In December 2002 and January 2003 satellite observations
of chlorophyll showed a strong coastal signal along the west African coast
between 10 and 22∘ N. In addition,
a wavelike pattern with a wavelength of about 750 km
was observed from 20 December 2002 and was detectable
for 1 month in the open sea, south-west of the Cape Verde Peninsula.
Such a pattern suggests the existence of a locally generated
Rossby wave which slowly propagated westward during this period. This hypothesis
was confirmed by analysing sea surface height provided by satellite altimeter during
this period. To decipher the mechanisms at play, a numerical study
based on a reduced-gravity shallow-water model has first been conducted. A wind burst,
broadly extending over the region where the offshore oceanic signal is observed,
is applied for 5 d. A Kelvin wave quickly develops along the northern edge
of the cape, then propagates and leaves the area in a few days.
Simultaneously, a Rossby wave whose characteristics seem similar
to the observed pattern forms and slowly propagates westward.
The existence of the peninsula limits the extent of the wave
to the north. The spatial extent of the wind burst
determines the extent of the response and correspondingly the timescale
of the phenomenon (about 100 d in the present case). When the wind burst
has a large zonal and small meridional extent, the behaviour
of a wave to the north of the peninsula differs
from that to the south.
These results are corroborated and completed by an analytical study of a linear
reduced-gravity model using a non-Cartesian coordinate system. This system is
introduced to evaluate the potential impact of the coastline shape.
The analytical computations confirm that a period of around 100 d
can be associated with the observed wave considering the value of the wavelength;
they also show that the role of the coastline remains moderate at such timescales.
By contrast, when the period becomes shorter (smaller than 20–30 d),
the behaviour of the waves is modified because of the shape of the coast.
South of the peninsula, a narrow band of sea isolated from the rest
of the ocean by two critical lines appears. Its meridional extent is about 100 km
and Rossby waves could propagate there towards the coast.
Introduction
Eastern boundary upwelling systems (EBUSs) –
such as the California, Humboldt,
Canary and Benguela upwelling systems – constitute a ubiquitous feature
of the coastal ocean dynamics, which has been extensively studied.
They are biologically very productive thanks to the transport of nutrients
from deep ocean layers to the surface,
which favours the bloom of phytoplankton. Consequently they
present a strong signature, which is detectable by ocean colour
satellite sensors (see, for example, Lachkar and Gruber, 2012, 2013).
The dynamics of EBUS were first studied with conceptual models.
Upwellings are created by alongshore equatorward winds
(Allen, 1976; McCreary et al., 1986) generating an offshore Ekman
transport, which is compensated for by a vertical transport at the coast in order to satisfy the mass conservation. The near-shore pattern
of the upwellings is affected by the baroclinic instability mechanism
which is associated with the coastal current system and produces
eddies and filaments (Marchesiello et al., 2003). Lastly
wind fluctuations modulate the upwelling intensity by generating
Kelvin waves propagating poleward (Moore, 1968; Allen, 1976;
Gill and Clarke, 1974; Clarke, 1977; 1983; McCreary, 1981).
At a given frequency, there is a critical latitude poleward of which Kelvin waves
no longer exist and are replaced
by Rossby waves propagating westward (Schopf et al., 1981; Clarke, 1983;
McCreary and Kundu, 1985). The critical latitude decreases when
the wave period shortens (Grimshaw and Allen, 1988) or when the
coastline angle with the poleward direction increases (Clarke and Shi, 1991).
This latter property suggests that the shape of the coast
has an impact on the upwellings.
Using a high-resolution 3-D numerical model, Batteen (1997) then
Marchesiello et al. (2003) confirmed that the shape of the coastline actually
plays a role in the upwelling pattern. However, they did not investigate by which
mechanisms they are driven. In particular
they did not try to compare their results with
the theoretical investigations of Crépon and Richez (1982) and Crépon et al. (1984), who
analysed the mechanisms responsible for this behaviour.
Using an f-plane model, these authors showed that a cape
modifies the characteristics of the upwelling, its intensity
being less on the upwind side of the cape than on the downwind side. However, they did not investigate what occurs in the open sea
up to 1000 km from the coast and which role the β effect could play.
The role of the forcing has also been investigated, both
from observational and theoretical viewpoints.
Enriquez and Friehe (1995) computed the wind stress
and wind stress curl off the California shelf from aircraft measurements and, thanks to numerical experiments with a two-layer
model and an analytical study, showed that a non-zero wind stress curl
expands the horizontal extent of upwelling offshore;
it increases from 20–30 to 80–100 km. The importance of the
wind stress curl, which generates a strong Ekman pumping,
was first emphasized by Richez et al. (1984). Later, Pickett and Paduan (2003) could establish that the Ekman pumping and the Ekman transport
due to the alongshore winds have a comparable importance
in the California Current area; Castelao and Barth (2006) found a similar result off Cabo Frio in Brazil.
These contributions did not study
what occurs beyond 100 km from the coast.
In the present paper, we study the role of wind stress and
coastline geometry in generating mesoscale anomalies offshore, up to a distance of
500–1000 km off the coast. Both a numerical and an analytical point of view
are adopted. The departure point is the observation of a wave-like pattern
on chlorophyll satellite observations off the Senegalese
coast, in the region of the Senegal–Mauritanian upwelling
(see Lathuillière et al., 2008, and Farikou et al., 2015, for further information about the
chl a variability and the upwellings off the west African coast,
Capet et al., 2017, for a recent analysis
of the small-scale variability close to the Senegal and Gambia coasts, and
Kounta et al., 2018, for a detailed study of the slope currents along west Africa).
Attention is focused on offshore mesoscale activity associated
with the upwelling, a recurrent feature
of upwelling systems (see Capet et al., 2008a, b). The alongshore activity, which has
received much more attention (see, for example, Diakhaté et al., 2016, and the
references therein) is not studied.
The paper is structured as follows. Observations of chlorophyll in 2002–2003
off the west African coast are shown and described in Sect. 2.
In Sect. 3, a numerical study with a non-linear reduced-gravity shallow-water
model on the sphere with a single active layer is conducted; it shows
that a wind stress anomaly active for a few days can generate a pattern
that seems very similar to the observed one.
The impacts of the wind anomaly extent and coastline geometry are also briefly studied.
In Sect. 4, a theoretical analysis of the wave dynamics
in the vicinity of a cape is conducted to confirm and enlarge upon the
results obtained in the previous section, using a linear shallow-water model
in a non-Cartesian coordinate system.
Observation of a wave off the Cape Verde Peninsula
from an ocean colour satellite sensor
The Senegal–Mauritanian upwelling off the west coast of Africa
forms the southern part of the Canary upwelling system.
This region has been intensively studied by analysis
of SeaWiFS (Sea-viewing Wide Field-of-View Sensor) ocean colour data and AVHRR (Advanced Very High Resolution Radiometer) sea surface temperature
as reported in Demarcq and Faure (2000) and more recently by Sawadogo et al. (2009), Farikou et al. (2013, 2015),
Ndoye et al. (2014), and Capet et al. (2017). These studies indicate that
the presence of an intense upwelling is attested
by ocean colour and sea surface temperature signals. Moreover
this upwelling shows a strong seasonal modulation.
