate. Vertically averaged orizontal velocity components (u, v) from te surface to a constant scale dept () of surface mixed layer are given by (Cu 8)

2 u K Λ u u, () t v K Λ v u, () t were τ τ x y Λ fv +, Λ fu +, () u E v E ρ ρ represent te residual between te Ekman transport and surface wind stress. Here, K is te eddy viscosity; f is te Coriolis parameter; ( τ, τ ) are te surface wind stress components; x y and (U E, V E ) are Ekman transports computed by E E () g g ( U, V ) ( u u, v v ) dz, were ( uv, ) are te vertically varying orizontal velocity components; and (u g, v g ) are te geostropic velocity components. Wit absence of orizontal pressure gradient, e.g., u g v g, Equations () and () reduce to te commonly used wind-forced slab model (Pollard and Millard 97). For te sake of convenience, we assume tat te residual between te Ekman transports (U E, V E ) and surface wind stress does not depend on te orizontal current vector (u, v). Away from te equator, tis approximation is similar to a small Rossby number approximation (Gill 98). If te forcing ( Λ, Λ ) is fluctuating u v around some mean value, Λ () t Λ + W () t Σ, u u () Λ () t Λ + W () t Σ, v v were te angle brackets represent ensemble mean and te fluctuations are taken to be isotropic and wite in time: W ( t ) W ( t ) δδ( t t ), () i j ij wit a strengt tat is represented by Σ. Note tat te Ekman transport is determined by te surface wind stress for time-independent case, and terefore te ensemble mean values of ( Λ, Λ ) are zero, u v Λ, Λ. (7) u v Substitution of ()-(7) into () and () gives u K () u + W t Σ, (8) t v K () v + W t Σ, (9) t wic is a set of stocastic differential equations for te surface current vector. Te joint PDF of (u, v) satisfies te Fokker-Planck equation, Σ p p p + t u v () K K + ( u) p + ( v) p, u v wic is a linear second-order partial differential equation wit te dept scale () taken as a constant. Transforming from te ortogonal coordinates (u, v) to te polar coordinates (w, ϕ ) respectively te current speed and direction, u wcos ϕ, v wsinϕ. () Te joint PDF of (u, v) is transformed into te joint PDF of (w, ϕ ), p( u, v) dudv p( u, v) wdwdϕ () pw (, ϕ) dwdϕ. Integration of () over te angle ϕ from to π yields te marginal PDF for te current speed alone, π pw ( ) pw (, ϕ ) dϕ. () For a constant eddy viscosity (K) at z -, te steady state solution of equation () is given by (, ) exp K puv A ( u + v) Σ, () were A is a normalization constant. Substitution of () into () and use of () yield te Rayleig distribution ( ) w exp w Σ pw, a, () a a K wit te scale parameter a. Te basic postulation of constant K may not be met always at te upper ocean. Hence we require a model tat can meet te twin objectives of (a) accommodating Rayleig distribution wenever te basic ypotesis (constant K) tat justifies it is satisfied and (b) fitting data under more general conditions. Tis requirement is supposed to be satisfied by te Weibull probability density function, b b w w pw ( ) exp, () a a a were te parameters a and b denote te scale and sape of te distribution. Tis distribution as been recently used in investigating te ocean model predictability (Ivanov and Cu, 7a,b).. Parameters of te Weibull Distribution

3 Te four parameters (mean, standard deviation, skewness, and kurtosis) of te Weibull distribution are calculated by (Jonson et al. 99), mean( w) aγ +, (7) b / std( w) a Γ + Γ +, (8) were Γ is te gamma function. Te parameters a and b can be inverted (Monaan ) from (7) and (8),.8 mean( w) mean( w) b, a. (9) std( w) Γ ( + / b) Te skewness and kurtosis are computed by skew( w) Γ + Γ + Γ + + Γ + b b b b / Γ + Γ + () N mean( w) w, i N i std( w) mean w mean( w), [ ] mean{[ w mean( w)] } skew( w), w std ( ) mean{[ w mean( w)] } kurt( w), std ( w) for te eac grid point. Te mean, standard deviation, skewness, and kurtosis fields of w estimated from te OSCAR data are displayed in Fig.. Large values of mean(w) occur in te western boundaries suc as in te Gulf Stream, Kurosio, and Somali Current, Malvinas Current and; secondary maxima in te equatorial zones especially in te western and central equatorial Pacific. Minima of mean(w) occur in te subtropical orse latitudes. Te standard deviation of w is also large near te western boundaries and in te equatorial zones. In general, w is positively skewed in te most part of te global oceans and negatively skewed in te equatorial zones and Soutern Ocean. Te kurtosis field is muc noisier tan tose of mean(w), std(w), or. Γ + Γ + Γ + b b b kurt( w) Γ + Γ + Γ + Γ + Γ + b b b +, Γ + Γ + () wic depend on te parameter b only [see () and ()] for te Weibull distribution. Te relationsip between te kurtosis and skewness can be determined from () and ().. Observational w-pdf Te data depicted in Section are used to investigate te statistical features of te global surface current speeds (w). Te four parameters (mean, standard deviation, skewness, and kurtosis) can also be calculated from te observational data (w) Fig.. First four parameters of te surface current speeds calculated from te OSCAR data (99-8).

