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VARIABLE EDDY VISCOSITIES IN THE
ATMOSPHERIC BOUNDARY LAYER FROM
AGEOSTROPHIC WIND-SPEED PROFILES
TONY LYONS
Abstract. We generate explicit height-dependent eddy viscosity
coefficients in the Ekman layer from convex wind speed profiles.
The solutions we obtain are parameterized in terms of the relative
deflection angle between the wind directions at the top and bottom
of the flow, as well as the geostrophic wind speed and a velocity
scale we interpret as the transfer rate of horizontal momentum in
the vertical direction. The solutions may be used to infer the thick-
ness of the Ekman layer for a variety of deflection angles different
from deflection angle of the classic Ekman spiral.
1. Introduction
A recent reformulation of the governing equations describing trans-
port of horizontal momentum in the planetary boundary layer (PBL)
is used in this paper to obtain eddy viscosity coefficients from Ekman
flows in this atmospheric layer. The reformulation of these governing
equations was first proposed in [5] as part of a broader investigation
of Ekman flows in the PBL. Surprisingly, the reformulation of the lin-
ear system of governing equations as a system of coupled, nonlinear
ordinary differential equations allows one to obtain the eddy viscos-
ity profile associated with the wind speed profile in the atmospheric
boundary layer. This is in contrast with many other approaches which
are often used to deduce the ageostrophic wind velocity starting with a
specific eddy viscosity profile. The aim of the work here is to consider
a general class of wind speed profiles which decay exponentially with
height, from which we deduce separable ordinary differential equations
to describe the wind speed and direction. In turn, the observations
made in [5] will allow us to derive the vertical profile of the eddy vis-
cosity coefficient. In the following, we show that this procedure allows
us to recover several wind speed profiles investigated in [5], and to
analyse several other wind velocity profiles and their associated eddy
viscosity coefficients.
2020 Mathematics Subject Classification. 76U05, 34B15.
Key words and phrases. Boundary layer Ekman flows, eddy viscosity coefficients,
wind speed profiles, exact solutions.
1
arXiv:2206.09712v1 [physics.flu-dyn] 20 Jun 2022
2 TONY LYONS
Within the Ekman layer turbulent airflow is understood as a balance
between pressure gradients, the Coriolis force and eddy viscosity [12,
19]. The eddy viscosity is governed by the flow structure of the Ekman
layer and is not a property of the fluid itself. Classical Ekman theory
assumes an eddy viscosity coefficient of the form K(z) = K0, where K0
is some constant. Under these assumptions the large scale atmospheric
flow in this atmospheric layer is described by the Ekman spiral, cf.
[10]. This solution was first derived as a model of wind-driven surface
ocean currents (see [6] for a generalisation to shallow-water Ekman
flows described using spherical coordinates valid at mid latitudes and
near the equator). The wind in the Ekman layer may be decomposed
into contributions from geostrophic and ageostrophic wind components
with a relative deflection of 45◦between the wind direction at the top
and bottom. Ascending the Ekman layer, the wind direction always
rotates clockwise in the northern hemisphere and its speed increases
monotonically, until it aligns with the geostrophic wind in the free
atmosphere above the PBL.
At mid-latitudes the absence of observational data in support of
the classical Ekman flow indicates the unsuitability of the constant
Kmodel, instead indicating eddy viscosities which vary with vertical
height. There are numerous models for the vertical profile of K(z),
with many featuring rapid vertical gradients near the base of the PBL
[12], while others incorporate a steady linear increase in the lower third
of the Ekman layer followed by an exponential decline towards the free
atmosphere. The widely used model due to O’Brien cf. [17] as well
as a similarly shaped profile due to Acker et al [2] both feature pro-
files with slow growth near the base of the Ekman layer with a turning
point followed by rapid decay towards the geostrophic layer above. In
contrast the works [16,14] investigate flows obtained from viscosity
profiles which decay steadily all along the Ekman layer. The models
considered in [11,3,23,26] analyse Ekman flows using perturbative
WKB approximations incorporating monotonic eddy viscosity coeffi-
cients which vary slowly with height in the PBL. In other cases, the
eddy diffusion coefficient is deduced numerically, for instance Dear-
dorff [7] has numerically integrated the nonlinear equations of motion
for mesoscale flows in the planetary boundary layer and derived the
eddy viscosity distribution a posteriori from these flows.
