OSOcean ScienceOSOcean Sci.1812-0792Copernicus PublicationsGöttingen, Germany10.5194/os-16-1089-2020The Ekman spiral for piecewise-uniform viscosityThe Ekman spiral for piecewise-uniform viscosityDritschelDavid G.david.dritschel@st-andrews.ac.ukPaldorNathanConstantinAdrianSchool of Mathematics and Statistics, University of St Andrews,
St Andrews KY16 9SS, UKThe Fredy & Nadine Herrmann Institute of Earth Sciences,
The Hebrew University, Jerusalem 9190401, IsraelDepartment of Mathematics, University of Vienna, Vienna 1090,
Austria
We re-visit Ekman's () 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.
Introduction
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 and constitutes a cornerstone of oceanography
see. 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:
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.
While near the Equator wind-drift currents move in the same direction
as the wind see the discussion in, 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, 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, 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.
This discrepancy is typically ascribed to the effect of a vertical
eddy viscosity that varies with depth. The explicit solution found by
, 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 tofor a survey of known
Ekman-type solutions. The challenging nature of the task is
highlighted by the recent analysis pursued in and 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.
Equations of motion and scaling
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:
ifU=1ρ∂τ∂Z-1ρ∇P+higher-order terms,
where U(Z)=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,
∇P=∂P/∂X+i∂P/∂Y is
the horizontal pressure gradient, τ(Z)=τx+iτy
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
U=Ug+Ue, we see from Eq. () that
the leading-order geostrophic and wind-driven flows separate,
with the linear equation
ifUe=1ρ∂τ∂Z
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),
τ=ρν∂Ue∂Z,
from Eq. () we obtain Ekman's equations for wind-driven ocean currents
ifUe=∂∂Z(ν∂Ue∂Z).
Let us now discuss the appropriate boundary conditions. At the
surface, the shear stress balances the wind stress, τ0:
τ0=ρν∂Ue∂ZonZ=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:
Ue→0asZ→-∞.
Letting τ0 denote the magnitude of the surface wind stress, we
non-dimensionalize the problem by scaling Ue by
2τ0/ρ and Z by 2τ0/ρ/f, since τ0/ρ
has units of L2/T2. The factor of 2 is introduced for
convenience below. Upon defining a dimensionless eddy viscosity
K=fρν/τ0, velocity u=Ue/2τ0/ρ
and depth z=Zf/2τ0/ρ, the equations transform to
7(Kψ′)′-2iψ=0forz<0,8ψ′(0)=1onz=0,9ψ→0asz→-∞,
where ψ=uK(0)
and a prime means a derivative with respect to zcf. Eqs. 14–16 in.
The scaling performed does not change the surface deflection angle
θ0, 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. (), noticing that K is real-valued.
Exact solution for piecewise-constant eddy viscosity
For piecewise-constant K, without loss of generality we can further
scale z so that K=1 in z∈[-h,0] while K=ℓ2 in
z∈(-∞,-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.
Constructing the solution
In each region, the complex velocity ψ satisfies a simple
constant-coefficient equation:
10ψ′′-2iψ=0for-h<z<0,11ℓ2ψ′′-2iψ=0for-∞<z<-h,
having exponential solutions
12ψ(z)=Ae(1+i)z+Be-(1+i)zfor-h<z<0,13ψ(z)=Ce(1+i)z/ℓfor-∞<z<-h,
where A, B and C are (generally complex) constants. The boundary
condition ψ→0 as z→-∞ has been used to eliminate the
growing solution in Eq. ().
At the discontinuity in K, at z=-h, we require continuity of ψ,
i.e. ψ(-h+)=ψ(-h-). Moreover, by integrating the equation
above across an infinitesimal region centred on z=-h, we obtain
ψ′(-h+)=ℓ2ψ′(-h-).
The upper surface boundary condition ψ′(0)=1 implies
(1+i)(A-B)=1
while continuity of ψ at z=-h implies
Ae-(1+i)h+Be(1+i)h=Ce-(1+i)h/ℓ
and finally the jump condition (Eq. ) on ψ′ at z=-h implies
Ae-(1+i)h-Be(1+i)h=Cℓe-(1+i)h/ℓ.
It follows that
A=12Ce-(1+i)h/ℓ(1+ℓ)e(1+i)handB=12Ce-(1+i)h/ℓ(1-ℓ)e-(1+i)h.
Applying the surface boundary condition (Eq. ) determines C as
C=(1-i)e(1+i)h/ℓ(1+ℓ)e(1+i)h-(1-ℓ)e-(1+i)h.
