the relation between the statistics of open ocean currents...

Climate Dynamics manuscript No.(will be inserted by the editor)

The relation between the statistics of open ocean1

currents and the temporal correlations of the2

wind-stress3

Golan Bel · Yosef Ashkenazy4

5

Received: date / Accepted: date6

Abstract We study the statistics of wind-driven open ocean currents. Using the7

Ekman layer model for the integrated currents, we investigate, analytically and8

numerically, the relation between the wind-stress distribution and its temporal9

correlations and the statistics of the open ocean currents. We find that temporally10

long-range correlated wind results in currents whose statistics is proportional to11

the wind-stress statistics. On the other hand, short-range correlated wind leads12

to Gaussian distributions of the current components, regardless of the stationary13

distribution of the winds, and therefore, to a Rayleigh distribution of the current14

amplitude, if the wind-stress is isotropic. We find that the second moment of the15

current speed exhibits a maximum as a function of the correlation time of the16

wind-stress for a non-zero Coriolis parameter. The results were validated using an17

oceanic general circulation model.18

Keywords Wind statistics · Ocean current statistics · Temporal correlations19

PACS 92.60.Cc · 91.10.Vr · 88.05.Np · 92.60.Gn20

1 Introduction21

Ocean currents are generated by local and remote forces and factors, including22

winds, tides, buoyancy fluxes and various types of waves. While many studies have23

investigated the distribution of the wind (Seguro and Lambert, 2000, Monahan,24

2006, 2010), focusing on its relevance to energy production, the distribution of25

ocean currents has received much less attention (Chu, 2008). Moreover, currently26

Golan BelDepartment of Solar Energy and Environmental Physics, Blaustein Institutes for Desert Re-search, Ben-Gurion University of the Negev, Sede Boqer Campus 84990, IsraelTel.: +972-8-6596845Fax: +972-8-6596921E-mail: [email protected]

Yosef AshkenazyDepartment of Solar Energy and Environmental Physics, Blaustein Institutes for Desert Re-search, Ben-Gurion University of the Negev, Sede Boqer Campus 84990, Israel

2 Bel and Ashkenazy

there is no accepted theory explaining the observed statistics of surface ocean27

currents.28

Here, we propose a simple physical theory for the distribution of wind-driven29

ocean currents and its relation to the spatially variable temporal correlations of30

the wind (see Fig. 1). We show that the distribution of wind-driven ocean currents31

strongly depends on the temporal correlations of the wind–when the wind exhibits32

long-range temporal correlations, the ocean current statistics is proportional to the33

wind-stress statistics, while for short-range correlations of the wind, the different34

components of the current vector follow Gaussian distributions. It was previously35

reported that the probability density function (PDF) of ocean currents follows the36

Weibull distribution (Gille and Smith, 1998, 2000, Chu, 2008, 2009, Ashkenazy37

and Gildor, 2011) (see Section 4.2 for the details of the Weibull distribution);38

We argue that this is not necessarily the case, even if the wind-stress magnitude39

follows the Weibull distribution.40

Oceans play a major and important role in the climate system, and ocean41

circulation underlies many climate phenomena, from scales of meters to thousands42

of kilometers and from scales of minutes to decades. The gap in our understanding43

of ocean current statistics leads to a lack in understanding the statistics of other44

related climate variables. Filling in this gap will be useful in many fields: it may45

help to predict extreme current events and, thus, may help to securely design46

maritime-associated structures. In light of the increasing efforts to find alternative47

sources of energy and the idea of using ocean currents as such a source, knowledge48

of the currents’ PDF may help to better estimate the energy production and49

to appropriately design ocean current turbines that will withstand even extreme50

current events. Moreover, knowledge regarding current statistics and, in particular,51

its relation to the wind-stress statistics may improve the parametrization of small-52

scale processes in state-of-the-art general circulation models (GCMs).53

The study of wind-driven ocean currents goes back more than 100 years, to the54

time when Ekman (1905) proposed his classical simple model to explain the effect55

of Earth’s rotation on upper ocean currents. His model predicted that the depth-56

integrated current vector is perpendicular to the wind vector, a prediction that57

was largely proven by observations. Since then, many studies have used Ekman’s58

model to propose more realistic models for ocean currents, as well as for surface59

winds (Gill, 1982, Cushman-Roisin, 1994). In what follows, we use Ekman’s model60

(Ekman, 1905) to study the statistics of wind-driven ocean currents.61

This paper is organized as follows. In section 2, we describe the Ekman layer62

model and provide a general (implicit) solution expressed in terms of the charac-63

teristics of the wind-stress statistics. In section 3, we consider two idealized cases64

of wind-stress statistics (step-like temporal behavior of the wind-stress in section65