It starts to intensify in October,
reaches its maximum in April and slows down in June.
Very high chlorophyll a concentrations are observed
near the coast where the maximum is reached. However, the concentration rapidly decreases
offshore (Farikou et al., 2013, 2015; Sawadogo et al., 2009),
suggesting that the upwelling extent and the eddy activity in this
region are less than in other upwelling systems like the Californian
upwelling system (Marchesiello and Estrade, 2009; Capet et al., 2017).
Chlorophyll observed by SeaWiFS at 4 different days.
A wave-like pattern (which is circled by a dashed line) is
visible between -22 and -18∘ in longitude
and 12 and 14∘ in latitude.
At the end of the period a westward propagation
seems to initiate. The chlorophyll concentration is given
in milligrams per cubic metre by the colour bar at the right of the maps.
From 20 December 2002 up to 8 January 2003, a strong
chlorophyll signal was observed along the African
coast on SeaWiFS satellite images, between 10 and 22∘ N,
indicating intense biological activity.
In addition, a well-defined “sine-like” pattern (circled in Fig. )
was observed on 10 images taken on 10 non-consecutive days,
which excludes a possible artefact due to image processing. This pattern
is located between 12 and 14∘ N, east of 20∘ W, and
extends offshore up to 22∘ W in a region
which is off the coastal upwelling zone. West
of 20∘ W the signal seems to have a larger meridional extent,
reaching 16–17∘ N (see panel “03 January 2003”).
This pattern, which broadly keeps the same form during a 20 d interval,
slowly progresses westward at a speed not exceeding 5 cm s-1 (a more precise
estimation of the speed from the observations is quite risky). After 8 January 2003, a cloudy period of several days occurred, which prevented satellite
observations. At the end of this episode (15 January)
the sine pattern was no longer visible. This episode might be the signature
of a Rossby wave propagating westward.
As this phenomenon lasts at least 1 month, its typical timescale
is expected to range between 1 and a few months.
Hovmöller diagrams of the SSH at
latitudes 12, 13.5, 15,
and 16.5∘ N (the SSH amplitude is given in centimetre by the colour bar at
the right of the diagrams).
Note the decrease in the amplitude at 16.5∘ N.
The phase velocity of the wave is about 4.5 cm s-1.
To corroborate this hypothesis, we analysed
the sea surface height (hereafter SSH) obtained from
AVISO (Archiving, Validation and Interpretation of Satellite Oceanographic data) satellite altimeter data for the corresponding period (December 2002–January 2003).
Hovmöller diagrams are shown at 12, 13.5, 15,
and 16.5∘ in Fig. ;
they clearly confirm the existence of a Rossby wave
propagating westwards with a velocity of about 4.5 cm s-1.
The amplitude of this wave becomes smaller northwards: it
peaked at 13 cm between 12 and 13.5∘
but did not exceed 7 cm at 16.5∘. The
wavelength is around 700 km, comparable with the extent of the chlorophyll signal.
An enhanced chlorophyll concentration, as seen by the satellite sensor, is
the signature of growing phytoplankton.
A local development of phytoplankton benefits from
an increase in the nutrient concentration in the surface layers of the ocean.
The latter may be generated by an enhanced mixing
in the surface layers associated with an increased
turbulence due to the Kelvin or Rossby wave activity and
by the vertical velocity associated with the divergent Rossby waves.
Another possible process explaining this growth could be the
advection linked with Rossby waves
– they create a current anomaly which can transport nutrients and
phytoplankton from the upwelling area where they are highly concentrated.
However, this mechanism is slow; for a current anomaly of 5 cm s-1, the transport
of a parcel of fluid over 500 km – less than the zonal extent of the offshore pattern –
would need about 1 year.
The signal along the coast is also modulated by coastal Kelvin
waves propagating northwards.
Clarke and Shi (1991) showed that they can propagate when their
angular frequency is larger than a critical angular frequency
ωc=cβcosΘ/2f0. In this formula
Θ is an angle taking into account the tilt of
the coastline with a meridian, c a typical velocity of a
baroclinic mode, and f0 and β the usual parameters linked to the
Earth's rotation. These authors found that ωc ranges
between 84.69 and 115.2 d around the Cape Verde Peninsula (see their Table 2a).
These values are close to the characteristic timescales we expect here.
The existence of a long period divergent baroclinic Rossby wave, evidenced by the AVISO
satellite altimeter data of the SSH, can
explain the offshore sine-like pattern described here. To investigate the
mechanism of emergence of this wave and analyse its properties,
we first describe numerical experiments made with a numerical shallow-water
model. They help us see how such a wave is created and elucidate its nature.
Then, a theoretical analysis is presented in order
to understand how the Kelvin and Rossby waves behave when the coast
presents a cape and to complete the interpretation of the numerical experiments.
A wide range of periods is explored, going from 10 d
to 1 year.
Numerical studyThe model
The numerical model is a reduced-gravity model on the sphere
with one active layer of thickness h. It extends over an infinite layer at rest. The
velocity v in the active layer and the thickness h
verify the equations
∂th+div(hv)=0
and
∂tv+(rotv+f)n×v=-gradΦ+τ0h-rhv+νΔHv,
where n is a vector normal to the Earth's surface and
rotv=(∇×v).n.
The function Φ is
equal to g⋆h+v2/2 where g⋆ is the reduced
gravity.
We assume for simplicity that the vector
τ0, which represents the surface wind stress divided by
the ocean density, derives from a potential ϕ0(x,y):
τ0=-gradϕ0
(implying an irrotational mean wind). This hypothesis allows us
to compute explicitly the obtained mean state. Indeed, v=0 and g⋆h02/2=-ϕ0+C0 are an obvious
solution of the previous system (the constant C0 is determined by using the
fact that the mean value of h0 remains unchanged during the integration).
It will also facilitate the analytical computations made in the next section. A more complex
set-up could be used for the numerical experiments, but it will be
seen below that this one suffices.
Mean wind velocity in December 2002 (maximum ≃8 m s-1)
and wind velocity anomaly on 3, 5, and 7 December (maximum ≃4 m s-1).
Response of the ocean to an anomalous wind stress applied for 5 d
(a) and corresponding evolution
of the active layer thickness for every 5 d from the
close of the wind stress anomaly up to 35 d after (b–i).
A Rossby wave is generated and a Kelvin wave quickly propagates along the coast.
The Rossby wave propagates westward and its amplitude is divided by
2 between (a) and (i). The 0 m isoline is indicated in bold; the isoline interval
is 0.5 m (blue: negative).
Numerical resolution
The model domain is closed and centred at 15∘ N,
the latitude of the Cape Verde Peninsula; it
has a latitudinal extent of 20∘ and a longitudinal extent of 30∘.
The peninsula is modelled as indicated in Fig.
in order to mimic the geometry of the coast of Senegal (simplified and smoothed).
The mean value of h0 is equal to 200 m and
the reduced gravity g⋆ to 0.02 m s-2.
Consequently the Rossby radius of deformation R0=g⋆h0/f0 at the
latitude of the Cape Verde Peninsula is equal to 53 km.
