Energy transfer of surface wind-induced currents to the deep ocean

via resonance with the Coriolis force

Yosef Ashkenazya,∗

aDepartment of Solar Energy and Environmental Physics, BIDR, Ben-Gurion University, MidreshetBen-Gurion, 84990, Israel

Abstract

There are two main comparable sources of energy to the deep ocean—winds and tides. However,

the identity of the most efficient mechanism that transfers wind energy to the deep ocean is

still debated. Here we study, using oceanic general circulation model simulations and analytic

derivations, the way that the wind directly supplies energy down to the bottom of the ocean

when it is stochastic and temporally correlated or when it is periodic with a frequency that

matches the Coriolis frequency. Basically, under these, commonly observed, conditions, one of

the wind components resonates with the Coriolis frequency. Using reanalysis surface wind data

and our simple model, we show that about one-third of the kinetic energy that is associated

with wind-induced currents resides in the abyssal ocean, highlighting the importance of the

resonance of the wind with the Coriolis force.

Keywords: Ekman layer, inertial oscillations, deep ocean, resonance, Coriolis force

1. Introduction1

Deep ocean mixing is an essential process that maintains the deep ocean circulation; with-2

out it, the abyssal ocean would be stagnant (Munk, 1966; Munk and Wunsch, 1998; Wunsch3

and Ferrari, 2004). Differential heating is insufficient to maintain a closed and steady ocean4

circulation as the oceanic heat source is, on average, above the cooling source (Sandstrom,5

1908; Defant, 1961; Huang, 1999; Wunsch and Ferrari, 2004). There are two main comparable6

sources of deep ocean energy: winds and tides (Munk and Wunsch, 1998). They both increase7

the vertical mixing by several orders of magnitudes, maintaining the meridional overturning8

circulation. If only molecular diffusion is taken into account, ocean circulation would be re-9

stricted to the upper few meters of the ocean (Munk and Wunsch, 1998). There are additional10

∗Corresponding author: ashkena@bgu.ac.il, Tel. +972 8 6596858, Fax +972 8 596921

Preprint submitted to Journal of Marine Systems November 14, 2016

sources of energy to the deep ocean, including bottom geothermal heating (Huang, 1999) and11

biomixing (Dewar et al., 2006). The different energy sources for the abyssal ocean all have one12

feature in common, namely, the large uncertainties associated with them (Munk and Wunsch,13

1998; Furuichi et al., 2008; Zhai et al., 2009).14

According to the seminal paper of Ekman (1905), wind-induced currents are restricted to15

the upper ocean (the first hundred meters or so) and decay exponentially with depth. However,16

this model is based on the simplistic assumption that the winds are constant, both in space17

and time. In fact, winds are stochastic by nature and vary on a wide range of temporal and18

spatial scales. They can generate internal waves, instabilities, and mesoscale eddies (Zhai et al.,19

2007; Danioux et al., 2008; Danioux and Klein, 2008) that eventually radiate into the abyssal20

ocean; interaction of the bottom topography with ocean circulation and eddies also contributes21

to ocean mixing (Munk and Wunsch, 1998). It was also shown that near-inertial energy is22

transported to the deep ocean via the “chimney effect” in which the energy is trapped within23

anticyclonic eddies and propagates downward to the deep ocean (Lee and Niiler, 1998; Zhai24

et al., 2005, 2007, 2009); see also recent review of Alford et al. (2016). Thus, although the25

Ekman layer model provides the basic understanding of the Coriolis force’s effect on surface26

currents, it is clear that the additional processes that are related to the wind must be taken27

into account.28

Winds are sometimes periodic. As such, when their frequency matches the inertial (Coriolis)29

frequency, they can resonate with the currents induced by them. This scenario was discussed in30

many previous studies (Pollard et al., 1973; Chang and Anthes, 1978; Price, 1981; Greatbatch,31

1983; Price, 1983; Greatbatch, 1984; Rudnick and Weller, 1993; Crawford and Large, 1996;32

Rudnick, 2003; Stockwell et al., 2004; McWilliams and Huckle, 2005; Mickett et al., 2010;33

Whitt and Thomas, 2015; Zhai, 2015), mainly in the framework of the depth-integrated Ekman34

layer model. Resonance conditions are satisfied when the winds at 30◦ latitude have a diurnal35

frequency, when the characteristic frequency of a storm matches the Coriolis frequency (Gill,36

1984; Crawford and Large, 1996), or when the tides’ frequency matches the Coriolis frequency.37

In addition, stochastic wind that has temporal correlations can lead to enhanced currents,38

depending on the strength of the correlation (Gonella, 1971; McWilliams and Huckle, 2005; Bel39

and Ashkenazy, 2013).40

Here we investigate the effect of the wind’s resonance with the Coriolis force on ocean cur-41

rents and, in particular, on the abyssal ocean’s kinetic energy (KE). We carry out oceanic42

general circulation model (GCM) simulations and provide analytic solutions and approxima-43

tions for the currents as a function of depth for a finite-depth ocean under the action of periodic44

2

and stochastic wind that exhibits temporal correlations. We show that in both cases, the wind-45

induced currents are significant in the deep ocean and contribute significant KE to the abyssal46

ocean. Unlike previous studies, we adopt here a simple and idealized approach that allows us to47

obtain analytic approximations for the currents as a function of depth, showing the relevance48

of the resonance in vast ocean areas and times. Yet, in our approach we ignore important49

processes that are related to the spatial variability of the wind (or to the wind stress curl) like50

the Sverdrup balance but this simplicity allows us to perform detailed analytic calculation.51