3.1 and exponentially decaying temporal wind-stress correlation in section 3.2) and66

present analytical expressions for the second moment (and fourth moment for the67

first case) of the currents’ distribution. In section 4, we provide a description of the68

simple numerical model and of the oceanic GCM (MITgcm) used to validate and69

extend the analytical results. The numerical tests involve the Weibull distribution;70

hence, a brief review of the distribution properties and the methods we used to71

generate correlated and uncorrelated Weibull-distributed time series is provided72

in 4.2. The numerical results are presented and discussed in section 5, followed by73

a brief summary in section 6.74

The relation between current and wind statistics 3

2 The Ekman model75

In spite of the simplicity of the Ekman model, it will enable us to start investigating76

the coupling between the wind-stress and the ocean currents. The time, t, and77

depth, z, dependent equations of the Ekman model (Ekman, 1905), describing the78

dynamics of the zonal (east-west) U and meridional (south-north) V components79

of the current vector, are:80

∂U

∂t= fV + ν

∂2U

∂z2

∂V

∂t= −fU + ν

∂2V

∂z2, (1)

where f = (4π/Td) sin(φ) is the local Coriolis parameter (Td is the duration of81

a day in seconds and φ is the latitude), and ν is the eddy viscosity coefficient,82

assumed to be depth-independent. In eqs. (1), we consider U = U−Ug, V = V −Vg83

where Ug, Vg are the bottom ocean geostrophic currents determined by the pressure84

gradient and U , V are the actual surface current components. In what follows, we85

consider the statistics of U , V , where the statistics of U , V can be obtained by a86

simple transformation. The boundary conditions are chosen such that the current87

derivative, with respect to the depth coordinate z, is proportional to the integrated88

current at the bottom of the layer described by our model and to the wind-stress89

vector (τx, τy) at the surface (Gill, 1982, Cushman-Roisin, 1994),90

∂U

∂z

∣∣∣z=−h

=r

νu(t);

∂U

∂z

∣∣∣z=0

=τxρ0ν

;

∂V

∂z

∣∣∣z=−h

=r

νv(t);

∂V

∂z

∣∣∣z=0

=τyρ0ν

, (2)

where,91

u ≡0∫

−h

U(z)dz; v ≡0∫

−h

V (z)dz. (3)

Here, we introduce the following notation: h is the depth of the upper ocean layer, r92

is a proportionality constant representing the Rayleigh friction (Airy, 1845, Pollard93

and Millard, 1970, Kundu, 1976, Kase and Olbers, 1979, Gill, 1982), (τx, τy) are the94

wind-stress components, and ρ0 is the ocean water density (hereafter assumed to95

be constant, ρ0 = 1028kg/m3). The value used for r in the numerical calculations96

is based on the empirical estimate outlined in Simons (1980), Gill (1982).97

By integrating eqs. (1) over a sufficiently deep layer (i.e., h �√ν/2f), we98

obtain the equations describing the integrated currents and their coupling to the99

wind-stress,100

∂u

∂t= fv − ru+

τxρ0

,

∂v

∂t= −fu− rv +

τyρ0

. (4)

4 Bel and Ashkenazy

In the derivation above, we did not explicitly consider the pressure gradient. Ex-101

plicit inclusion of the pressure gradient would result in constant geostrophic cur-102

rents, and equations (4) describe the dynamics of the deviation from the geostrophic103

currents.104

To allow a simpler treatment of these equations, we define w ≡ u + iv. It is105

easy to show that w obeys the following equation:106

∂w

∂t= −ifw − rw +

τ

ρ0, (5)

where τ ≡ τx + iτy. The formal solution of equation (5) is107

w(t) = w(0)e−(if+r)t +1

ρ0

t∫0

τ(t′)e−(if+r)(t−t′)dt′. (6)

This formal solution demonstrates that the currents depend on the history of the108

wind-stress, and therefore, the distribution of the currents depends, not only on109

the wind-stress distribution, but on all multi-time moments of the wind-stress.110

However, the two extreme limits are quite intuitive. When the correlation time111

of the wind-stress, T , is long (i.e., T � 1/r and T � 1/f), one expects that112

the currents will be proportional to the wind-stress since the ocean has enough113

time to adjust to the wind and to almost reach a steady state. In terms of eq.114

(6), in this limit, τ(t′) can be approximated by τ(t) since its correlation time is115

longer than the period over which the exponential kernel is non-zero. In the other116

limit, when the correlation time of the wind-stress is very short (i.e., rT � 1 and117

fT � 1), namely, the wind-stress is frequently changing in a random way, the118

ocean is not able to gain any current magnitude. In this case, the central limit119

theorem (Van-Kampen, 1981) implies that each component of the current vector120

is Gaussian distributed.121

It is useful to write the formal expression for the square of the currents ampli-122

tude in terms of the wind-stress temporal correlation function. Taking the square123

of eq. (6) and averaging over all realizations (with the same statistical properties)124

of the wind stress, one obtains:125

〈|w(t)|2〉 = |w(0)|2e−2rt (7)

+w(0)∗

ρ0e−(r−if)t

t∫0

〈τ(t′)〉e−(if+r)(t−t′)dt′

+w(0)

ρ0e−(r+if)t

t∫0

〈τ(t′)∗〉e−(r−if)(t−t′)dt′

+1

ρ20

t∫0

t∫0

〈τ(t′)τ(t′′)∗〉e−if(t′′−t′)e−r(2t−t′−t′′)dt′dt′′.

