I was thinking about what type of places people really enjoy knowing they could count on when it comes to self-care. Well, there's not doubt that manicure and pedicures are certainly a popular way to treat ourselves to something that goes into the self care category. Getting manicures and pedicures is a great thing to do for ourselves and is a fun way to spend time with our friends and loved ones too. But if you're new in Rocklin or you haven't nailed a favorite place (yes, I really said that), how do you find what's considered the best place? Word of mouth, right? So, in helping you decide your new favorite nail salon, I decided to do some research into what the buzz around town is on nail salons. And the number favorite salon in Rocklin is…I can't say because there are three that are VERY popular. So the good news is that you get to choose which of the three is going to be most convenient for you to try. Here's what you need to know to make that choice.
In researching (but of course I did) the factors considered most important to clients when using a nail salon, I didn't come away with any surprises. The first deciding factor named is cleanliness and sanitizing practices. The second is quality and professionalism related to the service and third is overall ambiance of salon. Now don't get me wrong, cost of service is listed but not before addressing the things that make the client feel safe and pampered. So with these factors in mind, read on to see how my top three picks for Rocklin measure up according to word of mouth by customers (a.k.a. YELP).
The first thing I noticed is that there were tons (125 out of 166 reviews) of five-star reviews for Sun Nails! Of all the nail salons in Rocklin, it had the most reviews which typically means that clients were impacted enough by the experience to want to tell others about it. The second thing I noticed was that clients mentioned they had moved from a city that caters to affluent clients and found Sun Nails to be comparable to what that level of service. Along those same lines, I noticed that several clients traveled from out of town just to have their nails done at Sun Nails! If you're like me, I really like using shops and services in my own community so for someone to travel up to an hour (yes, I read an hour) to use a salon, that speaks volumes. Anyone who uses a nail salon knows that there are many choices in most towns and cities in California. Although there were several favorable reviews mentioning more than one technician, Keven (the salon owner) received big shout outs for his professionalism, friendliness and technical skills. The salon ambiance included favorable reviews about air quality being excellent and of course, sanitary practices and cleanliness were given top marks.
From Sun Nails to Sky Nails, things keep looking great for nail salon choices in our lovely Rocklin. There's nothing like having choices so it appears as if, in the nail salon department, we have some heavy competition for "the best of the best". The same great things said of Sun Salon are said of Sky Nails. The owner, Kelsey, is mentioned many times and given five stars for her friendliness, technical knowledge and professionalism. Like Sun Nails, that seems to be true for her nail technicians as well. Many of the reviews advised potential clients to make an appointment because Sky Salon is always busy (a good sign). Many clients also mentioned how relaxing and quiet the salon is even when very busy.
If we're going to look at the whole picture, we have to say that the ultimate in pampering at a nail salon happens with pedicures. Now that spring is here, it's time to think flip flops and pedis, right? Now to be as clear as a top coat (get it?) Sky Nails and Sun Nails are both excellent in the manicure and pedicure department! But I was curious about finding a place that had a foot hold in the pedicure department and came across Organic Nails. While it didn't have the same number of reviews as Sky Nails and Sun Nails, it's only been in business a year. When I filed away at the numbers, I saw that its 33 reviews were proportionately as excellent as the other two salons because Organic Nails has only been in business a year!!! I learned that Organic Nails specializes on providing an amazing pedicure complete with all organic oils. Of course, the manicure reviews were excellent as well.
As you can see from what I learned regarding Sky Nails, Sun Nails and Organic Nails, they're outstanding in what they do. Here's what the numbers said. If you look at the number of years these three salons have been in business, all three have over 90 percent of their reviews being 5 and 4 stars. That's outstanding! No matter which one you choose to make your new favorite nail salon, you won't go wrong. Looking at what others say about these salons helped me decide which salons to try for myself and I look forward to visiting each salon for a wonderfully pampering experience.
Get Your Moving Crew Manicures and Pedicures!
Speaking of pampering, we know that moving day can be really hard on the hands and feet so why not treat yourself and your moving crew to an afternoon of manicures and pedicures? As you read above, Rocklin has great choices in salons. As your neighborhood self storage facility, Stanford Ranch Self Storage is Rocklin's newest self storage facility. We look forward to exceeding your self storage expectations and we're very excited to serve our community. We'd love to hear what you're excited about sharing so we can keep spreading the world about our great community of Rocklin.


text:
Taxikář (v americkém originále: Taxi Driver) je americký dramatický film z roku 1976. Režisérem filmu je Martin Scorsese. Hlavní role ve filmu ztvárnili Robert De Niro, Jodie Fosterová, Harvey Keitel, Cybill Shepherdová a Albert Brooks.

Ocenění
Jodie Fosterová za svou roli v tomto filmu získala dvě ceny BAFTA. Nominována byla také na Oscara. Robert De Niro byl za svou roli ve filmu nominován na Oscara, Zlatý glóbus a cenu BAFTA. Bernard Herrmann získal za hudbu k filmu cenu BAFTA. Film byl dále nominován na dva Oscary (kategorie nejlepší film a hudba), na Zlatý glóbus v kategorii nejlepší scénář a na tři ceny BAFTA (v kategorii nejlepší režie, film a střih).

Reakce
 aktuální k 7. srpnu 2014

Film získal mezi diváky na největších filmových databázích spíše velmi dobré hodnocení.

 csfd.cz: 84 %
 imdb.com: 8,4 z 10
 fdb.cz: 83,7 %

Obsazení

Atentát na Ronalda Reagana

Minimálně patnáctkrát navštívil promítání filmu v Hollywoodu John Hinckley, jenž se silně ztotožnil s jeho hlavní postavou Travisem
Bickleym. Jeho posedlost herečkou Jodie Fosterovou následně stála za atentátem na Ronalda Reagana z 30. března 1981. Během následujících let ji pronásledoval po celých Spojených státech. Navázat kontakt se mu však nepodařilo, což vyústilo v rozhodnutí provést velký manifestační čin – zavraždit prezidenta, kterým by se se stal historickou osobností rovnocennou Fosterové. Jednou z inspirací byl děj Taxikáře, v němž je plánován atentát na senátora kandidujícího na úřad prezidenta.

Odkazy

Reference

Externí odkazy

Taxikář (film)
Taxikář (film)
Taxikář (film)
Taxikář (film)
Filmy o prostituci
Filmy odehrávající se v New Yorku
Filmy studia Columbia Pictures
Filmy Martina Scorseseho
Filmy oceněné Zlatou palmou

redpajama_set_name: RedPajamaWikipedia


This is set to be an epic comeback.
February 2017 will see the return of explosive character Rosie Webster to the Coronation Street cobbles, following a five-year hiatus.
Actress Helen Flanagan is returning to her soap roots, having played the Corrie fave from 2000 until 2012 before leaving the show to "try something new".
Speaking about her return, she said: "I'm so excited to be returning to Coronation Street and being part of the Webster family again."
Although the actress' storyline has been kept firmly under wraps, her return has been described by bosses as "eventful".
What we do know is that she will surprise her parents when she shows up for a visit with her sister Sophie after a holiday in Miami.
Rosie left the cobbles in 2012.
Coronation Street producer Kate Oates said: "The Websters have missed Rosie and so have we. With Sophie away on holiday with her sister, it seemed like the ideal time to bring Rosie back home with her for a visit.
"We are delighted that Helen is as keen to explore what Rosie has been up to away from the cobbles as we are."
26-year-old Helen gave birth to daughter Matilda Jessica Sinclair in June 2015 with her footballer boyfriend Scott Sinclair.


text:
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- (UIInterfaceOrientationMask)application:(UIApplication *)application supportedInterfaceOrientationsForWindow:(UIWindow *)window {
    return UIInterfaceOrientationMaskAll;
}

redpajama_set_name: RedPajamaStackExchange


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HUBER HEIGHTS — The search for a new Superintendent of the Huber Heights City Schools began in earnest this week as representatives of the district's search firm held several focus group meetings.
Christopher S. Mohr and Dr. Dennis A. Leone of K-12 Business Consulting, Inc. led the focus group meetings in a discussion about the school district's strengths and weaknesses as well as traits that would be of value in a new Superintendent.
Current Superintendent Susan Gunnell announced her retirement earlier this month.
"We are trying to find the right match for this district, and having these focus groups and getting the search profiles filled out will help us do that," said Mohr.
The community is asked to visit the district's website where they can download the Superintendent Profile Questionnaire Deadline and return it to the the search company.
Under the timeline posted on the Huber Heights City Schools website, the search profiles are expected to go to the board of education on December 6. The position will be announced statewide on December 14 with applications being accepted through January 25.
First round interviews will occur the first week of February followed by a second round of interviews with finalists. A hire would be made next spring with the new Superintendent in place by August 1, 2019.
K-12 Business Consulting, Inc. is a recruitment source for school superintendents and treasurers all over the state. According to their website, recent superintendent searches have included Troy, Valley View, Bellbrook-Sugarcreek, Pickerington, Lakota Schools, and Springboro.


