These authors contributed equally to this work.

We re-visit Ekman's (

The motion of the near-surface ocean layer is a superposition of
waves, wind-driven currents and geostrophic flows. The basic theory of
wind-driven surface currents in the ocean, away from the Equator, is
due to

The surface current is deflected to the right and left of the prevailing wind direction in the Northern Hemisphere and Southern Hemisphere, respectively.

With increasing depth in the boundary layer, the current speed is reduced, and the direction rotates farther away from the wind direction following a spiral.

The net transport is at right angles to the wind direction, to the right and left of the wind direction in the Northern Hemisphere and Southern Hemisphere, respectively.

This discrepancy is typically ascribed to the effect of a vertical
eddy viscosity that varies with depth. The explicit solution found by

The important issue of a quantitative relation between the vertical
eddy viscosity and the magnitude of the deflection angle remains
open. The aim of this paper is to discuss this issue in cases when the
eddy viscosity is piecewise uniform. The in-depth analysis that can be
pursued in this relatively simple setting permits us to gain insight
into the way the turbulent parametrization (e.g. of general
circulation models) controls the deflection angle. This paper is
organized as follows: in Sect. 2 we present the Ekman equations for
wind-driven oceans having depth-dependent eddy viscosities, and we perform
a suitable scaling that reduces the number of parameters. In Sect. 3, an explicit solution is constructed and illustrated for an
infinitely deep ocean with two constant values of eddy viscosity.
This solution covers the full range of possibilities and exhibits
deflection angles covering the full range between 0 and 90

For a deep, vertically homogeneous ocean, of infinite lateral extent,
the horizontal momentum equation for steady flow takes the following
(complex) form under the

Let us now discuss the appropriate boundary conditions. At the
surface, the shear stress balances the wind stress,

Letting

For piecewise-constant

In each region, the complex velocity

At the discontinuity in

The surface current deflection angle,

First, we examine certain special cases.

When

As

As

For general

Surface deflection angle

A summary of the results in the

We have re-visited the famous problem originally posed by Nansen

Since then, a number of studies have sought to explain observed
discrepancies with Ekman's theory

This study makes a first step in this direction by considering the
case of piecewise-constant eddy viscosities for which analytical
solutions may be readily constructed and analysed. We have presented
results for the simplest situation of two regions having different
uniform viscosities in an infinitely deep ocean. (In fact the results
also apply when the two regions have different densities, such as a
mixed layer of density

In appropriate limits, we recover Ekman's classical solution, but
additionally the 45

The results obtained may help better formulate appropriate
parameterizations of eddy viscosities in global circulation models of
the ocean. For example, it is typical for the upper 100 m of the
ocean that solar heating quenches turbulence during the day

The results (in Fig.

All authors contributed equally to this work.

The authors declare that they have no conflict of interest.

The authors would like to thank the three anonymous referees for their helpful comments on our paper.

This research has been supported by the UK Engineering and Physical Sciences Research Council (grant no. EP/H001794/1).

This paper was edited by Neil Wells and reviewed by three anonymous referees.