The variance of the currents amplitude may be written as126

〈|w(t)|2〉 − |〈w(t)〉|2 = (8)

1

ρ20

t∫0

t∫0

C(t′, t′′)e−if(t′′−t′)e−r(2t−t′−t′′)dt′dt′′,


where we defined the temporal correlation function of the wind-stress as127

C(t′, t′′) = 〈τ(t′)τ(t′′)∗〉 − 〈τ(t′)〉〈τ(t′′)∗〉. (9)

To allow analytical treatment, we proceed by considering two special idealized128

cases in which the formal solution takes a closed analytical form. For simplicity,129

we only consider the case of statistically isotropic wind-stress (the direction of130

the wind-stress is uniformly distributed, Monahan, 2007), namely a case in which131

the wind-stress components are independent, identically distributed variables with132

identical temporal autocorrelation functions. The latter assumption is not always133

valid (see for example, Monahan, 2012); however, the generalization of our results134

to the case of non-isotropic wind stress is straightforward.135

3 Idealized cases136

3.1 Step-like wind-stress137

A simple way to model the temporal correlations of the wind is by assuming that138

the wind vector randomly changes every time period T while remaining constant139

between the “jumps.” By integrating the solution of w (eq. (6)) and taking into140

account the fact that the distribution of w at the initial time and final time should141

be identical, one can obtain the expressions for the second and fourth moments142

(ensemble average over many realizations of the stochastic wind-stress) of the143

current amplitude. To better relate our results to the outcome of data analysis,144

we also need to consider the time average since the records are independent of the145

constant wind-stress period. The double averaged second moment of the currents146

amplitude is defined as147

⟨|w|2

⟩≡

⟨1

T

T∫0

|w(t)|2dt⟩, (10)

and the angular brackets represent the average over different realizations of the148

wind-stress with the same statistical properties. Using eq. (7) and the fact that in149

this idealized case, the wind stress is constant over the period of the step (T ), we150

obtain for the double averaged second moment of the currents amplitude:151

⟨|w|2

⟩=

⟨|τ |2

⟩(1 + 1−A1

rT − 2 r(1−A1)+fA2

T (f2+r2)

)(f2 + r2) ρ20

, (11)

where we have introduced the notations A1 ≡ exp(−rT ) cos(fT ), and A2 ≡152

exp(−rT ) sin(fT ).153

The two extreme limits discussed in section 2 can be easily realized and under-154

stood. When the correlation time T is very long compared with 1/r, the second155

moment of the current is proportional to the second moment of the wind-stress,156

namely, 〈|w|2〉 = 〈|τ |2〉/[(f2 + r2)ρ2o]. This is exactly what one would expect. The157

long duration of constant wind-stress allows the system to fully respond and adjust158

to the driving force, and hence, the moments of the currents are proportional to159

the moments of the wind-stress. The coefficient of proportionality is given by the160

6 Bel and Ashkenazy

solution of eqs. (4) with constant wind-stress. The other limit is when rT, fT � 1161

and, in this case, 〈|w|2〉 ∼ 〈|τ |2〉T/(2rρ20).162

In this limit, the second moment of the current amplitude is very small and163

approaches zero as the correlation time approaches zero. This result is what one164

would expect for a rapidly changing wind that cannot drive significant currents.165

Using eq. (6) and the fact that wind-stress is constant over the step period (T ), we166

calculate the double averaged (over the step period and the ensemble of wind-stress167

realizations) fourth moment,168

⟨|w|4

⟩=

〈|τ |2〉2g2 (T ) + 〈|τ |4〉g4 (T )(f2 + r2)2 ρ40

, (12)

g2 (T ) =4 (D + 1− 2A1)

1−D×(

3−D − 2A1D

4rT− 2

3r (1−DA1) + fDA2

(f2 + 9r2)T

),

g4 (T ) = 1 +5− 3D + 2A1(A1 − 1−D)

