Generation of Rossby waves off the Cape Verde peninsula; role of the coastline

Abstract. In December 2002 and January 2003 satellite observations of Chl rophyll showed a strong coastal signal along the west african coast between 10◦ and22◦ N. In addition, a wavelike pattern with a wavelength of about 750 kms was observed from December2


Introduction
Eastern Boundary Upwelling Systems (EBUS) -let us quote the California, Humboldt, Canary and Benguela upwelling 20 systems -constitute an ubiquitous feature of the coastal ocean dynamics, which has been extensively studied. They are biologically very productive thanks to a transport of nutrients from deep ocean layers to the surface, which favors the bloom of phytoplanktons. Consequently they present a strong signature, which is detectable by ocean color satellite sensors (see for example Gruber 2012, 2013).
The dynamics of EBUS has first been studied with conceptual models. Upwellings are created by alongshore equatorward winds (Allen, 1976, or McCreary et al., 1986 generating an offshore Ekman transport, which is compensated 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 5 poleward (Moore, 1968;Allen, 1976;Gill and Clarke, 1974;Clarke, 1977Clarke, , 1983McCreary, 1981).
At a given frequency, there is a critical latitude at which the 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. 10 Using a high-resolution 3D numerical model, Batteen (1997) then Marchesiello et al. (2003) confirmed that the shape of the coastline actually plays a role on the upwelling pattern. However they did not investigate by which mechanisms they are driven. In particular they did not try to confront their results with the theoretical investigations of Crépon et al. (1982Crépon et al. ( , 1984 who analyzed 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 The role of the forcing has also been investigated, both from an observational and theoretical viewpoint. Enriquez and Friehe (1995) computed the wind stress and wind stress curl off the California shelf from aircraft measurements and showed, thanks to numerical experiments with a two layer model and an analytical study, that a non zero wind stress curl expands the horizontal extent of upwelling offshore; it increases from 20-30 km to 80-100 km. The importance of the wind stress curl, which generates 20 a strong Ekman pumping, were first emphasized by Richez et al., (1984) and later by Pickett and Paduan (2003) then Castelao and Barth (2006). They could establish that the Ekman pumping and the Ekman transport due to the alongshore winds have a comparable importance in the California Current area or off Cabo Frio in Brazil. These works did not study what occurs beyond 100 km from the coast.
In the present paper, we study the role of the wind stress and of the coastline geometry in generating mesoscale anomalies 25 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 Senagalese coast, in the region of the Sénégalo-Mauritanian upwelling (see Lathuillière et al., 2008, andFarikou et al., 2015 to find 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 30 along west Africa). Attention is focused on offshore mesoscale activity associated with the upwelling, a recurrent feature of upwelling systems (see Capet et al., 2008 a, b).
The paper is articulated as follows. Observations of Chlorophyll in 2002-2003 off the west African coast are shown and described in section 2. In section 3, a numerical study with a on linear reduced gravity shallow water model with a single active layer on the sphere is conducted; it shows that a wind stress anomaly active during a few days can generate a pattern that seems very similar to the observated one. The impacts of the wind anomaly extent and of the coastline geometry are also briefly studied. In section 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.
2 Observation of a wave off the Cape Verde peninsula from an ocean color satellite sensor. 5 The Senegalo-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 ocean-color data and AVHRR 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), andCapet et al. (2017). These studies indicate that the presence of an intense upwelling is attested by both ocean-color and sea-surface temperature signals. Moreover this upwelling shows a strong seasonal modulation. It starts to intensify in October, 10 reaches its maximum in April and slows down in June. Very high chlorophyll-a concentration are observed near the coast when the maximum is reached. However the concentration rapidly decreases offshore (Farikou et al., 2013(Farikou et al., , 2015Sawadogo et al., 2009), suggesting that the upwelling extent and the eddy activity in this region is less than in other upwelling systems like the Californian Upwelling system Estrade, 2009, Capet et al., 2017).
From December 20 th 2002 up to January 8 th 2003, a strong chlorophyll signal was observed along the African coast on 15 SeaWIFs satellite images, between 10 • N and 22 • N, indicating an intense biological activity. In addition, a well-defined "sinelike" pattern ( Figure 1) was observed on ten images taken at ten non-consecutive different days, which excludes a possible artifact due to image processing. This pattern is located between 12 • N 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 3). 20 This pattern, which broadly keeps the same form during a twenty day 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 hazardeous). After January 8 th 2003, a cloudy period of several days occurred, which prevented satellite observations. At the end of this episode (January 15 th ) 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 one month, its typical time-scale is expected to range between one and a few months.