A notable result from [5] is that instead of presupposing the ver-
tical dependence of the eddy viscosity, one may start from a given
wind speed profile and then derive the associated eddy viscosity func-
tion. The nonlinear system describing mesoscale flows proposed in [5]
is based on a re-parameterisation of the height variable combined with
a reformulation of the governing equations in polar form, which is then
used to obtain the deflection angle and eddy viscosity coefficients as-
sociated with a given wind speed profile. In this paper we do not start
VARIABLE EDDY VISCOSITY FROM AGEOSTROPHIC FLOWS 3
from explicitly prescribed wind speed profiles, instead we consider a
general class of exponentially decaying wind speeds, whose convexity
is used to deduce a general class of separable ordinary differential equa-
tions governing the height dependence of the wind speed. It is found
that these viscosity profiles are characterised by the value of the eddy
viscosity coefficient near the surface boundary layer and the relative
deflection between the wind velocity at the top and bottom the Ekman
layer. In some cases analytic expressions for the eddy viscosity may be
found in terms of the vertical height. In other cases, the monotonic re-
parameterisation of the height must be inverted numerically to obtain
the vertical dependence of the eddy viscosity coefficient. In all cases
considered it is found that the relative deflection between the wind di-
rections at the top and bottom of the Ekman layer may depart from
45◦as observed in the classical Ekman spiral. However, the clockwise
rotation of the wind direction with increasing height is reproduced in
each case. The last section of this paper considers a Riccati type equa-
tion governing the exponentially decaying wind speed, and outlines a
similar process for generating the vertical profile of the associated eddy
viscosity coefficient and wind velocity from this class of equations.
2. General features of Ekman flows with variable eddy
viscosity
Depending on atmospheric conditions, the Ekman layer may begin
between 20 and 100 meters above the surface layer with an upper
boundary in excess of 1000 meters and comprises approximately 90% of
earth’s atmosphere. The fluid motion in this layer is primarily governed
by pressure gradients within the fluid, along with frictional and Cori-
olis forces. The conventional governing equations for the atmospheric
flow generated by this system of forces is given by the system
(1)
f(u−ug) = d
dz Kdv
dz !
f(v−vg) = −d
dz Kdu
dz !
with u&vbeing the mean wind velocities in the zonal and meridional
directions respectively and Kis the eddy viscosity in the Ekman layer.
The constant f= 2Ω sin(φ) is the Coriolis parameter at latitude φ
while Ω = 7.29 ×10−5rad s−1is the rotation speed of the earth. The
interface between the Prandtl layer and the Ekman layer is denoted
by z= 0, and the no-slip the boundary condition between these two
layers is given by
(2) (u, v) = (0,0) at z= 0,
4 TONY LYONS
while the boundary condition
(3) (u, v)→(ug, vg) as z→ ∞,
ensures the wind achieves geostrophic balance above the PBL. The
solution of the system (1)–(3) with constant eddy viscosity Kis given
by
(4) u(z) = ug−e−γz [ugcos(γz) + vgsin(γz)]
v(z) = vg+e−γz [vgcos(γz)−ugsin(γz)] ,
where we introduce the parameter
γ=sf
2K.
This is the classic Ekman spiral wherein the wind direction rotates
clockwise and wind speed increases monotonically toward the geostrophic
wind speed, with increasing height. While the classical Ekman spiral is
rarely observed at mid-latitudes, [22] provides evidence for Ekman spi-
rals from wind-speed field data collected at Dome C on the Antarctic
Plateau, where atmospheric conditions may allow for wind patterns re-
sembling the classical Ekman spiral. At mid-latitudes a non-constant
eddy viscosity is more appropriate, making the system (1) consider-
ably more difficult to analyse, however recent progress has been made
in obtaining flows associated with various forms of K(z), cf. [4,9,13].
2.1. Re-parameterisation of the system. The solution (4), when
written in complex form is given by
(u(z)−ug) + i(v(z)−vg) = −e−(1+i)γz (ug+ivg).
This form of the classical Ekman solution shows that the wind velocity
(u(z), v(z)) spirals clockwise with increasing height z, until it aligns
with the geostrophic wind velocity (ug, vg) in the free atmospheric layer
above. The work in [5] extends the constant Kmodel to more general
systems (1)–(3) with varying eddy viscosity profiles K(z), subject only
to the conditions K: [0,∞)→[k−, k+] and limz→∞ K(z) = k∗∈
[k−, k+] where k−,k+and k∗are all positive. Using the fact K(z) is
positive allows one to re-parameterise the height according to
(5) z7→ s=Zz
0
1
K(σ)dσ,
where s∈[0,∞). Introducing the complex function
Ψ(s) = U(s) + iV (s),
where U=u−ugand V=v−vgare the components of the ageostrophic
wind velocity in the zonal and meridional directions, we may reformu-
late the system (1) according to
(6) d2Ψ
ds2=iα(s)Ψ(s) where α(s) = fK(z(s)).
VARIABLE EDDY VISCOSITY FROM AGEOSTROPHIC FLOWS 5
The boundary conditions (2)–(3) become
(7)
Ψ = −ug−ivgat s= 0
Ψ→0 as s→ ∞.
We note that since ds
dz >0, the monotone characteristics of K(z) are
preserved in α(s).