text:
Aung San Suu Kyi has no excuse for staying silent on the Rohingya crisis
Posted by Press Office on February 16, 2017 in Analysis · 0 Comments
Tom Smith, University of Portsmouth
Aung San Suu Kyi, the famed and much fawned-over former opposition leader of Myanmar, cannot be immune to criticism. Undoubtedly a force for good in opposition, I believe she has failed to live up to her promise since her release from house arrest in 2010.
While she was blocked from becoming president in the country's civilian elections, she nonetheless became de facto head of government as "state counsellor" – a position created just for her. She is far from a disempowered opposition voice and, after years of flexing her muscles in power, she faces quiet but mounting international disquiet. Many once-devoted onlookers worry she hasn't done enough to stand up to allegations that the military's actions may constitute crimes against humanity. Those alleged crimes include the violent persecution of Myanmar's 1.3m ethnic Rohingya Muslims.
Hundreds of thousands have fled to neighbouring Bangladesh, Thailand, Malaysia and Indonesia, many in overcrowded boats, starving to death in the process. Reports on the scale of the massacres and ethnic cleansing are evidenced by satellite images showing entire Rohingya villages destroyed, including 820 buildings in just eight days. Myanmar's authorities have denied the allegations of abuse.
And yet, Suu Kyi has remained largely silent, at least on the international stage. When she has occasionally been pressed on the topic of Rohingya suffering, often by the foreign media that once covered her own plight, she has disappointed with her answers. In 2013, she told the BBC that the violence stemmed from fear felt on both sides and rejected claims of ethnic cleansing. In December 2016, she even accused "the international community" of "concentrating on the negative side of the situation".
Meanwhile, the same Western leaders who lauded Suu Kyi as a beacon of reform and praised her for her dignity are now largely overlooking her apparent inaction. The task of raising the alarm over the Rohingya's plight has been left to human rights advocates around the world, among them the Dalai Lama, more than a dozen fellow Nobel Peace Prize winners, Malala Yousafzai, and Suu Kyi's friend and defender Desmond Tutu.
Since Suu Kyi's release, there have been rumblings of disapproval from the international media, whose support helped put her in power in the first place.
The shine rubs off
Warning signs of her sensitivity to public criticism began to appear while she was still under house arrest: back in 2008, a sharply critical Guardian article roused a complaint from Suu Kyi's family and a special apology from the newspaper, which corrected errors in the piece but declined to withdraw it.
Media criticism has intensified since Suu Kyi's release. Al-Jazeera's Medhi Hassan was among the first to openly condemn Suu Kyi: "It is well past time to take off the rose-tinted glasses," he wrote, "to see Suu Kyi for what she is: a former prisoner of conscience, yes, but now a cynical politician who is willing to put votes ahead of principles."
Other outlets, such as The New York Times and the Independent have also highlighted what they deem to be Suu Kyi's inadequate response to the violence, but this has as yet done little or nothing to ramp up international political pressure. In June 2015, The Daily Telegraph asked: "Is human rights heroine tarnished by her silence on persecution?" But the tarnishing (or not) of her reputation isn't the point, and nor are debates about revoking her Nobel Peace Prize or honorary degree from Oxford.
Even if Suu Kyi is powerless to stand in the military's way, she is clearly able to speak out in some capacity, just as she could and did when in opposition. The Rohingya need her voice: Human Rights Watch reports that they are dying in boats run by smugglers and traffickers and in interment camps, their cause has already been hijacked by jihadists, and ASEAN, a political bloc meant to bring South-East Asia's countries together, is toothless as ever.
The excuse normally given is that the issue requires "a very delicate balance" between democratic progress and reforming the junta. That's fair enough in itself; former Human Rights Watch researcher David Scott Mathieson is correct when he says "a more calibrated understanding of the difficulties" is needed. But Suu Kyi has yet to publicly advance any such understanding. She is running out of excuses, and her status as a heroine for all her people is surely in danger.
Tom Smith, Lecturer in International Relations, University of Portsmouth
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redpajama_set_name: RedPajamaCommonCrawl


text:
Un guardabosques es una figura que puede y suele ser representada por personas, pero también por personajes de mitología, leyenda o ficción y hasta por animales. En el ámbito personal y profesional designa a quien se dedica a la protección y conservación de áreas naturales. Estas pueden ser desde bosques o parques forestales hasta marismas, selvas, zonas yermas o altas montañas. Su ámbito incluye también los recursos arqueológicos de esas zonas y sus recursos culturales y educativos.

Consideración legal y denominaciones

Muy frecuentemente, los guardabosques son funcionarios de las administraciones para las que trabajan, suelen ir uniformados y pueden portar armas. Pueden llevar a cabo funciones de policía y apoyo a la justicia. En función de los lugares y tareas que desempeñen, los guardabosques pueden ser llamados también: guardaparques, guardafaunas, agentes medioambientales, agentes rurales, agentes forestales, rangers, guardas de coto, guardas de caza, etc.

Funciones

Independientemente de sus obligaciones regulares en cualquier disciplina, el objetivo de todos los guardabosques es el de conservar los recursos naturales y la biodiversidad para las futuras generaciones.

Entre las disciplinas y tareas que puede desempeñar un guardabosques se encuentran las de vigilancia, educación, administración, extinción de incendios, cumplimiento de la ley, respuesta ante emergencias, interpretación del patrimonio cultural, comunicación, reparación y mantenimiento.

La profesión de guardabosques puede ser de alto riesgo si se desarrolla en países donde existen conflictos armados o especies en peligro de extinción sometidas a tráfico ilegal. En algunos países, instituciones ajenas al gobierno preparan a los guardabosques como cuerpos especializados y disciplinados para responder rápidamente a los problemas que se generan en los parques nacionales.

Las tareas de los guardabosques son muy variadas, en correspondencia con las diferentes necesidades de las zonas que deben proteger, y pueden desempeñar servicios muy especializados, de tal forma que incluso a veces chocan con intereses de las administraciones públicas.

Origen de los guardabosques
En los bosques propiedad de los reyes de la Inglaterra medieval existía ya la figura de unos oficiales (Verderers en inglés), que tenían por cometido el salvaguardar el cumplimiento de las leyes forestales; estas leyes tienen su origen en la Carta Foresta de 1217. En este documento aparecía la palabra "regadatore" originaria del latín medieval y "ranger" fue su traducción posterior al inglés.

La historia de los guardabosques en España se inicia en el siglo XVII en tiempos de Carlos II, cuando surgió la preocupación por las talas abusivas en los bosques, pero no fue hasta 1877 cuando se crearon los primeros guardabosques, llamados Capataces de cultivo, figura que iría evolucionando hasta conformarse a principios del siglo XX la actual Guardería Forestal.

En la mitología, la leyenda y la ficción

Entre los griegos, el dios más relacionado con los asuntos de los bosques era Pan, cuyo equivalente en la mitología romana sería Fauno. También en esta última se adoraba al dios Silvano como protector de los bosques.

En la mitología ibérica se encuentran las figuras de Basandere y Basajaun (Vasconia) y Busgosu (Asturias), a las que se considera los primeros habitantes y Señores de los Bosques.

En el folclore brasileño existe la figura de Curupira, un joven de cabello rojizo y pies invertidos dedicado a deambular por los bosques evitando su maltrato. En Alemania existe una leyenda llamada El hijo del guardabosques, que narra una historia ocurrida en la Selva Negra.

En la serie estadounidense de dibujos animados de 1961 El show del Oso Yogui aparece el personaje John Francis Smith, guardabosques del Parque de Jellystone, donde suceden las aventuras de este oso.

Otro ejemplo de dibujos animados, en la serie británica Percy the Park Keeper (Percy el Guardabosques) de 1996 basada en los populares libros del autor británico Nick Butterworthcon protagonizada por Percy y muchos animales salvajes, como el tejón, el búho, el zorro, los ratones, el topo, el petirrojo, etc.

Desde la primera entrega de la saga de Harry Potter en 1997, aparece un personaje semigigante llamado profesor Rubeus Hagrid, que encarna al guardabosques del Colegio Hogwarts de Magia y Hechicería.

Desde 2005 existe un juego de rol llamado Guild Wars en el que uno de sus personajes es el guardabosques. En la película de animación japonesa Ōkami Kodomo no Ame to Yuki (Los niños lobo Ame y Yuki), estrenada en 2012, aparece un guardabosques que explica las labores que realizan.

La música relacionada con los guardabosques
El compositor alemán Carl Maria von Weber estrenó en 1821 la ópera Der Freischütz, en la que aparece la figura del jefe de los guardabosques del príncipe de Bohemia.
El cantautor argentino Horacio Guarany dedicó una canción a su padre, Pobre Tata, en la que se narra el trabajo duro y en soledad en el bosque.
El grupo de rock Pony Bravo tiene una canción llamada El Guarda Forestal.
El dúo Guardabarranco canta la canción Guardabosques con letra y música de Salvador Cardenal Barquero.

Literatura de y por guardabosques
En la obra El amante de Lady Chatterley, novela de D. H. Lawrence escrita en 1928, aparece un personaje que es guarda de coto.
En la obra In the Rukh de Rudyard Kipling aparece un guardabosques inglés que descubre a un hombre joven llamado Mowgli.
Un guardabosques estadounidense llamado Edward Abbey dejó escritas numerosas obras relacionadas con la profesión de guardabosques.
En 1958 se publicó la obra de Jack Kerouac Los vagabundos del Dharma, en la que una pareja de guardabosques acompañan al protagonista en su ascenso a la montaña donde debe trabajar de vigilante de incendios.
Con el título original de Born free: a lioness of two worlds la escritora Joy Adamson publicó en 1960 la historia de Elsa la Leona, en la que George Adamson, su marido, es el guardabosques que recoge a los cachorros de león. Esta obra fue posteriormente llevada a la pantalla en diferentes formatos.
Se publicó en 2008 el libro Oak, vivencias de un agente forestal por Antonio Gutiérrez Sánchez, guardabosques español que ha publicado dos volúmenes más de esta serie y otras dos novelas más.
Los tres hermanos Ruiz Díez, dos de ellos guardabosques españoles, publicaron en 2014 el libro Tres clanes, donde se relata la vida y costumbres del lobo ibérico en alta montaña.
Una mujer guardabosques protagoniza la obra El guardabosques de Yesica Prol Cid publicada en 2017.

Documentales, series y cine de guardabosques
La compañía Paramount Pictures produjo en 1942 la película The Forest Rangers (Los guardabosques) . De los años 50 se puede encontrar el documental que produjo el Ministerio de Agricultura de los Estados Unidos (USDA), sobre la figura del guardabosques: The story of a Forest Ranger (La historia de un guardabosques). De la misma manera son numerosas, especialmente en la filmografía americana, las apariciones de los guardabosques en la televisión. Véanse como ejemplos la serie de la perrita Lassie, en 1954, uno de cuyos dueños era un guardabosques llamado Corey Stuart. En Canadá, en 1963, se emitió la serie The Forest Rangers, que tuvo 104 episodios de 30 minutos.

La cadena italiana RAI emitió en 2011 la serie A un paso del cielo, donde el protagonista, Terence Hill, interpreta a un comandante del Cuerpo forestal del estado italiano.

En el programa Otros documentales en la serie Los primeros hombres, emitido por el canal español La 2 en 2018, en el capítulo de Australia, aparece una mujer aborigen guardabosques que trabaja en el parque nacional Mungo.

Asociaciones de guardabosques
Repartidas por toda la geografía mundial, las asociaciones de guardabosques se agrupan en la Federación Internacional de Rangers, fundada en 1992 en el Reino Unido. En España se agrupan en la Asociación Española de Agentes Forestales y Medioambientales, constituida en 1999.