2. MITgcm results52

2.1. Numerical model (MITgcm) setup53

We used the Massachusetts Institute of Technology general circulation model (MITgcm)54

(MITgcm-group, 2010) to study oceanic circulation under the action of periodic and stochastic55

wind stress. We used a simple rectangular domain with doubly periodic boundary conditions56

in the lateral x and y directions and with bottom no-slip (zero velocity) boundary conditions.57

The ocean floor was flat. The horizontal grid resolution was 100 m, and the vertical grid58

resolution increased monotonically from 4 m at the top layer to 100 m at a depth of 200 m;59

then a constant vertical grid spacing of 100 m was used down to the bottom of the ocean (which60

is 1.2 km in the key example discussed in the main text). There were 100 × 100 grid points61

in the horizontal direction (10×10 km). The Coriolis parameter was taken to be constant and62

equal to the diurnal frequency (f = 2π/86400 = 7.27× 10−5 s−1). The only driving force was63

the wind stress, which was temporally variable but spatially constant. We studied the ocean64

current both when the ocean was stratified and when it had uniform density.65

2.2. MITgcm results–uniform density66

We first considered the case of a “box-like” basin with a flat bottom at a depth of 1.2 km and67

constant density. The water surface was forced by a periodic wind stress in the x-direction—68

the wind’s frequency exactly matched the Coriolis frequency such that resonance between the69

Coriolis force and the wind stress was expected. The simulation was initiated from motionless70

water. The KE as a function of time is presented in Fig. 1a. Two cases were considered—71

simulations both with and without the advection terms. Without the advection terms, the72

equations of motion were linear and eddies could not develop, while with the advection terms,73

the dynamics was nonlinear and eddies could develop. At the beginning of the simulation,74

the KE increased monotonically with time (Fig. 1a), and the currents were limited to the top75

∼100 m of the water column (Fig. 1b), becoming strong and almost uniform with time (Fig. 1c).76

3

0 50 100 1500

1

2

3

4

Ek(m

3/s2)

t (days)

(a)

No advectionWith advection

0 0.05 0.1 0.15 0.2 0.25 0.3

−1

−0.8

−0.6

−0.4

−0.2

0

|q | (m/s)

z(k

m)

(b)

t=20 dayst=170 days

x (km)

y(k

m)

t=20 days

1 m/s

(c)

η(m

m)

2 4 6 8 10

1

2

3

4

5

6

7

8

9

10

−0.02

−0.01

0

0.01

x (km)

y(k

m)

t=170 days

0.4 m/s

(d)

η(m

m)

2 4 6 8 10

1

2

3

4

5

6

7

8

9

10

−8

−6

−4

−2

0

2

4

6

8x 10

−3

Figure 1: Results of a constant density ocean forced by a zonal periodic wind stress whose frequency matches

the Coriolis frequency (resonance conditions). (a) Depth-integrated KE, 12

∫ 0

−H|q|2dz, as a function of time,

without (black curve) and with (red curve) the nonlinear advection terms; q = u+ iv where u, v are the zonaland meridional current components. Note the “collapse” of the KE at t ≈ 21 days for the run with the advectionterms; this collapse is associated with the transition from non-turbulent surface-limited flow to turbulent flowalong the entire depth. (b) Current speed (|q| =

√u2 + v2) as a function of depth just before (blue curve) and

after (green curve) the transition shown in the red curve in panel (a). (c) Sea surface height (color), η, andsurface current field (arrows) just before the transition—note the almost regular pattern and almost uniformcurrent. (d) Same as (c), after the transition—note the much smaller current speed and the turbulent flowpattern. The current field is quite uniform horizontally and rotates clockwise with the Coriolis frequency.

4

At t ∼ 20 days, the nonlinear advection terms became dominant and led to the development of77

edge waves (Fig. 1c) (Vallis, 2006) and later to eddies (Fig. 1d), which mixed the entire water78

column and led an almost uniform speed with depth. This transition was associated with a79

drastic drop in the KE since the energy was spread over the entire water column, leading to a80

reduced mean current speed, and thus to reduced KE (as the KE is proportional to the square81

of the current speed). When the advection terms were excluded (black curve of Fig. 1a), the82

KE continued increasing to unrealistic values. Repeating the above numerical experiment, with83

the forcing frequency unequal to the Coriolis frequency, resulted in a much weaker flow that was84

limited to the upper ∼100 m of the water column. Still, also in this case, at the beginning of the85

simulation, the flow was spatially uniform and was limited to the upper 10-20 m of the water86

column. Then, after a few days of simulation, turbulent flow developed, the upper dynamical87

layer deepened to ∼ 100 m, and the current speed reduced. The surface (Ekman-like) layer88

in the run without the advection terms deepened monotonically with time until it reached the89

bottom. The numerical experiments described above indicate that when the forcing frequency90

matches the Coriolis frequency, a resonance occurs, resulting in an Ekman-like layer that spans91

the entire water column, and high values for the mean KE and the current speed.92

2.3. The role of density stratification in the effect of resonance93

Intuitively, stratification of the water column should limit vertical motion and, thus, should94

weaken the effect of the resonance discussed in the main text. When we started the simulation95

with a stratified water column (by initially specifying the water temperature to range from96

20◦C at the top to 10◦C at the bottom), after some time, eddies developed and eroded the97

stratification such that eventually the entire water column was mixed and had a uniform density.98