The previous equations are solved by finite differences on a C-mesh on
the sphere, the mesh size being equal to (1/12)∘ in longitudinal and
latitudinal directions. The spatial scheme preserves enstrophy,
following Sadourny (1975). No-slip boundary conditions are
applied everywhere (including along the artificial boundaries which limit the
open ocean). There is no added dissipation in the continuity
equations and mass is conserved by the numerical scheme.
The time integration is performed using a
leapfrog scheme with a time step of 300 s.
The viscosity ν and the coefficient r of Eq. ()
are respectively equal to 28 and
8×10-5 m s-1 (h/r≃1 month).
More details can be found in Février et al. (2007), where the
two-layer version of this model is described.
Numerical set-up of the model
In Fig. the mean wind for the considered period (December 2002–January 2003) is shown. It exemplifies the situation which is normally found in this region. The
wind regularly blows from the north–north-east with a velocity ranging from 4 to 8 m s-1. To take this into account, a constant
mean wind stress of amplitude equal to 0.06 N m-2 (corresponding to a
mean wind velocity of about 5 m s-1 and a value of τ0 equal
to 6×10-5 m2 s-2) and oriented
along a south–south-west direction is applied from rest
for 4 years until a stationary mean state, which verifies the theoretical
relation given in Sect. 3.1, is reached.
As shown by Fig. , a wind anomaly was active
when the wave of Fig. begins to be observed.
This anomaly is obviously transient, but to the
south of the Cape Verde Peninsula, it mainly points southwards. To represent this situation
in a simplified way, in a first experiment we defined a north–south wind stress anomaly which extends over
approximately 500 km and whose maximum
is still equal to 0.06 N m-2. This anomaly is applied
for 5 d (see Fig. a).
The integration is continued for 45 d, after the
anomaly has disappeared.
To explore the sensibility of the model response to the wind anomaly, others
wind anomalies were applied (see below, in particular Figs. 6 to 8). The
results obtained for these anomalies are discussed in the next section.
Numerical results
After the wind stress anomaly corresponding to the first experiment has vanished,
the subsequent states of the ocean are shown every
5 d in Fig. in terms of the active layer thickness.
A coastal Kelvin wave forms north of the cape and quickly
propagates along the coast. After 5 d, it has already gone
beyond 25∘ N and after 10 d only the remains of the wave are still visible.
South of the cape, a well-marked (Rossby) wave develops. Its size more or less matches the size of the wind stress anomaly. It slowly
propagates westwards with a velocity of about 4 cm s-1. The amplitude
of the wave decreases quickly because of the large value of h/r.
The minimum value of h is about -3.5 m when the wind
ceases (panel b) and reaches only -2 m after 25 d (panel g).
These characteristics are actually those of a Rossby wave
locally generated by a wind anomaly and then freely propagating in the open
ocean. The wavelength of this wave is about 750 km (kR≃0.84×10-5 m-1). Such a value is compatible with the theoretical
study presented in Sect. 4 when the period of the wave is about 100 d.
This modelled response to a wind stress anomaly also matches the satellite-observed
signal described in the previous section. It thus suggests that the latter is
the consequence of the existence of a Rossby wave generated by a wind burst.
Though the duration of the wind burst is short in our numerical experiment (5 d),
the response of the system privileges a much longer timescale, exceeding 2 months.
This result is not inconsistent. Indeed, the Fourier transform of a rectangular pulse
is the sine-cardinal function. It thus contains a significant amount
of energy at low frequencies and thus can generate a low-frequency response like the
one observed here.
Solution obtained for the same conditions as in Fig. for a
straight coastline. Note that the mean wind stress is added to the wind stress anomaly
in panel (a). A Rossby wave is still generated, but the halting of the signal
due to the cape is no longer observed. The latitudinal extent of the wave
east of 19∘ W is thus nearly twice larger than
in the previous case.
Figure shows the results obtained in a similar
experiment in which the cape is absent. The response of the model is
very similar: in the area of the wind anomaly, a Kelvin wave forms and then
quickly disappears, whereas a more persistent Rossby wave slowly propagates westward. However, the meridional extent of the Rossby wave is broader east of 18∘ W, in the area where the cape was previously. This suggests that
the cape simply limits the extent of the wave northward but does not
modify its dynamics. The theoretical study of Sect. 4 will confirm
that the role of the cape remains moderate at low frequency.
A question arises: why does the wave have such a wavelength?
Numerical experiments clearly show that the longitudinal wavelength
is defined by the spatial scale of the forcing anomaly.
This is first illustrated in Fig. , which shows
the response of the model to a wind burst whose extent is 4 times smaller
than the initial wind burst of Fig. . The latitudinal and longitudinal
extent of the model response are approximately divided by 2 as expected. No Kelvin wave
of significant amplitude is generated because there are no
longer any wind anomalies on the coast; indeed, the centre of the wind anomaly is
unchanged in comparison with the reference experiment.
We will show in Sect. 4 that the period associated with the wave is increased when the wavelength is reduced (reaching approximately 150 d).
The wind anomaly extent is
4 times smaller than the one used in Fig. and the position of the
centre is kept unchanged (a). It still acts for 5 d, and as previously
the active layer thickness is shown for every 5 d from the
close of the wind stress anomaly up to 35 d after.
A Rossby wave is generated and its extent is approximately
4 times smaller than previously. No Kelvin waves are created.
The 0 m isoline is indicated in bold; the isoline interval
is 0.5 m (blue: negative).
Two supplementary experiments (Figs. and )
were carried out, in which a wind burst of
large longitudinal (about 1000 km) and small latitudinal (about 100 km) extent is applied for 5 d; the anomaly is centred at 14∘ N in one case
and 17∘ N in the other (see Figs. a and a).
These anomalies create a Rossby wave with a large zonal extent. However, the response of the model differs in the two cases.
When the anomaly is located south of the cape, a Kelvin wave is generated
and a negative anomaly appears between
26 and 20∘ W. However, no positive anomaly with a comparable
amplitude can be seen closer to the coast. A weak signal appears
after day 20, but its extent is very small and its amplitude is 4 times
smaller than the amplitude of the anomaly
observed around 25–27∘ W. South of 14∘, a wave of small amplitude
is created and propagates southward. Its latitudinal wavelength is comparable with
the latitudinal extent of the wind anomaly.
When the wind anomaly is located north of the cape, a negative
anomaly appears between 26 and 20∘ W as previously;
besides, a positive anomaly can be seen from day 5 and its amplitude
is half the amplitude of the signal observed
around 25–27∘ W.
The wind anomaly now acts over a long (1000 km) and narrow (200 km) domain south
of the cape for 5 d (a). The solution after the close of the wind burst
is shown. A wave is generated, but the anomaly remains moderate near the coast.
The 0 m isoline is indicated in bold; the isoline interval is
0.1 m (blue: negative).
The wind anomaly now acts over a long (1000 km) and narrow (200 km) domain north
of the cape for 5 d (a). The solution after the close of the wind burst
is shown. A wave is generated and the anomaly near the coast has an amplitude comparable
with the anomaly around 25∘ W.
The 0 m isoline is indicated in bold; the isoline interval is 0.1 m (blue: negative).
