Ekman Layer Depth Calculator

Find the depth of the wind-driven Ekman layer, the Coriolis parameter and the inertial period from latitude and eddy viscosity, or from wind speed.

🌀 Ekman Layer Depth Calculator
Latitude (negative = south)
°
-9090
Vertical eddy viscosity (Az), typical 0.01 to 0.1
m²/s
0.0011
Latitude (negative = south)
°
-9090
Wind speed at 10 m (W)
m/s
040
Ekman depth (DE)
—
E-folding depth scale (D = DE/π)
—
Coriolis parameter (f)
—
Inertial period (2π/|f|)
—
Surface current deflection
—
Amplitude at DE (e-π)
—
Eddy viscosity (Az)
—

🌀 What is the Ekman Layer Depth?

The Ekman layer depth is the thickness of the surface layer of the ocean in which the wind-driven current is balanced between turbulent friction and the Coriolis force. When wind blows across the sea it drags the surface water along, and because Earth rotates the moving water is deflected, to the right of the wind in the Northern Hemisphere and to the left in the Southern Hemisphere. Each layer below is dragged by the one above and deflected further, so the current turns with depth and weakens. Plotted from above, the tips of the current vectors trace the Ekman spiral, first described by the Swedish physicist Vagn Walfrid Ekman in 1905 after Fridtjof Nansen noticed that Arctic sea ice drifted at an angle to the wind.

The Ekman depth DE is the depth where the current points opposite to the surface current. At that depth its speed is only e-π, about 4.3 percent, of the surface speed, so it is a practical bottom of the wind-driven layer. Oceanographers use it to size the Ekman transport that drives coastal upwelling, to judge how deep wind stress influences the water, to separate the wind-driven surface layer from the geostrophic interior, and to check whether the mixed layer is thinner or thicker than the layer the wind can organise. Typical results are tens to a few hundred metres.

Two depth scales appear in the literature and they are easy to confuse. The e-folding depth D = sqrt(2 Az / |f|) is where the amplitude has decreased by a factor of e. The Ekman depth DE = π D is where the direction has reversed. This calculator reports DE as the headline number and shows D beside it. Note also that the vertical eddy viscosity Az is not a property of seawater but of the turbulence, so values of 0.01 to 0.1 m²/s are typical yet highly uncertain. The wind-based mode avoids choosing Az by using an empirical relation in the wind speed, and then reports the Az that depth implies.

The calculator also returns the Coriolis parameter, the inertial period and the 45 degree surface deflection, and it plots Ekman depth against latitude with your location marked, next to a diagram of the spiral itself. Latitudes closer than 1 degree to the equator are rejected, because f tends to zero there and the depth becomes infinite.

📐 Formula

DE  =  π √(2Az / |f|)   with   f  =  2Ω sin(φ)
DE = Ekman depth (m), where the current reverses direction; D = DE / π = √(2Az / |f|) is the e-folding depth scale
Az = vertical eddy viscosity (m²/s), typically 0.01 to 0.1 but highly uncertain; default 0.05
f = Coriolis parameter (s-1), Ω = 7.2921159×10-5 rad/s, φ = latitude. The inertial period is 2π / |f|.
Wind-based mode: DE = 7.6 W / √(sin|φ|) metres, with W the wind speed at 10 m in m/s (Ekman 1905, as given by Pond and Pickard 1983). The implied eddy viscosity is Az = |f| DE² / (2π²).
Spiral: u = V0 eπz/DE cos(π/4 + πz/DE), v = V0 eπz/DE sin(π/4 + πz/DE), z negative downward, wind toward +y (mirrored in the Southern Hemisphere). At z = −DE the amplitude is e-π = 0.0432.
Example: Az = 0.05 m²/s at 45° gives f = 1.031×10-4 s-1 and DE = 97.8 m.

📖 How to Use This Calculator

Steps

1
Choose a mode. Pick Eddy viscosity to use DE = π sqrt(2 Az / |f|), or Wind-based estimate to use the empirical DE = 7.6 W / sqrt(sin|φ|).
2
Enter latitude and the driver. Type the latitude (negative for the Southern Hemisphere, at least 1 degree from the equator) and either the vertical eddy viscosity Az or the 10 m wind speed.
3
Read the Ekman depth and spiral. The calculator shows the Ekman depth, the e-folding depth, the Coriolis parameter, the inertial period and the 45 degree surface deflection, with a chart of depth against latitude (your latitude is the dashed line and dot) and a diagram of the spiral.