The current speed profile then converged to the almost uniform profile shown in Fig. 1b.99

To mimic real ocean stratification, we performed an additional simulation in which the100

temperature at the sides of the box was kept constant, imposing stratification at the boundaries101

of the box throughout the simulation. The results of this simulation are shown in Fig. 2 in102

which we imposed the linear temperature profile indicated by the red line in Fig. 2c. There103

are several important observations: (i) variations in the sea surface height are small (Fig. 2a),104

suggesting that the horizontal variations in the current field are small and that eddy activity105

is weak; (ii) the currents are practically uniform in the horizontal directions (Fig. 2a); (iii) u is106

perpendicular to v (Fig. 2a,b); (iv) the magnitude of u equals the magnitude of v (Fig. 2a,b);107

(v) u and v oscillate sinusoidally with time such that the current vector rotates clockwise with a108

frequency of one day (Fig. 2b); (vi) the current speed is almost constant with time; (vii) u and109

v (and hence the current speed) decrease linearly with depth (Fig. 2c) such that the current110

5

x (km)

y(k

m)

(a)

η(m

m)

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

−6

−5

−4

−3

−2

0 1 2 3 4 5−6

−4

−2

0

2

4

6

velocity

(m/s)

t (days)

(b)

uv

|q |

−1 0 1 2 3 4 5 6

−1

−0.8

−0.6

−0.4

−0.2

0

u, v (m/s)

z(k

m)

uv

10 12 14 16 18 20

−1

−0.8

−0.6

−0.4

−0.2

0

T (oC)

(c)

0 0.005 0.01 0.015 0.020

1

2

3

4

5

x 10−3

Tz (oC/m)

|q| z

(1/s)

(d)

MITgcmτ0

2ρ0ν

τ0

2ρ0ν(1 − e−

aHT0

Tz)

Figure 2: (a) Section of (1 × 1 km of the 10 × 10 km domain) sea surface height (color) and surface current(arrows) under stratification conditions. The white arrows indicate the current field after six hours of the blackarrows, indicating clockwise rotation of the current field with the Coriolis frequency. (b) u (blue), v (green) andcurrent speed (red) as a function of time. (c) u (blue open circle), v (blue full circle) as a function of depth undertemperature stratification conditions (red curve). (d) Gradient in current speed as a function of stratificationtemperature gradient for the MITgcm simulations (blue open circles) and for the exponential fit (red curve)similar to Eq. (1). The filled green circle indicates the theoretically predicted current speed, τ0/(2ρ0ν), undervery strong stratification (Tz → ∞). Parameter values: τ0 = 0.1 N m−2, ρ0=1028 kg m−3, ν = 0.01 m2 s−1,ω0 = f = 2π/86400 s−1, H = 1.2 km, T0 = 15 ◦C, and a = 5.25.

6

vector direction is constant with depth; and (viii) the slope of linear decrease of the current111

speed from (vii) converges to a constant value for sufficiently strong stratification (Fig. 2d).112

It is clear that for sufficiently strong imposed stratification, the developed flow will not be113

able to break the stratification as the Richardson number will be very large and the stratified114

shear flow will be stable (Cushman-Roisin, 1994). We thus expect the slope |q|z (where q = u+iv115

— see the next section for more details) to converge exponentially to a limiting value with the116

level of stratification Tz. The two limits of this dependence can be deduced from our analysis:117

when Tz → 0 also |q|z → 0 (Fig. 1), and when Tz → ∞, Eq. (15) holds and |q|z = τ02ρ0ν

(see118

below). Thus, if ρz = const, then119

|q|z =τ0

2ρ0ν

(1− e

aHρ0ρz), (1)

where a is a constant that quantifies the decay rate and can be estimated from the simulated120

currents. It is possible to obtain the interior ocean profile by integrating Eq. (1)121

|q(z)| = τ02ρ0ν

(z +H)− τ02ρ0ν

eaHρ0ρz (z +H − h0) , (2)

where h0 is an additional constant that can be estimated from the simulations.122

In the following we consider two cases–one of uniform density and one of very strongly123

stratified ocean. In reality neither of these cases is valid as in the more realistic case the124

stratification is finite and internal wave dynamics and geostrophic turbulence should be taken125

into account. The specific exponential fit shown in Fig. 2d [following Eqs. (1) and (2)] is valid126

for the specifications listed in Sec. 2.1 and is probably different under different specifications.127

3. Model and solution procedure128

After a sufficiently long time, the simulated currents are almost uniform in the horizontal129

directions, in both the barotropic (constant density) and the stratified water column cases;130

see 2.3. We thus assumed, in the following, that the current’s components (u, v) very weakly131

depend on the horizontal dimensions (x, y) such that it is possible to omit the advection terms132

and the horizontal viscosity terms from the governing equations. Then, one is left with the133

Ekman model (Ekman, 1905):134

ut − fv = νuzz, (3)

vt + fu = νvzz, (4)

7

where t and z are the time and depth coordinates, f is the (constant) Coriolis frequency, and ν is135

the eddy-parameterized vertical viscosity coefficient. u and v, can be regarded as the velocities136

around the interior geostrophic currents (if these do exist) and thus the omission of the pressure137

gradient terms in Eqs. (3), (4). It is possible to obtain a single equation qt + ifq = νqzz, using138

a complex velocity variable, q = u + iv; we then applied a (time) Fourier Transform (FT) to139

obtain an equation with respect to the depth variable z:140

qzz − i(f + ω)q/ν = 0, (5)

where ω is the frequency variable and the “hat” indicates the FT.141

To close the solution of Eq. (5), we applied no-slip boundary conditions at the bottom of142

the ocean, q(z = −H) = 0, and at the surface, we linked the wind stress τ = τx + iτy (τx, τy are143

the x, y components of the wind-stress vector) to the surface currents by qz(z = 0) = τ /(ρ0ν)144

