The above qualitative results were refined in a quantitative way as a last step. By following an inverse problem approach inspired from spectral geometry, we showed that some geometric features of the ocean basin may be inferred from the spectral irregularity, which is quantified by the normalised standard deviation of the frequency intervals separating consecutive relative maxima in the frequency spectrum of the tidal Love number. The sensitivity of this quantity to the geometrical parameters is characterised by threshold effects. As regards the position of the continent on the globe, the metric only allows the poles to be distinguished from other locations. Similarly, it seems to indicate that the signatures of continents with angular radii larger than $40^\degree$ are indistinguishable from each other.

Nevertheless, the harmonic analysis also suggests that a continent similar to South America (${\sim} 30^\degree$-angular radius) or smaller would not alter, qualitatively, the oceanic tidal response even if it is located at the equator, where the interference of the coastlines with tidal flows tends to be maximal. In such configurations, the tidally dissipated energy predicted by the harmonic analysis is approximately the same as in the absence of continent. Finally the metric appears to be sensitive to the transition between the quasi-adiabatic and frictional regimes, which occurs for $\fdrag \approx 3.2 \times 10^{-5} \ {\rm s^{-1}}$. This critical value is slightly greater than that estimated for Earth <cit.>, which is consistent with the frequency-resonant behaviour of the Earth's tidal dissipation rate inferred from Lunar laser ranging and geological data <cit.>.

Both the spectral method and the metric for continentality effects elaborated in the present work are highly relevant to studies dealing with the long-term tidal evolution of bodies with surface oceans, such as extrasolar rocky planets orbiting in the habitable zone of their host star. Particularly, the studied spectral irregularity of the tidal torque has direct implications on the history of planet-satellite systems. Tidal resonances cause irregular evolutions of the satellite's orbit and planet's rotation, thus leading to abrupt LOD or obliquity variations that might result in significant climatic effects. We will examine specifically the consequences of continentality on the long-term evolution of the Earth-Moon system in a separate dedicated study by using the theory established in the present work.

As this theory forms a basis for more sophisticated descriptions of continentality effects, we will consider refining the geometry in the future in order to account for the role played by the fractality of non-smooth coastlines, the complex bathymetry of the ocean basin, or the spatial dependence of energy dissipation. Also, while we have characterised the signature of one unique supercontinent in the planet's tidal response, the combined effects of several continents still needs to be elucidated. All these questions require further work improving meanwhile the theory and the quality of observational constraints on the past history of the Earth-Moon system. We will address them through several forthcoming studies.

The authors thank the anonymous referee for his/her insightful comments that significantly enhanced the manuscript. This project has been supported by the French Agence Nationale de la Recherche (AstroMeso ANR-19-CE31-0002-01) and by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (Advanced Grant AstroGeo-885250). This research has made use of NASA's Astrophysics Data System.

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§ NOMENCLATURE

The notations introduced in the main text are listed below in order of appearance.

§ ASSOCIATED LEGENDRE FUNCTIONS (ALFS)

The ALFs, usually denoted by $\LegF{\llat}{\mm}$, are the solutions of the equation

\begin{equation}
\label{helmholtz_col}
\frac{1}{\sin \col} \DD{}{\col} \left( \sin \col \DD{\hfunc}{\col} \right) + \left[  \llat \left( \llat + 1 \right) - \frac{\mm^2}{\sin^2 \col} \right] \hfunc = 0,
\end{equation}

which is found by separating the longitude ($\lon$) and the colatitude ($\col$) in the eigenvalues problems associated with the Laplace equation in spherical coordinates (see helmholtz). The function $\hfunc \left( \col \right) $ in helmholtz_col is assumed to be a regular function of the colatitude. The parameters $\llat$ and $\mm$ are commonly referred to as the degree and the order of the Legendre function, respectively. They may be integral, non-integral, or complex depending of the eigenvalues of the problem, $\mm^2$ and $\llat \left( \llat + 1 \right)$, which are determined both by the boundary conditions and the geometry of the domain of definition of the solutions on the unit sphere.

In the general case, the ALFs are defined, for $\zz \in \Cset$ such that $\left| \zz \right| < 1$ and arbitrary complex constants, $\llat$ and $\mm$, as <cit.>

\begin{equation}
\Plm \left( \zz \right) \define \frac{1}{\GammaF \left( 1 - \mm \right)} \left( \frac{1 + \zz }{1 - \zz} \right)^{\frac{\mm}{2}}  \hypergtwoFone \left( - \llat , \llat +1 ; 1 - \mm ; \frac{1 - \zz}{2} \right).
\label{alfgen}
\end{equation}

In the above equation, $\GammaF$ is the Gamma function introduced in imk2_solid_asymp, which is defined for $\Re \left(\zz \right) >0$ as <cit.>

\begin{equation}
\GammaF \left( \zz \right) \define \integ{t^{\zz-1} \expo{-t} }{t}{0}{\infty}.
\label{GammaF}
\end{equation}

The hypergeometric function $ \hypergtwoFone$ is defined as <cit.>

\begin{equation}
\label{hyperg2F1}
\hypergtwoFone \left( a , b ; c ; d \right) = \sum_{\jj = 0}^{\infty} \frac{\pochham{a}{\jj} \pochham{b}{\jj} }{ \pochham{c}{\jj}} \frac{\xx^{\jj}}{\jj !} ,
\end{equation}

where $\pochham{a}{\jj}$ designates the Pochhammer symbol <cit.>,

\begin{align}
& \pochham{a}{0} \define 1, \\
& \pochham{a}{\jj} \define a \left( a + 1 \right) \ldots \left( a + \jj  - 1 \right) = \frac{\GammaF \left(a + \jj \right)}{\GammaF \left( a \right)}.
\end{align}

The derivatives of the ALFs with respect to $\zz$ are given by

\begin{equation}
\dd{\LegF{\llat}{\mm}}{\zz} = \frac{1}{\zz^2 - 1} \left[ \zz \llat \LegF{\llat}{\mm} \left( \zz \right) - \left( \mm + \llat \right) \LegF{\llat-1}{\mm} \left( \zz \right)  \right].
\end{equation}

We note that the functions $\GammaF$ and $\hypergtwoFone$ in alfgen are defined only if their arguments satisfy certain conditions. In practice, the ALFs are computed for any values of these arguments by using the symmetry properties of the regularised hypergeometric function $\hypergtwoFonereg$, which is defined as <cit.>

\begin{equation}
\hypergtwoFonereg \left( a , b ; c ; \zz \right)  \define \frac{\hypergtwoFone \left( a , b ; c ; \zz \right) }{\GammaF \left( c \right)}.
\end{equation}

For $\mm \in \Zset$, $\llat \in \Rset$ and $\zz=\xx \in \Rset$ such that $-1 \leq \xx \leq1$, which corresponds to the case of the SCHs, the general definition given by alfgen simplifies to <cit.>

\begin{align}
\label{alfscha}
\Plm \left( \xx \right) = & \frac{\left( -1 \right)^{\frac{\mm + \abs{\mm}}{2}} }{2^{\abs{\mm}} \abs{\mm}!} \frac{\GammaF \left( \llat + \abs{\mm} +1 \right)}{\GammaF \left( \llat - \mm +1 \right) } \left( 1 - \xx^2 \right)^{\abs{\mm}/2}  \\
& \times \hypergtwoFonereg \left( \abs{\mm} - \llat , \llat + \abs{\mm} + 1; \abs{\mm} +1 ; \frac{1 - \xx}{2} \right). \nonumber
\end{align}

Here, the hypergeometric function is expressed as

\begin{equation}
\label{hyperg2F1reg_Plm}
\hypergtwoFonereg \left( \abs{\mm} - \llat , \llat + \abs{\mm} + 1;  \abs{\mm} +1 ; \frac{1 - \xx}{2} \right) = \sum_{\jj=0}^{\infty} \hyperci{\jj}  \left( \frac{1 - \xx}{2} \right)^\jj,
\end{equation}

with the coefficients $\hyperci{\jj}$ recursively defined as

\begin{align}
& \hyperci{0} = 1, \\
& \hyperci{\jj+1} = \frac{\left( \jj + \abs{\mm} - \llat \right) \left( \jj + \abs{\mm} + \llat + 1 \right)}{\left( \jj + 1 \right) \left( \jj + 1 + \abs{\mm} \right)} \hyperci{\jj}.
\end{align}

In the general case, the sum of alfscha converges only for $-1<\xx \leq 1$, which corresponds to the inequality $0 \leq \col < \pi$ in the change of coordinates $\xx = \cos \col$. However the sum of alfscha is not infinite any more when $\llat$ is an integer since $\hyperci{\jj+1} = 0$ as long as $\jj >  \llat - \abs{\mm} $. As a consequence it converges also for $\col = \pi$ in this case.

Since the degrees $\llat$ tend to integral values ($\llat = 0 , 1 , 2, \ldots$) as $\col \rightarrow \pi$, the generalised ALFs given by alfgenalfscha continuously converge towards the well-known ALFs of the SPHs, which are defined for $ -1 \leq \xx \leq 1 $, and $\llat$ and $\mm$ integers such that $\llat \geq \abs{\mm}$, as <cit.>

\begin{equation}
\Plm \left( x \right) \define \left(-1 \right)^\mm \left( 1 - x^2 \right)^{\mm/2}  \DDn{}{x}{\mm} \Pl \left( x \right),
\label{Plmstd}
\end{equation}

the $\Pl$ designating the Legendre polynomials, given by

\begin{equation}
\Pl \left( x \right) \define \frac{1}{2^\llat \llat !} \DDn{}{x}{\llat} \left[ \left( x^2 - 1 \right)^\llat \right].
\end{equation}

Owing to the large number of terms that is sometimes required to compute the sum of hyperg2F1reg_Plm at machine precision, the numerical evaluation of the generalised ALFs given by alfscha may be tricky in some cases. Particularly, the series converges very slowly when $\xx \approx -1$, which occurs if the ocean is almost global. In practice, the evaluation is achieved by making use of the build-in hypergeometric function $\hypergtwoFone$ of the  algebraic manipulator <cit.>.

§ SPHERICAL CAP HARMONICS (SCHS)

In the same way that the SPHs are the harmonics of the entire sphere, the SCHs designate the harmonics of the sphere truncated at a colatitude $\colbd$ <cit.>. Figure <ref> shows the two geometrical configurations. We note that the natural coordinate system of the SCHs is such that the z-axis is the axis going through the centre of the bounded ocean, which is the symmetry axis of the system, while the coordinate system of the SPHs does not need to be specified due to the spherical symmetry of the global ocean.

