Available online at www.sciencedirect.com
Journal of Marine Systems 68 (2007) 128 – 142
www.elsevier.com/locate/jmarsys
On the impact of wind curls on coastal currents
Wolfgang Fennel ⁎, Hans Ulrich Lass
Institut für Ostseeforschung Warnemünde an der Universität Rostock D-18119 Warnemünde, Germany
Received 21 July 2005; received in revised form 7 July 2006; accepted 14 November 2006
Available online 4 January 2007
Abstract
Studies of upwelling and coastally-trapped wave theory, as developed over the past thirty years, have largely neglected effects
of cross-shelf variation in wind stress and the resulting wind stress curl. However, recent satellite-based observations (QuikSCAT)
of global wind stress patterns show significant and persistent wind stress curls extending well offshore in some coastal regions
including the Benguela System. Motivated by this example, we use a relatively simple analytical model to investigate explicitly the
impact of cross-shelf variation in wind stress on the structure of the coastal currents.
The model is based on the linear Boussinesq equations of a stratified, flat bottomed coastal ocean on a f-plane (southern
hemisphere), bounded by a straight vertical wall. The model includes a wind mixed layer and a linear friction rate. The model
equations are solved using the method of Green's functions.
There are two mechanisms imposing divergencies of the Ekman transport, (1) coastal inhibition and (2) wind stress curl. In the
first case the coastal flows are affected significantly by Kelvin waves, due to the waveguide properties of boundaries. In the second
case, the wind stress curl generates vertical motion and hence horizontal pressure gradients, where the associated geostrophic flows
are limited by friction only. As a result, complex flow patterns with counter-currents can emerge. In order to highlight the role of
wind stress curls, the responses of the coastal ocean to different cross-shore variations of the alongshore wind stress are compared
with the baseline case of no wind curl.
© 2006 Elsevier B.V. All rights reserved.
Keywords: Wind stress curl; Coastal jets; Upwelling; Kelvin waves; Green's-functions; Benguela upwelling system
1. Introduction
Oceanic upwelling is generated by divergences of
Ekman-transport imposed by wind curls, coastal
boundaries or ice edges, e.g. McCreary and Chao
(1985), McCreary et al. (1987), Sjøberg and Mork
(1985), Fennel and Johannessen (1998). Coastal boundaries imply inhibition and, therefore, strong divergences
of the offshore transport for alongshore winds with the
⁎ Corresponding author.
E-mail address: [email protected]
(W. Fennel).
0924-7963/$ - see front matter © 2006 Elsevier B.V. All rights reserved.
doi:10.1016/j.jmarsys.2006.11.004
coast to the right/left at the northern/southern hemisphere. As a consequence, the near-surface isopycnals
slope upward and generate cross-shore pressure gradients which drive geostrophically balanced coastal jets,
e.g. Gill (1982). Owing to the waveguide properties of
boundaries, coastal upwelling can significantly be
reduced by coastally trapped waves. Coastally-trapped
wave theory, as developed over the past thirty years, has
largely neglected effects of cross-shelf variation in wind
stress and the resulting wind stress curl. For an idealized
model ocean of constant depth, bounded by a straight
wall, the relevant coastally trapped waves are Kelvin
waves, which propagate across the wind band and
W. Fennel, H.U. Lass / Journal of Marine Systems 68 (2007) 128–142
reduce the coastal upwelling, arrest the coastal jets,
generate alongshore pressure gradients and undercurrents, see e.g. McCreary (1981). Away from ocean
boundaries, the responses to wind curls are not affected
by waves due to the absence of waveguides. An
interesting question is how the coastal response is
modified when wind stress curls due to cross-shore
variation in wind stress near a boundary exist and the
two mechanisms interact.
A theoretical study of the role of wind stress curls on
the three-dimensional structure of the California Current
was presented in McCreary et al. (1987). The variation
of the wind curl was estimated from wind records near
the coast and 100 km offshore. It was indicated that the
wind curl might be a reason for coastal currents flowing
against the local winds, McCreary et al. (1987). The
influence of large scale wind curls on the Benguela
upwelling system was analyzed by Fennel (1999), using
wind stress curls estimated by Bakun and Nelson
(1991). In Bakun and Nelson (1991), maps of wind
stress curls were generated with 1° resolution for the
eastern ocean boundaries of the North and South
Atlantic and of the South Pacific, based on composites
of maritime data from a large number of years. The new
generation of the SeaWind scatterometer on the
QuikSCAT satellite, provided opportunities to map
spatial wind patterns with an unprecedented resolution
and sampling frequency. As shown by Chelton et al.
129
(2004), spatial structures of winds (divergences and
curls) are often surprisingly persistent.
This paper refers to the Benguela system as an
example region where the QuikSCAT observation show
a well established, persistent wind stress curl near the
ocean boundary. An example of the wind variations is
shown in Fig. 1, in terms of the monthly averaged
meridional wind stress of March 2003.
The typical patterns of the alongshore wind-stress
show the general features of the wind field in the
Benguela inferred from classical wind observations, e.g.
Shannon (1985), but reveal more detailed structures.
Basically, there are three centers of strong meridional
wind-stress, with insignificant cross-shore variations,
located near Cape Frio, (17°S), near Lüderitz, (27°S),
and off the area of the western Cape, (34°S). The centers
are separated by two bands of low meridional windstress near the coastal boundary. The bands are about
100 km wide and extend about 500 km alongshore.
Within these bands, strong wind-stress curls exist due to
the increasing meridional winds in offshore direction.
These patterns are remarkably persistent, apart from
some seasonal variations at its northern and southern
ramps, and some interannual variations in the overall
intensity of the wind-stress, see Hardman-Mountford
et al. (2003).