Animales guardabosques

A lo largo de su historia los Servicios Forestales han usado mascotas vestidas de guardabosques para difundir sus campañas educativas, en especial las de prevención de los incendios forestales, una de las labores más importantes de los guardabosques. El Servicio Forestal estadounidense creó en 1944 al oso Smokey. El Servicio Forestal español (ICONA) creó al Conejo Fidel en los años cincuenta. En 1976 el Servicio Forestal chileno (CONAF) creó a Forestín, un roedor parecido al castor y la nutria llamado coipo o coipú. En la isla de El Hierro en las Islas Canarias, un cuervo llamado Salvador y vestido de agente forestal hace las campañas de concienciación en la prevención de incendios desde el año 2017.

Siete especies de pájaros del género Lipaugus llevan el nombre de guardabosques con distintos apellidos. Tienen en común sus uniformes poco vistosos, el deambular solitario por los bosques tropicales de la América central y del sur, y sus frecuentes vocalizaciones de gran potencia.

Acceso a la profesión de guardabosques
La mayoría de las administraciones que emplean guardabosques en sus espacios naturales exigen la formación obligatoria, normalmente hasta los 16 años, y otra adicional en cursos de especialización. Normalmente habrá que superar pruebas escritas de conocimientos teóricos y prácticos, pruebas físicas, y puede haber requisitos adicionales como estar en posesión de licencia de conducir, manejo de armas y no padecer determinadas enfermedades. En la actualidad, dada la gran diversidad de conocimientos que exigen las obligaciones de los guardabosques, es muy frecuente acceder a la profesión desde distintos títulos universitarios y otras capacitaciones superiores.

Proverbios y dichos
 En el oficio de guardería, cuando hay hambre, es mediodía.
Fuente:anónimo.

Galería

Referencias

Enlaces externos

 Federación internacional de guardaparques
 El oso Smokey en la Wikipedia en inglés

Ocupaciones de la seguridad
Figuras o instrumentos de protección de espacios naturales
Conservacionismo


text:
\section{\label{sec:intro} Introduction}
The Very Long Baseline Interferometry (VLBI) technique allows us to establish with great accuracy the instantaneous spin axis of the Earth relative to an inertial frame built by observing radio signals from distant quasars. We also know with good accuracy the external torques acting on our spinning planet, mainly from the Sun and the Moon. Thus, knowing both the forcing and the planet's response, and assisted with other valuable data from seismology and geomagnetism, it is possible to infer reasonably accurate models of the Earth and its interior. This indeed is the case, see e.g. \citet{mathews2002}. The model's parameters, known as the Basic Earth Parameters (BEP), are obtained by fitting the VLBI observations to the models. Mantle elasticity, ocean loading and atmospheric effects make those models highly sophisticated and also successful in explaining most, but not all of the observations, for example as noted by  \citet{koot2010}. This might be related to the fact that the treatment of the liquid core flow is not nearly as sophisticated as other elements in the model. In fact, the core flow is modelled simply as an inviscid, incompressible flow with uniform vorticity, which for some specific purposes might be sufficient. On the other hand, there is a vast amount of experimental and numerical work on rapidly rotating fluids, where the motion of the bounding surface is prescribed (as steady rotation, including precession or libration), that has revealed surprising features and instabilities related to inertial modes (i.e. modes restored by the Coriolis force), see  \citet{lebars2015} for a review.

One of the problems of modelling planets with inviscid fluids is that although viscous effects might not be important a priori, the limit when viscosity tends to zero is not well behaved and inviscid solutions might become singular, a fact already noted by  \citet{rieutord2001}. In a spherical shell, like the Earth's fluid core, the only inertial modes that stay regular in the inviscid limit are the purely toroidal ones, also called planetary or Rossby modes. In astrophysical or geophysical applications the small but finite viscosity regularizes any singularities, which appear instead as detached shear layers  \citep{hollerbach1995,rieutord1997,rieutord2001}. There is consequently additional energy dissipation caused by these shear layers, but more importantly, in the case electrically conducting fluids, the ohmic dissipation is large in these layers and can become the main energy dissipation mechanism as the inviscid limit is approached, see  \citet{buffett2010} and  \citet{lin2017}.

Another aspect is the possible interaction between inertial modes and the rotational modes of the Earth as studied by  \citet{rogister2009}. This study is quite suggestive because it shows that avoided crossings can take place under certain conditions. However the limited resolution they could achieve precluded them to identify unambiguously their eigenmodes as inertial modes, apart from the fact that the fluid core is assumed inviscid, therefore limited to \emph{regular} (i.e. purely toroidal) inertial modes in their spherical shell configuration. The present study has a similar aim, namely to understand how the rotational modes like the Free Core Nutation (FCN) and the Chandler Wobble (CW) interact with inertial modes.

As a first step towards a better fluid dynamical treatment of the problem we consider a rigid spheroidal mantle and a completely fluid core, including viscosity. Instead of considering eigendisplacements as done by  \citet{rogister2009} we consider the Navier-Stokes equation (linear in velocity) to model with high resolution the core flow, including the motion of the mantle through the Coriolis and Poincar\'e forces. Simultaneously, we compute the motion of the mantle by considering the torques exerted by the fluid on the mantle (Euler equations). To deal with the spheroidal shape of the CMB, we adopt the same Taylor expansion method that we developed in  \citet{rekier2018}. We discuss in detail the model in Section \ref{sec:model} and the numerical method in Section \ref{sec:num}. Sections \ref{sec:res} through \ref{sec:disc} are devoted to the presentation and discussion of the results. In Section \ref{sec:end} we end with a summary and perspectives for future work. The appendix presents important technical details.

\section{The coupled inertial-rotational model}
\label{sec:model}
We consider a two-layer model planet with a completely fluid core enclosed by a rigid, oblate and axisymmetric ellipsoidal mantle. The basic (unperturbed) state of the planet is that of uniform angular speed $\Omega_0$ throughout and around the short axis of symmetry (the $\hat {\bm z}$ axis). We allow the mantle's spin $\vb{\Omega}(t)$, as referred to an inertial frame, to respond freely to torques exerted by the fluid core. We assume small perturbed motions compared to the basic state.

\subsection{The fluid core}
We model the core as an incompressible, homogeneous fluid with kinematic viscosity $\nu$ and density $\rho_f$. We describe the flow, $\vb{u}$, as observed from a reference frame attached rigidly to the mantle, where the boundary conditions can be prescribed more conveniently. With this choice, the dimensionless, \emph{linearized} Navier-Stokes equation describing the fluid's momentum balance reads
\begin{equation}
\partial_t \vb{u} + 2\,\vu{z} \cross \vb{u} + \partial_t \vb{\Omega} \cross \vb{r} = -\grad p+ E\,\nabla^2 \vb{u},
\label{eq:ns}
\end{equation}
where the unit time is $1/\Omega_0$, the unit length is the semimajor axis $a$ of the spheroidal CMB, $E$ is the Ekman number defined as $E=\nu/\Omega_0 a^2$, and $p$ is the \emph{reduced} pressure related to the physical pressure $P$ through
\begin{equation}
\label{eq:redp}
p = \frac{P}{\rho_f}-\left(\vb{\hat z}\cross\vb{r}\right)\cdot\left(\vb{M}\cross\vb{r}\right)+\Phi,
\end{equation}
where the total angular velocity of the mantle is $\vb{\Omega}=\vb{\hat z}+\vb{M}$, see Eq.~({\ref{eq:Mdef}}) further below, and $\Phi$ is the gravitational potential. The term $\partial_t \vb{\Omega} \cross \vb{r}$ in Eq.~(\mbox{\ref{eq:ns}}) is known as the \emph{Poincar\'e} or \emph{Euler} force, a ficticious force arising due to the mantle's unsteady rotation.
Now, to have a well defined problem we need to determine the mantle's angular velocity $\vb{\Omega}$ by considering the torque balance. The total torque exerted by the fluid core on the mantle is the sum of the pressure (i.e. topographic) and viscous torques (the gravitational torque vanishes since there is no mass redistribution):
\begin{equation}
\vb*{\Gamma} = \int \vb{r} \cross \grad P \dd{V} - \rho_f E \int \vb{r} \cross \nabla^2 \vb{u} \dd{V} \equiv \rho_f \vb*{\gamma},
\end{equation}
where the volume integrals extend over the whole fluid domain and $\vb*{\gamma}$ is dimensionless. The torque $\vb*{\Gamma}$ becomes itself dimensionless once we choose an appropriate unit for mass, which we do in such a way that $\rho_f=1$. The pressure torque involves the physical pressure, however it is desirable to have an expression for the torque that only involves $\vb{u}$. This can be accomplished once the poloidal-toroidal decomposition of the flow is introduced, see Section~\ref{sec:num}. The Ekman number $E$ constitutes one of the control parameters of the system.
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{diagram_twolayer.pdf}
\caption{Schematic diagram of the two-layer planet model. The fluid core has density $\rho_f$ and the mantle has density $\rho_m$. We choose the vertical $z$ axis in the mantle frame to coincide with the mantle's vertical figure axis. In our choice of units the semimajor axis $a$ of the axisymmetric CMB has unit length. The semimajor axis of the mantle's outer surface, $r_o$, together with the density ratio $\rho_m/\rho_f$ determines the control parameter $q$, see Eq.~\ref{eq:q}.}
\label{fig:schem}
\end{figure}

