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A note on ocean surface drift with application to surface velocities measured with HF Radar

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Abstract

The ocean drift current consists of a (local) pure drift current generated by the interaction of wind and waves at the sea surface, to which the surface geostrophic current is added vectorially. We present (a) a similarity solution for the wave boundary layer (which has been validated through the prediction of the 10-m drag law), from which the component of pure drift current along the direction of the wind (and hence the speed factor) can be evaluated from the 10-m wind speed and the peak wave period, and (b) a similarity solution for the Ekman layers of the two fluids, which shows that under steady-state neutral conditions the pure drift current lies along the direction of the geostrophic wind, and has a magnitude 0.034 that of the geostrophic wind speed. The co-existence of these two similarity solutions indicates that the frictional properties of the coupled air-sea system are easily evaluated functions of the 10-m wind speed and the peak wave period, and also leads to a simple expression for the angle of deflection of the pure drift current to the 10 m wind. The analysis provides a dynamical model for global ocean drift on monthly and annual time scales for which the steady-state neutral model is a good approximation. In particular, the theoretical results appear to be able to successfully predict the mean surface drift measured by HF Radar, which at present is the best technique for studying the near surface velocity profile.

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Notes

  1. The exact expressions for the components of mean velocity in the Ekman layer are, <u3> = u2 exp(−z′)sin(z′)/(z′) and <v 3> = v 2 [1 − exp(−z′) cos(z′)]/(z′), where z′ = z 3/z E. For z′ → 0, <u3> → u2 and <v3> → v2, and over the complete extent of the Ekman layer (z′ → ∞), the components of Ekman transport (U E = <u 3>z 3) are UE = 0, and from (37), V E = − (R − 1)w* z E/\( K_{\text{I}}^{ 1/ 2} \), which, on eliminating z E yields the classical expression, V E = −w 2* /f.

    The velocity profiles in the Ekman layer are shown in Chereskin (1995) and also discussed in Jenkins and Bye (2006).

Abbreviations

u(z):

Air velocity at height z

u(−z):

Water velocity at depth − z

oz :

Vertically upward axis, undisturbed sea surface, z = 0

ρ 1 :

Density of air

ρ 2 :

Density of water, ε = (ρ 1/ρ 2)1/2

τ s :

Surface shear stress

u * :

(u*, 0) friction velocity in air

w * :

(w*, 0) friction velocity in water

u L :

Wave induced velocity in air (the spectrally weighted phase velocity)

ε u L :

Wave induced velocity in water (the surface Stokes velocity), which occur at |z| = z R

u o :

Reference velocity for the air-sea boundary layer

u 1 :

u(z B) surface wind

u 2 :

u(−z B) surface current

z B :

gT 2/8π 2

T :

Peak wave period

g :

Acceleration of gravity

u S :

Surface drift velocity

u10 :

10 m wind speed, z 10 = 10 m

κ :

von Karman’s constant

K :

K(|z|) drag coefficient at |z| in both fluids

K I :

¼ K(z B) inertial drag coefficient

R :

Frictional parameter in the wave boundary layer, see Fig. 1

Fig. 1
figure 1

The velocity structure in the wave boundary layer reproduced from Fig. 1 of Bye and Wolff (2008). The definitions of the variables are summarized in the List of symbols

Full size image
k 1 :

1/2z R spectral high wave number cut-off

k o :

1/2z B wave number of the peak wave

a o :

Amplitude of the breaking wave

Q 1 and Q 2 :

Ekman layer frictional coefficients in air and water respectively

υ 1 and υ 2 :

Dynamic eddy viscosities in air and water respectively

f > 0:

Coriolis parameter

G :

Coefficient of non-dimensional eddy viscosity in the Ekman layers

u g :

Surface geostrophic wind

α = ½/(R − 1):

Non-dimensional frictional parameter in the Ekman layers

K o :

Geostrophic drag coefficient

γ :

Angle of turning of the surface geostrophic wind to the left hand side of the surface shear stress in the northern hemisphere

z s :

Roughness length for the velocity profiles at the sea surface

u d :

Mean drift current in the surface drift layer

z HF :

Depth sampled by the HF Radar measurements

z 3 :

Penetration depth of the HF Radar measurements into the Ekman layer

z E :

Ekman depth

φ = z HF/z B :

The proportion of the wave boundary layer sampled by the HF Radar measurements

u HF = (uHF, v HF):

HF Radar mean drift current

γ HF :

Angle of turning of the HF Radar mean drift current to the left hand side of the surface shear stress

[u HF ] :

Total HF Radar mean drift current

[γHF]:

Angle of turning of the total HF Radar mean drift current to the left hand side of the surface shear stress

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Acknowledgements

JATB gratefully acknowledges the award of a Fellowship at the Hanse-Wissenschaftskolleg, Delmenhorst, Germany in the first half of 2013 during which this study was carried out in the Institute for Chemistry and Biology of the Sea (ICBM), University of Oldenburg. Informed comments by Professor Yutaka Yoshikawa and two other Reviewers are also especially acknowledged.

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Authors and Affiliations

  1. School of Earth Sciences, The University of Melbourne, Victoria, 3010, Australia

    John A. T. Bye

  2. ICBM, Carl-von-Ossietzky Universität Oldenburg, Postfach 2053, 26111, Oldenburg, Germany

    Jörg-Olaf Wolff & Karsten A. Lettmann

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  1. John A. T. Bye
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  2. Jörg-Olaf Wolff
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  3. Karsten A. Lettmann
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Correspondence to John A. T. Bye.

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Bye, J.A.T., Wolff, JO. & Lettmann, K.A. A note on ocean surface drift with application to surface velocities measured with HF Radar. J Oceanogr 73, 491–502 (2017). https://doi.org/10.1007/s10872-017-0417-1

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