Observations of the three-dimensional coastal currents in the Benguela system are relatively rare. However,
Fig. 1. Monthly average of the northern wind stress component for March 2003, derived from QuikSCAT data.
130
W. Fennel, H.U. Lass / Journal of Marine Systems 68 (2007) 128–142
there is some observational evidence of surface flows
against the local winds, (e.g. G. Pitcher, personal
communication). Current observations along the crossshelf sections on the shelf off Walvis Bay were carried out
by LADCP casts with the r/v ‘Meteor’ in March 2003.
Two cross-shelf sections were recorded and the meridional current components are shown in Fig. 2. The
sections were repeated after seven days. The current
changes direction in different distances from the coast, in
the range of 15 and 90 nm. Since these survey are
snapshots of a variable system, it is not a priori clear
whether the bands of different directions reflect quasi
stationary features or are products of undersampled
variability, such as tides or inertial motions. However,
simulations with a numerical circulation model, based on
MOM 3 and forced with QuikSCAT winds show a similar
structure as observed, see Fig. 3, (M. Schmidt personal
communication. The model data are available on the web
site: http://las.io-warnemuende.de:8080/las_local/servlets/
dataset).
In this paper we show how flow reversals can be
forced by a wind curl next to the ocean boundary. To
elucidate how structures of wind fields can affect current
patterns of an upwelling system, we apply an analytical
theory. We consider the oceanic responses to alongshore
winds confined to a wind band of the width 2a along the
boundary and with cross-shore variations (wind curl) of
different strength.
The paper is organized as follows: in the next section
we give a brief outline of the model equation and sketch
how the solution can be obtained with a Green's
function method. In Section 3 we consider the solution
Fig. 2. Vertical cross-sections of the northern component of the current along 23°S. The two sections were worked at 22 to 23 March (upper panel) and
30 March 2003 (lower panel). The westernmost station is located 133 nm off the coast.
W. Fennel, H.U. Lass / Journal of Marine Systems 68 (2007) 128–142
131
Fig. 3. Ten day average of a vertical cross-sections of the modelled northern component of the current at 23°S. The model is based on MOM 3, and
was driven with QuikSCAT winds.
to a constant wind, i.e. no wind curl, as a reference case
and discuss the influence of the Kelvin wave on the
dynamics of upwelling, coastal jets and undercurrents in
some detail. Section 4 describes the solutions to wind
with cross-shelf variations, i.e. a wind stress curl. The
results are discussed in Section 5 and conclusions are
given in Section 6.
2. Model equations and formal solution
We consider a stratified, flat-bottomed f-plane ocean
on the southern hemisphere, bounded by a north–south
stretching straight, vertical wall. The wind enters the
ocean as a body force evenly distributed over a preexisting surface layer of thickness, Hmix. For simplicity
we use a linear friction rate. We note that in a more
sophisticated approach, friction rates can be related to
bottom stress, e.g. Brink (1982), Clarke and Brink
(1985). The model ocean can be described theoretically
by the linear, hydrostatic Boussinesq equations,
The subscripts x, y, z and t refer to partial differentiation.
The coast is along the y-axis, i.e. x = 0. The model
parameters, i.e., the depth, H, the thickness of the upper
mixed layer, Hmix, the inertial frequency, f, and the
BruntVäisälä Frequency N are chosen as: H = 1000 m,
Hmix = 60 m, f = 6 · 10− 5s− 1, and N = 0.01 s− 1, respectively. With these choices, the first mode internal
Rossby radius is R1 = 55 km. The linear friction rate is,
r = 0.01f.
The boundary conditions on u and w are
u ¼ 0 for x ¼ 0;
and jujbl for xY−l;
ð5Þ
and
w¼
pt
for z ¼ 0;
g
and
w ¼ 0 for z ¼ −H:
ð6Þ
The vertical co-ordinate, z, can be separated by
expanding the dynamical quantities into a series of
vertical eigenfunctions, Fn(z),
l
X
ut þ ru þ f v þ px ¼ X ;
ð1Þ
/ðx; y; z; tÞ ¼
vt þ rv−fu þ py ¼ Y ;
ð2Þ
pzt þ rpz −N 2 w ¼ 0;
ð3Þ
ux þ vy þ wz ¼ 0:
ð4Þ
where ϕ stands for u, v, and p. The Fn(z)′s are subject
to the vertical eigenvalue problem,
d 1 d
2
þ
k
n Fn ðzÞ ¼ 0;
dz N 2 dz
Here u, v and w are the cross-shore, alongshore, and
vertical current components, respectively, p is the
perturbation pressure divided by a reference density.
with the boundary conditions FnVð0Þ þ N2gð0Þ Fn ð0Þ p
¼ffiffiffiffi
0;
FnVð−HÞ p
¼ffiffiffiffiffiffi
0.ffi For a constant
have
F
¼
1=
H
;
rffiffiffiffiN we
0
nk
;
k
k0 ¼ 1= gH , and Fn ðzÞ ¼ H2 cos nkz
¼
;
ðn
¼
n
H
NH
/n ðx; y; tÞFn ðzÞ:
ð7Þ
n¼0
132
W. Fennel, H.U. Lass / Journal of Marine Systems 68 (2007) 128–142
1; 2; N Þ, LeBlond and Mysak (1978). Note that cn=
1 / λn = c1 / n and Rn = R1 / n.