Geometry of the land-ocean distribution in the model. Left: global ocean distribution where the eigenfunctions are the SPHs. Right: bounded circular ocean distribution where the eigenfunctions are the SCHs. In the bounded case, the frame of reference $\rframe{\iocean}{O}{\ex}{\ey}{\ez}$ introduced in ssec:geometry_ocean is such that $\ez$ corresponds to the symmetry axis going through the centre of the ocean basin, while it is not specified in the global case due to the spherical symmetry.

Values of the real degrees $\llati{\nk}$ of the SCHs with Neumann (SCHNs) or Dirichlet (SCHDs) boundary conditions for various angular radii of the supercontinent ($\colcont$).

$\colcont$ (deg)     $\left( 0,0 \right)$     $\left( 1,0 \right)$     $\left( 1,1 \right)$     $\left( 2,0 \right)$     $\left( 2,1 \right)$     $\left( 2,2 \right)$

7cNeumann boundary conditions (SCHNs)

0         0.000000         1.000000         1.000000         2.000000         2.000000         2.000000

10         0.000000         1.014533         0.984857         2.040535         1.957224         1.999318

20         0.000000         1.054431         0.945331         2.139648         1.889718         1.989826

30         0.000000         1.115646         0.900510         2.277970         1.885701         1.956707

40         0.000000         1.196823         0.867186         2.450068         1.948680         1.901867

50         0.000000         1.298769         0.851964         2.657126         2.062763         1.851211

60         0.000000         1.424123         0.856313         2.904340         2.220869         1.827443

70         0.000000         1.577470         0.880927         3.200567         2.423975         1.840734

80         0.000000         1.765876         0.927569         3.559164         2.679212         1.896224

90         0.000000         2.000000         1.000000         4.000000         3.000000         2.000000

100         0.000000         2.296162         1.105011         4.553241         3.408276         2.162574

110         0.000000         2.680266         1.254256         5.266546         3.939640         2.402474

120         0.000000         3.195691         1.467987         6.219529         4.654189         2.752588

130         0.000000         3.920704         1.783375         7.555648         5.660659         3.273969

140         0.000000         5.012005         2.275396         9.561941         7.176982         4.090469

150         0.000000         6.835398         3.119597        12.908284         9.712069         5.492825

160         0.000000        10.488504         4.843239        19.604445        14.793105         8.355251

170         0.000000        21.459763        10.083479        39.699467        30.056683        17.051968

7cDirichlet boundary conditions (SCHDs)

0         0.000000         1.000000         1.000000         2.000000         2.000000         2.000000

10         0.201220         1.286874         1.014533         2.359807         2.040535         2.000661

20         0.274502         1.424749         1.054431         2.561533         2.139648         2.009153

30         0.346184         1.568297         1.115646         2.777734         2.277970         2.037781

40         0.422281         1.726592         1.196823         3.019705         2.450068         2.095353

50         0.506293         1.905895         1.298769         3.296147         2.657126         2.186686

60         0.601509         2.112863         1.424123         3.616943         2.904340         2.315699

70         0.711801         2.355851         1.577470         3.994876         3.200567         2.487779

80         0.842253         2.646183         1.765876         4.447503         3.559164         2.711416

90         1.000000         3.000000         2.000000         5.000000         4.000000         3.000000

100         1.195606         3.441373         2.296162         5.690014         4.553241         3.374653

110         1.445566         4.008030         2.680266         6.576628         5.266546         3.869485

120         1.777288         4.762779         3.195691         7.758259         6.219529         4.542151

130         2.240037         5.818619         3.920704         9.412016         7.555648         5.497155

140         2.932276         7.401491         5.012005        11.892079         9.561941         6.944195

150         4.083687        10.038551         6.835398        16.024836        12.908284         9.373283

160         6.383235        15.311204        10.488504        24.289410        19.604445        14.255320

170        13.275607        31.126398        21.459763        49.081367        39.699467        28.946234

The values are given for the doublets $\left(\nk , \abs{\mm} \right) $ such that $ \nk=0,1,2$ and $\abs{\mm} \leq \nk $.

Real degrees $\llati{\kk}$ of the SCHs for doublets $\left( \kk , \abs{\mm} \right)$ with $0 \leq \abs{\mm} \leq \kk \leq 2$ as functions of the angular radius of the supercontinent. The set of basis functions obtained for the Neumann condition given by neucond (SCHNs) are designated by solid lines, and those obtained for the Dirichlet condition given by dircond (SCHDs) by dashed lines.

Eigenvalues of the SCHs with Neumann (SCHNs) or Dirichlet (SCHDs) boundary conditions for various angular radii of the supercontinent.

$\colcont$ (deg)     $\left( 0,0 \right)$     $\left( 1,0 \right)$     $\left( 1,1 \right)$     $\left( 2,0 \right)$     $\left( 2,1 \right)$     $\left( 2,2 \right)$

7cNeumann boundary conditions (SCHNs)

0         0.000000         2.000000         2.000000         6.000000         6.000000         6.000000

10         0.000000         2.043810         1.954799         6.204319         5.787950         5.996589

20         0.000000         2.166257         1.838982         6.717744         5.460750         5.949235

30         0.000000         2.360311         1.711429         7.467118         5.441569         5.785407

40         0.000000         2.629208         1.619197         8.452903         5.746033         5.518966

50         0.000000         2.985569         1.577808         9.717446         6.317753         5.278194

60         0.000000         3.452250         1.589586        11.339534         7.153126         5.166990

70         0.000000         4.065883         1.656959        13.444194         8.299630         5.229035

80         0.000000         4.884195         1.787952        16.226814         9.857388         5.491887

90         0.000000         6.000000         2.000000        20.000000        12.000000         6.000000

100         0.000000         7.568523         2.326061        25.285246        15.024623         6.839299

110         0.000000         9.864090         2.827415        33.003048        19.460404         8.174357

120         0.000000        13.408133         3.622974        44.902072        26.315661        10.329330

130         0.000000        19.292626         4.963802        64.643464        37.703724        13.992842

140         0.000000        30.132199         7.452826       100.992648        58.686045        20.822406

150         0.000000        53.558065        12.851483       179.532083       104.036347        35.663952

160         0.000000       120.497214        28.300199       403.938705       233.629059        78.165464

170         0.000000       481.981173       111.760030      1615.747119       933.460872       307.821577

7cDirichlet boundary conditions (SCHDs)

0         0.000000         2.000000         2.000000         6.000000         6.000000         6.000000

10         0.241710         2.942918         2.043810         7.928496         6.204319         6.003307

20         0.349854         3.454660         2.166257         9.122983         6.717744         6.045851

30         0.466027         4.027851         2.360311        10.493540         7.467118         6.190333

40         0.600603         4.707711         2.629208        12.138321         8.452903         6.485855

50         0.762626         5.538331         2.985569        14.160734         9.717446         6.968283

60         0.963323         6.577054         3.452250        16.699223        11.339534         7.678163

70         1.218462         7.905884         4.065883        19.953910        13.444194         8.676820

80         1.551642         9.648467         4.884195        24.227788        16.226814        10.063195

90         2.000000        12.000000         6.000000        30.000000        20.000000        12.000000

100         2.625081        15.284420         7.568523        38.066269        25.285246        14.762935

110         3.535228        20.072333         9.864090        49.828657        33.003048        18.842401

120         4.936042        27.446847        13.408133        67.948839        44.902072        25.173291

130         7.257803        39.674941        19.292626        97.998058        64.643464        35.715869

140        11.530519        62.183561        30.132199       153.313622       100.992648        55.166033

150        20.760187       110.811047        53.558065       272.820192       179.532083        97.231717

160        47.128920       249.744161       120.497214       614.264854       403.938705       217.469478

170       189.517352       999.979043       481.981173      2458.061916      1615.747119       866.830706

The values of $\eigenvali{\llat}$ are were computed from sch_eigenv for the doublets $\left(  \nk , \abs{\mm} \right) $ such that $\nk=0,1,2$ and $\abs{\mm} \leq \nk$ using the values of real degrees given by Table <ref>.

The SPHs and the SCHs are found by solving – for specific boundary conditions discussed further – the eigenvalue-eigenfunction problem described by the Helmholtz equation <cit.>,

\begin{equation}
\label{helmholtz}
\lap \ffunc = - \eigenval \ffunc,
\end{equation}

where $\lap$ designates the Laplace operator, defined for any function $\ffunc \left( \col, \lon \right)$ as

\begin{equation}
\lap \ffunc \define \frac{1}{\sin \col} \dd{}{\col} \left( \sin \col \dd{\ffunc}{\col} \right) + \frac{1}{\sin^2 \col} \ddd{\ffunc}{\lon}{\lon},
\end{equation}

the notation $\dd{}{\xx}$ or $\ddd{}{\xx}{\xx}$ referring to the first or second partial derivatives with respect to the $\xx$ coordinate. In both case, the variables $\col$ and $\lon$ can be separated, which allows the problem to be solved as two decoupled eigenvalue problems.

Since the functions $\ffunc \left( \col, \lon  \right)$ in helmholtz are defined for all $\lon$, the boundary conditions on $\lon$ are those of continuity,

\begin{align}
\ffunc \left( \col, \lon \right) & = \ffunc \left( \col , \lon + 2 \pi \right) , \\
\dd{\ffunc}{\lon} \left( \col , \lon \right) & = \dd{\ffunc}{\lon} \left( \col , \lon + 2 \pi \right),
\end{align}

which restricts the values of $\mm$ to be integral ($\mm \in \Zset$). These conditions hold both for the ordinary SPHs and for the SCHs. Analogously, the two sets of basis functions share the same regularity condition on $\col$ at the pole ($\col =0$),

\begin{align}
\dd{\ffunc}{\col} \left( 0 , \lon \right) = 0 && \mm=0, \\
\ffunc \left( 0 , \lon \right) = 0 && \mm \neq 0.
\end{align}

However they differ as regards the second boundary condition. The SPHs are obtained by solving the Helmholtz equation given by helmholtz on the entire sphere, $\sphere{:}\left( \col , \lon \right) \in \left[ 0, \pi \right] \times \left[ 0 , 2 \pi \right[$, with regularity boundary conditions at $\col=\pi$ similar to those applied at $\col=0$,

\begin{align}
\dd{\ffunc}{\col} \left( \pi, \lon \right) = 0 && \mm=0 , \\
\ffunc \left( \pi , \lon \right) = 0 && \mm \neq 0.
\end{align}

They are associated with integral degrees $\llat \in \Nset$ such that $\llat \geq \abs{\mm}$.