(Gill, 1982), where ρ0 is the ocean water density. The solution is then145

q =τ(1− i)d

2ρ0ν

e(1+i)(z+H)/d − e−(1+i)(z+H)/d

e(1+i)H/d + e−(1+i)H/d, (6)

where d =√

2ν/|f + ω| is the “spectral” Ekman layer depth.146

The FT of the second moment of the currents is given by147

|q|2 =|τ |2d22ρ20ν

2

cosh[2(z+H)

d

]− cos

[2(z+H)

d

]

cosh[2Hd

]+ cos

[2Hd

] . (7)

The frequency spectrum of the KE per unit area is then148

Ek =1

2

0∫

−H

|q|2dz =|τ |2d38ρ20ν

2

sinh(2Hd

)− sin

(2Hd

)

cosh(2Hd

)+ cos

(2Hd

) . (8)

The depth of the Ekman layer, d, diverges when ω = −f , and the wind-induced currents may149

reach the bottom of the ocean, depending on the frequency spectrum of the wind stress, τ .150

To explicitly find the currents, one must know the wind stress and must either apply an151

inverse FT or use Parseval’s theorem, i.e.,152

〈|q|2〉 =1

2π

∞∫

−∞

|q|2dω. (9)

8

We considered two cases, shown below: (i) a purely periodic wind stress, and (ii) stochastic153

wind stress with temporally decaying correlations.154

4. Periodic wind stress155

For simplicity, we chose the wind stress to be periodic only in the x direction such that156

τ(t) = τx,0 cos(ω0t) where τx,0 is the wind-stress amplitude in the x directions and ω0 is the157

frequency; below, for convenience, we drop the subscript x. The FT of the wind stress is then158

a sum of two δ functions and hence159

q(t) =

√2τ0

4ρ0ν

∑

j=+,−

d(jω0)sinh

[(1+i)(z+H)d(jω0)

]

cosh[(1+i)Hd(jω0)

] ei(jω0t−π4 ). (10)

The second moment of the currents is then160

〈q2〉 =τ 20

8ρ20ν2

∑

j=+,−

[d(jω0)]2cosh

[2(z+H)d(jω0)

]− cos

[2(z+H)d(jω0)

]

cosh[

2Hd(jω0)

]+ cos

[2H

d(jω0)

] , (11)

and the KE is161

Ek =τ 20

32ρ20ν2

∑

j=+,−

[d(jω0)]3

sinh[

2Hd(jω0)

]− sin

[2H

d(jω0)

]

cosh[

2Hd(jω0)

]+ cos

[2H

d(jω0)

] . (12)

Generally speaking, the depth of the Ekman layer, d, is much smaller than the depth of the162

ocean, H. In this case, the currents decay exponentially with depth and Eqs. (10)-(12) become163

independent of the ocean depth H; the current vector exhibits a periodic clockwise rotation164

with frequency ω0.165

In the case of “resonance,” the wind-stress frequency equals the Coriolis frequency (ω0 =166

−f) and the depth of the Ekman layer, d(ω0) =√

2ν/|f + ω0|, becomes infinite and, hence,167

much larger than the depth of the ocean. Thus, d(−f) → ∞ and d(f) =√ν/f . Then, based168

on Eq. (10), when the ocean is deep enough (i.e., H � d(f)), q may be approximated as169

q(t) =τ0

2ρ0ν

[d(f)√

2e(1+i)z/d(f)+i(ft−π/4) + (z +H)e−ift

](13)

such that both surface and bottom boundary conditions are satisfied. The current vector may170

9

be further approximated as171

[u(t), v(t)] = (z +H)τ0[cos(ft),− sin(ft)]/(2ρ0ν). (14)

Consequently, the current’s second moment and KE are172

|q|2 =τ 20

4ρ20ν2(z +H)2 (15)

Ek =τ 20

24ρ20ν2H3. (16)

Hence, under resonance conditions: (i) the current’s magnitude depends on the depth of the173

ocean and decreases linearly with depth to the bottom of the ocean, (ii) the current vector174

rotates clockwise with the Coriolis frequency, with a fixed magnitude, and (iii) the KE increases175

like H3.176

We validated the analytic solutions given above by comparing them with the MITgcm177

simulations. In these simulations, we focused on resonance conditions, excluding the spatial178

dependence. The results are shown in the left column of Fig. 3. The analytic solutions (black179

lines) are very close to the numerical ones (blue full circles), including (i) the clockwise rotation180

of the current vector with the Coriolis frequency (not shown), (ii) the linear decrease of the181

current speed with depth z (Fig. 3a), (iii) the linear increase of the surface current speed with182

the depth of the ocean, H (Fig. 3c), and (iv) the cubic dependence of the KE on the depth183

of the ocean, H (Fig. 3e). Under non-resonance conditions, the currents are limited to the184

surface Ekman layer and are independent of the depth of the ocean. It is important to note185

that when the current’s components u and v depend linearly on depth z, the viscosity terms186

in Eqs. (3),(4) vanish such that the equations take the form of the equations associated with187

the inertial oscillations that exhibit a clockwise rotation of the current vector with the Coriolis188

frequency.189

Resonance of the Coriolis force with other driving forces, such as diurnal winds or tides,190

is limited to a very narrow band of latitudes, questioning the overall effect of the resonance191

on real ocean circulation and energy. However, winds are stochastic in their nature such that192

one component of their frequency spectrum may resonate with the Coriolis force across the193

globe. This may lead to an enhancement of wind-induced currents over wide regions, possibly194

penetrating to the bottom of the ocean. We studied this hypothesis, as shown below.195