Clearly the response of the model close to the cape
depends on the location of the wind anomaly, north or south
of the cape. When the latter acts south of the cape,
the anomaly which forms around 15∘ W is small and
quickly disappears; a wave which propagates southward seems to prevent
its existence. By contrast, when the wind anomaly
acts north of the cape, an anomaly forms around 15∘W and
gets stuck in this place; the wave which propagates southward still
exists but its amplitude is about twice smaller than in the previous case.
In the next section, we analytically investigate this
dissymmetric behaviour in an idealized case.
Analytical study
In this section, we aim to understand whether the coastline
may influence the propagation of the Rossby waves
which are created close to the coast and propagate towards the open sea.
The impact of the coastline has been investigated by Crépon and Richez (1984),
Clarke (1977), and Clarke and Shi (1991) for the Kelvin waves using an analytical
approach. Here we focus on the Rossby waves; as we consider
an area which extends up to about 1000 km from the coast, we have
to generalize the approach followed by Clarke and Shi,
which introduced a local system of coordinates dependent of
the coastline to study Kelvin waves along an irregular coastline.
We try to answer the following questions:
Are there timescales for which the impact of the coastline
(small in the numerical experiments) becomes more important?
A dissymmetry between the response north and south of the cape
was visible in the numerical experiments; can this dissymmetry
be dependent on the existence of the cape?
The analysis begins by defining and building a system
of coordinates that permits us to follow the coastline geometry.
This procedure is a standard one in mathematics when
boundaries are complex; indeed, the boundary
conditions can be written simply, which constitutes a substantial
advantage. However, it has a drawback: the differential equations
which characterize the problem become slightly more
complex because they must include geometrical factors that
take into account the deformation associated with the new
system of coordinates. This drawback is small in comparison with the advantage.
When these new equations are established, straightforward
calculations are made to obtain a unique
partial differential equation (Eq. 7),
which characterizes the evolution of η (the thickness of the
active layer). This equation is a wave equation. Consequently
the ray theory (or equivalently the Wentzel–Kramers–Brillouin method) can be applied.
When the forcing terms are neglected, this yields
a first-order non-linear differential Eq. (11).
No new ideas are introduced after this. We just
rewrite Eq. (11) by introducing new notations,
in order to facilitate its study and the presentation of the results
(end of Sect. 4.2). We then describe the results when
the transport along the coast is much larger
than the transverse transport (Sect. 4.3).
A model for the Kelvin and Rossby waves
We consider a reduced-gravity shallow-water model
in the β plane forced by a constant
wind stress which derives from
a potential ϕ0(x,y) (τ0=-gradϕ0), as in Sect. 3.
Numerical integrations have shown that the exact solution
v=0 and g⋆h02/2=-ϕ0+C0 are actually
obtained after a few years' integration
(see Sect. 3.1 and 3.3).
If an anomaly (τx,τy) is added
to the mean forcing τ0, a perturbation is generated; it
is characterized by a depth anomalyh (the thickness of
the first layer is now h0+h) and a
velocity v. A linear approximation is
sufficient to study the first steps of
the evolution of the perturbation if the forcing anomaly remains
moderate. The anomaly (τx,τy) can be written in terms of a potential
ϕ(x,y,t) and a stream function ψ(x,y,t),
τx=-∂xϕ-∂yψ,τy=-∂yϕ+∂xψ,
so that the divergent part of the forcing is given by -Δϕ and
the rotational part by Δψ.
The equations verified by the anomaliesh and
v are thus
h0∂tu-fh0v+∂x(g⋆h0h+ϕ)=νh0ΔHu-ru-∂yψ,h0∂tv+fh0u+∂y(g⋆h0h+ϕ)=νh0ΔHv-rv+∂xψ,∂th+∂x(h0u)+∂y(h0v)=0.
The role of the diffusion and dissipation will not be considered below – a smoothing and damping of the
solution is expected when it is taken into account. The previous
system may be further simplified by introducing the zonal and meridional
transports Tx=h0u and Ty=h0v and a potential η equal to
g⋆h0h+ϕ. It becomes
∂tTx-fTy+∂xη=-∂yψ,∂tTy+fTx+∂yη=∂xψ,∂tη+c2[∂xTx+∂yTy]=∂tϕ,
where c=g⋆h0 is a function of x and y.
These equations apply inside the ocean domain, whatever its shape.
A boundary condition is added along the domain frontier (the normal transport
vanishes), but the latter is difficult to handle when the shape of the coast is complex.
Lastly, the spatial mean value
of the depth anomaly h remains null.
The propagation of a Kelvin wave along the eastern boundary and the possible
generation of Rossby waves can be studied by using a system (Eq. ).
Here, we consider an eastern boundary whose angle with the meridians varies smoothly, and we seek to understand how these variations may affect the wave dynamics. We
consider only the case of a cape, even though the method could be applied in other
cases. A coordinate change is made
in order to “straighten the coast” and therefore have a simple boundary condition;
Eq. is correspondingly
modified to match the new coordinates.
A new orthogonal system of coordinates X=X(x,y) and
Y=Y(x,y) is thus introduced
such that the eastern boundary is now defined by the
simple equation X=0 (rather than a complex one such as f(x,y)=0).
In the local orthonormal basis eX,eY associated
with this coordinates system, the line element dl reads
dl=adXeX+bdYeY,
where a and b are geometrical factors which convey the stretching
of the coordinates along orthogonal directions (note that the relations
a=(∂Xx)2+(∂Xy)2andb=(∂Yx)2+(∂Yy)2,
where the initial coordinates x and y are now functions
of the new coordinates X and Y, are used to compute a and b).
Example of orthogonal coordinates X and Y defined for a cape
protruding from the coast into the sea over a distance of 80 km.
Several values of X and Y have been indicated. Only one half of the
domain has been represented.
The coefficients a and b are given for the coordinate system of Fig.
(they have no dimension).
Such a coordinate change is illustrated in Fig. for a cape protruding
into the sea over a distance of 80 km. Only half of the symmetric domain is shown.
The initial coordinates x and y are the zonal and meridional coordinates;
the new coordinates X and Y are represented
in the original system and some of their values are indicated.
Other coordinate changes would be possible. The corresponding
geometrical factors a and b are shown in Fig. . They
differ from 1 in a close neighbourhood and to the west of
the cape (a is equal to 0.1 around the extremity of the cape, whereas b reaches
40 at 300 km west of the cape). A detailed study of this example
will be presented in Sect. 4.3.
Using the coordinates (X,Y), the system labelled as “Eq. (5)” () becomes
∂tTX-FTY+a-1∂Xη=-b-1∂Yψ,∂tTY+FTX+b-1∂Yη=a-1∂Xψ,∂tη+(C2/(ab))[∂X(bTX)+∂Y(aTY)]=∂tϕ,
where F, C, TX and TY are functions of X and Y
corresponding to f, c, Tx and Ty
(the notations for η, ϕ and ψ have not been changed
to enhance the readability, but these functions also
depend on X and Y).
