3.1 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 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 $\mathrm{rot}\mathit{v}=(\mathrm{\nabla }\times \mathit{v}).\mathit{n}$. The function Φ is equal to $g^{\star }h+{\mathit{v}}^{\mathrm{2}}/\mathrm{2}$ 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): ${\mathit{\tau }}_{\mathrm{0}}=-\mathrm{grad}{\mathit{\varphi }}_{\mathrm{0}}$ (implying an irrotational mean wind). This hypothesis allows us to compute explicitly the obtained mean state. Indeed, v=0 and $g^{\star }{h}_{\mathrm{0}}^{\mathrm{2}}/\mathrm{2}=-{\mathit{\varphi }}_{\mathrm{0}}+C_{\mathrm{0}}$ 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.

Figure 3Mean 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).

Figure 4Response 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).

3.2 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. 4 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 $R_{\mathrm{0}}=\sqrt{g^{\star }{h}_{\mathrm{0}}}/f_{\mathrm{0}}$ 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 $(\mathrm{1}/\mathrm{12}){}^{\circ }$ 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. (2) are respectively equal to 28 and $\mathrm{8}\times {\mathrm{10}}^{-\mathrm{5}}$ m s⁻¹ ($h/r\simeq \mathrm{1}$ month). More details can be found in Février et al. (2007), where the two-layer version of this model is described.

3.3 Numerical set-up of the model

In Fig. 3 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 $\mathrm{6}\times {\mathrm{10}}^{-\mathrm{5}}$ 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. 3, a wind anomaly was active when the wave of Fig. 1 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. 4a). 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.

3.4 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. 4 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 ($k_R\simeq \mathrm{0.84}\times {\mathrm{10}}^{-\mathrm{5}}$ 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.

Figure 5Solution obtained for the same conditions as in Fig. 4 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 5 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. 6, which shows the response of the model to a wind burst whose extent is 4 times smaller than the initial wind burst of Fig. 4. 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).

Figure 6The wind anomaly extent is 4 times smaller than the one used in Fig. 4 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. 7 and 8) 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. 7a and 8a). 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.

Figure 7The 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).

Figure 8The 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.

4.1 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) (${\mathit{\tau }}_{\mathrm{0}}=-\mathrm{grad}{\mathit{\varphi }}_{\mathrm{0}}$), as in Sect. 3. Numerical integrations have shown that the exact solution v=0 and $g^{\star }{h}_{\mathrm{0}}^{\mathrm{2}}/\mathrm{2}=-{\mathit{\varphi }}_{\mathrm{0}}+C_{\mathrm{0}}$ are actually obtained after a few years' integration (see Sect. 3.1 and 3.3).

If an anomaly (τ_x,τ_y) is added to the mean forcing τ₀, a perturbation is generated; it is characterized by a depth anomaly h (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 (τ_x,τ_y) can be written in terms of a potential $\mathit{\varphi }(x,y,t)$ and a stream function $\mathit{\psi }(x,y,t)$,

so that the divergent part of the forcing is given by −Δϕ and the rotational part by Δψ.

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 T_x=h₀u and T_y=h₀v and a potential η equal to $g^{\star }{h}_{\mathrm{0}}h+\mathit{\varphi }$. It becomes

where $c=\sqrt{g^{\star }{h}_{\mathrm{0}}}$ 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 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. 5). 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. 5 is correspondingly modified to match the new coordinates.

A new orthogonal system of coordinates $X=\mathcal{X}(x,y)$ and $Y=\mathcal{Y}(x,y)$ is thus introduced such that the eastern boundary is now defined by the simple equation X=0 (rather than a complex one such as $f(x,y)=\mathrm{0}$). 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 relations

where the initial coordinates x and y are now functions of the new coordinates X and Y, are used to compute a and b).

Figure 9Example 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.

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Figure 10The coefficients a and b are given for the coordinate system of Fig. 9 (they have no dimension).

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Such a coordinate change is illustrated in Fig. 9 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. 10. 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)” (5) becomes

where F, C, T_X and 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).

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:

We use the complete system (6) 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 $(\mathrm{1}/b^{\mathrm{2}})({\partial }_Y\mathit{\theta }{)}^{\mathrm{2}}$ and $i{F}^{\mathrm{2}}{\partial }_Y(a/(b{F}^{\mathrm{2}}\left)\right){\partial }_Y\mathit{\theta }$ 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. 4–8: the wind stress anomaly that had created the depth anomalies has ceased to exist. With these hypotheses, system (6) can be reduced to the following equation, which characterizes the evolution of η:

where $R_{\mathrm{0}}=C/F$ 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 A.

When a and b are close to 1 and the coast has a south–north orientation (see Fig. 10), the new coordinate system is nearly similar to the original one; indeed F mainly depends on Y≃y only, and in those regions Eq. (7) thus simplifies to

We recognize the equation characterizing the propagation of waves in the β plane ($\mathit{\beta }={\partial }_{Y}F$) 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. (9) can be further simplified. It becomes ${\partial }_t\mathit{\eta }-R_{\mathrm{0}}^{\mathrm{2}}{\partial }_{Y}F{\partial }_X\mathit{\eta }=\mathrm{0}$, which simply models the westward propagation of long Rossby waves.