The vertical current is related to the pressure by
1 A
A
þr
pðx; y; z; tÞ
wðx; y; z; tÞ ¼ − 2
N At
Az
1 X AFn ðzÞ A
þ r pn ðx; y; tÞ: ð8Þ
¼− 2
N n
Az
At
A
kn
With − N −2 Fn ðzÞ ¼
Az
N
write
l
1X
wðx; y; z; tÞ ¼
N n¼1
rffiffiffiffi 2
nk z , we may
sin
H
H
rffiffiffiffi 2
nk z wn ðx; y; tÞ;
sin
H
H
ð9Þ
Fig. 4. Sketch of the cross-shore structure of alongshore wind profiles
associated with different strength of the wind curls. The parameter ν
controls the strength of the curl.
implying wn
wn ¼
A
þ r kn pn ðx; y; tÞ:
At
ð10Þ
The system is forced by a meridional, i.e. alongshore,
wind band, acting as volume force over an upper layer
of thickness Hmix,
X ¼ 0; and Y ¼
hðz þ HmixÞ
T ðtÞQðyÞPðxÞ:
Hmix
Here the step function describes the vertical structure,
Q( y) and Π(x) the alongshore and cross-shore variations, and T(t) the time behavior of the forcing function.
After expansion into vertical eigenfunctions with the
Fourier coefficients,
rffiffiffiffi kn
1
1
1
2 sin H Hmix
¼ pffiffiffiffi ; and
¼
;
h0
hn
H kn
H
H Hmix
where n ¼ 2lkcurl . The scale of the wind stress curl, lcurl, is
related tol bylcurl =νl. The parameterν varies from 1 to ∞ and
controls the strength of the curl within the coastal strip. For
ν → ∞ the wind is uniform, i.e., the curl vanishes, while the
strongestcurlfollowsforν =1.ExamplesareshowninFig.4.
For simplicity, we look on the example of a switchedon forcing, i.e.,
T ðtÞ ¼ hðtÞ:
The technical details of the solution to the problem are
describedintheAppendixA.Forsufficientlylargetimes,i.e.
when t is much larger than the time the Kelvin waves need to
cross the wind band, see below, the response becomes,
2
pn ðx; y; tÞ ¼
we have
v2
Yn ðx; y; tÞ ¼ ⁎ T ðtÞQð yÞPðxÞ:
hn
wind curl
ð12Þ
3
1
x þ xV 7
7
þ Kn ðy; tÞP⁎ exp
7
Rn
Rn 5
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
ð11Þ
For the alongshore variation we assumed a top hat
structure, Q( y) = θ(a − | y|) = θ(a − y) − θ(− a − y). For the
width of the wind band, we choose 2a = 500 km which
amounts to 2a ≈ 10R1.
For the cross-shore variation of the alongshore wind
we assume that the wind decreases towards the coast
within a coastal strip of the width l, i.e., there is a wind
stress curl in the vicinity of the coast. We choose the
following analytical shape to describe the wind stress curl,
PðxÞ ¼ hð−x−lÞ þ hðx þ lÞcosðnðx þ lÞÞ;
6f
v2⁎
⁎
hða−jyjÞ6
4 r Gnx V P
hn
|fflfflfflfflffl{zfflfflfflfflffl}
ð13Þ
Kelvin wave affected
2
vn ðx; j; tÞ ¼
6 1
v2⁎
hða−jyjÞ6
4− r Gnxx V⁎P
hn
|fflfflfflfflffl{zfflfflfflfflffl}
wind curl
3
1
x þ xV 7
7
− Kn ðy; tÞkn P⁎ exp
7:
Rn
Rn 5
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
Kelvin wave affected
ð14Þ
W. Fennel, H.U. Lass / Journal of Marine Systems 68 (2007) 128–142
In the frame of the long-wave and low frequency
(subinertial) approximation, the alongshore current is in
geostrophic balance with the cross-shore pressure
gradient, i.e., pnx = −fvn. For the un component we find
un ðx; j; tÞ ¼
v2⁎
1
T ðtÞhða−jyjÞ 2 Gn ⁎P:
Rn f
hn
ð15Þ
In the Eqs. (13) (14) and (15) occur convolution
integrals of the form,
Z 0
1
x þ xV
dx V
x þ xV
P⁎ exp
PðxVÞexp
¼
;
Rn
Rn
Rn
−l Rn
ð16Þ
Z
Gn ⁎P ¼
0
−l
dx VGn ðx; x VÞPðx VÞ;
ð17Þ
where Gn(x;x′) is the Green's-function
Gn ðx; xVÞ ¼
Rn ðxþx ÞV =Rn −jx−x jV=Rn
ðe
−e
Þ;
2
ð18Þ
which is written for the subinertial frequency range and
in the frame of the long wave approximation.
As indicated in Eqs. (13) and (14), terms of the form
Gnx′ ⁎ Π refer to contributions due to wind stress curls.
Note that, Gnx′ ⁎ Π = − Gn ⁎ Πx′. Moreover, the expression Λn( y, t) was introduced, which is calculated
explicitly in the Appendix B. As indicated in Eqs. (13)
and (14), Λn ( y, t) in conjunction with the convolution
V
P⁎ R1n exp xþx
Rn , is related to the coastal jet and its
modification by Kelvin waves. The term is explicitly,
133
According to Eq. (10) the vertical current component
is
wn fkn
A
þ r Kn ð y; tÞ;
At
where
A
þ r Kn ð y; tÞ
kn
At
¼ −hða−jyjÞ þ hða−yÞe−rkn ða−yÞ hðt−kn ða−yÞÞ
−hð−a−yÞe−rkn ð−a−yÞ hðt−kn ð−a−yÞÞÞ
We consider the processes associated with the n-th
mode at a location y inside the wind band, a N y N − a.