10

0 0.02 0.04 0.06 0.08 0.1−5

−4

−3

−2

−1

0

z(k

m)

√〈|q |2〉(m/s)

(a)

MITgcmTheory

−5

−4

−3

−2

−1

0

0 0.01 0.02 0.03 0.04

z(k

m)

√〈|q|2〉(m/s)

(b)

error−barMITgcmTheory 1Theory 2

0 1 2 3 4 50

0.05

0.1

H (km)

√〈|q|2〉(m/s)

(c)

MITgcmTheory

0 1 2 3 4 50

0.5

1

1.5

2

2.5

H (km)

3√E

k(m

/s2

/3 ) (e)

MITgcmTheory

0 1 2 3 4 5

8

9

10

11

x 10−4

H (km)

〈|q|2〉(m

2/s2) (d)

MITgcmTheory

0 1 2 3 4 50

0.05

0.1

0.15

0.2

H (km)

Ek(m

3/s2)

(f)

MITgcmTheory

Figure 3: Summary of the MITgcm 1d simulations for periodic wind-stress forcing under resonance conditionswith the Coriolis frequency (left column) and stochastic, temporally correlated wind stress (right column). (a)Current speed as a function of depth z under the action of periodic wind stress. The blue dots indicate theMITgcm results and the black line indicates Eq. (15). (b) Same as (a) for (stochastic) correlated wind stress.The black line indicates the results obtained by numerical integration of Eq. (9) using Eq. (7) and the dashedlines indicate the approximations (A.8)-(A.10). Note that the analytic approximations fall within the error barsof the simulations (gray shading). (c) Surface current speed as a function of ocean depth, H, when the windstress is periodic. The blue dots indicate the MITgcm results and the black line indicates Eq. (15). (d) Thesquare of the surface current speed as a function of ocean depth, H, for (stochastic) correlated wind stress;

the black line indicates Eq. (A.11). (e) Depth-integrated KE cube root, E3/2k , as a function of ocean depth,

H, under the action of periodic wind stress. The blue dots indicate the MITgcm results and the black lineindicates Eq. (16). (f) Depth integrated KE for stochastic and temporally correlated wind stress where theblack line indicates Eq. (A.13). In all panels, the MITgcm simulations are indicated by the blue dots while theanalytic solutions/ approximations are indicated by the solid and dashed lines. The error bars indicate mean± 1 standard deviation of 20 MITgcm realizations. Parameter values: τ0 = 0.005 (0.1) N m−2 for periodic(stochastic) wind stress, ρ0=1028 kg m−3, ν = 0.1 m2 s−1, γ = 10−5 s−1, and ω0 = f = 2π/86400 s−1.

11

5. Stochastic wind stress196

To study the effect of stochastic wind on ocean currents, we forced the model with a stochas-197

tic wind, whose temporal correlations decay exponentially,198

〈τ(t)τ(t+ t)〉 =τ 202e−γ|t| (17)

where τ0 is a parameter that quantifies the magnitude of the wind stress, and γ is a parameter199

that quantifies the decay rate of the temporal correlations (Bel and Ashkenazy, 2013; Ashkenazy200

et al., 2015). Similar to the above, for simplicity, we restricted the winds to be along the x201

direction. The FT of the wind stress (17) is202

|τ |2 =γτ 20

γ2 + ω2. (18)

We first used this wind stress to simulate currents using the MITgcm. The configuration was203

the same as the example presented above, i.e., a domain of 100 × 100 horizontal grid points204

with a resolution of 100 m and a 1.2-km-deep ocean, with doubly periodic boundary conditions.205

This configuration and forcing resulted in (spatially) almost uniform currents. We thus used the206

spatially independent Eqs. (3),(4) and the implicit solution (7) to gain greater understanding207

of the system’s dynamics. We developed, in Appendix A, analytic approximations for the208

dependence of the second moments of the current speed |q|2 on depth z, for the dependence209

of the surface current speed on ocean depth H, and for the dependence of the KE on ocean210

depth, Ek(H).211

We show that when the wind stress is stochastic with exponentially decaying temporal212

correlations, the current speed is not limited to the upper ocean but decreases linearly in the213

lower part of the ocean and vanishes only at the ocean floor; see Fig. 3b and Eq. (A.8). This is214

due to the “resonance” of the Coriolis force with the wind stress at the Coriolis frequency. At215

this frequency, the Ekman layer becomes infinite, leading to the penetration of wind-induced216

currents to the bottom of the ocean. The decrease is faster at the upper part of the ocean217

[Fig. 3b and Eqs. (A.9),(A.10)]. The MITgcm simulations exhibit a similar decrease with218

depth (full circles in Fig. 3b).219

The second moment of the surface current speed, |q|2 as a function ocean depth, H, is shown220

in Fig. 3d. The analytic approximation (A.11) indicates a logarithmic increase of |q|2 with H221

and is in good agreement with the MITgcm simulations (Fig. 3d). The analytic approximation222

of the KE, Eq. (A.15), points to a linear increase with the ocean depth, H. The agreement with223