4 Te Weibull parameters (a, b) were calculated from mean(w) and std(w). Te distribution of te parameter a over te global oceans (Fig. a) is quite close to te distribution of mean(w), i.e., wit large values in western boundaries and equatorial zone. Te distribution of te Weibull parameter b is sown in Fig. b. Tus, a four-parameter dataset as been establised eac location. Te scatter diagrams were drawn for global oceans during all or different time periods. Latitude Latitude o N o N o S o S o N o N o S o S o E o E 8 o W o W Longitude Fig.. Same as in Fig., but for Weibull parameters (a, b). Te relationsips between and (representing te parameter b) and between te and te may be used to identify te fitness of te Weibull distribution for observational w-pdfs. Te solid curve on tese figures sows te relationsip for a Weibull variable. Fig. sows te kernel density estimates of joint PDFs of versus and versus for January (left panels) and July (rigt panels) OSCAR data from 99 to 8. Fig. sows te similar items from te OSCAR data during (a) (b) 998-, and (c) -7. Te contour intervals are logaritmically spaced. Te tick black line is te teoretical curve for a Weibull variable. For te observational surface current speeds (w), te is evidently a concave function of te ratio (te same as te Weibull distribution), suc tat te teoretical function is positive for small values of tis ratio and negative for large values. However, for te core of te kernel wit te joint probability iger tan., is always less tan. and is always positive (Fig., upper panes). Similarly, te relationsip between and in te observations is similar to tat for a Weibull variable (Fig., lower panels) wit smaller kurtosis. Tese features are almost te same between January (Fig. a) and July (Fig. b), and among tree time periods: (Fig.a), 998- (Fig. b), and -7 (Fig. c)..... oscar Jan oscar Jul... Fig.. Kernel density estimates of joint PDFs of (top) and and (bottom) and for (left) for (a) January and (b) July OSCAR data from 99 to 8. Te contour intervals are logaritmically spaced. Te tick black line is te teoretical curve for a Weibull variable. (a) oscar 998 to oscar 99 to (b) (c) oscar to 7... Fig.. Kernel density estimates of joint PDFs of (top) and and (bottom) and for (left) from te OSCAR data during (a) (b) 998-, and (c) -7. Te contour intervals are logaritmically spaced. Te tick black line is te teoretical curve for a Weibull variable

5 Te agreement between te moment relationsips in te OSCAR data and tose for a Weibull variable reinforces te conclusion tat tese data are Weibull to a good approximation.. Conclusions Tis study as investigated te probability distribution function of te surface current speeds (w), using long-term (99-8) day Ocean Surface Current Analyses Real Time (OSCAR) (OSCAR) data; and teoretically, using a stocastic model derived using upper boundary layer pysics. Te following results were obtained. () probability distribution function of te global surface current speeds (w) approximately satisfies te two-parameter Weibull distribution. In te upper ocean wit a constant eddy viscosity K, te probability distribution function satisfies a linear second-order partial differential equation (i.e., te Fokker-Planck equation) wit an analytical solution te Rayleig distribution (special case of te parameter Weibull distribution). () Four moments of w (mean, standard deviation, skewness, kurtosis) ave been caracterized. It was found tat te relationsips between and and between and from te data are in fairly well agreement wit te teoretical Weibull distribution for te upper ( - m) tropical Pacific for te wole period Te OSCAR data also sow tat te ratio is generally less tan. and te skewness is generally positive for te wole global oceans. () Te Weibull distribution provides a good empirical approximation to te PDF of global ocean surface current speeds wit little seasonal and interannual variations. Tis may improve te representation of te orizontal fluxes tat are at te eart of te coupled pysical biogeocemical dynamics of te marine system. References Cu, P.C., Probability distribution function of te upper equatorial Pacific current speeds. Geopysical Researc Letters,, doi:.9/8gl9, 8. Galanis, G., P. Louka, P. Katsafados, G. Kallos I. Pytaroulis, Applications of Kalman filters based on non-linear functions to numerical weater predictions. Ann. Geopys.,, -,. Gill, A.E., Atmospere-Ocean Dynamics. Academic Press, San Diego, pp, 98. Ivanov, L.M., and P.C. Cu, On stocastic stability of regional ocean models to finiteamplitude perturbations of initial conditions. Dynamics of Atmospere and Oceans,, 99-, doi:./j.dynatmoce.7.., 7a. Ivanov, L.M., and P.C. Cu, On stocastic stability of regional ocean models wit uncertainty in wind forcing. Nonlinear Processes of Geopysics,, -7, 7b. Jonson, N., S. Kotz, and N. Balakrisnan, Continuous Univariate Distributions. Vol.. Wiley, 7 pp, 99. Lozano, C. J., A.R. Robinson, H.G. Arrango, A. Gangopadyay, Q. Sloan, P. J. Haley, L. Anderson, and W. Leslie, An interdisciplinary ocean prediction system: assimilation strategies and structured data models. In: Modern Approaces to Data Assimilation in Ocean Modeling, edited by P. Malanotte-Rizzoli, -, Elsevier, Amsterdam, 99. Monaan, A.H., Te probability distribution of sea surface wind speeds. Part-: teory and SeaWinds observations. Journal of Climate, 9, Pollard, R.T., and R. C. Millard, Comparison between observed and simulated wind-generated inertial oscillations. Deep Sea Researc, 7, 79-8, 97. Acknowledgments Tis researc was supported by te Office of Naval Researc, Naval Oceanograpic Office, and Naval Postgraduate Scool.