2rT

− 43r (1−DA1) + fDA2

(f2 + 9r2)T

+r(1−A2

1) + rA22 + 2fA1A2

(f2 + r2)T

− 4r(1−A1) + fA2

(f2 + r2)T,

where D ≡ exp(−2rT ). Here, we again use the fact that the integrals of the time169

averaging are easily carried due to the fact that the wind-stress is constant during170

the averaged period.171

While at the limit T → 0, both the second and the fourth moments vanish, the172

ratio between the fourth moment and the square of the second moment remains173

finite and is equal to 2. This corresponds to the Rayleigh distribution which orig-174

inates in the facts that the components of the currents are independent and each175

one has a Gaussian distribution with a zero mean and the same variance.176

One can easily understand the origin of the Gaussian distribution by consid-177

ering the central limit theorem, which can be applied in this limit. Each of the178

current components is the integral of many independent, identically distributed179

random variables (the wind-stress at different times). Based on the second and180

fourth moments in the limit of T → 0, we conjecture that the overall PDF of the181

ocean current speed, in this case, is given by the Rayleigh distribution.182

In Gonella (1971), the case of constant wind-stress over some duration was183

investigated using the depth-dependent Ekman model. The existence of a wind-184

stress duration for which the current amplitude is maximal was found. The results185

of this section generalize these results by considering the randomness of the wind-186

stress and its continuity. In addition, we consider here the empirical Rayleigh187

friction.188

In order to better understand the existence of constant wind-stress duration,189

T , for which the average current amplitude is maximal, we present, in Fig. 2, two190

typical trajectories of the current velocity components u, v. The dynamics of the191

velocity field may be understood as follows. When the friction, r, is smaller than the192


Coriolis frequency, the velocity field undergoes circular motion and returns, with193

frequency f , to a point in the velocity space which is very close to the initial point.194

The initial current is irrelevant since even a small amount of friction ensures that195

the initial point is close to the origin at the long-time limit. The above dynamics196

is valid for any value of the temporal wind-stress. Therefore, if the wind-stress197

changes with a frequency equal to the Coriolis frequency, the current amplitude198

cannot be driven far from the origin, and its average remains small (this explains199

the observed minimum for T ≈ 2π/f). On the other hand, when the wind-stress200

changes with a frequency equal to half the Coriolis frequency, the current velocity201

reaches the largest amplitude when the wind-stress changes and the new circle202

starts around a different initial point (possibly far from the origin). Therefore,203

in the latter case, the average current speed is maximal. To better illustrate this204

picture, we present, in Fig. 3, typical time series of the current velocity components205

and amplitude. The dots represent the times when the wind-stress changed. It can206

be seen that for T = π/f , the typical velocity amplitude when the wind-stress207

changes (the dots in Fig. 3), is significantly larger than the one for T = 2π/f .208

The intuitive explanation above relies on the fact that r � f . To elucidate209

the role of the friction, we present, in Fig. 4, the analytically calculated second210

moment (eq. (11)) of the current versus the constant wind-stress duration for211

different values of r. When the viscosity is large, there is no maximum but rather212

a monotonic increase of the average current amplitude as T increases.213

3.2 Exponentially decaying temporal correlations of the wind-stress214

The second case we consider is the Orenstein-Uhlenbeck wind-stress. This pro-215

cess results in a Gaussian wind-stress with an exponentially decaying temporal216

correlation function. It is important to note that, due to the Gaussian nature of217

the force, the first and the second moments of the wind-stress, together with its218

two-point correlation function, provide all the information on the driving force.219

The correlation function of the wind-stress components is given by220

〈τi(t)τj(t′)〉 = δij〈τ2i 〉 exp(−γi|t− t′|). (13)

The long-time limits of the current components’ averages are:221

〈u〉 ∼ r〈τx〉+ f〈τy〉ρ0 (f2 + r2)

; 〈v〉 ∼ r〈τy〉 − f〈τx〉ρ0 (f2 + r2)

. (14)

The long-time limit of the current amplitude variance, s2 ≡ 〈|w|2〉 − 〈u〉2 − 〈v〉2,222

is given by223

s2 ∼∑

i=x,y

〈τ2i 〉 (r + γi)

ρ20r(f2 + (r + γi)

2) . (15)

Under the assumption of isotropic wind-stress, the variance may be expressed as224

s2 ∼ 〈|τ |2〉 (r + γ) /(ρ20r

(f2 + (r + γ)2

)). Note that in deriving the expression225

above, we assumed that the autocorrelation function of the wind-stress is isotropic.226

In general, this is not the case (Monahan, 2012); however, for simplicity, we present227

here the results for this special case, and the generalization to the more realistic228