Note that we use a more complex system than the traditional one which is used
for the study of the Kelvin waves and which neglects the variations in the transport
perpendicular to the coast:
-FTY+a-1∂Xη=-b-1∂Yψ∂tTY+FTX+b-1∂Yη=a-1∂Xψ∂tη+(C2/(ab))[∂X(bTX)+∂Y(aTY)]=∂tϕ.
We use the complete system () because we study Rossby waves far from the coast, for
which these hypotheses do not hold. Indeed, if the previous
simplified system is used, the two terms (1/b2)(∂Yθ)2 and iF2∂Y(a/(bF2))∂Yθ must be removed from Eq. (11) below. These terms
include the geometrical factors a and b due to coordinate change,
and we try to investigate which is the
impact of such terms.
We now concentrate on processes whose timescale is much larger than 1 d. Moreover
we consider only the evolution of free waves.
This situation corresponds to the case numerically investigated in Sect. 3
and illustrated in Figs. –: the wind stress anomaly that had created the depth anomalies
has ceased to exist. With these hypotheses, system () can be reduced to the following equation, which characterizes the evolution of η:
∂tη-R021a2∂XX2η+1b2∂YY2η-……R02abF2∂XbaF2∂tX2η+∂YF∂Xη+F2∂YabF2∂tY2η-∂XF∂Yη=0,
where R0=C/F is the Rossby radius.
The boundary condition at the eastern coast now reads
atX=0,b∂tX2η+aF∂Yη=0
for all t>0 and Y. The details of the computations can be found in Appendix A.
When a and b are close to 1 and the coast has a south–north orientation (see Fig. ), the new coordinate system is nearly similar to the original one; indeed
F mainly depends on Y≃y only, and
in those regions Eq. () thus
simplifies to
∂tη-R02(∂XX2η+∂YY2η)-R02∂YF∂Xη+F2∂Y(1F2)∂tY2η=0.
We recognize the equation characterizing
the propagation of waves in the β plane
(β=∂YF) for a shallow-water model.
For a tilted coast the dependance of the Coriolis parameter as a
function of X should be still taken into account.
For spatial scales much larger than R0 and at low frequency, Eq. () can be further simplified. It becomes
∂tη-R02∂YF∂Xη=0, which simply models the westward propagation of long Rossby waves.
The coefficient b decreases as one gets closer to
the cape (see Fig. ). An impact of the cape in the open
sea may therefore be expected because of the terms proportional to
∂XbaF2∂tX2η and ∂XF∂Yη
in Eq. ().
Ray theory
When a wave propagates in a medium whose properties change spatially,
it does not follow straight lines but more complex paths.
Ray theory – or the Wentzel–Kramers–Brillouin approximation – is used to determine the paths followed
by the waves in such a medium. It applies when the wavelength is smaller than the typical
scale at which the properties of the medium vary.
The spatial variations in the components k and l of
the wave vectors are taken into account by introducing a
complex function θ(X,Y)=θR+iθI such as
k=∂Xθ and l=∂Yθ.
The potential η is then
equal to
η=η0(X,Y)exp[i(ωt+θ(X,Y))],
where it is assumed that
|η0-1∂Xη0|≪|∂Xθ|,
|k-1∂Xk|≪|∂Xθ|,
|η0-1∂Yη0|≪|∂Yθ|,
|l-1∂ll|≪|∂Yθ|.
As required by the theory, these inequalities ensure that the wavelength is smaller than the
typical scale of variation in the system, here conveyed by R0, F, and
the coefficients a and b.
Except in a very close vicinity of the cape (distance smaller than around 10 km) and for the meridional wavelength in the area located between -50 and 50 km
north and south of the cape and beyond 200 km west of the cape, these inequalities mean
that several wave patterns must be visible in the considered domain. This condition
is verified for the waves observed in the numerical experiments and for the
waves considered below.
Considering these hypotheses, η is given from Eq. ()
in the neighbourhood of a point M0
of coordinates (X0,Y0) by the approximate expression
η(X,Y)=η0(X0,Y0)exp[i(ωt+k(X-X0)+l(Y-Y0))],
where k=∂Xθ|M0 and l=∂Yθ|M0.
The physical meaning of this solution is explicated by taking its
real part:
ℜ(η)=η0(X0,Y0)e-kI(X-X0)-lI(Y-Y0)cos[ωt+kR(X-X0)+lR(Y-Y0)].
The solution must not increase westward since it cannot become infinite, which
implies kI<0. Note, however, that kI>0 is possible if this occurs only in
a (small) bounded domain. For kR>0, the wave propagates westwards.
The values of k and l are obtained by computing the function θ
from Eqs. () and (). After simplifying them by using the hypotheses above,
we find that θ verifies the approximate (eikonal or Hamilton–Jacobi) equation
1+R02[1a2∂Xθ2+1b2∂Yθ2]-……R02abiF2∂XbaF2∂Xθ+iF2∂YabF2∂Yθ+∂YFω∂Xθ-∂XFω∂Yθ=0,
with the boundary condition
ib∂Xθ+a(F/ω)∂Yθ=0
at X=0.
To make the computations clearer we set
z1=(∂Xθ)R0/a=kR0/a,
z2=(∂Yθ)R0/b=lR0/b,
w1=(R0/2b)∂Y(Fω)+iF2∂X(baF2),
w2=(R0/2a)-∂X(Fω)+iF2∂Y(abF2),
and the problem () associated with
the boundary condition () is rewritten as the following
system:
13∂Yaz1R0=∂Xbz2R0,141+z12+z22-2z1w1-2z2w2=0,15z2=-iωFz1atX=0.
At the boundary, Eqs. () and () permit us to
determine the values of z1 and z2 and hence of k and l.
Indeed, the introduction of Eq. () in Eq. () leads
to the equation
1+1-ω2F2z12-2z1w1-iωFw2=0.
Setting ξ=1-(ω/F)2, this equation can be rewritten as
J(ξz1)=1ξw1-iωFw2=WR+iWI,
where
J(z)=12(z+1z)
is the Joukowsky transform of z;
WR and WI are given by the relations
WR=R0F2ξ×Fω-ξ21b∂Y1F+1-ξ21a∂YabF,WI=R0F2ξ×1b∂XbaF.
Equation () is the dispersion relation of the wave at the boundary.
The Joukowsky transform of ξz1 (or equivalently
of iFωξz2) depends on
the frequency ω, the mean state (through R0),
the latitude (through F) and the geometry of the coast (through a and b).
In (a) the half complex plane ℑ(z)≤0 with z=ξz1 is shown.
The domain is restricted to complex numbers with a negative imaginary part
since kI=aℑ(z)/(ξR0) must be negative. The half circle |z|=1 and
a few straight lines have been drawn. In (b), the Joukowsky transform of the
previous half plane W=J(z) is shown. The segment [-1,1] is
the image of the half circle. The upper (lower) half plane corresponds to the image
of its interior (exterior). The image of the straight lines
is correspondingly represented.
The sketch in Fig. , which shows the half complex plane
ℑ(z)<0 (panel a) and the complex plane W=J(z) (panel b),
explains how the Joukowsky transformation works. The inferior
half plane has been chosen since we expect kI<0 or in other words
ℑ(z)<0.
When WI=ℑ(W) is positive, the
complex number ξz1 such as J(ξz1)=W=WR+iWI
has a norm smaller than 1 (it corresponds to the grey areas in Fig. ).
