On the Geothermal Gradient and Heat Production in the Inner Core

Examensarbete vid Institution för geovetenskaper ISSN 1650-6553 Nr 136

On the geothermal gradient and heat production in the inner core

Peter Schmidt

Abstract

In this thesis I have investigated the upper bounds on the possible presence of radiogenic heat sources in the inner core of the Earth, using both steady state and transient models, built during this work. Necessary theory for this and model descriptions are collected into appendices at the end of this work. In addition, the published literature is reviewed for various formation scenarios, modelling of the inner core, theoretical and experimental values of relevant thermodynamic parameters. A general expression for the upper thermodynamical bounds on the initial heat source abundance at onset of inner core soldiﬁcation is derived, which in the range of the published values of the thermodynamical parameter space yields upper bounds of 0.20 ± 0.15 wt% initial abundance of 40K, the most favoured radiogenic candidate in the inner core. Alternatively the expression can be used to set an upper limit to the age of the inner core given that we know the present abundance of heatsources and thermal parameters. E.g. assuming a heat transfer coefﬁcient of k = 80 W m−1 K−1, a melting temperature of iron of 5500 K at the inner core boundary, and a value of the thermodynamical Gruneisen¨ parameter of γth,ICB = 1.5, it is found that if the core is older than 0.9 Gyr the inner core 40K abundance has to be lower than 0.142 wt% (the constraint set by cosmochemical arguments) and if the inner core is older than 2.52 Gyr the upper bound is less than 0.058 wt% (upper limit as set by high pressure experiments). Several geotherms for the inner core in subspaces of the parameter space are also presented. A comparison between the steady state and transient models is also performed, with the result that steady state models generally underestimates the temperatures and are not suitable for the inner core geotherm, mainly due to the transient nature of inner core formation and evolution. Finally the nickel-silicide/georeactor inner core model, as proposed by Herndon is investigated. It is found that this would generate a large molten region at the centre of the inner core, which has not been observed today. Hence it is concluded that a georeactor can not be operational at the centre of the Earth today. Sammanfattning

I detta examensarbete har jag studerat de ovre¨ granserna¨ for¨ forekomsten¨ av radioaktiva nukleider i Jordens inre karna¨ med hjalp¨ av sav˚ al¨ jamvikts¨ som evolutionara¨ modeller, vilka har utvecklats i detta arbete. Den teoretiska bakgrunden sav˚ al¨ som modell beskrivningar finns samlade i olika appendix i slutet av detta dokument. Utover¨ detta revideras tillanglig¨ publicerad litteratur over¨ olika formations scenarier, modellering av jordens karna,¨ teoretiska och experimentella varden¨ pa˚ for¨ problemet relevanta termodynamiska parametrar. Ett generellt analytiskt uttryck for¨ den termodynamiskt begransade¨ ovre¨ gransen¨ for¨ koncentrationer av varmek¨ allor¨ vid tiden for¨ den innersta karnans¨ initiala stelning ar¨ harlett,¨ baserat pa˚ detta ar¨ det funnet att inom publicerade parametervarden¨ ligger den ovre¨ gransen¨ pa˚ 0.20 ± 0.15 wt% 40K, vilken ar¨ den troligaste radionukliden i den inre karnan.¨ Alternativt kan uttrycket anvandas¨ for¨ att satta¨ en ovre¨ grans¨ pa˚ aldern˚ av den inre karnan,¨ givet att dagens koncentration av varme¨ kallor¨ och termala parametrar ar¨ kanda.¨ Till exempel, antagande en varmelednings¨ koefficient pa˚ k = 80 W m−1 K−1, en smalt¨ temperatur for¨ jarn¨ pa˚ 5500 K vid gransen¨ till den inre karnan,¨ och ett varde¨ pa˚ den termodynamiska Gruneisen¨ parametern pa˚ γth,ICB = 1.5, sa˚ ar¨ det funnet at om den inre karnan¨ ar¨ aldre¨ an¨ 0.9 miljarder maste˚ koncentrationen av 40K vara lagre¨ an¨ 0.142 wt% (ovre¨ gransen¨ baserad pa˚ kosmokemiska argument) och om den inre karnan¨ ar¨ aldre¨ an¨ 2.52 miljarder ar˚ maste˚ koncentrationen vara lagre¨ an¨ 0.058 wt% (ovre¨ gransen¨ satt av hortrycks¨ experiment). Ett flertal mojliga¨ geotermer for¨ den inre karnan¨ ar¨ presenterad i olika subrum av parameterrumet. Vidare gors¨ en jamf¨ orelse¨ mellan jamvikts¨ och de evolutionara¨ modellerna, vilken ger vid handen att jamvikts¨ modeller overlag¨ tenderar att ge for¨ laga˚ temperaturer och darf¨ or¨ inte kan anses vara anvandbara¨ vid studier av geotermer i jordens inre karna.¨ Orsaken till detta star˚ huvudsakligen att finna i den transienta naturen av den inre karnans¨ tillblivelse och utveckling. Slutligen har en modell foreslagen¨ av Herndon, i vilken den inre karnan¨ bestar˚ av nickel-silicid med en central georeaktor, blivit studerad. Det ar¨ funnet att modellen skulle resultera i en relativt stor uppsmalt¨ region i centrum av den inre karnan,¨ vilket inte har blivit observerat. Slutsatsen ar¨ dragen att en dylik georeactor inte kan vara aktiv i karnan¨ idag. Contents

1 Introduction 1

2 Formation of the Solar system and the Earth 1 2.1 Star and circum stellar disk formation ...... 2 2.2 Gravitational instability model ...... 3 2.3 Core accretion model ...... 3 2.4 Formation of terrestrial planets and core-mantle differentiation ...... 4

3 The inner structure of the Earth 5 3.1 The Mantle ...... 6 3.2 The core ...... 6 3.3 The PREM model ...... 7

4 Probing the Earth’s interior 8 4.1 Mass and density distribution ...... 9 4.2 Elastic properties ...... 9 4.3 Composition ...... 9 4.4 Heat ...... 10 4.5 Laboratory measurements ...... 10 4.6 Theoretical ...... 11

5 Fe at high Pressure and Temperature 11 5.1 Fe phase diagram and melting curve ...... 12 5.2 Thermal and elastic properties ...... 13

6 Models of the Earth’s Core 15 6.1 Thermal models ...... 15 6.2 Earlier core models ...... 16 6.3 Inner core solidiﬁcation ...... 17 6.4 Radionuclides in the core ...... 18 6.5 Alternative core models ...... 19

7 Constraints on the inner core geotherm and heat production 20 7.1 Inner core heat sources ...... 20 7.2 Inner core geotherm ...... 22 7.3 Constraining the inner core temperature proﬁle ...... 28 7.4 Transient models ...... 32

8 Summary and Discussion 37

9 Acknowledgements 40

A PREM 45

B The heat equation 47 B.1 Spherical symmetry ...... 48 B.2 Solving the heat equation in spherical coordinates ...... 49

C Thermodynamics and Mechanics 51 C.1 Thermodynamic fundamentals ...... 51 C.2 Thermoelastic coupling ...... 52 C.3 Lattice vibrations and the Debye approximation ...... 53 C.4 Gruneisen¨ parameters ...... 54 C.5 Adiabatic temperature ...... 55 C.6 Melting curves ...... 57 D COMSOL Multiphysics 59 D.1 Spherical symmetry using the Heat module ...... 59 D.2 Constants, Expressions, Variables, and Functions ...... 63 D.3 Meshing and Solvers ...... 66 D.4 Evaluating the output of COMSOL ...... 67

E Transient heat transfer model of the inner core 69 E.1 Transient solution to the heat equation ...... 69 E.2 Inner core growth model ...... 70 E.3 Numerical issues ...... 71 E.4 Testing ...... 76 E.5 CoreT.m ...... 77 1 Introduction

The Earth is truly a remarkable place, where ever you look at its surface, from the deepest oceans to the highest mountains, from the coldest polar regions to the hottest desserts, you will find life. And so far, our home planet is the only place in the universe where we know for sure that life once evolved and continues to exist today. So what makes our planet so suitable for life? For one thing, the composition and evolution of the Earth has offered the proper conditions for life. Thus, an understanding of life as we know it demands knowledge of the geology of the Earth, as well as its geological history. A second criterion is the presence of liquid water at the surface, which is made possible not only by the Earth’s distance to the sun, but also by the atmosphere. It needs to be realized that had it not been for the (so commonly referred to) greenhouse effect, the surface temperature of the Earth would have varied by more than hundred degrees during the course of a day, with a mean temperature below the freezing point of water. Now, the relatively close proximity to the sun is not only beneficial, as a result the Earth is under a constant heavy bombardment of charged high energy particles, emanating from the sun. Exposed to such flux levels, most life forms on our planet would not stand a chance to survive, neither would the atmosphere. Fortunately the Earth is shielded from the lethal stream by its relatively strong magnetic field, originating in the liquid outer core. However, as we know form our geological archives, the magnetic field is not static in time, neither in strength nor in direction. Even if life has shown to be able to survive the numerous polar reversal and field variations in our past, the high technological society of mankind that has evolved over the last decades is highly sensitive to such events. Therefore it is of greatest importance that we understand the operation of the geodynamo. This in turn demands knowledge of the energy supplies available in the core, in which the inner core plays an essential part, as well as the possible presence of radionuclides in the core.

In this thesis work I have investigated the upper bounds for the presence of radionuclides in the inner core and the associated thermal proﬁles (geotherms) as constrained by the fact that the inner core is solid. In order to do this I have reviewed the literature (mainly of the last decade) for our present knowledge of the inner core. However, as it will turn out, the presences of radionuclides in the core is dependent on how the Earth once formed. Therefore this report will start with a review of our present understanding of the formation of the solar system, leading up to the formation and differentiation of the Earth in chapter 2. I will then present the interior structure of the Earth in chater 3 as well as a seismic model that will be used later. In chapter 4 various methods availible for measuring the Earth’s interior will be presented, whilst chapter 5 will present the high pressure, high temperature behaviour of iron, believed to be themain constituent of the core. Chapter 6 will then present different models of the Earths core, including sections on the inner core solidiﬁcation and the presence of radionuclides in the core. My own work will be presented in chapter 7, followed by a discussion of the method and summary in chapter 8. In order to increase the readability of this thesis, I have tried to compile the relevant physics and mathematical derrivations in appendices at the end of this report. In addition, an introduction to the COMSOL multiphysics software used for modelling as well as a presentation of my models can be found in one of the appendices.

2 Formation of the Solar system and the Earth

The foundations for today’s scientiﬁc models of the Solar system formation was laid in 1775 by the German philosopher Immanuel Kant (1724-1804)1. In what is known as the Nebula Hypothesis, Kant had come to realize that stars are born in the gravitational collapse of slowly rotating giant clouds. During the collapse the rotation of the cloud increases, due to conservation of angular moment, giving rise to an increasing centrifugal force. The gravitational force is spherically symmetric and directed toward the centre of mass, whilst the centrifugal force is cylindrically symmetric about the rotational axis, directed outwards. Thus the cloud will contract toward a disk shaped structure with a central bulge. In the central bulge, the gravitational force will out win the centrifugal force

1This has been contested by some historians, claiming that the idea originated from the Swedish scientist, philosopher, seer and theologian Emanuel Swedenborg (1688-1772) who had published it in his book Arcana Caelestia (1749). It is said that Kant who was one of only three persons who purchased this expensive work during Swedenborgs lifetime. Arcana Caelestia, which was Swedenborgs attempt to interpret spiritual meaning of every verse of the bible, covers Genesis and part of Exodus. Swedenborg is said to have claimed that the work had been revealed to him by angels, a provenance that according to some creationists disproves today’s scientiﬁc view. It should also be noted that a similar model was independently proposed in 1796 by the French mathematician and astronomer Pierre-Simon Laplace (1749-1827).

1 and the contraction will continue. However, due to the contraction, the temperature and gas-pressure will increase, eventually enough to balance the gravitational force and halting the infall, a central star has been born. In the disk, instabilities will cause local regions to start to gravitationally contract into the planets.

The main features of the Nebula Hypothesis have survived into the modern day theory of stellar formation which is summarized in section 2.1. Sections 2.2 and 2.3 will then present two rivalling scenarios for the formation of giant gas planets2, whilst the formation of terrestrial planets and two different core formation models will be presented in section 2.4

2.1 Star and circum stellar disk formation

Today we believe that stars are born in giant interstellar molecular clouds (e.g. the Orion nebula). Due to perturbations, local regions in the cloud becomes gravitationally instable3 and starts to contract. In addition the contracting region is likely to fragment into individually contracting regions during the collapse [87, 112]. Hence most stars are born in groups and preferentially as double or multiple stellar systems.

When the cloud contracts the temperature in the cloud will rise, which increases the gas pressure. In fact, unless energy is lost during the contraction the gas pressure will reach values high enough to stop the gravitational collapse long before any star can form. This is the reason why molecules are needed for star formation today4, as the dissociation of molecules provides a means for removing energy from the cloud. In addition to gas and molecules, the clouds also contains dust, typically gas to dust ratios are of the order of 1/100 [44], and the vaporisation of this dust also removes some energy from the cloud. Measured by astronomical standards, the time from onset of collapse to ignition of the ﬁrst fusion reactions inside the proto-star (< 1 My) is very short, and once the star has lightened up the radiation pressure from the star will start to clear its surrounding disk from gas and dust. Observations of T Tauri stars (young stellar objects of approximately solar mass) has yielded that ∼ 30 % of the objects has cleared their disks within 1 My, while the rest has accomplished this within 10 My. Thus if planets are formed around these objects, planetary formation has to be a fairly rapid process.

During the formation of the disk structure, temperatures might reach values high enough to fully vaporise the dust content in the inner parts of the disk. But, as the disk has a large surface to volume ratio coupled to the low release of gravitational energy, the disk quickly cools by radiation. Minerals and metals can therefore re-condense rather quickly into small grains and dust particles, starting with high temperature condensates such as hibbonite (CaAl12O19), corundum (Al2O3) and grossite (CaAl4O7).

As the disk evolves so will the dust content of the disk, by coagulation the dust size will increase from sub micron up to mm sized as evident from observations [115]. Larger grains will settle toward the mid plane of the disk, thus creating a vertical stratiﬁcation of grain sizes [91]. Observations has yielded that a signiﬁcant part of the dust is of crystalline form, compared to the dust found in molecular clouds and the Inter Stellar Medium (ISM). This is generally attributed to a thermal annealing of the amorphous material in the very inner regions of the disk, from where it later on can be redistributed to the outer parts of the disk by radial mixing [139, 71, 43]. Observations of 5 the dust chemistry has indicated the presence of silicates such as amorphous olivine ((Mg,Fe)2SiO4), forsterite (Mg2SiO4), enstatite (MgSiO3), pyroxenes (both amorphous and crystalline, Mg0.8−0.9Fe0.2−0.1SiO3) and quartz (SiO4) [65, 48], as well as Polycyclic Aromatic Hydrocarbons (PAH, [50]).

Further out from the star, beyond the so called frost line, temperatures are low enough for ices like water, ammonia

2Although this is really of no importance for this work, these sections has been included for the completeness of this work 3The criterion for gravitational instability is really a local condition stating that the gravitational force is locally larger then the force exerted by the gas pressure. Consider then a giant molecular cloud where gas is being moved into new regions by e.g. the stellar wind from a newly born star or an old star going of as a supernova. This might cause nearby regions to become gravitationally unstable. 4 The ﬁrst stars formed did not have access to molecules (except possibly H2) since in principle no elements heavier than He where formed in the Big Bang. As a reason for this the ﬁrst generation stars are believed to have been extremely massive stars, but their formation is still today not fully understood. 5 Note that olivine is the name of a series of minerals where forsterite is one of the end members (the other being fayalite, Mg2SiO4). The reason for dividing between amorphous olivine and forsterite here lies entirely in the degree of crystal structure, giving rise to different spectra.

2 and methane (H20, NH3, CH4) [104] to condense. Increasing the mass fraction of solids by a factor of 2-3 [21]. Today two rivalling models for planetary formation in circum stellar disks exists, the core accretion model and the gravitational instability model. However these models are really concerned with the formation of gas giants, and the formation for terrestrial planets6 are in principle the same in both models.

2.2 Gravitational instability model

In the gravitational instability model, ﬁrst proposed by Kuiper 1957 [75] and later discussed by Cameron [35] and Boss [26, 25], giant gas planets are formed from local gravitational instabilities in a circum stellar disk. As a result of this, the formation time of a planet is short (∼ 100 - 1000 years). Modelling has shown that planets are likely to form with on eccentric orbits, as observed in extra solar planetary systems.

As a planet starts to accumulate mass, non-negligible torques and drag forces are exerted on it from the surrounding disk, forcing the planet to migrate inwards (type I migration). However, once the planet has become large enough, it will clear a gap in the disk7 reducing the inwards migration (type II migration). In general type I migration occurs on timescales shorter than the disk lifetime, whereas type II migration is a slower process. As the planets of the gravitational instability scenario quickly grows large enough to clear gaps in their host disk, type I migration does not exist. Therefore the model can explain the presence of giant planets close to their stars.

However, the gravitational instability scenario demands very massive disks, approaching the upper end of what has been observed. In addition, gravitationally instability does not necessary mean that the disk fragments into gas giants. Planets formed are also very massive (approximately 10 Jupiter masses), making it hard to explain the gas giants of the Solar system as well as most extra solar planets observed. A further obstacle is that the model has problems explaining the presence of solid cores in gas giants.

2.3 Core accretion model

In the core accretion model, originally proposed by Safronov 1969 [118], the formation of gas giants can be split into three stages:

1. Accretion of solids and dust to form a core of a few Earth masses with a thin gas envelope 2. Continued accretion of dust and gas, but with higher accumulation rate of gas, resulting in a faster mass increase of the gas envelope. This proceeds until the mass of the core and the gas envelope approximately equals. 3. Runaway gas accretion.

As pointed out above, larger grains will settle toward the mid-plane of the disk. It is believed that during their settling the grains will also coagulate into even larger boulders (∼ 0.1 - 1 m). It is not fully understood today how these boulders then accumulate further into planetesimals, but various suggestions involves gravitational instabilities [54], secular instabilities due to gas drag [55] and ”sticky” particle collisions [39]. Once the planetesimals has reached sizes of ∼ 1-10 km they decouple from the gas and move in keplerian orbits. Continued accretion of dust, gas and collisions leads to the formation of the terrestrial planets and solid cores of gas giants. As the mass of the planetary embryos increases so does their feeding zone. Gas capture becomes an increasingly important mechanism, and a gas envelope builds up around the cores. During this stage an energy balance is established between the release of gravitational energy from accreting solids, and emergent radiation from the proto planet. However, once the mass of the gas envelope becomes approximately equal to the mass of the solid core, the balance can not

6By terrestrial planets we refer to solid planets, as opposed to the gas giants. 7A criterion for this can be set as when the Roche lobe of the planet grows larger than the scale height of the disk

3 be supported by the accumulation of solids only. The gas envelope starts to contract, increasing the rate of gas accumulation, which in turn increases the rate of contraction and so on. Eventually a runaway gas accretion onto the proto-planet occurs, resulting in the build up of a gas giant.

Two mechanisms has been discussed for the termination of the runaway gas accretion phase, disk dispersal and the opening of a gap in the disk (see footnote 7). Where the ﬁrst might be important in order to explain the presence and structure of Neptune and Uranus, the second scenario seems critical in order to explain Jupiter mass planets and above. Finally follows a stage of further contraction until a hydrodynamical equilibrium of the planet is established.

Among the strength’s of the core accretion model are the mass range of planets that can be formed as well as the initial disk masses needed. The presence of solid cores in the gas giants are also naturally explained. However, the formation times are rather long due to the relatively slow second (and ﬁrst) phase. A process that might well exceed the disk lifetime, also the orbits of the planets formed generally have low eccentricities. A ﬁnal obstacle is that type I migration will be important, so unless the gas giants can form close to their stars it is hard to explain observed presence of hot Jupiter’s.

2.4 Formation of terrestrial planets and core-mantle differentiation

In principal the formation schemes for terrestrial planets are the same in the gravitational instability model and the core accretion model. The difference being that the formation of planetesimals precedes the formation of gas giants in the core accretion model, whilst they are assumed to form simultaneously or even after in the gravitational instability model. The following description of the formation of terrestrial planets assumes the core accretion model, but could as well be used for the gravitational instability model.

Not all planetesimals would have been accumulated into the giant gas planets, nor accumulated enough mass to trigger runaway gas accretion. Hence, at the time the proto-sun started to clear its nebula from gas, a swarm of planetary embryos would be present. This would be a highly unstable situation due to numerous close encounters and perturbations from the newly formed outer giant planets. Collisions would be frequent, ultimately leading to the ﬁnal accumulation of the inner terrestrial planets [140]. It is estimated that the time of formation of the terrestrial planets is of the order of 40-200 My [102, 142], as compared to accumulation times of the gaseous giants8

From gravity and moments of inertia considerations it is known today that the terrestrial planets have a metallic core surrounded by a silicate mantle. Thus, during the formation or early evolution of the terrestrial planets a core-mantle differentiation has to have occurred. Radiometric dating (Pb-isotopes) of carbonaceous chondrites9 10 +0.002 and achondrites has yielded a time span from the formation of the Solar nebula (4.566−0.001 Ga) to the presence of magmatic activity and partial melting on planetesimals of only 1-8 My [9, 15]. While U/Pb ratios of the continental crust and 129Xe excess in the Earth’s atmosphere has yielded an age, at end of Earth accretion and early differentiation, approximately 100 My younger than the formation of the Solar nebula [9]. It is therefore likely that silicate/metal differentiation occurred early on in small planetesimals, so that the earth accumulated from already differentiated bodies. This is also supported by recent high precision W isotopic data from iron meteorites [81, 123], indicating the presence of differentiated bodies at very early ages. In fact some studies even

8As noted above, disk life times are expected to be below 10 My, thus the accumulation of gaseous giants has to occur within this time. 9Chondrites are one of two major types of stony meteorites, composed mainly of iron and magnesium bearing minerals. As chondrites originates from bodies that has never undergone melting or differentiation (asteroids) their overall elemental abundance (except for the H and He) is believed to reﬂect the original composition of the Solar nebula, at the site they formed. Thus chondrites can be used to infer the chemical stratiﬁcation of the Solar nebula. In addition, nearly all chondrites contains chondrules, little round droplets of olivine and pyroxene that condensed during the cooling of the newly formed Solar nebula (before planetary formation), making them suitable for age determination of the Solar nebula. The chondrites is usually subdivided into subgroups of common properties, among these are the carbonaceous chondrites which are believed to be the most pristine of the chondrites. 10Achondrites are the second major type of stony meteorites and are made of rock that crystallised from a melt. Mainly composed of one or more of the minerals plagioclase, pyroxene and olivine, most achondrites are chemically similar to basalts, and are believed to have originated as melts on large celestial bodies that had either undergone or where in the process of differentiation.

4 suggest that the formation of differentiated bodies occurred earlier then the formation of the parental bodies of chondrites [72].

For differentiation to have occurred early in the relatively weak gravitational ﬁelds of the planetesimals rather high temperatures are necessary (> 1000 K [119]). Even though impacts will contribute with some energy this heat source would not be sufﬁcient to sustain the temperatures needed given the short timescales. Instead suggestions has been raised that the heating was due to the decay of short-lived radio nuclides injected into the young Solar nebula by a nearby supernova11. In particular the isotopes 26Al and 60Fe have been discussed and modelling has yielded that differentiation is likely to occur given an early formation of planetesimals and reasonable estimates of isotope fractions [145, 92].

The formation of the Earth’s core from the planetesimals could then have followed two extremes. On one side the colliding planetesimals results in a relatively homogeneous mixing of the material. Accumulation of material onto the proto-earth proceeded by collisions with increasingly larger bodies, as also the planetesimals grew. A signiﬁcant fraction of the gravitational energy released in these collisions was converted into heat. When the radius of the proto-Earth had grown to ∼ 2000-3000 km melting is likely to have occurred. Models have indicated that a completely molten magma ocean could have formed at shallow depths (< 300 km) while a partly molten magma ocean could have continued almost all the way down to the centre [70]. In the melt, hydrodynamically stable droplets of Fe (< 1 cm) would have been gravitationally transported toward the centre. Additionally, lakes of molten Fe could have accumulated on rheological boundaries, resulting in a gravitational overturn, upon which the Fe would have been transported deeper into the Earth and eventually all the way to the centre.

In the other extreme, as suggested by Saxena et al. [121], the collisions of differentiated bodies could have stripped of the outer layers of the planetesimals, leaving a primitive metallic proto-core (Fe-Ni-FeS-C). Whereas the proto- mantle could have accreted later on, partly by infall of the stripped material but also by accumulation of less dense chondritic planetesimals. As the proto-earth continued to accumulate mass, temperatures and pressures increased in the inner parts, with the result that silicates broke down into oxides. At pressures above 80 GPa perovskite transforms into a magnesiowustite¨ and a silica phase. This reaction depleted the lower mantle from silica, and the magnesiowustite¨ could dissociate into a MgO-rich and a FeO-rich phase. The core could then grow by reactions between FeO and the proto-core metal, accumulating even more Fe into the core along with O. The main-stream scenario today is rather the initially homogeneous Earth model than the proto-core model.

3 The inner structure of the Earth

The interior of the Earth is not homogeneous, therefore it is customary to radially divide the Earth into different regions (spherical shells) where bulk properties are alike. This can be done in two different ways (at least), one originating from a seismic velocity point of view (crust, mantle and core) and one originating from a mechanical point of view (Lithosphere, asthenosphere, mesosphere and core). However, the radial differences between the regions of the two schemes are small, and mainly important when considering the outermost part of the earth (lithosphere vs. crust). So as we are mainly interested in the inner parts we need not worry about this duality. Traditionally, when dealing with energy transport, the seismic vocabulary has been employed (mantle and core) and so shall also I.

In this chapter I will give a brief presentation of the mantle in section 3.1, and of the core in section 3.2, as well as presenting a seismological model of the Earth’s interior in section 3.3.

11Possibly also triggering the onset of the Solar nebula formation.

5 3.1 The Mantle

The mantle comprises the region from the Core Mantle Boundary (CMB, about 3480 km from the centre of the Earth) up to the thin crust (some tens of km thick). Its main constituent is believed to be magnesian silicates, in the forms of olivine, spinel and perovskite (ranging from lower to higher pressure). The mantle can further be divided into two sub regions, the upper and the lower mantle, separated at a depth of 660-700 km below the surface of the earth. This depth corresponds to the transition between the spinel and perovskite structure and can easily be seen in S-wave velocity proﬁles. Thus the olivine and spinel phases belong to the upper mantle, whilst the perovskite phase belongs to the pressures and temperatures of the lower mantle.

Over geological time the mantle behaves as a high viscous ﬂuid. Calculations based postglacial uplifts has yielded viscosity values of the order of 1021 Pa s for the upper mantle and 1021 − 1023 Pa s for the lower mantle [49]. With a critical Rayleigh number12 of the order of 103 in the mantle, and an estimated value of the mantle Rayleigh number above 106 [136], it is clear that the mantle is convecting. This implies that the temperature proﬁle of the mantle is close to adiabatic. Whether or not this convection occurs separately in the upper and lower mantle or throughout the full mantle is still debated though.

At the lower most 200 km of the mantle a sharp increase of S-wave velocity of 3-4% indicates a chemically peculiar layer, also refer to as the D”-layer. The chemistry and physics of the D”-layer are still poorly understood [63], but recent ﬁndings of a post-perovskite phase transition [93] could explain some of the features observed [14]. The temperature proﬁle of the D”-layer is assumed to be very step, estimates ranges in the interval ∆T = 1000 - 1800 K over the layer [31, 130], indicating that conduction is the major transport mechanism of heat over the D”-layer.

3.2 The core

As in the case of the mantle, also the core can be divided into two sub regions, the outer and the inner core. The liquid outer core stretches from the Inner Core Boundary (ICB, at a radius of 1221 ± 1 km) out to the CMB, just below the D”-layer. Equivalently this corresponds to about 15.6 % of the Earth’s volume or 30.8 % of the Earth’s mass due to the high density of the core. It is generally believed that the outer core mainly consists of a Fe-Ni mixture13 (8 ± 7 wt% Ni [68]), alloyed with small amounts of lighter elements [19]. Based on cosmochemical abundances, various suggestions for the lighter elements covers S, Si, Mg, O, C and H [3, 20, 113, 137], although today S, Si and O are considered to be the top candidates. In addition it has been claimed that the core contains more than 90% of the Earth’s entire inventory of Highly Sideroﬁle Elements (HSE: Ru, Re, Rh, Pd, Os, Ir, Pt and Au) [108].

Estimates of the viscosity of the outer core spans over the order of 14 magnitudes [125], but with an upper limit of the order of 1012 Pa s, the viscosity is still much lower than in the mantle. Hence the outer core is vigorously convecting, generating the Earth’s magnetic ﬁeld that shields life on the surface from a lethal bombardment of charged particles from space. Due to the convection, the outer core is generally considered to be adiabatic (isentropic) and well mixed, with a hydrostatic pressure gradient. It is not known at what time the georeactor became active. However, the earliest records of the geomagnetic activity dates at least 3.5 Ga [90]. Thus, a sufﬁciently large portion of the core must have been liquid by that time.

In contrast to the outer core, the inner core is a solid body, comprising approximately 0.7 % of the Earth’s volume or equivalently 1.7 % of the mass. The consensus is that the present inner core has solidiﬁed over time from the

12The Rayleigh number is a dimensionless measure of a ﬂuids stability to thermal convection. As the quantity is not normalised and depends on the ﬂuids thermal and thermomechanical properties, as well as geometry and thermal gradient, the critical Rayleigh number marks the boundary between stable conditions (no convection for lower Rayleigh numbers) and unstable conditions. 13An alternative proposal for the composition of the Earth’s core was put forward in 1979 by Herndon [60]. Based on the observation of oxygen poor enstatite chondrites Herndon came to the conclusion that the core should consist of nickel-silicide (Ni2−3Si) instead. However, as Ni is about 5 % as abundant as Fe [86] this would lead to an apparent lack of Fe and over-abundance of Ni in the Earth, as compared to Solar system abundances.