8 Bel and Ashkenazy

case is trivial. Due to the Gaussian nature of the wind-stress and the linearity229

of the model, the current components have Gaussian distributions that are fully230

characterized by the mean and variance. Note that the result of eq. (15) holds for231

any distribution of the wind-stress components, as long as the two-point correlation232

function is given by eq. (13). Equation (15) is easily derived by substituting the233

two-point correlation function (eq. (13)) in eq. (8). Similar to the first idealized234

case discussed above, it is clear that the second moment vanishes at the limit of a235

short wind-stress correlation time, (γ � r and γ � f).236

4 Details of the numerical models237

We performed two types of numerical simulations to validate the analytical deriva-238

tions presented above. The first type was a simple integration of eqs. (4)–these239

numerical solutions are provided, both to validate the analytical results and to240

present the solutions of cases not covered by the analytical solutions. In the second241

type of simulation, we used a state-of-the-art oceanic GCM, the MITgcm (MIT-242

gcm Group, 2010), to test the applicability of our analytical derivations when the243

spatial variability of the bottom topography and the nonlinearity of the ocean244

dynamics (Gill, 1982) are taken into account.245

4.1 MITgcm setup and details246

The MITgcm solves the primitive equations (MITgcm Group, 2010) and is imple-247

mented here using Cartesian coordinates with a lateral resolution of 1 km (or 10248

km) with a 50 × 50 grid points. We consider open (periodic) and closed physical249

boundaries of the domain (the results of the closed boundary setup are presented250

only in the last figure). There is one vertical level that expresses a 2D parabolic251

basin with a maximum depth of 90m, as shown in Fig. 5. The basin is situated in a252

plateau of 100m depth. The use of the partial cell option of the MITgcm enables us253

to handle depth variability, even with a single vertical level. The integration time254

step is 10s, and the overall integration time is two years. Water temperature and255

salinity are kept constant. The horizontal viscosity is 1m2/s, and the vertical one256

is 1×10−4m2/s. We use the linear bottom drag option of the MITgcm, and the ef-257

fective drag coefficient is depth dependent where the mean bottom drag coefficient258

is 1× 10−5s−1. We also use the implicit free surface scheme of the MITgcm.259

There are different ways to introduce heterogeneity to the water flow, for ex-260

ample, by forcing the water surface with spatially variable wind stress. Changes261

in density due to changes in temperature and salinity, as a result of spatially and262

temporally variable surface heat and freshwater fluxes, can also enrich water dy-263

namics. The main goal of the MITgcm numerical experiments presented here is264

to validate the analytical results presented above when advection and horizontal265

viscosity are taken into account. To enrich water dynamics, we chose to vary the266

water depth when including (excluding) the boundaries of the domain–there is no267

particular reason for choosing this way over another. We find the depth variability268

simpler, from a numerical point of view, but still rich enough to allow us to study269

the boundary effects on the water dynamics and to compare the numerical results270

with the derived analytical expressions.271


4.2 The Weibull distribution272

It is widely accepted that the PDF of ocean currents follows the Weibull distribu-273

tion (Gille and Smith, 1998, 2000, Chu, 2008, 2009, Ashkenazy and Gildor, 2011);274

therefore, we use this distribution in our numerical tests. For consistency, we pro-275

vide here a brief review of its properties and the methods we used to implement276

temporally correlated and uncorrelated time series. The Weibull PDF is defined277

for positive values of the variable, x > 0, and is characterized by two parameters,278

k and λ:279

Wk,λ(x) =k

λ

(xλ

)k−1exp(− (x/λ)k). (16)

where λ and k are both positive. λ and k are usually referred to as the scale280

and shape parameters of the distribution. The cumulative Weibull distribution281

function is given by:282

Fk,λ(x) = 1− e−(x/λ)k . (17)

It is possible to express the moments of the Weibull distribution, 〈xm〉, using k283

and λ:284

〈xm〉 = λmΓ(1 +

m

k

), (18)

where Γ is the Gamma function. Thus, it is sufficient to calculate the first and285

second moments of a time series and, from them, to find the k and λ that char-286

acterize the Weibull distribution (assuming that we have a priori knowledge that287

the series is Weibull distributed). The k parameter can be calculated using any288

two different moments (n, m) of the time series by solving (numerically) a tran-289

scendental equation 〈xn〉m/n/〈xm〉 = (Γ (1 + n/k))m/n /Γ (1 + m/k), and λ can290

be estimated by using the first moment (the mean) and the estimated k. It is also291

possible to calculate the k parameter from the slope of the hazard function of292

the Weibull distribution W (x)/(1−F (x)) = (k/λ)(x/λ)k−1 when a log-log plot is293

used. In this manuscript, we use the second and the fourth moments to derive the294

values of k and λ. Previous studies used other approximations to estimate k and295