\subsection{The rigid mantle}
For simplicity we assume that the flattening $\alpha$ (defined as $(a-b)/a$ where $a$ and $b$ are the semimajor and semiminor axes, respectively) of the core-mantle boundary is small and coincides with the flattening of the mantle's outer surface. See Fig.~\ref{fig:schem} for a schematic. If the mantle's moment of inertia around the short axis of symmetry is represented by $\rho_m C$, and the moment of inertia around an equatorial axis by $\rho_m A$ ($\rho_m$ being the mantle's density) then, up to third order in $\alpha$, the \emph{dynamical} flattening is
\begin{equation}
f(\alpha)\equiv\frac{C-A}{A}=\alpha+\frac{1}{2}\,\alpha^2-\frac{1}{4}\,\alpha^3+\order{\alpha^4}.
\end{equation}
The mantle's spin $\bm \Omega$ is a vector relative to an inertial reference frame, however for our purposes it is convenient to express its components using the mantle's body axes, namely
\begin{equation}
\label{eq:Mdef}
\vb{\Omega} = M_x\,\vu{x}+ M_y\,\vu{y}+ (M_z+1)\,\vu{z}=\vb{M} + \vu{z}
\end{equation}
with $M_x,\,M_y,\,M_z$ being time-dependent and small compared to one. The Liouville equations of motion representing the angular momentum balance in an inertial reference frame reduce to the Euler equations for the motion of the rigid mantle. To first order in $\vb{M}$ they are
\begin{align}
\partial_t{M_x} + f(\alpha)\, M_y & = \frac{\rho_f}{\rho_m A}\gamma_x  \label{eulx}\\
\partial_t{M_y} - f(\alpha)\, M_x & = \frac{\rho_f}{\rho_m A}\gamma_y  \label{euly}\\
\partial_t{M_z} &= \frac{\rho_f}{\rho_m C}\gamma_z.\label{eulz}
\end{align}

The moment of inertia $\rho_m A$ depends on the outer semimajor axis of the mantle $r_0$ and also on the flattening $\alpha$ (the inner semimajor axis of the mantle being unity in our choice of units). To third order in $\alpha$ we can write
\begin{equation}
\frac{\rho_f}{\rho_m A}=\frac{15}{16\pi(r_0^5-1)}\frac{\rho_f}{\rho_m}\left(2+4\alpha+5\alpha^2+5\alpha^3\right),
\end{equation}
consequently, we define a control parameter $q$ as
\begin{equation}
\label{eq:q}
q\equiv\frac{1}{(r_0^5-1)}\frac{\rho_f}{\rho_m},
\end{equation}
which is essentially the inverse of the mantle's mean moment of inertia.
Using recent data for Mercury as an example, we estimate $q\approx0.269$. For Enceladus' icy crust $q\approx 1.6$ and for Earth's mantle $q\approx 0.026$. We have then a total of three control parameters: $E,\,q$ and $\alpha$. Setting $q=0$ will render the torques $\gamma_x,\gamma_y,\gamma_z$ ineffective and as a result the mantle's rotation becomes steady in time. In this case the spectrum of eigenmodes would correspond to the spectrum of inertial modes of a steadily rotating spheroidal cavity. If $\alpha=0$ the boundaries are spherical and therefore only the pressure torque would vanish.

\subsection{Boundary conditions}

We employ a no-slip boundary condition at the CMB and require regularity at the center. Given the near-spherical shape of the CMB it is possible to prescribe the no-slip condition using a Taylor expansion  \citep{rekier2018} as follows:
\begin{equation}
	\label{bc}
	\left.\vb{u}\right|_\textsc{\tiny CMB} = \left.\vb{u}\right|_{r=1} +
    (r_\textsc{\tiny CMB}-1)\,  \left. \partial_r   \vb{u}\right|_{r=1} +
	{\frac{1}{2}}(r_\textsc{\tiny CMB}-1)^2\,\left. \partial_r^2 \vb{u}\right|_{r=1} +
	{\frac{1}{6}}(r_\textsc{\tiny CMB}-1)^3\,\left. \partial_r^3 \vb{u}\right|_{r=1} = 0,
\end{equation}
where $r_\textsc{\tiny CMB}$ describes the shape of the spheroidal CMB. Up to third order in $\alpha$, it satisfies
\begin{equation}
	\label{eq:cmb1}
	r_\textsc{\tiny CMB}(\theta,\phi)-1 = a_0(\alpha)+a_2(\alpha)\,Y_2^0+a_4(\alpha)\,Y_4^0+a_6(\alpha)\,Y_6^0,
\end{equation}
where $Y_l^m$ are the familiar spherical harmonics and $a_i(\alpha)$ are polynomials of order three in $\alpha$ (see the appendix for their explicit form).
The advantage of this approach is that it is compatible with spherical harmonic expansions for the flow field $\vb{u}$. However, this technique is inadequate if the flattening is comparable or larger than the typical thickness $\sqrt{E}$ of the viscous boundary layer where the components of $\vb{u}$ undergo large spatial variations. In such a situation, the spherical surface $r=1$ would be too far from the Ekman boundary layer for a polynomial series expansion to describe accurately the exponentially decaying flow near the CMB. Therefore we limit our study to combinations of $E$ and $\alpha$ such that $\alpha\ll\sqrt{E}$. This is probably not an issue if stress-free boundary conditions are used since in that case the viscous boundary layers would be mostly suppressed.
We have chosen to extend the Taylor series up to $\order{\alpha^3}$ in order to allow a range as large as possible in the flattening $\alpha$ for a given Ekman number. The resolution of the Taylor expansion is $\sim (\alpha/\sqrt{E})^k$ and should be $\ll 1$ For comparison, a first order calculation ($k=1$) would only allow us to use flattening only up to $\alpha\sim\sqrt{E}/20$, while a third order calculation ($k=3$) allows us to go confidently up to $\alpha\sim\sqrt{E}/5$. A fourth order calculation would only bring a very small increase from this range while greatly increasing the computational cost.

The regularity condition at the origin is better described using the symmetry characteristics of the spherical harmonics, which we introduce below.

\section{Numerical method}
\label{sec:num}

\subsection{Radial and angular discretization}
We deal with oscillatory and possibly damped motions, therefore we write the time dependence as
\begin{equation}
\vb{u}(\vb{r},t) =\vb{u}_0(\vb{r})\text{e}^{\lambda t},\,\vb{M}(t) = \vb{M}_0\text{e}^{\lambda t},
\end{equation}
where $\vb{u}_0,\,\vb{M}_0$ and $\lambda=\sigma+i\omega$ are complex valued. We assume an incompressible fluid, hence a divergenceless flow. A poloidal-toroidal decomposition for $\vb{u}_0$ is then adequate:
\begin{equation}
\vb{u}_0(\vb{r}) = \curl{\curl{\mathcal P\vb{r}}}+\curl{\mathcal T\vb{r}}.
\end{equation}
The scalar functions $\mathcal P,\mathcal T$ are in turn expanded into spherical harmonics:
\begin{align}
\label{eq:poltor}
\mathcal P(r,\theta,\phi) &= \sum_{\ell=1}^{\ell_\text{max}} \sum_{m=-\ell}^{\ell}  P_{\ell m}(r)\,Y_\ell^m(\theta,\phi), \\
\mathcal T(r,\theta,\phi) &= \sum_{\ell=1}^{\ell_\text{max}} \sum_{m=-\ell}^{\ell}  T_{\ell m}(r)\,Y_\ell^m(\theta,\phi).
\end{align}

For the radial discretization of the components $P_{\ell m}$ and $T_{\ell m}$ we employ a fast spectral method devised by  \citet{olver2013}.  This method represents the unknown variables using Chebyshev polynomials and Gegenbauer polynomials for their derivatives, its most distinguishing feature is that the resulting matrices are \emph{sparse} as opposed to common spectral collocation methods where the resulting matrices are dense. Explicitly, the Chebyshev expansions for a given spherical harmonic component are
\begin{align}
P_{\ell m} = \sum_{k=0}^N P_{\ell m}^k\,t_k(r),\\
T_{\ell m} = \sum_{k=0}^N T_{\ell m}^k\,t_k(r),
\end{align}
where $t_k$ is the Chebyshev polynomial of degree $k$. The coefficients $P_{\ell m}^k,\,T_{\ell m}^k$ constitute the unknowns specifying the core flow once the pressure has been eliminated, see section 3.3. The total number of such coefficients is determined by the truncation levels $N$ and $\ell_\text{max}$.

\subsection{Symmetries}
To comply with the regularity condition at the origin we extend first the radial physical domain from $r\in[0,1]$ to $r\in[-1,1]$ in order to match the natural domain of the Chebyshev polynomials. There is a consideration that must be taken care of, however; for any given radius $r$, $T_{\ell m}(r)$ and $T_{\ell m}(-r)$ are connected by a reflection through the origin and consequently they should possess the same symmetry (parity) as the corresponding spherical harmonic $Y_\ell^m$, i.e.
\begin{equation}
Y_\ell^m(\pi-\theta,\pi+\phi)=(-1)^\ell Y_\ell^m(\theta,\phi).
\end{equation}
Therefore we use only even or odd Chebyshev polynomials depending on whether $\ell$ is even or odd. Obviously the same consideration goes for $P_{\ell m}$.

The problem specified by Eq.~(\ref{eq:ns}) cleanly decouples in the azimuthal wave number $m$ after carrying out the expansions explained above, which is a consequence of the CMB being axisymmetric. This is not the case for different angular degrees $\ell$ that end up coupled by the Coriolis force and by the boundary condition at the CMB.

The solutions for $\vb{u}$ possess a well defined equatorial symmetry: they are either equatorially symmetric, in which case the poloidal scalars $P_{\ell m}$ are such that $\ell-m=$ even while the toroidal scalars $T_{\ell m}$ fulfil $\ell+m=$ odd; or vice versa if the solutions are equatorially antisymmetric. This, together with the symmetry requirements for the Chebyshev polynomials, reduces the size of the problem to one fourth of the original.
In this study we focus mainly with flows associated with nutations, which are equatorially antisymmetric with $m=1$.

\subsection{Pressure torque as a function of flow velocity}
We still need a way to compute the topographic torque as a function of $\vb{u}_0$ in order to solve simultaneously the motion of the fluid core and the rigid mantle. The goal is to find an expression for the spherical harmonic component $\Pi_{\ell m}(\vb{r})$ of the physical pressure involving only $M_0$ as well as the poloidal and toroidal scalars $\mathcal P(\vb{r})$ and $\mathcal T(\vb{r})$, which may contain spherical harmonic components additional to $\ell$ due to the coupling induced by the Coriolis force. This can be accomplished by taking the \emph{consoidal} $\ell, m$ component (i.e. proportional to $\grad Y_l^m$) of Eq.~(\ref{eq:ns}) after expanding all terms in spherical harmonics and then solving for $\Pi_{\ell m}(\vb{r})$. Since these steps are algebraically very cumbersome, we perform them with the help of \classname{TenGSHui}, a symbolic tensor calculus package developed by  \citet{trinh2018}. The resulting expressions are listed in Appendix \ref{sec:apb}.