6 liquid outer core, and that the growth continues today as the inner core cools14. By density considerations, it is believed that its composition is similar to that of the outer core, but contains even smaller amounts of lighter (than Fe, Ni) elements.

The inner core shows some degree of anisotropy. It has been known since the early eighties [107] that P-waves travelling through the core parallel the rotational axis of the Earth has a shorter travel time than P-waves travelling in the equatorial plane. More surprising is then the recent observation that there exists an anisotropy between the eastern and western hemisphere. It appears that P-waves travelling parallel to the rotational axis in the western hemisphere arrive about 4 s faster than P-waves travelling in the equatorial plane, while P-waves travelling in the eastern hemisphere only shows an anomaly of about 1 s. Several explanations has been put forward to the directional anisotropy ranging from solid-state convection, magnetic alignment to anisotropic growth (for a review see [135]).

Another peculiarity of the inner core is that the rotation of the inner core seems to be somewhat faster, than the rotation of the mantle [135, 149]. This is consistent with predictions from numerical models of the geodynamo, even though the measured super-rotation is somewhat slower than the predicted one. Numerical values ranges from about 1 deg/year to about 0.2 deg/year.

3.3 The PREM model

The most widely used model of the Earth’s present interior structure in terms of density, pressure, seismic velocities and mechanical properties, is the Preliminary Reference Earth Model [46] (PREM). PREM is an azimuthally averaged model, based on an extensive Earth data set15 where the Earth is radially divided into sub regions, separated by seismological discontinuities. In each region the parameters of the model are either given as low- order polynomials or tabulated (see appendix A). It should be noted that even though the Earth is recognised as anisotropic, tabulated values are given for the ”equivalent” isotropic Earth. Meaning that the tabulated model has approximately the same bulk and shear modulus as the anisotropic model, not that it provides an equivalent, or satisfactory, ﬁt to the data16. It should also be noticed that the Earth is dispersive, i.e. seismic wave speeds are frequency/period dependent, therefore the PREM model are given for two reference periods, 1 and 200 s. For other periods, T, the velocities must be modiﬁed according to the equations:

lnT V (T ) = V (T = 1s) 1 − Q−1 s s π µ (3.1) lnT V (T ) = V (T = 1s) 1 − (1 − E)Q−1 + EQ−1 p p π K µ where 4 V 2 E = s 3 Vp

And QK and Qµ are the bulk and shear quality factor.

14Note however that in the proto-core scenario, the present inner core could be a remnant (partly or fully) of the initial primitive proto-core. In the extreme scenario we could also have a situation where the inner core even diminish in size due to chemical reactions at the ICB. This is not a likely scenario though. 15including total mass, moments of inertia, free oscillations/normal modes, normal mode Q-values, long period surface waves and body waves 16The anisotropy is recognised in the upper mantle, in the region between 24.4 km to 220 km depth. As the model is azimuthally averaged, it is effectively the spherical equivalent of transverse anisotropy, with the symmetry axis along the vertical (radial) direction.

7 (a)

(b)

Figure 1: Some of the tabulated PREM model parameters for a reference period of 1 s

4 Probing the Earth’s interior

In order to understand the past, present and future of our planet we need to know its interior. However, the interior of the Earth is inaccessible to us for in situ explorations and measurements. Today, the deepest we have penetrated into the Earth is about 12 km (The Kola Superdeep Borehole, TKSB), which in comparison with the Earth’s radius of about 6371 km means that we have only scratched the surface, and this is not likely to change in the future. Thus we need to have methods by which we can measure the interior from a distance. In this chapter I will brieﬂy present different methods used to investigate the interior of the earth. But it should be recognized that the methods presented here does not present a full coverage of the methods available.

8 4.1 Mass and density distribution

From Newton’s law of gravity we know that we can measure that mass, M, of the Earth by measuring its gravity which today can be done by careful laboratory experiments, alternatively we can infer the Earth’s mass from the orbits of satellites or the acceleration of spacecrafts. What is actually measured is not the Earth mass though, but rather the product GM, where G is Newton’s gravitational constant. And so the major uncertainty in today’s value of the Earth’s mass, 5.977 x 1024 kg, resides in the determination of G, which is much harder to measure17. However, this will not give us any information on the density distribution in the deep interior of the Earth. A measure of the density distribution can be found from the moment of inertia of the Earth (about 8.0 x 1037 [116]), which can be estimated from the astronomical precession constant and the second degree zonal harmonics of the geo potential. However the uncertainty in the numerical value is large, resulting in a large uncertainty of the Earth interior.

The best methods available are instead seismological. As wave speeds through a material are inversely proportional to the density of the material we here have an opportunity to get detailed information on the density distribution of the Earth’s interior. To be able map the interior of the Earth, the waves naturally needs to pass through the interior, hence a sufﬁciently powerful source is needed in order to generate these waves. In general the seismic sources of the deepest parts of the Earth are naturally occurring Earthquakes (although underground tests of nuclear weapons have also served as sources.). As a rule of thumb the longer wavelength the deeper penetration of the wave is possible. As we are mainly interested of the inner core in this study the wavelength of the waves needed, are of the order of 10-100 km. This sets an upper limit to the resolution that can be achieved in the inner core, since the waves will not be sensitive to the existence of lateral heterogeneities on scales much below its wavelength. However, due to the fact that the sources cannot be controlled by the scientists, the actual resolution is generally worse. In addition to seismology, studies of the Earth’s eigen (normal) modes, triggered by large earthquakes, yields further constraints on the density distribution [59, 135].

4.2 Elastic properties

In addition to the density, both seismic waves and normal modes are dependent on the elastic properties of the Earth’s interior. Seismic waves can be divided into S- and P-waves depending on the particle motion18. Whereas the velocity of P-waves are dependent on both the bulk modulus and the shear modulus of the material it propagates through, the S-wave velocity only depends on the shear modulus, and so as the shear modulus vanishes for a liquid, S-waves can not propagate through liquids. Due to this S-waves can not travel through the outer core. However, when a seismic wave impinges on a boundary at an angle other than 90 degrees, a phenomenon known as mode conversion occurs, resulting in at least two waves, one S-wave and one P-wave. Therefore S-waves will exist in the inner core. Now, S- generally have a smaller propagation speed than P-waves, hence the S-waves will arrive to the observer after the P-wave. A seismometer measuring waves that have travelled through the Earth will therefore detect several wave trains stemming from the same event. This way seismologists have the possibility to measure the properties of the inner core.

4.3 Composition

The composition of the crust and mantle can be studied geologically on the surface of the Earth, as different geological processes has transported material from these regions up to the surface. Unfortunately this is not the case for the core. Instead the composition of the core has to be inferred from mass constraints and elemental abundances of meteorites, believed to be the ancestors of our home planet. It is generally believed today that the bulk composition is close to that of chondrites (see footnote 9)

17The value recommended by the Committee on Data Science and Technology (CODATA) from 1998 has an uncertainty of 0.15 %, however more recent measurements [124] yields uncertainties on the ppm level. 18For the P-wave the particle motion is parallel to the propagation of the wave

9 4.4 Heat

In investigating the geotherm of the Earth it is important to know the heat flow of the Earth. At the surface this can simply be measured, yielding a current value of 44.2 TW [106]. The heat flow over internal surfaces of the Earth are not known, but reasonable estimates of the heat flow at the CMB ranges in between 5-10 TW [38, 97, 100], whilst the heat flow at the ICB is generally believed to be well below 1 TW. However these numbers are highly sensitive to the presence of radiogenic isotopes in the core. Now, knowing the heat fluxes in the Earths interior yields values of the energy available for driving various motions (e.g. mantle convection giving rise to plate tectonics, or outer core convection driving the geodynamo). It is therefore of great importance to know the distribution of heat sources throughout the Earth. One way to do this could be by detecting the neutrinos produced in the decay [110] as we already today posses detectors appropriate for this [45, 52, 110], in fact the first reports on successful detections of 28 geo-neutrino has already been published [13].

4.5 Laboratory measurements

As we move further into the Earth’s interior, both pressure and temperatures increases, hence a study of the interior of the Earth is a study of the behaviour of material properties under high pressures and temperatures. In order for us to be able to relate the results from the techniques mentioned above to each other, and to the geology of the Earth, we need to now this behaviour in advance. E.g. where we to assume that the density of solids does not change as pressure and temperatures increase, we might be tempted to believe that the inner core (ρmean ≈ 12.9 g/cm3) consisted mainly of Hf (ρ ≈ 13.31 g/cm3) or some even heavier element mixed with some lighter element. Even if we somehow knew that the major constituent should be Fe, our assumption would force us to mix in at least 34 % of elements denser19 than Hf, which would be very hard to accept from an elemental abundance point of view.

It is therefore of great importance that we study the behaviour of different materials under high pressures and temperatures. There are two different techniques used today to achieve this, shock wave (SW) experiments and Diamond Anvil cell’s (DAC)

4.5.1 Shock wave experiments

In SW experiments, the pressure in the studied target is almost instantaneously increased to high values, generating a shock wave that propagates through the target in fractions of a second. Several methods of achieving this exists, including, contact explosives, high speed impact of solid bodies, and rapid deposition of energy by high power laser irradiation [143]. The quantities measured during the experiments are, shock wave velocity, particle velocity, and temperature and pressure, which can then be related to other mechanical and thermodynamical quantities of the studied sample (for a review of different measurement techniques see Ahrens [2]). The peak shock states of several SW experiments, starting from equal initial conditions and reaching different pressures, are embodied in Hugoniot curves, or Hugoniots. It is of importance to recognize that these curves does not represent thermodynamical paths followed by the material under the influence of increasing pressure. The actual thermodynamical path followed by the material under specific experiment is instead a straight line from the initial state to the final state, referred to as a Rayleigh line. Likewise the Hugoniots are neither isotherms20 nor isentropes (or adiabats21), but isotherms and isentropes can be derived from the Hugoniot.

The main advantage of shock wave experiments over static pressure experiments is the pressure range available. In SW experiments pressures beyond TPa (1012 Pa) have been reached (e.g. Ragan [109] used neutrons from an

19There are 17 naturally occurring elements heavier than the mean density of the inner core, Hf, Ta, W, Re, Os, Ir, Pt, Au, Hg, Pa, U, Np, Pu, Am, Cm, Bk, and Cf. The densest of these, Os, would demand a presence of 34 %. 20Higher temperatures are reached for higher pressure experiments. 21Some heat is always dissipated through viscosity, plastic deformation and other dissipative processes.

10 underground nuclear explosion to investigate the behaviour of molybdenum at 2 TPa). However, disadvantages include difﬁculties in measuring the temperature, as well as the question whether the material behaves equally under shock conditions as under static conditions.

4.5.2 Diamond Anvil Cells

In a DAC the sample is uni-axially compressed between two diamonds. By keeping the ﬂat culet22 surface of the diamonds small (0.05 to 1 mm depending on the pressure range of interest) high pressures are easily achieved. The sample is then placed in an even smaller hole (∼ 0.3 - 0.5 µm) of a metal gasket, placed in between the diamonds culets, where after the pressure is increased. It is possible to reach pressures of the order of 350 GPa using DAC.

In addition to the obvious advantages of being able to keep the sample under high pressure for an extended period of time, diamonds are transparent to a large portion of the electromagnetic spectrum. This means that we can directly observed our sample, as well as examine it using techniques such as mossbauer¨ spectroscopy and various diffraction methods. It is also possible to alter the temperature of the sample using directed laser beams or electrical resistive heating. The uni-axial pressure can be transformed into a hydrostatic pressure by surrounding the sample by some ﬂuid pressure medium, or into a quasi-hydrostatic stress state by the use of a soft solid pressure-transmitting medium23.

4.6 Theoretical

The theoretical work relating to the interior of the Earth can coarsely be divided into three parts

• Macroscopic behaviour of the Earth. Including geotherms, convection, and the geodynamo. The results of such work relating to this work will mainly be dealt with in sect.6 • Behaviour of the Earths constituents under high pressures and temperatures. In addition to laboratory measurements theoretical studies of materials behaviour under high pressure, has yielded new insights and constraints on the interior of our planet. The results from such studies regarding Fe will be presented in sect. 5. • Extrapolation of material properties. As the measured high pressure properties only covers a limited pressure range, an extrapolation of the values to higher pressures might be needed for the deeper parts of the Earth, e.g. melting curves. A solid theoretical basis, i.e. an Equation Of State (EOS) is then necessary for appropriate extrapolation. For a collection of various EOS used for extrapolation of different quantities see Poirier [105]

5 Fe at high Pressure and Temperature

As the Earth’s core is believed to mainly consist of Fe a lot of effort has been put into examining the high pressure and temperature behaviour of elemental Fe. Using DAC, SW experiments, and theoretical modelling the phase diagram, thermal and elastic properties of Fe has been revised several times over the last two decades, although large discrepancies still exists. In this chapter we will review the results from such studies relevant for this work.

22The culet on a diamond is the tiny ﬂat facet that is formed by polishing off the tip at the bottom, where all the facets of the pavilion (bottom part of the diamond, below the girdle) otherwise come to a point. 23All known liquid pressure media freeze at pressures above ∼ 16 GPa at room temperature [42], hence at higher pressures a hydrostatic pressure is not achievable.

11 5.1 Fe phase diagram and melting curve

At ambient conditions Fe have a body centred cubic (bcc) structure, also known as α-Fe. Increasing the temperature, this transforms into a face centred (fcc) structure at 911 oC, also known as γ-Fe, and at even higher temperatures again to a bcc structure (δ-Fe) at 1394 oC, before melting at 1808 oC. Increasing the pressure transforms the Fe-structure to hexagonal close-packed (hcp), also known as -Fe. It has been suggested that at increasing temperature, the -Fe undergoes a transformation to a β-phase with a double hexagonal close-packed structure [120, 40] or an orthorombicly distorted hcp structure [12]. However, the existence of the β-phase has been debated [74, 111, 126] and its presence is still not settled. In addition it has been indicated from SW experiments that an additional phase, possibly a bcc structure, develops at pressures above 200 GPa [29]. Hence a large uncertainty is still associated with the phase diagram of Fe (see ﬁgure 2)

(a) (b)

Figure 2: Phase diagram of Fe. Left figure from Boehler [24], triangle and dots indicate individual measurements of Fe melting temperatures (see original paper for details). Right figure modified from Nguyen & Holmes [99], triangles, diamonds and dots indicate Fe-melting data from SW experiments, (see original paper for references), dotted lines for the location of the CMB and ICB has been added to the original figure and references in the figure has been removed as they do not agree with the numbering in this work.

Neglecting the solid phases of Fe and concentrating on the melting temperature only, we ﬁnd a scatter at the ICB of about 1700 degrees between 5400 (± 400, [80]) K to 7100 K [18] (see ﬁgure 3). In general it can be said that SW [30, 141, 144]24 experiments yields somewhat higher melting temperatures than DAC measurements [23, 122, 126], whilst theoretical experiments scatter all over the temperature range [5, 18, 80] (in fact both the upper and lower limit are set by theoretical considerations). It needs to be remembered though that not all experiments has reached ICB-pressures (this especially holds for DAC measurements) and the actual ICB melting temperatures presented are relatively sensitive to the extrapolation performed.

As pointed out above the composition of the core is not pure Fe, but rather an Fe-Ni alloy with some lighter elements, Si, S, and O being the most favoured candidates. This will also affect the melting temperature of the core material. It is generally considered that the inclusion of Ni will not have a major effect on the melting temperatures, which can not be said for the lighter elements. Using ab initio simulations of chemical potentials at core conditions, constrained by core densities, Alfe et al. [4, 6] came to the conclusion that the outer core contains 8.5 ± 2.5 molar % S and/or Si plus 10.0 ± 2.5 molar % O, while the inner core contains 0.2 ± 0.1 molar % S and/or Si plus 8.0 ± 2.5 molar % O, resulting in a melting temperature depression of -700 ± 100 K. Whereas Andersson [10] considers several mixtures of S, Si and O, yielding ICB melting point depressions in the range -700 to -2271 K. However, DAC measurements of Boehler [22, 23] indicates that the melting depression observed in the Fe-FeO-FeS systems diminishes at increasing pressure, which could possibly affect the theoretical results.

24Note that the Williams [141] used a combination of SW and DAC

12 Figure 3: Comparison of Fe melting curves from theoretical (heavy solid, long dashed, dotted, and light solid curves as well as ﬁlled circles)and experimental results, including both DAC measurements (chained and short dashed curves, and open diamonds and stars) and SW experiments (open squares, open circle and full diamond). reprinted from Alfe [8] (see original paper for references).

5.2 Thermal and elastic properties

Table 5.2 summarizes values found in the literature of the last decade, for a number of thermal and elastic parameters of hcp-Fe at inner core pressures. Although some of the values are based on results from high pressure experiments, it should be remembered that most of the quantities are not directly measurable, but rather rely on the measurement of some other quantity which they can be related to. In addition, most of the parameters are both pressure and temperature dependent [7], and so an interpolation of experimental values to inner core pressures and temperatures is necessary. Thus, a number of possible sources of uncertainty exists in the presented numerical values, including:

• Data in all measured data there is always a minimum level of uncertainty. This will naturally set the minimum uncertainty of any parameter derived from that data. Unfortunately not all values are presented with their estimated uncertainties as they really should be. • Theory The relation between a measured quantity and a derived quantity goes via some theory, hence the derived quantity will only be as good as the theory. • Interpolation I Today, only SW experiments has the capability to ”easily” reach inner core pressures and beyond25. But they do so on Hugoniots, so unless the initial conditions are precisely the right, the temperatures in SW experiments at inner core pressures will differ from inner core temperatures. In general, interpolations are needed both with respect to pressure and temperature. Now, to do an interpolation one needs to have an EOS relating the interpolated quantity to the variables. In the geological high pressure community a number of such EOS ﬂourish (for a recent review and discussion see [132, 130]). The use of two different EOS will generally give some variations in the derived quantity. Now, the EOS is derived from some theory, however, this is not always the same theory used to relate the derived quantity to the measured quantity, or possibly a very simpliﬁed version, which is the main reason why we split theory and interpolation here.

25Although DAC experiments is closing in, e.g. [88, 41]

13 • Interpolation II To do an interpolation we need at least one known data point with known coordinates26 as well as the coordinates of the desired data point. Now as discussed in sect. 5.1, we do not know the inner core temperatures very precisely. In addition we might not know the coordinates of known data points to a high precision either. So unless the interpolated quantity is relatively insensitive to variations in the coordinate values, this could seriously affect the presented data. It is therefore important to check what assumptions that has gone into the interpolation.

Anderson [10] uses the Wiederman-Franz law and the electrical conductivity found by Matassov [89], to ﬁnd the −1 −1 electronic thermal conductivity, ke, for the core to be 39 W m K . Adjusting for the lattice contribution, kl, (assumed to be about 4 W m−1 K−1), he presents a value of k of 43 W m−1 K−1. Although, it is recognized that values of Matassov might need correction that could lead to a value of k of the order of 60-70 W m−1 K−1. Stacey [131] also uses the data of Matassov but imposes corrections on them, interpolates to core values and compares the values to those measured by Brigman [27], resulting in a value of k of 79 W m−1 K−1 in the inner core. It should be noted though that Stacey considers an Fe-Ni-Si alloy, and Anderson considers a Fe-Si alloy, both papers ﬁnds k(140 GPa) to be 43 W m−1 K−1.

−1 −1 −1 −1 −5 −1 k [W m K ] γ Cp [J kg K ] α [10 K ] KT [GPa] T [K] Ref. - 1.45 - 1.2 - 6000I [7] - 1.27 - - - 5100II [10] - 1.52 - - - 5709 [11] - - - - - 7100 [18] - 1.28III - - - - [41] 60 1.5 715 1.95 - 1.02IV - 5500 [57] - - - - - 5400 [80] 79V - - - - 4971V [131] - 1.38VI 860VI 1.32VI - 4500VI [133] - 1.27VII 826VII 1VII 1194VII 4500VIII [129] - 1.5 - 1.2 900-1177IX 6000X [138]

Table 1: Various values of thermal and elastic properties for Fe at inner core pressures. The column T refers to the temperature at which the tabulated value is given. A - sign indicates that no values are given for this parameter. I Values for T = 4000 and 2000 K can be found in the original paper. II It is assumed that γ is independent of T, otherwise this temperature should be revised. III Value given at a radius of 1400 km. IV Values given from ICB-pressures to centre of Earth. V Values given for Fe-Ni-Si alloy matching inner core density (10 % Ni). VI Tabulated values are given at P = 243 GPa (mid core) VII Values given at 1400 km radius VIII Temperature given at 2400 km radius IX values tabulated for densities 12.37 and 13.31 g cm−3 of hcp-Fe, approximately equal to expected hcp-Fe densities at ICB and centre of Earth. X Values for T = 4000 and 2000 K can be found in the original paper.

Anderson [10] also finds the pressure derivative of the bulk modulus at zero pressure, K’0, from PV of hcp-Fe data of Mao et al. [88] and the Birch-Murnagahan EOS. Using a thermodynamical relation for hcp-Fe, derived by Stacey [129] he then finds γ at zero pressure. He finally interpolates to ICB pressure by assuming27 that γ ∝ ρ−0.7, yielding a value of 1.27. Where as in a later paper Anderson [11] relates γ to the vibrational and electrical Specific heat at constant pressure, CV,vib and CV,el. Using data by Stixrude [134] he then finds a value of γ of 1.53. Other examples of experimentally determined quantities comes from Dubrovinsky et al. [41], who used high-quality powder X-ray diffraction data, collected in DAC’s up 285GPa, to determine the γ to be 1.28 at a radius of 1400 km (i.e. about 180 km above the ICB)28

26I.e. numerical values of the variables, e.g. temperature and pressure in our case 27the value 0.7 is constrained from earlier theoretical estimates of γ by Stacey, [129] and 28Unfortunately the derived value of γ rests on the use of a thermal equation of state of hcp-Fe, from an unpublished manuscript by Dubrovinsky et al. I have not been able to ﬁnd out whether this paper has been published afterward.

14 In addition to experimental measurements, the advancements in both theory and computer technology has made it possible to perform thermodynamical ab initio simulations of the high pressure and temperature behaviour of materials. E.g. Gubbins et al. [57] and Alfe et al. [7] both computes Helmholtz free energy of hcp-Fe using Density Functional Theory (DFT), upon which a number of parameter values are derived. An alternative route is taken by. Stacey [129] who combines results from molecular dynamical calculation by Barton and Stacey [17] with thermodynamical relations and derivatives to ﬁnd numerical values for a number of parameters.

6 Models of the Earth’s Core

Any model describing the Earth’s core demands a knowledge of the temperature profile, as this relates to both material properties and the energy present. Now, in a self consistent model the temperature should therefore always be a parameter solved for, even if the primary target is not the temperature profile. An alternative approach would be to use a known temperature profile, e.g. resulting from a self consistent modelling. The question is then whether we can construct a simple self consistent model that will return only the temperature profile. To answer this question let us have a look at the requirements for a thermal model, starting with the inner core only (as this is the region of interest of this thesis).

6.1 Thermal models

As it turns out, building a thermal model of the inner core is not a trivial problem. A closer look at the problem yields that we are facing the problem of solving the non-homogeneous transient heat equation in spherical coordinates. Non-homogeneous, since that even if we do not have any radiogenic isotopes in the inner core, we still some contributions to the heat budget due to the release of gravitational energy when the inner core contracts upon cooling. Transient since the nature of the possible inner core heat sources are transient by nature. However, we know that the inner core is solid, i.e. we need not worry about convection. To simplify the problem even further, we can assume the inner core to be spherically symmetric, hence we can settle with a 1D solution over the radius of the inner core. So what are then the boundary conditions? At the centre of the core our assumption of spherical symmetry implies that the heat ﬂux should be equal to zero.

∂T k = 0 (6.1) ∂r r=0 However, at the ICB we have a moving boundary, so we are facing a Stefan problem, but even worse, the boundary condition at the ICB is dependent of the position of the ICB, which in turn is dependent on the heat flow through the outer core. Thus we also need to solve the heat equation in the outer core, but as this is fluid and vigorously convecting we can no longer confine ourselves to a 1D solution. In addition, as convection means conversion of heat into kinetic energy, that might be lost due to dissipation (e.g. the generation of a magnetic field), we can no longer only solve the heat equation, the problem has become a multi-physics problem.

And so it continues as we realize that we cannot solve the heat equation in the outer core without knowledge off the heat ﬂow over the CMB, which is dependent on the heat transported through the mantle etc. In fact we have to continue all the way to the surface of the Earth (or even the upper layers of the atmosphere) before we can ﬁnd a rigid boundary condition. It should be evident by now that no analytical solution can be found to the problem, hence we are left with numerical schemes. Now, building a self consistent full Earth model is not a feasible task. What we must do is to make some physical assumptions of the boundary conditions at a proper radius, and construct a model using these assumptions.

15 6.2 Earlier core models

Several core models has been found in the literature. Labrosse et al. [78, 79] considered models constrained by the CMB heat flux, Labrosse and Macouin [77] and Labrosse [76] constructed models constrained by ohmic dissipation. Gubbins two models compares different core composition (one including compositional convection [56] and one without [57]) to investigate the heat fluxes needed to drive the geodynamo. Where as Butler [33], Costin [38], Nakagawa [97], Nimmo [100], and Yukutake [148] included mantle models to constrain the heat flow at the CMB. The most striking result from the different models is the young age of the inner core. With the exception of Gubbins models, these all fall in the range of about 1-2 Gyr (e.g. see figure 4(a)). Likewise, the models solving for the CMB heat flow all displays values in the range 8-9.5 TW, and ICB heat flows in the range of 0.2-0.5 TW (e.g. see figure 4(b)). Another interesting feature is that many of the papers present several models with different contents of radiogenic isotopes, but this shall be discussed further in section 6.4 below.

Figure 4: Time evolution of the inner core radius after onset of inner core solidiﬁcation, and various contributions to the heat balance of the outer core. Reprinted from Labrosse et al. [78], for model parameters see original paper.

In the models presented above, it is generally assumed that the inner core geotherm is close to isothermal. I have only found one paper, by Yukutake [147], that actually considers the inner core geotherm. Unfortunately this paper was found in a very late stage of my work, when all my work was done and this report almost ﬁnished. The aim of the paper is to investigate the claim by Jeanloz and Wenk [69] that the inner core should poses thermal convection. One criterion for thermal convection to occur, is that the gradient of geotherm should be steeper than the adiabatic gradient, and so in order to investigate the claim, Yukutake sets up an evolutionary model of the inner core geotherm. The model assumes pressure induced freezing of the inner core (see section 6.3) and an adiabatic temperature proﬁle of the outer core. As initial condition the adiabatic temperature equals the melting temperature at the centre of the Earth, where after the adiabatic temperature is assumed to decrease linearly with time.

For every time step, δt, the radius of the inner core is initially evaluated where after the temperature proﬁle is evolved, using an analytical expression29 given by Carlslaw and Jaeger [37]30 for the transient heat equation with a varying boundary temperature. As a minimum age of the inner core 1.5 Gyr used and a maximum presence of 100 ppm 40K is considered. For all models it is found that the inner core geotherm gradient is below the adiabatic gradient, or equivalently a maximum temperature difference today between the ICB and centre amounts to 129

29The analytical expression is found using Green’s functions (see eq.(B.23) in appendix B), resulting in an inﬁnite series solution. 30Reference taken from paper [147], as I could not gain access to the referred book.

16 degrees, which is less than the adiabatic temperature difference of 143 degrees. Hence the conclusion is drawn that the inner core geotherm is always sub-adiabatic.

6.3 Inner core solidiﬁcation

Inside the Earth the pressure monotonically increases with depth, reaching a maximum value of about 364 GPa at the centre. This will increase the melting temperature of any material. We know that the geodynamo has been operational for at least 3.5 Gyr [90] which implies that the liquid outer core has been convecting throughout this period, which in turn implies that the geotherm of the outer core has been close to adiabatic throughout most of Earth’s lifetime. It can be shown that for reasonable values of the Gruneisen¨ parameter, the gradient of the core adiabat will be less step than the melting temperature of the inner core. Consider an initially molten core, the minimum difference between the core geotherm and the melting temperature of the core will then be at the centre of the Earth. As time elapse, the Earth cools and eventually the core geotherm will be equal to the meting temperature of the core. The inner core then starts to solidify. As time goes on the Earth cools further and the intersection between the core geotherm and the melting curve will progress toward larger radii, causing the inner core to grow. At every instant having a temperature at the ICB equal to the melting temperature of the core material (see left panel in ﬁgure 5). I.e. the inner core solidiﬁes from inside out. From now on we shall refer to this scenario as pressure induced freezing.

Even though pressure induced freezing is the most frequently encountered model for inner core growth, this is a relatively simplified picture. The reason being that the core is not of elemental composition, but rather a mixture. When a mixture freeze (unless of eutectic composition) it will not have a clearly defined melting temperature, but rather a melting temperature interval (see figure 23(b) in section C.6), under which the composition of the solid phase will differ from the composition of the liquid phase. In the case of the outer/inner core this is evident from the increased density of the inner core, which can not be explained from a volume change upon solidification only. Thus the interpretation is that the fraction of lighter elements is lower in the inner core than in the outer core. And so the temperature of the ICB is bounded in the freezing interval of the core material, with upper limit set by the melting temperature of the inner core composition (see right panel in figure 5). This has certain implications for the solidification of the inner core. To start with the solidification of the inner core will not only take place at the ICB, but rather in an extended region, stretching out into the outer core. This will enrich the solidification region in lighter elements, giving rise to a compositional convection in the region, transporting lighter elements out into the outer core, and accumulating more material onto the inner core. In addition it is likely that the ICB is not a distinct interface but rather a mixture of solid and liquid material, with an inwards increasing fraction of solids, i.e. a mushy layer will develop on top of the core. In fact, some indications for the existence of such a layer has emerged from recent seismological studies of the inner core [36, 73, 146], for one thing, instabilities in such a mushy layer could explain the growth of the ICB by 1.37 ± 0.38 km between 1993 to 2003 as observed by Lianxing [85]. Note that the enrichment of lighter elements in the solidification zone will alter the melting temperature interval of the liquid fraction, possibly leading to a state of super cooling, however, a recent study by Shimizu [128] has indicated that the cooling rate of the Earth, and hence growth rate of the inner core is far to low for this to happen.