The general solution of the system (6) may be written according to
Ψ(s) = C−Ψ−(s) + C+Ψ+(s),
where the basis solutions Ψ±(s) are defined according to their asymp-
totic behaviour
(8) Ψ±(s)'e∓(1+i)λ0sas s→ ∞,
where the conditions (8) are deduced from the asymptotic form of the
system (6) itself, namely
d2Ψ
ds2'ifk∗Φ as s→ ∞.
The Wronskian of this basis is given by
W[Ψ+,Ψ−]=Ψ+
dΨ−
ds −dΨ+
ds Ψ−
and satisfies
d
dsW[Ψ+,Ψ−]=0,
as a result of (6), in which case we may use the asymptotic form of
Ψ±(s) to obtain
W[Ψ+,Ψ−] = 2(1 + i)λ06= 0.
Hence, the Wronskian of the solution basis is non-zero, meaning this
basis is linearly independent. The boundary condition Ψ →0 as s→ ∞
ensures the physically relevant solution to the system (6) is of the form
Ψ(s) = C+Ψ+(s),
which remains stable as s→ ∞.
2.2. The nonlinear formulation. Reformulating the system (6) us-
ing polar coordinates, namely
Ψ(s) = ρ(s)eiτ (s)
with
U(s) = ρ(s) cos(τ(s)) V(s) = ρ(s) sin(τ(s)).
6 TONY LYONS
a variety of solutions for the eddy viscosity K(z) become accessible, cf.
[5]. We now find Ψ00 =iα(s)Ψ may be alternatively written in terms
of real and imaginary parts, according to
ρ00 cos(τ)−2ρ0M0sin(τ)−ρρ00 sin(τ)−ρρ02cos(τ) = −αρ sin(τ)(9a)
ρ00 sin(τ)+2ρ0ρ0cos(τ) + ρρ00 cos(τ)−ρρ02sin(τ) = αρ cos(τ).(9b)
We combine the above equations according to (9a)×cos(τ)+(9b)×
sin(τ) and (9a)×(−ρsin(τ)) + (9b)×(ρcos(τ)) to yield the following:
ρ00 −ρτ02= 0(10a)
(ρ2τ0)0−αρ2= 0,(10b)
where equations (10a) & (10b) correspond to Ψ00 =iαΨ whenever
ρ6= 0. Introducing
ug+ivg=ρgeiτg,
the boundary conditions (7) become
τ(∞) = τg
τ(0) = τg+π.
As shown in [5], the modulus ρ(s) is convex and satisfies
(11) ρ(s)>0
ρ0(s)<0)for s > 0,
while the argument satisfies
(12) τ0(s)<0 for s≥0.
Using equation (10a) to eliminate τ0(s) yields
(13) τ0(s) = −v
u
u
t
ρ00(s)
ρ(s),
and substituting this expression for τ0(s) into equation (10b), we infer
(14) α(s) = −3ρ0ρ00 +ρρ000
2ρ√ρρ00 .
Thus we may obtain the eddy viscosity profile from the wind speed
alone.
3. A separable ODE for the ageostrophic wind speed
Given an appropriate ageostrophic wind speed ρ(s), restricted by the
conditions (11) and the requirement that ρ(s) be a convex function of
s(cf. equation (13)), we may reconstruct the eddy viscosity profile
associated with this Ekman flow. In [5] the authors choose explicit
examples of weakly decaying and exponentially decaying wind speed
VARIABLE EDDY VISCOSITY FROM AGEOSTROPHIC FLOWS 7
profiles and outline the process of obtaining the deflection angle and
associated eddy diffusion coefficient associated with these flows.
In this paper we wish to analyse a general class of convex, exponen-
tially decaying ageostrophic wind speeds of the form
ρ(s) = eµ(s)for 0 ≤s < ∞,
subject to the conditions (11) and α(s)>0. We note that the condition
ρ0(s)<0 also requires µ0(s)<0 for s > 0, while convexity of ρ(s) means
ρ00(s)>0, which written in terms of µ(s) becomes
(15) µ0(s)2+µ00(s)eµ(s)>0.
We propose a separable ordinary differential for µ0(s) of the form
(16) µ00 +µ02=γ(s)µ02.
where γ(s)>0 for s > 0,thus ensuring (15) is preserved. Moreover,
since ρ00
ρ=µ02+µ00,
we find that equation (13) may be reformulated according to
(17) τ0(s) = µ0(s)qγ(s),
which agrees with condition (12) when µ0(s)<0 and γ(s)>0 for s > 0.
We observe that for the general class of wind speed profiles governed by
equation (16), the relationship between α(s) and ρ(s) given by equation
(14) may be reformulated as
(18) α(s) = −d
dsqµ02+µ00−2µ0qµ02+µ00 =− d
ds + 2µ0!qγ(s)µ02,
which will prove useful in what follows.