λ (Monahan, 2006, Chu, 2008).296

A special case of the Weibull distribution, called the Rayleigh distribution, is297

obtained when k = 2. It can be associated with the distribution of the magnitude298

of a vector whose components are two independent, Gaussian-distributed, random299

variables; i.e., if x and y are Gaussian-distributed independent random variables,300

then the distribution of s =√x2 + y2 is a Rayleigh distribution (Weibull distri-301

bution with k = 2).302

It is fairly easy to generate an uncorrelated Weibull-distributed time series303

(using a simple transformation rule). However, below we use time series that are304

both Weibull distributed and temporally correlated. Such time series are generated305

as follows:306

(i) Generate uncorrelated, Gaussian-distributed, time series.307

(ii) Introduce temporal correlations by applying Fourier transform to the time308

series from (i), multiply the obtained power spectrum by the power spectrum309

corresponding to the desired correlations (in our case, exponentially decaying310

temporal correlations) and apply an inverse Fourier transform.311

(iii) Generate uncorrelated Weibull-distributed time series.312

10 Bel and Ashkenazy

(iv) Rank order the time series from step (iii) according to the time series of step313

(ii).314

The resulting time series is both temporally correlated and Weibull distributed.315

5 Results316

The analytical results presented above for the idealized cases highlight the impor-317

tant role played by the temporal correlations of the wind-stress in determining318

the statistics of surface ocean currents. The behaviors at the limits of long-range319

temporal correlations (rT � 1) and short-range temporal correlations (T → 0)320

are intuitive, once derived. The existence of an optimal correlation time, at which321

the average current amplitude is maximal, is less trivial. A similar maximum was322

reported by Gonella (1971), as described above. McWilliams and Huckle (2005)323

(right panel of Fig. 11 of their paper) studied the Ekman layer rectification using324

the K-profile-parameterization and observed a maximum in the depth-averaged325

variance of the current speed as a function of the Markov memory time. In what326

follows, we present the results of the numerical tests.327

In Fig. 6, we show the second moment of the current amplitude for the isotropic,328

step-like wind-stress model. The analytical results (eq. (11)) are compared with329

the numerical solution of the Ekman model (eqs. (4)) and the MITgcm modeling330

of the currents in a simple artificial lake (details of which are provided above, using331

open boundaries with 1 km resolution). One can see that the dependence of the332

average current amplitude on the constant wind-stress duration is non-monotonic333

(due to the fact that the Coriolis effect is significant – |f | > r) and that there is334

an excellent agreement between all the results. At the equator (not shown), where335

f = 0, the second moment increases monotonically to 〈|τ |2〉/(r2ρ20

)as a function336

of T .337

It was previously argued that, under certain conditions, the wind-amplitude338

(directly related to the wind-stress) PDF is well approximated by the Weibull339

distribution (Monahan, 2010); we thus chose, in our demonstrations, a Weibull340

distribution of wind-stress. In Fig. 7, we present the Weibull kcurrent parameter341

(eq. (16)) of the current distribution versus the constant wind-stress duration for342

two different values (kwind = 1 and kwind = 2) of the wind-stress Weibull kwind343

parameter (again, we only present here the results of the isotropic case). The an-344

alytical value of kcurrent was found based on the ratio between the fourth moment345

and the square of the second moment; we note, however, that the current’s PDF is346

not necessarily a Weibull PDF, and we chose to quantify the current’s PDF using347

the k parameter of the Weibull distribution for presentation purposes only. The348

findings of Monahan (2008) are consistent with the lower range of kwind values349

used here. One can see that, for short correlation times, the current amplitude350

exhibits a Rayleigh distribution (kcurrent = 2), independent of the wind-stress dis-351

tribution. This corresponds to Gaussian distributions of the current components.352

In the other limit of long constant wind-stress periods, kcurrent converges to kwind353

(not shown for kwind = 2).354

The above results were obtained for the maximal value of the Coriolis param-355

eter, i.e., f at the pole. In Fig. 8(a,b), we show kcurrent versus kwind for different356

values of the constant wind-stress duration, T , and the two limiting values of the357

Coriolis parameter, f , at the equator and at the pole. Here again, one observes an358


excellent agreement between the numerical and the analytical results for both val-359

ues of f . The limits of short and long temporal correlations of the wind, at which360

kcurrent = 2 and kcurrent = kwind, correspondingly, are clearly demonstrated.361

In Figs. 9 and 10, we present similar results for the case of an exponentially362

decaying correlation function of the wind-stress. As mentioned in Sec. 3.2, the363

expression for the current’s second moment (eq. (15)) holds for any distribution364

of the wind-stress (given that the temporal autocorrelation follows eq. (13)), and365

thus it is possible to obtain analytically the current’s second moment in the case366

of Weibull-distributed wind-stress whose k value does not necessarily equal 2.367

We generated a Weibull- distributed time series that has exponentially decaying368

temporal correlations, following the procedure described in Section 4.2. In Fig.369