It should be recognized though, that the expected magnitude of the difference in the melting temperatures of the inner and outer core alloys is small, at maximum a few tenth’s of degrees. Given then that the adiabatic gradient at the ICB is of the order of 0.37 K km−1 (see ﬁgure 22) the region of inner core solidiﬁcation will be at maximum about 100 km wide.

Finally we shall mention a third effect that will influence the solidification of the inner core. As recognized by Gubbins et al. [57] the solidification of the inner core is associated with a small volume decrease, which will cause a small contraction of the core and hence an small increase of pressure. This in turn will increase the melting temperature, and therefore also increase the size of the inner core. Even though one would naively consider the effect to be small, the model constructed by Gubbins et al. indicates that this could account for as much as 10-20 % of the inner core growth. However, it should be said that the model presented by Gubbins et al. displays several peculiarities, like an inner core age of only a few hundred Myr and heat fluxes of the order of 30-50 % of the

17 Earth’s current heat ﬂux at the surface. In addition it is not fully transparent (at least not for me) exactly how Gubbins et al. evaluates the effect.

Figure 5: Models of inner core solidiﬁcation. Blue line indicates the core geotherm, red line indicates the melting temperature of the core. Left panel displays the pressure induced freezing scenario and right panel displays the scenario with different melting temperature of the inner and outer core alloy.

6.4 Radionuclides in the core

The partitioning of K into the core was suggested on geochemical grounds in the early seventies [53, 58, 84], especially if S was the main light constituent of the core. In the beginning of the nineties this idea was opposed by results from simulations and high pressure experiments [67, 127]. But, as time went on and experiments was conducted at even higher pressures, evidence for the possibility of K partitioning into the core accumulated [51, 95, 103], culminating with the estimate by Lee and Jeanloz [82] that up to 0.7 wt.% of K, could be incorporated into hcp-Fe at core pressures, yielding a present day abundance of about 580 ppm. Assuming the initial composition of the Earth to be chondritic, and that no K was lost during its accumulation and evolution, an upper limit is set to about 0.142 wt.% in the core today [114]. On the other hand, given the age of the Earth and the half-life of 40K (see table 2), we have that this equals approximately 1/12 of the initial 40K. Since 0.142 wt.% 40K in the core would produce a total of about 9.56 TW today (see table 3), this would imply a very high heat production in the early days, which could be hard to defend.

Other indications for the presence of radionuclides in the core comes from different models of the core and the operation of the geodynamo [33, 38, 97]. The reason is that most core models indicate a relatively young age of the inner core (see section 6.2), as well as cooling of the core being an insufﬁcient energy source for the geodynamo. Thus these models more or less demands the presence of radionuclides for the geodynamo to be operational over geological time. Estimates on the amount of radionuclides necessary ranges from 100 to 600 ppm.

In addition to K, also U and Th has been considered as possible candidates in the core [127, 94]. Even though the general opinion seems to be against U and Th in the core, recent experiments has indicated that small amounts, of the order of a few tenth’s ppb, could be present in the core [16, 96]. Table 2 lists the suggested radionuclides for the core including natural abundances, half life’s and heat production rates for pure substances.

The question is then whether or not radionuclides could be present in the inner core as well. Unfortunately the high pressure experiments performed so far can not answer this question as the pressure range of interest has been restricted to pressures relevant for different core-formation scenarios (i.e. well below 50 GPa). However if the presence of radionuclides in the core is associated with the presence of lighter elements, and especially S, the deficiency of lighter elements in the inner core (relative to the outer core) could indicate a partitioning coefficient between liquid/solid core in favour of the outer core. On the other hand, if, as indicated by Parker et al. [103], the presence of K is associated with the presence of Ni in the core, the same argument would lead to a partitioning coefficient in favour for the inner core. A further consideration could be the atomic weight of the radionuclides

18 considered, both U and Th are very dense atoms that could possibly have tendency of gravitational migration toward the centre of the Earth, making them likely constituents of the inner core. Where as in the case of K, being an alkali metal and therefore having a relatively low density, the situation could be the reversed depending on the exact conﬁguration of (elemental or molecular) the core K content.

Finally it needs to be pointed out that the presence of radionuclides in the core, as indicated by high pressure experiments is highly sensitive to the exact formation scenario of the inner core. It should be remembered that most of the experiments has investigated the partitioning coefficient between various Fe-alloys (mainly iron-sulfides) and Silicates. And, as indicated by the experiments, the partitioning coefficient increases in favour of the Fe-alloy with increasing pressure. Hence the results can only be transferred into core values if the Earth’s core formed at chemical equilibrium at depth in an initially homogeneous Earth, or at least in the interior of a relatively large proto-planetary body.

Nuclide Abundancea Half lifeb Heat productionb [%] [Gyr] [W kg−1] 40K 0.0117 1.2511 ± 0.002 2.966 x 10−5 232Th 100 14.01 ± 0.08 2.556 x 10−5 235U 0.7200 0.7038 ± 0.00048 5.749 x 10−4 238U 99.2745 4.4683 ± 0.0024 9.166 x 10−5

Table 2: Possible candidates for core radionuclides. Abundances in present day natural abundance. Heat production assumes the decay chain to have reached steady state, and values are given for pure substances, i.e. 100 % of the speciﬁed isotope. a from Nordling & Osterman¨ [101] b from Rybach [117]

6.5 Alternative core models

Even though the consensus in the scientific community is that the core is composed of an Fe-Ni alloy with the addition of some lighter elements, other alternatives have been proposed. In a series of paper by Herndon [60, 61, 62, 64] it has been claimed that composition of the inner core should be nickel-silicide31 rather than an Fe-Ni alloy. In addition it is proposed that the inner core should poses an actinide sub-core of a few km radius at its centre, surrounded by a shell of decay and fission products (see figure 6).

In Herndon’s model most of the inner parts of the Earth are in a highly reduced state and the radius of the inner core is constant (although it is allowed to initially have grown rapidly from a liquid state). Based on the natural occurrence of Ni2Si in oxygen-poor enstatite chondrites it is then concluded that the inner core composition should be that of nickel-silicide. The model makes the assumption that the melting temperature of nickel-silicide at core conditions is much higher than the inner core temperatures. Thus making it possible to have a ﬁxed size inner core throughout the lifetime of the Earth. Also found in highly reduced enstatite chondrites are traces of uranium, U, and thorium, Th, associated with the nickel-silicide. It is therefore argued that U and Th will be present in the inner core as well. As a consequence of the higher densities of U and Th the model postulates the precipitation of a high-density, high-temperature sub core within the inner core, in which the concentration of actinides is high enough for self-sustaining nuclear ﬁssion reactions to occur. This is referred to as the georeactor, producing an estimated effect of 3 TW presently. Taking into account also the natural decay of 235U and 238U would increase the effect to a present day value of 3.5 TW.

Over time, as the georeactor operates, a build-up of decay products will occur, halting the ﬁssion reactions thus lowering the effect. However, the decay products are lighter, and will over time be removed by gravitationally driven diffusion, and so the effect of the georeactor will again increase. This behaviour could then explain the variability and reversibility of the geomagnetic ﬁeld, as it is dependent on the power supply in the core. Another evidence of the georeactor comes from 3He/4He ratios released to the oceans at the mid oceanic ridges, which

31 nickel-silicide is a group name of several Ni-Si compounds, e.g. Ni3Si, Ni2Si, NiSi

19 is about eight times greater then in the atmosphere. 4He is naturally produced by the decay of U and Th in the mantle, but so far no mechanism deep within the Earth that can account for a substantial 3He production has been known. Instead the deep-Earth 3He has been assumed to be of primordial origin. Simulations by Herndon has indicated that the observed levels could well be explained by the presence of a georeactor.

Figure 6: The model of the Earths inner core structure as proposed by Herndon. Reprinted from [61]

7 Constraints on the inner core geotherm and heat production

In this chapter I have studied temperature proﬁles of the inner core using COMSOL multiphysics. Before we go in to this let’s quickly recall our knowledge of the core from previous chapters (for a discussion of each claim, see the preceding chapters). The core of the Earth has an outer radius of 3480 ± 5 km, and comprises a total of about 16.3 % of the Earth’s volume. We can divide the core into two regions, the solid inner core with a radius of 1221 ± 1 km, and the liquid outer core where the Earth’s magnetic ﬁeld is generated. The core consist mainly of a Fe-Ni alloy mixed with some lighter elements, O, S and Si being the top candidates. It is further recognised that the amount of lighter elements is higher in the outer core than in the inner core. In addition it is also possible that the core contains fractions of long lived radionuclides, in particular 40K has been considered. Since the composition of the inner core is different from the composition of the outer core, it is possible that the melting temperatures of the inner and outer core differs by a few tenth of degrees. The melting temperature of elemental Fe at ICB pressures is estimated to lie within the interval 5400 to 7100 K, which relates to the core alloys via a melting depression of a few hundred up to 2000 K. For the inner core it is believed that k ∈ [60; 80] W m−1 K−1, γ ∈ −1 −5 −1 [1.27; 1.53], Cp ∈ [715; 800] J kg K, α ∈ [1; 2] x 10 K .

7.1 Inner core heat sources

There are only two possible heat sources in the inner core, release of gravitational energy and decay of radionuclides. However, it has been argued [32] that the release of gravitational energy, due to shrinking of the cooling inner core, is fully converted into compressional (strain) energy in the inner core and therefore does not contribute to the heat equation. In any case, if radionuclides are present in the inner core, the gravitational contribution will be small in comparison and can be neglected.

20 It is most likley that the core has been melted throughout at some stage of the Earth’s evolution. Given the timescales involved in inner core formation and the fact that the outer core is convecting it should then be safe to assume that the inner core formed at chemical equilibrium with the outer core. Therefore, any distribution of radionuclides in the inner core is initially set by the partition coefﬁcients of the radionuclides present. Although a redistribution may have occurred with time, due to gravitational migration. In any case, from the processes involved we should expect any distribution to be a spherically symmetric smooth function (i.e. continuously differentiable).

From a physical point of view we know that the function describing the distribution also has to be non-singular32 inside the inner core. It follows then that any source distribution can be described by a power series, i.e. a polynomial in only positive powers of the radius, r.

X n Q = cnr (7.1) n

Where cn are some constants. Now if radionuclides are present in the core they have to be present in both the inner and outer core, with relative abundances initially set by the partition coefficient, Dsolid/melt, of each speciem. Unless the partition coefficient is heavily biased toward partitioning into the solid inner core we should expect any reasonable function, describing the distribution of radionuclides in the inner core, to be dominated by its zeroth order constant, i.e. by c0 in equation (7.1). The reason being that the inner core comprises only about 4.3 % of the total volume (or equivalently about 5% of the total mass) of the core, and so the levels of radionuclides in the outer core can, to a first order approximation, be considered to be independent of the radius of the inner core.

Even if the initial distribution is not constants in the inner core, the decay of the radionuclides will drive the distribution towards an evenly dispersed (i.e. constant) with time (see ﬁgure 7.1 below). Of the radionuclides considered as possible candidates for the inner core, only 40K have a half-life of the order of the estimated age of the inner core33. On the other hand 40K is the top candidate of the suggested core radionulides and therefore also for the inner core. Hence any initial deviation from a constant distribution should have diminished to about half its value over the life time of the inner core.

Figure 7: Time evolution of the inner core distribution of radionuclides due to partitioning coefficient and decay. A = abundance, r = Distance from the Earth’s centre. Solid line indicates distribution due to the behaviour of Dsolid/melt, left panel displays a increasing trend with increasing pressure/density, middle panel displays constant value and right panel an decreasing trend with increasing pressure/density. Dotted line adds the effect of decay of radionuclides in outer core, dashed line adds the effect of decay also in the inner core. Note that the decay tends to level out any non constant behaviour of the distribution. For simplicity the distribution are displayed as linear profiles, although it should be recognized that they in reality are not (e.g. decay is inversely exponential) this does not affect the final conclusion.

So what about a redistribution due to gravitational migration. First of all we need to realize that the gravity of the inner core is low, reaching a maximum value of about 4.36 m s−2 at the ICB (see ﬁgure 7.1 below), and so any gravitational redistribution will be slow. In addition, entropy driven diffusion34 will oppose any tendency away from a constant value of the heat source distribution, including gravitational migration.

32I.e. it can not go to inﬁnity 33This is strictly not true since the half life of 235U is about half that of 40K. On the other hand 235U has a natural abundance of only 0.72 % today, in comparison to 99.27 % for 238U. And there is really no reason why this should not also be the case for the inner core. Hence the Half life of 235U is not important in this discussion 34Diffusion driven by a non-zero gradient of the concentration.

21 Figure 8: Gravitational acceleration force inside the inner core.

In light of these arguments we can think about the feasibility of an actinide core as proposed by Herndon (see sect. 6.5 above). In the discussion above it has been assumed that the core initially was completely liquid and well mixed. If this holds true the formation of an actinide core could only have proceeded via a partitioning coefﬁcient heavily biased toward the solid inner core composition. Otherwise the low gravitational ﬁeld and entropy diffusion would have effectively hindered the formation of an actinide core. Relaxing the well mixed assumption in the innermost parts of the core, does give some possibilities for the accumulation of radionuclides before the formation of the core, however being liquid, entropy driven diffusion will be more effective, thus the actinide core scenario still seems rather unfeasible. In Herndon’s view the inner core need not have been liquid throughout the life time of the Earth, hence the only way for an actinide core to have formed, would have been if it formed elsewhere than in the centre of the Earth and then became accumulated into the inner core upon the formation of the Earth. However this only transfers the problem to another location. Thus the formation of an actinide core seems a highly unlikely scenario.

Several radio nuclides has been discussed as constituents of the core (see table 2 of sect. 6.4). In order to reduce the complexity of the inner core geotherm, I shall in what follows speak in terms of integrated inner core heat production, rather than abundance of a particular element, unless the discussion is dependent on the exact choice of heat sources. Any presented heat production can the be related back to a particular abundance of a particular element via table 3 below, e.g. for an inner core heat production of 0.5 TW we would need a K abundance of about 0.147 % and assuming this abundance to hold throughout the whole core we would have a heat production in the core of about 9.88 TW.

Element Abundance Volumetric Inner core Core heat heat generation heat generation generation [10−9 W m−3] [TW] [TW] K 0.1 % 44.80 0.34076 6.7307 Th 0.1 ppm 33.00 0.25102 4.9581 U 0.1 ppm 122.81 0.93413 18.451

Table 3: Examples of heat production in the inner core and the whole core for a given elemental abundance. ppm = 10−6

7.2 Inner core geotherm

To estimate the geotherm of the inner core for various distributions of radionuclides I have used steady state models, assuming that the age of the inner core is great enough for this to be a valid assumption. Using a steady state model means that we can ﬁnd an analytical solution for the heat equation (see eq.(B.21) of appendix B.2).

1 1 Z Z A T (r) = Qr2dr − Qrdr − + B (7.2) k r kr

22 Where Q is heat sources and sinks per volume35, [W m−3], k is the thermal conductivity, [W m−1 K−1], and A and B are constants determined by the boundary conditions. Note that in the derivation it has been assumed that k is constant. If not so we have to use the form Z 1 Z A T (r) = B − Qr2dr dr − (7.3) kr2 kr

Given the boundary conditions of the inner core we find that A = 0, as dT/dr|r=0 = 0, and that B is related to the temperature at some radius36 of the inner core37. Hence, we see that the temperature profile is linearly dependent on the temperature at the reference radius. Changing the reference temperature by an amount ∆T will change the whole temperature profile by an amount ∆T . This means that we can work with relative temperatures rather than absolute, i.e. we do not need to know the temperature at any point of the core to find a solution38. In what follows I shall denote the relative temperature by ∆T , rather than T , to indicate that the presented geotherms can be scaled to any temperature of the chosen reference radius by simple addition. For convenience I shall always use the ICB as the reference radius, i.e. ∆Ticb = 0 in what follows.

It is very instructive to study the behaviour of the geotherm and its gradient, dT/dr, due to the source distribution and thermal conductivity under steady state conditions. Therefore we shall do so before I present any geotherms for speciﬁc combinations of Q and k, with the aim that the discussion of any presented geotherm can be related this study. It needs to be stressed that this study assumes steady state conditions, hence conclusions drawn are not necessarily valid under non-steady state conditions

First we shall consider the quantity Q itself. Let’s start with the sign. According to our deﬁnition above (see footnote 35) a positive sign means that heat is being produced, a source, and a negative sign means that heat are being consumed, a sink. Now, inside a simply connected volume39 the only way heat can be consumed is by conversion into some other energy form or by a decrease of the temperature. However, in steady state the temperature can not change by deﬁnition, and if heat is directly transformed into some other energy form it must not enter into the heat equation. Thus we can not have any internal sinks inside the inner core.

To investigate the geotherms behaviour due to Q we shall start by assuming k to be constant. Consider a sphere of radius r0 with no internal heat sources. This means that the temperature proﬁle has to be isothermal (i.e. constant). Now add an external layer of thickness ∆r, containing some heat source distribution Q, to the surface of the sphere. Since integrals are linear operators, we can then split eq.(7.2) into two integrals over the continuously connected regions, 0 to r0 and r0 to r0 + ∆r. The inner region (i.e. the integral over 0 to r0) will not contain any heat sources, hence the temperature will still be isothermal here. In the outer region, we will have a thermal gradient < 0. Let’s add one more layer to our sphere, this time with no internal heat sources. From energy conservation we know that the heat produced inside of this layer has to diffuse outwards through the layer. Denote the integrated heat produced inside of r0 + ∆r by P (r0 + ∆r), we then have that the heat ﬂow per unit surface, q, at a radius r > r0 + ∆r is P (r0 + ∆r) q(r) = 4πr2 and so from Fourier’s law in spherical coordinates we have that

0 dT q P (r + ∆r) = − = − 2 (7.4) dr r>r0+∆r k 4πkr

35I will refer to this term as heat sources in what follows, equating heat sinks as negative heat sources 36referred to as reference radius here after 37This can easily be realized by considering T (r = r0) = B + C(r’), where C(r’) becomes a numerical value that might be 6= 0 depending on the distribution of heat sources. 38Note that the above stated not necessarily holds true if k is a function of temperature. On the other hand, if that is the case, the given solutions (eq.(7.2) and eq.(7.3)) will not be applicable anymore. We shall therefore in what follows neglect any temperature dependence of k 39Simply connected means that we can continuously shrink the volume to a single point, i.e. no ”holes” are allowed inside the volume. Thus we can consider the inner core to be continuously connected.

23 To summarise we have that

 0  0 0 ≤ r ≤ r If Q(r) = 6= 0 r0 ≤ r ≥ r0 + ∆r  0 ≥ r0 + ∆r ≤ r  0 0 ≤ r ≤ r0 dT  Then = < 0 r ≥ r0 dr  ∝ −r−2 ≥ r0 + ∆r ≤ r

Thus the gradient of the geotherm at a given radius, will not depend on heat sources outside of that radius. The more heat sources inside, the steeper gradient. Note that if Q is constant inside of some radius r0 then q(r < r0) is proportional to r, hence the gradient of the geotherm will then be proportional to −r inside of r0. To relate the gradient to the geotherm we need to choose some reference radius where we keep the temperature ﬁx. By our convention stated above we choose then the ICB. We see then that concentrating the heat sources toward the centre of the core will increase the central temperatures, whilst moving them further toward the ICB will ﬂatten the geotherm.

We shall note another important aspect that follows from the linear behaviour of an integral with respect to its integrand. Given a complex source distribution that can be described as a sum of several simple source distributions, i.e. Q = Q1 + Q2 + Q3 + ... we can always split the integrals of eq.(7.2) into several integrals. Deﬁne the operator I(x) as 1 1 Z Z I(x) = xr2dr − xrdr (7.5) k r We can then write eq.(7.2) as X T (r) = B + I(Q) = B + I (Qn) (7.6) n and so the geotherm can be expressed as the sum of several simpler geotherms. Since we have argued for that any source distribution of the inner core has to be continuously differentiable and that it can not posses any singularities, we already know that we can express any inner core source distribution in the desired way. In addition we have argued that any source distribution function will be dominated by its zeroth term coefﬁcient. Using eq.(7.1) above we have that

X n T (r) = B + I(c0) + I (cnr ) ≈ B + I(c0) (7.7) n>0

And so the steady state gradient of the inner core geotherm in the presence of internal heat sources are to a ﬁrst order approximation proportional to −r, i.e. the geotherm is proportional to −r2.

We then move on to the response of the inner core geotherm to k, given some arbitrary source distribution. Again we denote the integrated heat sources inside of some radius r by P (r). Even if k is not a constant function of the radius of the sphere we shall assume it to spherically symmetric. Inspecting the Fourier’s law in spherical coordinates, we see that for a given source distribution, the thermal gradient at any radius is proportional to k−1 at that radius. Thus, assume that we change k at some radius, r0 of the sphere, this will only affect the geotherm gradient locally, or equivalently geotherm inside of r0 if the reference temperature is outside of r0 and vice versa. Hence, whilst the geotherm at r is affected by all heat sources inside of r it is only affected by k at r.

To conclude, since we have set the reference radius to the ICB, we need to know the source distribution over the whole inner core. But this we have argued for to be constant to a ﬁrst order approximation. If we are only interested in the temperature at some speciﬁc radius inside the core we only need to know k in between this radius and the ICB.

Figure 9 displays various relative geotherms for the inner core assuming evenly distributed heat sources and a constant heat transfer coefficient. Note that the shape of the profiles are proportional −r2 as discussed above, likewise we see the k−1 behaviour of the geotherms. Now we could expect k to increase somewhat with pressure and temperature, i.e. inwards. Hence from our study above, the given profiles should be considered to be upper

24 limits. If we assume the inner core contains 40K as only radionuclide present, with an abundance equal to the maximum allowed on cosmochemical grounds (0.142 wt%) the present inner core heat productions amounts to about 0.5 TW. Using the upper estimate of the thermal conductivity (80 W m−1 K−1) we ﬁnd a maximum temperature difference over the inner core of about 204 K, where as a thermal conductivity of 60 W m−1 K−1 would yield a maximum temperature difference of about 272 K.

(a) (b)

Figure 9: Relative geotherms of the inner core for evenly distributed heat sources with an integrated effect of P , and constant heat transfer coefficient, k. Figure 9(a) displays profiles for k = 80 W m−1 K−1 with Q varying in the interval [0 2] TW , figure 9(b) displays profiles for Q = 0.5 TW with k varying in the interval [20 100] W m−1 K−1, figure 9(c) shows the central relative temperatures for a number of models in P − k space, and figure 9(d) shows the temperatures at a radius of 500 km in P − k space. White solid lines superimposed on the surface indicate isotherms for every 100 K for the upper two figures, for every 150 K for the lower left figure and for every 50 K for the lower right figure.

The above used term evenly distributed heat sources refers to a constant amount of radionuclides per volume, i.e. the same amount of heat is generated in every cm3 of the entire core. However, what is commonly used in the literature is weight percentage, i.e. evenly distributed in that sense should then be constant per mass, i.e. the same amount of heat is generated in every kg of the entire core. We know that the density of the inner core increases inwards and so defining a constant weight percentage will mean an increased number of radionuclides per unit volume toward the centre of the core. And from our study above we know that this in turn implies a higher central temperature. On the other hand, as the volume of a spherical shell is proportional to the radius squared, this effect will be small unless the density changes drastically over the radius. Anyway, it can be instructive to investigate the difference between the two scenarios. Assuming the PREM model (see table 4 of appendix A) the density over the inner core increases by less than 2.6 percent. Using a heat transfer coefficient of 80 W m−1 K−1 and the PREM model density distribution we find that the temperature increase at the centre (where the effect is largest, see figure 10) is about 1.54 K or 0.03 % (assuming a temperature at the ICB of 5500 K) for the constant weight percentage scenario with an integrated inner core heat production of 0.5 TW. Increasing the heat production to 2 TW yields an increase at the centre of 6.16 K or 0.1 %. Given that the magnitude of the temperature change between the two models is so small, we conclude that our use of constant heat production per unit volume is acceptable.

25 (a) (b)

Figure 10: Temperature difference over the inner core between models assuming constant internal heat generation per unit volume and unit mass for k = 80 W m−1 K−1. Figure 10(a) shows the difference in degrees K whilst figure 10(b) shows the difference in percentage assuming the ICB temperature to be fixed at 5500 K. Note that the difference goes to zero just before the ICB, this means that both models has the same thermal gradient at the ICB which should be the case since both models generate an equal amount of heat inside of the ICB. White solid lines super imposed onto surface indicate isotherms for every 1 K in figure 10(a) and for every 0.02 % in figure 10(b)

In sect.6.5 I reviewed an alternative model of the core proposed by Herndon [60, 62], in which a georeactor of a few km radius would be operating at the centre of the core. I have shown above this would imply a very high temperature at the centre of the core, thus it is now time to investigate the thermal proﬁles of Herndon’s model.

Figure 11 displays some geotherms over the inner core for the Herndon georeactor model, as well as temperatures at the centre and at a radius of 500 km. The models assumes that all heat is being evenly produced inside a radius of 12 km of the inner core. Note how the temperature profiles outside of the georeactor drops off like r−2 as proven above and that the central temperatures are proportional to P/k. Clearly the resulting central temperatures are ridiculous. E.g. for the proposed georeactor output of 3.5 TW the central temperatures reach as high as 423 500 K for a thermal coefficient of 80 W m−1 K−1. Even if we increase the thermal coefficient to 300 W m−1 K−1 we still reaches a value of 112 930 K. Clearly something must be wrong.

We do not really know the thermal coefficient of nickel-silicide at core pressures and temperatures, but at room temperature and pressure it is about40 1/8 to 1/4 of that of Fe [98]. Let’s estimate the magnitude of the thermal conductivity needed to reach some reasonable central temperatures. From the known behaviour of the geotherm we find that the relative central temperatures are given by a function on the form ∆Tc(P, k) = aP/k. Using the −6 results of my modelling we find that a = 9.68 x 10 , hence assuming P = 3.5 TW and ∆Tc = 10 000 K, we find that we need to have a constant value of k of 3388 W m−1 K−1 through out the entire inner core. This is about 42 times the estimated thermal conductivity of the Fe at core pressures. It seems very unlikely that this should be the case. Now at high temperatures we should expect the thermal conductivity to increase, especially at the extreme temperatures found in Herndon’s model. But from our discussion above it should be clear that if we tried to lower the thermal conductivity to reasonable values near the ICB, the central temperatures would increase even further. To compensate for this we would then have to increase the thermal conductivity in the inner parts to values even higher than 3388 W m−1 K−1. Thus increasing the thermal conductivity in the inner core does not solve the problem.

At the temperatures found we should expect the centre of the inner core to be liquid and convective. Now, we did argue in section 7.1 that the gravitation is low in the inner core (see ﬁgure 7.1), and this will especially be true at the centre of the Earth. Therefore buoyancy forces would be small. Still, the tremendous temperatures would lead to convection in the liquid region. The convection has to be relatively fast and extend out to a radius of at least a few tenths (or even a few hundreds depending on model, see ﬁgure 11), of km from the core. Otherwise the temperature of the inner regions will still reach very high temperatures41. But this implies that the inner most

40As nickel-silicide is a group name for several Ni-Si compounds the exact value of the thermal conductivity of the inner core will be set by the exact composition. E.g. k(Ni3Si) = 18.2 W/m/K, k(NiSi) = 10.3 W/m/K 41This is naturally dependent on the heat capacity of nickel-silicide (45 J K−1 mol−1 at room temperature [1]), but as this is about the same

26 region of the core should be low viscous to allowed for an effective enough convection. This is on the other hand compatible with high temperatures. The big issue is then how the georeactor could have survived over time, and not have become distributed out over the convective region?

(a) (b)

Figure 11: Relative geotherms of the inner core assuming Herndon’s georeactor model. figure11(a) shows profile for various effects of the georeactor assuming k = 80 W m−1 K−1, figure 11(b) shows various models for different values of k, assuming a constant georeactor effect of 3.5 TW, figure 11(c) shows the central relative temperatures for a range of models in P − k space, and figure 11(d) shows temperatures at a radius of 500 km in P − k space. Isotherms are indicated by white solid lines at every 100 000 K for the upper two figures, extra isotherms has been added for the relative temperatures 500, 1000, 2000, 5000, and 10 000 K, for the lower left figure isotherms are displayed at an interval of 100 000 K with extra lines at 50 000 and 25 000 K, and for the lower right figure isotherms are displayed at an interval of 1000 K with extra lines at 500 and 250 K.

Unfortunately the central convection that must arise in Herndon’s model makes it impossible for us to find the central temperature from our steady state solid models, we can only conclude that they will be high. But what about the temperatures further out. Since we know the source distribution over the entire core, we can find geotherms for the outer part where temperatures are low enough for us to give a reasonable estimate of the thermal conductivity. It seems reasonable to believe that if the high pressure thermal conductivity of nickel-silicide is larger that of Fe, then it is at maximum a few times (say 3) larger. Let us then assume that we have a temperature span of a few thousands of degrees for ∆T with only negligible changes in the thermal conductivity, at what distance from the would we then find a relative temperature of say 2000 K, or maybe 5000 K?

Figure 12 displays Isothermal surfaces in the P − k space for T = 2000 K and T = 5000 K. We can see that even if we increase the thermal conductivity to 300 W m−1 K−1 and lower the georeactor output to 0.5 TW, the temperature in the core would be 5000 K at a radius of 26.8 km and 2000 K at a radius of 64.7 km , where as in Herndon’s scenario (for k = W m−1 K−1) we would have a temperature of 5000 K at about 162.3 km radius and 2000 K at about 336.7 km radius. So at what temperatures would then nickel-silicide melt in the inner core? At ambient pressures Ni2Si has about the same melting temperature [60] as the Fe-alloys that has been suggested for the core composition. Given that Ni and Fe have relatively similar properties and that Si also is a likely constituent

as for Fe, although this is not an assurance that so is the case also at high pressures and temperatures I assume this to be the case.