Following [Varshalovich et al., 1988], we use, for the complex SPHs, the convention given by <cit.>

\begin{equation}
\label{sphc}
\Ylm{\llat}{\mm} \left( \col , \lon \right) = \sqrt{\frac{\left( 2 \llat + 1 \right) \left( \llat - \mm \right)!}{4 \pi \left( \llat + \mm \right)!}} \LegF{\llat}{\mm} \left( \cos \col \right) \expo{\inumber \mm \lon},
\end{equation}

or, equivalently,

\begin{equation}
\Ylm{\llat}{\mm} \left( \col , \lon \right) \define \left( -1 \right)^{\frac{\abs{\mm} - \mm}{2}}  \sqrt{\frac{\left( 2 \llat + 1 \right) \left( \llat - \abs{\mm} \right) !  }{4 \pi \left( \llat + \abs{\mm} \right)! }} \LegF{\llat}{\abs{\mm}} \left( \cos \col \right) \expo{\inumber \mm \lon},
\end{equation}

where the the $\Plm$ are the standard ALFs defined in Plmstd. The eigenvalues associated with the SPHs only depend on the degree $\llat$, and are expressed as

\begin{equation}
\eigenvali{\llat} = \llat \left( \llat + 1 \right).
\end{equation}

In the used convention (sphc), the SPHs form a set of orthonormal basis functions through the scalar product defined, for any complex functions $\ffunc$ and $\gfunc$, as

\begin{align}
\label{sprodYlm}
\scal{\ffunc}{\gfunc} &  \define \integ{\conj{\ffunc} \gfunc }{\surface}{\sphere}{}, \\
& = \integ{\integ{ \conj{\ffunc \left( \col , \lon \right)} \gfunc \left( \col , \lon \right) \sin \col \,}{\col}{\col=0}{\pi} }{\lon}{\lon=0}{2 \pi}. \nonumber
\end{align}

the notation $\infvar{\surface}$ referring to an infinitesimal surface element of the unit sphere, and $\conj{\ffunc}$ to the conjugate of $\ffunc$. This scalar product allows the $\ltwonorm$-norm of the space of functions of $\sphere$ to be defined,

\begin{equation}
\label{ltwonorm}
\normtwo{f} \define \sqrt{\scal{\ffunc}{\ffunc}} .
\end{equation}

Hence, for any pair of SPHs $\Ylm{\llat}{\mm}$ and $\Ylm{\pp}{\qq}$,

\begin{equation}
\label{sphscal}
\scal{\Ylm{\llat}{\mm}}{\Ylm{\pp}{\qq}} = \kron{\llat}{p} \kron{\mm}{\qq}.
\end{equation}

The SCHs are obtained by solving the Helmholtz equation given by helmholtz on the domain filled by the spherical cap, $\sphcap:\left( \col , \lon \right) \in \left[0 , \colbd \right] \times \left[0 , 2 \pi \right[$, together with the boundary conditions specified at $\sphcapbd$, namely $\col = \colbd$. In the present model, the applied boundary conditions are either Neumann (fixed value for $\dd{\ffunc}{\col} $ at $\col= \colbd$) or Dirichlet (fixed value for the function $\ffunc$ itself) conditions, which are respectively formulated as

\begin{align}
\label{neucond}
\dd{\ffunc}{\col} \left( \colbd , \lon \right) = 0  && \mbox{(Neumann condition)},  \\
\label{dircond}
\ffunc \left( \colbd, \lon \right) = 0  && \mbox{(Dirichlet condition)}.
\end{align}

Each condition leads to a set of orthogonal basis functions. These sets are referred to as the `SCHNs' for neucond and the `SCHDs' for dircond.

As the Helmholtz equation is the same for both the SCHs and the SPHs, solving the eigenfunction-eigenvalue problem defined by helmholtzneucond (or dircond) amounts to finding the set of ALFs (see alfscha) that satisfy the specified boundary conditions for a given $\mm \in \Zset$. As a consequence, the eigenfunction-eigenvalue problem becomes a root-finding problem where one computes the series of real degrees $\llat $ such that, alternately,

\begin{align}
\label{neucond_plm}
\DD{\LegF{\llat}{\mm} }{\col} \left( \cos \colbd \right) = 0, && \mbox{(Neumann condition)} , \\
\label{dircond_plm}
\LegF{\llat}{\mm} \left( \cos \colbd \right) = 0 && \mbox{(Dirichlet condition)}.
\end{align}

This can be achieved numerically using a standard Newton-Raphson algorithm <cit.> since the ALFs defined by alfscha and their derivatives are regular functions of $\llat$.

For each condition (neucond or dircond), we end up with the degrees of the distorted ALFs, $\llati{\nk} \left( \mm \right)$, which are subscripted with indices $\nk \in \Nset$ such that $\nk \geq \abs{\mm}$. These indices[Other conventions exist for the indices $\nk$. For instance, [Haines], 1985] introduces a convention where the parity of $\kk - \abs{\mm}$ allows the two sets of basis functions (with Neumann or Dirichlet conditions) to be distinguished. In this convention, the roots of neucond_plm (Neumann condition) are designated by even $\nk - \abs{\mm}$, while those of dircond_plm (Dirichlet condition) are designated by odd $\nk - \abs{\mm}$. correspond to the integral degrees of the SPHs towards which the SCHs converge for the $\ltwonorm$-norm defined by ltwonorm in the global ocean limit: $\llati{\nk} \rightarrow \nk$ as $\colbd \rightarrow 180^\degree$ independently of the applied boundary condition. Similarly as integral degrees, the real degrees $\llati{\nk} $ are ranked in ascending order. For any $\nk$ and $\jj$ such that $\nk < \jj$, the inequality $\llati{\nk} < \llati{\jj}$ is verified. The values of the real degrees are given by Table <ref> for $0 \leq \abs{\mm} \leq 2$ and various values of angular radius of the continent, $\colcont = 180 - \colbd$ (in degrees). They are plotted on fig:degrees_size.

We note that $\llati{\nk} \geq \abs{\mm}$ when Dirichlet conditions are applied, similarly as in the case of SPHs <cit.>. However, when the Neumann conditions are applied, this is only valid for $\colbd \leq 90^\degree$ since there exists degrees $\llati{\abs{\mm}} < \abs{\mm}$ as long as $\colbd > 90^\degree$. Asymptotic scaling laws may be used to estimate the values of the degrees $\llati{\nk} \gg \abs{\mm}$ as functions of the indices $\nk$ and the angle $\colbd$ <cit.>. Particularly, these laws show that $\llati{\nk} \scale \colbd^{-1}$ for $\colbd \ll 1$ consistently with what may be observed in fig:degrees_size. This results from the fact that the degree $\llat$ of an harmonic scales as the inverse of the corresponding wavelength, which is determined by the size of the spherical cap.

The resulting SCHs are defined on the unit sphere, for $\abs{\mm} = 0, \ldots , + \infty$ and $\llat = \llati{\abs{\mm}}, \llati{\abs{\mm}+1}, \ldots$, as

\begin{equation}
\label{schc}
\Hlm{\llat}{\mm} \left( \col , \lon \right) \define
\left\{
\begin{array}{ll}
\alm{\llat}{\mm} \LegF{\llat}{\abs{\mm}} \left( \cos \col \right) \expo{\inumber \mm \lon} & \mbox{if} \ \left( \col , \phi \right) \in \sphcap, \\
0 & \mbox{if} \ \left( \col , \phi \right) \in \sphere \setminus \sphcap
\end{array}
\right.
\end{equation}

where the normalisation coefficients $\alm{\llat}{\mm} \in \Rset$ are to be specified. The corresponding eigenvalues $\eigenvali{\llat}$ are expressed as

\begin{equation}
\label{sch_eigenv}
\eigenvali{\llat} \left( \mm \right) = \llat \left( \mm \right) \left[ \llat \left( \mm \right) + 1 \right],
\end{equation}

similarly as in the case of SPHs, though they now also depend on the order $\mm$ through $\llat$. The eigenvalues associated with the SCHNs and SCHDs are given in Table <ref> for $0 \leq \abs{\mm} \leq \nk \leq 2$ and various values of the continental angular radius ($\colcont$). They are computed straightforwardly from the values of the real degrees given in Table <ref> using the expression given by sch_eigenv.

The SCHs obtained for a given boundary condition are orthogonal basis functions of the space of regular functions on $\sphcap$ through the scalar product introduced in sprodYlm. Since the harmonics are set to zero for $\col>\colbd$, this scalar product reads

\begin{equation}
\label{scal_sch}
\scal{\ffunc}{\gfunc}  = \integ{\integ{ \conj{\ffunc \left( \col , \lon \right)} \gfunc \left( \col , \lon \right) \sin \col \,}{\col}{\col=0}{\colbd} \, }{\lon}{\lon=0}{2 \pi}.
\end{equation}

For any pair of SCHs belonging to the same set, $\Hlm{\llat}{\mm}$ and $\Hlm{\pp}{\qq}$, the property given by sphscal for the SPHs,

\begin{equation}
\label{schortho}
\scal{\Hlm{\llat}{\mm}}{\Hlm{\pp}{\qq}} = \kron{\llat}{\pp} \kron{\mm}{\qq},
\end{equation}

is enforced by defining the normalisation coefficients $\alm{\llat}{\mm} \in \Rset$ as

\begin{equation}
\label{normcoeff}
\alm{\llat}{\mm} = \left( -1 \right)^{\frac{\abs{\mm} - \mm}{2}}  \left[ 2 \pi \integ{\left[ \LegF{\llat}{\abs{\mm}} \left( \cos \col \right) \right]^2 \sin \col \,}{\col}{0}{\colbd} \right]^{-1/2}.
\end{equation}

However, the functions in one set are not orthogonal to those in the other since <cit.>

\begin{equation}
\scal{\Hlm{\llat}{\mm}}{\Hlm{\pp}{\qq}} = \kron{\mm}{\qq} \frac{2 \pi \sin \colbd \LegF{\llat}{\mm} \left( \cos \colbd \right) }{\left( \pp - \llat \right) \left( \pp + \llat + 1 \right)} \DD{\LegF{\pp}{\qq}}{\col} \left( \cos \colbd \right) ,
\end{equation}

for $\Hlm{\llat}{\mm}$ and $\Hlm{\pp}{\qq}$ taken in the sets of SCHNs and SCHDs, respectively. In the general case, $\LegF{\llat}{\mm} \left( \cos \colbd \right) \neq 0 $ and $\DD{\LegF{\pp}{\qq}}{\col} \left( \cos \colbd \right) \neq 0$, which implies $\scal{\Hlm{\llat}{\mm}}{\Hlm{\pp}{\mm}} \neq 0$. This highlights the fact that the SCHDs and the SCHNs are two distinct complete sets of basis functions for the space of functions defined on $\sphcap$.