3.1. The slowly decaying solution. Obviously the simplest case to
analyse is of the form
γ(s) = 1 + a > 0,
where ais constant. It is straight forward to show that
(19) µ0(s) = −b
1 + abs,
where we introduce the integration constant µ0(0) = −b < 0. Given a
sufficiently large swe observe that
µ0(s)' − 1
as <0,
in which case the condition µ0(s)<0 for all s > 0 requires a > 0.
Integrating equation (19), we find
µ(s) = µg+ ln(1 + abs)−1
a,
8 TONY LYONS
with µgdefined according to ρg=eµg, where ρgis the geostrophic wind
speed. Hence the speed profile is given by
(20) ρ(s) = ρg(1 + abs)−1
a.
while the associated deflection angle is given by
(21) τ0(s) = −b√1 + a
1 + abs ⇒τ(s) = τg+π−√1 + a
aln (1 + abs)
with τgthe direction of the geostrophic wind at the top of the Ekman
layer. Equations (18)–(19) yield an eddy viscosity coefficient given by
(22) α(s) = α0
(1 + abs)2,
and with α0>0 it follows that α(s)>0 for all s > 0, in line with the
conditions for K(z) proposed in [5].
3.1.1. The slowly decaying solution in physical variables. It follows from
equation (5) that d
ds =Kd
dz , and using the notation F0=dF
ds and
˙
F=dF
dz for any function F, we may interpret the coefficients aand b
in terms of ρg=ρ(0), K0=K(0) and the vertical gradients ˙ρ0= ˙ρ(0)
and ˙τ0= ˙τ(0). Using this notation, we evaluate equations (19) and
(21) at s=z= 0 to yield
(23) b=−K0˙ρ0
ρg
,−b√1 + a=K0˙τ0,
and since b,K0and ρgare all positive, it indicates the ageostrophic
wind speed decreases with height near the bottom, as expected. Equa-
tion (23) yields
(24) a=ω2−1, ω ≡ρg˙τ0
˙ρ0
where this definition of ωwill be used throughout. Since we require
a > 0 we must impose ω2>1 for the weakly decaying model. Lastly,
we also note from equations (6) and (22) that α0=fK0.
To re-write the expressions for ρ,τand Kin terms of the vertical
coordinate z, we observe from equation (5) that dz =α(s)
fds. Using
equation (22) we may integrate explicitly to find z(s), thereby allowing
us to deduce its inverse
(25) s(z) = z
K0−abz =z
K01−z
h,
where we introduce the height-parameter h=−ρg
˙ρ0(ω2−1) >0.
Equations (20) and (23)–(25) allow us to write
ρ(z) =
ρg1−z
h1
ω2−1for z∈[0, h)
0 for z∈[h, ∞).
VARIABLE EDDY VISCOSITY FROM AGEOSTROPHIC FLOWS 9
The corresponding deflection angle is of the form
(26) τ(z) =
τg+π+ω
ω2−1ln 1−z
h,for z∈[0, h)
τgfor z∈[h, ∞)
which is a monotonically decreasing function of z, meaning the ageostrophic
wind direction rotates clockwise with increasing height, as expected.
Moreover, the associated eddy viscosity profile is simply given by
K(z) =
K01−z
h2,for z∈[0, h)
0 for z∈[h, ∞),
and so clearly we have K(z)→0 and dK
dz →0 as z→h, properties
shared with eddy viscosity profiles previously investigated in [17] for
example.
3.1.2. The relative deflection angle & the height of the Ekman layer.
The relative deflection of the the flow is the angle between the wind
direction at height sand the direction of the geostrophic wind and is
given by
β(s) = arctan ρ(s) sin(τ(s)−τg)
ρ(s) cos(τ(s)−τg) + ρg!,
cf. [5]. This deflection angle may be calculated at the base of the
Ekman layer s= 0 using l’Hˆopital’s rule to give
β(0) = arctan ρgτ0(0)
ρ0(0) != arctan ρg˙τ0
˙ρ0!= arctan(ω).
Thus we see the significance of the parameter ω, it is the tangent of
the relative angle between the wind direction at the bottom of the PBL
and the geostrophic wind at the top. Given that we require ω > 1 for
the weakly decaying case (cf. equation (24)), it follows this model is
only appropriate when the angle between the wind directions at the
bottom and top of the atmospheric boundary layer exceeds 45◦. Such
scenarios are known to arise, for instance field data from the Tibetan
plateau reveal relative deflections above 50◦(see [25]).
A useful definition for the height of the Ekman layer is as the smallest
value z=Hwhere the wind-direction is aligned with the geostrophic
wind-direction, cf. [5]. It follows that τ(H) = τgand equation (26)
means this height His explicitly given by
H=h1−e−π
ω(ω2−1).