27 of the outer core, one should not expect the melting temperatures of nickel-silicide to differ much from that of the Fe-alloy of the outer core42, hence an inner core of nickel-silicide should be melted in regions where the temperature was more than 2000 K above Ticb. But this would then give an central liquid region of radius 380 km today. It seems very unlikely that this would still be undetected. Even if we assume that the inner core can survive in solid form to temperatures of 5000 K above Ticb the liquid region would still be very large and likely to be detected by seismics. Now even a georeactor output of 0.5 TW only, would still produce relatively large liquid regions.

(a) (b)

Figure 12: Isothermal surfaces in the P −k space for the georeactor model of Herndon with k = 80 W m−1 K−1. Left panel shows iso-surfaces at T = 2000 K and right panel at T = 5000 K. White solid lines indicates the distance to the centre for every hundred km, extra lines has been added for the distances 50 and 25 km.

Having said this two things needs to be realised. First of all the central parts of the inner core are not easily accessed. The only reliable data with sensitivity to these regions are absolute travel times from antipodal distances, which ammounts to about 3000 unevenly spread measurments. Secondly the central parts of the inner core does display some peculiarities. In a study by Ishii and Dziewonski [66], using the availible data set, indications was found of seismically distinct region at the center of the inner core, with a radius of 300 km. ”although higher quality data are required to draw a ﬁrm conclusion.” Note however, that this is a solid region, although if one whished to support the Herndon model one could interpret this as evidence of a different chemistry, or perhaps the remnants of a melt that solidiﬁed inside the inner core.

Turning our attention back to the geotherms of the mainstream models we wish to ﬁnd ways to constrain these. As in the case of Herndon’s models we shall try to do so again by looking at the melting temperature of the inner core material.

7.3 Constraining the inner core temperature proﬁle

From eq.(7.3) above we can compute the relative temperature profile of the inner core, however we wish to somehow constrain the possible set of temperature profiles of the inner core. If we assume that the inner core is solid throughout we can use the melting temperature in the inner core to put an upper limit on the possible temperature profiles. At the high temperatures and pressures present of the core we can assume Debye theory to hold43. We can then estimate the melting temperature over the inner core from the Lindemann law of melting (where we for simplicity have assumed q = 1, see eq.(C.45) of appendix C.6)

ρ 2/3 ρ T (r) = T ICB exp 2γ 1 − ICB (7.8) m m,ICB ρ(r) th,ICB ρ(r)

42It is known from high pressure experiments [47] that at least up to 60 GPa the melting temperature of elemental Ni is lower than that of elemental Fe. 43The molar volume of hcp-Fe is expected to be less than 4.8 cm3/mol at inner core pressures [10], which implies that debye theory holds, see appendix C.3

28 The subscript ICB indicates quantities measured at the ICB, and γth is the thermodynamical gruneisen¨ parameter. It should be recognized that the Lindemann law of melting is derived for elemental substances and should be applicable for Fe at core pressures. The inner core is not pure Fe but rather a Fe-Ni mixture including some lighter elements, and this will affect the melting temperature toward lower temperatures. Still experiments has shown that the Lindemann law is applicable if the molecule complexity is low [105], and so as the core predominately consists of Fe with Ni and some lighter elements as minor constituents, we will assume that the Lindemann law is valid.

As we do not know TICB with acceptable accuracy we would like to work with relative temperatures as we did with the temperature proﬁles. However note from eq.(7.8) that we can not express the Lindemann law in terms of relative temperatures. We could express it as a ratio, but this is on the other hand not possible with the temperature distribution (see eq.(7.3)). In addition we have that the melting law is dependent on the gruneisen¨ parameter. Let us therefore start by examining the behaviour of the Lindemann law. For the density proﬁle we shall use the

PREM model (see sect.3.3 above). To generalise we form the dimensionless melting temperature ratio QTm

Tm(r) Qtm (r) = (7.9) Tm,ICB

As can be seen in figure 13, for a reasonable range of γth,ICB values, QTm (r) can be assumed to display a linear behaviour with respect to γth,ICB with slope given by figure 14. Also note that the curves could be relatively well 44 2 fitted by a function QTm = a − br , where a and b are linear functions of γth,ICB.

(a) (b)

Figure 13: Melting temperature ratios, QTm (r) = Tm(r)/Tm,ICB , for various values of γth, left panel shows proﬁles over the inner core for γth = 1.2 to 1.8 in steps of 0.1, right panel shows the ratio between melting temperatures at the ICB and at the centre of the Earth, as a function of γth. The model assumes PREM densities.

Returning back to the inner core geotherm we see that if the geotherm at some radius, r0, coincides with the 0 0 melting temperature, i.e. TIC (r ) = Tm(r ), the temperature gradient locally has to be proportional to the radial coordinate to the power of at least 1, i.e.

0 0 0 dTIC (r ) n if TIC (r ) = Tm(r ) ⇒ ∝ r ; n ≥ 1 dr r=r0±

where is some small number. The reason being that dTm/dr ∝ r, and so if the condition above is not fulﬁlled 0 0 TIC (r + ∆r) > TIC (r + ∆r) and the inner core will re-melt. Note that the criterion covers nearby regions on both sides of r0. As argued above, the likely inner core distribution of radionuclides should be constant, which was shown to imply a thermal gradient proportional to r, therefore fulﬁlling our criterion. Consider then if the geotherm coincides with the melting temperature at some radius. We then have two possibilities, since both the

44 2 This is so since we have used the PREM density, which can be described by an analytical function on the form ρ(r) = ρc − dr (see 2 2 table 5) QTm = a − br of appendix A). Thus QTm = a − br is simply the first terms in a Taylor series expansion. Using figure 13 and −8 2 −4 14 one can find the fit to be QTm = 0.9806 + 0.0525γth,ICB − 10 (−1.3035 + 3.5273γth,ICB )r , which is accurate to within 10 for γth,ICB ∈ [1.2 1.8]. N.B. r is in units of km.

29 geotherm gradient and the melting temperature gradient are proportional to r the temperatures either can coincide at one radius only or at all radius. If they coincides at one radius only this can only be at the ICB, otherwise we can not have a solid inner core, or alternatively the whole core has to be solid. Since we are interested in constraining the possible distributions of heat sources in the inner core, we are interested in the case where the geotherm coincides with the melting temperature at the centre. Recognizing this, we can then use ﬁgure 13(b) and 9(c) to constrain the possible total heat generation of the inner core.

Figure 14: dQTm /dγth as a function of radius in the region γth ∈ [1.21.8]

(a) (b)

Figure 15: Maximum inner core heat production. ∆T is the difference between the geotherm and the inner core melting temperature at the ICB. Upper left figure displays maximum inner core heat production assuming evenly distributed heat sources, TICB = 5500 K, and γth,ICB = 1.5. Upper right figure displays the change in % if we lower γth,ICB by 0.1. Lower left figure displays the percentage change if we lower TICB by 500 K, and lower right figure displays the percentage change if we lower TICB by 500 K and γth,ICB by 0.1.

Figure 15(a) displays the maximum inner core heat production as constrained by the Lindemann melting law, −1 −1 assuming evenly distributed heat sources, TICB = 5500 K, and γth,ICB = 1.5. We see that for k = 80 W m K , the pressure induced freezing model has an upper limit of about 0.80 TW, or about 105.3 W m−3, whilst allowing

30 for difference of 100 K between the geotherm and the inner core melting temperature at the ICB, increases this to about 0.98 TW. From the other ﬁgures of ﬁgure 15 we also see that lowering TICB by 500 K, or γth,ICB by 0.1 units, lowers the possible inner heat production by about 9.5 % for the pressure induce freezing model, whilst the effect is about 7 % if the difference between the geotherm and the inner core melting temperature at the ICB is 100 K.

However, we need to realize that the inner core heat productions presented in figure 15 are primordial in the sense that they lack the time aspect of inner core formation. Only if the inner core were very young, the current day heat generation could correspond to that constrained by the melting temperature, other wise more energy would have been produced at earlier times, increasing the temperature above the melting temperature. As we discussed in sect. 7.1 above we need to compensate for the decay of the radionuclides present. We also came to the conclusion that the effect of decay would be to smooth the source distribution toward an evenly distributed (see figure 7.1). If we integrate equation (7.2) above we find that the temperature difference between the ICB and the centre at any time t is equal to Qe−λt ∆T = R2 (t) (7.10) 6k ICB Where we have taken into account the effect of decay of the radio nuclides via the decay constant, λ, and the time dependence of the radius of the inner core. The melting temperature can within a reasonable interval of the gruneisen¨ parameter be approximated by (see footnote 44)

−14 2 Tm = Tm,ICB 0.9806 + 0.0525γth,ICB − 10 (−1.3035 + 3.5273γth,ICB)r (7.11)

Where Tm,ICB is the melting temperature at the inner core boundary today. Hence we ﬁnd that the temperature difference between the ICB at time t and the centre of the Earth is given by

−14 2 ∆Tm = Tm,ICB ((−1.3035 + 3.5273γth,ICB)) 10 RICB(t) (7.12)

Thus for all t it has to hold that Qe−λt R2 (t) < T (−1.3035 + 3.5273γ ) 10−14R2 (t) 6k ICB m,ICB th,ICB ICB −λt −14 (7.13) ⇔ Qe < 6kTm,ICB (−1.3035 + 3.5273γth,ICB) 10 −14 ⇒ Q < 6kTm,ICB (−1.3035 + 3.5273γth,ICB) 10

Where the last step follows since the left hand side is at maximum for t = 0. This relation can be used in two ways, if we know the age of the inner core we use it to find an upper limits on the present day heat generation inside the inner core. On the other hand, if we know the present day inner core heatsources we can use it to put an upper limit to the age of the inner core. Figure 16 displays both wievpoints. From figure 16(a) we see that for the inner core to meet the max abundance as set by cosmochemical arguments, the inner core has to be younger than 0.65 Gyr, whilst the upper limits as set by high pressure experiments demands an inner core older than 2.27 Gyr. Plots over the maximum initial heat production at onset of inner core solidification can be found in figures 19(a) - 19(b)

So what about the scenario where the inner and outer core alloys display different melting temperatures at equal pressures. Well, unless the inner core started to grow from a super-cooled solution this will not alter the constraining amounts of heat sources, as given by equation (7.13), since the inner core alloy will start to separate from the liquid core in solid form, as soon as the temperature has dropped below its melting temperature. Thus if the heat sources does not full ﬁll equation 7.13 the inner core alloy will heat up and re-melt. The only difference will be that Tm in equation 7.13 will be substituted by the actual temperature at the ICB, which is bounded in between the melting temperatures of the inner and outer core.

31 (a) (b)

Figure 16: Maximum inner core age for various present day 40K abundances assuming k = 80 W m−1 K−1 (left) and maximum present day inner core heat production for various ages and thermal conductivities of the inner core (right). The plots assumes a gruneiesen¨ 40 parameter, γth, of 1.5, a temperature, TICB , of 5500 K at the ICB, and K as only heat source. Changes of the presented values due to different values of γth and Tm can be found in ﬁgures 15(b), 15(c), and 15(d). Iso-lines for every 0.1 TW are superposed onto the surface in ﬁgure 16(b).

7.4 Transient models

So how god are then steady state models for the inner core geotherm? To investigate this I constructed a simple transient diffusion model of the inner core, following the recipe given by Yukutake [147]. The model assumes that the inner core grows by pressure induced freezing, i.e. the ICB temperature will be set by the melting temperature at the ICB pressure. It is further assumed that the volume of the inner core has increased at constant speed (see appendix E for a more thorough description of the model). The final assumption of the model is that volumetric decrease of heat sources inside the inner core is only due to radiogenic decay, implying that the partition coefficient of present radio nuclides is constant over the inner core pressures and that the volumetric amount of radio nuclides in the outer core is unaffected by the partitioning into the inner core. Let us start by assuming a model with present day ICB temperature of 5500 K, a thermal conductivity of 80 W m−1 K−1, a heat capacity of 860 J kg−1 K−1, a mean density of 12909 kg m−3, and a gruneisen¨ parameter at the ICB of 1.5. For simplicity we shall consider 40K as the only heat source. From figure 15(a) or eq. 7.13 we find that the maximum initial amount of heat sources in the inner core at time of formation equals about 105.3 W m−3.

Figure 17 displays the evolution of the inner core geotherm assuming an age of 1.5 Gyr and an initial heat production of 105.3 W m−3. Also displayed in figure 17 are the equivalent steady state model, the difference between the steady state and transient model, and the evolution of the ICB heat flux of the models. In the other extreme we find an inner core void of heat sources. Figure 18 displays the equivalents of figure 17 of the no internal heat sources case, using the same properties as the first model. Clearly we have a discrepancy between the transient and steady state model. For the no internal heat sources case this amounts to about 115 degrees at the centre of the Earth, whilst for the model with maximum initial amount of heat sources the difference amounts to about 90 degrees.

32 (a) (b)

Figure 17: Evolution of inner core geotherm, assuming a present day ICB temperature of 5500 K, a thermal conductivity of 80 W m−1 K−1, a heat capacity of 860 J kg−1 K−1, a mean density of 12909 kg m−3, and a gruneisen¨ parameter at the ICB of 1.5. The model further assumes an inner core age of 1.5 Gyr and 40K as only heat source, with initial heat production of 105.3 W m−3. Figure 17(a) displays the resulting geothermal gradient, figure 17(b) displays the equivalent steady state model, figure 17(c) displays the difference between the transient and the steady state model, and figure 17(d) displays the evolution of the ICB heat flow for both models. Note that direction of the time axes differs in the figures. Also note the initial constant appearance of the heat flow of the transient model. This is due to the fact that the temperature profile is not being evolved at t = 0

In terms of absolute temperatures the deviations between the steady state models and the transient models found in figure 17 and 18 are small. However, absolute temperatures are not an appropriate scale for this study, e.g. consider a ICB temperature of 5500 K, a temperature difference of 330 degrees then amounts to 6 % in absolute temperatures which is relatively small. On the other hand assuming γ = 1.5, we have that the melting temperature difference differs by about 326 degrees over the inner core for TICB = 5500. And so an a positive error of 6 % in absolute temperatures, leads to an inner core that can not be solid (as the lowest temperature is reached at the ICB temperatures internal of this has to be higher). Instead of absolute temperatures we need to work with relative temperatures (relative to the ICB temperature), and so the error immediately grows to unacceptable levels. In the upper limit of internal heat generation we then find that an error in the steady state temperatures of about 61 % at the centre of the Earth, whilst in the no internal heat sources model the error approaches infinity. Thus, the presented steady state inner core geotherms are not good enough.

Note however that as for the steady state geotherms, also the transient geotherms are approximately proportional45 to −r2 . Therefore, in terms of inner core geotherms, it sufﬁces the give the central and ICB temperature since the behaviour of the geotherm in between is known. Figure 19 below displays maps of the central and ICB temperatures of the inner core for several combinations in the thermal parameter space. This does on the other hand not alter the criteria initial amount of heat sources allowed in the inner core as given by relation (7.13). Even if the initial derivation assumes a steady state geotherm, the ﬁnal expression has to apply to all t, and especially at t = 0, where right hand term is at its largest value. Now at t = 0, the steady state geotherm is equivalent to the transient geotherm, and therefore relation (7.13) also has to hold for the transient model. Thus the maximum

45for one thing this can be realised from the difference between the steady state and transient models, as this also is proportional to −r2

33 initial amounts of heat sources are model independent.

(a) (b)

Figure 18: Evolution of inner core geotherm, assuming a present day ICB temperature of 5500 K, a thermal conductivity of 80 W m−1 K−1, a heat capacity of 860 J kg−1 K−1, a mean density of 12909 kg m−3, and a gruneisen¨ parameter at the ICB of 1.5. The model further assumes an inner core age of 1.5 Gyr and no internal heat sources. Figure 18(a) displays the resulting geothermal gradient, figure 18(b) displays the equivalent steady state model, figure 18(c) displays the difference between the transient and the steady state model, and figure 18(d) displays the evolution of the ICB heat flow. Note that direction of the time axes differs in the figures. Also note the initial decrease of the heat flow of the transient model. This is due to the fact that the temperature profile is not being evolved at t = 0

Another thing to notice from figure 17(d) and 18(d) are the elevated ICB heat flows, due to the elevated geotherms. In the no heat source case we find that given the assumed model parameters the ICB heat flow is about 0.23 TW where as in the maximum initial heat source model the heat flow displays an additional 0.17 TW as compared to the steady state model, reaching a present day value of about 0.51 TW. This can be compared to the release of latent heat, L, at the ICB, adopting a value of 118 J kg−1 K−1 [79] for the entropy change, ∆S, upon inner core 28 solidification, we find that over the life time of the inner core a total of Ltot = ∆S MIC T¯m ≈ 6.4 x 10 J has been released at the ICB (where MIC is the mass of the present inner core and T¯m is the mean melting temperature, assumed to be 5500 K). Assuming an inner core age of 1.5 Gyr we then find an upper limit to L as 1.35 TW at present, thus the possible contribution from radiogenic heat sources inside the inner core ranges in the interval of 13 - 37 %. The exact number will naturally change if we consider other ages for the inner core or different inner core thermal parameters, but will generally be of the same order.

From the relative temperatures displayed in ﬁgure 19 we can conclude that the relative temperatures of the inner core are highly sensitive to the Age and thermal conductivity of the inner core with lower relative temperatures for higher values of both parameters. On the other and the relative temperatures are relatively insensitive to the heat capacity of the inner core, as well as to some extent the thermodynamical gruneisen¨ parameter, with lower relative temperatures for lower parameter values. Note however from the deﬁnition of the thermodynamical gruneisen¨ parameter (see equation C.26) that this is some what contradictory to what is expected since γth is inversely proportional to Cp.

34 (a) (b)

(e) (f)

(g) (h)

35 (i) (j)

(k) (l)

(m) (n)

(o) (p)

36 (q) (r)

Figure 19: Plots of possible present day inner core central relative temperatures in the subspaces of the thermal parameter space. Figures 19(a) - 19(c) displays upper constraints to the initial abundance of inner core 40K. Default values assumes an 40 −1 −1 initial abundance of inner core K A = 0.243 wt%, a thermal conductivity k = 80 W m K , a heat capacity Cp −1 −1 = 860 J kg K , the melting temperature at the inner core boundary Tm 5500 K, the thermodynamical gruneisen¨ parameter γth = 1.5, and an Inner core age of 1.5 Gyr. Estimated numerical errors in the given relative temperatures are of th order of 2 K. Note that for high values of Tm, γ, and k, the relative temperatures displayed exceeds the melting temperature of the inner core, as an artefact of A being dependent on these parameters.

8 Summary and Discussion

Can we then understand the difference between the steady state models and the transient models. A Commonly referred to concept when dealing with diffusion is the diffusion time, τ, which gives a measure of the timescales involved for a perturbation to propagate through the body. For the spherical geometry of the inner core the diffusion time can found from the transient solution of the heat equation (see equation (E.5) in appendix E)

r2 τ = m (8.1) κπ2

where κ = k/(ρcp). More precisely, it can be shown that for a cooling sphere, where the initial temperature proﬁle is proportional to the radius squared, that this gives the time for temperature difference between the centre and the boundary to have dropped to about 45 % of its original value in the absence of internal heat sources. For the present day inner core and the thermal parameters given in table 5.2, τ is of the order of 0.5 to 0.8 Gyr, which is shorter than, or of the order of, the various estimated ages of the inner core. However, for the inner core τ is not a good measure by three reasons. First of all the radius of the core is not constant and so the diffusion time will change over time, becoming longer as the inner core grows. The effect is dependent on the growth history of the ICB radius, but will generally lower the temperature difference between the ICB and the centre. Note that a growing radius will not affect the present day steady state temperature (since the present day ICB radius is not a variable), and so the difference between a steady state model and a transient model will be lower as compared to a core with a non-evolving radius.

Also note that a rapid initial growth of the inner core (as argued for in section 7.4) will increase the ICB-Centre temperature difference as compared to a slow initial growth. In fact, as the thermal diffusion time scales as the radius squared a growth of ICB radius proportional to tn, where n < 1/2 will give that the age of the inner core at early times is lower than the diffusion time. This is the case for assumed growth history of the inner core, presented in this paper. E.g. In the case of an inner core age of 1.5 Gyr the diffusion time is less then the age of the inner core until the ICB radius has reached a size of about 540 km (or equivalently an age of 0.13 Gyr, see ﬁgure 20(a)), where as a greater age of the inner core would lower these values.

The second reason is that the ICB temperature is not constant, reaching lower values as time goes on. The heat ﬂow is proportional to the local temperature gradient. Hence if the temperature decreases slower at the centre than the ICB temperature, this means that absolute temperature drop in degrees is less at early times. However

37 if central temperatures decreases faster the opposite will occur. Thus the response is dependent on the diffusion time, and can both increase and decrease the difference between the steady state and the transient model. Note that this will not affect the steady state geotherm (since the present day ICB temperature is not a variable).

(a) (b)

Figure 20: Figure 20(a) displays the diffusion time and age of the inner core for an inner core growth at constant volume increase, assuming −1 −1 a present day age of 1.5 Gyr. Figure 20(b) displays the temporal evolution of the same model assuming k = 80 W m K , cp = 860 J kg−1 K−1, ρ = 12909 kg m−3, and no internal heat sources.

Since the diffusion time is coupled to the growth history of the inner core we have that the two effects accounted for above really are coupled46.. In the model used here it holds that even if the inner core is as old as the Earth itself, the central temperature has decreased at a slower rate then the ICB temperature. Which can also be understood from a physical point of view, given that the inner core grows as a result of a pressure induced elevation of the inner core alloy melting temperature (pressure decreases outwards) and the cooling of the outer core. Hence the effect of a growing sphere is to some extent opposed by the effect of a dropping ICB temperature. In fact, assuming that the core initially was molten, the inner core must have started from an almost isothermal state. So since the diffusion time constantly increases, whilst the ICB temperature constantly decreases the difference between the ICB temperature and the temperature at the centre of the Earth will continue to increase (at least until the entire core has solidiﬁed and the ICB temperature is approaching the temperature of interplanetary space).

Finally we have come to the last but most important aspect. The question of the presence of heat sources in the inner core. In the presence of radiogenic heat sources the solution to the heat equation gains an additional time dependent part (see equation (E.6) in appendix E) containing the decay term, e−λt, which makes it in principle impossible to derrive an analytical expression for the diffusion time. However, if we assume a sphere containing radio nuclides, initially at steady state, we can ﬁnd a measure of the diffusion time as

r2 t τ = m + 1/2 (8.2) κπ2 log(2)

Where t1/2 is the half life of the radio nuclide. Hence, if we assume the inner core radiogenic heatsources to be 40K, the diffusion time increases by approximately 1.8 Gyr, yielding diffusion times larger than the inner core age throughout the evolution of the inner core. From equation 8.2 we can then understand the orign of the discrespancy between the transient and steady state models. If we are to speak of a diffusion time of the steady state models this would equal t1/2, whereas in the transient case the diffusion time is t1/2 multiplied by a factor of 1/log(2) ≈ 1.44 plus the diffusion time of the equivalent sphere with no internal heatsources. And so the difference between the steady state and transient model will grow over a time equal to the longer of the two diffusion times (i.e. the

46Note that unless the inner core contains heat sources the growth of the inner core alone will not change the evolution of the inner core geotherm, it is the dropping ICB temperature that changes the inner core geotherm. This follows from the assumption of an initially molten inner core, hence the initial inner core geotherm upon solidiﬁcation will be isothermal, so if the core can grow with constant ICB temperature the inner core geotherm will stay isothermal as will the steady state geotherm. However, this is a purely philosophical argument as the whole Earth is cooling and so also the inner parts (other wise the inner core would not grow in the ﬁrst place)

38 diffusion time of the transient model). In addition the combined effect of a growing radius and decreasing ICB temperature will increase the difference between the models.

Steady state models has also been used to study the thermal effects of the georeactor scenario suggested by Herndon, as well as the suggested nickel-silicide composition of the inner core. Now as we have spent a lot of time to downgrade the steady state models above this needs some explanation. The major result from our comparison between that transient model and the steady state models is that the steady state models will always underestimate the geotherm in the case of diffusion only, thus we can use the steady state models to put a lower bound on the geotherms resulting from a proposed model.

In Herndon’s models it was found that the suggested output of the georeactor (3 TW + 0.5 TW from radiogenic deccay) would imply a large molten central region of the inner core. Even if we lowered the georeactor output to 0.5 TW we still end up with a relatively large molten central region. However, it should be noted that for the self sustained georeactor proposed, 3 TW is about the lower bound, and so lowering the ammounts of radionuclides would effectively halt the proposed reactor and leave us with the 0.5 TW from radiogenic deccay. In addition the dependence of the georeactor on gravitational migration, initially for the formation of the actinide sub core, and later for the removal of decay products from the georeactor, is highly unlikely, due to the low gravitational acceleration at the centre of the Earth. A problem with my models lies in the fact that no information has been found for the high pressure, high temperature behaviour of nickel-silicide. But a comparison with elemental Fe at shows similar behaviour at ambient conditions, hence we should not expect any major differences at core pressures. In fact from the comparison we might actually expect the relevant parameters (e.g. thermal conductivity and melting temperature) to be slightly lower than those of Fe. However, given the fact that the central 300 km of the inner core appears to display a distinct seismological anisotropy, one could argue that a georeactor once was operational in the interior. But we can atleast conclude that it is highly unlikley that a georeactor is operational at the center of the inner core at present.

In addition, given the low gravitational ecceleration in the inner core and entropy driven diffusion it seems very unlikely that a georeactor could have formed in the ﬁrst place.

Finally we need to emphasise that the transient model used here is really a very simple model, with its own ﬂaws. For one thing the growth of the inner core has to be imposed onto the model. But, as the growth of the inner core releases heat into the outer core, both from cooling as well as from release of latent heat, this will affect the heat budget of the outer core, which in turn controls the growth of the inner core. For simplicity we have assumed that the inner core has grown by a constant volume increase, however, a careful analysis might give at hand that this is not the case. So as the growth history affects the resulting relative temperatures the presented geotherms are likely to change if another growth history is chosen. It is possible to relatively easy implement a different growth rate into the model although this has not been done, given the level of this work.

Another obvious ﬂaw of the transient model is assumption of diffusion only. As it turns out, if we assume that the inner core started to grow at maximum initial intenal heatproduction, as given by equation 7.13, the models predict that the entire inner core would be super-adiabatic if younger than 1.55 Gyr (see ﬁgure 21). Only if the inner core is older than about 2 (2.5) Gyr it will be fully sub-adiabatic through out its interior. Although we should be aware that super adiabatic conditions inside a solid does not neccesary means that (solid state) convection will occur. However it needs to be remembered that this work is not trying to claim that the age of the inner core is coupled to the possible radiogenic abundance, it merely investigates the upper bounds on the possible inner core heat source abundance as constrained from a thermodynamical point of view. In any case the presented transient geotherms can be considered as upper bounds, as the actual geotherm can not be higher.

Further ﬂaws with the used transient model involves the assumption of constant thermal parameters, as well as neglecting the density variation (we have instead used the mean density) of the inner core when evolving the geotherms. However this is not expected to have a great inﬂuence on the geotherms.

Finally it should be recognised that the model does not fully conserve energy (as discussed in appendix E.4), translated into a mean temperature this accumulates to about +2 K over the full life time of the model, which

39 in the light of the models simplicity is considered acceptable. It is believed that the problem resides in a slight numerical error near and on the boundary of the domain (i.e. than ICB), giving rise to slightly underestimated heat ﬂow. On the other, the internal temperatures has shown to be very robust, thus for the given model the geotherms can be considered to be good in most parts, whilst some precaution needs to taken when considering the gradient at the boundary. This is also the reason why I have chosen not to put to much emphasis on the ICB heat ﬂow in the results.

(a) (b)

Figure 21: Comparison of the inner core geothermal gradient vs the adiabatic gradient (see equation C.42). For various ages of the inner core. Positive values means sub-adiabatic conditions and negative values menas super-adiabaic conditions. The models assumes −1 −1 an initial heat production at onset of inner core solidiﬁcation in accordance with equation 7.13, and k = 80 W m K , cp = 860 J kg−1 K−1, ρ = 12909 kg m−3, and 40K as only heat source. Left panel displays difference vs. the inner core age and right panel displays the same ﬁgure but plotted against the resulting present day 40K abundance.

9 Acknowledgements

Thanks goes out to Professor Peter Lazor at the department of Earth science/solid Earth geology at Uppsala university, for giving me the opportunity to do this work, as well as being patient with me when the work dragged out on time. Secondly, recongnition goes to associate Professor Christoph Hieronymus at the department of Earth science/his help Geophysics at Uppsala university, for his role as co-supervisor for this work. Tuna Eken is recognised for the help with various articles concerning seismic imaging/probing of the core. Dario‘ Alfe and Jean-Paul Poirier is thanked for cleariﬁcations of parts of their publications. Frank D Stacey is thanked for some mail correspondance as well as sending me a preprint of his often cited 1995 paper [129], which I could not access other wise. recognision also goes out to my friends Peter Dahlin (Geologist) and Zuzana Konopkova (Phd-student in solid Geochemistry) for comments regarding parts of the content of this work and spell checking. Finally I would like to thank the litle green monkey occupying the space under my bed for not storing ”Knackebr¨ od”¨ under my pillow anymore.