\subsection{The generalized eigenvalue problem}
Our unknowns are the set of coefficients $P_{\ell m}^k,\,T_{\ell m}^k$ and the components of $\vb{M}_0$. For the fluid core part we take first the curl of both sides of Eq.~(\ref{eq:ns}), compute the projection along the position vector $\vb{r}$, and multiply by $r^2$ (to avoid singularities at the origin). Reorganizing the terms a bit this is
\begin{equation}
\label{eq:curl1}
r^2\,\vb{r}\cdot \curl{\left( E\nabla^2\vb{u}_0 -2\vu{z}\cross\vb{u}_0   \right)} = \lambda\,r^2\,\vb{r}\cdot \curl{\left(\vb{u}_0+\vb{M}_0\cross\vb{r}\right)}.
\end{equation}
Once the spherical harmonic expansion is carried out the expression above will involve the toroidal coefficients $T_{\ell m}^k$ for the most part plus some poloidal coefficients brought in by the Coriolis force. Taking the curl twice and projecting along $r^4\vb{r}$ brings us to
\begin{equation}
\label{eq:curl2}
r^4\,\vb{r}\cdot\curl\curl\left(E\nabla^2\vb{u}_0 -2\vu{z}\cross\vb{u}_0\right) = \lambda\,r^4\,\vb{r}\cdot\curl\curl\left(\vb{u}_0\right),
\end{equation}
involving mostly the poloidal coefficients $P_{\ell m}^k$ and some toroidal ones. Lastly, we write the angular momentum balance more conveniently as
\begin{align}
M_{+}\left[\lambda^\dagger +i\,f(\alpha)\right] &= \frac{\rho_f}{\rho_m A}\gamma_{+},\\
M_{-}\left[\lambda -i\,f(\alpha)\right] &= \frac{\rho_f}{\rho_m A}\gamma_{-},\label{eq:lio}
\end{align}
where $M_\pm\equiv M_x\pm i\,M_y$, $\gamma_\pm\equiv \gamma_x\pm i\,\gamma_y$, and $(^\dagger)$ denotes the complex conjugate. The reason for this choice is that the quantity $\gamma_+$ depends only on the $m<0$ components of $\vb{u}_0$, therefore if we restrict to $m>0$ we need to consider $M_-$ and $\gamma_-$ only. The torque $\gamma_z$ vanishes as long as $m\ne0$, and since we are focusing on antisymmetric $m=1$ modes, we do not need to consider $M_z$.
The vector of unknowns $\vb{x}$ is then composed with the set of coefficients $\{P_{\ell m}^k\}$, $\{T_{\ell m}^k\}$ together with $M_-$. The problem represented by Eqs.~(\ref{eq:curl1}), (\ref{eq:curl2}) and (\ref{eq:lio}) becomes a generalized eigenvalue problem of the form
\begin{equation}
\label{eq:evp}
\vb{A}\,\vb{x} = \lambda\,\vb{B}\,\vb{x}
\end{equation}
where $\vb{A}$ and $\vb{B}$ are complex square matrices. The boundary condition represented by Eq.~(\ref{bc}) is included by replacing appropiate rows in both $\vb{A}$ and $\vb{B}$, see  \citet{olver2013} for further details. We solve Eq.~(\ref{eq:evp}) numerically using the open-source packages SLEPc  \citep{dalcin2011,slepc-toms,slepc-manual} and MUMPS  \citep{MUMPS01,MUMPS02} employing a \emph{shift and invert} strategy.

\section{Overview of the results}
\label{sec:res}

\subsection{The mode spectrum}
We begin by presenting an overview of the spectrum of the least damped equatorially antisymmetric eigenmodes with $m=1$ when $q=0.1$, $\alpha=10^{-4}$ and $E=10^{-6}$. Figure~\ref{fig:first} shows the kinetic energy of the mantle compared to the total kinetic energy of the fluid core as observed from a reference frame that rotates with the mean rotation rate of the planet, i.e. the steadily rotating frame (SRF).
In Appendix \ref{sec:apc} we provide details on how to transform the flow field as seen from the mantle frame (the frame used in the eigenvalue problem) to the flow field as seen from the SRF.

\begin{figure}
\centering
\includegraphics[width=1.15\linewidth]{firstfig2.pdf}
\caption{Ratio of mantle's kinetic energy to core kinetic energy in the steadily rotating frame (SRF). Horizontal axis is the mode frequency $\omega={\mathcal Im}(\lambda)$, the color scale corresponds to the damping $\sigma={\mathcal Re}(\lambda)$ scaled by $\sqrt{E}$.
The spectrum of our two-layer planet model exhibits naturally inertial modes, including the Free Core Nutation (FCN) and the Chandler Wobble (CW). The latter being the least-damped mode of all.}
\label{fig:first}
\end{figure}
\noindent Two eigenmodes stand out, the Chandler Wobble (CW) and the Free Core Nutation (FCN). The CW, with a small prograde (i.e. negative in our convention) frequency $\omega$, has most of its kinetic energy as an oscillation of the mantle's spin axis and a small fraction of the energy in the fluid core flow. It is by several order of magnitudes the least damped mode of all and it has no counterpart in the spectrum of purely inertial eigenmodes. The FCN on the other hand, is retrograde with frequency $\omega\sim 1$ and about 10\% of its total kinetic energy is taken by the mantle's motion. This fraction is dependent primarily on $q$ as we discuss further below. In an inertial reference frame the FCN and the CW can be described as mostly a uniformly rotating core flow with an axis slightly different from the mantle's spin axis. The FCN mode resembles the well-known `spin-over' inertial mode with the main difference being that the mantle here is executing an oscillatory motion in addition to steady rotation. The rest of the modes have only a very small fraction of their energy in the mantle's spin oscillation, particularly the prograde ones. All modes, with the exception of the CW, have a counterpart in the spectrum of the purely inertial eigenmodes of a steadily rotating spheroid.

Other features evident in Fig.~\mbox{\ref{fig:first}}, such as the `voids' in the frequency distribution of the eigenmodes (e.g. near $\omega\sim\pm \sqrt{2}$), or their semingly ordered distribution (e.g. around the frequency band $-0.25<\omega<0.25$), are most likely related to the 'quasi-regular' character of some modes. These features, albeit very interesting in themselves, involve many subtleties that are well beyond the scope of our study. Interested readers are encouraged to consult the recent and very detailed discussion by \mbox{\citet{rieutord2018}}.

\begin{figure}
\centering
\subfloat[]{\includegraphics[width=0.5\textwidth]{omega_qramp2.pdf}}
\subfloat[]{\includegraphics[width=0.5\textwidth]{sigma_qramp2.pdf}}\\
\subfloat[]{\includegraphics[width=0.5\textwidth]{kin_ratio_qramp.pdf}}
\subfloat[]{\includegraphics[width=0.5\textwidth]{k11_qramp.pdf}}\\
\caption{As the inverse moment of inertia represented by $q$ is varied, the FCN induces a profound rearrangement of neighbouring modes. The FCN's frequency grows approximately linearly with $q$ when $q\lesssim 1$ as seen in panel (a), while it grows more damped as shown in (b). To ease mode identification, the curves across all panels are colored according to the scaled damping $\sigma/\sqrt{E}$ shown in (b). Here the Ekman number is $E=10^{-6}$ and the flattening is $\alpha=10^{-4}$. See main text for more details.}
\label{fig:main}
\end{figure}

\subsection{Mode interactions}
Now we want to explore the behavior of the modes in the vicinity of the FCN as the control parameter $q$ is varied. We keep the same flattening and Ekman number as before and vary $q$ starting from zero as shown in Fig.~\ref{fig:main}. When $q\sim 0$ the FCN is essentially identical to the spin-over mode and is the second least-damped mode, with the CW being the least-damped one. The toroidal $\ell=1,m=1$ component $T_{11}$ of the core flow is dominant as evidenced by Fig.~\ref{fig:main}(d), which is a typical feature of a flow undergoing mostly a solid body rotation. Both the FCN and the CW exhibit this characteristic. As $q$ continues increasing the FCN's frequency goes up at the same time that it becomes more damped. From Fig.~{\ref{fig:main}}(b) we see that at $q\sim 2$ the FCN is already more damped than the mode at $\omega\approx1.017$, which in turn becomes the second least damped mode after the CW. The fraction of the total energy involved in the FCN's mantle oscillation increases and reaches a maximum around $q\sim 6$ as Fig.~\ref{fig:main}(c) shows.
Interestingly, around $q\sim 8$, the FCN begins interacting with nearby modes and it undergoes `avoided crossings' in frequency while `crossing' in damping, or vice versa. At $q\sim 11$ the toroidal $T_{11}$ component of the FCN is not anymore dominant and therefore the mode cannot be described any longer as a solid body rotation flow. In the range $10<q<30$ other modes undergo similar interactions. In this range frequency crossings take place, while avoided crossings in the damping occur that could rather be described as consecutive `mode bumpings'. The FCN mutates into a mode that is not affected by changes in $q$ any more. Around $q\sim 26$ a mode crossing in damping takes place (near $\sigma/\sqrt{E}\sim -80$) along with a corresponding avoided crossing in frequency. It is possible that similar dynamics can take place again at higher $q$ and more negative damping values.

\begin{figure}
\centering
\includegraphics[width=0.65\textwidth]{complex.pdf}
\caption{Mode trajectories in the $(\omega,\sigma)$ plane as $q$ is varied (indicated by the color scale). It is possible to establish a path from the FCN mode at $q=0$ leading to the most damped mode of Fig.~\ref{fig:main}(b) linking intermediate damping modes and varying $q$ back and forth. Modes performing loops do not lose their identity.}
\label{fig:complex}
\end{figure}

The FCN mode induces a substantial rearrangement of the modes as $q$ is varied. At the same time its own identity is lost in the process and when $q\sim15$ it has taken the place of the mode that initially had $\sigma/\sqrt{E}\sim -30$ when $q=0$. This latter mode in turn morphed into the one that had initially $\sigma/\sqrt{E}\sim -36$ when $q=0$. This exchange appears to continue successively as the modes `bump' each other into more negative damping values. In Fig.~\ref{fig:complex} we show the trajectory that the modes describe in the $(\omega,\sigma)$ plane as $q$ changes. Interestingly, it is indeed possible to follow a path connecting the FCN mode at $q=0$ to the most damped mode visible in Fig.~\ref{fig:main}(b) at $q\sim6$. The modes that form isolated loops in Fig.~\ref{fig:complex} are the ones that do not lose their identity throughout the interactions induced by $q$. We also note that even though the modes at $q=0$ seem to match the `morphed' modes at large $q$, the eigenmodes are not identical since the mantle's kinetic energy is different as Fig.~\ref{fig:main}(c) shows. Only their fluid part match.