12

the MITgcm simulations is satisfactory and better for shallower depths (Fig. 3f). In summary,224

under the action of temporally correlated stochastic wind stress, the currents are not confined225

in the upper ocean, as in the classical Ekman layer model, but extend to the bottom of the226

ocean.227

6. Implementation to the real ocean228

As the wind is stochastic and exhibits temporal correlations (Bel and Ashkenazy, 2013),229

the resonance of stochastic wind stress with the Coriolis force is relevant in extended regions230

of the globe. To get a rough idea of the effect of the resonance on the real ocean, we used the231

six-hourly winds of the NCEP-DOE reanalysis 2 (Kanamitsu et al., 2002). Yet, the analysis232

we present below does not meant to be realistic and is only aimed in highlighting the possible233

importance of Ekman resonance in the real ocean; this is unlike previous studies (e.g., Price234

et al., 1986) that presented realistic analysis on the effect of diurnal forcing on the ocean.235

Global-coverage (∼2.5◦) surface (10 m) winds of 36 years (1979-2014) were first transformed236

into wind stress based on Large and Yeager (2004):237

(τx, τy) = 10−3ρa(0.142U + 0.076U2 + 2.7)(ua, va), (19)

where ρa = 1.25 kg m−3 is the air density, U is the wind speed, and ua, va are zonal and238

meridional wind components. Then, we assumed, based on the data, that the auto-correlation239

function of the wind stress components follows:240

〈τx,y(t)τx,y(t+ t)〉 =1

2τx,y0 cos(ω0t) +

1

2τx,y1e

−γx,y |t|, (20)

where ω0 is the diurnal frequency. [We also estimated the KE based on the power spectrum of241

the wind stress and by directly numerically integrating Eq. (8) and obtained results that are242

very similar to the results described below. The advantage of using the present approach is243

the ability to estimate the contribution of the different components of the wind stress, namely244

the periodic and stochastic components of the wind stress (τx,y0 and τx,y1 respectively), and245

verify their relative importance.] Both τx,y0 and τx,y1 are calculated form the data: we first246

estimated τx,y0 from the FT of τx,y, then found τx,y1 from Eq. (20) at lag t = 0, then estimated247

the exponents γx,y from the auto-correlation function of the first four lags.248

The estimated exponents, γx,y, are shown in Fig. 4, and the two exhibit similar patterns.249

Generally speaking, the low latitudes (30◦S to 30◦N) are more correlated than the higher250

latitudes that are weakly correlated in regions of enhanced storm activity (e.g., the storm251

13

Figure 4: Temporal correlation exponents, (a) γx and (b) γy, in s−1, estimated based on the NCEP reanalysis2 surface (10 m) winds.

tracks and high mountains). The mean values of the exponents over the ocean are very similar252

to the mean value over the land and are 〈γx〉 = 10−5 s−1, 〈γy〉 = 1.2× 10−5 s−1.253

The different components of the wind stress are shown in Fig. 5. The correlated noise254

component, τ1 (Fig. 5a), is high in the regions of enhanced wind activity, such as the storm255

track and high mountains, and it roughly matches the regions of high correlation exponents256

γx,y shown in Fig. 4. τ1 is larger than the periodic wind component τ0 (Fig. 5b) by more257

than an order of magnitude. The periodic component is often large in regions of a large noise258

component, indicating that the periodicity is, in part, due to higher wind energy (power) in259

these regions. The mean wind-stress component, τm, is high only in the Southern Ocean and260

along the edge of Antarctica. The global mean values of τ1, τ0, τm over the ocean is 0.26,261

6.6 × 10−3, and 0.075 N m−2, respectively, and are very similar to the wind-stress values over262

14

Table 1: Summary of the different components of KE for NCEP-DOE and ERA-INTERIM reanalysis (in Joules).

NCEP-DOE ERA-INTERIMglobal mean deep ocean mean global mean deep ocean mean

Ek,1 7.27× 1017 2.49× 1017 2.66× 1017 9.16× 1016

Ek,0 1.4× 1017 6.04× 1016 5.23× 1016 2.51× 1016

Ek,m 6.44× 1015 2.4× 1012 3.06× 1015 1.31× 1011

land.263

Given the wind-stress components, τx,y0,1,m, it is possible to calculate the KE that is asso-264

ciated with each of the components, based on the ocean depth at each grid point and based265

on Eqs. (7), (9). The KE of the entire water column is shown in Fig. 6. The KE associated266

with temporal correlation, Ek,1, is much larger than the other two components associated with267

the diurnal periodicity, Ek,0, and the mean wind stress, Ek,m. The global mean of the three268

components of the KE are listed in Table 1. [Analysis of the ERA-Interim (Berrisford et al.,269

2011) 6 hourly wind data yielded 2-3 times smaller KE; see Table 1.] While Ek,1 is significant270

in vast ocean regions, Ek,0 is very strong, as expected, around latitude 30◦. The estimated271

KE is obviously not realistic as it ignores all spatial dynamics and is about 1/5 of the realistic272

estimation of the global ocean KE ∼ 5.2× 1018J (vonStorch et al., 2012). The most important273

parameter is the viscosity coefficient, which was chosen to be 0.1 m2 s−1. We note that a smaller274

viscosity coefficient like 0.01 m2 s−1 (Cushman-Roisin, 1994) or even smaller (Gill, 1982) would275

yield much larger KE and the KE is inversely proportional of the viscosity coefficient; see Eq.276

A.15. It is reasonable that ν is much smaller at a depth where the eddy activity is weaker277

