**Research article**
18 Sep 2020

**Research article** | 18 Sep 2020

# The Ekman spiral for piecewise-uniform viscosity

David G. Dritschel Nathan Paldor and Adrian Constantin

^{1,},

^{2,},

^{3,}

**David G. Dritschel et al.**David G. Dritschel Nathan Paldor and Adrian Constantin

^{1,},

^{2,},

^{3,}

^{1}School of Mathematics and Statistics, University of St Andrews, St Andrews KY16 9SS, UK^{2}The Fredy & Nadine Herrmann Institute of Earth Sciences, The Hebrew University, Jerusalem 9190401, Israel^{3}Department of Mathematics, University of Vienna, Vienna 1090, Austria^{}These authors contributed equally to this work.

^{1}School of Mathematics and Statistics, University of St Andrews, St Andrews KY16 9SS, UK^{2}The Fredy & Nadine Herrmann Institute of Earth Sciences, The Hebrew University, Jerusalem 9190401, Israel^{3}Department of Mathematics, University of Vienna, Vienna 1090, Austria^{}These authors contributed equally to this work.

**Correspondence**: David G. Dritschel (david.dritschel@st-andrews.ac.uk)

**Correspondence**: David G. Dritschel (david.dritschel@st-andrews.ac.uk)

Received: 12 Apr 2020 – Discussion started: 27 May 2020 – Revised: 23 Jul 2020 – Accepted: 10 Aug 2020 – Published: 18 Sep 2020

We re-visit Ekman's (1905) classic problem of wind-stress-induced
ocean currents to help interpret observed deviations from Ekman's
theory, in particular from the predicted surface current deflection
of 45^{∘}. While previous studies have shown that such
deviations can be explained by a vertical eddy viscosity varying
with depth, as opposed to the constant profile taken by Ekman,
analytical progress has been impeded by the difficulty in solving
Ekman's equation. Herein, we present a solution for
piecewise-constant eddy viscosity which enables a comprehensive
understanding of how the surface deflection angle depends on the
vertical profile of eddy viscosity. For two layers, the
dimensionless problem depends only on the depth of the upper layer
and the ratio of layer viscosities. A single diagram then allows
one to understand the dependence of the deflection angle on these
two parameters.

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 Ekman (1905) and constitutes a cornerstone of oceanography (see Vallis, 2017). Ekman dynamics is due to the balance between Coriolis and the frictional forces generated by the wind stress. Its main features, consistent with observations of steady wind-driven ocean currents, are the following:

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

- ii.
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.

- iii.
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.

While near the Equator wind-drift currents move in the same direction
as the wind (see the discussion in Boyd, 2018), away from the Equator
a deflection of steady wind-driven currents with respect to the
prevailing wind direction occurs in a surface boundary layer, whose
typical depth is tens of metres. Ekman's pioneering solution
(see Ekman, 1905), derived for a constant vertical eddy viscosity, captures
the general qualitative behaviour, but differences of detail between
observations and Ekman theory were recorded in the last decades. While
the characteristics (ii)–(iii) hold for any depth-dependent vertical
eddy viscosity (see Constantin, 2020), there is a need to explain the
occurrence of surface currents at an angle in the range 10–75^{∘}
to the wind (rather than the 45^{∘} predicted by Ekman), with
large variations depending on the regional and seasonal climate
(see the data in Röhrs and Christensen, 2015; Yoshikawa and Masuda, 2009).

This discrepancy is typically ascribed to the effect of a vertical
eddy viscosity that varies with depth. The explicit solution found by
Madsen (1977), for a vertical eddy viscosity that varies linearly with
depth, leads to a plausible, although somewhat low, surface current
deflection angle of about 10^{∘}. The avenue of seeking explicit
solutions is not very promising, since only a few are available and
the intricacy of the details makes it difficult to extract broad
conclusions (we refer to Constantin and Johnson, 2019; Grisogno, 1995, for a survey of known
Ekman-type solutions). The challenging nature of the task is
highlighted by the recent analysis pursued in Bressan and Constantin (2019) and Constantin (2020) where
asymptotic approaches, applicable for eddy viscosities that are
small perturbations of a constant, revealed the convoluted way in which
the eddy viscosity influences the deflection angle: while a slow and
gradual variation of the eddy viscosity with depth results in a
deflection angle larger than 45^{∘}, the typical outcome of an
eddy viscosity concentrated in the middle of the boundary layer is a
deflection angle below 45^{∘}. A better understanding of the
deflection angle is important theoretically but also for operational
oceanography, e.g. in the context of search-and-rescue operations or
in remedial action for oil spills.