9, we present the second moment of the current amplitude versus the correlation370

time (1/decay rate) of the correlation function. The existence of an optimal decay371

rate, for which the average current amplitude is maximal, is demonstrated for this372

case as well. One notable difference is the absence of the secondary maxima points373

which appeared in the step-like model. For this case, one may easily find that the374

maximal average amplitude of the currents is obtained for γ = |f | − r, assuming375

that |f | ≥ r.376

In Fig. 10, we present kcurrent versus kwind. Here again, we obtain an excellent377

agreement between the predicted and the numerically obtained limiting behav-378

iors. It is important to note that the good agreement between the MITgcm results379

and the analytical results has been proven to be valid only for the setup described380

above, using a fine resolution of 1 km and open boundaries. Different behavior may381

occur for different scenarios, such as regions close to the boundary of the domain,382

spatially variable wind, complex and steep bottom topography, and vertically and383

spatially variable temperature and salinity. Using a setup with closed boundaries,384

we found that close to the boundaries of the artificial lake, the results showed a385

significant deviation due to the boundary effects that were neglected in the ana-386

lytical model. The results of the MITgcm model with closed boundary conditions387

and coarser resolution of 10 km are shown in Fig. 11. All the other figures show388

the MITgcm results for the case of open boundaries. Moreover, we used a spatially389

uniform wind-stress in our simulation and have not considered the more realistic390

case of non-uniform wind-stress.391

6 Summary392

In summary, we have shown, in two idealized cases, that the PDF of wind-driven393

ocean currents depends on the temporal correlations of the wind. For short-range394

correlations, the current speed approaches zero, and the PDF of its components395

is Gaussian. For long-range temporal correlations of the wind, the currents’ PDF396

is proportional to the wind-stress PDF. The different cases considered here and397

the MITgcm simulations described above suggest that the existence of a maximal398

current speed as a function of the temporal correlation time is not unique to the399

cases studied here, and may also be relevant in more realistic types of temporal400

correlations and setups. Analysis that is based on the space-dependent model401

(either in the horizontal or the vertical dimensions or both) is a natural extension402

of the current study and will allow us to compare the analytical results to altimetry-403

based surface currents and to study non-local phenomena.404


Acknowledgements The research leading to these results has received funding from the Eu-405

ropean Union Seventh Framework Programme (FP7/2007-2013) under grant number [293825].406

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Fig. 1 A map of the correlation time (in days) of the wind-stress magnitude. We define thecorrelation time as the time at which the normalized auto-correlation function first dropsbelow 1/e. Clearly, there is a large variability in the correlation time. One can also notice theremarkably shorter correlation time overland compared with that over the ocean (except forthe polar regions). The map is based on the NCEP/NCAR reanalysis six-hourly wind data forthe period of 1993-2010 (Kalnay et al, 1996). The spatial resolution of the data is 2.5◦ × 2.5◦.Other definitions of the correlation time yield qualitatively the same results. The black contourline indicates a one-day correlation time, and the white line indicates the coast line. A grid(light blue lines) of 30◦ × 30◦ was superimposed on the map.

-2 -1 0 1 2u

-2

-1

0

1

2

v

T=π/f (1/4 day)

-2 -1 0 1 2u

-2

-1

0

1

2

v

(a) (b)

T=2π/f (1/2 day)

Fig. 2 Typical trajectories of the current velocity components for the step-like wind stress.In the left panel, (a) the constant wind-stress duration, T = π/f , corresponds to changesin the wind-stress after the velocity vector completes ∼ 1/2 circle. On the right panel, (b),T = 2π/f , that is, the wind-stress changes after a full circle of the velocity field. In bothpanels, the Coriolis parameter, f , was set to its value at the pole. The symbols indicate thetimes at which the wind-stress was changed.


-3

-2

-1

0

1

2

3

uv|w|

0 2 4 6 8 10 12Time (days (4π/f))

-3

-2

-1

0

1

2

3

Inte

gra

ted v

eloci

ty (

m2/s

)

(a)

(b)

T=π/f

T=2π/f

Fig. 3 Typical time series of the current velocity components and amplitude. In the top panel,(a), the constant wind-stress duration, T = π/f . In the bottom panel, (b), T = 2π/f . In bothpanels, the Coriolis parameter, f , was set to its value at the pole. The symbols indicate thetimes at which the wind-stress was changed.

|w|2

(f2

+r

2)ρ

2 0

|τ|2

r=0.1f

r=0.25f

r=0.5f

r=0.75f

r=f

2π 10π 12π

6

4

14π8π0

2

0

T×f4π 6π

Fig. 4 The second moment of the current amplitude (eq. (11)) versus the constant wind-stressduration, T . The different lines correspond to different values of the Rayleigh friction, r.


x (km)

y (

km

)

−40 −30 −20 −10 0 10 20 30 40

−40

−30

−20

−10

0

10

20

30

40

100

110

120

130

140

150

160

170

180

Fig. 5 Contour plot of the bathymetry used in the MITgcm simulations.