25
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 of 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 30 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 one 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 ω c = cβ cosΘ/2f 0 . 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 5 f 0 and β the usual parameters linked to the Earth's rotation. These authors found that ω c ranges between 84.69 and 115.2 days around the Cape Verde peninsula (see their table 2a). These values are close to the characteristic time scales we expect here.
The existence of a long period divergent baroclinic Rossby wave could explain the offshore sine-like pattern described here.
To investigate this hypothesis, we first present 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 10 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 days to one year.

The model
The numerical model is a reduced gravity model on the sphere with one active layer of thickness h. It extends over an infinite 15 layer at rest. The velocity v in the active layer and the thickness h verify the equations and where n is a vector normal to the Earth's surface and rotv = (∇ × v).n. The function Φ is equal to the reduced gravity.
We assume for simplicity that the vector τ 0 , which represents the surface wind stress divided by the ocean density, derives from a potential φ 0 (x, y): τ 0 = −gradφ 0 (implying an irrotational mean wind). This hypothesis allows us to compute explicitely the obtained mean state. Indeed v = 0 and g ⋆ h 2 0 /2 = −φ 0 + C 0 is an obvious solution of the previous system (the constant C 0 is determined by using the fact that the mean value of h 0 remains unchanged during the integration). It will 25 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.

Numerical resolution
The model domain is closed and centered 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 modeled as indicated in Fig. 2 in order to mimic the geometry of the coast of Senegal (simplified and smoothed). The mean value of h 0 is equal to 200 m and the reduced gravity g ⋆ to 0.02 m s −2 .
Consequently the Rossby radius of deformation R 0 = √ g ⋆ h 0 /f 0 at the latitude of the Cape Verde is equal to 53 km.

5
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. 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 seconds. The viscosity ν and the coefficient r of equation (2) are respectively equal to 28 m 2 s −1 and 8 × 10 −5 m s −1 (h/r ≃ 1 month).

Numerical results
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 τ 0 equal to 6 × 10 −5 m 2 s −2 ) and oriented along a south-south-west direction is applied during four years from a rest state. A stationary mean state, which verifies the theoretical relation given in section 3.1, is at that time reached. A north-south wind stress anomaly which extends over approximately 500 km and whose maximum is still equal to 0.06 N m −2 is then 15 applied during five days (see Fig. 2 first panel). The integration is continued during 45 days, after the anomaly has disappeared.
After the wind stress anomaly has vanished, the subsequent states of the ocean are shown every five days in Fig. 2  study presented in the next section when the period of the wave is about 100 days. This modeled 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 Rossby wave generated by a wind burst.
Though the duration of the wind burst is short in our numerical experiment (5 days), the response of the system privileges a much longer time scale, exceeding 2 months. This result is not inconsistent. Indeed the Fourier transform of a rectangular 30 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 as the one observed here. Figure 3 shows the results obtained in a comparable experiment when the cape is absent. The response of the model is very similar: in the area of the wind anomaly, a Kelvin wave forms then quickly desappears 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 previously was the cape. This suggests that the cape simply limits the extent of the wave northward but does not modify its dynamics. The theoretical study of section 4 will confirm that the role of the cape remains moderate at low frequency.