In the case of real eigenfunctions defined as

\begin{equation}
\Hlm{\llat}{\mm} \left( \col , \lon \right) \define \left\{
\begin{array}{ll}
\alm{\llat}{\mm} \LegF{\llat}{\abs{\mm}} \left( \cos \col \right) \cos \left( \mm \lon + \lonm \right) & \mbox{if} \ \left( \col , \phi \right) \in \sphcap, \\[0.1cm]
0 & \mbox{if} \ \left( \col , \phi \right) \in \sphere \setminus \sphcap,
\end{array}
\right.
\end{equation}

with $\lonm =0$ if $\mm \geq 0$ and $\lonm = \pi/2$ if $\mm<0$, the orthogonality property given by schortho still holds and the normalisation coefficients defined in normcoeff become

\begin{equation}
\alm{\llat}{\mm} = \left( -1 \right)^{\frac{\abs{\mm} + \mm}{2}} \left[ \left( 1 + \kron{\mm}{0} \right) \pi \integ{\left[ \LegF{\llat}{\abs{\mm}} \left( \cos \col \right) \right]^2 \sin \col \,}{\col}{0}{\colbd} \right]^{-1/2}.
\end{equation}

We note that the factors $\left( - 1 \right)^k$ with $k = \left( \abs{\mm} - \mm \right)/2 $ or $k = \left( \abs{\mm} + \mm \right)/2$ in the expressions of the normalisation coefficients are introduced to make the SCHs consistent with the SPHs in the used convention (sphc).

As $\colbd$ increases, the ALFs smoothly evolve from the Bessel functions ($\colbd \rightarrow 0$) to the standard ALFs of integer parameters ($\colbd \rightarrow \pi$). Similarly, the SCHs evolve from the cylindrical harmonics, where there is no curvature effect (small-sea configuration), to the SPHs given by sphc (global-ocean configuration). Nevertheless, we emphasise that both the eigenvalues and the eigenfunctions depend on the specified boundary condition through the degrees $\llat$, different conditions leading to different sets of eigenfunctions. Besides, the limit $\colbd \rightarrow \pi$ is not strictly equivalent to the global-ocean configuration since boundary conditions are still applied at $\sphcapbd$, instead of the periodic conditions of the SPHs. In other words, the presence of a small island still alters the eigenmodes of the oceanic tide locally although this effect is negligible at a global scale. The impact of the island on the horizontal structure of tidal flows is quantified by the differences of degrees, $\abs{\llati{\nk} - \nk}$, and overlap coefficients, $\abs{1-\scal{\Hlm{\llati{\nk}}{\mm}}{\Ylm{\nk}{\mm}}} $, both of them tending to zero as the size of the island decreases.

§ EULER ROTATION MATRIX

We consider the frame of reference $\rframe{}{O}{\eX}{\eY}{\eZ} $ having the planet's centre of gravity as origin and associated with the Cartesian coordinates $\left( \XX, \YY, \ZZ \right)$. As shown by fig:euler_angles, any rotation $\rframe{}{O}{\eX}{\eY}{\eZ} \rightarrow \rframe{\irotation}{O}{\ex}{\ey}{\ez}$ of this frame of reference may be performed by three rotations about the coordinate axes <cit.>: (i) a rotation about the $\ZZ$-axis through an angle $\angalp$ such that $0 \leq \angalp < 2 \pi$, (ii) a rotation about the new $\YY^\prime$-axis through an angle $\angbet$ such that $0 \leq \angbet \leq \pi$, and (iii) a rotation about the new axis $\zz $ through an angle $\anggam$ such that $0 \leq \anggam \leq 2 \pi$. The frame of reference $\rframe{\irotation}{O}{\ex}{\ey}{\ez}$ is associated with the Cartesian coordinates $\left( \xx, \yy , \yy \right)$. The three rotation angles, $\angalp$, $\angbet$, and $\anggam$, are the so-called Euler angles.

Euler angles corresponding to the standard Euler rotation matrix defined by eulermat.

Introducing the notations $\cosi{\angrot} = \cos \angrot$ and $\sini{\angrot} = \sin \angrot$ for $\angrot = \angalp, \angbet, $ or $\anggam$, the standard Euler rotation matrix is defined in terms of the Euler angles as <cit.>

\begin{equation}
\eulermat_{ \angalp , \angbet , \anggam} \define
\begin{bmatrix}
\cosi{\angalp} \cosi{\angbet} \cosi{\anggam} - \sini{\angalp} \sini{\anggam} &
- \cosi{\angalp} \cosi{\angbet} \sini{\anggam} - \sini{\angalp} \cosi{\anggam} &
\cosi{\angalp} \sini{\angbet} \\
\sini{\angalp} \cosi{\angbet} \cosi{\anggam} + \cosi{\angalp} \sini{\anggam} &
- \sini{\angalp} \cosi{\angbet} \sini{\anggam} + \cosi{\angalp} \cosi{\anggam} &
\sini{\angalp} \sini{\angbet} \\
- \sini{\angbet} \cosi{\anggam} &
\sini{\angbet} \sini{\anggam} & \cosi{\angbet}
\end{bmatrix},
\label{eulermat}
\end{equation}

which is formed by the product of three rotation matrices,

\begin{align}
\eulermat_{ \angalp , \angbet , \anggam} = &  \rotmati{3}{\angalp} \rotmati{2}{\angbet} \rotmati{3}{\anggam} \\
= &
\begin{bmatrix}
\cosi{\angalp} &  - \sini{\angalp}  & 0 \\
\sini{\angalp} & \cosi{\angalp} & 0 \\
0 & 0 & 1
\end{bmatrix}
\begin{bmatrix}
\cosi{\angbet} & 0 &  \sini{\angbet}  \\
0 & 1 & 0 \\
- \sini{\angbet} & 0 & \cosi{\angbet}
\end{bmatrix}
\begin{bmatrix}
\cosi{\anggam} & - \sini{\anggam}  & 0 \\
\sini{\anggam} & \cosi{\anggam} & 0 \\
0 & 0 & 1
\end{bmatrix}.
\nonumber
\end{align}

The inverse rotation matrix $\eulermat^{-1}$ is given by

\begin{equation}
\eulermat^{-1}_{ \angalp, \angbet, \anggam} = \transp{\eulermat}_{\angalp, \angbet, \anggam}= \eulermat_{- \anggam, - \angbet , - \angalp}.
\end{equation}

The Euler rotation matrix and its inverse correspond to the change of basis matrices between the initial and rotated vectorial bases, $\left( \eX, \eY , \eZ \right) \rightarrow \left( \ex, \ey , \ez \right)$, which is expressed as

\begin{align}
\label{change_basis_rotation}
\begin{bmatrix}
\ex \\
\ey \\
\ez
\end{bmatrix}
= \eulermat_{\angalp,\angbet,\anggam}^{-1}
\begin{bmatrix}
\eX \\
\eY \\
\eZ
\end{bmatrix},
\begin{bmatrix}
\eX \\
\eY \\
\eZ
\end{bmatrix}
= \eulermat_{\angalp,\angbet,\anggam}
\begin{bmatrix}
\ex \\
\ey \\
\ez
\end{bmatrix}.
\end{align}

Besides the rotation $\Uvect \rightarrow \Uvectrot $ of any vector $\Uvect $ through the Euler angles $\angalp, \angbet,$ and $\anggam$ in the Cartesian coordinate system $\coordsini \left\{ \XX , \YY , \ZZ \right\}$ is expressed as $\Uvectrot = \eulermat_{\angalp,\angbet,\anggam} \Uvect $.

In spherical coordinates, the transformation relations are deduced from change_basis_rotation by making use of trigonometric identities. We denote by $\left( \rrpla, \colpla , \lonpla \right)$ the spherical coordinates associated with the initial frame of reference, $\framestd$, where $\rrpla$ designates the radial coordinate, $\colpla$ the colatitude, and $\lonpla$ the longitude. The analogous coordinates in the rotated frame of reference, $\framerotation$, are denoted by $\left( \rr, \col , \lon \right)$, with $\rr = \rrpla$. In the forward transformation, $\left( \rrpla, \colpla , \lonpla \right) \rightarrow \left( \rr, \col , \lon \right) $, the relations between angles $\left( \colpla,\lonpla \right)$ and $\left( \col,\lon \right)$ are given by <cit.>

\begin{align}
\cos \col & = \cos \colpla \cos \angbet + \sin \colpla \sin \angbet \cos \left( \lonpla - \angalp \right) , \\
\cot \left( \lon + \anggam \right) & = \cot \left( \lonpla - \angalp \right) \cos \angbet - \frac{\cot \colpla \sin \angbet}{ \sin \left( \lonpla - \angalp \right)},
\end{align}

while the inverse relations are

\begin{align}
\cos \colpla & = \cos \col \cos \angbet - \sin \col \sin \angbet \cos \left( \lon + \anggam \right) , \\
\cot \left( \lonpla - \angalp \right) & = \cot \left( \lon + \anggam \right) \cos \angbet + \frac{\cot \col \sin \angbet}{\sin \left( \lon + \anggam \right)}.
\end{align}

As an example, we consider the rotation generating the coordinate system of the ocean basin from the geocentric coordinate system rotating with the planet. From the definition of the ocean centre in the usual geocentric frame of reference, we obtain

\begin{equation}
\ez = \cos \lonoc \sin \coloc \eX + \sin \lonoc \sin \coloc \eY + \cos \coloc \eZ.
\label{ez1}
\end{equation}

Besides, by considering the same vector as the image of $\eZ$ through a rotation of Euler angles $\angalp$, $\angbet$, and $\anggam$, we can write $\ez = \eulermat_{\angalp, \angbet, \anggam} \transp{\left( 0 , 0 , 1 \right)}$, where we made use of the standard Euler rotation matrix defined in eulermat. It follows

\begin{equation}
\ez = \cos \angalp \sin \angbet \eX + \sin \angalp \sin \angbet \eY + \cos \angbet \eZ.
\label{ez2}
\end{equation}

By comparing ez1ez2, we identify $\angbet = \coloc$ and $\angalp = \lonoc$. Considering the inverse transformation, $\eZ = \transp{\eulermat_{\angalp, \angbet, \anggam}} \transp{\left( 0 , 0 , 1 \right)}$, we get

\begin{equation}
\eZ = - \sin \angbet \cos \anggam \ex + \sin \angbet \sin \anggam \ey + \cos \angbet \ez,
\end{equation}

which shows that $\eZ$ does not depend on the rotation through the angle $\angalp$ in the oceanic frame of reference.