We can distinguish two phases: before the Kelvin front
arrives, i.e., t b λn(a − y)and after the passage of the wave
front, i.e. t N λn(a − y). Before the arrival it follows inside
the band that wn ∼ 1. This describes the well known
coastal Ekman upwelling, which is independent of the
alongshore coordinate, y. Behind the wave front, t N λn
(a − y), it follows, wn ∼ 1 − e−rλn(a−y). Thus, the upwelling contribution of the n-th mode is decreased by the
action of the corresponding Kelvin wave, and declines
with increasing distance from the northern edge of the
wind band, y = a. In the inviscid case, r → 0, it follows
that the vertical component of the n-th mode stops
completely, wn=0, i.e. the Kelvin wave switches off the
upwelling of the considered mode.
In the steady state case, t → ∞, which is established
when all Kelvin waves modes have crossed the wind
band, it follows,
1
ðhða−jyjÞð1−e−rkn ða−yÞ Þ
rkn
þ hð−a−yÞðe−rkn ð−a−yÞ Þ−e−rkn ða−yÞ ÞÞ:
Kn ðyÞ ¼ −
coastal jet
hðtÞ zfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflffl{
ððe−rt −1Þhða−jyjÞ
rkn
Kelvin wave inside the wind band
zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{
− hða−yÞðe−rt −e−rkn ða−yÞ Þhðt−kn ða−yÞÞ
Kn ð y; tÞ ¼
þ hð−a−yÞðe−rt −e−rkn ð−a−yÞ Þhðt−kn ð−a−yÞÞ Þ:
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
Kelvin wave south of the wind band
ð19Þ
The Kelvin waves emerge in terms of the propagating
step functions, which start at the edges of the wind band.
The phase and group speed of the n-th mode is λn− 1 and
the alongshore spatial scale of the range of the wave is
set by the e-folding distance ðrkn Þ−1 ¼ rRn
f . The signals
starting at y = a, cross the wind band and are then
exported outside the band. Similarly, Kelvin waves
starting at the southern edge, y = − a, propagate outside
the wind band to the south.
ð20Þ
For the inviscid case this amounts to Λn(y) = −(θ(a −
|y|)(a − y) + θ(− a − y)2a). Thus, even without friction,
the response of the coastal ocean is bounded by the
effects of the Kelvin waves.
After the switch-on of the forcing, the system needs
some time to adjust to the steady state. The adjustment
time is characterized by the travel time of the Kelvin
waves through the wind patch. The first mode baroclinic
Kelvin wave propagates with the speed of c1 = λ1− 1 =
f R1 ≈ 280 kmd− 1 , i.e., it takes about two days to
cross the wind patch.
3. Zero wind-stress-curl
As reference case we summarize the response of the
coastal ocean to an alongshore wind band without crossshore variation, i.e. Π(x) = 1. Then, it follows
134
W. Fennel, H.U. Lass / Journal of Marine Systems 68 (2007) 128–142
Z 0
Gnx V⁎P ¼
dxVGnx Vðx; x VÞ ¼ 0;
−l
Z 0
1 Rx dx VGn ðx; x VÞ ¼
e n −1 ;
Gn ⁎P ¼
Rn
−l
with the dynamical balance
fvn þ
Avn
−fun ¼ Yn ;
At
Aun
Apn
þ k2n
¼ 0:
Ax
At
For t+ = λn(a + ϵ), we find
and
P⁎
Apn
¼ 0;
Ax
1
x þ xV
x
exp
¼ exp
:
Rn
Rn
Rn
Yn x 1−eRn ;
f
x
pn ¼ −Yn ða−yÞeRn ;
un ¼ −
This implies
with the dynamical balance
x Yn x v2⁎
T ðtÞhða−jyjÞ eRn −1 ¼
eRn −1 ;
hn f
f
v2
x
;
pn ðx; y; tÞ ¼ ⁎ Kn ðy; tÞexp
Rn
hn
1A
v2
x
pn ¼ − ⁎ kn Kn ðy; tÞexp
vn ¼ −
:
f Ax
Rn
hn
un ðx; j; tÞ ¼
fvn þ
To elucidate the role of Kelvin waves a little further
we consider the dynamical balance of a certain mode, n,
near the middle, y ≈ 0 of the wind band, stretching from
− a b y b a. It is, in particular, illuminating to consider Λ
for the limit of zero friction, i.e.,
coastal jet
zfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflffl{
t
Kn ðy; tÞ ¼ hðtÞð − ða−jyjÞ
kn
Kelvin wave inside the wind band
zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{
t
þ hða−yÞ
−ða−yÞ hðt−kn ða−yÞÞ
kn
t
− hð−a−yÞ
þ ð−a−yÞ hðt−kn ð−a−yÞÞ Þ:
kn
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
Kelvin wave south of the wind band
ð21Þ
Before the corresponding Kelvin arrives at y = 0, say
at the time t_ = λn(a − ϵ), with ϵ being a small positive
quantity, we have
Kn ð y; t− Þjy¼0 ¼ −
This switch of Λ implies for t b t_ = λn(a − ϵ),
x
vn ¼ Yn teRn ;
−fun þ
Apn
¼ Yn ;
Ay
Aun Avn
þ
¼ 0:
Ax
Ay
Thus, the regime of a purely Ekman driven
accelerating coastal jet and upwelling switches to a
balance where the upwelling has stopped and alongshore gradients of vn and pn have developed. In this
regime, the divergence of the Ekman transport maintains
the alongshore gradient of the alongshore flow. This is
closely connected with the development of an undercurrent, but this can only be seen if the summation over
the vertical modes is carried out.