The surface current deflection angle, θ0,
measured clockwise, is determined from
tanθ0=-I(ψ(0))R(ψ(0))=-I(A+B)R(A+B).
But given C above in Eq. (), we have
A+B=12(1-i)(1+ℓ)e(1+i)h+(1-ℓ)e-(1+i)h(1+ℓ)e(1+i)h-(1-ℓ)e-(1+i)h.
Introducing the real values α=(1+ℓ)eh and β=(1-ℓ)e-h
enables us to write
A+B=12(1-i)αeih+βe-ihαeih-βe-ih,
which, after multiplying top and bottom by the complex conjugate of the
denominator, simplifies to
A+B=12(1-i)α2-β2-2iαβsin(2h)α2+β2-2αβcos(2h).
Hence, taking the (negative of the) ratio of the imaginary to real
parts of this, we obtain
tanθ0=α2-β2+2αβsin(2h)α2-β2-2αβsin(2h).
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 tanθ0=1,
i.e. θ0=45∘ 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, tanθ0
reduces to
tanθ0=sinh(2h)+sin(2h)sinh(2h)-sin(2h),
which has a non-trivial dependence on h. The maximum value
is attained as h→0; then tanθ0→∞ or
θ0→90∘.
As ℓ→∞, corresponding to an extremely viscous lower
layer, tanθ0 reduces to the inverse of the previous expression,
i.e.
tanθ0=sinh(2h)-sin(2h)sinh(2h)+sin(2h).
The minimum occurs for h→0 and there tanθ0→0 or
θ0→0.
For general ℓ, there are also values of h for which
tanθ0=1. These occur when the numerator and the denominator
of the general expression above for tanθ0 are equal. But
this means αβsin(2h)=0 or (1-ℓ2)sin(2h)=0. One
solution is the classical Ekman spiral with ℓ=1 noted above. But
we also have h=nπ/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 θ0 exhibits a non-monotonic
dependence on h for fixed ℓ. In fact, tanθ0 exhibits a
decaying oscillation about a value of unity.
Surface deflection angle θ0 (in degrees) as a
function of the lower-layer non-dimensional viscous length
ℓ and the non-dimensional depth of the upper layer h.
A summary of the results in the ℓ–h plane is provided
in Fig. .
Along any line ℓ= constant (excluding ℓ=1), θ0 reaches
a minimum or maximum in h when the following relation holds:
1-ℓ1+ℓ=±e2hcos(2h)-sin(2h)cos(2h)+sin(2h)
obtained by setting the partial derivative of tanθ0 with respect to h
equal to zero. The first extremum with increasing h occurs for h<π/8
(when h=π/8 the above equation yields ℓ=1). Note that as h→0,
(1-ℓ)/(1+ℓ)→±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 θ0 about 45∘, diminishing like e-nπ for positive
integers n. When n=1, the maximum excursion in tanθ0 is
approximately 0.05735.
Conclusions
We have re-visited the famous problem originally posed by Nansen
see the discussion in and solved by to
understand wind-driven currents in the ocean. By balancing viscous
and Coriolis forces, and assuming a constant vertical eddy viscosity,
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, 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 , in particular
deflection angles significantly different from the 45∘
prediction . 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 ρ1 overlying a denser deep layer of
density ρ2. In that case the lower-layer dimensionless viscosity
ℓ2 includes the density ratio ρ1/ρ2.) 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 θ0.
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 and . Indeed, writing K(z)=ℓ2+εK1(z) for z≤0, with ε=|1-ℓ2| and
K1(z)=(1-ℓ2)/ε,z∈[-h,0],0,z<-h,
the perturbative approach developed in and shows that a
positive and negative value of the integral
1-ℓ2ε∫-h0e2ssin(2s+π4)ds,
corresponds to a deflection angle larger and smaller than 45∘. respectively. The relation
∫-h0e2ssin(2s+π4)ds=122e-2hsin(2h)
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. 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. The same reasoning applies to the large
seasonal variations of the deflection angle observed at some locations
see the data in 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. 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.
Data availability
The results (in Fig. ) are easily generated from the simple equation derived, Eq. ().
Author contributions
All authors contributed equally to this work.
Competing interests
The authors declare that they have no conflict of interest.
Acknowledgements
The authors would like to thank the three anonymous referees for their
helpful comments on our paper.
Financial support
This research has been supported by the UK Engineering and Physical Sciences Research Council
(grant no. EP/H001794/1).
Review statement
This paper was edited by Neil Wells and reviewed by three anonymous referees.
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