5
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 Figure 4, which shows the response of the model to a wind burst whose extent is four times smaller than the initial wind burst of Figure Figures 5 and 6). These anomalies create a Rossby wave with a large zonal extent. 15 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 four 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 20 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.
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 desappears; a wave which propagates southward seems to prevent its existence. On the contrary, when the wind anomaly acts north of the cape, an 25 anomaly forms around 15 • W and gets stuck in this place; the wave which propagates southward still exists but its amplitude is about twice samller 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 generalize the approach followed by Clarke and Shi (1991) in order to study the response of the ocean at satellite data). We thus suppose that the eastern coast presents a cape and analyze how the free waves are influenced by its existence.
A new system of coordinates is introduced over the entire domain. It allows us to establish equations valid over the whole basin but with a very simple boundary condition on the domain frontier where the cape is located. By using the WKB approximation and making the hypothesis that the mass transport parallel to the coast is much larger than the perpendicular one, we 5 can investigate the impact of the curvature of the coastline on the Rossby wave dynamics.

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 φ 0 (x, y) (τ 0 = −gradφ 0 ), as in section 3. Numerical integrations have shown that the exact solution v = 0 and g ⋆ h 2 0 /2 = −φ 0 + C 0 is actually obtained after a few year integration (see section 3.1 and 3.3).

10
If an anomaly (τ x , τ y ) is added to the mean forcing τ 0 , a perturbation is generated; it is characterized by a depth anomaly h (the thickness of the first layer is now h 0 + 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 (τ x , τ y ) can be written in terms of a potential φ(x, y, t) and a stream function ψ(x, y, t), so that the divergent part of the forcing is given by −∆φ and the rotationnal part by ∆ψ.
The equations verified by the anomalies h and v are thus: 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 where c = √ g ⋆ h 0 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 25 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 system (5). Here, we consider an eastern boundary whose angle with the meridians smoothly varies 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; equations (5) are correspondingly modified to match the new coordinates.

5
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 e X , e Y associated with this coordinates system, the line element dl reads where a and b are geometrical factors which convey the stretching of the coordinates along orthogonal directions (note that the where the initial coordinates x and y are now functions of the new coordinates X and Y are used to compute a and b). west of the cape). A detailed study of this example will be presented in subsection 4.3.
Using the coordinates (X, Y ), system (5) becomes where F , C, T X , T Y are functions of X and Y corresponding to f , c, T x and T y (the notations for η, φ and ψ have not been changed to enhance the readability, but these functions also depend on X and Y ).
We now concentrate on processes whose time scale is much larger than one day. Moreover we consider only the evolution of free waves. This situation corresponds to the case numerically investigated in section 3 and illustrated in Figures (2 -6): 25 the wind stress anomaly that had created the depth anomalies has ceased to exert. With these hypotheses, system (6) can be reduced into the following equation which characterizes the evolution of η: where R 2 0 = C 2 /F 2 is the Rossby radius. The boundary condition at the eastern coast now reads: for all t > 0 and Y . The details of the computations can be found in Appendix 1.
When a and b are close to 1 (see Figure 8) the new coordinates system is nearly similar to the original one; indeed F no longer depends on X but only on Y ≃ y, and in those regions where this condition is verified equation (7) simplifies: We recognize the equation characterizing the propagation of waves in the β plane (β = ∂ Y f ) for a shallow water model. For spatial scales much larger than R 0 and at low frequency, equation (9) can be further simplified. It becomes ∂ t η−R 2 0 ∂ Y F ∂ X η = 0 which simply models the westward propagation of long Rossby waves.
The coefficient b decreases as one gets closer to the cape (see Figure 8). An impact of the cape in the open sea may therefore 10 be expected because of the terms proportional to ∂ X ( b aF 2 )∂ 2 tX η and ∂ X F ∂ Y η in equation (7).