A similar regime follows for a viscid system. With
friction the switching of the dynamical regimes is
smoother, because the Kelvin waves are damped while
propagating alongshore. From Eq. (19) we find that
before the corresponding Kelvin arrives at y = 0, say at
the time t_ = λn(a − ϵ),
Kn ðy; t− Þjy¼0 ¼ −
1−e−rt
hða−jyjÞjy¼0 ;
rkn
while after the passage of the Kelvin wave, say at t+ =
λn(a + ϵ),
Kn ðy; tþ Þjy¼0 ¼ −
1−e−rkn ða−yÞ
hða−yÞjy¼0 :
rkn
Yn x Yn
x
1−eRn ; vn ¼ ð1−e−rt ÞeRn ;
f
r
x
Yn
ð1−e−rt ÞeRn ;
pn ¼ −
rkn
un ¼ −
Kn ð y; tþ Þjy¼0 ¼ ða−yÞjy¼0 :
Yn x 1−eRn ;
f
Apn
¼ 0;
Ax
This implies for t b t_ = λn(a − ϵ),
t
hða−jyjÞjy¼0 ;
kn
while after the passage of the Kelvin wave, say at t+ =
λn(a + ϵ),
un ¼ −
x
vn ¼ Yn kn ða−yÞeRn ;
pn ¼ −Yn
t Rx
e n;
kn
with the dynamical balance
Apn
A
þ r vn −fun ¼ Yn ;
fvn þ
¼ 0;
Ax
At
Aun
2 A
þ r pn ¼ 0:
þ kn
At
Ax
W. Fennel, H.U. Lass / Journal of Marine Systems 68 (2007) 128–142
For t N t+=λn(a + ϵ) we find,
Yn x Yn
x
1−eRn ; vn ¼ ð1−e−rkn ða−yÞ ÞeRn
un ¼ −
f
r
pn ¼ −
135
where
x
x−l
WðxÞ ¼ ðcosðnlÞ þ nRn sinðnlÞÞeRn þ n2 R2n e Rn :
x
Yn
ð1−e−rkn ða−yÞ ÞeRn ;
rkn
with the dynamical balance,
Apn
Apn
¼ 0; rvn −fun þ
¼ Yn ;
Ax
Ay
Aun Avn
þ
þ k2n rpn ¼ 0:
Ax
Ay
fvn þ
This dynamical regime is illustrated as the baseline
case in the Figs. 5–10, (upper left panels). The principle
structure of this scenario was sketched in, e.g. McCreary
(1981) and Philander and Yoon (1982).
With Eq. (10) the vertical component, wn, can be
derived from Eq. (23). In the stationary case, it follows
wn(x, y) = rλnpn(x, y). Thus, friction affects the coastal
upwelling only in terms of the Kelvin wave contributions, where r occurs in the exponential terms, while the
upwelling due to the wind curl is independent of the
linear friction rate.
The alongshore current is geostrophically adjusted to
the cross-shore pressure gradient, vn = −pnx / f, and it
follows
vn ðx; y; tÞ ¼ −
4. Non-zero wind-stress-curl
½
v2⁎
1
hða−jyjÞ
rhn 1 þ n2 R2n
x
In this section we look at examples of wind stress-curls,
generated by cross-shore variations of the alongshore wind.
The wind decreases towards the coast within a coastal stripe
of the width l, as defined in Eq. (12), i.e., at y=l, the
alongshore winds starts to decrease. For simplicity we
consider only the case of a wind switched on at t =0 and
being constant thereafter.
We sketch briefly the solution to the problem, a
description of calculations of the involved convolution
integrals is given in the Appendix A and B. The
expressions for the cross-shore current, un, is
v2
un ¼ ⁎ hða−jyjÞT ðtÞ
fhn
ð
−PðxÞ−
½
n2 R2n
1 x−l
hðx þ lÞcosðnðx þ lÞÞ− e Rn
2 2
2
1 þ n Rn
Þ
−jx−lj
1
cosðnlÞ Rx x
þ signðx þ lÞe Rn þ
en :
2
1 þ n2 R2n
ð22Þ
For the pressure we obtain, using the involved convolution integrals, Eqs. (25) and (26), which are calculated in
Appendix B,
pn ðx; yÞ ¼
v2⁎
1
hn rkn ð1 þ n2 R2n Þ
½
ð
x
hða−jyjÞ −nRn hðx þ lÞsinðnðx þ lÞÞ−cosðnlÞeRn
−
jxþlj
n2 R2n x−l
e Rn þ e− Rn þ e−rkn ða−yÞ WðxÞÞ
2
−hð−a−yÞðe−rkn ð−a−yÞ −e−rkn ða−yÞ ÞWðxÞ ;
ð23Þ
ð−n2 R2n hðx þ lÞcosðnðx þ lÞÞ−cosðnlÞeRn
−
jxþlj
n2 R2n ðx−lÞ
e Rn −signðx þ lÞe− Rn þ e−rkn ða−yÞ WðxÞÞ
2
−hð−a−yÞðe−rkn ð−a−yÞ −e−rkn ða−yÞ ÞWðxÞ :
For the visualization of the solution we have to
perform the sums over the vertical eigenfunctions. This
was done numerically. We found that the sum over 200
modes suffices. Tests with increasing the number of
modes showed that the results are not changed by
inclusion of more modes.
5. Results and discussion
We start the discussion with the cross-shore current,
u, which is, in the frame of our approximation,
independent of the alongshore coordinate but restricted
to the area of the wind band. Hence the structures shown
in Fig. 5 apply for the whole width, 2a, of the alongshore
wind band. The cross-circulation is not affected by the
linear friction parameter and consists of the offshore
Ekman transport in the upper surface layer and the
onshore coastal rectification flow, which is imposed by
the coastal inhibition of the Ekman transport.