💡 Example Calculations

Example 1 — Mid-Latitude Ocean

Latitude 45°N, Az = 0.05 m²/s

1
f = 2 × 7.2921159×10-5 × sin 45° = 1.031×10-4 s-1, so the inertial period is 2π / f = 16.92 h
2
D = √(2 × 0.05 / 1.031×10-4) = 31.1 m, and DE = π D
Ekman depth DE = 97.8 m, surface current 45° to the right of the wind
Try this example →

Example 2 — Low Latitude

Latitude 10°N, Az = 0.05 m²/s

1
f = 2 × 7.2921159×10-5 × sin 10° = 2.533×10-5 s-1, inertial period 68.92 h
2
D = √(0.1 / 2.533×10-5) = 62.8 m, then DE = π D
Ekman depth DE = 197.4 m, about twice the mid-latitude value because f is much smaller
Try this example →

Example 3 — High Latitude

Latitude 70°N, Az = 0.05 m²/s

1
f = 2 × 7.2921159×10-5 × sin 70° = 1.370×10-4 s-1, inertial period 12.74 h
2
D = √(0.1 / 1.370×10-4) = 27.0 m, then DE = π D
Ekman depth DE = 84.9 m, the shallowest of the three because f is largest
Try this example →

Example 4 — Wind-Based Estimate

Wind 10 m/s at 40°N

1
sin 40° = 0.6428, so √(sin 40°) = 0.8017. DE = 7.6 × 10 / 0.8017 = 94.8 m
2
With f = 9.375×10-5 s-1 (inertial period 18.62 h), the implied eddy viscosity is Az = f DE² / (2π²) = 0.0427 m²/s, inside the typical range
Ekman depth DE = 94.8 m (e-folding depth 30.2 m)
Try this example →

❓ Frequently Asked Questions

What is the Ekman layer depth?+
The Ekman layer depth D_E is the depth at which the wind-driven current in the Ekman spiral points opposite to the surface current. It equals pi times sqrt(2 A_z / |f|), where A_z is the vertical eddy viscosity and f is the Coriolis parameter. Typical values are tens to a few hundred metres.
What is the formula for Ekman depth?+
D_E = pi sqrt(2 A_z / |f|), with f = 2 Omega sin(latitude) and Omega = 7.2921159e-5 rad/s. For A_z = 0.05 m2/s at 45 degrees latitude, f is 1.031e-4 per second and D_E is about 97.8 m.
What is the difference between D_E and the Ekman depth scale?+
The e-folding depth scale is D = sqrt(2 A_z / |f|), where the current amplitude has dropped by a factor of e. The Ekman depth D_E = pi D is where the current reverses direction and has decayed to e^(-pi), about 4.3 percent. This calculator reports D_E as the headline and D as a second result.
Why is the surface current 45 degrees from the wind?+
In the classical Ekman solution the balance of wind stress, friction and the Coriolis force turns the surface current 45 degrees to the right of the wind in the Northern Hemisphere and to the left in the Southern Hemisphere. The deflection increases with depth, giving the spiral. Observed surface angles are often smaller than 45 degrees.
What eddy viscosity should I use for the ocean?+
Typical textbook values for the upper ocean are 0.01 to 0.1 m2/s, and the default here is 0.05. Real values change with wind speed, wave breaking and stratification by orders of magnitude, so the resulting depth is only an estimate.
How is the wind-based Ekman depth calculated?+
The empirical form D_E = 7.6 W / sqrt(sin|phi|) metres, with W the 10 m wind speed in m/s, goes back to Ekman (1905) and is quoted by Pond and Pickard (1983). A 10 m/s wind at 40 degrees gives about 94.8 m.
What is the Coriolis parameter and how is it calculated?+
The Coriolis parameter f = 2 Omega sin(latitude) is twice the local vertical component of Earth's rotation. It is about 1.03e-4 per second at 45 degrees, 1.46e-4 per second at the poles and zero at the equator.
What is the inertial period?+
The inertial period is 2 pi / |f|, the time a free current takes to complete an inertial circle. It is about 16.9 hours at 45 degrees, about 12 hours at the poles and tends to infinity near the equator.
Why does the calculator reject latitudes near the equator?+
At latitude 0, f is zero and the formula gives an infinite Ekman depth, so the theory does not apply. The calculator shows an error for any latitude with an absolute value below 1 degree.
Does the Ekman spiral actually occur in the ocean?+
Fully developed spirals are rarely seen because the wind changes, the water is stratified and eddy viscosity varies with depth. Observations of drifters and current meters show a clear turn to the right of the wind and a net transport 90 degrees from the wind, but the full spiral is usually shallower and flatter.
Does the Ekman depth change in the Southern Hemisphere?+
No. The depth depends on the absolute value of f, so 45 degrees south gives the same D_E as 45 degrees north. Only the turning direction changes: the surface current is deflected 45 degrees to the left of the wind and the spiral rotates the other way.
Tags: Ekman Layer Depth Calculator Ekman Depth Ekman Spiral Coriolis Parameter Calculator Eddy Viscosity Inertial Period Wind Driven Current Calculator Formula Free Online