Ray theory
When a wave propagates in a medium whose properties spatially change, it does not follow straight lines but more complex paths. Ray theory -or WKBJ 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 15 of the components k and l of the wavevectors are taken into account by introducing a complex function θ(X, Y ) = θ R + iθ I such as k = ∂ X θ and l = ∂ Y θ. The potential η is then equal to where it is assumed that As required by the theory, these inequalities ensure that the wavelength is smaller than the typical scale of variation of the system, here conveyed 20 by R 0 , F and the coefficients a and b. Excepted in a very close vicinity of the cape (distance smaller than around ten kilometers) 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 equation (10) in the neighbourhood of a point M 0 of coordinates (X 0 , Y 0 ) by 25 the approximate expression where k = ∂ X θ| M0 and l = ∂ Y θ| M0 . The physical meaning of this solution is explicited by taking its real part: Ocean Sci. Discuss., https://doi.org/10.5194/os-2019-22 Manuscript under review for journal Ocean Sci. Discussion started: 4 April 2019 c Author(s) 2019. CC BY 4.0 License.
The solution must not increase westward since it cannot become infinite, which implies k I < 0. Note however that k I > 0 is possible if this occurs only in a (small) bounded domain. For k R > 0, the wave propagates westwards.
The values of k and l are obtained by computing the function θ from equations (7) and (8). After simplifying them by using the hypotheses, we find that θ verifies the approximate (eikonal or Hamilton-Jacobi) equation with the boundary condition at X = 0.
To make the computations clearer we set 2 ) and the problem (11) associated with the boundary condition (12) is rewritten as the following system :

20
At the boundary, equations (14) and (15) permit to determine the values of z 1 and z 2 , hence of k and l. Indeed, the introduction of (15) in (14) leads to the equation Setting ξ = 1 − (ω/F ) 2 , this equation can be rewritten where J(z) = 1 2 (z + 1 z ) is the Joukowsky transform of z; W R and W I are given by the relations Equation (16) is the dispersion relation of the wave at the boundary. The Joukowsky transform of ξ z 1 (or equivalently of i F ω ξ z 2 ) depends on the frequency ω, the mean state (through R 0 ), the latitude (through F ) and the geometry of the coast (through a and b).
The sketch in Figure 9, which shows the half complex plane ℑ(z) < 0 (upper panel) and the complex plane W = J(z) (lower panel), explicits how the Joukowsky transformation works. The inferior half plane has been chosen since we expect k I < 0, or 5 in other words ℑ(z) < 0.
-When W I = ℑ(W ) is positive, the complex number ξ z 1 such as J(ξ z 1 ) = W = W R + i W I has a norm smaller than 1 (it corresponds to the gray areas in Figure 9). The wavelength is thus large (λ > 2πξR 0 /a). If W R = ℜ(W ) is positive, the phase speed is negative (westward propagation).
-When W I = ℑ(W ) is negative, the complex number ξ z 1 such as J(ξ z 1 ) = W = W R + i W i has a norm larger than 1 (it 10 corresponds to the white areas). The wavelength is thus small. If W R = ℜ(W ) is positive, the phase speed is still negative.
-When W I vanishes, a situation which occurs when the coast is a straightline, the unique complex solution previously found can cease to exist. Indeed, if |W R | is smaller than 1, there is one complex solution whose norm is equal to 1 (on the half circle in bold in Figure 9); and if |W R | is larger than 1, two real solutions are obtained (between ] − 1, 1[ and outside this interval). The case (W I = 0) is detailed in the next subsection. 15 At the coast, relation (15) determines z 2 when z 1 is known. It implies that For low frequencies ω/F is much smaller than 1 and l I and l R can be ignored. An approximate expression of the waves is and the dynamics is controlled by the westward propagation of Rossby waves. For shorter periods, the situation may be more Knowing z 1 and z 2 along the coast, it is possible to continue the resolution of the problem (13-14) and compute explicitely the rays characterizing the propagation of the waves. An algorithm which fulfils this objective is described in appendix 2.