We note that the angle $\anggam$ describes a rotation of coordinates around the axis defined by $\ez$. Since the ocean is circular, the geometry remains unchanged through this rotation. The angle $\anggam$ can therefore be set to $\anggam = 0$, which allows the position of the North pole in the ocean's coordinate system to be simplified to $\eZ = - \sin \angbet \ex + \cos \angbet \ez$. The coordinates of the North pole in the coordinate system defined from the centre of the ocean basin are thus given by $\left( \colnp , \lonnp \right) = \left( \angbet , \pi \right)$. At the end, the ocean's coordinate system is simply obtained from the usual geocentric coordinate system through a rotation of Euler angles $\left( \angalp , \angbet , \anggam \right)= \left( \lonoc , \coloc , 0 \right)$. We shall emphasise that the gyroscopic coefficients of the tidal theory only depend on the colatitude of the supercontinent's centre since $\angbet$ is the only angle necessary to define the position of the spin axis in the coordinate system of the ocean basin. Moreover, the rotation of angle $\angalp$ can be ignored owing to the periodicity of the tidal response across the longitudinal coordinate, $\lonpla$, which implies that the longitude of the supercontinent's centre can be set to $\lonscont=0$ without any loss of generality.

§ ROTATION OF SPHERICAL HARMONICS

We consider a rotation $\rframe{}{O}{\eX}{\eY}{\eZ} \rightarrow \rframe{\irotation}{O}{\ex}{\ey}{\ez}$ described by the Euler angles introduced in app:euler_rotation, which are denoted by $\angalp, \angbet$, and $\anggam$ (see fig:euler_angles). The initial frame of reference, $\framestd$, is associated with the spherical coordinate system $\left( \rrpla, \colpla, \lonpla  \right) $, while the rotated frame of reference, $\framerotation$, is associated with the coordinate system $\left( \rr , \col , \lon \right)$. In the convention defined by sphc, the complex SPHs of the rotated coordinate system are expressed as functions of those defined in the initial coordinate system <cit.>,

\begin{equation}
\Ylm{\llat}{\mm} \left( \col , \lon \right) = \sum_{\qq=-\llat}^\llat \wignerD{\qq}{\mm}{\llat} \left( \angalp, \angbet, \anggam \right) \Ylm{\llat}{\qq} \left(  \colpla, \lonpla \right),
\label{Ylm_Ylq}
\end{equation}

and, conversely, the SPHs of the initial coordinate system are expressed as

\begin{equation}
\Ylm{\llat}{\mm} \left( \colpla , \lonpla \right) = \sum_{\qq=-\llat}^\llat \conj{\wignerD{\mm}{\qq}{\llat}} \left( \angalp, \angbet, \anggam \right) \Ylm{\llat}{\qq} \left( \col, \lon \right).
\label{Ylq_Ylm}
\end{equation}

Rotation matrix $\wignermat$ of wignermat for Euler angles $\left( \angalp, \angbet, \anggam \right) = \left(120^\degree, 60^\degree, 270^\degree \right) $. Left: Modulus of the matrix coefficients. Right: Argument of the matrix coefficients (degrees). The truncation degree of the set of SPHs is set to $\lmax = 4$. The SPHs are sorted in ascending order of degrees ($\llat=0,1,2,3,4$), with orders ($-\llat \leq \mm \leq \llat$) grouped together.

The notation $\wignerD{\qq}{\mm}{\llat}$ refers to the Wigner D-functions <cit.>,

\begin{equation}
\wignerD{\qq}{\mm}{\llat} \left( \angalp, \angbet, \anggam \right)  \define \expo{- \inumber \qq \angalp} \wignerd{\qq}{\mm}{\llat} \left( \angbet \right) \expo{- \inumber \mm \anggam},
\end{equation}

where $\wignerd{\qq}{\mm}{\llat}$ is a real function expressed as <cit.>

\begin{align}
\wignerd{\qq}{\mm}{\llat} \left( \angbet \right) = & \left( -1 \right)^{\llat - \mm} \sqrt{\left( \llat + \qq \right)! \left( \llat - \qq \right)! \left( \llat + \mm \right) ! \left( \llat - \mm \right)! } \\
& \times \sum_{\kk} \left( -1 \right)^\kk \frac{ \left( \cos \frac{\angbet}{2} \right)^{\qq + \mm + 2 \kk} \left( \sin \frac{\angbet}{2} \right)^{2 \llat - \qq - \mm - 2 \kk} }{\kk ! \left( \llat - \qq - \kk \right) ! \left( \llat - \mm - \kk \right) ! \left( \qq + \mm + \kk \right) ! }. \nonumber
\end{align}

In the above equation, $\kk$ runs over all integer values for which the factorial arguments are non-negative, namely $\max \left( 0 , - \qq - \mm \right) \leq \kk \leq \min \left( \llat - \qq , \llat - \mm \right)$.

The Wigner D-functions introduced in Ylm_YlqYlq_Ylm are the matrix elements of the rotation operator,

\begin{equation}
\label{wignermat}
\wignermat \define
\begin{bmatrix}
\wignerDmati{0}{0}{0} & 0 & \ldots & \ldots & \ldots & 0  \\
0 & \wignerDmati{\qq}{\mm}{1} &  \ddots & & & \vdots   \\
\vdots  &  \ddots & \ddots & \ddots & &\vdots  \\
\vdots &  & \ddots & \wignerDmati{\qq}{\mm}{\llat} & \ddots & \vdots \\
\vdots &  & & \ddots & \ddots & 0  \\
0 & \ldots & \ldots & \ldots & 0 & \wignerDmati{\qq}{\mm}{\lmax}
\end{bmatrix},
\end{equation}

where $\lmax$ is the chosen truncation degree, and $\wignerDmati{\qq}{\mm}{\llat} $ is the matrix defined as

\begin{equation}
\wignerDmati{\qq}{\mm}{\llat} \define
\begin{bmatrix}
\wignerD{-\llat}{-\llat}{\llat} & \ldots & \wignerD{-\llat}{\mm}{\llat} & \ldots &  \wignerD{-\llat}{\llat}{\llat} \\
\vdots & & \vdots & & \vdots \\
\wignerD{\qq}{-\llat}{\llat} & \ldots & \wignerD{\qq}{\mm}{\llat} & \ldots & \wignerD{\qq}{\llat}{\llat} \\
\vdots & & \vdots & & \vdots \\
\wignerD{\llat}{-\llat}{\llat} & \ldots & \wignerD{\llat}{\mm}{\llat} & \ldots & \wignerD{\llat}{\llat}{\llat}
\end{bmatrix} .
\end{equation}

Thus, the set of SPHs associated with the rotated coordinate system, denoted by $\vectSPHrot \define \transp{\left[ \Ylmi{1}, \ldots ,\Ylmi{\nsph}  \right]}$, is expressed as a function of the initial set, $\vectSPHini \define \transp{\left[ \Ylmrefi{1}, \ldots ,\Ylmrefi{\nsph}  \right]}$, through the algebraic relation

\begin{equation}
\vectSPHrot = \transp{\wignermat } \vectSPHini,
\end{equation}

which is analogous to the change of basis equation given by change_basis_rotation. The inverse transformation is given by

\begin{equation}
\vectSPHini = \conj{\wignermat } \vectSPHrot.
\end{equation}

We remark that $\transp{\wignermat }\conj{\wignermat } = \idmat$, the symbol $\idmat$ referring to the identity matrix. As a consequence, the rotation transformation is conservative: going back and forth through the transformation does not induce any loss of information. Figure <ref> shows the rotation matrix obtained for a rotation of Euler angles $\left( \angalp, \angbet, \anggam \right) = \left(120^\degree, 60^\degree, 270^\degree \right) $ with a truncation degree set to $\lmax = 4$. As previously observed in Ylm_Ylq, the rotated SPHs are expressed in terms of the non-rotated SPHs having the same degree ($\llat$). The orders ($\mm$) only are mixed up by the rotation of the coordinate system.

§ TRANSITION MATRICES

In this appendix, we establish the general transition relations between sets of basis functions of various types  (SPHs, SCHNs, SCHDs; see ssec:helmholtz_decomposition and app:sch), and associated with different coordinate systems. These relations are used to compute the overlap coefficients between the forcing gravitational tidal potential and the tidal response in the calculation of the tidal torque.

SCHNs   SCHDs    SPHs

$\kkb$  $\kkb$  $\kkb$
Transition matrices between the SCHNs, SCHDs and SPHs for an ocean basin of angular radius $\colbd = 130^\degree$. The transition matrices are defined in transmat. The SCHs (SCHNs and SCHDs) are calculated from the expressions given by schc, and the SPHs from the expression given by sphc, using the same coordinate system for all the sets of basis functions and the truncation degree $\lmax = 4$. In every set, the basis functions are sorted in ascending order of orders ($\mm=-4,-3,\ldots , 3,4$), with degrees ($\llat \geq \abs{\mm}$) grouped together.

Original function ($\Hlm{\llati{3}}{1}$)  Expanded in SPHs  Expanded in rotated SPHs

$\lmax = 6$

$\lmax = 8$

$\lmax = 16$

Real part of complex SCH of order $\mm = 1$ and degree $\llati{3} = 6.806057$ corresponding to an ocean basin of angular radius $\colbd = 80^\degree$ (i.e. $\colcont = 100^\degree$) with Dirichlet boundary conditions (dircond). Left: Function plotted in its coordinate system using the expression given by schc. Middle: Function expanded in SPHs up to the truncation degree $\lmax = 6,8, 16$ (from top to bottom) in the same coordinate system using the transition relation of Yab_Ya. Right: Function expanded in rotated SPHs ($\angalp = 120^\degree$, $\angbet = 60^\degree$, $\anggam=270^\degree$) following the relation given by vfuncAB. All functions are plotted in polar coordinates, the radial coordinate corresponding to the colatitude of the considered coordinate system, and the angular coordinate to its longitude (in degrees).