0 2 4 6 8 10 12 14

T (104 s)

0

0.5

1

1.5

2

2.5m

2(m4 /s

2 )

kwind

=1, numerick

wind=2, numeric

kwind

=1, analytick

wind=2, analytic

kwind

=1, MITgcmk

wind=2, MITgcm

Fig. 6 The second moment of the current amplitude versus the constant wind-stress duration,T . The analytical results (solid lines) are compared with the numerical solution of the Ekmanmodel (empty symbols) and with the MITgcm modeling of the currents in a simple artificiallake with open boundaries using a 1 km resolution. The wind-stress amplitude was drawnfrom a Weibull distribution with two different values of the kwind parameter as specified inthe figure. The Coriolis frequency, f ≈ 1.45 × 10−4s−1, was set to its value in the pole, andthe constant wind duration associated with this Coriolis frequency is indicated by the verticaldashed line. The Rayleigh friction parameter is r = 10−5s−1.


0 2 4 6 8 10 12 14

T (104 s)

1

1.5

2

k curr

ent

kwind

=1, numerick

wind=2, numeric

kwind

=1, analytick

wind=2, analytic

Fig. 7 The Weibull kcurrent parameter of the current distribution versus the constant wind-stress duration, T . The analytical results (solid lines) are compared with the numerical solutionof the Ekman model (empty symbols). The wind-stress was drawn from a Weibull distribu-tion with two different values of the kwind parameter as specified in the figure. The Coriolisfrequency, f , was set to its value in the pole and the corresponding duration is indicated bythe vertical dashed line.


1

1.2

1.4

1.6

1.8

2

2.2

kcu

rren

t

1 1.2 1.4 1.6 1.8 2k

wind

1

1.2

1.4

1.6

1.8

2

2.2

kcu

rren

t

lines - analyticsymbols - numeric

T=106

s

T=104 s

T=103 s

T=5x104 s

T=104

s

T=103 s

T=106

s

T=5x104 s

(a)

(b)

Fig. 8 The Weibull kcurrent parameter of the current distribution versus the kwind parameterof the step-like wind-stress distribution. The analytical results (solid lines) are compared withthe numerical solution of the Ekman model (symbols). The Coriolis parameter, f , in panel (a),is 0, corresponding to its value at the equator. In panel (b) , f ≈ 1.45×10−4s−1, correspondingto its value at the pole and is larger than r = 10−5s−1. The different lines correspond todifferent constant wind-stress durations, T , as indicated in the figure.


0 2 4 6 8 10

1/γ (104 s)

0.5

1

1.5

2

m2 (

m4 /s

2 )

numericanalyticMITgcm

Fig. 9 The second moment of the current amplitude for the case of wind-stress with expo-nentially decaying temporal correlations. The analytical results (solid line) are compared withthe results of numerical integration of Ekman equations (red circles) and with the currentssimulated by the MITgcm model (blue triangles) in a simple lake with open boundaries (seethe description in section 4). The wind-stress amplitude was drawn from a Weibull distribu-tion with kwind = 1.5. The Coriolis frequency, f ≈ 7.27 × 10−5s−1, corresponds to its valueat lat = 30◦ and r = 10−5s−1. Note that the second moment is plotted versus 1/γ, to alloweasier comparison with the results of the step-like wind-stress case.


1 1.2 1.4 1.6 1.8 2 2.2k

wind

1

1.2

1.4

1.6

1.8

2

2.2

k curr

ent

1/γ=104 (s)

1/γ=105 (s)

1/γ=106 (s)

Fig. 10 The Weibull kcurrent parameter of the current amplitude distribution versus the kwind

parameter of the wind-stress with exponentially decaying temporal correlations for differentcorrelation times (corresponding to 1/γ as indicated in the figure). The results were obtainedusing different realizations of the wind-stress. The solid lines are drawn to guide the eye. Thevalues of the Coriolis parameter, f and the Rayleigh friction parameter, r, are the same as inFig. 9.


0 2 4 6 8 10

1/γ (104 s)

0

0.5

1

1.5

2

m2

theorycenterperiphery

Fig. 11 Comparison of the analytical results for the second moment of the current amplitudewith the results of the detailed MITgcm model with closed boundaries and a coarse resolutionof 10 km. It is shown that for a wide range of the wind-stress correlation time (the exponen-tially decaying temporal correlations were considered), the MITgcm results agree qualitativelywith the analytical results of the over-simplified model. On the other hand, the results in theperiphery show a significant deviation. These results, combined with the fact that for openboundaries, there was a good agreement between the analytical and the simulations results,suggest that, as expected, the simple model fails to capture the boundary effects.

the relation between the statistics of open ocean currents...

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