25
However, under the supplementary assumption that |T X | remains much smaller than |T Y | up to a distance from the coast of about a few Rossby radius of deformation, approximate expressions for z 1 and z 2 can be obtained analytically. This also permits to initialize the algorithm described in appendix 2.

Analytical study for |T
In this case, equation (15) which is exact at the coast X = 0 can also be used on a band of a few hundred kilometers off the  (7). 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 -let us mention Richez et al. (1984), 5 Grimshaw and Allen (1988), Clarke and Shi (1991) (16) has two real solutions. If W R ≫ 1 these solutions are close to 2W R ≃ R 0 β/ω and 1/2W R ≃ ω/(βR 0 ). Consequently, the wave number k is equal to either β/ω or ω/(βR 2 0 ). This result proves the existence of Rossby waves, whose wavelength is either short or long. When the frequency increases, W R decreases and eventually reaches the critical value 1. When W R becomes smaller than 1, there are two complex conjugate solutions. The only acceptable 20 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 k I is represented as a dashed curve in Figure 10 (bottom panel). When the wave frequency is close to the critical value, k I vanishes and consequently the Kelvin wave is no longer trapped along the coast. The westward propagation of a Rossby wave with a significant amplitude can occur. 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/day. On the other hand, as l R 25 vanishes with k I , 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 is characterized by the coefficients W R and W I which 30 convey the effect of the coastline on the propagation of the wave. The coefficient W R can be splitted into two terms; the first ω is similar to the term obtained from a straight coastline whereas the second one 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 days 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 of 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. On the contrary, at low frequency, the lengthening or shortening of the path 5 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 of the coastline geometry also prevent the existence of two distinct solutions at low frequency. Indeed, W I differs from 0 and consequently two complex solutions are obtained as explicited in Figure 9. If W I is strictly negative (grey area), the solution z is inside the half unit disk and k R is small. If W I is strictly positive, the solution is outside and k R is large. 10 The degeneracy of the equation thus desappears and a selection of the wavelength operates. Figure 11 shows W R for T = 10, 20, 50, and 100 days and makes visible an interesting property. When the period T is equal to 10 days, a dissymmetry around the cape occurs: W R is negative south and positive north of the cape. This dissymmetry weakens when the period increases. For T = 20 days, the area where W R is negative is strongly reduced and for T = 50 days, it has vanished. Since the signs of W R and k R are similar (see Figure 9 and the associated comments), a negative k R is expected 15 in this area and actually appears (see Figure 13) Since W I is nearly independent of the period in the considered frequency range, a single map suffices to describe it ( Figure   12). W I is positive everywhere except in an narrow area west of the cape (the isoline -0.2 is indicated and the bold line corresponds to W I = 0). Figure 9 shows that the corresponding values of ξz 1 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 Figure 13. 20 The maps of k R and k I (see Figures 13 and 14) show properties in agreement with the previous analysis. For periods shorter than 20 days, k R becomes negative south of the cape. Consequently the waves no longer propagate westward towards the open sea, but eastward towards the coast. On the contrary, north of the cape the propagation occurs always westward, whatever the frequency. At lower frequency this phenomenon is not observed. The coefficient k I 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 k I is 25 larger -but it still exists and propagates northwards.
Lastly, note that the order of magnitude predicted in Figure 10 for a straight coastline with a south north orientation are 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 days. Now, figure 13 shows that this wavelength are obtained in a large part of the domain for a period equal to 100 days. Note that this value is close to the values predicted by Clarke and Shi (1991) at the coast 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. We investigated this possibility and the potential role of the cape, by first doing numerical experiments with a forced nonlinear 10 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. Simultaneously, a Kelvin wave formed along the coast and quickly propagated northwards. In our experiments a longshore wind burst which lasted 5 days was used to generate the oceanic response. We showed that the spatial scale of the latter matches the spatial scale 15 of the forcing. The time scale of the reponse, controlled by the wavelength (see below), is not that of the forcing (5 days) but much longer around 100 days 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 20 the cape, the anomaly which forms around 15 • W is small and quickly desappears; a wave which propagate southward seems to prevent its existence. On the contrary, 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 exist but desappears quickly.
The analytical study, which extends the method suggested by Clarke and Shi (1991) to the open sea up to a distance of about 1000 km away from the coast, helps us interpret the numerical results and gives futher results. It first shows that a time scale 25 around 100 days 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 time scales.
On the contrary, when the period becomes shorter (smaller than 20-30 days), the waves behaves differently north and south 30 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 -depends on the geometry of the coastline for time scales shorter than one month.
These changes no longer matter for longer time scales. Note that the behaviour difference predicted by the theory are not so important in the numerical experiments. This is not surprising since the geometry of the system is different in the numerical experiments.
Lastly this study suggests that offshore upwellings can be created or enhanced by Rossby waves. An example of such a phenomenom 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 (see for example their Figure 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 structure by satellite requires several condition which are seldom occuring together. First we need a long period of observations without 10 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.