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44 A PREM

Region Radius Vp Vs ρ Ks µ ν P g [km] [m/s] [m/s] [kg/m3] [GPa] [GPa] [Gpa] [m/s2] 0.0 11262.20 3667.80 13088.48 1425.3 176.1 0.4407 363.850 0.0000 200.0 11255.93 3663.42 13079.77 1423.1 175.5 0.4408 362.900 0.7311 400.0 11237.12 3650.27 13053.64 1416.4 173.9 0.4410 360.030 1.4604 600.0 11205.76 3628.35 13010.09 1405.3 171.3 0.4414 355.280 2.1862 800.0 11161.86 3597.67 12949.12 1389.8 167.6 0.4420 348.670 2.9068 1000.0 11105.42 3558.23 12870.73 1370.1 163.0 0.4428 340.240 3.6203 1200.0 11036.43 3510.02 12774.93 1346.2 157.4 0.4437 330.050 4.3251 Inner core 1221.5 11028.27 3504.32 12763.60 1343.4 156.7 0.4438 328.850 4.4002 1221.5 10355.68 0.00 12166.34 1304.7 0.0 0.5000 328.850 4.4002 1400.0 10249.59 0.00 12069.24 1267.9 0.0 0.5000 318.750 4.9413 1600.0 10122.91 0.00 11946.82 1224.2 0.0 0.5000 306.150 5.5548 1800.0 9985.54 0.00 11809.00 1177.5 0.0 0.5000 292.220 6.1669 2000.0 9834.96 0.00 11654.78 1127.3 0.0 0.5000 277.040 6.7715 2200.0 9668.65 0.00 11483.11 1073.5 0.0 0.5000 260.680 7.3645 2400.0 9484.09 0.00 11292.98 1015.8 0.0 0.5000 243.250 7.9425 2600.0 9278.76 0.00 11083.35 954.2 0.0 0.5000 224.850 8.5023 2800.0 9050.15 0.00 10853.21 888.9 0.0 0.5000 205.600 9.0414 3000.0 8795.73 0.00 10601.52 820.2 0.0 0.5000 185.640 9.5570 3200.0 8512.98 0.00 10327.26 748.4 0.0 0.5000 165.120 10.0464 3400.0 8199.39 0.00 10029.40 674.3 0.0 0.5000 144.190 10.5065 Outer core 3480.0 8064.82 0.00 9903.49 644.1 0.0 0.5000 135.750 10.6823 3480.0 13716.60 7264.66 5566.45 655.6 293.8 0.3051 135.750 10.6823 3600.0 13687.53 7265.75 5506.42 644.0 290.7 0.3038 128.710 10.5204 D”-Layer 3630.0 13680.41 7265.97 5491.45 641.2 289.9 0.3035 126.970 10.4844 3630.0 13680.41 7265.97 5491.45 641.2 289.9 0.3035 126.970 10.4844 3800.0 13447.42 7188.92 5406.81 609.5 279.4 0.3012 117.350 10.3095 4000.0 13245.32 7099.74 5307.24 574.4 267.5 0.2984 106.390 10.1580 4200.0 13015.79 7010.53 5207.13 540.9 255.9 0.2957 95.760 10.0535 4400.0 12783.89 6919.57 5105.90 508.5 244.5 0.2928 85.430 9.9859 4600.0 12544.66 6825.12 5002.99 476.6 233.1 0.2898 75.360 9.9474 4800.0 12293.16 6725.48 4897.83 444.8 221.5 0.2864 65.520 9.9314 5000.0 12024.45 6618.91 4789.83 412.8 209.8 0.2826 55.900 9.9326 5200.0 11733.57 6563.70 4678.44 380.3 197.9 0.2783 46.490 9.9467 5400.0 11415.60 6378.13 4563.07 347.1 185.6 0.2731 37.290 9.9698 5600.0 11065.57 6240.46 4443.17 313.3 173.0 0.2668 28.290 9.9985 5600.0 11065.57 6240.46 4443.17 313.3 173.0 0.2668 28.290 9.9985 Lower mantle 5701.0 10751.31 5945.08 4380.71 299.9 154.8 0.2798 23.830 10.0143 5701.0 10266.22 5570.20 3992.14 255.6 123.9 0.2914 23.830 10.0143 5771.0 10157.82 5516.01 3975.84 248.9 121.0 0.2909 21.040 10.0038 5771.0 10157.82 5516.01 3975.84 248.9 121.0 0.2909 21.040 10.0038 5871.0 9645.88 5224.28 3849.80 218.1 105.1 0.2924 17.130 9.9883 5971.0 9133.97 4932.59 3723.78 189.9 90.6 0.2942 13.350 9.9686 5971.0 8905.22 4769.89 3543.25 173.5 80.6 0.2988 13.350 9.9686 6061.0 8732.09 4706.90 3489.51 163.0 77.3 0.2952 10.200 9.9361 Transition zone 6151.0 8558.96 4643.91 3435.78 152.9 74.1 0.2914 7.110 9.9048 6151.0 7989.70 4418.85 3359.50 127.0 65.6 0.2797 7.110 9.9048 6221.0 8033.70 4443.61 3367.10 128.7 66.5 0.2796 4.780 9.8783 Low velocity Zone 6291.0 8076.88 4469.53 3374.71 130.3 67.4 0.2793 2.450 9.8553 6291.0 8076.88 4469.53 3374.71 130.3 67.4 0.2793 2.450 9.8553 LID 6346.6 8110.61 4490.94 3380.76 131.5 68.2 0.2789 0.604 9.8394 6346.6 6800.00 3900.00 2900.00 75.3 44.1 0.2549 0.604 9.8394 6356.0 6800.00 3900.00 2900.00 75.3 44.1 0.2549 0.337 9.8332 6356.0 5800.00 3200.00 2600.00 52.0 26.6 0.2812 0.337 9.8332 Crust 6368.0 5800.00 3200.00 2600.00 52.0 26.6 0.2812 0.300 9.8222 6368.0 1450.00 0.00 1020.00 2.1 0.0 0.5000 0.300 9.8222 Ocean 6371.0 1450.00 0.00 1020.00 2.1 0.0 0.5000 0.000 9.8156

Table 4: Excerpt from the PREM model table for a reference period of 1 s.

45 Region Radius ρ Vp Vs Qµ QK [km] [kg/m3] [m/s] [m/s] Inner core 0- 13088.5 11262.2 3667.8 0.0846 1.3277 1221.5 -8838.1x2 -6364x2 -4447.52

Outer core 1221.5- 12581.5 11048.7 0 0 57.823 3480.0 -1263.8x -4036.2x -3642.6x2 +4802.3x2 -5528.1x3 -13573.2x3

Lower Mantle 3480.0- 7956.5 15389.1 6925.4 0.312 57.823 3630.0 -6476.1x -5318.1x +1467.2x +5528.3x2 +5524.2x2 -22083.4x2 -3080.7x3 -2551.4x3 +978.3x3

3630.0- 7956.5 24952.0 11167.1 0.312 57.823 5600.0 -6476.1x 40467.3x 13781.8x -5528.3x2 +51483.2x2 +17457.5x2 -3080.7x3 -26641.9x3 -9277.7x3

5600.0- 7956.5 29276.6 22345.9 0.312 57.823 5701.0 -6476.1x -23602.7x 17247.3x +5528.3x2 +5524.2x2 -2083.4x2 -3080.7x3 -2551.4x3 +978.3x3

Transition 5701.0- 5319.7 19095.7 9983.9 0.143 57.823 zone 5771.0 -1483.6x -9867.2x -4932.4x

5771.0- 11249.4 39702.7 22351.2 0.143 57.823 5971.0 8029.8x -32616.6x -18585.6x

5971.0- 7108.9 20392.6 8949.6 0.143 57.823 6151.0 -3804.5x -12256.9x -4459.7x

LVZ* vertical 6151.0- 2691.0 831.7 5858.2 0.080 57.823 6291.0 +692.4x +7218.0x -1467.8x

horisontal 3590.8 -1083.9 +4617.2x +5717.6x

LID* vertical 6291.0- 2691.0 831.7 5858.2 0.600 57.823 6291.0 +692.4x +7218.0x -1467.8x

horisontal 3590.8 -1083.9 +4617.2x +5717.6x

Crust 6346.6- 2900 6800 3900 0.600 57.823 6356.0

6356.0- 2600 5800 3200 0.600 57.823 6368.0

Ocean 6368.0 1020 1450 0 0 57.823 6371.0

Table 5: Coefﬁcients of the polynomials describing the PREM for a reference period of 1 s. The variable x is the normalised radius: x = r/a where a = 6371 km. The particular distribution of of bulk and shear Q values, QK and Qµ, sohuld only be understood as a way to lower the Q of radial modes in order to make them more compatible with observations.

*The region between 24.4 and 220 km depth (LID and LVZ) is transversely isotropic with a vertical/radial symmetry axis. The effective isotropic velocities over this interval can be approximated by: Vp = 4187.5 + 3938.2x Vs = 2151.9 + 2348.1x

46 B The heat equation

Consider a thin rod of length δx, insulated everywhere except at the end points. From Fourier’s law ∂T q = −k (B.1) ∂x we know that if the temperature, T, of the rod is not equilibrated with its surroundings, or a temperature gradient exists over the rod, a heat flow, q [W m−2], proportional to the thermal gradient times the thermal conductivity, k [W m−1 K−1], will occur over the boundaries of the rod. We shall denote this heat flow over the end points of the rod by q(x) and q(x + δx), where q > 0 for a heat flow in the positive x-direction. Inside the rod, heat sources and sinks might also exist, we shall denote these per unit length by Q [W m−1], where Q > 0 indicates a source. From energy considerations in steady state, i.e. the temperature of the rod does not change, it must therefore hold that Qδx = q(x + δx) − q(x) (B.2) A Taylor expansion of q(x + δx) gives

∂q (δx)2 ∂2q q(x + δx) = q(x) + δx + ... (B.3) ∂x 2 ∂x2 For small δx we can linearize this Taylor expansion to yield ∂q q(x + δx) ≈ q(x) + δx (B.4) ∂x Taking the second derivative of Fourier’s law with respect to x yields ∂q ∂ ∂T = − k (B.5) ∂x ∂x ∂x Combining eq.(B.3) and (B.5) into eq.(B.2) gives ∂ ∂T Qδx = −δx k (B.6) ∂x ∂x or equivalently ∂ ∂T Q + k = 0 (B.7) ∂x ∂x In a non steady state, transient, situation, we also need to account for a varying temperature inside the rod. Con- sider a local temperature increase in the rod, this means that heat is transferred and accumulated at the location of the temperature increase. I.e. we can view a temperature increase as a heat sink. In the same manner we can consider a temperature decrease as a heat source. Let us therefore split Q into two parts, a true source/sink part, QS and a part due to temperature changes, QT .

Q = QS + QT (B.8)

−1 −1 If the rod has a density per unit length of ρl [kg/m] and a heat capacity of cp [J kg K ], we can express QT as dT Q = −ρ c (B.9) T l p dt And so eq.(B.7) becomes ∂ ∂T dT Q + k = ρ c (B.10) S ∂x ∂x l p dt Note that the time derivative of the temperature is not expressed as a partial derivative. For a solid body we need not worry about this. However consider the effect of a viscous body. In general, heating a substance results in a change of volume (thermal expansion) and for most substances the volume increases with increasing temperature. Thus the density of the substance decreases with temperature. In a viscous body this gives rise to buoyancy forces, i.e. the interior of the body will tend to redistribute to achieve a force-equilibrium (a phenomenon known

47 as convection). In our 1D model this means that we need to allow the x-coordinates of the body to be time dependent, x = x(t) (B.11) Hence the time derivative of the temperature can be written as:

dT (t, x[t]) ∂T ∂x ∂T ∂T ∂T = + = + u (B.12) dt ∂t ∂t ∂x ∂t ∂x Where u is the velocity in the positive x-direction. Equation (B.10) then becomes

∂ ∂T ∂T ∂T Q + k = ρ c + u (B.13) S ∂x ∂x l p ∂t ∂x

Equation (B.13) can be easily generalized into 3D to yield

dT Q + ∇ · k∇T = ρc + u∇T (B.14) p dt

Where ρ and Q now are density and sources and sinks per volume, [kg m−3] and [W m−3], and u is the three dimensional velocity vector.

B.1 Spherical symmetry

As a ﬁrst order approximation we can assume the cores of a celestial body to be spherical. It is therefore beneﬁcial to work with spherical coordinates when formulating the physics of such objects. Explicitly writing out eq.(B.14) in spherical coordinates yields 1 ∂ ∂T 1 ∂ ∂T 1 ∂ ∂T dT Q + kr2 + k sin θ + k = ρc (B.15) r2 ∂r ∂r r2 sin θ ∂θ ∂θ r2 sin θ ∂φ ∂φ p dt If we also assume the cores to be spherically symmetric with respect to the centre (r = 0), we can simplify eq.(B.15) to 1 ∂ ∂T dT Q + kr2 = ρc (B.16) r2 ∂r ∂r p dt In most cases k are considered to be geometry independent and can therefore be moved out of the left hand side equation. We can then rewrite eq.(B.16) to yield another common form of the heat equation:

2k ∂T ∂2T dT Q + + k = ρc (B.17) r ∂r ∂r2 p dt Finally, in the homogeneous case (i.e. no sources or sinks, Q = 0) we express the equation for a solid body as

∂2rT ∂rT κ = (B.18) ∂r2 ∂t where we have combined ρ, cp and k into the thermal diffusivity, κ k κ = (B.19) ρcp i.e. we are still assuming k to be constant throughout the body.

48 B.2 Solving the heat equation in spherical coordinates

Unfortunately the heat equation can only be solved in special cases due to fact that the constants Q, k, ρ and cp generally not are constants but rather functions of the position (and temperature). Thus, in general numerical methods are necessary. However, under the assumption of at least constant material constants a number of solutions can be found. In the steady state scenario with Q = Q(r), eq.(B.16) becomes ∂ ∂T kr2 = −Qr2 (B.20) ∂r ∂r Which integrates to yield 1 2 Z Z A T (r) = Qd3r − Qd2r − + B k r kr 1 1 Z Z A = Qr2dr − Qrdr − + B (B.21) k r kr Z 1 Z A = B − Qr2dr dr − kr2 kr where A and B are some arbitrary constant and dnr indicates the dimension of the integral (double and triple in our case). In the homogeneous transient case (i.e. Q = 0) several particular solutions can be found T (r, t) = A + B 2κtr−1 + r " n # X (2n + 1)(2n) ... (2n − 2i + 2) T (r, t) = A + B r2n + (κt)ir2n−2i i! i=1 T (r, t) = A + Br−1exp(κµ2t ± µr) B r2 T (r, t) = A + exp − 3/2 t 4κt (B.22) B r2 T (r, t) = A + √ exp − r t 4κt T (r, t) = A + Br−1exp(−κµ2t)cos(µr + C) T (r, t) = A + Br−1exp(−µr)cos(µr − 2κµ2t + C) B r T (r, t) = A + erf √ r 2 κt Where A, B, C and µ are arbitrary constants and i is a positive integer. However, note that all of these posses at least one singularity as r → 0 or t → 0. For a solution bounded at r = 0 or t = 0 we can instead make use of Green’s functions. This is also the way to solve the non-homogeneous transient heat equation (Q = Q(r, t), diffusion only, constant coefﬁcients and spherical symmetry as before). Consider a problem in the interval 0 ≤ r ≤ R with the initial condition T = f(r) at t = 0 Subjected to homogeneous boundary conditions. The solution can then be written in terms of a Green’s function, G, as Z R Z t Z R T (r, t) = A + f(ξ)G(r, ξ, t)dξ + Q(ξ, τ)G(r, ξ, t − τ)dξτ (B.23) 0 0 0 E.g. given a Dirichlet boundary condition that T (R, t) = 0

Yields the Green’s function and constant A A = T (R, 0) ∞ 2ξ X nπr nπξ κn2π2t (B.24) G(r, ξ, t) = sin sin exp − Rr R R R2 n=1

49 Where n is a positive integer. As an example of a Neumann boundary condition we assume

∂T = 0 ∂r r=R Yielding the Green’s function and constant A as

A = 0 3 ∞ 2 2 3ξ 2ξ X µ + 1 µnr µnξ κµ t (B.25) G(r, ξ, t) = + n sin sin exp − n R3 Rr µ2 R R R2 n=1 n

Where µn are the roots of the transcendental equation tanµ − µ = 0. The ﬁrst ﬁve roots are

µ1 = 4.4934, µ2 = 7.7253, µ3 = 10.9041, µ4 = 14.0662, µ1 = 17.2208

The Green’s functions can also be used in problems involving moving boundaries (i.e. Stefan problems) for an example of the thawing of a spherical body see Leung et al. [83]. In problems involving both a moving boundary and a time dependent boundary condition (e.g. the growth of the inner core) one can instead use perturbation methods that yields approximate analytical expressions [34]

50 C Thermodynamics and Mechanics

In this appendix is given a quick review of thermodynamic and mechanical theory necessary for the work presented in this thesis. The treatment mostly follows Poirier [105], where a more extended discussion of the topic can be found.

C.1 Thermodynamic fundamentals

Thermodynamics are governed by four laws47:

Zeroth law: If two thermodynamic systems are in thermal equilibrium with a third, they are also in thermal equilibrium with each other. First law: The increase in the internal energy of a thermodynamic system is equal to the amount of heat energy added to the system minus the work done by the system on the surroundings. Second law: Heat cannot of itself pass from a colder to a hotter body.

or:

The entropy of an isolated system not at equilibrium will tend to increase over time, approaching a maximum value. Third law: As a system approaches absolute zero of temperature all processes cease and the entropy of the system approaches a minimum value.

In thermodynamics one studies the effect of changes in temperature, T, pressure, P, entropy, S, and volume, V, on physical systems at the macroscopic scale48. These quantities can be divided into extensive (S and V) and intensive (T and P) quantities49 or grouped into conjugate quantities50, hence we can define a set of potentials (energies), for a given system, as linear combinations of the products of the systems conjugate quantities. The most familiar ones in thermodynamics being: Internal Energy E (C.1) Enthalpy H =E + PV (C.2) Helmholtz free energy F =E − TS (C.3) Gibbs free energy G =H − TS (C.4) Using the first law of thermodynamics: dE = δQ − δW = T dS − P dV (C.5) Where dE is the infinitesimal change of internal energy in a system, δQ are the infinitesimal amount of heat added to the system, and δW the infinitesimal amount of mechanical work done by the system on its surroundings51.

47It might seem somewhat awkward to number the laws by zero to three instead of one to four, the reason being that the need to state it explicitly as a law was not perceived until the ﬁrst third of the 20th century. By that time the ﬁrst three laws had already been recognised for a long time. The term: the zeroth law, was coined by Ralph H. Fowler in recognition of it, in many ways, being more fundamental than the other three laws. However, it should be recognised that there is still some discussion about its status in relation to the other three laws 48Note that we are not including the chemical potential, µ, and number of spices, N, here as we will assume N to be constant through out most of this section. 49An extensive quantity is size dependent, whereas an intensive is not. Alternatively extensive quantities are also called generalized dis- placements whilst intensive quantities are known as generalised forces. 50That an extensive and an intensive quantity are conjugate quantities means that their product has the dimension of energy. 51 P Note that we are assuming that the number of particles in the system is preserved in eq.(C.5), if not an additional term i µidNi should be added to the system (see footnote 48). Also note the use of δ rather then d to mark that δQ and δW are not exact differentials, meaning that they are generally not path independent.

51 The thermodynamic potentials are state functions i.e. the variation of a given potential depends only on its initial and ﬁnal state, hence we can express the differentials of the above potentials as exact differentials dH = T dS + V dP (C.6) dF = −SdT − P dV (C.7) dG = −SdT + V dP (C.8)

In a system deﬁned by N extensive, ek, and N intensive, ik quantities, the differential increase in energy, dU, per unit volume due to a change of ek is given by N X dU = ikdek (C.9) k=1 Using this we can deﬁne the intensive quantities of a system as ∂U ik = (C.10) ∂ek For the same system we can express Gibbs free energy as

N X G = U − ikek (C.11) k=1 N X ⇒ dG = − ekdik (C.12) k=1 and so the extensive quantities can be deﬁned as ∂G ek = − (C.13) ∂ik Thus we can express the intensive and extensive quantities in thermodynamics as ∂E ∂H T = = (C.14) ∂S V ∂S P ∂F ∂G S = = − (C.15) ∂T V ∂T P ∂E ∂F P = = − (C.16) ∂V S ∂V T ∂H ∂G V = = (C.17) ∂P S ∂P T

C.2 Thermoelastic coupling

Since the thermodynamic differentials are exact differentials, differentiation of the potentials with respect to the independent variables are commutative. Using this we can derive the Maxwell’s relations: ∂S ∂V − = (C.18) ∂P T ∂T P ∂S ∂P = (C.19) ∂V T ∂T V ∂T ∂V = (C.20) ∂P S ∂S P ∂T ∂P = − (C.21) ∂V S ∂S V (C.22)

52 Finally we shall recall the chain rule for partial derivatives of a function f(x, y, z) = 0:

∂x ∂y ∂z = −1 (C.23) ∂y z ∂z x ∂x y Combined with the Maxwell’s equation the chain rule offers derivatives with respect to non-conjugate quantities, offering a coupling between thermodynamics and mechanics (see table 6 below for some derivatives).

∂S = CP ∂S = −αV ∂S = CV ∂S = CP ∂P V αKS T ∂P T ∂T V T ∂T P T

∂S CP ∂S ∂V 1 ∂V αV T = = αKT = = ∂V P αV T ∂V T ∂S T αKT ∂S P CP

∂V = − CP ∂V = αV ∂V = − V ∂V = − V ∂T S αKS T ∂T P ∂P T KT ∂P S KS

∂T = T ∂T = T ∂T = 1 ∂T = − αKS T ∂S V CV ∂S P CP ∂V P αV ∂V S CP

∂T 1 ∂T αV T ∂P ∂P CP = = = αKT = ∂P V βP ∂P S Cp ∂T V ∂T S αV T

∂P = − KS ∂P = − KT ∂P = − 1 ∂P = αKS T ∂V S V ∂V T V ∂S T αV ∂S V CP

Table 6: Partial derivatives of the thermodynamic extensive (S,V) and intensive (T,P) quantities. Cx denotes heat capacity at constant x, α is thermal expansion, Kx is the bulk modulus at constant x and β is...

C.3 Lattice vibrations and the Debye approximation

The temperature of a crystal can be related to high-frequency vibrations of the atoms about their equilibrium positions in the crystal lattice. Hence, from an energy point of view a crystal can be considered as a collection of oscillators whose global properties govern the thermal behaviour of the solid. In addition the low-frequency part of the vibrational spectrum corresponds to elastic waves, and so a vibrational treatment of solids naturally offers a physical basis to thermoelastic coupling. In ﬁnite crystals, the lattice vibrations/normal modes are quantized and behave as quasi-particle, also known as phonons. In particular it is of interest to determine

• The normal modes of vibration of the crystal • The dispersion relation, i.e. the relation ω = f(k) between the frequency ω and the wave vector k • The vibrational energy

However, the equations arising can be rather complex and some basic assumptions are needed. Debye’s approximation consists in assuming that a linear dispersion curve52, ω =vk ¯ (v¯ is an average velocity of sound waves), holds for the whole vibrational spectrum. This effectively means that all the modes are considered to be acoustic, with the same average velocity, and an upper cut-off frequency in the vibrational spectrum, referred to as the De- 53 bye frequency, ωD. An expression for ωD can be derived assuming the Brillouin zone to have the simple shape of a sphere and that all the different atoms play equivalent mechanical roles in the vibrations:

ρ 1/3 ω = (6π2N )1/3 v¯ (C.24) D A M¯ 52The linear dispersion curve is also known as long wave or continuum approximation.) 53The ﬁrst Brillouin zone is the primitive cell in the reciprocal lattice in momentum space. Where the reciprocal lattice is the set of all vectors k such that eik·R = 1 for all lattice point position vectors R. There are also second, third, etc., Brillouin zones, corresponding to a sequence of disjoint regions (all with the same volume) at increasing distances from the origin, but these are used more rarely. As a result, the ﬁrst Brillouin zone is often called simply the Brillouin zone.

53 where M¯ is the mean atomic mass, ρ is the density, NA Avogadro’s number and the mean velocity can be found from the formula " #1/3 1 1 −1/3 v¯ = 3 3 + 3 (C.25) vp vs where vp and vs is the p- and s- wave velocities. Hence for the Debye approximation to be valid most of the actual vibrational spectrum should correspond to frequencies lower than ωD. As it turns out, the Debye approximation is reasonable for all elements, simple close-packed substances as well as close-packed minerals with a volume per atom smaller than 5.8 cm3/mol, e.g. stichovite, corundum, periclase and the dense lower-mantle perowskite form.

C.4 Gruneisen¨ parameters

The most commonly encountered Gruneisen¨ parameter is the thermodynamical, γth, which is a dimensionless ratio between elastic and thermodynamical quantities

αV KT αV KS γth ≡ = (C.26) CV CP

Alternatively we can express γth as

αV KS γth = CP ∂T ∂P γth = − = ∂V S ∂S V ∂lnT ∂lnT γth = − = (C.27) ∂lnV S ∂lnρ S 1 CP γth = − 1 αT CV ∆P γ = V th th ∆E

where ∆Pth and ∆E are the change in thermal pressure (caused by heating a solid a constant temperature) and internal energy explicitly. However the above deﬁnition of the Gruneisen¨ parameter is not unique, instead a large family of Gruneisen¨ parameters, or gammas as they are also referred to, exists in the literature, based on different approximation and normally named after their author(s). We shall only go through a few of them here which are of importance for this thesis, others can be found in Poirier [105] together with their underlying assumptions and derivations.

Let us start by inspecting Gruneisen’s¨ own deﬁnition based on the volume dependence of the frequency, ωi of the ith vibrational mode of a crystal lattice ∂lnω γ = − i (C.28) i ∂lnV Under Debye’s approximation all the ωi’s have the same volume dependence and only one Debye cut-off frequency exists we can then write ∂lnω ∂lnΘ γ = − D = − D ≡ γ (C.29) ∂lnV ∂V D where γD is the Debye-Gruneisen¨ parameter and ΘD is the Debye temperature, deﬁned by

¯hωD ΘD = (C.30) kB

where ¯h and kB are the reduced Planck’s (Dirac’s) constant and Boltzmann’s constant respectively. Now, the Debye-Gruneisen¨ parameter originates in the derivation of the Mie-Gruneisen¨ equation of state, valid under the Debye approximation dE E P = − c + γ D (C.31) dV D V

54 where Ec is the cohesive energy of the system at T = 0 K and ED is the internal energy of the phonon-gas. Assuming that the mode frequencies are temperature independent we can differentiate eq.(C.31) with respect to T ∂P γD ∂ED γD = = CV (C.32) ∂T V V ∂T V V where the last equality follows from the ﬁrst law of thermodynamics and table 6 above. Using table 6 we also see that ∂P = αKT (C.33) ∂T V hence αV KT γD = = γth (C.34) CV

The last gamma we shall discuss is Slater’s gamma, γsl. Using Gruneisen’s¨ original deﬁnition of γ (eq.(C.28)), Debye’s approximation and the assumption that Poisson’s ratio is independent of the volume he derived an expression for Gruneisen’s¨ parameter as 1 1 dK γ = − + (C.35) sl 6 2 dV Now, the assumption that Poisson’s ratio is independent of the volume is implicit in Debye’s theory (Jean Paul Poirier, private correspondence), so γsl is in fact equal to γD. Thus if Debye theory holds and mode frequencies are temperature independent we have that γth = γD = γsl (C.36)

C.5 Adiabatic temperature

An adiabatic process means that the no heat is transferred between a system and its surroundings, i.e. δQ = 0. Normally when referring to an adiabatic process people also assumes the process to be reversible, meaning that the entropy of the system is ﬁxed54, dS = 0 for the process. Using this assumption an expression for the adiabatic temperature gradient as a function of pressure can be found from table 6

∂T αV T αT = = (C.37) ∂P s Cp ρcp

Where cp is the heat capacity per unit mass. Convection can be consider to be almost an adiabatic process or rather an quasi-adiabatic process, since the diffusion time, td, of an individual volume element of the convective media generally is longer than the timescale on which convection proceeds, tconv.: td > tconv.. Hence the mean temperature of a vigorously convecting cell will be very close to an adiabat. As major parts of the interior of the Earth are convective, it is therefore of interest to know the adiabat of the Earth. To derive an expression we start from eq.(C.37) above, assuming the Earth to be spherically symmetric with respect to temperature and mass distribution we are interested in ﬁnding Tad(r). Noting that the pressure gradient inside the Earth can be written as ∂P = −ρg(r) (C.38) ∂r we get ∂T ∂P αgT = − ∂P ∂r cp αg ⇒ ∂lnT = − ∂r (C.39) cp Z r αg ⇒ Tad(r) = T (r0)exp − ∂r r0 cp

54Known as a isentropic process

55 This is the most commonly encountered form of the adiabatic temperature. Using eq.(C.27) above we can ﬁnd an 55 equivalent expression. If we assume that the ratio αKT /cV is constant with respect to ρ at high temperatures we have that

αKT V αKT γth = = CV ρcV (C.40) ⇒γth,0ρ0 = γthρ

More generally we can write

γthρ = constant ρ q (C.41) ⇒ γ = γ 0 th th,0 ρ

where q is some positive constant. Inserting this into eq.(C.27) we get q ∂lnT ρ0 = γth,0 ∂lnρ S ρ Z Tad Z ρ ρq ∂lnT 0 = γ 0 dρ0 th,0 ρ0q+1 T0 ρ0 (C.42) T γ ρ q ln ad = th,0 1 − 0 T0 q ρ γ ρ q T (ρ) = T exp th,0 1 − 0 ad ρ0 q ρ

However, the effect of q in eq.(C.42) is relatively small (see ﬁgure 22(a)), thus for most applications the choice of q = 1, will yield a reasonable answer.

(a) (b)

Figure 22: Adiabatic temperature profiles of the core from equation (C.42) assuming the PREM density profile. All profiles are fixed at an assumed temperature of 6000 K at the centre of the Earth. Figure 22(a) displays five profiles with γth,0 = 1.5, and q varying in the interval [0.2 1] in steps of 0.2. Within this interval of q the adiabatic temperature varies by only 22 K at the ICB or 116 K at the CMB (steepest gradient corresponding to highest q value). Figure 22(b) displays five profiles with q = 1, and γ varying in the interval [1.3 1.7] in steps of 0.1. Within this interval of γ the adiabatic temperature varies by 59.7 K at the ICB or 416.6 K at the CMB (steepest gradient corresponding to highest γ value). Assuming γ = 1.5 and q = 1 the adiabatic gradient is about 0.37 K km−1 at the ICB, with a mean value of about 0.2 K km−1 in the inner core, or equivalently an adiabatic temperature difference of 242 K over the inner core, or 1900 K over the whole core.