As already mentioned, the FCN mode at $q=0$ corresponds to the viscous spin-over mode in a steadily rotating spheroid. Apart from the viscous boundary layer and internal shear layers, this mode corresponds to the analytical (inviscid) mode in a full sphere with wavenumbers $(n=2,k=1,m=1)$ using Greenspan's notation  \citep{greenspan1968}. As it turns out, the other modes (at $q=0$) appearing in Fig.~\ref{fig:main}(a) can be linked also to their spherical inviscid counterpart where their $n$ wavenumber differ by \emph{six} between two consecutive modes. This is, they form a family for which $n=6j+2$ with $j=0,1,2,\dots$ and $m=1$ ($m$ being the azimuthal wavenumber) and $k$ is such that the frequency $\omega$ is closest to one, see also Figs.~\ref{fig:Eramp} and \ref{fig:schm}. As $n$ increases, the mode's structure becomes more complex and hence more damped. Interestingly enough, the first three of these modes ($n=8,14,20$) are precisely the ones studied by  \citet{schmitt2006}. According to that study, as the flattening increases the spin-over frequency and the modes in question can undergo frequency crossings, resulting in a similar viscous interaction as in our case.

\begin{figure}
\centering
\subfloat[]{\includegraphics[width=0.5\textwidth]{omega_Eramp2.pdf}}
\subfloat[]{\includegraphics[width=0.5\textwidth]{sigma_Eramp2.pdf}}
\caption{The frequency (a) and scaled damping (b) of eigenmodes near the FCN as the Ekman number is varied. Here $\alpha=10^{-5}$ and $q=5$. Colors indicate the scaled damping $\sigma/\sqrt{E}$. The black horizontal lines in (a) mark the frequency of the theoretical inviscid modes of a sphere for the family of modes with $n=2+6j,\,j=1,2,\dots$ (See main text for further details). The FCN on panel (a) starts near $\omega\sim1.000$ at $E=10^{-8}$ and on panel (b) it starts near $\sigma/\sqrt{E}\sim -16$ at $E=10^{-8}$. As $E$ increases the FCN undergoes a crossing in frequency and an avoided crossing in the damping near $E\sim 10^{-5}$.}
\label{fig:Eramp}
\end{figure}

\subsection{Reducing the Ekman number}
If we reduce the Ekman number, the correspondence explained above can be made more evident since we expect the solutions to approach their inviscid value. We do so in Fig.~\ref{fig:Eramp} where we reduce the Ekman number down to about $E=10^{-7.9}$. However, we need to reduce as well the flattening in order to stay in the range of validity of the spheroidal boundary condition in Eq.~\ref{bc}, therefore we set $\alpha=10^{-5}$ and also set $q=5$. As we discuss later in Section \ref{sec:FCN}, the slope $\partial\omega/\partial q$ of the FCN reduces as $E$ decreases and therefore mode interactions will take place at comparatively higher $q$ values. For instance in Fig.~\ref{fig:Eramp}, where $q=5$, we see already a frequency crossing (with a corresponding avoided crossing in damping) near $E\sim 10^{-5}$. Taking the flattening aside momentarily, this is to be compared with the example at $E=10^{-6}$ in Fig.~\ref{fig:main} where modes begin interacting at $q\sim 8$. Therefore we do not expect mode interactions of this kind to occur at realistic values of $q$ ($q\lesssim 1$) when considering more astrophysically relevant Ekman numbers ($E\lesssim10^{-12}$).
Already at $E=10^{-8}$ the FCN is essentially noninteracting even though $q=5$. In fact, at that Ekman number the frequencies of the $n=2+6j$ inviscid mode family are reassuringly in good agreement with the frequencies of our model as Fig.~\ref{fig:Eramp}(a) shows.

As opposed to the FCN, the CW mode does not appear to interact with any nearby modes in frequency because its damping is very different from its neighbors', which are several orders of magnitude more damped. As we have seen, interactions occur when modes get close enough to each other in the $(\omega,\sigma)$ plane. In fact, although the CW's frequency does change with $q$ and crosses other modes' frequencies, its damping remains small even for moderate $q$, so no interaction occurs. We present a more detailed study of the CW's frequency and damping as functions of $q,\,\alpha$ and $E$ in Section \ref{sec:CW}.

\section{The Free Core Nutation (FCN) mode}
\label{sec:FCN}

We turn now to the dependence of the FCN mode on the parameters $q,\,\alpha$ and $E$ away from the interaction region, which is more relevant for planets. The frequency $\omega_\textsc{\tiny fcn}$ and damping $\sigma_\textsc{\tiny fcn}$ are approximately linear with $q$ as long as there are no other modes with similar $(\omega,\sigma)$ that could interact with the FCN. This translates to $q\lesssim 0.1$ for Ekman numbers in the range $10^{-8}<E<10^{-4}$, and flattenings in the range $10^{-7}<\alpha<10^{-4.6}$. Within those limits we perform linear fits against $q$ following:
\begin{equation}
\begin{split}
\omega_\textsc{\tiny fcn}-\omega_* &= b_\omega (\alpha,E) + q\,m_\omega(\alpha,E),\\
\sigma_\textsc{\tiny fcn} &= b_\sigma (\alpha,E)+ q\,m_\sigma(\alpha,E),
\end{split}
\label{eq:fcn1}
\end{equation}
where $\omega_*$ is the theoretical inviscid spin-over mode frequency in a spheroid with flattening $\alpha$ (computed to third order in $\alpha$). The intercepts $b_\omega$ and $b_\sigma$ represent the frequency and the damping of the viscous spin-over mode, therefore suitable for direct comparison with the viscous corrections computed by  \citet{zhang2004}.

In turn, the slopes and intercepts $m_\omega, b_\omega$ and $m_\sigma, b_\sigma$ appear to be either linear or quadratic in $\alpha$ if $E$ is fixed,
 so we perform the following additional fits:
\begin{equation}
\begin{split}
m_\omega(\alpha,E) &= m_{0,\omega}(E) + \alpha\,m_{1,\omega}(E) + \alpha^2\,m_{2,\omega}(E),\\
b_\omega(\alpha,E) &= b_{0,\omega}(E) + \alpha\,b_{1,\omega}(E),
\end{split}
\label{eq:fcn2}
\end{equation}
together with
\begin{equation}
\begin{split}
m_\sigma(\alpha,E) &= m_{0,\sigma}(E) + \alpha\,m_{1,\sigma}(E) + \alpha^2\,m_{2,\sigma}(E),\\
b_\sigma(\alpha,E) &= b_{0,\sigma}(E) + \alpha\,b_{1,\sigma}(E).
\end{split}
\label{eq:fcn3}
\end{equation}
Lastly, we establish approximate scalings of the coefficients $m_{i,\omega},b_{i,\omega}$ and $m_{i,\sigma},b_{i,\sigma}$ $(i=0,1,2)$ with
respect to the Ekman number $E$. The resulting scaling laws for $\omega_\textsc{\tiny fcn}$ and $\sigma_\textsc{\tiny fcn}$ are shown
in Fig.~\ref{fig:omega_fcn} and Fig.~\ref{fig:sigma_fcn}, respectively.

\begin{figure}
\centering
\subfloat[]{\includegraphics[width=0.5\textwidth]{fcn_omega_m.pdf}}
\subfloat[]{\includegraphics[width=0.5\textwidth]{fcn_omega_b.pdf}}\\
\caption{Fit coefficients for $m_\omega$ (a) and $b_\omega$ (b) corresponding to the FCN mode frequency $\omega_\textsc{\tiny fcn}$, see Eqs.(\ref{eq:fcn1}) and (\ref{eq:fcn2}). Dots represent the numerical results (after performing linear fits against $q$) and dashed lines are the power law fits against $E$. Inset panel in (a) is a zoom to reveal more detail on $m_{1,\omega}$. The continuous lines in panel (b) represent the theoretical viscous corrections of a steadily rotating spheroid as computed by  \citet{zhang2004}.}
\label{fig:omega_fcn}
\end{figure}

A few comments are in order. First, from Fig.~\ref{fig:omega_fcn}(a), we note that $m_{0,\omega}(E)$ follows approximately a power law once $E$ is below $\sim10^{-6}$, therefore we perform a power law fit (quoted in the figure) for $E$ in the range $10^{-8}<E<10^{-6}$. In Fig,~\ref{fig:omega_fcn}(a), $m_{1,\omega}$ seems to level off at a value very close to unity, which is expected as we discuss in the appendix. The behaviour of $m_{2,\omega}$ is rather surprising since it scales with a negative power of the Ekman number, which means that the magnitude of $m_{2,\omega}$ \emph{increases} as the $E$ decreases, while we expected $m_{2,\omega}$ to at least level off to a constant value near $-1/2$ for asymptotically small $E$. This scaling is likely valid in only the limited range we used for the flattening and the Ekman number as we discuss further below. The power law fits for $b_\omega$ shown in Fig.~\ref{fig:omega_fcn}(b) are in reasonable agreement with the asymptotically vanishing viscosity correction of the spin-over frequency computed by  \citet{zhang2004}. Summarizing, we approximate the FCN's frequency through $m_\omega$ and $b_\omega$ as
\begin{equation}
\begin{split}
\omega_\textsc{\tiny FCN} &=m_\omega\,q + b_\omega,\,\textrm{where}\\
m_\omega &\approx 0.599\,E^{0.543}+\alpha-0.427\,\alpha^2\,E^{-0.499},\,\textrm{and}\\
b_\omega &\approx 0.558\,E^{0.539}+2.915\,\alpha\,E^{0.527}.
\label{eq:omega_fcn}
\end{split}
\end{equation}

\begin{figure}
\centering
\subfloat[]{\includegraphics[width=0.5\textwidth]{fcn_sigma_m.pdf}}
\subfloat[]{\includegraphics[width=0.5\textwidth]{fcn_sigma_b.pdf}}\\
\caption{Fit coefficients for $m_\sigma$ in panel (a) and $b_\sigma$ in panel (b) corresponding to the FCN mode damping $\sigma_\textsc{\tiny fcn}$, see Eqs.(\ref{eq:fcn1}) and (\ref{eq:fcn3}). Dots represent the numerical results (after performing linear fits against $q$) and dashed lines are the power law fits against $E$. In panel (b) the continuous lines represent the expectations from  \citet{zhang2004}, only with the sign changed. Our results for $b_{0,\sigma}$ are in excellent agreement with the theoretical expectation while $b_{1,\sigma}$ only begins approaching the theoretical scaling towards the lower Ekman numbers we explored.}
\label{fig:sigma_fcn}
\end{figure}