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^{∘}.
Various special or limiting cases are highlighted. Finally, Sect. 4
offers our conclusions.

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

where $\mathit{U}\left(Z\right)=U+iV$ is the complex horizontal velocity in
the (*X*,*Y*) plane, *Z* is the depth below the mean surface *Z*=0, *f*
is the Coriolis parameter, *ρ* is the (constant) density,
$\mathrm{\nabla}P=\partial P/\partial X+\mathrm{i}\phantom{\rule{0.125em}{0ex}}\partial P/\partial Y$ is
the horizontal pressure gradient, $\mathit{\tau}\left(Z\right)={\mathit{\tau}}_{x}+i{\mathit{\tau}}_{y}$
is the shear stress due to molecular and turbulent processes, and the
higher-order terms, representing interactions between the variables,
are presumed to be small. Decomposing the horizontal velocity into
pressure-driven (geostrophic) and wind-driven (Ekman) components
$\mathit{U}={\mathit{U}}_{\mathrm{g}}+{\mathit{U}}_{\mathrm{e}}$, we see from Eq. (1) that
the leading-order geostrophic and wind-driven flows separate,
with the linear equation

governing the dynamics of the wind-driven flow. By relating the stress
vector within the fluid, ** τ**, to the shear profile through a
turbulent eddy viscosity coefficient

*ν*(

*Z*),

from Eq. (2) we obtain Ekman's equations for wind-driven ocean currents

Let us now discuss the appropriate boundary conditions. At the
surface, the shear stress balances the wind stress, *τ*_{0}:

The “bottom” boundary condition expresses the vanishing of the wind-driven current with depth (necessary to keep the total kinetic energy finite), where the flow is essentially geostrophic:

Letting *τ*_{0} denote the magnitude of the surface wind stress, we
non-dimensionalize the problem by scaling *U*_{e} by
$\sqrt{\mathrm{2}{\mathit{\tau}}_{\mathrm{0}}/\mathit{\rho}}$ and *Z* by $\sqrt{\mathrm{2}{\mathit{\tau}}_{\mathrm{0}}/\mathit{\rho}}/f$, since *τ*_{0}∕*ρ*
has units of *L*^{2}∕*T*^{2}. The factor of 2 is introduced for
convenience below. Upon defining a dimensionless eddy viscosity
$K=f\mathit{\rho}\mathit{\nu}/{\mathit{\tau}}_{\mathrm{0}}$, velocity $\mathit{u}={\mathit{U}}_{\mathrm{e}}/\sqrt{\mathrm{2}{\mathit{\tau}}_{\mathrm{0}}/\mathit{\rho}}$
and depth $z=Zf/\sqrt{\mathrm{2}{\mathit{\tau}}_{\mathrm{0}}/\mathit{\rho}}$, the equations transform to

where *ψ*=*u**K*(0)
and a prime means a derivative with respect to *z*
(cf. Eqs. 14–16 in Gill, 1982).
The scaling performed does not change the surface deflection angle
*θ*_{0}, equal to the argument of the complex vector *ψ*(0),
even if the scaling results in an orientation of the horizontal axes
such that the surface wind stress points in the positive
*x* direction. Finally, we note that this formulation is appropriate
for the Northern Hemisphere where *f*>0. The formulation for the
Southern Hemisphere is obtained by taking the complex conjugate in
Eq. (7), noticing that *K* is real-valued.

For piecewise-constant *K*, without loss of generality we can further
scale *z* so that *K*=1 in $z\in [-h,\mathrm{0}]$ while *K*=ℓ^{2} in
$z\in (-\mathrm{\infty},-h)$, where *h* is the dimensionless depth of the upper
layer. Note that ℓ is the ratio of the lower-layer to upper-layer
viscous lengths. The analysis below can be readily extended to any
number of regions of constant *K*, but the simplest to understand is
two regions, since then the solution depends on only two dimensionless
parameters, ℓ and *h*.