55 In the high temperature limit CV → 3nR, further experiments indicates that αKT is constant at high T for a number of crystals, Poirier [105], p.56

56 C.6 Melting curves

Melting of a solid means a lost of the solids crystalline long-range order and resistance to shear. It is a ﬁrst order phase transition, meaning that the process exhibits discontinuities in the ﬁrst derivatives of the free energy (e.g. Helmholtz, Gibbs or Enthalpy, see eq.(C.2)-(C.4) and (C.6)-(C.8) above). In essence melting results in a change in volume, ∆V and entropy ∆S, consider then the product of the conjugate quantities T and S under the such a process, T ∆S, as we know this is equal to an energy, also known as latent heat, L. Thus L = T ∆S (C.43)

For an elemental solid the melting temperature, Tm, is fixed at a given pressure Tm(P ). Unfortunately this is generally not the case for non-elemental solids, such as minerals and alloys, where melting and differentiation might occur over a range of temperatures depending on the solids initial composition. Several cases may arise and are best understood via phase diagrams56. E.g. the phase of an elemental substance might be displayed in a P-T diagram (see figure (23(a))), whilst a two component system might be displayed in a T-composition diagram (i.e. at fixed P, see figure (23(b))). Upon melting a binary mixture can display either congruent melting (melt and solid has the same composition) or incongruent melting (melt and solid has different composition) in addition at a given composition the phase diagram might display one or several eutectic point(s) where melting occurs at a fixed temperature. Covering all the possible scenarios is really out of the scope of this thesis. We shall therefore not try to delve further into the subject and in what follows only consider elemental substances.

We are interested in ﬁnding an expression for Tm(P ) at high pressures and temperatures. Assuming a harmonic solid with sinusoidally vibrating atoms as well Debye’s approximation and Slater’s approximation (Poisson’s ratio constant along the melting curve) an expression for Tm(P ), also known as the Lindemann’s law of melting can be found as (see [105] p.126-132): dlnT 1 m = 2 γ − (C.44) dlnρ sl 3

Using γsl = γth (since we already have assumed Debye’s approximation to be valid, see sect. C.4 above) we can make use o eq.(C.41) above and integrate eq.(C.44) to yield

Z Tm Z ρ q 0 ρ0 1 0 dlnTm = 2 γth,0 0 − dlnρ Tm,0 ρ0 ρ 3 T 2γ ρ q 2 ρ ln m = th,0 1 − 0 + ln 0 (C.45) Tm,0 q ρ 3 ρ ρ 2/3 2γ ρ q T = T 0 exp th,0 1 − 0 m m,0 ρ q ρ

(a) (b)

Figure 23: Examples of phase diagrams of an elemental substance (left panel) and a binary mixture (right panel)

56A phase diagram displays the state of a substance i.e. solid (including different crystal structures), liquid, gas)

57 58 D COMSOL Multiphysics

COMSOL Multiphysics (formerly known as FEMLAB) is a commercial finite element code with a Graphical User Interface (GUI). As the name indicates, COMSOL has the capability to cross-couple equations without any knowledge of numerical analysis from the user. In its basic form COMSOL offers modelling of acoustic, convection and diffusion, electromagnetic, fluid dynamic, heat transfer, and structural mechanics problem. There are possibilities to use deformed mesh’s and general PDE’s. Depending on application, COMSOL can offer several types of finite elements ( Lagrange ,linear to quintic, Hermitian ,linear to quintic, and Argyris ,quintic57,) and coordinate systems (Cartesian, 1D-3D, and cylindrical, 1D or 2D). For highly complex models it is also possible to use different coordinate systems (of different dimension) for different parts of the model. For the construction of the model and model constants one can define own equations and functions to be solved simultaneously with the PDE’s of the model. In addition to the default modes of COMSOL one can add several specialized modules, including modules for Heat transfer, Earth science, chemical engineering etc. COMSOL can be run together with MATLAB although this demands that the user has access to MATLAB. Finally COMSOL has its own script- language (one of the additional modules) from which COMSOL can be run. All in all it can be said that COMSOL Multiphysics offers great capabilities for a quick construction of advanced models.

All modelling in COMSOL follows seven steps

1. Choosing equations to be solved and coordinate systems. 2. Defining the geometry of the model 3. Defining constants, expressions, variables, and functions of the model 4. Specifying boundary conditions and material coefficients on each sub domain 5. Choosing solver and constructing a mesh over the model 6. Solving the model 7. Post processing the results

In this thesis work I have used the heat transfer module to compute steady state geotherms over the inner core. Under the assumption that the inner core is spherically symmetric this condenses to solving the 1D steady state diffusion equation in spherical coordinates. Now, COMSOL does not offer any spherical coordinate system, but it is possible to use the 1D Cartesian coordinate system as a pseudo-1D-spherical coordinate system, I will explain two ways this can be done in the heat module in sect.D.1, but the schemes presented are easily generalised to other modules. I will then present the use constants, expressions, variables, and functions in COMSOL, as well as presenting the ones used in my models in sect.D.2. In sect.D.3 I will brieﬂy touch upon meshing and the different solvers of COMSOL. Finally I will discuss how I have performed the post processing of the results in sect.D.4.

Note that this appendix is not meant to be a manual for COMSOL Multiphysics and will therefore not cover all applications and options of COMSOL. The intention is to give a short introduction to the parts of COMSOL that has been used in the models built in this thesis work, as well as presenting the models built.

D.1 Spherical symmetry using the Heat module

By default the heat module only offers possibilities to model, using Cartesian or cylindrical coordinates, so in order to use spherical coordinates this we need to be careful how we deﬁne the constants and boundary conditions used in our equations. We shall start by choosing a 1D Cartesian coordinate system for our model and then use

57Hermitian and Argyris elements are only available when modelling using PDE’s

59 appropriate factors to transform this into a 1D spherical coordinate system for the heat equation (see eq.(B.16)). Note that the heat equation in spherical coordinates contains a factor 1/r2 which might lead to a singularity at r = 0. As a ﬁrst step we therefore multiply through by a factor58 of r2

∂T ∂ ∂T ρc r2 − kr2 = r2Q (D.1) p ∂t ∂r ∂r

In the Heat transfer module the conductive heat equation is deﬁned as eq.(B.14) above, which in 1D for a solid body can becomes as eq.(B.10). Comparison eq.(D.1) to eq.(B.10) shows that we can use the 1D Cartesian coordinate system as a spherical coordinate system by using the quantities

ρ´ = x2ρ

k´ = x2k Q´ = x2Q in eq.(D.1) (x is the 1D Cartesian coordinate accessible as the independent variable x in COMSOL59). We shall do one additional modiﬁcation of the equations before implementing them. Consider the dimensionless radial coordinate rˆ, deﬁned via the relations: r ∂ 1 ∂ rˆ = ; = (D.2) R ∂r R ∂rˆ

If we let R be equal to the inner core radius, Ricb, we will always ﬁnd the inner core boundary (ICB) at rˆ = 1. The advantage of doing this is that we can easily change the physical radius of the inner core whit out changing the geometry, so in an evolutionary model we can simply let Ricb become Ricb(t). Substituting relation D.2 into eq.(D.1) yields: 2 2 ∂T ∂ rˆ ∂T 2 ρcprˆ − k =r ˆ Q (D.3) ∂t ∂rˆ Ricb ∂rˆ or expressed in the 1D Cartesian coordinate x as

2 2 ∂T ∂ x ∂T 2 ρcpx − k 2 = x Q (D.4) ∂t ∂x Ricb ∂x Hence, in order to use the 1D Cartesian coordinate system of COMSOL as a dimensionless 1D spherical coordinate system we want to use the quantities

1. ρ˜ = x2ρ

2. 2 ˜ x k = 2 k Ricb 3. Q˜ = x2Q

Next we consider the boundary conditions, if we use a Dirichlet boundary condition, i.e. the temperature is known for all points along the boundary at a given time, t, no difference exists between Cartesian and spherical coordinates, however if we use a Neumann condition, i.e. the heat ﬂux, q0, in a direction normal to the boundary is known at a given t, we need to be cautious. Since we have assumed the temperature to be spherically symmetric

58This is not really necessary to do in COMSOL multiphysics since the solvers use automatic variable scaling when necessary to get well-conditioned problems. 59Alternatively we can, upon deﬁning the settings of the model, use the multiphysics alternative and add a 1D geometry where we can assign other names to the space variables e.g. rho theta and phi, in which case we can access the 1D independent variable as rho in COMSOLE. On the other hand we will have to add a factor of rho2 instead of r2 since we are still using a Cartesian coordinate system.

60 the mathematical formulation of the Neumann boundary condition is equal to the 1D Fourier’s law in spherical coordinates ∂T q = −k (D.5) 0 ∂r Due to the transformation used above in order to use the 1D Cartesian coordinate system of COMSOL as a 1D spherical coordinate system eq.(D.5) becomes

x2 x2 ∂T q0 2 = − 2 k (D.6) Ricb Ricb ∂x

This can be realized in two ways, first of all we want to use the quantity k˜ rather than k in our equations, thus to 2 2 do this we also need to multiply a factor of r /Ricb to the left hand side of the equation. Secondly, as we are using a 1D coordinate system we need to compensate for the increase of the surface area of the sphere, which is done 2 2 by multiplying the normal surface heat flux by a factor r /Ricb. The Neumann boundary condition is defined in COMSOL as −n · (−k∇T ) = q0 + h(Tinf − T ) (D.7) Where −nˆ is the outward normal to the boundary and h is the heat transfer coefficient [W m−2 K−1]. In our models we will not use h but for completeness we note that also this quantity needs to be modified by the multiplication 2 2 of a factor r /Ricb. Thus we need to add two more relations to our transformation scheme

4. x2 q˜0 = q0 2 Ricb 5. 2 ˜ x h = h 2 Ricb

Note that the above two relations are valid on boundaries only, while the previous relations are valid on sub domains.

There are at least two schemes in which the above transformations can be implemented in COMSOL. The first being to simply multiply the appropriate transformation terms when specifying the physical quantities in the sub domain and boundary settings of the model (according to the five relations given above, e.g. see fig.24).

The second scheme is to directly manipulate the formulation of governing equation system of the underlying PDE, by adding the appropriate transformation terms (see fig.25). COMSOL specifies the heat equation in the sub domain as ∂2T ∂T ea 2 + da + ∇Γ = F ∂t ∂t (D.8) ∂Γ ∂F ∂F ∂Γ c = − , a = − , y = Γ, β = − , o = − , f = F ∂∇T ∂T ∂∇T ∂T To transform the equation system to a 1D spherical coordinate system according to relation 1 to 3 above, the coefficients needs to be defined according to table 7

Using this approach we also need to modify the equation system on the boundaries. In COSMSOL this is deﬁned as

∂RT − n · (Γ1 − Γ2) = G + µ ∂u (D.9) ∂G ∂R g = G, q = − , r = R, h = − ∂u ∂u

61 A comparison to eq.(D.7) and transformation relation 4 and 5 above, gives that the modification needed is a multiplication of the g and q coefficients60 by a factor of x∧2/Ricb∧2. Note that in order to modify the g and q coefficient we first need to impose Neumann boundary conditions in the boundary settings of our model (other wise g and q will appear as 0). After the q and g coefficients have been modified we can change the boundary conditions.

Coefﬁcient Default Transformed

ea 0 0 ∧ da Dts htgh*rho htgh*C htgh Dts htgh*rho htgh*C htgh*x 2 c -diff(-k htgh*Tx,Tx) -diff(-k htgh*Tx*x∧2/Ricb∧2,Tx) a -diff(Q htgh,T) -diff(Q htgh*x∧2,T) y -k htgh*Tx -k htgh*Tx*x∧2/Ricb∧2 β -diff(Q htgh,Tx) -diff(Q htgh*x∧2,Tx) o -diff(-)k htgh*Tx,T) -diff(-k htgh*Tx*x∧2/Ricb∧2,T) f Q htgh Q htgh*x∧2

Table 7: Appropriate coordinate transformations in the sub domain settings of the equation system in the heat module of COMSOL when modelling radial conductive heat transfer using a 1D Cartesian coordinate system as a spherical coordinate system. N.B. The default expressions are given for a transient model with internal heating, and will therefore differ from the default settings of a steady state model or a model with no internal heating.

Figure 24: Example of appropriate sub domain settings in COMSOL when using a 1D Cartesian coordinate system (independent variable x) as 1D spherical coordinate system (spherically symmetric). k, rho, cp, Q and Ric are pre-deﬁned as constants.

60This holds true as long as we are only considering conduction. Also note that there exists two expressions for the q coefficient, T and J. The first being temperature and the second being radiosity (defined as the sum of reflected and emitted radiation), so as we are not considering radiation we only need to modify the T-expression

62 Figure 25: Example of an appropriate modiﬁcation of the governing sub domain equations for the heat transfer module of COMSOL in the case of a transient conduction model

Finally it shall be noted that when using the second scheme and several geometries (sub domains) are being used the given transformations has to be appended on all geometries and boundaries. For my models I have chosen to use the ﬁrst scheme presented as the implementation of this is much easier. However I have also constructed a model using the second scheme in order to verify it (it was approved ; ).

D.2 Constants, Expressions, Variables, and Functions

To model the physics, COMSOL offers ﬁve possibilities to specify the properties of the model. In the following sections, any word written in italic refers to an actual choice or menu in COMSOL.

• Directly in the sub domain and boundary settings • Constants: A constant is a fix value, valid globally in the model. It can be specified using other constants or mathematical function, upon which the value of the constant is evaluated directly when you click Apply or Ok in the constants dialog box. For a list of the constants used in the models of this thesis work see table 8 • Expressions: Expressions are being evaluated during the simulation and can depend on any variable (e.g. space variables, solutions components) and constants as well as other expression. When defining an expression you need to choose on which parts of the model the expression is valid on. Therefore up to six different classes of expressions exists (depending on the dimensions of your model): – Global expressions: Valid on all parts in all geometries of the model – Scalar expressions: Valid on all parts of the specified geometries of the model

63 – Sub domain expressions: Valid on specified sub domains – Boundary expressions: Valid on specified Boundaries – Edge expressions: Valid on specified edges. This class is only available for 3D geometries – Point expressions: Valid on specified points. this class is only available for 3D and 2D geometries. In the models built in this thesis work I have only used Global expressions. For a list of the expressions used see table 9 • Coupling variables Used to transfer information between different parts of the model, or to whit draw additional information from the models. Three different classes are available – Integration coupling variables Evaluates the integral of an expression over a set of domains, thus produces a scalar value. – Projection coupling variables Evaluates a series of line integrals over a set of domains – Extrusion coupling variables Maps values from a source domain to a destination domain, if both the source and destination domain are of equal dimensions the mapping is one-to-one. If the dimension of the destination domain is higher than the source domain, each point value from the source domain is extruded into higher dimensions. In the models built for this thesis work I have used Integration coupling variables in order to evaluate various quantities (like volume, mass and heat production) over the inner core. This could as well have been done using the sub domain integration option under the Postprocessing menu. For larger models this is probably wise since the evaluation of the coupling variables will extend the computational time of the model. For a list of the Integration coupling variables used in the models see table 10 • Functions Used to define new functions. Any function defined has to accept matrix in/output, and it is not possible to use defined constants, expressions or coupling variables when defining the function. Functions can be based on either analytical expressions or interpolation from data. For the interpolation functions the data can be supplied either directly in a table when specifying the function, as a COMSOL script or MATLAB function, or be defined in an ascii-file. If an interpolating functions has more than one argument (a maximum of three arguments are allowed for interpolating functions), it cannot be defined directly in a table and the only available interpolation schemes are linear or nearest neighbour. Data defined in a ascii-file has to be defined as follows: The first row has to be a blank row or start with a %. You are allowed to make any comments after the % sign. The rows directly after the first row can either be a number of blank rows or comment rows (starting with a %). This shall be followed by as many rows as there are arguments (dimensions) of the function. E.g. if the function has three arguments, the trailing three rows shall be specified as follows, first row should contain all the coordinates along the first dimension (argument) separated by blanks/spaces. Second row should contain all the coordinates along the second dimension (argument) separated by blanks/spaces. Third row should contain all the coordinates along the third dimension (argument) separated by blanks/spaces. Note that the second and third dimension are optional, and hence also the corresponding input. It is however important that the coordinates are specified in an ascending order. After the coordinates has been specified there has to be at least one row starting with a % sign. Again you can comment on this row as well as specifying additional rows starting with % or blank rows. Finally the function values shall be specified on the trailing rows. Each row giving all the values along the first dimension (argument) for a fixed value along the other dimensions. First row should correspond to the first coordinate of the second and third dimension. The trailing rows should then be organized in such way that the coordinates along the second dimension increases fastest. In the models built for this thesis work three functions has been defined: PREMrho(x), PREMP(x), and LindemannMeltingLaw(rhox). The first two functions are linear interpolation functions using the density (PREMrho(x)) and pressure (PREMP(x)) of PREM (see appendix A), as tabulated in ascii-files. Extrapo- lation outside the range of the tabulated values are chosen to be constant61. The argument x is the radial coordinate and should be specified in units of km upon call to the functions. The third function is based on an analytical expression:

61This is really not an issue for my models since the tables covers the entire interior of the Earth. To view the proﬁles in COMSOL, open the Plot parameters dialog box from the Postprocessing menu and specify the expression PREMrho(x*6371) or PREMP(x*6371) on the Line

64 LindemannMeltingLaw(rhox) = Tm0*exp(2*gamma*(1-PREMrho(R0/1000)/rhox)+2/3*log(PREMrho(R0/1000)/rhox))

where rhox is the density in units of kg m−3

When defining Constants, Expressions, Variables, and Functions in COMOSOL the syntax used is in principle the same as the syntax of MATLAB, thus any one familiar with MATLAB will not have any problems formulating these quantities. It is also possible to use model specific quantities, these are normally appended by: ****, where **** is an abbreviation for the application being used. E.g. quantities in the heat module will be appended by: htgh. For instance to access the density used by COMSOL upon evaluation just specify: rho htgh, for the source term specify: Q htgh. The coordinate space variables are always accessible under their specified names, e.g. x, y, and z for default 3D Cartesian coordinates. Likewise the time variable is accessible as: t, and the dependent variables under their specified names, e.g. for the heat transfer module the dependent variables are temperature and radiosity, accessible as: T, and J, respectively. It is also possible to use derivatives of the dependent variables, e.g. to use the spatial derivative of T with respect to x use: Tx. Finally COMSOL offers several predefined operators, e.g. for the models of this thesis work I have been using the flc1hs(x’,dx’) function (see the Global expression: step in table 9), which is a step function with continuous first derivative, equal to 1 for x > x’+dx’/2, and 0 for x < x’-dx’/2 <.

Constant Value Comment Ricb 1220.49e3 Present radius of inner core [m] k 80 Thermal conductivity of the inner core [W/m/K] Ticb 5500 Temperature at the ICB [K] cp 860 Specific heat at constant pressure of the inner core [J/kg/K] gamma 1.38 Gruneisen parameter at the ICB year 3600*24*365.25 Seconds per year [s] HK40 2.966e-5 Heat production of K40 [W/kg] HTh232 2.566e-5 Heat production of Th232 [W/kg] HU235 5.749e-4 Heat production of U235 [W/kg] HU238 9.166e-5 Heat production of U238 [W/kg] tK40 1.25e9*year Halflife of K40 [s] tTh232 1.40e10*year Halflife of Th232 [s] tU235 7.03e8*year Halflife of U235 [s] tU238 4.46e9*year Halflife of U238 [s] aK40 1.17e-4 Natural abundance of K40 today [K40/K] aTh232 1 Natural abundance of Th232 today [Th232/Th] aU235 2.7e-3 Natural abundance of U235 today [U235/U] aU238 0.9927 Natural abundance of U238 today [U238/U] cK40 1 Mass concentration of K40 cTh40 1 Mass concentration of Th232 cU 1 Mass concentration of U P 7.3e11 Total amount of heat generated in the inner core [W] V 4/3*pi*Ricbˆ3 Inner core volume [m3] Tm Ticb Melting temperature at the inner core boundary [K] Age 1.6e9*year Age of the inner core tmodel 0 Time of the model [s]

Table 8: Constants tab. Note that since the geometry is modelled from 0 to 1 we need scale the dependent variable x by the Earth’s radius, we could naturally choose some other scale factor if we were only interested in the proﬁle of some other region. E.g. PREMrho(x*1221) would only plot the density structure of the inner core.

65 Name Expression Comment rho PREMrho(x*Ricb/1000) PREM density profile of the inner core [kg/m] Pre PREMP(x*Ricb/1000) PREM pressure profile of the inner core [Pa] Qk40 HK40*aK40*cK*exp(-tmodel*log(2)/tK40) Model dependent energy production from decay of K40 [W/kg] QTh232 HTh232*aTh232*cTh*exp(-Te*log(2)/tTh232) Model dependent energy production from decay of Th232 [W/kg] QU235 HU235*aU235*cU*exp(-Te*log(2)/tU235) Model dependent energy production from decay of U235 [W/kg] QU238 HU238*aU238*cU*exp(-Te*log(2)/tU238) Model dependent energy production from decay of U238 [W/kg] Lindemann LindemannMeltingLaw(rho) Lindemanns melting law for the model [K] dV 4*pi*xˆ2*Ricbˆ3 Differential volume of centered subshell [m2] Step flc1hs(0.01-x,0.001) Stepfunction with continuous first derivative, = 1 for x < 0.009, 0< Step <1 for 0.009 < x < 0.011, = 0 for x > 0.011 Qcvol P/V Model dependent Constant heat generation per unit volume [W/m3] Qcweight P*rho/Mass Model dependent constant heat production per unit mass [W/kg] QHerndon P*Step/VHerndon Inner core heat production in georeactor [W/m3] kVariabel 2*k-k*(T/Ticb)*flc1hs(T-10000,1000) Linearly variable Heat transfer coefficient, 2k at 10000 k and k at ICB F exp(-Age*(1-x)ˆ3*log(2)/tK40) Source distribution decay scaling factor

Table 9: Global expressions

Name Expression Integration Global Comment order Pgenerated Q htgh*Ricbˆ3*4*pi 4 yes Total heat production [TW] PK40 qK40*xˆ2*Ricbˆ3*4*pi 4 yes Total heat production from decay of 40K [TW] PTh232 qTh232*xˆ2*Ricbˆ3*4*pi 4 yes Total heat production from decay of 232Th [TW] PU235 qU235*xˆ2*Ricbˆ3*4*pi 4 yes Total heat production from decay of 235U [TW] PU238 qU238*xˆ2*Ricbˆ3*4*pi 4 yes Total heat production from decay of 238U [TW] Mass rho htgh*xˆ2*Ricbˆ3*4*pi 4 yes Inner core mass [Kg] VHerndon Step*xˆ2*Ricbˆ3*4*pi 4 yes Herndon’s georeactor volume [m3] G 3*xˆ2*F 4 yes Decay scaling constant

Table 10: Integration coupling variables

D.3 Meshing and Solvers

Having speciﬁed the model it is time to generate a mesh over the model. This is easily done under the Mesh menu of COMSOL. For models with several sub domains and boundaries, different meshes can be deﬁned on different parts of the model. For the models used in this thesis work I have chosen to use 2000 elements over the inner core (this is really much more than necessary for the problem).

Next up is choosing the solver. COMSOL offers four different classes of solvers depending on the nature of the model:

• Stationary Either linear or non-linear

66 • Time dependent • Eigenvalue

• Parametric Either linear or non-linear

For every class several different solvers are offered:

• Direct (UMFPACK) • Direct (SPOOLES) • GMRES • Conjugate gradients • Geometric multigrid an for every solver several different options are available. However, I do not have the intention to present them here. However I will mention two more aspects/features that might be useful when modelling using COMSOL. First of all, for large models COMSOL tends to use a lot of memory when solving the models, possibly leading to the hanging of your computer if not enough memory is available. An alternative might then be to store the solution on file during evaluation time, this reduces the use of internal memory, and COMSOL automatically deletes the solution files when done evaluating the model. The option to store the solution to file can be found on the advanced tab, in the Solver Parameters dialog box, under the Solve menu. Secondly, it is possible to use the results of earlier simulation as initial conditions for later simulations. One application of this could be to a steady state model as initial condition for a transient model, other applications include highly complex nonlinear models where convergence problems exists due to the non-linear nature of the problem. The scheme is then to first solve for only one or a few of the coupled quantities, use the solution as initial condition for the next simulation which is solved for the next coupled quantity, and so on. This procedure can be performed automatically using a small script. The specification of the initial value is set on the Initial Value tab, in the Solver Manger dialog box, under the Solve menu. Scripting is done on the Script tab, in the Solver Manger dialog box. It is possible to auto generate a script for the settings of the model already done.

For the models used in this thesis I have used the stationary and parametric classes of solvers, using the default solver (Direct (UMFPACK)).

D.4 Evaluating the output of COMSOL

Once the evaluation of he model is done it is time to think about how to interpret the output of COMSOL. COM- SOL offers several possibilities to plot the result using both line and different surface plots. It is also possible to plot various expression involving model quantities and coordinates. In addition one can evaluate expression in given speciﬁc points of the model and integrate expressions over the model domains. However, I have chosen to export all data to ascii-ﬁles and then use a MATLAB script instead to make the post processing, the reason being that I then can easily reproduce all the plots by running the MATLAB script again. Other wise one would have to rerun the model in COMSOL. Further I have found that even though the post processing possibilities in COMSOL are relative generous, they did not fully cover my wishes.

Before any post processing can be done we need to think a bit about the physical dimensions of the output of our COMSOL model. The discussion that follows is specialised to the models I have used in this thesis work, but can serve as a guidance when using the 1D Cartesian coordinate system of COMSOL as a 1D spherical coordinate system, as described in sect.D.1 above.

67 Effectively what we have done in our coordinate transformations is to eliminate the length dimension of the dependent variable. However, as the coefﬁcients of our equation are not dimension less (a closer look at eq.(B.16) above gives the PDE a dimension of [w/m]) we need to be cautious when converting the output of COMSOL into physical quantities.

For this study we are mainly interested in the temperature profile/gradient, the heat flow and internal heat generation. We will always need to rescale the x coordinates by a multiplication of the factor Ricb. The temperature profile, not having the dimension of length does not need to be rescaled. The temperature gradient has a dimension of [K/m] in SI units, since it is defined as the derivative of T [K] with respect to the space variable [1/m]. As the dependent variable is dimension less in our models, this means that the temperature gradient found in the model will have a dimension of [K], thus to get back to SI units we needed to scale the temperature gradient by 1/Ricb [1/m]. The heat flux [W/m2] is equal to minus the temperature gradient times the thermal conductivity coefficient, k [W/m/K]. When compensating for the use of spherical coordinates we multiplied this coefficient by a factor of 1/Ricb2 (i.e. [1/m2]). Hence the computed heat flux will have a dimension of [W/m3] so in order to get back to SI units we must multiply the computed Heat flux by a factor of Ricb [m]. Finally the internal heat generation is already expressed in terms of [W/m3] as should be.

Finally a word about integration. As we have mentioned above it is possible to integrate quantities over the sub domain, either during the evaluation of the model as an Integration coupling variable, or in post processing mode. Now if we want to perform a volume integral we need to realize that since we are using a 1D model (the radial coordinate only) we need to manually multiply the expression by the integrated solid angle, 4π. Further we need to remember the integrating factor r2 for volume integrals in spherical coordinates. If we integrate a model speciﬁc quantity (those appended by htgh in the heat transfer module, see sect.D.2 above) this factor is already integrated in the quantity via the transformations we have done. However if we integrate any other quantity over the sub domain we also need to multiply a factor of x2 to the integrand (x being our spatial coordinate). Finally we need to compensate for the use of non-dimensional spatial coordinates, thus we have to multiply the integrand (or the integrated value) by a factor Ricb3 where Ricb2 comes from the integrating factor x2 and an additional factor Ricb comes from the differential dx, as can be seen from eq.D.2 above.

68 E Transient heat transfer model of the inner core

In the ﬁnal stage of the work presented in this report, a paper by Yukutake [147] was found in which an analytical solution to the transient heat equation was given. The paper also brieﬂy describes how this solution can be used to build a thermal model of the inner core. In this appendix I describe how I after having found the paper used the solution and the proposed scheme to build a model of my own.

E.1 Transient solution to the heat equation

Starting from the transient heat equation inside a sphere of constant radius;

1 ∂T 1 ∂ ∂T A = r2 + (E.1) κ ∂t r2 ∂r ∂r K

constant boundary temperature, and known initial temperature distribution:

T (r, 0) = To(r) at t = 0 (E.2) T (rm, t) = Tm at r = rm, t > 0

Assuming a heat generation with an exponential decay

−λt A(t) = Aoe (E.3)

A solution can be found using Green’s functions as presented by equations (B.23) and (B.24) in appendix B62:

T (r, t) = u(r, t) + w(r, t) (E.4) where

∞ 2 2 2 nπr − κn π t X r2 u(r, t) =Tm + sin e m r rm n=1 (E.5) Z rm 0 n 1 0 0 nπr 0 (−1) x r To(r ) sin dr + rmTm rm o rm nπ and

 λ 1/2  κA rm sin r κ  w(r, t) = o e−λt − 1 Kλ λ 1/2 r sin rm  κ (E.6) ∞ 3 n κn2π2 2rmAo X (−1) nπr − 2 t + sin e rm 3 2 rπ K 2 λrm rm n=1 n n − κπ2

u(r, t) solves the homogenous heat equation (i.e. A = 0) with the boundary conditions

u(r, 0) = To(r) at t = 0

u(rm, t) = Tm at r = rm, t > 0

and w(r, t) solves the non-homogenous heat equation, with the boundary conditions

w(r, 0) = 0 at t = 0

w(rm, t) = 0 at r = rm, t > 0

62Note that it is assumed that the density, thermal conductivity and heat capacity are constant through out the sphere.

69 Likewise we can ﬁnd the thermal gradient by taking the derivative of eq.(E.4), yielding ∂T ∂u ∂w = + (E.7) ∂r ∂r ∂r where

∞ κn2π2 ∂u 2 X nπ nπr 1 nπr − 2 t = cos − sin e rm ∂r r rm rm r rm n=1 (E.8) Z rm 0 n 1 0 0 nπr 0 (−1) x r To(r ) sin dr + rmTm rm o rm nπ and

 λ 1/2 λ 1/2 λ 1/2  rm cos r rm sin r ∂w κAo −λt  κ κ κ  = e − − 1 ∂r Kλ λ 1/2 2 λ 1/2  r sin rm κ r sin rm κ  2 2 (E.9) ∞ − κn π t 3 n r2 2r Ao X (−1) e m nπr nπr 1 nπr + m cos − sin 3 2 rπ K 2 λrm rm rm r rm n=1 n n − κπ2

E.2 Inner core growth model

Now at a ﬁrst glance the above equations are rather hideous creatures, but they can be relatively easily implemented into some computer code. Where after the series terms can be computed and added until a desired accuracy has been reached. However, in the case of the inner core we can not do this directly as we know that the radius of the inner core as well as the boundary (ICB) temperature are not constant quantities. Instead we need to use an iterative scheme where the radius and boundary temperature is altered between each iteration.