(similar to the effect of weak stratification on the diffusion coefficient, (Gargett, 1984)) such278

that the resultant currents will be stronger, leading to a larger KE.279

We have also calculated the KE of the deep ocean (deeper than 1 km) and found that280

a significant part is stored in the deep ocean. Fig. 7 depicts the ratio, r, between the deep281

ocean KE and the entire water column KE and, consistent with our predictions, the stochastic282

(Fig. 7a) and periodic (Fig. 7b) components of the wind stress account for tens of percentage283

points of the total KE. The deep ocean KE that resulted from the mean forcing (Fig. 7c) is,284

as expected, negligible; see Table 1. The ratio of global mean KE to the global mean deep285

ocean KE of the different components is r1 = 34%, r0 = 44% and rm = 0.04% indicating that286

indeed significant part of the KE is stored in the deep ocean; see Table 1. The ratio captures287

the pattern of the deep ridges as the KE depends on ocean depth H.288

The above calculations were based on the assumption that the ocean is strongly stratified.289

However, the deep ocean is only weakly stratified, such that one expects weaker currents and KE290

15

Figure 5: Wind-stress coefficient (in N m−2) of the (a) stochastic, τ1 =√τ2x,1 + τ2y,1, (b) periodic, τ0 =

√τ2x,0 + τ2y,0, and (c) mean, τm =

√〈τx〉2 + 〈τy〉2, components of the wind stress. Estimated based on the

NCEP reanalysis 2 surface (10 m) winds.

16

Figure 6: KE wind-stress coefficient (in N m−2) of the (a) stochastic, τ1, (b) periodic, τ0, and (c) mean, τm,components of the wind stress. Estimated based on the NCEP reanalysis 2 surface (10 m) winds.

17

Figure 7: Ratio between the KE of the deep ocean (deeper than 1 km) and the entire water column KE forthe (a) stochastic, (b) periodic, and (c) mean components of the wind stress. Estimated based on the NCEPreanalysis 2 surface (10 m) winds and the numerical integration of Eq. (8).

18

(see Fig. 2). The MITgcm simulations with an open boundary restoration of the temperature291

and salinity of the global and temporal mean of the Levitus atlas (Levitus, 1982) resulted in292

a linearly decreasing current speed in the few upper kms of the ocean and an almost uniform293

current speed in the abyssal ocean. However, the eddy viscosity coefficient may be smaller294

under such conditions due to the weaker eddy activity, probably leading to stronger currents295

and, hence, larger KE. Thus, the values given above only very roughly estimate the ocean’s296

KE.297

Following the above, our calculations indicate that the contribution of the stochastic wind298

stress is relevant in vast ocean regions and is much larger than the periodic wind stress compo-299

nent that is restricted to the 30◦ latitude. In addition, 34% of the energy due to the resonance300

of the Coriolis force with the stochastic part of the wind is absorbed in the deep ocean (deeper301

than 1 km). The contribution associated with the mean wind stress is much smaller than302

the contribution due to the stochastic and periodic parts of the wind stress, and is negligible303

in the deep ocean. While these estimations are based on realistic wind stress, they are very304

rough due to the simplistic assumptions of our approach, in particular due to the choice of the305

eddy-parameterized viscosity coefficient, ν, which depends on the eddy activity. In addition,306

we assumed that the eddy viscosity coefficient is constant both in time and space while, in fact,307

this is probably not the case. For example, the viscosity coefficient may drastically increase308

when a storm frequency matches the inertial frequency and the mixed layer deepens (Pollard309

et al., 1973). Moreover, dispersion, due to the variations in the Coriolis force (Anderson and310

Gill, 1979; D’Asaro, 1989) or other dispersion effects (Gill, 1984), also makes our estimation311

less accurate. In addition, it was shown that (a) stochastic winds in the Southern Ocean do312

not directly energize the deep ocean and that it is the work associated with the pressure field313

(Weijer and Gille, 2005) and (b) the wind energy input to the deep ocean may be drastically314

decreased due to dissipation processes and conversion to potential energy in the upper ocean315

(Furuichi et al., 2008; Zhai et al., 2009). Yet, our results indicate the penetration of wind316

energy to the deep ocean, even if the viscosity coefficient is large, and highlight the possible317

importance of the wind’s temporal variability on the transformation of wind KE to the deep318

ocean.319

7. Summary and conclusions320

In summary, we studied the role of the wind’s variability on deep ocean currents. We first321

showed, using the MITgcm, that when the wind is periodic with a frequency that matches322

the Coriolis frequency and when starting from rest, turbulent flow develops that eventually323

19

leads to the mixing of the entire water column and to a constant interior ocean current speed.324

Under stratification conditions imposed at the sides of the domain, the current speed decreases325

linearly with depth z and vanishes only at the bottom of the ocean. In all cases, the current326

field is almost laterally uniform, justifying the omission of the advection and lateral viscosity327

terms. We then provided an analytic solution for the currents under the action of periodic328

wind stress and found that the current speed decreases linearly with depth z and only vanishes329

at the ocean floor. The surface current speed depends linearly on the ocean depth, and the330

KE increases as a function of the cube of the ocean depth H. Under more realistic stochastic331

and temporally correlated wind stress, one of the wind’s frequencies resonates with the Coriolis332

force, causing the currents to reach the bottom of the ocean. We show, numerically and via333

analytic approximations, that the current speed in the deep ocean decreases linearly with depth334

z, that the second moment of the surface current increases logarithmically with the ocean depth335