The coefficient b decreases as one gets closer to the cape (see Fig. 10). An impact of the cape in the open sea may therefore be expected because of the terms proportional to ${\partial }_X\left(\frac{b}{a{F}^{\mathrm{2}}}\right){\partial }_{tX}^{\mathrm{2}}\mathit{\eta }$ and ${\partial }_{X}F{\partial }_Y\mathit{\eta }$ in Eq. (7).

4.2 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 $\mathit{\theta }(X,Y)={\mathit{\theta }}_R+i{\mathit{\theta }}_I$ such as $k={\partial }_X\mathit{\theta }$ and $l={\partial }_Y\mathit{\theta }$. The potential η is then equal to

where it is assumed that $\left|{\mathit{\eta }}_{\mathrm{0}}^{-\mathrm{1}}{\partial }_X{\mathit{\eta }}_{\mathrm{0}}\right|\ll \left|{\partial }_X\mathit{\theta }\right|$, $\left|k^{-\mathrm{1}}{\partial }_{X}k\right|\ll \left|{\partial }_X\mathit{\theta }\right|$, $\left|{\mathit{\eta }}_{\mathrm{0}}^{-\mathrm{1}}{\partial }_Y{\mathit{\eta }}_{\mathrm{0}}\right|\ll \left|{\partial }_Y\mathit{\theta }\right|$, $\left|l^{-\mathrm{1}}{\partial }_{l}l\right|\ll \left|{\partial }_Y\mathit{\theta }\right|$. 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. (10) in the neighbourhood of a point M₀ of coordinates (X₀,Y₀) by the approximate expression

where $k={\partial }_X\mathit{\theta }{|}_{M_{\mathrm{0}}}$ and $l={\partial }_Y\mathit{\theta }{|}_{M_{\mathrm{0}}}$. The physical meaning of this solution is explicated by taking its real part:

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 Eqs. (7) and (8). After simplifying them by using the hypotheses above, 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

• $z_{\mathrm{1}}=\left({\partial }_X\mathit{\theta }\right)R_{\mathrm{0}}/a=k{R}_{\mathrm{0}}/a$,

• $z_{\mathrm{2}}=\left({\partial }_Y\mathit{\theta }\right)R_{\mathrm{0}}/b=l{R}_{\mathrm{0}}/b$,

• $w_{\mathrm{1}}=(R_{\mathrm{0}}/\mathrm{2}b)\left[{\partial }_Y\left(\frac{F}{\mathit{\omega }}\right)+i{F}^{\mathrm{2}}{\partial }_X\left(\frac{b}{a{F}^{\mathrm{2}}}\right)\right]$,

• $w_{\mathrm{2}}=(R_{\mathrm{0}}/\mathrm{2}a)\left[-{\partial }_X\left(\frac{F}{\mathit{\omega }}\right)+i{F}^{\mathrm{2}}{\partial }_Y\left(\frac{a}{b{F}^{\mathrm{2}}}\right)\right]$,

and the problem (11) associated with the boundary condition (12) is rewritten as the following system:

At the boundary, Eqs. (14) and (15) permit us to determine the values of z₁ and z₂ and hence of k and l. Indeed, the introduction of Eq. (15) in Eq. (14) leads to the equation

Setting $\mathit{\xi }=\sqrt{\mathrm{1}-(\mathit{\omega }/F{)}^{\mathrm{2}}}$, this equation can be rewritten as

where $J\left(z\right)=\frac{\mathrm{1}}{\mathrm{2}}(z+\frac{\mathrm{1}}{z})$ 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 z₁ (or equivalently of $i\frac{F}{\mathit{\omega }}\mathit{\xi }{z}_{\mathrm{2}}$) depends on the frequency ω, the mean state (through R₀), the latitude (through F) and the geometry of the coast (through a and b).

Figure 11In (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 $k_I=a\mathrm{\Im }\left(z\right)/\left(\mathit{\xi }{R}_{\mathrm{0}}\right)$ must be negative. The half circle $\left|z\right|=\mathrm{1}$ 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 $[-\mathrm{1},\mathrm{1}]$ 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.

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The sketch in Fig. 11, 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 k_I<0 or in other words ℑ(z)<0.

• When W_I=ℑ(W) is positive, the complex number ξ z₁ such as $J\left(\mathit{\xi }{z}_{\mathrm{1}}\right)=W=W_R+i{W}_I$ has a norm smaller than 1 (it corresponds to the grey areas in Fig. 11). The wavelength is thus large ($\mathit{\lambda }>\mathrm{2}\mathit{\pi }\mathit{\xi }{R}_{\mathrm{0}}/a$). 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 z₁ such as $J\left(\mathit{\xi }{z}_{\mathrm{1}}\right)=W=W_R+i{W}_i$ has a norm larger than 1 (it 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 straight line, the unique complex solution previously found can cease to exist. Indeed, if $\left|W_R\right|$ is smaller than 1, there is one complex solution whose norm is equal to 1 (on the half circle in bold in Fig. 11); and if $\left|W_R\right|$ is larger than 1, two real solutions are obtained (between $]-\mathrm{1},\mathrm{1}[$ and outside this interval). The case (W_I=0) is detailed in the next subsection.