Let be two sets of basis functions, $\Abasisref$ and $\Bbasisref$, associated with the same coordinate system, $\framestd$, but not necessarily the same definition domains[By definition domain, we refer to the domain where the functions are non-zero, the SCHs being defined on the full unit sphere similarly as the SPHs in the used convention (see schc).], denoted by $\sphcapAref$ and $\sphcapBref$, respectively. Both $\Abasisref$ and $\Bbasisref$ can be either the SPHs, SCHNs, or SCHDs, whose expressions are given by sphcschc. The sets undergo the rotation transformations $\Abasisref \rightarrow \Abasis$ and $\Bbasisref \rightarrow \Bbasis$, where $\Abasis$ and $\Bbasis$ are assumed to have the same definition domains as $\Abasisref$ and $\Bbasisref$, respectively. The expressions of the standard Euler rotation matrices $\rmati{\Abasis}$ and $\rmati{\Bbasis}$ describing these transformations are given by eulermat, and the corresponding Wigner rotation matrices for the functional bases, $\wignermatA$ and $\wignermatB$, are given by wignermat. We note that the second Euler angle can only be set to $\angbet =0$ in the case of the SCHs since the definition domains of the initial and rotated function would not overlap otherwise in that case. However, the rotation matrix of the SPHs still holds for the rotations of angles $\angalp$ and $\anggam$ since these angles just introduce a phase lag in the $\expo{\inumber \mm \lon}$ component of the SCHs.

The initial and rotated functional basis vectors are denoted by $\vfuncAref$, $\vfuncBref$, $\vfuncA$, and $\vfuncB$, the vector $\vfunci{\anybasis}$ of any set $\anybasis$ being expressed as

\begin{equation}
\vfunci{\anybasis} = \transp{\left[ \funci{\anybasis}{1}, \funci{\anybasis}{2}, \ldots, \funci{\anybasis}{\kk} , \ldots , \funci{\anybasis}{\nbasisi{\anybasis}}  \right]},
\end{equation}

where the $\funci{\basis{H}}{\kk}$ designate the orthogonal functions of set $\anybasis$, and $\nbasisi{\anybasis}$ its total number of elements, which depends on the truncation degree $\lmaxi{\anybasis}$ introduced in wignermat. To lighten expressions, we assume that the sets of  basis functions $\Abasisref$ and $\Abasis$ contain $\nbasisA$ elements, and that $\Bbasisref$ and $\Bbasis$ contain $\nbasisB$ elements, so that $\nbasisi{\Abasisref} = \nbasisi{\Abasis} = \nbasisA $ and $\nbasisi{\Bbasisref} = \nbasisi{\Bbasis} = \nbasisB$. Using the above notations, we can write down the relations

\begin{equation}
\label{Y_Yrot}
\begin{array}{ll}
\vfuncA = \transp{\wignermatA} \vfuncAref, & \vfuncB = \transp{\wignermatB} \vfuncBref,  \\[0.3cm]
\vfuncAref = \conj{\wignermatA} \vfuncA, & \vfuncBref = \conj{\wignermatB} \vfuncB.
\end{array}
\end{equation}

As a next step, we establish the relationship between the bases $\Abasisref$ and $\Bbasisref$, which share the same coordinate system. The functions $\funcBrefi{\jj} $ and $\funcArefi{\kk} $ are expressed in terms of each other through the relations

\begin{align}
\label{Yj_series}
& \funcBrefi{\jj} = \sum_{\kk=1}^{\nbasisA} \frac{\scal{\funcArefi{\kk}}{\funcBrefi{\jj}} }{\scal{\funcArefi{\kk}}{\funcArefi{\kk}}} \funcArefi{\kk} + \resfi{\Bbasisref}{\jj} ,  \\
\label{Yk_series}
& \funcArefi{\kk} = \sum_{\jj=1}^{\nbasisB} \frac{\scal{\funcBrefi{\jj}}{\funcArefi{\kk}} }{\scal{\funcBrefi{\jj}}{\funcBrefi{\jj}}} \funcBrefi{\jj} + \resfi{\Abasisref}{\kk} .
\end{align}

In the above equations, the functions $\resfi{\Bbasisref}{\jj} $ and $\resfi{\Abasisref}{\kk}$ are the residuals of the functions $\funcBrefi{\jj}$ or $\funcArefi{\kk}$ when the latter are expanded in terms of the functions of set $\Abasisref$ or $\Bbasisref$, respectively. These residuals result both from the truncation of the sets $\Abasisref$ and $\Bbasisref$ – which are mathematically defined as infinite-dimension sets of basis functions for the definition domains $\sphcapAref $ and $\sphcapBref$ –, and from the fact that $\sphcapAref $ and $\sphcapBref$ are different from each other in the general case.

For instance, if $\sphcapAref \subsetneq \sphcapBref$, the functions of set $\Bbasisref$ can be expressed as linear combinations of the elements of set $\Abasisref$ on $\sphcapAref$ only. This is the case for SPHs, which cannot be approximated using SCHs while the opposite is true. Therefore, the function $\funcBrefi{\jj}$ tends to be fully described by the functions of set $\Abasisref$ as $\nbasisA \rightarrow \infty$ if $\sphcapBref \subset \sphcapAref$, and consequently $\normtwo{\resfi{\Bbasisref}{\jj} } \rightarrow 0$ in that case. Conversely, $\normtwo{\resfi{\Bbasisref}{\jj} }>0$ for $\nbasisA = \infty$ in the general case if $\sphcapAref \subsetneq \sphcapBref$. The residual function $\resfi{\Abasisref}{\kk}$ behaves in a similar way if the domains $\sphcapAref $ and $\sphcapBref$ are interchanged: $\normtwo{\resfi{\Abasisref}{\kk}} \rightarrow 0$ as $\nbasisB \rightarrow \infty$ if $\sphcapAref \subset \sphcapBref$, while $\normtwo{\resfi{\Abasisref}{\kk}} >0$ if $\sphcapBref \subsetneq \sphcapAref$.

Since the functions of the two sets are assumed to be orthonormal by construction (see app:sch), $\scal{\funcArefi{\kk}}{\funcArefi{\kk}} = \scal{\funcBrefi{\jj}}{\funcBrefi{\jj}} = 1$ for all $\kk \in \left\{1 , \ldots , \nbasisA  \right\}$ and $\jj \in \left\{ 1 , \ldots , \nbasisB \right\}$. As a consequence, the expressions given by Yj_seriesYk_series simplify to

\begin{align}
\label{Yj_Ybak}
& \funcBrefi{\jj} = \sum_{\kk=1}^{\nbasisA} \scal{\funcArefi{\kk}}{\funcBrefi{\jj}} \funcArefi{\kk} + \resfi{\Bbasisref}{\jj} & \mbox{on} \ \sphcapBref,  \\
\label{Yk_Yabj}
& \funcArefi{\kk} = \sum_{\jj=1}^{\nbasisB} \scal{\funcBrefi{\jj}}{\funcArefi{\kk}} \funcBrefi{\jj} + \resfi{\Abasisref}{\kk} & \mbox{on} \ \sphcapAref,
\end{align}

which, introducing the vectors

\begin{align}
\resmBref & = \transp{\left[ \resfi{\Bbasisref}{1}, \ldots , \resfi{\Bbasisref}{\jj} , \ldots , \resfi{\Bbasisref}{\nbasisB}  \right]},  \\
\resmAref& = \transp{\left[ \resfi{\Abasisref}{1}, \ldots , \resfi{\Abasisref}{\kk} , \ldots , \resfi{\Abasisref}{\nbasisA}  \right]}  ,
\end{align}

may be rewritten as

\begin{align}
\label{Yab_Ya}
& \vfuncBref = \transp{\transmati{\Abasisref}{\Bbasisref}} \vfuncAref + \resmBref,&  \vfuncAref = \conj{\transmati{\Abasisref}{\Bbasisref}} \vfuncBref + \resmAref.
\end{align}

In the above equations, the transition matrix is expressed as

\begin{equation}
\label{transmat}
\transmati{\Abasisref}{\Bbasisref} \! \define \!
\begin{bmatrix}
\scal{\funcArefi{1}}{\funcBrefi{1}} & \ldots & \scal{\funcArefi{1}}{\funcBrefi{\jj}} & \ldots & \scal{\funcArefi{1}}{\funcBrefi{\nbasisB}} \\
\vdots &  & \vdots & & \vdots \\
\scal{\funcArefi{\kk}}{\funcBrefi{1}} & \ldots & \scal{\funcArefi{\kk}}{\funcBrefi{\jj}} & \ldots & \scal{\funcArefi{\kk}}{\funcBrefi{\nbasisB}} \\
\vdots &  & \vdots & & \vdots \\
\scal{\funcArefi{\nbasisA}}{\funcBrefi{1}} & \ldots & \scal{\funcArefi{\nbasisA}}{\funcBrefi{\jj}} & \ldots & \scal{\funcArefi{\nbasisA}}{\funcBrefi{\nbasisB}}
\end{bmatrix} \! \!,
\end{equation}

where $\scal{\funcArefi{\kk}}{\funcBrefi{\jj}}$ is the scalar product of the $\kk$-th element of $\Abasisref$ with the $\jj$-th element of $\Bbasisref$ following the formulation given by sprodYlm. We note that the integral of the scalar product is actually performed on the intersection of the domains associated with $\Abasisref$ and $\Bbasisref$, namely $\sphcap = \sphcapAref \cap \sphcapBref$.

With the used normalisations, the transition matrix satisfies the properties $\transmati{\Bbasisref}{\Abasisref} = \transp{\conj{\transmati{\Abasisref}{\Bbasisref}}} $, $\transmati{\Abasisref}{\Abasisref} = \idmati{\nbasisA} $, and $\transmati{\Bbasisref}{\Bbasisref} = \idmati{\nbasisB}$, the notation $\idmati{\nbasisA}$ referring to the identity matrix of size $\nbasisA$. However, one should pay attention to the fact that $\transmati{\Abasisref}{\Bbasisref} \transmati{\Bbasisref}{\Abasisref} \neq \transmati{\Abasisref}{\Abasisref} $ and  $\transmati{\Bbasisref}{\Abasisref} \transmati{\Abasisref}{\Bbasisref} \neq \transmati{\Bbasisref}{\Bbasisref} $ due to the inability of the sets of basis functions to fully describe the space of functions defined on the unit sphere, as discussed above. In other words, going back and forth between the two sets necessarily induces a loss of information contrary to the rotation operator of the SPHs detailed in app:rotation_sph, which is conservative.