The wavelength is thus large (λ>2πξR0/a).
If WR=ℜ(W) is positive, the phase speed is
negative (westward propagation).
When WI=ℑ(W) is negative, the
complex number ξz1 such as J(ξz1)=W=WR+iWi
has a norm larger than 1 (it corresponds to the white areas). The wavelength is thus small.
If WR=ℜ(W) is positive, the phase speed is still negative.
When WIvanishes, a situation which occurs when the coast is a straight line,
the unique complex solution previously found can cease to exist. Indeed,
if |WR| is smaller than 1, there is one complex solution
whose norm is equal to 1 (on the half circle in bold in Fig. );
and if |WR| is larger than 1, two real solutions are obtained
(between ]-1,1[ and outside this interval). The case (WI=0) is
detailed in the next subsection.
At the coast, relation () determines z2
when z1 is known. It implies that
lI=ωFbakRandlR=-ωFbakI.
For low frequencies ω/F is much smaller than 1
and lI and lR can be ignored. An approximate expression of the
waves is
η(X,Y,t)=η0(X0,Y0)e-kI(X-X0)cos(ωt+kR(X-X0)),
and the dynamics are controlled by the westward propagation of Rossby waves.
For shorter periods, the situation may be more complex because
the ratio ωb/(aF) may be close to 1. This effect is in agreement with the
results obtained in Sect. 3, where a southward propagation of waves was
observed in the open ocean, simultaneously with the westward propagation of Rossby waves.
Knowing z1 and z2 along the coast,
it is possible to continue the resolution of
the problem (Eqs. –) and compute explicitly
the rays characterizing the propagation of the waves. An algorithm
which fulfils this objective is described in Appendix 2.
However, under the supplementary assumption that |TX|
remains much smaller than |TY| up to a distance from the coast of about
a few Rossby radiuses of deformation, approximate expressions for z1
and z2 can be obtained analytically. This also permits us to initialize the algorithm
described in Appendix B.
Analytical study for |TX|≪|TY|
In this case, Eq. (), which is exact on the coast
X=0, can also be used on a band of a few hundred kilometres off the coast and
yields a good approximation of the solution (for a more detailed discussion,
see Appendix 2). Consequently, Eq. () becomes valid
over a domain which extends far off the coast in the open ocean.
In this section, the consequences of this relation are briefly presented
for a straight coastline and then investigated in detail
for the cape shown in Fig. (). In agreement with the hypotheses of the
previous subsection, we assume that ω≪F, but the results and graphics will
be produced up to the limit value ω=F.
Case of a straight coastline
This case has been extensively studied in the literature
– for example, Richez et al. (1984), Grimshaw and Allen (1988), Clarke and Shi (1991),
McCalpin (1995), and Liu et al. (1998); each article stresses a particular
issue. Using the previously established equations, we here summarize known results
about the existence of critical frequencies along an ocean boundary.
For a straight coastline
following the south–north direction, we have X=x and Y=y;
consequently, a=b=1 and F depends only on Y. Thus WI=0 and
2WR=R0βω2ξ2-1ξ,
where β=∂YF.
(a)WR is shown for latitudes ranging from 5∘ N (left segment)
to 40∘ N (right segment).
The critical value WR=1 is reached at 15∘ N for a period
of about 125 d. (b) Wavenumber k as a function of the period T
for three different latitudes (10∘ (blue),
15∘ (red), and 20∘).
In Fig. a,
the coefficient WR is shown at different latitudes as a function of the period.
At 15∘ N, the critical value
WR=1, which ensures the transition from a complex solution to
two real solutions, is reached when the period is 140 d.
Panel b shows the wavenumber as a function
of the wave period computed from WR for
10∘ N (black), 15∘ N (grey) and
20∘ N (light grey). When WR>1, a condition which is always
fulfilled at low frequency (characteristic time longer than 125 d at 15∘),
Eq. () has two real solutions. If WR≫1, these solutions are close to
2WR≃R0β/ω and 1/2WR≃ω/(βR0).
Consequently, the wavenumber k is equal to either
β/ω or ω/(βR02) (see Eq. ).
This result proves the existence of Rossby waves,
whose wavelength is either short or long. When the frequency increases,
WR decreases and eventually reaches the critical
value of 1. When WR becomes smaller than 1, there are two complex conjugate solutions.
The only acceptable solution has a negative imaginary part and conveys the
existence of a Kelvin wave, trapped along the coastline and propagating
northward. The absolute value of this imaginary part kI is represented as a dashed curve
in Fig. b. When the wave frequency is
close to the critical value, kI vanishes
and consequently the Kelvin wave no longer exists.
A Rossby wave with a significant amplitude can propagate westwards.
At 15∘ N its wavelength will be around 300 km. The zonal
velocity (phase speed) of this non-dispersive wave is about 2.5 km d-1.
On the other hand, as lR vanishes with kI, the meridional
velocity becomes infinite.
Similar results are obtained when the coast presents a constant angle θ
with a meridian. The only change is that the critical frequency
for which the wave regime changes is modified and depends on the tilt
of the coast as indicated in Clarke and Shi (1991).
Case of a cape
The wave dynamics in a neighbourhood of the cape are characterized
by the coefficients WR and WI which convey the effect of the
coastline on the propagation of the wave. The coefficient
WR can be split into two terms; the first one
R02b2ξ2-1ξ∂YFω is
similar to the term obtained from a straight coastline, whereas the
second one R02a1-ξ2ξ∂Yab
explains the role of the coast. It is large when ξ is small
– in the frequency range where the model is valid, this means for
wave periods going from ∼10 d to a month – and when the
deformation of the coastline is large.
The existence of such a term was expected. When the angle of the coastline
with a meridian increases, the impact of the
latitudinal variations in the Coriolis parameter decreases along the path
followed by the wave. It even vanishes when the coast becomes parallel to the
Equator. These changes are taken into account by this term. By contrast, at low frequency,
the lengthening or shortening of the path followed by the wave becomes
negligible because it occurs over a time which remains short in comparison
with the period of the wave.
The variations in the coastline geometry also prevent the existence of
two distinct solutions at low frequency. Indeed, WI differs from
0, and consequently two complex solutions are obtained as explicated in
Fig. . If WI is strictly negative (grey area), the solution
z is inside the half unit disc and kR is small.
If WI is strictly positive, the solution is outside and kR is large.
The degeneracy of the equation thus disappears, and a selection of the
wavelength operates.
Dimensionless coefficient WR for periods equal to 10,
20, 50 and 100 d.
The bold black line corresponds to WR=0. When the period of the
wave shortens (smaller than 20 d), WR becomes negative in a
narrow band south of the cape.
For period longer than 100 d, the diagram is nearly symmetric.
Dimensionless coefficient WI. It is independent of the
period of the wave. Note that the domain is reduced in comparison with
the previous figure.
Figure shows WR for
T=10, 20, 50 and 100 d and
makes visible an interesting property. When the period T
is equal to 10 d, a dissymmetry
around the cape occurs: WR is negative south and positive north of the cape.