We move on now to the linear fits of the damping $\sigma_\textsc{\tiny FCN}$ against $q$. We show the resulting slopes $m_\sigma$ and intercepts $b_\sigma$ in Fig.~\ref{fig:sigma_fcn}. As there is no prediction of what the damping of the FCN mode might be, theoretically or otherwise, we can only compare the intercepts $b_\sigma$ as they should reduce to the damping of the spin-over mode in a spheroid.  Regarding $m_\sigma$, in Fig.~\ref{fig:sigma_fcn}(a), we find a similar situation as before, the $m_{2,\sigma}$ coefficient seems to follow an inverse power law scaling with $E$, albeit with a smaller magnitude exponent and amplitude compared to $m_{2,\omega}$. Below $E\sim10^{-7}$ we start seeing the effects of numeric round-off errors, which affect more high-order quantities such as $m_{2,\sigma}$ compared to $m_{0,\sigma}$, for example. For this reason we perform fits for $m_{1,\sigma}$ and $m_{2,\sigma}$ in the range $10^{-7.5}<E<10^{-6}$ only. In Fig.~\ref{fig:sigma_fcn}(b) the fit for $b_{0,\sigma}$ matches the theoretical asymptotic expectation rather satisfactorily, however the observed power law for $b_{1,\sigma}$ only begins to approach the asymptotic expectation at the lowest end of our Ekman number range. At the higher end, above $E\sim10^{-4.5}$, our numerical result for $b_{1,\sigma}$ has the opposite sign compared to expectation, which seems to be also the case in the study by  \citet{schmitt2006}, see their Figure 10(a). In sum, we find that the damping of the FCN is approximately
\begin{equation}
\begin{split}
\sigma_\textsc{\tiny FCN} &=  m_\sigma\,q + b_\sigma,\,\textrm{where}\\
m_\sigma &\approx -2.705\,E^{0.502}+11.58\,\alpha\,E^{0.553} +0.088\,\alpha^2\,E^{-0.37},\,\textrm{and}\\
b_\sigma &\approx -2.630\,E^{1/2}-25.40\,E + \alpha\,\left(-0.566\,E^{1/2}+299.3\,E\right).
\label{eq:sigma_fcn}
\end{split}
\end{equation}

\section{The Chandler Wobble (CW) mode}
\label{sec:CW}
There appears to be no mode interactions between the CW and other modes as $q$ is varied. In fact, there are other modes with similarly low frequencies as the CW, but the only ones with potentially low damping that could interact happen to be equatorially symmetric, as can be seen in Fig.~\ref{fig:schm}(a), hence completely decoupled from the CW. The frequency and damping of the CW appear to be directly proportional to the flattening $\alpha$ if $q$ and $E$ are fixed, at least for the parameter range we tested, so it is convenient to perform linear fits against $\alpha$ first:
\begin{equation}
\begin{split}
\omega_\textsc{\tiny cw} &= m_\omega (q,E)\,\alpha,\\
\sigma_\textsc{\tiny cw} &= m_\sigma (q,E)\,\alpha.
\end{split}
\label{eq:cw1}
\end{equation}
The proportionality constants $m_\omega$ and $m_\sigma$ themselves turn out to be approximately linear or quadratic in $q$:
\begin{equation}
\begin{split}
m_\omega(q,E) &= m_{0,\omega}(E) + q\,m_{1,\omega}(E),\\
m_\sigma(q,E) &= q\,m_{1,\sigma}(E) + q^2\,m_{2,\sigma}(E).
\end{split}
\label{eq:cw2}
\end{equation}
Here we do analogously to the FCN case by establishing the approximate scalings of the coefficients $m_{i,\omega}$ and $m_{i,\sigma}$ $(i=0,1,2)$ appearing in Eq.~(\ref{eq:cw2}) with respect to the Ekman number $E$. Figure \ref{fig:m_cw} shows the results.

\begin{figure}
\centering
\subfloat[]{\includegraphics[width=0.5\textwidth]{cw_omega.pdf}}
\subfloat[]{\includegraphics[width=0.5\textwidth]{cw_sigma.pdf}}\\
\caption{Fits for $m_\omega$ (a) and $m_\sigma$ (b) corresponding to the Chandler Wobble mode frequency $\omega_\textsc{\tiny CW}$ and damping $\sigma_\textsc{\tiny CW}$, respectively. See Eqs.(\ref{eq:cw1}) and (\ref{eq:cw2}). In panel (b) $m_{2,\sigma}$ and its corresponding fit have been deliberately multiplied by $1/2$ in order to distinguish it visually from $m_{1,\sigma}$.  }
\label{fig:m_cw}
\end{figure}

As the Ekman number decreases the CW frequency approaches the theoretical inviscid value, i.e., the proportionality constants $m_{0,\omega}$ and $m_{1,\omega}$ get both close to $-1$ following approximately a familiar $E^{1/2}$ scaling. The same scaling applies to the damping as Figure~\ref{fig:m_cw}(b) shows.
Summarizing, we find that the Chandler Wobble frequency is
\begin{equation}
\begin{split}
\omega_\textsc{\tiny CW}=\alpha &\left[-1-1.02\cdot10^{-5}-4.75\cdot10^{-3}\,E^{1/2}+0.237\,E \right. \\
                                      &\,\, \left. +q\left(-1+1.21\cdot10^{-5}+3.292\,E^{1/2}+12.53\,E \right) \right],
\label{eq:omega_cw}
\end{split}
\end{equation}
and its damping is
\begin{equation}
\sigma_\textsc{\tiny CW}=\alpha\,q\,\left[-3.579\,E^{1/2}-17.37\,E+q\left(-3.579\,E^{1/2}+7.511\,E \right) \right].
\label{eq:sigma_cw}
\end{equation}

\section{Discussion}
\label{sec:disc}

By allowing the mantle to react to torques exerted by the fluid we extend the spectrum of purely inertial eigenmodes to include the Chandler Wobble (CW), which induces a flow in the bulk of the fluid core, however small in magnitude compared to the mantle's motion. Additionally, the Free Core Nutation (FCN) appears as an extension of the spin-over mode as soon as the mantle's moment of inertia ($1/q$) is considered finite. The FCN has a smaller mantle/fluid kinetic energy ratio than the CW, while other modes have even smaller ratios. As $q$ is varied the FCN may interact with the $n=8,14,20,\dots$ family of modes when they become close enough in the $(\omega,\sigma)$ plane, in a conceptually similar way as they interact with the spin-over mode as shown by  \citet{schmitt2006}, the difference being that in our case the relevant parameter is $q$ while in Schmitt's case it is the flattening. Figure~\ref{fig:schm} helps understanding why these modes are prone to either kind of interaction. In Fig.~\ref{fig:schm}(a) we show the inviscid eigenfrequency distribution for modes with azimuthal wave number $m=1$ for a spheroid with flattening $\alpha=0.0202$. We see that the equatorially antisymmetric $n=8,14,20,\dots$ modes are indeed the closest in frequency to the spin-over mode. Note that we can always find modes arbitrarily close in frequency to the spin-over mode, given that the inviscid spectrum is dense. However, those modes will have very complex structure, i.e. they will have high $n$ numbers, and would be very damped in the viscous case preventing them effectively from any interaction with the spin-over mode. In our case, with the FCN replacing the spin-over mode, increasing $q$ from zero causes the FCN frequency to go up at the same time that the damping increases in magnitude, resulting in complex interactions as shown in Fig.~\ref{fig:main}. In Schmitt's case, increasing the flattening causes the spin-over frequency to cross with the $n=8,14,20,\dots$ modes frequencies as well, although their damping is affected somewhat differently, \citep[see][their Figure 4]{schmitt2006}.

\begin{figure}
\centering
\subfloat[]{\includegraphics[width=0.5\textwidth]{schmitt2.pdf}}
\subfloat[]{\includegraphics[width=0.5\textwidth]{schmitt3.pdf}}\\
\caption{The spin-over mode in a spheroid can interact with nearby modes if the flattening is large enough. Panel (a) shows the distribution of theoretical (inviscid) eigenfrequencies at $\alpha=0.0202$), the red dashed line indicating the spin-over ($n=2$) mode frequency. Panel (b) shows the distance in the frequency axis between the spin-over mode and other nearby modes (antisymmetric ones indicated by continuous lines) as the flattening is varied. For details on the ensuing viscous interaction see  \citet{schmitt2006}.}
\label{fig:schm}
\end{figure}

Figure~\ref{fig:schm}(a) also explains why the CW does not interact with any nearby modes. The only modes with small enough prograde frequencies (i.e. negative in our convention) like the CW, and possibly small enough damping, are equatorially symmetric, thus unable to interact.

Nothing stops both effects from acting simultaneously if the mantle is free to react to torques and the flattening is $\alpha\gtrsim 0.08$, in which case an astrophysically small Ekman number would not prevent the FCN to interact with the other modes as it does in our case. This is indeed an interesting perspective, however such a large flattening is beyond our reach given the Taylor expansion method that we use to implement the spheroidal boundary condition.

The width of the viscous boundary layer scales as $E^{1/2}$, so we expect that our Taylor expansion method is valid as long as the flattening is such that $\alpha/E^{1/2}\ll 1$. This might help a bit in understanding the somewhat surprising scaling of the FCN's $m_{2,\omega}$ and $m_{2,\sigma}$ coefficients. We can rearrange slightly the expression for the FCN's $m_\omega$ given by Eq.~(\ref{eq:omega_fcn}) as
\begin{equation}
m_\omega \approx E^{1/2}\left[ 0.599\,E^{0.043} + \left(\frac{\alpha}{E^{1/2}}\right) -0.427 \left(\frac{\alpha}{E^{1/2}}\right)^2 E^{0.001}\right].
\end{equation}
We see that the powers of $\alpha/E^{1/2}$ within the square brackets form a second order power series with coefficients that are either independent of $E$ or decrease slightly as $E$ decreases. Clearly, this series would not converge if the flattening is comparable or larger than the width of the viscous boundary layer.