The modifications of the cross circulation due to the
different intensities of the wind curl are shown in the
different panels of Fig. 5. The differences are relatively
small and effects can mainly be identified in the upper
layer, where the signal is directly related to the spatial
variations of the Ekman transport. With increasing
intensity of the wind stress curl, see Fig. 4, the onshore
136
W. Fennel, H.U. Lass / Journal of Marine Systems 68 (2007) 128–142
Fig. 5. Structure the cross-shore currents for the different strength of the wind stress curls, as sketched in Fig. 4. The area of non-zero wind curl
stretches between the coast, x = 0, and the dashed line, x = − l. Solid lines correspond to positive speeds, dotted lines to negative currents.
subsurface current tends to become weaker at a given
distance from the coast compared to the reference case
of zero wind curl. However, small differences in the
Ekman transport and its coastal rectification below the
upper layer, imply significant effects on the alongshore
jets and vertical flows.
The alongshore current system consists of two
contributions, the coastal jet and the flow driven by
the wind curl. The coastal jet is affected by the
southward propagating Kelvin waves, while the curldriven part is limited only by friction. The Kelvin waves
are excited at the northern edge of the wind band and
Fig. 6. The cross-shore structures of the alongshore currents at the center of the wind strip, y = 0, for the different strength of the wind stress curls, as
sketched in Fig. 4. Solid lines correspond to positive speeds, dotted lines to negative currents.
W. Fennel, H.U. Lass / Journal of Marine Systems 68 (2007) 128–142
137
Fig. 7. The cross-shore structures of the alongshore currents 200 km south of the center of the wind strip, y = − 200, for the different strength of the
wind stress curls, as sketched in Fig. 4. Solid lines correspond to positive speeds, dotted lines to negative currents.
propagate southward. Behind the wave front, the coastal
jet is arrested and a coastal undercurrent is generated.
The range of the Kelvin waves depend on the friction.
Consequently, for strong friction the Kelvin waves are
greatly damped when they arrive in the southern part of
the wind band and, hence, their effect on upwelling and
coastal jet is small compared to the case of low friction
rates.
In order to indicate the alongshore variation of the
coastal flow, two sets of cross-shore sections of the
Fig. 8. The cross-shore structures of the vertical flows along the center of the wind strip, y = 0, for the different strength of the wind stress curls, as
sketched in Fig. 4. The area of non-zero wind curl stretches between the coast, x = 0, and the dashed line, x = −l.
138
W. Fennel, H.U. Lass / Journal of Marine Systems 68 (2007) 128–142
Fig. 9. The cross-shore structures of the vertical flows 200 km south of the center of the wind strip, y = − 200, for the different strength of the wind
stress curls, as sketched in Fig. 4.
alongshore flow, in the middle of the wind band, at y = 0,
and more southward at y = − 200 km, are shown in
Figs. 6 and 7. The shown flow structures are significantly modified for varying strengths of the wind curl.
For a zero wind curl, ν → ∞, it follows the typical
picture of a coastal jet and the opposite undercurrent
below. With increasing strength of the curl, the undercurrent reaches the surface and bands of currents and
Fig. 10. The cross-shore structures of the vertical flows 200 km south of the center of the wind strip, y = − 200, as in Fig. 9, but for a tenfold enhanced
friction parameter, r = 0.1f.
W. Fennel, H.U. Lass / Journal of Marine Systems 68 (2007) 128–142
counter-currents emerge. For the strongest wind curl, the
countercurrent dominates a substantial part of the
coastal area. By comparison of Figs. 6 and 7 it can
clearly be seen that the coastal jet shows alongshore
variations, but not the curl driven part of the flow.
The divergence of the Ekman transport, as imposed
by the inhibition at the ocean boundary, generates
upwelling within a coastal strip of the width of a
baroclinic Rossby radius. The wind curl generated
divergency of the Ekman transport, drives significant
vertical velocity signals over an area, matching the scale
of the wind curl. Two sets of cross-shore sections of
the vertical flow are shown in Figs. 8 and 9 for y = 0 and
y = − 200 km, respectively. Since the upwelling due to
the wind-stress curl is not affected by friction, see
Eq. (23) and note that wn = rλnpn, the overall vertical
velocity driven by winds dropping towards the coast is
clearly stronger than a pure coastal upwelling for
vanishing wind curl. We note that close to the coastline
near z = − Hmix, there is a small cell of relatively high
vertical flow, which is however due to the geometry of
the model and not a realistic feature.
While the upwelling due to wind stress curl is
independent of the friction parameter, the coastal Ekman
upwelling is controlled by friction through the Kelvin
waves. For small friction rates the Kelvin waves
propagate virtually undamped alongshore and switch
off the coastal upwelling. For high friction the Kelvin
waves are rapidly damped and affect only the upwelling
in a small fraction of the coastal part of the wind patch.
Thus, for strong friction the effects of the Kelvin waves
are reduced and the upwelling increases to the south
while for weak friction the Kelvin waves propagate
through the wind patch and reduce the upwelling
significantly.
An example of the effect of strong friction, r = 0.1f,
on the vertical flow is shown in Fig. 10. Due to the
strongly damped Kelvin waves, the coastal upwelling at
y = − 200 km, is relatively strong for the four cases
shown in Fig. 10. The effect of decreasing Ekman
transport near the ocean boundary is replaced by the
effect of an increasing wind curl. This illustrates again
the crucial role of coastally trapped waves, i.e., in the
case of a flat bottomed ocean the Kelvin waves.
6. Summary and conclusions
The QuikSCATS products provide wind data of high
spatial resolution and give clear evidence for a relatively
intense wind stress curl in various parts of the ocean and
in particular for the eastern ocean boundaries, among
them the Benguela upwelling system, known as one of
139
the most intense upwelling regions. These unprecedented satellite observations allow the construction of
theoretical cases of wind stress curls for analytical
studies. In this study we looked at the modification of
upwelling, alongshore and cross-shore circulation
imposed by wind curls near an ocean boundary.