A useful feature of this definition for the height of the Ekman layer is
that it may be obtained from ground based measurements of the flow
and the geostrophic wind speed, which is essentially constant above
this height.
10 TONY LYONS
We note that K(z)→0 as z→h, which appears to contradict the
condition K: [0,∞)→[k−, k+] with k±positive constants. However,
as applied in [5] it appears this condition is a sufficient condition for the
existence of a solution of (6). Furthermore, in [18] the authors develop
analytic solutions for atmospheric Ekman flows with slowly varying
eddy viscosity profiles where K(z)≥0, so a vanishing eddy viscosity
appears to be physically reasonable also. In [5] the authors proposed
an ab-initio ageostrophic wind speed profile of the form
(27) ρ(s) =
˜
b
1+˜as ,for 0 ≤s≤s0
˜
b
1+˜as0,for s>s0,
with ˜a,˜
band s0all positive constants. Thus we see that (20) and (27)
to a large extent are the same wind speed profile when
(28) a= 1, b = ˜a, ρg=˜
b, s0=s(H) = H
K01−H
h
The wind speed profile (20) is a generalisation of the profile (27) in the
sense that the decay rate of the profile (20) may be altered by varying
the value of the parameter a(or equivalently the physical parameter
ω). On the other hand the eddy viscosity profiles K(z) associated with
the speed profiles (20) and (27) are basically the same when we impose
(28), with both profiles decaying quadratically as zincreases. In figure
1the graphs of the wind speed ρ(z), the relative deflection β(z) and
the eddy diffusion K(z) are shown in the top three panels, while the
lower panel is the hodograph of the wind velocity (the graph of v(z)
vs. u(z) for z∈[0,H)) for the slowly decaying model.
3.2. The exponentially decaying solution. In this case we consider
the ageostrophic wind speed governed by the separable ODE
µ00 +µ02= 1−λ2
(as +b)2!µ02for s > 0,with µ0
0=−ab
λ,
where a,band λare positive constants, while the condition b2≥λ2
ensures γ(s) = 1 −λ2
(as+b)2>0 for all s > 0. Separation of variables
allows us to integrate to obtain
(29) ρ(s) = ρge−a2s2
2λ2−ab
λ2s,
in which case the profile investigated in [5] is reproduced when λ=√2
and b= 2. Equations (17) and (29) yield
(30) τ0(s) = −a
λ2q(as +b)2−λ2,
whose integral we will compute later.
VARIABLE EDDY VISCOSITY FROM AGEOSTROPHIC FLOWS 11
2 4 6 8 10 12 14
(
z
)(ms 1)
0
20
40
60
80
100
120
140
160
z
(m)
0 10 20 30 40
(
z
)( )
0
20
40
60
80
100
120
140
160
z
(m)
012345
K
(
z
)(m2s1)
0
200
400
600
800
z
(m)
0 2 4 6 8 10 12 14 16
u
(
z
) (ms 1)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
v
(
z
) (ms 1)
Figure 1. The wind speed ρ(z), relative deflection
β(z) and eddy viscosity K(z) as well as the hodograph
of the slowly decaying model. The parameters used
in this model are ρg= 15 ms−1, ˙ρ0=−0.25 s−1and a
maximum relative deflection β0= 46◦. The eddy viscos-
ity coefficient at the bottom of the PBL is K0= 5m2s−1.
The profile of the eddy viscosity coefficient shows the
decay over the Ekman layer 0 < z .163 m (solid line).
We apply equation (14) to obtain the eddy viscosity coefficient
α(s) = a2(as +b) [2(as +b)2−3λ2]
2λ4q(as +b)2−λ2.
and to ensure α(s) is positive for all s > 0 it is clear that we actually
require b2≥3
2λ2, which also ensures α(s) is bounded for all s≥0.
Integrating with respect to s, we have
(31) z(s) = a
6fλ4h√x2−λ22x2−3λ2ix=as+b
x=b.
Since α(s)>0⇒z0(s)>0 for s > 0 when b2>3
2λ2with z(s) also
being a continuous function of sfor all s∈[0,∞), an inverse function
s(z) for all z≥0 is ensured, under an appropriate choice of parameters
a,band λcf. [8].
3.2.1. Exponential decay in physical variables. Applying the notation
from section 3.1.1 to equations (29) and (30), we evaluate ρ0(0) and
τ0(0), to find
−ab
λ2=K0˙ρ0
ρg
,−a√b2−λ2
λ2=K0˙ρ0,
12 TONY LYONS
Multiplying and dividing these expressions separately, it may be de-
duced that
a2
λ2=K2
0˙ρ2
0(1 −ω2)
ρ2
g≡w2
0,b2
λ2=1
1−ω2,
where ω=ρg˙τ0
˙ρ0. The velocity scale w0is interpreted as the rate at
which horizontal momentum is transferred vertically near the bottom
of the flow at z= 0. The condition α(0) = f K0combined with these
relations yields
w2
0=2fK0ω(1 −ω2)
(3ω2−1) ,
which relates this vertical velocity scale to latitude via the Coriolis
parameter. The conditions 3ω2−1>0 and 1−ω2>0 ensure the model
is only valid when the relative deflection angle between the geostrophic
wind direction and the wind at the bottom of the PBL is constrained by
30◦< β(0) <45◦, which agrees with available field data (cf. [20,22]).