## 3.1 Constructing the solution

In each region, the complex velocity *ψ* satisfies a simple
constant-coefficient equation:

having exponential solutions

where *A*, *B* and *C* are (generally complex) constants. The boundary
condition *ψ*→0 as $z\to -\mathrm{\infty}$ has been used to eliminate the
growing solution in Eq. (13).

At the discontinuity in *K*, at $z=-h$, we require continuity of *ψ*,
i.e. $\mathit{\psi}(-{h}^{+})=\mathit{\psi}(-{h}^{-})$. Moreover, by integrating the equation
above across an infinitesimal region centred on $z=-h$, we obtain

The upper surface boundary condition ${\mathit{\psi}}^{\prime}\left(\mathrm{0}\right)=\mathrm{1}$ implies

while continuity of *ψ* at $z=-h$ implies

and finally the jump condition (Eq. 14) on *ψ*^{′} at $z=-h$ implies

It follows that

Applying the surface boundary condition (Eq. 15) determines *C* as

The surface current deflection angle, *θ*_{0},
*measured clockwise*, is determined from

But given *C* above in Eq. (19), we have

Introducing the real values $\mathit{\alpha}=(\mathrm{1}+\mathrm{\ell})\phantom{\rule{0.125em}{0ex}}{e}^{h}$ and $\mathit{\beta}=(\mathrm{1}-\mathrm{\ell})\phantom{\rule{0.125em}{0ex}}{e}^{-h}$ enables us to write

which, after multiplying top and bottom by the complex conjugate of the denominator, simplifies to

Hence, taking the (negative of the) ratio of the imaginary to real parts of this, we obtain

## 3.2 Results

First, we examine certain special cases.

When ℓ=1, there is no discontinuity in eddy viscosity. Since in
this case *β*=0, we have tan *θ*_{0}=1,
i.e. ${\mathit{\theta}}_{\mathrm{0}}={\mathrm{45}}^{\circ}$ in agreement with the classical Ekman spiral
solution.

As ℓ→0, the eddy viscosity vanishes in the lower layer, and the
flow field *ψ* must also vanish. In this case, tan *θ*_{0}
reduces to

which has a non-trivial dependence on *h*. The maximum value
is attained as *h*→0; then tan *θ*_{0}→∞ or
*θ*_{0}→90^{∘}.

As ℓ→∞, corresponding to an extremely viscous lower
layer, tan *θ*_{0} reduces to the inverse of the previous expression,
i.e.

The minimum occurs for *h*→0 and there tan *θ*_{0}→0 or
*θ*_{0}→0.

For general ℓ, there are also values of *h* for which
tan *θ*_{0}=1. These occur when the numerator and the denominator
of the general expression above for tan *θ*_{0} are equal. But
this means *α**β*sin (2*h*)=0 or $(\mathrm{1}-{\mathrm{\ell}}^{\mathrm{2}})\mathrm{sin}\left(\mathrm{2}h\right)=\mathrm{0}$. One
solution is the classical Ekman spiral with ℓ=1 noted above. But
we also have $h=n\mathit{\pi}/\mathrm{2}$ for non-negative integers *n*. When *n*=0,
the upper layer vanishes and the eddy viscosity is uniform throughout the
entire depth. The classical Ekman spiral is expected in this case.
The other special depths imply *θ*_{0} exhibits a non-monotonic
dependence on *h* for fixed ℓ. In fact, tan *θ*_{0} exhibits a
decaying oscillation about a value of unity.

A summary of the results in the ℓ–*h* plane is provided
in Fig. 1.
Along any line ℓ = constant (excluding ℓ=1), *θ*_{0} reaches
a minimum or maximum in *h* when the following relation holds:

obtained by setting the partial derivative of tan *θ*_{0} with respect to *h*
equal to zero. The first extremum with increasing *h* occurs for $h<\mathit{\pi}/\mathrm{8}$
(when $h=\mathit{\pi}/\mathrm{8}$ the above equation yields ℓ=1). Note that as *h*→0,
$(\mathrm{1}-\mathrm{\ell})/(\mathrm{1}+\mathrm{\ell})\to \pm \mathrm{1}$, implying either ℓ→0 or ℓ→∞
as noted previously. Extrema also occur for larger *h* since the function
in the square root above is periodic, but these involve much weaker variations
in *θ*_{0} about 45^{∘}, diminishing like *e*^{−nπ} for positive
integers *n*. When *n*=1, the maximum excursion in tan *θ*_{0} is
approximately 0.05735.