An important aspect of this study is the comparison of
the effects of the divergence of Ekman transports due to
coastal inhibition and due to wind stress curls. We considered a band of alongshore wind with a wind stress
curl, due to a decrease of winds towards the coast, and
used the case of no curl as ’control experiment’. Coastal
trapped waves, i.e., Kelvin waves in the case of a flatbottomed ocean, propagate through the wind band, arrest
coastal jets and reduce the coastal upwelling. While for
zero wind curls upwelling is restricted to a narrow strip
of the width of the first internal Rossby radius, the
upwelling driven by non-zero wind curls can spread
much wider offshore. The wind curls drive alongshore
flows only limited by friction and the related upwelling is
completely balanced by the divergence of the Ekman
transport due to the wind curl. Thus, strong wind stress
curl can substantially intensify upwelling, even if alongshore winds decrease towards the coast and, therefore,
the offshore Ekman transport is much smaller than for
constant winds with zero curl. The decrease of the
Ekman transport can be overcompensated by the effect
of the divergence due to the wind curl.
The alongshore current shows significantly changing
cross-shore structures due to the effects of the wind curl,
in particular, coastal flow against the local winds can be
maintained. This is in a qualitative agreement with the
observed feature in the Benguela system where the
poleward undercurrent seems to surface and to emerge as
a surface countercurrent. A further finding of relevance
for the Benguela system, is that the onshore Ekman
recirculation below the upper layer becomes weaker for a
stronger wind curl, implying a weaker ventilation of
anoxic bottom waters near the coastal boundary as
compared to a situation with vanishing wind curl.
We are aware that the analytical model studied here is
dynamically too simple to be able to explain all
properties of the Benguela upwelling system. For
example, the model ignores important features, such
as a realistic shelf, alongshore variations of the forcing
field, and non-linear effects. Modern numerical circulation models include most of these effects and simulate
the process with an impressing realism. Nevertheless,
the simplified analytical theory is able to simulate major
features of the circulation, suggesting that they contain
much of the fundamental dynamics involved. The
theory can be useful to analyze results of numerical
140
W. Fennel, H.U. Lass / Journal of Marine Systems 68 (2007) 128–142
circulation models and observation. The study shows in
particular effects of the wind curl by comparison with
the resulting flows for no cross-shelf variation of the
wind, i.e. zero wind curl.
Essential parts of the analytical calculations as
outlined in the appendix can be applied to other
examples in a straightforward manner.
were introduced, and Gn(x;x′) is the Green's-function
Gn ðx; xVÞ ¼
1 an ðxþx ÞV −an jx−x jV
ðe
−e
Þ;
2an
with
a2n ¼ k2n ð f 2 −x̄ 2 Þ þ j2 :
Acknowledgements
The QuikSCAT data were produced by Remote
Sensing Systems and sponsored by NASA Ocean Vector
Winds Science Team. The data are available at www.
remss.com. We thank Dr. Martin Schmidt for providing
data of the numerical circulation model of the Benguela
system. We are grateful to two anonymous referees for
constructive comments.
The Green's-function obeys the symmetry relation
Gn(x;x′) = Gn(x′;x). Due to the structure of the forcing
function (11) the convolution integrals amount to
integrals over Π(x). Noting that X = 0 and using the
obvious relationship G n ⁎ Π x′ = − G nx′ ⁎ Π we can
rewrite Eq. (24) as
pn ðx; j; xÞ ¼
Appendix A. The general formal solution
In order to solve Eqs. (1)–(4) we use the Fourier
transforms with respect to y and t
Z l
dj dx ijy−ixt
/n ðx; y; tÞ ¼
e
/n ðj; x; xÞ;
2k
2k
−l
where ϕ stands for u, v, p, X, and Y. In the Fourier
domain the Eqs. (1), (2) and (4) have the form
ð
v2⁎
i
T ðxÞQðjÞ 2 2
P⁎ jdðx−x VÞ
hn
x̄ kn −j2
þ
Þ
j
x̄kn
Gn −
Gnx þ jGnxx V−j2 f x̄Gnx V ;
2
Rn
Rn
or, after partial fraction decompensation
pn ðx; j; xÞ¼
½
v2⁎
f
Gnx V⁎P
T ðxÞQðjÞi
x̄
hn
þ
ð
1
Gn
P⁎d x−xVÞ þ 2 þ Gnxx V
2ðx̄kn −jÞ
Rn
−i x̄un þ fvn þ pnx ¼ 0;
−i x̄vn −fun þ ijpn ¼ Yn ;
unx þ ijvn −i x̄k2n pn ¼ 0;
−
with the notation ω̄ = ω + ir. The complete formal
solution is
j
un ðx; j;x̄Þ ¼ k2n fGn ⁎Yn þ Gn ⁎Ynx V;
x̄
Gn
Gnx þ Gnx V
⁎ dðx−xVÞ þ 2 þ Gnxx V þ
Rn
Rn
:
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl
ffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
ð
i
x̄k2
vnðx; j;x̄Þ¼ 2 2
x̄k2n Yn þ 2n Gn ⁎Yn −k2n
Rn
x̄ kn −j2
2
j
þ k2n f jGn ⁎Ynx V− Gnx ⁎Ynx V ;
ð
x̄
f jGnx ⁎Yn
Þ
i
pn ðx; j;x̄Þ ¼ 2 2
jYn − x̄k2n f Gnx ⁎Yn −jGnx ⁎Ynx V
x̄ kn −j2
þ jk2n f 2 Gn ⁎Yn þ
Þ
j2 f
Gn ⁎Ynx V :
x̄
Convolution integrals of the form, e.g.,
Z
Gn ⁎Yn ¼
0
l
dxVGn ðx; xVÞYn
¼0
That, as indicated, the last term vanishes, follows
from the properties of the Green's function. We look at
the sub-inertial frequency range, ω̄≪ f, and employ the
long wave approximation, κRn ≪ 1, we have αn ≈ fλn =
Rn− 1 to find
pn ðx; j; xÞ ¼
ð24Þ
Þ
Gnx þ Gnx V
1
P
−
2ðx̄kn þ jÞ
Rn
½
v2⁎
T ðxÞ
Gnx V⁎P
i fQðjÞ
x̄
hn
QðjÞ
1
x þ xV
P⁎ exp
:
−T ðxÞ
Rn
Rn
x̄kn þ j
The expression for pn has singularities in the complex
frequency plane at ω = − ir and ω = − ir − κ / λn, which
W. Fennel, H.U. Lass / Journal of Marine Systems 68 (2007) 128–142
correspond to the steady motion and to Kelvin waves.