The damping of the FCN presents a similar but slightly milder issue. The coefficient $m_{\sigma}$ shown in Eq.~\ref{eq:sigma_fcn} can be written
\begin{equation}
m_{\sigma} \approx E^{1/2}\left[ -2.705\,E^{0.002} + 11.58\,\left(\frac{\alpha}{E^{1/2}}\right)\,E^{0.553} +0.088 \left(\frac{\alpha}{E^{1/2}}\right)^2 E^{0.13} \right],
\end{equation}
which shows the coefficient of $\alpha^2/E$ within the square brackets decreasing faster as $E$ decreases compared to the same coefficient in $m_\omega$'s case, i.e., $E^{0.13}$ vs. $E^{0.001}$. These `anomalous' scalings effectively prevent us from extrapolating the FCN's frequency and damping to larger flattenings. On the other hand, the Chandler Wobble appears free of such a complication and there is in principle no obstacle to extrapolate Eqs.~(\ref{eq:omega_cw}) and (\ref{eq:sigma_cw}) to higher values of the flattening. This is also true for the FCN's $b_\omega$ and $b_\sigma$ (that reduce to the spin-over mode frequency and damping respectively) where the scalings we found are not restricted to small $\alpha$, of course up to where Schmitt's mode interactions start becoming noticeable.

\begin{figure}
\centering
\subfloat[]{\includegraphics[width=0.5\textwidth]{Po_dis_ratio.pdf}}
\subfloat[]{\includegraphics[width=0.5\textwidth]{torque_ratio.pdf}}
\caption{In panel (a) we can see that the power $D_P=|\partial_t\vb{M}\cdot\vb{L}|$ associated with the Poincar\'e force can be many times larger than the viscous dissipation $D_\text{kin}=|\mathcal{D}_\nu|$, thus dominating the overall damping rate. The pressure torque to viscous torque ratio in panel (b) shows that the viscous torque is dominant over the range of $q$ that we explored ($0\le q \le 30$). In both panels the parameters are identical to those of Fig.~\ref{fig:main}, color scale indicate the scaled damping $\sigma/\sqrt{E}$.}
\label{fig:ratios}
\end{figure}

The magnitude increase of the FCN's damping rate with $q$ is closely related to the fact that not all the damping is due to viscous dissipation processes. In fact, the Poincar\'e force is capable of performing work (as opposed to the Coriolis force) and such work must be taken into account in the overall power balance of the system. Starting from the linear momentum balance in Eq.~(\mbox{\ref{eq:ns}}), taking the dot product with $\vb{u}$ and integrating over the whole fluid volume it is straightforward to show that the damping rate $\sigma$ of any given mode fulfills
\begin{equation}
2\,\sigma\,K_f = \mathcal{D}_\nu - \left(\partial_t \vb{M}\right) \cdot \vb{L},
\end{equation}
where $K_f$ is the total kinetic energy of the fluid (in the mantle frame), $\vb{L}$ is the total angular momentum of the fluid (relative to the mantle frame) and $\mathcal{D}_\nu$ is the viscous dissipation computed as
\begin{equation}
\mathcal{D}_\nu=E\int \vb{u}\cdot \nabla^2\vb{u}\dd{V}
\end{equation}
over the whole fluid volume. For the FCN, as $q$ increases the mantle's motion also goes up (because its moment of inertia decreases), see Fig.~\ref{fig:main}(c), along with the ratio $|\partial_t\vb{M}\cdot\vb{L}|/|\mathcal{D}_\nu|$ as shown in Fig.~\ref{fig:ratios}(a). Near the maximum at $q\sim8$ the power associated with the Poincar\'e force is almost seven times as large as the power associated to viscosity. Beyond this value of $q$ mode interactions dominate and the FCN loses its character as a mainly rotational (toroidal) motion.

We should remark that the mode coupling we witness in this system is essentially viscous in nature for the sole reason that pressure torques exerted on the spheroidal mantle are much smaller than the viscous torques. Figure~{\ref{fig:ratios}}(b) shows that pressure torque to viscous torque ratio sits near 4{\%} when the mode interactions take place. That being said, it is the mixing among poloidal and toroidal components of different angular degrees induced by the torques that is at the heart of the mode interaction mechanism. However different in magnitude, both viscous and pressure torques induce similar mixing among components and so there is nothing in principle preventing pressure and gravitational torques, if they were to be dominant over viscous ones, from leading to similar resonant coupling among modes. This would be the case for a more realistic Earth model, where the Ekman number is very small and the pressure torques dominate. Note that additionally, a relatively large flattening such as Earth's, would bring the FCN's mode frequency closer to the region where it might interact resonantly with neighbouring modes regardless of the pressure torques, just as is the case for the spin-over mode as shown by \mbox{\citet{schmitt2006}}. Unfortunately, our Taylor expansion method prevents us from exploring such regime.

Lastly, other dissipation mechanisms such as Joule heating (in an electrically conducting core) might lead to similar resonant interactions. In such case magnetic torques would play a similar role as the viscous torque. This is particularly relevant for the Earth where we know that pressure, gravitational and magnetic torques dominate over viscous torques. It is then perfectly possible that resonant mode interaction, conceptually similar to the one presented in this study, might take place using more realistic (Earth or planetary) parameters.

\section{Summary and outlook}
\label{sec:end}
We have implemented an efficient spectral method  \citep{olver2013} to obtain numerically the eigenmodes of a rapidly rotating two-layer planet model in which the rigid mantle is free to react to torques exterted by the viscous, incompressible fluid core. The aim of this model is to understand at least qualitatively the relationship between inertial modes as studied in the fluid dynamics literature, where the rotational motion of the container (the `mantle') is prescribed, and the rotational modes of a planet (Free Core Nutation, Chandler Wobble) as studied in Geodesy where the motion of the fluid core is generally taken as a simple Poincar\'e flow. We can accomodate a flattened core-mantle boundary by means of a Taylor expansion technique as long as the flattening is smaller than the typical width of the Ekman boundary layer. In the special case of $q=0$ (i.e. a steadily rotating mantle) our model reproduces satisfactorily the viscous inertial eigenmodes in rotating spheroids as reported by previous studies  \citep{schmitt2006,zhang2004}.

When $q\ne0$ we find a rich system where the Free Core Nutation, which reduces to the spin-over mode when $q=0$, interacts with a family of nearby modes ($n=8,14,20,\dots$) and induces a substantial reordering. This family of modes that interact with the FCN include, incidentally, the three modes that  \citet{schmitt2006} describes as entering in resonance with the spin-over mode for large enough values of the flattening. We thus uncover a common underlying mechanism in which a varying parameter can bring modes' eigenvalues close enough in the complex plane, i.e. the $(\omega,\sigma)$ plane, leading to avoided crossings. If a crossing in frequency occurs, an avoided crossing in the damping takes place and vice versa.

The spin-over mode transitions into the FCN mode, increasing its frequency and becoming more damped as $q$ is increased. The other modes in the $n=8,14,20,\dots$ family are basically immune to changes in $q$ as long as the frequency and damping of the FCN is still away from the frequency and damping of any member of that family. The FCN, as $q$ keeps going up, gradually morphes into one of the members of the family, losing its own identity. The mode the FCN morphes into, with its identity taken away by the FCN, transforms itself into the next more damped mode of the family, and so on `bumping' successively each other into more negative values of $\sigma$. A clear explanation for this remarkable reordering still evades us.

The Chandler Wobble, just as the FCN, is affected by varying $q$, with the difference that the CW is not able to interact with other modes, the reason being that the very small damping of the CW is a difficult match for any low-order, equatorially antisymmetric modes with similar frequencies. The only modes that could match the CW's damping and frequency are low-order equatorially symmetric ones, but their interaction is precluded by their differing symmetry.

Away from the interaction region, when $q\lesssim0.1$, the FCN's frequency and damping seem to be approximatly linear with $q$. In the case of the CW, the frequency and damping appeared directly proportional to the flattening $\alpha$. We established the frequency and damping of both modes in this regime on the three control parameters $q,\alpha$ and $E$. We find that the theoretical inviscid expectation for the FCN frequency, to first order in $\alpha$, is recovered when $E\to 0$ but only as long as $\alpha E^{-1/2}\ll1$. We recover as well the inviscid CW frequency when $E\to 0$ this time without restrictions on $\alpha$.

The model we present in this study can be readily modified to include magnetic fields, in which case magnetic torques and ohmic dissipation are likely to be important if the Ekman number is reduced  \citep{buffett2010,lin2017}. Additionally, the inclusion of an inner core together with gravitational torques will bring two new modes, namely the Free Inner Core Nutation (FICN) and the Inner Core Wobble (ICW). The mere presence of an inner core will also impact significantly the modes and will probably increase the computational cost.

We acknowledge that our study is far removed from Earth's parameters and therefore it is not able to address quantitatively the shortcomings of current nutation models in relation to observations. In a speculative realm, our model might be of some relevance in the dynamics of rapidly rotating neutron neutron stars where rigid crusts are very thin (and with low density compared to the dense fluid core) and thus their $q$ parameter might be larger than unity, leading possibly to resonant mode interactions. In a less exotic note, however, our work provides a proof of concept indicating that inertial modes are relevant for Earth's nutations and that their study, including an inner core and magnetic fields, is well worth the effort. These additional features will render our model much more realistic in relation to the Earth or other planets and moons. They are relatively straighforward to implement and will certainly add surprises to the rich dynamics we already discovered.

\begin{acknowledgments}
We would like to thank Rapha\"el Laguerre, Ping Zhu and J\'er\'emy Vidal for many useful conversations and suggestions. We thank as well Ankit Barik for proofreading the article and giving constructive feedback. Also we express our gratitude to the PETSc and SLEPc team for their prompt help using their solver. The funding for this research is provided by the European Research Council through the Advanced ERC Grant `Rotation and Nutation of a Wobbly Earth' (RotaNut) No. 670874.
\end{acknowledgments}

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