3.2.2. The inverse map. To reformulate equation (31) in physical vari-
ables we introduce the notation
(32) q(as +b)2−λ2=aq(s+s1)2−s2
0≡aξ(s)
where we define s0≡1
w0and s1≡1
w0√1−ω2and
a
6fλ2√b2−λ2h2b2−5λ2i=w0ω(3ω2+ 2)
6f√1−ω2≡z0.
Hence, we may recast equation (31) in the form of a cubic polynomial
ξ3−3s2
1
2ξ−η(z)
2= 0, η(z) = 5f
w4
0
(z−z0),
whose only real root is given by
ˆ
ξ(z) =
3
r2η+qη2−2s6
1+3
r2η−qη2−2s6
1
2.
The function ˆ
ξ(z) is the function ξ(s), parameterised with respect to
z, as opposed to s. Transposing equation (32), and using ξ(s)≡ˆ
ξ(z),
can write the inverse of z(s) as follows:
s(z) = qˆ
ξ(z)2+s2
0−s1.
VARIABLE EDDY VISCOSITY FROM AGEOSTROPHIC FLOWS 13
3.2.3. The height of the boundary layer. Written in terms of the pa-
rameter s, we find that the ageostrophic wind speed may be written
as
ρ(s) = ρgexp "−w2
0s2
2−w0s
√1−ω2#,
Given the restriction 1
3< ω2<1, it is clear that ρ(s) is convex for all
s > 0, as expected. The height of the boundary layer is defined as the
smallest value s=Ssuch that τ(S) = τg, and so equation (30) yields
π+1
2ln h√x2−1 + xi
S+s0
s1
x=s1
s0−1
2hx√x2−1i
S+s0
s1
x=s1
s0
= 0.
An explicit expression for Sin terms of s0,s1is obviously not avail-
able from this implicit definition, however a numerical value is always
assured for appropriate values of s0and s1as a consequence of the
implicit function theorem. The ageostrophic wind speed ρ, the rel-
ative deflection βand the eddy viscosity K(z) for the exponentially
decaying model are shown in the top three panels of figure 2, while
the lower panel of this figure shows the hodograph of the ageostrophic
wind velocity in the PBL.
0 2 4 6 8 10 12 14
(
z
)(ms 1)
0
200
400
600
800
1000
1200
1400
z
(m)
0 5 10 15 20 25 30 35
(
z
)( )
0
200
400
600
800
1000
1200
1400
z
(m)
5 10 15 20 25 30
K
(
z
)(m2s1)
0
200
400
600
800
1000
1200
1400
z
(m)
0 2 4 6 8 10 12 14 16
u
(
z
) (ms 1)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
v
(
z
) (ms 1)
Figure 2. The wind speed ρ(z), relative deflection
β(z) and eddy viscosity K(z) as well as the hodograph
of the exponentially decaying model. These profiles
correspond to the choice of parameters ρg= 15 ms−1,
w0= 0.02 ms−1and a maximum relative deflection β0=
35◦and at a latitude of 52◦north. Under this choice of
parameters, the eddy viscosity coefficient at the bottom
of the PBL is K0= 2.82 m2s−1and the Ekman layer has
height H'1379 m.
14 TONY LYONS
3.3. Slow exponential decay. In this example we consider the sep-
arable ODE give by
µ00 +µ02=1−e−(as+b)µ02, µ0
0=−a
e−b+ 1,
where aand bare positive constants. Integrating we find
ρ(s) = ρg 1 + eb
1 + eas+b!.
while the associated deflection angle is given by
τ(s) = τg+π+ ln
√1−x−1√1−x+√2√2
√1−x+ 1√1−x−√2√2
x=e−(as+b)
x=e−b.
The eddy viscosity coefficient α(s) is given by
(33) α(s) = a2eas+b4e2(as+b)−7eas+b+ 1
2 (1 + eas+b)2√e2(as+b)−eas+b,
and using dz =α(s)
fds we obtain after integration
z(s) = "√2 ln √1−x+√2
√1−x−√2!+ ln √1−x−1
√1−x+ 1!#x=e−(as+b)
x=e−b.
Since we require α(s)>0 for all s > 0 and the polynomial 4ξ2−7ξ+1 =
0 has roots at 7±√33
8, we infer from equation (33) that bmust satisfy
the inequality
b≥ln 7 + √33
8!,
This in turn ensures z(s) is a monotonic function of sand therefore
invertible in principle. As an exact expression for the inverse of this
function cannot be found we calculate the inverse numerically once an
appropriate choice is made for the parameters aand b.