We start out by assuming the radius of the inner core to be 0 at t0 = 0 and Ricb1 at t1. We also need to 0 assume some initial temperature distribution, T0. Since we know t0 and t1 we can then use eq.(E.4) to ﬁnd the 63 0 geotherm , T1, at t1, by evolving T0 over a time t = dt1 = t1. The next step will then be to evolve the radius 0 of the sphere to Ricb2 at t2. As an initial geotherm, T1,for r ≤ Ricb1 we then simply use T1 where as for Ricb1 < r ≤ Ricb2 we again have to assume some initial geotherm. T2 at t2 is then found by the use of eq.(E.4) again over t = dt2 = t2 − t1. However we now need to adjust the initial levels of heat sources by the use of eq.(E.3) and t1 as we are not starting at t = t0 but at t = t1. This scheme is then repeated until the desired age of the has been reached. Likewise we can solve for the geothermal gradient by the same scheme, using eq.(E.7) instead of (E.4).

The given scheme will at each time step return slightly to high temperatures for a cooling core (where as the temperatures will be slightly to low for a heating core), as the effect of a continuously varying radius is lost. However the this will mostly affect the outer parts of the sphere and the error can be estimated by comparing different models with different time steps between each evaluation of the geotherm. Anyway, given the size of the inner core and the characteristic diffusion time (τ = Ricb2/κ ≈ 1017 s today) we ﬁnd that this should only be a matter of concern at early times when the radius of the core is small.

What will control the result is the initial geotherm, as well as the growth history and age of the inner core, and so it is important that these are chosen appropriately. For simplicity we shall assume that the ”freezing” of the inner core is purely pressure induced as described in section 6.3. If we then assume that the melting temperature of the inner core is governed by the Lindemann law of melting (see eq.(C.45)) we have our equation for the initial temperature. Now, the Lindemann law ﬂattens out towards the center of the Earth, hence it is reasonable to assume that the growth rate of the inner core radius was highest when the inner core was newly born, and then successively

63Naturally we will also have to assume the heat capacity, thermal conductivity, density, and internal heat generation of the sphere.

70 decreased. For simplicity then, we will assume that the inner core has grown at constant volume per time unit, hence the radius at time t will be given by

t 1/3 r(t) = R0 (E.10) t0

where R0 is the present day inner core radius and t0 is the age of the inner core (see ﬁgure 26). Note that this demands an assumed age of the inner core. Equation (E.10) can be justiﬁed by the results of numerical models of the inner core growth (e.g. [78]) whose output indicates a rapid growth of the inner core at early times.

Figure 26: Assumed evolution of the ICB radius according to eq.(E.10)

E.3 Numerical issues

Evaluating eq.(E.4) and (E.7), some caution must be taken due to the integral in equation (E.5) and (E.8).

Z rm 0 0 0 nπr 0 r To(r ) sin dr o rm As the frequency of the sinusoidal will increase for increasing summation index. Let us start by examining why this P would be a problem and how it would manifest in the solution. In principle we write u as: Tm+ n an sin(bnr)/r, P 2 and ∂w/∂r as: an(bn cos(bnr)/r − sin(bnr)/r ). If we check the limit as r → 0 we ﬁnd that u becomes P n Tm + anbn whilst ∂w/∂r becomes 0 at the center of the Earth (i.e. u attains a maximum value at the center). n P Likewise if we take the limit as r → rm we ﬁnd that u becomes Tm whilst ∂w/∂r becomes n anbn/rm at the ICB (rm). Lets us then investigate the behaviour of an, we have that

2 2 rm 0 n − κn π t 1 Z nπr (−1) r2 0 0 0 an = 2e m r To(r ) sin dr + rmTm (E.11) rm o rm nπ Now given the same initial temperature and radius the expression inside the square brackets will only depend on the summation index, whereas the exponential is also dependent on t. We know the exponential is bounded in the interval ]0 1] with lower values for higher n. Hence it will damp out the terms for increasing summation index. However note that if we decrease t the damping will be less efﬁcient, and therefore the convergence of u will also become slower for lower t. This means that we will have to increase the number of computed terms in the series to reach a given accuracy. Let us deﬁne the decay number of the exponential as the value of n when the exponential

71 has decreased to 1/e of its original value (i.e. at n = 1). Some elementary algebra then gives at hand that r r2 rτ n = m = (E.12) κπ2t t

Where τ is a measure of the diffusion time. Lets estimate the value of n for the present day inner core, we have 6 −6 2 that rm ≈ 1.22 x 10 m and κ ≈ 7 x 10 m /s. Specifying t in years instead of seconds then gives us an estimate on n as r 7 x 108 ryr n ≈ IC t yr which can be used as a very crude estimate on the number of terms needed to reach an acceptable accuracy. Hence we see that if we choose t = one hundred thousand years (1 x 105) we would need to compute of the order of the ﬁrst 80 terms of the series. This is a very crude estimate, but gives us an order of magnitude of terms needed.

0 Consider then the evaluation of the expression within the curly brackets. First of all we shall notice that if To(r ) = 0 Tm the expression will be equal to zero. Whereas if To(r ) > Tm the expression will be positive for n = 1 whereas the sign of following terms depends on the exact shape of initial temperature profile (see figure 27). Generally it holds though that if the Initial temperature profile is smooth and just slightly to the boundary temperature, the sign will oscillate, being positive for odd n a negative for even n. In any case the value of the integral will approach the value of the other term (referred to as boundary term from now on) inside the brackets as n increases. Meanwhile the magnitude of the integrals value decreases, however, this decrease is slower than the decrease of the difference between the integral and the boundary term. But this means that we need to solve the integral with an increasingly higher accuracy as n increases. If we assume that we need to sum 80 terms to reach a high enough accuracy his 0 means that the last term involves an integral over sin(80πr /rm). The problem here is that we have the initial temperature profile sampled at a number of discrete points, so it is not possible to evaluate the integral with a higher accuracy without adding extra points to our grid. Before we do this, let us stop for a moment and consider why we need to have an accurate value at all. Surly every term becomes progressively smaller so what harm cold the then posse.

Figure 27: r’sin(nπr’) for n ∈ [1 5]

As the frequency increases the last maxima/minima of the sinusoidal shifts progressively closer to the boundary. This means that unless the higher frequencies get properly damped out, they will have the possibility to have a great non-physical affect at the boundary on the ﬁnal temperature proﬁle. I.e. Temperatures and gradients in the vicinity of the boundary will not be physical. In the scheme used in this model we need to evolve the temperature

72 proﬁle several times, thus such an error can potentially propagate further into the interior of the model, as the evolved temperature proﬁle will serve as the input for next time step. In fact, until the problem was realised, the initial version of the model displayed thermal gradients near the boundary with a decay > r2, indicating that energy from the interior parts had to be deposited in a thin layer just inside of the boundary. Which is not physical given the problem investigated. Thus, for the model to be successful we need to be able to solve the integral with high accuracy.

There are at least three ways to solve the integral with higher accuracy. The first and most simple would be to increase the number of radial grid points, however this increases the computational time enormously. Tests with the early version of the model indicated that a radial grid of the order of 106 grid points was necessary to approach a reasonable solution, resulting in computational times of the order of a few hours, in addition this puts rather high demands in terms of memory and is not recommended. A second approach would be use some type of adaptive grid, it would be possible to either feed in a very fine grid and initially only use, say every 100/n grid point for the evaluation of the integral, i.e. as n increases a larger sub sample of the grid points would be used. Alternatively one could start with a coarse grid and as n increases interpolate onto an increasingly finer grid. This would save some time swell as some memory, however not enough still yielding computational times of the order of hours. The third alternative is then to try to derive some analytical approximation, in which the error can be controlled, without the necessity of a ridiculously fine grid. And this is exactly what I shall do. Assume then that we have a temperature profile

T = [T (r0),T (r1),T (r2), ..., T (rm)] (E.13) is sampled at m + 1 radial coordinates r = [r0, r1, r2, ..., rm] (E.14) where we deﬁne

r0 = 0

rm = rICB(t)

For the given problem we know that T (r) can for all t be describe by a smooth function, hence we can locally in between ri and ri+1 approximate T (r) by a polynomial of degree w as

w X k T (r) = ck,ir ri ≤ r ≤ ri+1 (E.15) k=0

Note that T (r) is a continuous function, but generally not continuously differentiable. As integrals are linear operators we can then write our integral as

m−1 w Z ri+1 X nπr X k+1 sin ck,ir dr (E.16) rm i=0 ri k=0 We can integrate this expression by parts to yield

m−1 " w w ! X rm nπri X k+1 nπri+1 X k+1 cos ck,iri − cos ck,iri+1 nπ rm rm i=0 k=0 k=0 (E.17) w # Z ri+1 0 rm nπr X k + cos (k + 1)ck,ir dr nπ rm ri k=1 which in turn can be integrated by parts again to yield

73 m−1 " w w ! X rm nπri X k+1 nπri+1 X k+1 cos ck,iri − cos ck,iri+1 nπ rm rm i=0 k=0 k=0 2 w w ! rm nπri+1 X k nπri X k + sin (k + 1)ck,iri+1 − sin (k + 1)ck,iri (E.18) nπ rm rm k=0 k=0 w # 2 Z ri+1 rm nπr X k−1 + cos k(k + 1)ck,ir dr nπ rm ri k=2 which in turn can be integrated by parts, yielding an even worse expression. Continuing to integrate by parts and rearranging the ﬁnal results we ﬁnd that we can write the solution on the form m−1 X nπri+1 nπri F 0(r ) cos − F 0(r ) cos i i+1 r i i r i=0 m m (E.19) 1 nπri+1 1 nπri +Gi (ri+1) sin − Gi (ri) sin rm rm

where

(w+o)/2  2j  2j+1 w " # X  rm X Y  F a(r) = (−1)j+1 c rk−2j+1 (k + 2 − l) i nπ k,i j=a  k=2j−1 l=1  (E.20) (w−o)/2+1  2j−1  2j w " # X  rm X Y  Gb(r) = (−1)j+1 c rk−2(j−1)k (k + 2 − l) i nπ k,i j=b  k=2(j−1) l=1 

and ( 1 if w is odd o = (E.21) 0 if w is even and we have used the convention that a negative summation index is equal to 0. For the given problem we wish to solve we know that the temperature behaves nicely, with a gradual but slow decrease from the center to the ICB. And so, to a first order approximation we can assume that the geotherm varies linearly in between the sampled points, i.e. we have that  0 if j > 1   T (ri+1)−T (r1) cj,i = if j = 1 (E.22) ri+1−r1   T (ri) − c1,iri if j = 0 Hence using eq.(E.19) and (E.20) and w = 1, we find that the integral can be approximated by m−1 rm X nπ nπ c r + c r2 cos r − c r + c r2 cos r nπ o,i i 1,i i r i o,i i+1 1,i i+1 r i+1 i=0 m m rm nπ nπ (E.23) + (co,i + 2c1,iri+1) sin ri+1 − (co,i + 2c1,iri) sin ri nπ rm rm 2 rm nπ nπ + 2c1,i cos ri+1 − cos ri nπ rm rm However we can simplify eq.(E.23) even further, first of all we note that according to relation (E.22) above, the input temperature will be described by T (r) = c0,i + c1,iri in the interval ri ≤ r ≤ ri+1, and so the first two cosine terms in eq.(E.23) can be written as nπ nπ riT (ri) cos ri − ri+1T (ri+1) cos ri+1 rm rm

74 which is on the form Hi − Hi+1 Likewise we can write the sinusoidal components as rm nπ nπ (T (ri+1) + c1,iri+1) sin ri+1 − (T (ri) + c1,iri) sin ri nπ rm rm rm nπ nπ = −T (ri) sin ri + T (ri+1) sin ri+1 nπ rm rm rm nπ nπ + c1,i ri+1 sin ri+1 − ri sin ri nπ rm rm

Where we again can write the ﬁrst parts on the form Hi − Hi+1. Consider then the summation over such a series m−1 X Hi − Hi+1 = H0 − H1 + H1 − H2 + H2 − H3 + ... + Hm−2 − Hm−1 + Hm−1 − Hm = H0 − Hm i=0

Further we have that H0 = 0, in the cosines case due to the multiplying r0 = 0, and in the sinus case due to sin(0) = 0. Whereas for Hm we ﬁnd that sinus term vanishes since sin(nπ) = 0, whilst the cosine term can be written as cos(nπ) = (-1)n. Hence our approximation of the integral becomes

n 2 m−1 (−1) rm X nπ nπ − r2 T (r ) + c r sin r − r sin r nπ m m nπ 1,i i+1 r i+1 i r i i=0 m m 3 m−1 (E.24) rm X nπ nπ + 2 c cos r − cos r nπ 1,i r i+1 r i i=0 m m

n where we have used ro = 0 = sin(0) = sin(nπ), and -cos(nπ) = -(-1) . Going back to eq.(E.11) above we ﬁnd that the expression inside of the curly brackets can be approximated by

n m−1 rm(−1) rm X nπ nπ (Tm − T (rm)) + 2 c1,i ri+1 sin ri+1 − ri sin ri nπ (nπ) rm rm i=1 (E.25) m−1 2r2 X nπ nπ + m c cos r − cos r (nπ)3 1,i r i+1 r i i=1 m m where c1,i is equal to the thermal gradient at radius ri. However note that by the boundary conditions given above that T (rm) = Tm for t > 0, hence if this also holds at t = 0 the first term is equal to zero and we are left with the series term only. Now eq.(E.25) holds if the initial thermal profile can be locally approximated by a first order polynomial. However, we could as well have assumed it to be locally approximated by a polynomial of order w given by eq.(E.15), using the same steps as in the 1st order case we then find a general solution as

n m−1 rm(−1) nπ X 1 nπ 1 nπ (Tm − T (rm))+ 2 riFi (ri+1) sin ri+1 − riFi (ri) sin ri nπ rm rm rm i=0 (E.26) m−1 1 X nπ nπ + F 1(r ) cos r − F 1(r ) cos r r i i+1 r i+1 i i r i m i=0 m m 1 0 0 Note the use of Fi as compared to Fi in eq.(E.19), this is due to the fact that the j = 0 term of Fi goes into T (rm).

Now even if eq.(E.24) looks somewhat complicated it is actually relatively simple to implement into some code or script, and it has the clear benefit of being relatively sensitive to the sampling interval as the frequency of the sinusoidal increase for increasing n. Now it could be tempting to use a higher order polynomial to fit the temperature profile with. However, as we are using the polynomial coefficients, and a higher order polynomial always posses a risk of over shooting this might induce a large error source in our approximation. Instead what should be done is to choose a larger number of radial grid points for the problem, thus increasing the validity of the local linear approximation of the thermal profile. Note also that the radial coordinates does not need to be equally spaced. Thus, an alternative approach if the initial temperature has some ”sharp” feature would be to locally increase the sampling in the vicinity of this feature.

75 E.4 Testing

To validate the output from the model two different schemes has been used, the first one is to match the output to that generated by an equivalent model using COMSOL multiphysics. As I have not figured out a neat way to implement a growing geometry in COMSOL64 the models compared has had a fixed radius with either constant, linearly varying, or quadratic varying boundary temperature (always decreasing). For the models with fixed boundary temperature it is found the absolute differences are of the order of 10−3 K at maximum. In fact the COMSOL output converges towards the solution given by my implementation of the analytical expression. For the models with varying boundary temperature the fit is somewhat worse, but generally less than 1 K over a time span of 1.5 Gyr (It should be noted that converted to percentage of temperatures above the ICB temperature this translates into less than 0.5 %). However, it should be noted here that for some models the fit gets worse at decreasing time steps. Unfortunately I do not really understand why this should be so (logically the fit should be better as the time step decrease), and this is rather worrying.

(a) (b)

Figure 28: Comparison between output of transient model and equivalent COMSOL model of radius 1220 km. All models assumes an initial temperature profile ∝ −r2 with a temperature difference between the centre and boundary of 279 K, or equivalently corresponding to a steady state model with evenly distributed heat sources of 9 x 10−8 W m−3. Relevant −1 −1 −1 −1 common thermal parameters for the models are k = 80 W m K ,Cp = 860 J kg K , and ρ = 12909 kg m3. Figure 28(a) and 28(b) displays comparisons between models with constant boundary temperature and no internal heating, 28(a), or evenly distributed heat sources of 9 x 10−8 W m−3, 28(a). Both models are sampled and evolved on equally spaced grids with dr = 1.22 km and dt = 30 Myr. Computational times of the models are approximately 15 s. Figure 28(c) and figure 28(d) has the same amount of internal heat sources as figure 28(b), but with a boundary temperature varying linearly, 28(c), or quadratic, 28(d), with -300 K over the model evolution. Both models are sampled and evolved on equally spaced grids with dr = 1.22 km and dt = 1.5 Myr. Computational times of the models are approximately 150 s.

64Otherwise I would have built the model in COMSOL

76 Also note the sharp difference in ﬁgure 28(c) and 28(d) in the vicinity of the boundary, as discussed above. Hence the model is not suitable for thermal gradients near the boundary.

For the final models, where also the radius is being evolved, an energy scheme has been employed instead. The idea is to compute the primordial heat, the internal heat, the radiogenic heat, and the total heat flow, and compare these, see figure 29(a). The primordial heat is then defined as the initial temperature of the inner core (i.e. set by the melting temperature) integrated over the volume of the inner core and multiplied by mass and heat capacity. Likewise the internal heat is the final thermal profile integrated over the volume of the inner core and multiplied by mass and heat capacity. The radiogenic heat is the heat released by radiogenic decay inside the inner core over the lifetime of the inner core, whereas the total heat flow is the time and surface integrated heat flow out of the core. Ideally the primordial and radiogenic heat should add up to the same value as the negative heat flow and the internal heat. However, being a numerical scheme where we nowhere actually adjust the solution for energy conservation we should expect some deviations from this. Expressed in temperatures it is found that the temperature of the final models are about 1 degrees to high (if we disperse the excess energy evenly over the inner core, see figure 29(b)), which is acceptable given the simplicity of the model.

(a) (b)

Figure 29: Energy balance of transient model. Figure 29(a) displays the various contributions to the energy balance, ﬁgure 29(b) displays the resulting average temperature residual of the model (positive temperatures indicates excess energy). Model assumes same thermal parameters as the models in ﬁgure 28 as well as γth = 1.5 and a boundary temperature of 5500 K. The model is sampled on a grid of dr = 1.2 km, dt = 0.5 Myr. approximate computational time of model = 420 s.

E.5 CoreT.m function [error Tout rout dTout] = CoreT(k,Cp,rho,gamma,A,tau,Ricb,Tm,... Age,dt,coords_int,soln,... coords_soln,LogName,ModelName,... dTproced,Progress,Err,tol); if(nargin == 0) % ======% Program CoreT evaluates temperature profiles over the inner core given an % initial ammount of heat produced per unit volumen with an exponential % decay. % % The model assumes an initial temperature equal to the melting % temperature, as calculated using the Lindemann law of melting, assuming % PREM densities. No redistribution of the density profile due to inner % core formation is considered. % % The inner core is assumed to grow at constant volume per unit time.

77 % % The model can be run in two different ways: % % 1 All model parameters are set below in the model parameter % space and the model is started by typing: CoreT % in the MATLAB terminal % 2 All model parameters are specified as arguments in the call % to model, i.e.: CoreT(k,Cp,rho,...,Popt) % either directly from the MATLAB terminal or via a MATLAB % script. This way several models can be run directly after % each other whitout the need of manually changing model % parameters inside this file, further this might be % desirable if other postproccesing than the offered is % wanted % % If used in the second way output variables are % % ouput: size: Comment/value: % error scalar = 0 if Err = 0, else = -1 if thermal % gradient becomes negative locally, = -2 % central temperature increases, = -3 if both % thermal gradient becomes locally negative % central temperature increases. % Tout (ceil(Age/soln) + 1) Every row contains a single geotherm, first % X coords_soln row corresponds to initial getherm guess % as constrained by the Lindemann law, last % row contains final geotherm at t = Age. % Note that the time step between the last % and the second last row might be less than % soln. % rout (ceil(Age/soln) + 1) Every row contains radial coordinates % X coords_soln corresponding to the temperatures given by % Tout or dTout % dTout (ceil(Age/soln) + 1) Same construction as Tout above, but for % X coords_soln the getheral gradient % % A small script is given below that calls the model and plots the % geotherm and gradient as 3D surfaces. % ======% % MODEL PARAMETERS % % ======% Model parameter Description Units Comments % (see below) k = 80 ;% Thermal conductivity [W/m/K] 1 Cp = 860 ;% Heat capacity [J/kg/K] 2 rho = 12909 ;% Density [kg/mˆ3] 3 gamma = 1.5 ;% Gruneisen parameter at ICB 4 A = 105.27e-9 ;% Heat generation at t = 0 [W/kg] 5 tau = 1.25e9 ;% Half life [year] 6 Ricb = 1.22e6 ;% Present day inner core radius [m] 7 Tm = 5500 ;% Melting temperature at ICB [K] 8 Age = 1.5e9 ;% Age of inner core [year] 9 dt = 1.5e7 ;% Timestep of model evolution [year] 10 coords_int = 2e2 ;% Number of radial gridpoints 11 soln = 1.5e7 ;% Timestep of output [year] 12 coords_soln = 200 ;% Number of radial gridpoints in % output 13 LogName = ’none’; ;% Name of logfile 14

78 ModelName = ’none’; ;% Name of file to store output in 15 dTproced = 1 ;% dT evaluation procedure 16 Progress = 1 ;% Progress monitoring switch 17 Err = [0] ;% Error control 18 tol = 1e-6 ;% T tolerance in each timestep and ;% coordinate 19

% ======% Comments % % 1 Constant % % 2 Constant % % 3 Constant % % 4 Constant % % 5 Constant, N.B. Initial value, present day value can be found using %A*exp(-l*Age). 1e-9 is approximately equal to 22.37 ppm K40, or % 0.03 ppm U, or 0.0081 ppm Th232 % % 6 Constant. l(K40) = 1.2511e9, l(U238) = 4.4683e9, l(Th232) = 1.401e10 % % 7 Constant % % 8 Constant % % 9 Constant % % 10 Constant, Note that a large dt will lead to a large uncertainty in the % model, whilst a small dt will increase the computational time. It is % not recomended to choose dt < 1e4 % % 11 Constant value, can be lower than coords_soln % % 12 Constant, time interval between stored solutions in output, if % smaller than dt extra solutions will be found by interpolation % % 13 Constant, number of equally spaced radial grid points in output, if % larger the coords_soln (10) extra grid points will be found by % interpolation. % % 14 The log file gives the number of iterations for every solution step. % No log file will be written if specified as ’none’ % % 15 The model file is an ascii file written with a reader friendly format. % No log file will be written if specified as ’none’. A small script % is given below that will read the variables and model into MATLAB. % % 16 This parameter sets the scheme for evaluation of the geogradient. % availible options are: % % dTproced = Evaluation scheme % 0 No evaluation of dT. N.B. if dTproced is set to 0, Popt % equal to 2,3,5,6,10,11,18,19, or 20 will not generate any % plots % 1 Linear interpolation using evaluated temperature profile % and radial grid % 2 Analytical expression. N.B. the evaluation using an

79 % analytical expression will double the computational time % of the model. The thermal profile is evaluated to a % tolerance = tol*1e-4 % % 17 Determines wheter or not the progress of the model evaluation is % displayed in the terminal window. % % Progress = Action % 0 Progress NOT displayed in terminal window % 1 Progress displayed in terminal window % % 18 Two different error controls are availible during the evluation of the % model. If violated the program will halt and produce asked for plots % (see 14 below) after which control is return to the keyboard. % Availible error controls are % % Err = Action % 0 No Error control % 1 Halts if thermal gradient becomes negative locally % 2 Halts if central temperature increases % 3 Combines Err = 1 and Err = 2 % % 19 The analytical expression for each timestep involves two fourier series % the terms of each series is evluated until the minimum deviation at a % single coordinate is less than tol. % % ------% SCRIPT FOR READING IN THEW MODEL FILES INTO MATLAB % (see model file for more information on content) % ------%% fid = fopen(’TestModel15e6.txt’); %% param = textscan(fid,’%n’,10,’commentStyle’,’%’); %% k = param{1}(1); %% cp = param{1}(2); %% rho = param{1}(3); %% gamma = param{1}(4); %% A = param{1}(5); %% tau = param{1}(6); %% Tm = param{1}(7); %% Age = param{1}(8); %% dt = param{1}(9); %% soln = param{1}(10); %% tmp = textscan(fid,’%n’,2,’commentStyle’,’%’); %% nn = textscan(fid,’%d’,2,’commentStyle’,’%’); %% %% for i = 1:nn{1}(2) %% tt = textscan(fid,’%n’,1,’commentStyle’,’%’); %% t(i,1) = tt{1}(1); %% for j = 1:nn{1}(1) %% rt = textscan(fid,’%n’,1,’commentStyle’,’%’); %% r(i,j) = rt{1}(1); %% end %% for j = 1:nn{1}(1) %% Tt = textscan(fid,’%n’,1,’commentStyle’,’%’); %% T(i,j) = Tt{1}(1); %% end %% for j = 1:nn{1}(1) %% dTt = textscan(fid,’%n’,1,’commentStyle’,’%’); %% dT(i,j) = dTt{1}(1); %% end %% end

80 %% clear(’param’,’tmp’,’n’,’tt’,’rt’,’Tt’); %% fclose(fid); % ------% % SCRIPT FOR CALLING THE MODEL AND PLOTTING THE RESULTING % GEOTHERM AND GRADIENT AS 3D SURFACES % % ------%%% Model parameter Description Units %% k = 80 ;% Thermal conductivity [W/m/K] %% Cp = 860 ;% Heat capacity [J/kg/K] %% rho = 12909 ;% Density [kg/mˆ3] %% gamma = 1.5 ;% Gruneisen parameter at ICB %% A = 90e-9 ;% Heat generation at t = 0 [W/kg] %% tau = 1.25e9 ;% Half life [year] %% Ricb = 1.22e6 ;% Present day inner core radius [m] %% Tm = 5500 ;% Melting temperature at ICB [K] %% Age = 1.5e9 ;% Age of inner core [year] %% dt = 1.5e7 ;% Timestep of model evolution [year] %% coords_int = 2e2 ;% Number of radial gridpoints %% soln = 1.5e7 ;% Timestep of output [year] %% coords_soln = 200 ;% Number of radial gridpoints in output %% LogName = ’none’ ;% Name of logfile %% ModelName = ’none’ ;% Name of file to store output in %% dTproced = 2 ;% dT evaluation procedure %% Progress = 0 ;% Progress monitoring switch %% Err = [0] ;% Error control %% tol = 1e-6 ;% T tolerance in each timestep and coordinate %% %% [error Tout rout dTout] = CoreT(k,Cp,rho,gamma,A,tau,Ricb,Tm,... %% Age,dt,coords_int,soln,... %% coords_soln,LogName,ModelName,... %% dTproced,Progress,Err,tol,... %% Popt,Palt); %% %% year = 365.25*24*3600; %% tgrid = [-Age:Age/ceil(Age/soln):0]./1e9; %% rgrid = [0:Ricb/(coords_soln-1):Ricb]; %% Tgrid(1:(ceil(Age/soln) + 1),1:coords_soln) = 0.; %% for i = 1:(ceil(Age/soln) + 1) %% Tgrid(i,:) = interp1(rout(i,:),Tout(i,:),rgrid); %% end %% %% dTgrid(1:(ceil(Age/soln) + 1),1:coords_soln) = 0.; %% for i = 1:(ceil(Age/soln) + 1) %% dTgrid(i,:) = interp1(rout(i,:),dTout(i,:),rgrid); %% end %% %% figure(’color’,’w’) %% surf(rgrid/1e3,tgrid,Tgrid,’linestyle’,’none’) %% xlabel(’R [km]’) %% ylabel(’Age [Gyr]’) %% zlabel(’T [K]’) %% set(gca,’XDir’,’rev’,’YDir’,’rev’) %% %% figure(’color’,’w’) %% surf(rgrid/1e3,tgrid,dTgrid*1e3,’linestyle’,’none’) %% xlabel(’R [km]’) %% ylabel(’Age [Gyr]’) %% zlabel(’dT/dr [K kmˆ{-1}]’) %% set(gca,’XDir’,’rev’,’YDir’,’rev’)

81 % ======% ************************************************************************* % ======% % DO NOT EDIT THE LINES BELOW % % ======end % set number of coordinates used by the solver ------

% set inital parameters ------year = 365.25*24*3600; % seconds in a year Age = Age*year; % ======dt = dt*year; % Converting specified times to units of seconds soln = soln*year; % ======decC = log(2)/(tau*year);% Decay constant t = 0; % time of current solution SolnCount = 1; % Stored solution counter Prog = 0; % Progres of model counter error = 0; % Error control parameter

% Find the number of time steps of the model ------Fstep_mod = floor(Age/dt); % Number of full time steps to evaluate res_mod = Age-Fstep_mod*dt; % Resudial timestep if(res_mod == 0); step_mod = Fstep_mod; % Total number of timesteps to evaluate else step_mod = Fstep_mod+1; % Total number of timesteps to evaluate end

% Model progress counter ------if(Fstep_mod > 1000); Prog_time_fact = 0.01; % Time scale factor dProg = 1; % Model progress increment else Prog_time_fact = 0.1; % Time scale factor dProg = 10; % Model progress increment end Prog_time = Age*Prog_time_fact;% Time variabel for model progress counter teval = 0; % Time estimate control variabel da = 0; % Time estimate variabel