This dissymmetry weakens when the period increases. For T=20 d,
the area where WR is negative is strongly reduced and for T=50 d,
it has vanished. Since the signs of WR and kR are similar
(see Fig. and the associated comments), a negative
kR is expected in this area and actually appears (see Fig. ).
Coefficient kR for periods equal to 10,
20, 50 and 100 d.
The bold black line separates positive from negative values.
Where WR is negative, kR is negative (for periods shorter than 20 d,
in the area located to the south of the cape).
Coefficient kI for periods equal to 10 and 100 d.
This coefficient depends weakly on the period of the wave.
Since WI is nearly independent of the period in the considered frequency range,
a single map suffices to describe it (Fig. ).
WI is positive everywhere except in a narrow area west of the cape (the isoline -0.2 is indicated and the
bold line corresponds to WI=0). Figure shows that the corresponding
values of ξz1 are smaller than 1. This suggests that this area is
occupied by waves whose wavelength is longer than everywhere else, a
result which appears in Fig. .
The maps of kR and kI (see Figs. and )
show properties in agreement with the previous analysis.
For periods shorter than 20 d,
kR becomes negative south of the cape. Consequently the waves no longer
propagate westward towards the open sea but eastward towards the coast.
By contrast, north of the cape the propagation always occurs westward, whatever the frequency. At lower frequency this
phenomenon is not observed. The coefficient kI shows a smaller dependence
as a function of the period. The shape of the Kelvin wave is modified
close to the tip of the cape – its offshore extent is smaller since kI is
larger – but it still exists and propagates northwards.
Lastly, note that the order of magnitude predicted in Fig. for a straight
coastline with a south–north orientation is noticeably changed when the shape of the
coast is taken into account. For a wavelength of about 700 km, the corresponding period
was equal to approximately 150 d. Now, Fig. shows that this
wavelength is obtained in a large part of the domain for a period equal to
100 d. Note that this value is close to the values predicted by Clarke and Shi (1991) at the coast (between 84.69 and 115.2 d). Note also that,
in a small area around the extremity of the cape (blue area),
larger wavelengths are compatible with periods of about 100 d.
Conclusions
The analysis of SeaWiFS satellite observations of chlorophyll showed
a well-marked signal along and off the west African coast, between 10
and 22∘ N, in winter (December to April). Along the coast
the high concentration of chlorophyll is associated with the offshore Ekman
drift generated by the equatorward component of the trade wind,
which forces an upward motion. Its variability
is modulated by Kelvin waves propagating northwards.
In December 2002 and January 2003, we observed a
wavelike pattern in the open sea, which extends far away offshore,
up to a distance of about 800 km off the coast. This signal
was visible from 20 December 2002 and was detectable for approximately
1 month, south of the Cape Verde Peninsula.
This pattern suggested the existence of locally generated
Rossby waves, which slowly propagated westward. Indeed, such a wave
can generate an elevation of the lower layers of the ocean corresponding to an upwelling
of nutrient-rich water. The existence of this wave was confirmed by the study of the
SSH signal coming from AVISO altimeter data. It evidenced a wave propagating westward with
a velocity of about 4.5 cm s-1. The existence of a chlorophyll signal far from the
coast – here extending up to 750 km west to the Cape Verde Peninsula – has to our knowledge
never been described. This strongly differs from the coastal signals associated with
Kelvin waves, which have been previously carefully
analysed (see Diakhaté et al., 2016, and the references therein).
In this study we thus investigated the mechanisms which could lead to the existence of
such a wave and analysed the potential role of the cape, by first doing
numerical experiments with a forced non-linear model, then by analytically studying
a linear reduced-gravity model.
The numerical study, based on a reduced-gravity shallow-water model,
showed that a Rossby wave similar to the observed pattern
could be created by a wind burst broadly extending
over the region where the oceanic signal was seen. This agreed with wind reanalysis
of this period.
In our experiments a longshore wind burst which lasted 5 d was used to generate
the oceanic response. We showed that the spatial scale of the latter
matches the spatial scale of the forcing. The timescale of the response, controlled
by the wavelength (see below), is not that of the forcing (5 d)
but much longer around 100 d for the first experiment.
The cape does not seem to modify the basic features of the wave dynamics. It
mainly limits the extent of the wave to the north. However,
when the wind burst has a large zonal extent (of about 1000 km)
and a small meridional extent (not exceeding 100 km),
the response of the model close to the cape depends on the location
of the wind anomaly. When the latter acts south of the cape,
the anomaly which forms around 15∘ W is small and
quickly disappears; a wave which propagates southward seems to prevent
its existence. By contrast, when the wind anomaly
acts north of the cape, an anomaly forms and remains around 15∘ W, without
moving; a secondary wave which propagates southward still
exists but disappears quickly.
The analytical study is new and extends the study of Clarke and Shi (1991)
to the open sea up to a distance of about 1000 km away from the coast. It
helps us interpret the numerical results and gives further information.
It first shows that a timescale around 100 d can be associated with the observed wave,
considering the value of the wavelength (around 700 km).
This value matches the critical value predicted by Clarke and Shi
along the coast of Senegal, even though their model
applied only at the coast when the angle defining the tilt of the coastline
is not too large. It also shows that the role of the cape does
not dramatically modify the dynamics of the system at such timescales.
By contrast, when the period becomes shorter (smaller than 20–30 d),
the waves behave differently north and south of the cape,
as suggested by the numerical experiments. For the studied set-up,
Rossby waves can propagate eastwards, in a narrow band of the ocean whose latitudinal
extent is about 100 km. We verified that this property
vanished when the cape flattened (the period of the wave progressively
becoming shorter). This strongly suggests that the wave dynamics in the vicinity of
a cape – and the associated upwelling – depend on
the geometry of the coastline for timescales shorter than 1 month. These changes no longer matter for longer timescales.
Note that the behaviour difference predicted by the theory is not so important
in the numerical experiments. This is not surprising since the geometry
of the system is different in the numerical experiments.
This study thus suggests that offshore upwellings can be created or enhanced by Rossby waves.
An example of such a phenomenon has been observed in the region off Senegal. This example is probably not unique. For instance Kounta et al. (2018) show patterns which
provide clear evidence of an important Rossby wave activity to the south of the Cape Verde Peninsula
(see, for example, their Fig. 10, which shows a climatology of the volume meridional
transport). Observations will have to be pursued in this region and in
other EBUS region to determine the importance of such events. However, the observations of such a structure by satellite require several conditions which seldom occur together. First we need a long period of observations without clouds, second a
typical wind event able to generate the Rossby wave, and third the existence of nutrients
in the subsurface layers, which could enrich the surface layers.
An important problem is the detectability of these waves.
Dealing with a reduced-gravity model whose characteristics fit the observations,
we found that the elevation of the interface probably does not exceed a few metres.
The interface elevation facilitates the nutrient enrichment of the surface layers
and consequently favours phytoplankton blooms. As the elevation of the interface
is relatively small, the phytoplankton bloom is likely
to occur only under very specific conditions such as a relatively small
average thermocline depth or the presence of phytoplankton species
capable of rapid growth, with a strong chlorophyll signature like diatoms.