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What is the Ekman layer depth?

The Ekman layer depth D_E is the depth at which the wind-driven current in the Ekman spiral points opposite to the surface current. It equals pi times sqrt(2 A_z / |f|), where A_z is the vertical eddy viscosity and f is the Coriolis parameter. Typical values are tens to a few hundred metres.

What is the formula for Ekman depth?

D_E = pi sqrt(2 A_z / |f|), with f = 2 Omega sin(latitude) and Omega = 7.2921159e-5 rad/s. For A_z = 0.05 m2/s at 45 degrees latitude, f is 1.031e-4 per second and D_E is about 97.8 m.

What is the difference between D_E and the Ekman depth scale?

The e-folding depth scale is D = sqrt(2 A_z / |f|), where the current amplitude has dropped by a factor of e. The Ekman depth D_E = pi D is where the current reverses direction and has decayed to e^(-pi), about 4.3 percent. This calculator reports D_E as the headline and D as a second result.

Why is the surface current 45 degrees from the wind?

In the classical Ekman solution the balance of wind stress, friction and the Coriolis force turns the surface current 45 degrees to the right of the wind in the Northern Hemisphere and to the left in the Southern Hemisphere. The deflection increases with depth, giving the spiral. Observed surface angles are often smaller than 45 degrees.

What eddy viscosity should I use for the ocean?

Typical textbook values for the upper ocean are 0.01 to 0.1 m2/s, and the default here is 0.05. Real values change with wind speed, wave breaking and stratification by orders of magnitude, so the resulting depth is only an estimate.

How is the wind-based Ekman depth calculated?

The empirical form D_E = 7.6 W / sqrt(sin|phi|) metres, with W the 10 m wind speed in m/s, goes back to Ekman (1905) and is quoted by Pond and Pickard (1983). A 10 m/s wind at 40 degrees gives about 94.8 m.

What is the Coriolis parameter and how is it calculated?

The Coriolis parameter f = 2 Omega sin(latitude) is twice the local vertical component of Earth's rotation. It is about 1.03e-4 per second at 45 degrees, 1.46e-4 per second at the poles and zero at the equator.

What is the inertial period?

The inertial period is 2 pi / |f|, the time a free current takes to complete an inertial circle. It is about 16.9 hours at 45 degrees, about 12 hours at the poles and tends to infinity near the equator.

Why does the calculator reject latitudes near the equator?

At latitude 0, f is zero and the formula gives an infinite Ekman depth, so the theory does not apply. The calculator shows an error for any latitude with an absolute value below 1 degree.

Does the Ekman spiral actually occur in the ocean?

Fully developed spirals are rarely seen because the wind changes, the water is stratified and eddy viscosity varies with depth. Observations of drifters and current meters show a clear turn to the right of the wind and a net transport 90 degrees from the wind, but the full spiral is usually shallower and flatter.

📌 Quick Tips

💡Vertical eddy viscosity A_z is the least certain input. Values from 0.01 to 0.1 m2/s are typical for the upper ocean, but measured values range far wider, so treat the depth as an order of magnitude estimate.
💡Use the wind-based mode when you know the wind speed but not A_z. It also shows the eddy viscosity that the empirical depth implies, which is a useful sanity check against the 0.01 to 0.1 range.
💡Ekman depth grows quickly toward the equator because f goes to zero there. The calculator rejects latitudes closer than 1 degree to the equator, where the Ekman theory is not meaningful.
💡Enter negative latitudes for the Southern Hemisphere. The depth is the same as at the mirror latitude, but the surface current turns 45 degrees to the left of the wind instead of the right.
💡The e-folding depth sqrt(2 A_z / |f|) is the depth scale in the equations, and the Ekman depth D_E is pi times larger. Do not mix the two when comparing with published values.