H, and that the KE increases linearly as a function of ocean depth H. Thus, when considering336

the infinite depth Ekman layer model, an additional friction term must be included to avoid337

the singularity of the ocean currents (Kim et al., 2014; Ashkenazy et al., 2015).338

Under realistic global ocean bathymetry and realistic wind stress forcing, the contribution339

of the wind stress’s stochastic component is much larger than the contribution of the wind340

stress’s periodic component and the (temporal) mean wind stress, which is negligible. The341

stochastic component of the wind stress is significant in extended ocean areas, unlike the pe-342

riodic component which is significant only at 30◦ latitude, where the Coriolis force resonates343

with the diurnal winds. Moreover, around one-third of the KE is stored in ocean depths deeper344

than 1 km. While this is a very rough estimate due to the simplicity of the model, it indicates345

that the stochastic nature of the wind plays an important role in the energy budget of the deep346

ocean.347

8. Acknowledgment348

We thank Golan Bel, Georgy Burde, Hezi Gildor, Avi Gozolichiani, and Eli Tziperman for349

helpful discussions.350

Appendix A. Temporally correlated stochastic wind stress351

Given the FT of the of temporally correlated wind stress, Eq. (18), it is possible to find the352

FT of the second moment of the currents, using Eq. (7). The second moment of the currents353

is obtained by using the Parseval’s relation, Eq. (9). Eq. (7) is singular when ω = −f and the354

20

integral (9) around this point can be approximated as355

I2 =τ 20 (H + z)2

2πρ20ν2

ω2∫

ω1

γ

γ2 + ω2dω. (A.1)

Far from the point of singularity, Eq. (7) can be approximated as356

|q|2 ≈ |τ |2

ρ20ν

e2z/d

|f + ω| ≈|τ |2ρ20ν

1

|f + ω| . (A.2)

Then, when ω � −f , integral (9) can be approximated as357

I1 = − τ 202πρ20ν

ω1∫

ωl

γ

γ2 + ω2

1

f + ωdω, (A.3)

and when ω � −f , it can be approximated as358

I3 =τ 20

2πρ20ν

ωr∫

ω2

γ

γ2 + ω2

1

f + ωdω. (A.4)

The limits ω1,2 of I1,2,3 connect the different regions of the integral and are found based on the359

numerical results:360

ω1,2 = ∓ πνH2− f. (A.5)

The limits ωl,r are found such that the linear approximation of the exponent in Eq. (A.2) is361

zero,362

ωl,r = ∓ ν

2z2− f. (A.6)

Given the above limits, I2 is found to be363

I2 =τ 20 (H + z)2

2πρ20ν2

[tan−1

(πν

γH2− f

γ

)+ tan−1

(πν

γH2+f

γ

)]. (A.7)

I2 can be further approximated as364

I2 ≈τ 20 (H + z)2

ρ20ν

γ

f 2 + γ2. (A.8)

21

We find that I2 reproduces fairly well the second moment of the current speed close to the365

bottom.366

Close to the surface of the ocean, the second moment of the currents can be approximated367

by I1 + I3, which can be expressed, given the limits ωl,1,2,r, as follows:368

I1 =τ 20

4π(γ2 + f 2)ρ20ν[2f

(tan−1

(πν

γH2+f

γ

)− tan−1

(ν

2γz2+f

γ

))+

γ lnγ2H4 + (πν +H2f)2

4π2γ2z4 + (ν + 2fz2)2

](A.9)

I3 =τ 20

4π(γ2 + f 2)ρ20ν[2f

(tan−1

(ν

2γz2− f

γ

)− tan−1

(πν

γH2− f

γ

))+

γ lnγ2H4 + (πν −H2f)2

4π2γ2z4 + (ν − 2fz2)2

]. (A.10)

Thus, following Eq. (A.8), close to the bottom, the current speed decreases linearly with depth,369

z. A simple analysis of I1 + I3 of Eqs. (A.9),(A.10) indicates that close to the surface of the370

ocean, the decrease of current speed with depth z is approximately logarithmic.371

Based on I1 + I3 of Eqs. (A.9),(A.10), we approximate the second moment of the surface372

current (z = 0) as373

|q|2 ≈ τ 20π(γ2 + f 2)ρ20ν

[f tan−1

(f

γ

)+

γ ln

(√γ2 + f 2

ν

)+ 2γ lnH

]. (A.11)

Thus, |q|2 grows logarithmically with the ocean depth, H.374

It is possible to approximate the total KE in a way similar to the above treatment. Based375

on Eq. (8), the FT of the KE close to the singularity point (ω = −f) is approximated as376

Ek ≈|τ |2H3

6ρ20ν2, (A.12)

22

and given Eqs. (18), (9), the KE is then377

Ek ≈τ 20H

3

12πρ20ν2

ω2∫

ω1

γ

ω2 + γ2dω =

τ 20H3

12πρ20ν2

[tan−1

(3πν

2γH2− f

γ

)+ tan−1

(3πν

2γH2+f

γ

)], (A.13)

where the limits ω1,2 are378

ω1,2 = ∓ 3πν

2H2− f. (A.14)

As the depth of the ocean is usually much larger than the depth of the Ekman layer (i.e.,379

H2 � 3π4

2νf

), the KE can be further simplified,380

Ek ≈γτ 20

4ρ20ν(γ2 + f 2)H. (A.15)

Thus, when the wind stress is stochastic with exponentially decaying correlations, the KE381

increases linearly with the depth of the ocean H.382

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