Figure <ref> shows the transition matrices $\transmati{\Abasisref}{\Bbasisref}$ between the SCHNs, SCHDs, and SPHs for an ocean of angular radius $\colbd = 130^\degree$ (i.e. a continent of angular radius $\colcont = 50^\degree$). The basis functions are sorted in ascending order of degrees and orders in every set, with degrees grouped together, which differs from the SPHs of fig:rmat, where orders are grouped together. Only basis functions with the same order overlap, due to the fact that the SCHs and SPHs have the same longitudinal component. However, some functions belonging to one set are poorly described by the basis functions of the other sets.

By combining together the relations given by Y_YrotYab_Ya, we finally obtain

\begin{align}
\label{vfuncAB}
& \vfuncB = \transp{\transmati{\Abasis}{\Bbasis}} \vfuncA + \transp{\wignermatB} \resmBref , & \vfuncA = \conj{\transmati{\Abasis}{\Bbasis}} \vfuncB + \transp{\wignermatA} \resmAref ,
\end{align}

where the transition matrix $\transmati{\Abasis}{\Bbasis}$ is given by

\begin{equation}
\label{transmatAB}
\transmati{\Abasis}{\Bbasis} = \transp{\conj{\wignermatA}} \transmati{\Abasisref}{\Bbasisref} \wignermatB.
\end{equation}

A given function $\ffunc$ defined on the sphere can thus be written both in terms of the basis functions of $\Abasis$ and those of $\Bbasis$ as

\begin{align}
& \ffunc = \transp{\vcoorA}  \vfuncA + \resci{\Abasis}{\ffunc}, &  \ffunc = \transp{\vcoorB}  \vfuncB + \resci{\Bbasis}{\ffunc},
\end{align}

where we have introduced the residual functions $\resci{\Abasis}{\ffunc}$ and $\resci{\Bbasis}{\ffunc}$, and the coordinate vectors of $\ffunc$ in the two sets of basis functions, $\vcoorA$ and $\vcoorB$, defined as

\begin{align}
\vcoorA \define & \transp{\left[  \coorAi{1}, \ldots , \coorAi{\kk} , \ldots , \coorAi{\nbasisA} \right]} , \\
\vcoorB \define & \transp{\left[  \coorBi{1}, \ldots , \coorBi{\jj} , \ldots , \coorBi{\nbasisB} \right]} .
\end{align}

The vectors $ \vcoorA$ and $ \vcoorB$ contain the coefficients of the expansion of $\ffunc$ on the set of basis functions $\Abasis$ or $\Bbasis$, respectively. They are related to each other by the transition matrix $\transmati{\Abasis}{\Bbasis}$ introduced in transmatAB through the relation equations

\begin{align}
& \vcoorA = \transmati{\Abasis}{\Bbasis} \vcoorB, & \vcoorB = \transp{\conj{\transmati{\Abasis}{\Bbasis}}} \vcoorA.
\end{align}

Figure <ref> shows the three steps of the transition from SCHs to rotated SPHs through the example of the eigenfunction $\Hlm{\llati{3}}{1}$ obtained with Dirichlet conditions (SCHD). First the function is evaluated in its associated coordinate system using the expression given by schc (left panels). We note that the region where the function is set to zero is not represented in the plot. Second, the function is expanded in series of SPHs in the same coordinate system (middle panels). Finally, we apply an Euler rotation to the SPHs, as described by Ylm_Ylq, and $\Hlm{\llati{3}}{1}$ is evaluated in terms of the rotated SPHs in the rotated coordinate system (right panel). These three steps are shown for various truncation degrees of the SPHs, $\lmax = 6,8,16$ (from top to bottom). We observe that the discontinuity of the gradient of the function at $\col = \colbd$ gives rise to `ringing' when $\Hlm{\llati{3}}{1}$ is expanded in series of SPHs. This effect also alters the expansions of the SCHNs due to their discontinuous transition at $\col = \colbd$. However it vanishes as the truncation degrees $\lmax$ increases.

§ GYROSCOPIC COEFFICIENTS

In this section, we detail the calculation of the gyroscopic coefficients that couple the LTEs. Following earlier studies <cit.>, these coefficients are denoted by $\gyrocoeff$. They are defined as

\begin{align}
\label{gyro1}
\gyroci{\jjb}{\kkb} \define & - \integ{\cos \colpla \, \er \dotp \left( \conj{\grad \fpoti{\jjb} } \crossp \grad \fpoti{\kkb} \right) }{\surface}{\sphcap}{}, \\
\label{gyro2}
\gyroci{\jjb}{- \kkb} \define & \ \ \ \ \integ{\cos \colpla  \left( \conj{\grad \fpoti{\jjb} } \dotp \grad \fstreami{\kkb} \right)}{\surface}{\sphcap}{}, \\
\label{gyro3}
\gyroci{-\jjb}{\kkb} \define &  - \integ{\cos \colpla \left( \conj{\grad \fstreami{\jjb}} \dotp \grad \fpoti{\kkb} \right)}{\surface}{\sphcap}{}, \\
\label{gyro4}
\gyroci{-\jjb}{-\kkb} \define & - \integ{\cos \colpla \, \er \dotp \left(  \conj{\grad \fstreami{\jjb}} \crossp \grad \fstreami{\kkb} \right)}{\surface}{\sphcap}{},
\end{align}

where the $\fpoti{\jjb}$ and $\fstreami{\jjb}$ are the basis functions generating the potential and stream functions, respectively (see ssec:helmholtz_decomposition and app:sch).

$\Im \left( \gyroci{\jjb}{\kkb} \right)$   $\Re \left( \gyroci{\jjb}{-\kkb} \right)$  $\Re \left( \gyroci{-\jjb}{\kkb} \right)$  $\Im \left( \gyroci{-\jjb}{-\kkb} \right)$

$\coloc = 0^\degree$

$\coloc = 45^\degree$

$\coloc = 90^\degree$

$\kkb$  $\kkb$  $\kkb$  $\kkb$
Gyroscopic coefficients of an ocean basin of angular radius $\colbd=130^\degree$ (i.e. $\colcont = 50^\degree$) for various values of the colatitude of the ocean's centre. Top: $\coloc=0^\degree$ (polar ocean). Middle: $\coloc=45^\degree$ (mid-latitude ocean). Bottom: $\coloc = 90^\degree$ (equatorial ocean). The gyroscopic coefficients are computed from the expressions given by gyro1_expgyro4_exp with the truncation degree set to $\lmax=4$.  The basis functions of indices $\jjb$ and $\kkb$ are sorted in ascending order of orders ($\mm = -4,-3, \ldots,3,4$), with degrees ($\llat \geq \abs{\mm}$) grouped together. For convenience, the minimum and maximum of the colour bar are set to 6 in absolute value although the maximum values reached by gyroscopic coefficients are around ${\sim}$15 in this case.

We emphasise that $\jjb$ and $\kkb$ are the indices of the functions in their respective sets. One shall be cautious not to mix up these indices with the orders and degrees of the functions. The third coefficient, given by gyro3, is straightforwardly deduced from the second one, given by gyro2, since $\gyroci{-\jjb}{\kkb} = - \conj{\gyroci{\kkb}{- \jjb}}$. In some particular cases, such as the hemispherical ocean configuration considered in previous studies <cit.>, the recurrence relations of the associated Legendre  functions can be used to obtain the analytical expressions of the gyroscopic coefficients <cit.>. Unfortunately, these recurrence relations do not hold for non-integral degrees, which requires to evaluate the integrals of gyro1gyro4 numerically. However, it is still possible to write down these integrals as products of separated coordinates.

Introducing the coordinates of the North pole in the ocean's coordinate system, $\left( \colnp , \lonnp \right) = \left( \coloc , \pi \right)$, and using trigonometric identities, we express $\cos \colpla$ as a function of the colatitude of the ocean's centre ($\coloc$) and the ocean's coordinate system,

\begin{align}
\cos \colpla & = \cos \colnp \cos \col + \sin \colnp \sin \col \cos \left( \lon - \lonnp \right) \\
&  = \cos \coloc \cos \col - \sin \coloc \sin \col \cos \lon.
\end{align}

Considering the definitions of the SPHs and SCHs given by sphcschc, the functions $\fpoti{\jjb}$ and $\fstreami{\jjb}$ are expressed in the general case as

\begin{align}
\fpoti{\jjb} \left( \col , \lon \right) & = \fpotci{\jjb}  \phicolfi{\jjb} \left( \col \right) \expo{\inumber \mmi{\jjb} \lon} , \\
\fstreami{\jjb} \left( \col , \lon \right) & = \fstreamci{\jjb} \psicolfi{\jjb} \left( \col \right)  \expo{\inumber \mmi{\jjb} \lon},
\end{align}

where $ \phicolfi{\jjb} \left( \col \right) = \LegF{\llati{\jjb}}{\mmi{\jjb} } \left( \cos \col \right)$ designates the ALF of the $\jjb$-th function of the set of SCHNs (Neumann conditions) used to expand the potential function, and $ \psicolfi{\jjb} \left( \col \right) = \LegF{\llati{\jjb}}{\mmi{\jjb} } \left( \cos \col \right)$ the ALF of the $\jjb$-th function of the set of SCHDs (Dirichlet conditions) used to expand the stream function, as detailed in app:sch. The gyroscopic coefficients given by gyro1gyro4 are therefore expressed as

\begin{align}
\gyroci{\jjb}{\kkb} = & - \inumber 2 \pi \fpotci{\jjb} \fpotci{\kkb}   \left\{  \cos \coloc \kron{\mmi{\jjb}}{\mmi{\kkb}} \mmi{\jjb}   \!\! \integ{ \! \! \! \! \cos \col \left[ \DD{\phicolfi{\jjb}}{\col} \phicolfi{\kkb} + \phicolfi{\jjb} \DD{\phicolfi{\kkb}}{\col}  \right] \!}{\col}{0}{\colbd}   \right. \nonumber \\
\label{gyro1_exp}
& \left. - \frac{\sin \coloc}{2}  \kron{\abs{\mmi{\kkb} - \mmi{\jjb}}}{1} \!\! \integ{\! \! \! \! \sin \col \left[ \mmi{\kkb} \DD{\phicolfi{\jjb}}{\col} \phicolfi{\kkb} + \mmi{\jjb} \phicolfi{\jjb} \DD{\phicolfi{\kkb}}{\col}  \right] \! }{\col}{0}{\colbd}    \right\},
\end{align}