3.3.1. Physical parameters of slow exponential decay. Using the nota-
tion from section 3.1.1, we find that the parameters aand bare related
to the physical parameters K0, ˙ρ0, ˙τ0and ρgby the conditions
−a
1 + e−b=K0˙ρ0
ρg
,−a√1−e−b
1 + e−b=K0˙τ0,
which may be solved to yield
a2√1−e−b
(1 + e−b)2=w2
0, e−b= 1 −ω2,
VARIABLE EDDY VISCOSITY FROM AGEOSTROPHIC FLOWS 15
where ω=ρg˙τ0
˙ρ0and w2
0=K2
0˙ρ0˙τ0(2−ω2)
ρgω2. The condition α(0) = f K0now
gives
w2
0=2fK0ω(2 −ω2)
(ω4+ 5ω2−2) ,
where again we interpret w0as the rate at which horizontal momen-
tum is transported in the vertical direction due to the effects of eddy
viscosity.
Again we define the height of the atmospheric boundary layer as
the smallest value s=Swhere the condition τ(S) = τgis first sat-
isfied, with Sobviously being the height of the PBL in terms of the
s−parameterisation. This condition defines Simplicitly according to
0 = π+ ln
√1−x−1√1−x+√2√2
√1−x+ 1√1−x−√2√2
x=e−(aS+b)
x=e−b
,
and since it is not possible to find an exact expression for Sin terms for
aand b, we revert to numerical methods to determine a value for this
height after appropriate values of aand bare chosen. The condition
w2
0>0 requires 2 −ω2>0 and ω4+ 5ω2−2>0 which restricts the
relative deflection angle between the geostrophic wind and the wind at
the bottom of the Ekman layer according to 31◦< β(0) <55◦. The
vertical profile of the wind speed, relative deflection, eddy viscosity and
hodograph of the wind velocity are shown in figure 3.
3.4. Square root decay. The final case we consider is governed by
the separable ODE
µ00 +µ02= 1 + 1
√as +b!µ02, µ0
0=−a
2√b
where aand bare positive constants. Integrating we find
(34) ρ(s) = ρge−√as+b+√b.
Equations (17) and (34) yield
(35) τ(s) = τg+π+"1
2ln √x+ 1 −√x
√x+1+√x!−√x2+x#x=√as+b
x=√b,
where we use the substitution u=q1 + 1
√as+bto integrate. Meanwhile,
equations (14) and (34) yield the eddy viscosity coefficient
α(s) = a2(4x2+ 6x+ 1)
8x3√x2+x, x =√as +b,
which is clearly positive for all s > 0 with any choice of positive con-
stants aand b. Using the condition dz =α(s)
fds, we may integrate this
16 TONY LYONS
0 2 4 6 8 10 12 14
(
z
)(ms 1)
0
25
50
75
100
125
150
175
z
(m)
0 5 10 15 20 25 30 35
(
z
)( )
0
25
50
75
100
125
150
175
z
(m)
123456
K
(
z
)(m2s1)
0
25
50
75
100
125
150
175
z
(m)
0 2 4 6 8 10 12 14 16
u
(
z
) (ms 1)
0
1
2
3
4
v
(
z
) (ms 1)
Figure 3. The wind speed ρ(z), relative deflection
β(z) and eddy viscosity K(z) as well as the hodograph
of the slow exponential decay model. These pro-
files correspond to ρg= 15 ms−1,w0= 0.02 ms−1and
a maximum relative deflection β0= 35◦and at a lati-
tude of 52◦north. Under this choice of parameters, the
eddy viscosity coefficient at the bottom of the flow is
K0= 0.613 m2s−1and the Ekman layer has height H'
734 m.
expression to obtain
z(s) = a
4f"4 ln 2qx(x+ 1) + x+ 1−2√x+ 1 (16x+ 1)
3x3
2#x=√as+b
x=√b
.
Given dz
ds =α(s)
fand α(s)>0 for all s > 0, it is clear that zis a mono-
tonically increasing function of s, in which case the implicit function
theorem ensures there exists a function s(z) which is the inverse of z(s)
given above. While in this case no analytic expression is available, we
may always calculate its inverse numerically.