We have re-visited the famous problem originally posed by Nansen
(see the discussion in Huntford, 2002) and solved by Ekman (1905) to
understand wind-driven currents in the ocean. By balancing viscous
and Coriolis forces, and assuming a constant vertical eddy viscosity,
Ekman (1905) predicted that the surface current is deflected by
45^{∘} to the right and left of the prevailing wind direction in the
Northern Hemisphere and Southern Hemisphere, respectively. Moreover, Ekman (1905) found that the net
fluid transport is 90^{∘} to the right and left of the wind direction.

Since then, a number of studies have sought to explain observed
discrepancies with Ekman's theory (Röhrs and Christensen, 2015; Yoshikawa and Masuda, 2009), in particular
deflection angles significantly different from the 45^{∘}
prediction (Madsen, 1977; Grisogno, 1995; Bressan and Constantin, 2019; Constantin and Johnson, 2019; Constantin, 2020). The main conclusion is that
these discrepancies can be explained by vertically varying eddy
viscosities. However, due to the mathematical difficulty in
constructing exact or asymptotic solutions, no general scenario has yet
emerged relating the deflection angle to the profile of eddy
viscosity.

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 *ρ*_{1} overlying a denser deep layer of
density *ρ*_{2}. In that case the lower-layer dimensionless viscosity
ℓ^{2} includes the density ratio *ρ*_{1}∕*ρ*_{2}.) By an
appropriate scaling of the governing equations, the solutions can be
shown to depend on only two parameters: the ratio of the
lower-to-upper viscous lengths ℓ and the dimensionless depth of
the upper layer *h*. This permits one to see at a glance how both
ℓ and *h* determine the surface deflection angle *θ*_{0}.

In appropriate limits, we recover Ekman's classical solution, but
additionally the 45^{∘} deflection angle may *also* occur
for arbitrary ℓ, when *h* assumes special values. In general, for
*h* sufficiently small and ℓ<1 (a less viscous lower layer), the
deflection angle exceeds 45^{∘} (and can reach nearly 90^{∘}
for ℓ≪1). When ℓ>1 (a more viscous lower layer), the
deflection angle is less than 45^{∘} and tends to zero as
ℓ→∞ for *h*≪1.
For ℓ∼1 our conclusions are in agreement with the results
obtained recently in Bressan and Constantin (2019) and Constantin (2020). Indeed, writing $K\left(z\right)={\mathrm{\ell}}^{\mathrm{2}}+\mathit{\epsilon}\phantom{\rule{0.125em}{0ex}}{K}_{\mathrm{1}}\left(z\right)$ for *z*≤0, with $\mathit{\epsilon}=|\mathrm{1}-{\mathrm{\ell}}^{\mathrm{2}}|$ and

the perturbative approach developed in Bressan and Constantin (2019) and Constantin (2020) shows that a positive and negative value of the integral

corresponds to a deflection angle larger and smaller than 45^{∘}. respectively. The relation

shows that this is consistent with our conclusions.

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 (see
the discussion in Woods, 2002). Our model captures these changes:
during the day we set ℓ>1, with ℓ<1 during the night,
thus explaining the observation that often the deflection angle
exceeds 45^{∘} during the day, and is below 45^{∘} during the
night (see Krauss, 1993). The same reasoning applies to the large
seasonal variations of the deflection angle observed at some locations
(see the data in Yoshikawa and Masuda, 2009) and explains why one observes angles
below 45^{∘} in arctic regions, where the ice cover quells the
turbulence near the ocean surface. On the other hand, the regularity
of strong winds in the Drake Passage makes the assumption of a uniform
eddy viscosity reasonable (i.e. ℓ=1) so that in this region
the deflection angle is typically close to 45^{∘}
(see the data in Polton et al., 2013; Roach et al., 2015). We are not aware of detailed
observational studies relating the deflection angle to the vertical
profile of eddy viscosity, but we hope that our work will serve
as a guide.

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.

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