\begin{align}
\gyroci{\jjb}{-\kkb} = &  \ \pi \fpotci{\kkb} \fstreamci{\jjb} \left\{  \cos \coloc \kron{\mmi{\jjb}}{\mmi{\kkb}} \! \!   \integ{ \!\!\!\! \sin 2 \col  \left[ \DD{\phicolfi{\kkb}}{\col} \DD{\psicolfi{\jjb}}{\col} + \mmi{\jjb}^2 \frac{\phicolfi{\kkb} \psicolfi{\jjb}}{\sin^2 \col} \right] \! }{\col}{0}{\colbd}  \right. \nonumber \\
& \left. - \sin \coloc  \kron{\abs{\mmi{\kkb} - \mmi{\jjb}}}{1} \!\! \integ{ \!\!\!\! \sin^2 \col  \left[ \DD{\phicolfi{\kkb}}{\col} \DD{\psicolfi{\jjb}}{\col} + \mmi{\jjb} \mmi{\kkb} \frac{\phicolfi{\kkb} \psicolfi{\jjb}}{\sin^2 \col} \right] \! }{\col}{0}{\colbd}  \! \right\}\! ,
\end{align}

\begin{equation}
\gyroci{-\jjb}{\kkb} = - \gyroci{\kkb}{-\jjb},
\end{equation}

\begin{align}
\gyroci{-\jjb}{-\kkb} = & - \inumber 2 \pi \fstreamci{\jjb} \fstreamci{\kkb} \!  \left\{ \!   \cos \coloc \kron{\mmi{\jjb}}{\mmi{\kkb}} \mmi{\jjb} \! \! \integ{ \! \! \! \! \! \cos \col \! \left[ \! \DD{\psicolfi{\jjb}}{\col} \psicolfi{\kkb} + \psicolfi{\jjb} \DD{\psicolfi{\kkb}}{\col} \! \right] \! }{\col}{0}{\colbd}   \right. \nonumber \\
\label{gyro4_exp}
& \left. - \frac{\sin \coloc }{2} \kron{\abs{\mmi{\kkb} - \mmi{\jjb}}}{1} \! \! \integ{\! \! \! \!  \sin \col \left[ \mmi{\kkb} \DD{\psicolfi{\jjb}}{\col} \psicolfi{\kkb} + \mmi{\jjb} \psicolfi{\jjb} \DD{\psicolfi{\kkb}}{\col}  \right] \! }{\col}{0}{\colbd}    \right\} \!.
\end{align}

Interestingly, the coefficients $\gyroci{\pm \jjb}{\pm \kkb}$ are zero if $\abs{\mmi{\kkb}-\mmi{\jjb}} >1  $, meaning that Coriolis effects couple the eigenmodes having either the same order or neighbouring orders in the general case. If the ocean is polar ($\coloc=0,\pi$), the gyroscopic coefficients only mix up eigenmodes with the same order ($\mmi{\kkb}=\mmi{\jjb}$). In this configuration, $\gyroci{\pm\jjb}{\pm\kkb} \left( \coloc = \pi \right) = - \gyroci{\pm\jjb}{\pm\kkb} \left( \coloc = 0 \right)$. If the ocean is equatorial ($\coloc = \pi/2$), the gyroscopic coefficients only mix up eigenmodes of neighbouring orders ($\abs{\mmi{\kkb}-\mmi{\jjb}}=1$). Between these two extremal configurations, the gyroscopic coefficients are the sum of one component in $\cos \coloc$ associated with eigenmodes having the same order, and one component in $\sin \coloc$ associated with eigenmodes having neighbouring orders. The gyroscopic coefficients that do not mix up potential and stream functions ($\gyroci{\jjb}{-\kkb} $ and $\gyroci{-\jjb}{-\kkb} $) are pure imaginary numbers. Conversely, the coefficients that mix up the functions ($\gyroci{\jjb}{-\kkb}$ and $\gyroci{-\jjb}{\kkb} $) are pure real numbers.

By comparing the expressions given by gyro1_expgyro4_exp, we remark that the $\gyroci{\jjb}{\kkb} $ and $\gyroci{-\jjb}{-\kkb}  $ converge towards the same asymptotic values as $\colbd \rightarrow 180^\degree$ since  both the SCHNs and SCHDs converge towards the standard SPHs for the $\ltwonorm$-norm introduced in ltwonorm, as illustrated by fig:degrees_size. To our knowledge, no recurrence formula exist between the ALFs of real degrees of the SCHNs and SCHDs except in the specific case of the hemispherical ocean, where the degrees are integers <cit.>. As a consequence, the integrals in gyro1_expgyro4_exp are evaluated numerically in practice.

Figure <ref> shows the gyroscopic coefficients characterising an ocean basin of angular radius $\colbd = 130^\degree$ for three typical configuration: $\coloc = 0^\degree$ (polar ocean), $\coloc = 45^\degree$ (mid-latitude ocean), and $\coloc = 90^\degree$ (equatorial ocean). The eigenmodes of indices $\jjb$ and $\kkb$ are sorted in ascending order of degrees and orders in the displayed plots, which emphasises the fact that gyroscopic coefficients only couple eigenmodes with the same or neighbouring orders.

The coefficients $\cstreamsphi{\jjb}{\qq}$ introduced in cstreamkq are evaluated similarly as the gyroscopic coefficients. These coefficients are defined as

\begin{equation}
\cstreamsphi{\jjb}{\qq} \define \integ{ \! \! \er \! \dotp \! \left( \conj{\gradn \fstreami{\jjb}} \crossp \gradn \Ylmi{\qq}  \right) \!}{\surface}{\oceancap}{},
\end{equation}

where the SPHs $\Ylmi{\qq} $ are written as products of functions of separated variables,

\begin{equation}
\Ylmi{\qq} \left( \col , \lon \right) = \sphci{\qq} \sphcolfi{\qq} \left( \col \right) \expo{\inumber \mmi{\qq} \lon }.
\end{equation}

After an integration by part, the coefficients are simply expressed as

\begin{equation}
\cstreamsphi{\jjb}{\qq} = \inumber 2 \pi \fstreamci{\jjb} \sphci{\qq}  \mmi{\jjb}  \kron{\mmi{\jjb}}{\mmi{\qq}} \left[ \psicolfi{\jjb} \left( \colbd \right) \sphcolfi{\qq} \left( \colbd \right) - \psicolfi{\jjb} \left( 0 \right) \sphcolfi{\qq} \left( 0 \right) \right] .
\end{equation}

We note that the $\cstreamsphi{\jjb}{\qq}$ are zero if $\mmi{\jjb} = 0$. Besides, if $\abs{\mmi{\jjb}}>0$, the Dirichlet condition satisfied by the $\fstreami{\jjb}$ at $\colbd$ implies that $\psicolfi{\jjb} \left( \colbd \right) =0$, and the polar condition of $\Ylmi{\qq}$ implies that $\sphcolfi{\qq} \left( 0 \right) = 0$. As a consequence, $\cstreamsphi{\jjb}{\qq} = 0$ for all $\jjb$ and $\qq$.

§ SOLID TIDAL LOVE NUMBERS

The gravitational and load Love numbers introduced in tiltdtiltg describe the tidal response of the planet's solid part in the general case as far as the latter is spherically symmetric. In the present study, these parameters are formulated analytically from the prescriptions given by [Bolmont et al., 2020] for solid Earth (see Table 2 of the article). This formulation is based on the tidal model described in [Tobie et al., 2005], which integrates the equations of the elasto-gravitational theory <cit.> for realistic radial profiles of background quantities. As a first step, [Bolmont et al., 2020] use this tidal model with the background profiles computed from the internal structure model of [Sotin et al., 2007]. As a second step, they fit the parameters of simplified models to the obtained solutions. These simplified models describe the rheology of equivalent bodies with uniform interiors, for which the Love numbers can be explicitly formulated in terms of the degrees of the associated SPHs <cit.>.

The degree-$\llat$ Love numbers of a homogeneous solid body of mass $\Mcore$, radius $\Rcore$, and uniform shear modulus $\mupla$, are expressed as <cit.>

\begin{equation}
\left\{ \kl , \hl , \kloadl, \hloadl \right\} \! = \! \frac{1}{1+\Cmul} \! \left\{ \! \frac{3}{2 \left( \llat - 1 \right)} , \frac{2 \llat +1}{2 \left( \llat - 1 \right)} , - 1 , -  \frac{2 \llat +1}{3} \!\right\} \! .
\label{love_solid}
\end{equation}

In the above equation, the degree-$\llat$ dimensionless shear modulus $\Cmul$ is defined as

\begin{equation}
\Cmul  \define \Al \Cmu,
\label{Cmul}
\end{equation}

where $\Cmu$ designates the dimensionless visco-elastic rigidity accounting for the rheology of the material, and $\Al $ the dimensionless parameter given by <cit.>

\begin{equation}
\Al \define \frac{4 \left( 2 \llat^2 + 4 \llat + 3 \right) \pi \Rcore^4 \mupla}{3 \llat \Ggrav \Mcore^2}.
\label{Al_Cmu}
\end{equation}

Since the oceanic layer is thin in the present study, the mass and radius of the solid part can be approximated by the planet mass and radius, respectively, which yields $\Mcore \approx \Mpla$ and $\Rcore \approx \Rpla$.

The normalised rigidity $\Cmu$ is a complex parameter that describes the rheological behaviour of the solid part in the frequency domain. This parameter is determined by the adopted rheological model. For standard models such as the Maxwell visco-elastic rheology <cit.>, it is expressed as a simple function of the tidal frequency, $\ftide$. Following [Bolmont et al., 2020], we opt for an Andrade rheology. Similarly as the Maxwell rheology, the Andrade model describes the visco-elastic deformation of the material. However, it additionally accounts for the unrecoverable creep forming the anelastic component of the response, which affects the frequency-dependence of tidal dissipation in the high-frequency range  <cit.>. In the Andrade rheology, $\Cmu$ is expressed as <cit.>

\begin{equation}
\Cmu = \frac{1}{1 + \left( \inumber \ftide \tauA \right)^{- \alphaA} \GammaF \left( 1 + \alphaA \right) + \left( \inumber \ftide \tauM \right)^{-1}},
\label{Andrade_mu}
\end{equation}

where $\tauM$ designate the Maxwell time, $\tauA$ the Andrade time, $\GammaF$ the Gamma function introduced in GammaF, and $\alphaA$ the parameter determining the duration of the transient response in the primary creep <cit.>.