3.4.1. Physical parameters of the square-root decay model. Using the
notation of section 3.1.1 and equations (34)–(35), the vertical gradients
of ρand τevaluated at s= 0 satisfy
−a
2√b=K0˙ρ0
ρg
,−s1 + 1
√b
a
2√b=K0˙ρ0,
VARIABLE EDDY VISCOSITY FROM AGEOSTROPHIC FLOWS 17
which combine to yield
a2=4K0˙ρ0˙τ0
ρgω2≡w2
0and 1
√b=ω2−1,
where ω=ρg˙τ0
˙ρ0as usual. Using the condition α(0) = fK0and the
above relations for aand b, we find
w2
0=8K0fω
(ω2−1)2(ω4+ 4ω2−1),
and so the conditions ω4+ 4ω2−1>0 and ω2−1<0 restrict the rela-
tive deflection angle between the geostrophic wind and the ageostrophic
wind at the bottom of the PBL according to 26◦< β(0) <45◦, ap-
proximately. This deflection angle, along with the wind speed, the
eddy diffusion coefficient and the wind velocity hodograph are shown
in figure 4.
0 2 4 6 8 10 12 14
(
z
)(ms 1)
0
50
100
150
200
z
(m)
0 5 10 15 20 25 30 35
(
z
)( )
0
50
100
150
200
z
(m)
0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20
K
(
z
)(m2s1)
0
50
100
150
200
z
(m)
0 2 4 6 8 10 12 14 16
u
(
z
) (ms 1)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
v
(
z
) (ms 1)
Figure 4. The wind speed ρ(z), relative deflection
β(z) and eddy viscosity K(z) as well as the hodograph
of the square-root decay model. These profiles are
obtained from the choice of parameters ρg= 15 ms−1,
w0= 0.02 ms−1and a maximum relative deflection β0=
35◦and at a latitude of 52◦north. Under this choice of
parameters, the eddy viscosity coefficient at the bottom
of the Ekman layer is K0= 0.614 m2s−1and the Ekman
layer has height H'193 m.
4. A Riccati equation for the wind speed
The profiles µ(s) generated in Section 3are restricted by the require-
ment they satisfy a separable ODE of the form (16). Of course, the
18 TONY LYONS
requirements for an exponential speed profile of the form ρ(s) = eµ(s)
are µ0<0 and µ00 +µ02>0, in which case we may also consider models
of the form
(36) µ00(s) + µ0(s)2=γ(s), γ(s)>0 for s > 0,
which is a Riccati type ODE for µ0(s). The reader is also referred
to the recent work [15], were Riccati equations for the velocity profile
Φ(s) = eRs
0h(ξ)dξ are explored. While solutions µ0(s) of equation (36)
are highly contingent on the form of γ(s), explicit solutions for µ0(s)
are known to exist for an extensive range of functions γ(s), see [21] for
instance. However, in contrast to the method presented above, finding
an explicit expression for the map z(s) appears to be more challenging
under the approach adopted here.
As a simple example, we consider the Riccati equation
(37) µ00 +µ02=as +b a, b > 0
where as +b > 0 for all s > 0 as required. Upon applying the Riccati
transformation µ0=y0
y, and introducing the change of variable ξ=
1
3
√a2(as +b), equation (37) now becomes
(38) yξξ −ξy = 0,
where yξdenotes dy
dξ . Equation (38) is the well known Airy equation
(see [24] for example), whose general solution is of the form
y(ξ) = c1Ai(ξ) + c2Bi(ξ),
cf. [21], where c1,2are arbitrary integration constant while Ai and Bi
are the Airy functions of the first and second kind respectively (see
[1]). Given that Bi(ξ) grows without limit as ξ→ ∞, it follows that
the physically stable solutions are of the form
µ(s) = µg+ ln Ai(ξ(s))
Ai(ξ(0))!⇒ρ(s) = ρgAi(ξ(s))
Ai(ξ(0)) .
The associated deflection angle is given by
τ0(s) = −qξ(s)⇒τ(s) = τg+π−23
√a
3qξ(s)3−qξ(0)3.
Given that d
ds =1
3
√a
d
dξ and Aiξξ (ξ) = ξAi(ξ), it follows from equation
(14) that
α(s) = −1
3
√a
1
2qξ(s)+ 2qξ(s)Aiξ(ξ(s))
Ai(ξ(s))
,
which cannot be integrated in closed form to give z(s). Nevertheless,
we may easily perform this integration numerically to determine z(s),
and given α(s)>0 for all s > 0, the implicit function theorem (cf. [8])
VARIABLE EDDY VISCOSITY FROM AGEOSTROPHIC FLOWS 19
ensures there exists a unique inverse s(z), which may also be calculated
easily using numerical methods.
Acknowledgments. The author is grateful to the organisers of the
workshop “Mathematical Aspects of Geophysical Flows,” held at the
Erwin Sch¨odinger Institute for Mathematics and Physics, Vienna, Aus-
tria, January 20–24, 2020. The author would also like to thank the
anonymous referees for several helpful comments.
Conflict of interests. The author declares there is no conflict of in-
terest with this manuscript.
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Department of Computing & Mathematics, Waterford Institute of
Technology,
Waterford, Ireland
Email address:tlyons@wit.ie