In fact, phytoplankton pigment retrieval from ocean colour satellite observation
shows that the chlorophyll signal we observed is dominated by fucoxanthin,
which is a signature of diatoms (Khalil et al., 2019)
Data availability
The water-leaving reflectances were obtained from the SeaWiFS daily
reflectances provided by NASA/GSFC/DAAC (https://oceancolor.gsfc.nasa.gov/data/seawifs/, last access: December 2019)
observed at the top of
the atmosphere (TOA). The atmospheric correction was reprocessed at
LOCEAN (Farikou et al., 2015).
The reflectances are available at the following website:
http://poacc.locean-ipsl.upmc.fr/ (last access: April 2019).
The altimeter products are available from AVISO (Archiving, Validation,
and Interpretation
of Satellite Oceanographic) at https://www.aviso.altimetry.fr/ (last access: December 2019). We used
gridded sea level heights and derived variables (types of dataset:
Ssalto/Duacs gridded
multi-mission altimeter products). The Ssalto/Duacs altimeter products are
produced and distributed by the Copernicus Marine and Environment
Monitoring Service
(CMEMS) (http://marine.copernicus.eu; last access: July 2019).
The wind data are available at the following address:
https://www.ecmwf.int/en/forecasts/datasets/reanalysis-datasets/era-interim (last access: December 2019).
They come from
the global atmospheric reanalysis that is available from 1 January 1979
to 31 August 2019
(see the report by Berrisford, P., Dee, D.P., Poli, P., Brugge, R.,
Fielding, M., Fuentes, M.,
Kållberg, P.W., Kobayashi, S., Uppala, S., published in 2011 at
https://www.ecmwf.int/node/8174; last access: July 2019).
The data can also be made available upon request to the authors.
Since we consider processes whose timescale is larger than a day,
Eq. can be simplified. We define
a characteristic frequency ω0 and a daily frequency
Fd and assume that they verify ϵ=ω0/Fd≪1.
Setting ω0t=τ and F=FdF0, the first two equations of system () become
ϵ∂τTX-F0TY=RX/Fd,ϵ∂τTY+F0TX=RY/Fd,
with RX=-a-1∂Xη-b-1∂Yψ and
RY=-b-1∂Yη+a-1∂Xψ.
An elementary computation leads to the following relations:
(ϵ2∂ττ+F02)TX=(ϵ∂τRX+F0RY)/Fd,(ϵ2∂ττ+F02)TY=(ϵ∂τRY-F0RX)/Fd.
Considering our hypothesis, the terms of order 1
and ϵ can be kept in the previous
equations and the terms of order ϵ2 can be neglected. Consequently we
can use the approximate relations
TX=F-2(∂tRX+FRY),TY=F-2(∂tRY-FRX).
(We used the fact that
ϵFd-1∂τ=F-1∂t and
F0Fd=F.) Note that the terms ∂tRX and ∂tRY
are of order ϵ in comparison with the terms FRY and FRX.
We now introduce these equations in the last equation
of system (). This leads to a new equation:
∂tη+C2ab∂XbF2∂tRX+FRY+∂YaF2∂tRY-FRX=∂tϕ
or equivalently
∂tη-R021a2∂XX2η+1b2∂YY2η-R02abF2∂XbaF2∂tX2η+∂YF∂Xη+F2∂YabF2∂tY2η-∂XF∂Yη=∂tϕ+Rψ,
where R02=C2/F2 is the Rossby radius and Rψ contains the
forcing terms depending on ψ:
Rψ=-C2ab∂X1F2∂tYψ+∂Y1F2∂tXψ+∂XbaF∂Xψ-∂YabF∂Yψ.
The vanishing of the velocity orthogonal to the coordinates
is easily obtained from system (). With the same approximations,
the condition TX=0 at X=0 implies ∂tRX+FRY=0
or equivalently
b∂tX2η+aF∂Yη=-a∂tY2ψ+bF∂Xψ
for all t>0 and Y.
When the forcing terms can be neglected, the previous relations simplify. They become
∂tη-R021a2∂XX2η+1b2∂YY2η-……R02abF2∂XbaF2∂tX2η+∂YF∂Xη+F2∂YabF2∂tY2η-∂XF∂Yη=0,
and at X=0, for all t>0 and Y,
b∂tX2η+aF∂Yη=0.
A method to solve system (–) with the boundary condition () is presented here. It has been shown that the values of z1 and
z2 are known at the boundary thanks to Eq. (). We set
z1=z10+z̃1andz2=z20+z̃2,
where z10 and z20 verifies Eq. (), and the condition
z20=-i(ω/F)z10 everywhere. Consequently, z10 verifies Eq. () everywhere,
J(ξz10)=WR+iWI,
and can be computed on the ocean domain. The results of this computation
are presented in Sect. 4.3 (z20 is also known thanks to the
relation z20=-i(ω/F)z10).
Since z1 and z2 are known on the coast and z10 and z20 everywhere,
z1̃ and z2̃ are known on the coast X=0.
It is now easy to write the equations verified by z1̃ and z2̃.
B1∂Yaz̃1R0-∂Xbz̃2R0=∂Yaz10R0-∂Xbz20R0=Z0,B2z̃12+2z̃1z10-w1+z̃22+2z̃2z20-w2=0,
with z̃1=0 and z̃2=0 for X=0.
The variables z1̃ and z2̃ can be computed on a grid
(-iΔX,jΔY) for i=0,1,…,N,
j=-P,…,-1,0,1,…,P. They are known for
X=0 (i=0). We suppose that they have been computed for
i>0 (values z̃1,i,j and z̃2,i,j) and show how
z̃1,i+1,j and z̃2,i+1,j can be computed.
Equation () can be discretized in the following way:
bz̃2,i+1,jR0=bz̃2,i,jR0-ΔX2ΔYaz̃1,i,j+1R0-bz̃1,i,j-1R0+ΔXZ0,i,j,
the error being proportional to ΔX. A boundary condition
(for example, z̃2,i,-P=0) is prescribed to end the computation.
Knowing z̃2,i+1,j, the value of z̃1,i+1,j is obtained by solving
z̃1,i+1,j2+2z̃1,i+1,jz10-w1+z̃2,i+1,j2+2z̃2,i+1,jz20-w2=0.
The computation of z10 and z20 would correspond to a solution such as
TX=0 everywhere. The correction brought by z̃1 and z̃2
is associated with a mass transport perpendicular to the coast. As
the latter in general is much smaller than the transport TY parallel to the coast,
it is expected that z̃1 and z̃2 are small in comparison with
z10 and z20 (they are null at X=0 and increase
proportionally to the distance to the coast). In Sect. 4.3, the approximate
value of k associated with the solution z10 is fully described.
Author contributions
MC provided the satellite analysis. JS made the numerical simulation. The paper benefitted from input from all the authors
Competing interests
The authors declare that they have no conflict of interest.
Acknowledgements
We thank Xavier Capet for fruitful discussions, for his careful reading
of the paper and pertinent comments. We thank Rémi Tailleux and
Jacyra Soares for their precise and constructive reviews, which permitted us to
improve this article greatly.
Review statement
This paper was edited by Neil Wells and reviewed by Remi Tailleux and Jacyra Soares.
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