Appendix A: Appendix A
Since we consider processes whose time scale is larger than a day, equations (6) can be simplified. We define a characteristic frequency ω 0 and a daily frequency F d and assume that they verify ǫ = ω 0 /F d ≪ 1. Setting ω 0 t = τ and F = F d F 0 , the first 15 two equations of system (6) become An elementary computation leads to the following relations Considering our hypothesis, the terms of order 1 and ǫ can be kept in the previous equations and the terms of order ǫ 2 can be 20 neglected. Consequently we can use the approximate relations (We used the fact that ǫF −1 d ∂ τ = F −1 ∂ t and F 0 F d = F ). Note that the terms ∂ t R X and ∂ t R Y are of order ǫ in comparison with the terms F R Y and F R X .
We now introduce these equations in the last equation of system (6). This leads to a new equation or equivalently where R 2 0 = C 2 /F 2 is the Rossby radius and R ψ contains the forcing terms depending on ψ: The vanishing of the velocity orthogonal to the coordinates is easily obtained from system (A3). With the same approximations, for all t > 0 and Y .
When the forcing terms can be neglected, the previous relations simplify. They become respectively and at X = 0, for all t > 0 and Y b ∂ 2 tX η + aF ∂ Y η = 0 (A8)

Appendix B: Appendix 2 15
A method to solve system (13-14) with the boundary condition (15) is presented here. It has been shown that the values of z 1 and z 2 are known on the boundary thanks to equation (16). We set z 1 = z 0 1 +z 1 and z 2 = z 0 2 +z 2 where z 0 1 and z 0 2 verifies equation (14) and the condition z 0 2 = −i(ω/F )z 0 1 everywhere. Consequently, z 0 1 verifies equation (16) everywhere 20 J(ξ z 0 1 ) = W R + iW I and can be computed on the ocean domain. The results of this computation are presented in section 4.3 (z 0 2 is also known thanks to the relation z 0 2 = −i(ω/F )z 0 1 ).
Since z 1 and z 2 are known on the coast and z 0 1 and z 0 2 everywhere,z 1 andz 2 are known on the coast X = 0. It is now easy to write the equations verified byz 1 andz 2 .
Knowingz 2,i+1,j , the value ofz 1,i+1,j is obtained by solving z 2 1,i+1,j + 2z 1,i+1,j (z 0 1 − w 1 ) +z 2 2,i+1,j + 2z 2,i+1,j (z 0 2 − w 2 ) = 0 (B3) The computation of z 0 1 and z 0 2 would correspond to a solution such as T X = 0 everywhere. The correction brought byz 1 and z 2 is associated with a mass transport perpendicular to the coast. As the latter in general is much smaller than the transport T Y 15 parallel to the coast, it is expected thatz 1 andz 2 are small in comparison with z 0 1 and z 0 2 (they are null at X = 0 and increase proportionally to the distance to the coast). In section 4.3, the approximate value of k associated with the solution z 0 1 is fully described.