At the coast, relation (15) determines z₂ when z₁ 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 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. 13–14) 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 $\left|T_X\right|$ remains much smaller than $\left|T_Y\right|$ 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.

4.3 Analytical study for $\left|T_X\right|\ll \left|T_Y\right|$

In this case, Eq. (15), 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. (16) 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. (9). 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.

4.3.1 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, $a=b=\mathrm{1}$ and F depends only on Y. Thus W_I=0 and

$$\mathrm{2}{W}_R={\displaystyle \frac{R_{\mathrm{0}}\mathit{\beta }}{\mathit{\omega }}}{\displaystyle \frac{\mathrm{2}{\mathit{\xi }}^{\mathrm{2}}-\mathrm{1}}{\mathit{\xi }}},$$

where $\mathit{\beta }={\partial }_{Y}F$.

Figure 12(a) W_R is shown for latitudes ranging from 5^∘ N (left segment) to 40^∘ N (right segment). The critical value W_R=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^∘).

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In Fig. 12a, the coefficient W_R is shown at different latitudes as a function of the period. At 15^∘ N, the critical value W_R=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 W_R for 10^∘ N (black), 15^∘ N (grey) and 20^∘ N (light grey). When W_R>1, a condition which is always fulfilled at low frequency (characteristic time longer than 125 d at 15^∘), Eq. (16) has two real solutions. If W_R≫1, these solutions are close to $\mathrm{2}{W}_R\simeq R_{\mathrm{0}}\mathit{\beta }/\mathit{\omega }$ and $\mathrm{1}/\mathrm{2}{W}_R\simeq \mathit{\omega }/\left(\mathit{\beta }{R}_{\mathrm{0}}\right)$. Consequently, the wavenumber k is equal to either β∕ω or $\mathit{\omega }/\left(\mathit{\beta }{R}_{\mathrm{0}}^{\mathrm{2}}\right)$ (see Eq. 16). 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 of 1. When W_R 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 k_I is represented as a dashed curve in Fig. 12b. When the wave frequency is close to the critical value, k_I 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 l_R 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).

4.3.2 Case of a cape

The wave dynamics in a neighbourhood of the cape are characterized by the coefficients W_R and W_I which convey the effect of the coastline on the propagation of the wave. The coefficient W_R can be split into two terms; the first one $\frac{R_{\mathrm{0}}}{\mathrm{2}b}\frac{\mathrm{2}{\mathit{\xi }}^{\mathrm{2}}-\mathrm{1}}{\mathit{\xi }}{\partial }_Y\frac{F}{\mathit{\omega }}$ is similar to the term obtained from a straight coastline, whereas the second one $\frac{R_{\mathrm{0}}}{\mathrm{2}a}\frac{\mathrm{1}-{\mathit{\xi }}^{\mathrm{2}}}{\mathit{\xi }}{\partial }_Y\frac{a}{b}$ 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, W_I differs from 0, and consequently two complex solutions are obtained as explicated in Fig. 11. If W_I is strictly negative (grey area), the solution z is inside the half unit disc and k_R is small. If W_I is strictly positive, the solution is outside and k_R is large. The degeneracy of the equation thus disappears, and a selection of the wavelength operates.

Figure 13Dimensionless coefficient W_R for periods equal to 10, 20, 50 and 100 d. The bold black line corresponds to W_R=0. When the period of the wave shortens (smaller than 20 d), W_R becomes negative in a narrow band south of the cape. For period longer than 100 d, the diagram is nearly symmetric.

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Figure 14Dimensionless coefficient W_I. It is independent of the period of the wave. Note that the domain is reduced in comparison with the previous figure.

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Figure 13 shows W_R 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: W_R is negative south and positive north of the cape. This dissymmetry weakens when the period increases. For T=20 d, the area where W_R is negative is strongly reduced and for T=50 d, it has vanished. Since the signs of W_R and k_R are similar (see Fig. 11 and the associated comments), a negative k_R is expected in this area and actually appears (see Fig. 15).

Figure 15Coefficient k_R for periods equal to 10, 20, 50 and 100 d. The bold black line separates positive from negative values. Where W_R is negative, k_R is negative (for periods shorter than 20 d, in the area located to the south of the cape).

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Figure 16Coefficient k_I for periods equal to 10 and 100 d. This coefficient depends weakly on the period of the wave.

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Since W_I is nearly independent of the period in the considered frequency range, a single map suffices to describe it (Fig. 14). W_I is positive everywhere except in a narrow area west of the cape (the isoline −0.2 is indicated and the bold line corresponds to W_I=0). Figure 11 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. 15.

The maps of k_R and k_I (see Figs. 15 and 16) show properties in agreement with the previous analysis. For periods shorter than 20 d, k_R 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 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 larger – but it still exists and propagates northwards.

Lastly, note that the order of magnitude predicted in Fig. 12 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. 15 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.