Further singularities can be introduced by the choice of
the forcing, T(ω). The branch points defined by αn = 0
are filtered out by the long wave and small frequency
approximation.
In a similar manner the expressions for vn and un can
be found:
vn ðx; j; xÞ ¼
½
v2⁎
T ðxÞ
Gnxx V⁎P
i −QðjÞ
x̄
hn
QðjÞ
1
x þ xV
P⁎ exp
;
þ T ðxÞkn
Rn
Rn
x̄kn þ j
and,
1
j
un ðx; j;x̄Þ ¼ Yn ⁎ 2 Gn − Gnx V :
Rn f
x̄
Note that Gnx Vf R1n Gn , then the second term can be
neglected because of κRn ≪ 1
1
jRn
≫ 2 :
R2n f
Rn x̄
141
The calculation is straightforward and gives the
expression (19). The estimation of the convolution
Gn ⁎ Π can easily be done with the aid of the differential
equation of the Green's function, which can be written
as
G¼
1
ðGxx −dðx−x VÞÞ:
a2
Then we have explicitly,
Z
Gn ⁎P¼
−l
−l
−l
Z 0
dxVGn ðx; xVÞþ dxVGn ðx; xVÞcosðnðx þ
V lÞÞ
Z
¼
ð
−l
1
dxV 2 −hð−x−lÞ þ Gnx xV Vðx; x VÞ
an
−l
Z 0
þ
dx VGn ðx; x VÞcosðnðxVþ lÞ
Þ
−l
1
Gnx Vðx; −lÞ
hð−x−lÞ þ
a2n
a2n
Z 0
þ
dx VGn ðx; xVÞcosðnðxVþ lÞÞ:
¼−
−l
We note that vn is geostrophically balanced by the
cross-shore pressure gradient pnx.
Appendix B. Calculation of some integrals
The inverse Fourier transformation of cross-shore
current, un, with respect to y is trivial in the frame of the
long-wave approximation,
un ðx; y;x̄Þ ¼
v2⁎
1
T ðxÞhða−jyjÞ 2 Gn ⁎P:
Rn f
hn
For the alongshore current, vn, an the pressure, pn,
we have to compute the integral
Z l
dj QðjÞeijy
;
Jn ðx̄; yÞ ¼
−l 2k x̄kn þ j
The integral in the last term can be solved by repeated
partial integration, and we find
2
Z 0
n
þ
1
dx VGn ðx; x VÞcosðnðxVþ lÞÞ
a2n
−l
1
1
¼ − 2 hðx þ lÞcosðnðx þ lÞÞ þ 2 ðcosðnlÞGnx Vðx; 0Þ
an
an
−Gnx Vðx; −lÞÞ;
which gives after applying the long wave approximations,
1
1
Gn ⁎P ¼ −PðxÞ þ
ðn2 R2n hðx þ lÞ
R2n
1 þ n2 R2n
x
cosðnðx þ lÞÞ þ cosðnlÞeRn
þ n2 R2n Gnx Vðx; −lÞÞ:
which can be estimated with the aid of the convolution
theorem as,
1
Jn ðx̄; yÞ ¼
ðhða−jyjÞ þ hð−a−yÞexpðix̄kn ð−a−yÞÞ
x̄kn
−hða−yÞexpðix̄kn ða−yÞÞÞ:
Analogously we find for the other convolution
integrals
Then we have to find the inverse Fourier transforms
of T (ω) Jn (ω̄, y) to find the expression Λ as
Z l
dx
ð−iÞT ðxÞJn ðy;x̄Þe−ixt :
Kn ð y; tÞ ¼
2k
−l
Gnx V⁎P ¼
a2n
þ
1
þ n2
ð
A
PðxÞ þ nsinðnlÞean x
Ax
Þ
n2 an ðx−lÞ −an jxþlj
ðe
−e
Þ ;
2an
ð25Þ
142
W. Fennel, H.U. Lass / Journal of Marine Systems 68 (2007) 128–142
To calculate the contributions of the Kelvin wave, we
have to compute the convolution integrals
Z −l
dx V
x þ xV
dx V
x þ xV
PðxÞexp
exp
¼
Rn
Rn
−l Rn
−l Rn
Z 0
dxV
x þ xV
−
cosnðxVþ lÞexp
R
Rn
n
−l
exp x−l −ðcosðnlÞ þ nR sinðnlÞÞexp x
n
x−l
Rn
Rn
¼ exp
−
Rn
1 þ n2 R2n
ðcosðnlÞ þ nRn sinðnlÞÞexp Rxn þ n2 R2n exp x−l
Rn
¼
1 þ n2 R2n
W
¼
;
ð26Þ
1 þ n2 R2n
Z
0
with
x
W ¼ ðcosðnlÞ þ nRn sinðnlÞÞexp
Rn
þ
n2 R2n exp
x−l
;
Rn
which gives the expression (23).
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