% Find the number of solutions to store ------m = floor(Age/soln); % Number of full solution times res_soln = Age-m*soln; % Resudial timestep of solution if(res_soln == 0) step_soln = m+1; % Total number of solutions (including initial step) else step_soln = m+2; % Total number of solutions (including initial step) end

% Initiate variables to store solution in ------tsoln(1:step_soln) = 0; % Solution time array rsoln(1:step_soln,1:coords_soln) = 0; % Radial coordinates solution array Tsoln(1:step_soln,1:coords_soln) = 0; % Temperature solution array if(dTproced ˜= 0) dTsoln(1:step_soln,1:coords_soln) = 0; % Temperature gradient else % solution array

82 dTsoln = 0; end

% Connect logfile and model files ------if(strcmp(LogName,’none’)) logf = -1; else logf = fopen(LogName,’w’); end if(strcmp(ModelName,’none’)) Modf = -1; else Modf = fopen(ModelName,’w’); end

% Write model information to model file ------if(Modf ˜= -1) fprintf(Modf,[’\n%% This file contains the result from a run by ’... ’the program CoreT.\n%% All values are given in SI units except’... ’ temporal quantities\n%% which are given in units of years, ’... ’and the thermal gradient\n%% which is given in units of K/km.\n’]); fprintf(Modf,’\n%% Model not fully solved .\n’); fprintf(Modf,[’\n%% Input parameters (see comments in CoreT for ’... ’more information):\n’]); fprintf(Modf,[’%% k Cp rho Gamma A tau’... ’ Tm Age dt soln\n’]); fprintf(Modf,[’%6.2f %7.2f %9.2f %5.2f %10.2e %10.2e %8.2f %10.2e ’... ’%10.2e %10.2e\n’],k,Cp,rho,gamma,A,tau,Tm,Age/year,dt/year,... soln/year); fprintf(Modf,’\n%% Solver parameters:\n’); fprintf(Modf,’%% Radial gridpoints Temporal gridpoints\n’); fprintf(Modf,’%12i %18i\n’,coords_int,step_mod); fprintf(Modf,[’\n%% Solutions in this file\n’...’ ’%% Radial Temporal\n%8i %9i\n’],coords_soln,step_soln); end

% Initiate solver and timing of solver ------if(Progress == 1) display(’Solving model:’) end tic % Initiate timing of model evolution

% Solve full time steps of model ------for i = 1:Fstep_mod if(i == 1) % Initial thermal profile ------[Ttemp,r] = LindTExt(-1,coords_int,gamma,Tm,Age,Ricb,dt); T = Ttemp; r(1) = 0.01; % Set first coordinate to 1cm to avoid % singularities in solver

% Store initial thermal profile and write to model file ------tsoln(SolnCount) = t; rsoln(SolnCount,:) = [r(1):(r(coords_int)-r(1))/... (coords_soln-1):r(coords_int)]; Tsoln(SolnCount,:) = interp1(r,T,rsoln(SolnCount,:)); if(dTproced ˜= 0) dT = (T(1:coords_int-1)-T(2:coords_int))./... (r(1:coords_int-1)-r(2:coords_int)); dTr = r(1:coords_int-1)+(r(1:coords_int-1)-r(2:coords_int))./2;

83 dTsoln(SolnCount,2:coords_soln) = ... interp1(dTr,dT,rsoln(SolnCount,2:coords_soln),... ’cubic’,’extrap’); end if(Modf ˜= -1) fprintf(Modf,’\n%% Initiating profile at t:\n’); fprintf(Modf,’%6i\n’,t/year); fprintf(Modf,’\n%% r:\n’); fprintf(Modf,’% 6.2f’,rsoln(SolnCount,:)); fprintf(Modf,’\n\n%% T:\n’); fprintf(Modf,’% 8.4f’,Tsoln(SolnCount,:)); if(dTproced ˜= 0) fprintf(Modf,’\n\n%% dT:\n’); fprintf(Modf,’% 8.4f’,dTsoln(SolnCount,:)*1000); end end else % Increase the radius ------[Ttemp,r] = LindTExt(T,r,gamma,Tm,Age,Ricb,dt); end

% Store T-profile before temporal evolution and prepare logfile ------if(logf ˜= -1) fprintf(logf,’\n t = %f year\n’,t/year); end T_old = T; % Temperature profile before temporal evolution

% Evolve the thermal profile ------if(logf ˜= -1) T = Tspht(1,Ttemp,r,k,rho,Cp,r(coords_int),Ttemp(coords_int),... t,dt,A,decC,tol,logf); if(dTproced == 2) dT = Tspht(-1,Ttemp,r,k,rho,Cp,r(coords_int),... Ttemp(coords_int),t,dt,A,decC,tol,logf); end else T = Tspht(1,Ttemp,r,k,rho,Cp,r(coords_int),Ttemp(coords_int),... t,dt,A,decC,tol); if(dTproced == 2) dT = Tspht(-1,Ttemp,r,k,rho,Cp,r(coords_int),... Ttemp(coords_int),t,dt,A,decC,tol); end end t = t+dt; % Increase the time variabel

% Error control ------

if((Err == 1) | (Err == 3)) % Negative thermal gradient dTmod = T(1:coords_int-1)-T(2:coords_int); if(min(dTmod) < 0) display([’Solution halted at t = ’ num2str(t)... ’ due to a negative temperature gradient’]) error = -1; break end end

if((Err == 2) | (Err == 3)) % Increasing central temperature if(T(1) > T_old(1)) display([’Solution halted at t = ’ num2str(t)... ’ due to an increasing central temperature’])

84 error = error-2; break end end

if(error < 0) break end

% Store solution ------if((t-SolnCount*soln) >= 0) SolnCount = SolnCount+1; % Adjust solution counter tsoln(SolnCount) = t; rsoln(SolnCount,:) = [r(1):(r(coords_int)-r(1))/... (coords_soln-1):r(coords_int)]; Tsoln(SolnCount,:) = interp1(r,T,rsoln(SolnCount,:)); if(dTproced == 1) dT = (T(1:coords_int-1)-T(2:coords_int))./... (r(1:coords_int-1)-r(2:coords_int)); dTr = r(1:coords_int-1)+(r(1:coords_int-1)-r(2:coords_int))./2; dTsoln(SolnCount,2:coords_soln) = ... interp1(dTr,dT,rsoln(SolnCount,2:coords_soln),... ’cubic’,’extrap’); elseif(dTproced == 2) dTsoln(SolnCount,:) = interp1(r,dT,rsoln(SolnCount,:)); end if(Modf ˜= -1) fprintf(Modf,’\n\n%% Profile at t:\n’); fprintf(Modf,’%6i\n’,t/year); fprintf(Modf,’\n%% r:\n’); fprintf(Modf,’% 6.2f’,rsoln(SolnCount,:)); fprintf(Modf,’\n\n%% T:\n’); fprintf(Modf,’% 8.4f’,Tsoln(SolnCount,:)); if(dTproced ˜= 0) fprintf(Modf,’\n\n%% dT:\n’); fprintf(Modf,’% 8.4f’,dTsoln(SolnCount,:)*1000); end end end

% Display progress of model ------

if(Progress == 1) if(t > Prog_time) Prog = Prog + dProg; Prog_time = Prog_time+Age*Prog_time_fact; display([num2str(Prog) ’% of model solved’]) teval = teval+1; if(teval == 2) da = toc; tic display([’Estimated evaluation time for model is ’... num2str(da/(2*Prog_time_fact)) ’ s’]); end end end end

% Evaluate resudial time step ------

85 if(res_mod ˜= 0) [Ttemp,r] = LindTExt(T,r,gamma,Tm,Age,Ricb,res_mod); if(LogName == ’none’) T = Tspht(1,Ttemp,r,k,rho,Cp,Ricb,Tm,t,res_mod,A,decC,tol); else T = Tspht(1,Ttemp,r,k,rho,Cp,Ricb,Tm,t,res_mod,A,decC,tol,logf); end if(dTproced == 2) if(LogName == ’none’) dT = Tspht(-1,Ttemp,r,k,rho,Cp,Ricb,Tm,t,dt,A,decC,tol); else dT = Tspht(-1,Ttemp,r,k,rho,Cp,Ricb,Tm,t,dt,A,decC,tol,logf); end end end

% Store resudial model ------if((res_soln ˜= 0) | ((SolnCount) < step_soln)) SolnCount = SolnCount+1; tsoln(SolnCount) = t; rsoln(SolnCount,:) = [r(1):(r(coords_int)-r(1))/... (coords_soln-1):r(coords_int)]; Tsoln(SolnCount,:) = interp1(r,T,rsoln(SolnCount,:));

if(dTproced == 1) dT = (T(1:coords_int-1)-T(2:coords_int))./... (r(1:coords_int-1)-r(2:coords_int)); dTr = r(1:coords_int-1)+(r(1:coords_int-1)-r(2:coords_int))./2; dTsoln(SolnCount,2:coords_soln) = ... interp1(dTr,dT,rsoln(SolnCount,2:coords_soln),... ’cubic’,’extrap’); elseif(dTproced == 2) dTsoln(SolnCount,:) = interp1(r,dT,rsoln(SolnCount,:)); end if(Modf ˜= -1) fprintf(Modf,’\n\n%% Profile at t:\n’); fprintf(Modf,’%6i\n’,t/year); fprintf(Modf,’\n%% r:\n’); fprintf(Modf,’% 6.2f’,rsoln(SolnCount,:)); fprintf(Modf,’\n\n%% T:\n’); fprintf(Modf,’% 8.4f’,Tsoln(SolnCount,:)); if(dTproced ˜= 0) fprintf(Modf,’\n\n%% dT:\n’); fprintf(Modf,’% 8.4f’,dTsoln(SolnCount,:)*1000); end end end

% Display solution time of model and adjust model file ------time = toc; if(Progress == 1) display([’Model solved in ’ num2str(time+da) ’seconds, plotting’]); end fclose(’all’); if(Modf ˜= -1) Modf = fopen(ModelName,’r+’); fprintf(Modf,[’\n%% This file contains the result from a run by ’... ’the program CoreT.\n%% All values are given in SI units except ’...

86 ’temporal quantities\n%% which are given in units of years, and ’... ’the thermal gradient\n%% which is given in units of K/km.\n’]); fprintf(Modf,’\n%% Model solved in %f seconds’,time); fclose(’all’); end

% Assign output values if(nargout == 2) Tout = Tsoln; elseif(nargout == 3) Tout = Tsoln; rout = rsoln; elseif((nargout == 4) & (dTproced ˜= 0)) Tout = Tsoln; rout = rsoln; dTout = dTsoln; elseif((nargout == 4) & (dTproced == 0)) Tout = Tsoln; rout = rsoln; dTout = NaN; end

% ======% % SUB FUNCTIONS % % ======% Function that evaluates the temperature profile or temperature gradient % over a sphere. % % [T] = Tspht(T0,r,k,rho,Cp,R,TR,t,dt,A,decC,logf) % % Input parameters: % % type if = 1,the temperature profile is evaluated, if ˜= 1, the % temperature gradient is evaluated. % T0 Initial radial temperature profile [K] % r Radial coordinates at which T0 is given [m] % k Thermal conductivity [W/m/K] % Cp Heat capacity [J/kg/K] % rho Density [kg/mˆ3] % R Radius of the sphere [m] % TR Boundary temperature [K] % t Time at which T0 is given [s] % dt Timestep at which to evaluate new temperature profile [s] % A Heat generation at t = 0 [W/kg] % decC Decay constant [1/s] % tol tolerance [K] % log logfileID % % Output parameter: % % T Temperature or gradient at time t + dt % ------function T = Tspht(type,T0,r,k,rho,Cp,R,TR,t,dt,A,decC,tol,logf) n = size(r,2); if(r(n) == R) m = n; else m = 1; end

87 i = 0; j = 0; U(1:n) = 0; U1(1:n) = 0; W(1:n) = 0; W1(1:n) = 0; diffU = 10; diffW = 10; kap = k./(rho.*Cp); f = 2.*R.ˆ3.*A.*exp(-decC.*t)./(r.*pi.ˆ3.*k); c1 = (T0(2:n)-T0(1:n-1))./(r(2:n)-r(1:n-1)); while((diffU > tol)); i = i+1; a = R.*(-1).ˆi.*(TR-T0(m))./(i.*pi) +... R/(i*pi)ˆ2*sum(c1.*(r(2:n).*sin(i.*pi.*r(2:n)./R)-... r(1:n-1).*sin(i.*pi.*r(1:n-1)./R)))+... 2*Rˆ2/(i*pi)ˆ3*sum(c1.*(cos(i.*pi.*r(2:n)./R)-... cos(i.*pi.*r(1:n-1)./R))); if(type == 1) U1 = 2./r.*a.*sin(i.*pi.*r./R).*exp(-kap.*(i.*pi).ˆ2.*dt./R.ˆ2); U = U+U1; diffU = max(abs(U1)); if(nargin == 14) ; fprintf(logf,[’T Iteration: %4f, a = %9.4f, diffU = %9.4f, ’... ’Min(U1) = %9.4f, Max(U) = %9.4f\n’],i,a,diffU,min(U1),max(U1)); end else U1 = 2./r.*a.*exp(-kap.*(i.*pi).ˆ2.*dt./R.ˆ2).*... (i.*pi./R.*cos(i.*pi.*r./R)-sin(i.*pi.*r./R)./r).*1e6; U = U+U1; diffU = max(abs(U1)); if(nargin == 14) ; fprintf(logf,[’dT Iteration: %4f, a = %9.4f, diffU = %9.4f, ’... ’Min(U1) = %9.4f, Max(U) = %9.4f\n’],i,a,diffU,min(U1),max(U1)); end end end if(type == 1) U = U+TR; else U = U./1e6; end while((diffW > tol)) j = j+1; if(type == 1) W1 = f.*(-1).ˆj./(j.ˆ3-j.*decC.*R.ˆ2./(kap.*pi.ˆ2)).*... sin(j.*pi.*r./R).*exp(-kap.*(pi.*j).ˆ2.*dt./R.ˆ2); W = W+W1; diffW = max(abs(W1)); if(nargin == 14); fprintf(logf,[’T Iteration: %4f, diffW = %9.4f, ’... ’Min(W1) = %9.4f, Max(W) = %9.4f\n’],j,diffU,... min(W1),max(W1)); end else W1 = f.*(-1).ˆj./(j.ˆ3-j.*decC.*R.ˆ2./(kap.*pi.ˆ2)).*... exp(-kap.*(pi.*j).ˆ2.*dt./R.ˆ2).*(-sin(j.*pi.*r./R)./r+... j.*pi./R.*cos(j.*pi.*r./R)).*1e4; W = W+W1; diffW = max(abs(W1)); if(nargin == 14);

88 fprintf(logf,[’T Iteration: %4f, diffW = %9.4f, ’... ’Min(W1) = %9.4f, Max(W) = %9.4f\n’],j,diffU,... min(W1),max(W1)); end end end if(type == 1) W = W + A.*exp(-decC.*(dt+t))./(rho.*Cp.*decC).*... (R.*sin(r.*sqrt(decC./kap))./(r.*sin(R.*sqrt(decC./kap)))-1); else W = W./1e4 + A.*exp(-decC.*(dt+t))./(rho.*Cp.*decC).*... (sqrt(decC./kap).*R.*cos(r.*sqrt(decC./kap))./... (r.*sin(R.*sqrt(decC./kap)))-R.*sin(r.*sqrt(decC./kap))./... (r.ˆ2.*sin(R.*sqrt(decC./kap)))); end T = U+W; clear(’A’,’Cp’,’R’,’T0’,’TR’,’U’,’U1’,’W’,’W1’,’a’,’ans’,’c1’,... ’decC’,’diffU’,’diffW’,’dt’,’f’,’i’,’j’,’k’,’kap’,’logf’,... ’m’,’n’,’r’,’rho’,’t’,’tol’,’type’) % ======% Function that extrapolates the radius and temperature distribution of % a growing sphere, assuming the sphere grows at constant volume. The % temperature in the new regions is set equal to the melting temperature of % the sphere, assuming Lindemanns law of melting and PREM densities. % % [T,r] = LindTExt(T0,r0,gamma,Tm,tau,R,dt) % % Input parameters: % % T0 Initial temperature profile [K] % r0 Radial coordinates for T0 and [m] % gamma Gruneisen parameter at radius R0 (see below) % TmR0 Melting temperature of sphere at radius R0 (see below)[K] % tau Growthtime of sphere to reach radius R0 (see below) [s] % R0 Radius of sphere at t = tau [m] % dt Timestep of growth [s] % logf logfileID % % Output paramters: % % T New temperature profile [K] % r Radial coordinates of new temperature profile [m] % % Note that T0 and r0 should be column vectors, % i.e. size(T0) = size(r0) = [1 n]; n > 1; % The output T and r will be column vectors of equal size as T0. Atleast % two coordinates will be whitin the radii dr = r-r0. The interpolation of % T from T0 is performed using linear interpolation. % % LindTExt can also be used to produce a melting temperature profile if T0 % is set < 0, in which case r0 should be the number of radial coordinates % in r and T. Note that in this case size(T0) = size(r0) = [1 1] % ------function [T,r] = LindTExt(T0,r0,gamma,TmR0,tau,R0,dt,logf) if(size(T0,2) > 1) n = size(T0,2); r(1:n) = 0; T(1:n) = 0; R = r0(n); t = tau*(R/R0)ˆ3+dt; Rn = R0*(t/tau)ˆ(1/3);

89 if(2*Rn/(n-1) < (Rn-R)); r(1:n-2) = [r0(1):(R-r0(2))/(n-3):R]; r(n-1) = R+(Rn-R)/2; r(n) = Rn; m = n-2; else r = [r0(1):(Rn-r0(1))/(n-1):Rn]; m = ceil(R*(n-1)/(Rn-r0(1))); end

j1 = 1; for i1 = 1:m; while(r(i1) > r0(j1)); j1 = j1+1; end if(r(i1) == r0(j1)) T(i1) = T0(j1); else T(i1) = T0(j1-1)+(r(i1)-r0(j1-1))*(T0(j1)-T0(j1-1))/... (r0(j1)-r0(j1-1)); end end ratio = (13088.5-8838.1.*(R0./6.371e6).ˆ2)./(13088.5-8838.1.*... (r(m+1:n)./6.371e6).ˆ2); T(m+1:n) = TmR0.*exp(2.*gamma.*(1-ratio)+2./3.*log(ratio));

if(nargin == 8) fprintf(logf,[’\n Extrapolation of radius and temperature ’... ’profile\n’]); fprintf(logf,[’Input radius = %f, output radius = %f, new ’... ’radial coordinates = %f\n\n’],R,Rn,m); fprintf(logf,’rnew = %11.3e’,transpose(r)); fprintf(logf,’Tnew = %13.2f’,transpose(T)); end else Rn = R0*(dt/tau)ˆ(1/3); r = [0:Rn/(r0-1):Rn]; ratio = (13088.5-8838.1.*(R0./6.371e6).ˆ2)./(13088.5-8838.1.*... (r./6.371e6).ˆ2); T = TmR0.*exp(2.*gamma.*(1-ratio)+2./3.*log(ratio)); end q = size(r,2); rt = r; Tt = T; clear(’T’,’r’); r = [rt(1):(rt(q)-rt(1))/(q-1):rt(q)]; T = interp1(rt,Tt,r); clear;

90 Tidigare utgivna publikationer i serien ISSN 1650-6553

Nr 1 Geomorphological mapping and hazard assessment of alpine areas in Vorarlberg, Austria, Marcus Gustavsson

Nr 2 Verification of the Turbulence Index used at SMHI, Stefan Bergman

Nr 3 Forecasting the next day’s maximum and minimum temperature in Vancouver, Canada by using artificial neural network models, Magnus Nilsson

Nr 4 The tectonic history of the Skyttorp-Vattholma fault zone, south-central Sweden, Anna Victoria Engström

Nr 5 Investigation on Surface energy fluxes and their relationship to synoptic weather patterns on Storglaciären, northern Sweden, Yvonne Kramer

Nr 6 Förekomst och utlakning av bly från Tudors industriområde i Nol, Anna Bohlin

Nr 7 Partial Melting in Granulite Gneisses from Hornslandet, SE Central Sweden, Sofia Carlsson

Nr 8 Högupplösande processering av seismiska data insamlade nära Gardermoens flygplats, Marie Larsson

Nr 9 Sedimentundersökning i Lillsjön och Vikern Gyttorp, Jan Sävås

Nr 10 Integrering av MIFO och Grundvattenmodeller, Marit Brandt

Nr 11 GIS-baserad förstudie till MKB för behandling av förorenade jordmassor i Stockholms respektive Södermanlands län, Borka Medjed-Hedlund

Nr 12 Groundwater Chemistry in the Area East of the Dead Sea, Jordan, Alice Nassar

Nr 13 Bly i morän och vatten i delar av Småland, naturligt eller antropogent?, Karin Luthbom

Nr 14 Metanflöde mellan väldränerad, subarktisk tundra och atmosfären -betydelsen av markens vattenhalt och kemiska egenskaper, Gunilla Nordenmark

Nr 15 Effects of isothermal versus fluctuating temperature regimes on CO2 efflux from sub-arctic soils, Pär Eriksson

Nr 16 En dagvattenmodell för beräkning av avrinning och transport av kväve och fosfor i Flatendiket i södra Stockholm, Sara-Sofia Hellström

Nr 17 Långsiktiga effekter av odlingsinriktning på förändringar i markens humusförråd - en fallstudie, Helena Näslund

Nr 18 Dynutveckling längs kusten utanför Halmstad, under senare hälften av 1900-talet. Ingrid Engvall Nr 19 Humidity Structures in the Marine Atmospheric Boundary Layer, Andreas Svensson

Nr 20 The Influence of Waves on the Heat Exchange over Sea, Erik Sahlée

Nr 21 Åska längs Sveriges kuster, Ulrika Andersson

Nr 22 En enkel modell för beräkning av tjäldjup, Johan Hansson

Nr 23 Modelling the Wind Climate in Mountain Valleys Using the MIUU Mesoscale Model, Nikolaus Juuso

Nr 24 Evaluation of a New Lateral Boundary Condition in the MIUU Meso-Scale Model, Anna Jansson

Nr 25 Statistisk studie av sambandet mellan geostrofisk vind och temperatur i södra Sverige, Jonas Höglund

Nr 26 A comparison of the temperature climate at two urban sites in Uppsala, Paulina Larsson

Nr 27 Optiska djupet för atmosfäriska aerosolpartiklar över södra Sverige, Jonas Lilja

Nr 28 The wind field in coastal areas, Niklas Sondell

Nr 29 A Receiver Function Analysis of the Swedish Crust, David Mawdsley

Nr 30 Tjäldjupsberäkningar med temperatursummor, Malin Knutsson

Nr 31 Processing and Interpretation of Line 4 Seismic Reflection Data from Siljan Ring Area, Daniela Justiniano Romero

Nr 32 Turning Ray Tomography along deep Reflection Seismic Line 4 from the Siljan Ring Area, Anmar C. Dávila Chacón

Nr 33 Comparison of two small catchments in the Nopex research area by water balance and modelling approaches, Ulrike Kummer

Nr 34 High resolution data processing of EnviroMT data, Tobias Donner

Nr 35 Paleoclimatic conditions during late Triassic to early Jurassic, northern North Sea: evidence from clay mineralogy, Victoria Adestål

Nr 36 Controlled Source Magnetotellurics - The transition from near-field to far-field Hermann Walch

Nr 37 Soil respiration in sub-arctic soils – controlling factors and influence of global change, Evelina Selander

Nr 38 Miljöeffekter av Triorganiska tennföreningar från antifoulingfärg – med avseende på sedimentologi, ekotoxikologi och hydrogeologi, Sara Berglund Nr 39 Depth distribution of methanotroph activity at a mountain birch forest-tundra ecotone, northern Sweden, Erik Melander

Nr 40 Methyl tert-Butyl Ether Contamination in Groundwater, Linda Ahlström

Nr 41 Geokemisk undersökning av vattnet i Västerhavet Med avseende på metallhalter och 129I, Anette Bergström

Nr 42 Fracture filling minerals and the extent of associated alteration into adjacent granitic host rock, Erik Ogenhall

Nr 43 Bi-Se minerals from the Falun Copper mine, Helena Karlsson

Nr 44 Structures and Metamorphism in the Heidal-Glittertindarea, Scandinavian Caledonides, Erik Malmqvist

Nr 45 Structure and isotope-age studies in Faddey Bay region of central Taymyr, northern Siberia, Robert Eriksson

Nr 46 Stabilitetsindex – en stabil prognosmetod för åska?, Johan Sohlberg

Nr 47 Stadsklimateffekter i Uppsala, Andreas Karelid

Nr 48 Snow or rain? - A matter of wet-bulb temperature, Arvid Olsen

Nr 49 Beräkning av turbulenta flöden enligt inertial dissipationsmetoden med mätdata från en specialkonstruerad lättviktsanemometer samt jämförelse med turbulenta utbytesmetoden, Charlotta Nilsson

Nr 50 Inverkan av det interna gränsskiktets höjd på turbulensstrukturen i ytskiktet, Ulrika Hansson

Nr 51 Evaluation of the Inertial Dissipation Method over Land, Björn Carlsson

Nr 52 Lower Ordovician Acritarchs from Jilin Province, Northeast China, Sebastian Willman

Nr 53 Methods for Estimating the Wind Climate Using the MIUU-model, Magnus Lindholm

Nr 54 Mineralogical Evolution of Kaolinite Coated Blast Furnace Pellets, Kristine Zarins

Nr 55 Crooked line first arrival refraction tomography near the Archean-Proterozoic in Northern Sweden, Valentina Villoria

Nr 56 Processing and AVO Analyses of Marine Reflection Seismic Data from Vestfjorden, Norway, Octavio García Moreno

Nr 57 Pre-stack migration of seismic data from the IBERSEIS seismic profile to image the upper crust, Carlos Eduardo Jiménez Valencia

Nr 58 Spatial and Temporal Distribution of Diagenetic Alterations in the Grés de la Créche Formation (Upper Jurassic, N France), Stefan Eklund Nr 59 Tektoniskt kontrollerade mineraliseringar i Oldenfönstret, Jämtlands län, Gunnar Rauséus

Nr 60 Neoproterozoic Radiation of Acritarchs and Environmental Perturbations around the Acraman Impact in Southern Australia, Mikael Axelsson

Nr 61 Chlorite weathering kinetics as a function of pH and grain size, Magdalena Lerczak and Karol Bajer

Nr 62 H2S Production and Sulphur Isotope Fractionation in Column Experiments with Sulphate - Reducing Bacteria, Stephan Wagner

Nr 63 Magnetotelluric Measurements in the Swedish Caledonides, Maria Jansdotter Carlsäter

Nr 64 Identification of Potential Miombo Woodlands by Remote Sensing Analysis, Ann Thorén

Nr 65 Modeling Phosphorus Transport and Retention in River Networks, Jörgen Rosberg

Nr 66 The Importance of Gravity for Integrated Geophysical Studies of Aquifers, Johan Jönberger

Nr 67 Studying the effect of climate change on the design of water supply reservoir, Gitte Berglöv

Nr 68 Source identification of nitrate in a Tertiary aquifer, western Spain: a stable-isotope approach, Anna Kjellin

Nr 69 Kartläggning av bly vid Hagelgruvan, Gyttorp, Ida Florberger

Nr 70 Morphometry and environmental controls of solifluction landforms in the Abisko area, northern Sweden, Hanna Ridefelt

Nr 71 Trilobite biostratigraphy of the Tremadoc Bjørkåsholmen Formation on Öland, Sweden, Åsa Frisk

Nr 72 Skyddsområden för grundvattentäkter - granskning av hur de upprättats, Jill Fernqvist

Nr 73 Ultramafic diatremes in middle Sweden, Johan Sjöberg

Nr 74 The effect of tannery waste on soil and groundwater in Erode district, Tamil Nadu, India A Minor Field Study, Janette Jönsson

Nr 75 Impact of copper- and zinc contamination in groundwater and soil, Coimbatore urban areas, Tamil Nadu, South India A Minor Field Study, Sofia Gröhn

Nr 76 Klassificering av Low Level Jets och analys av den termiska vinden över Östergarnsholm , Lisa Frost

Nr 77 En ny metod för att beräkna impuls- och värmeflöden vid stabila förhållanden, Anna Belking Nr 78 Low-level jets - observationer från Näsudden på Gotland, Petra Johansson

Nr 79 Sprite observations over France in relation to their parent thunderstorm system, Lars Knutsson

Nr 80 Influence of fog on stratification and turbulent fluxes over the ocean, Linda Lennartsson

Nr 81 Statistisk undersökning av prognosmetod för stratus efter snöfall, Elisabeth Grunditz

Nr 82 An investigation of the surface fluxes and other parameters in the regional climate model RCA1during ice conditions, Camilla Tisell

Nr 83 An investigation of the accuracy and long term trends of ERA-40 over the Baltic Sea, Gabriella Nilsson

Nr 84 Sensitivity of conceptual hydrological models to precipitation data errors – a regional study, Liselotte Tunemar

Nr 85 Spatial and temporal distribution of diagenetic modifications in Upper Paleocene deep- water marine, turbiditic sandstones of the Faeroe/Shetland basin of the North Sea, Marcos Axelsson

Nr 86 Crooked line first arrival refraction tomography in the Skellefte ore field, Northern Sweden, Enrique Pedraza

Nr 87 Tektoniken som skulptör - en strukturgeologisk tolkning av Stockholmsområdet och dess skärgård, Peter Dahlin

Nr 88 Predicting the fate of fertilisers and pesticides applied to a golf course in central Sweden, using a GIS Tool, Cecilia Reinestam

Nr 89 Formation of Potassium Slag in Blast Furnace Pellets, Elin Eliasson

Nr 90 - Syns den globala uppvärmningen i den svenska snöstatistiken?Mattias Larsson

Nr 91 Acid neutralization reactions in mine tailings from Kristineberg, Michal Petlicki och Ewa Teklinska

Nr 92 Ravinbildning i Naris ekologiska reservat, Costa Rica, Axel Lauridsen Vang

Nr 93 Temporal variations in surface velocity and elevation of Nordenskiöldbreen, Svalbard, Ann-Marie Berggren

Nr 94 Beskrivning av naturgeografin i tre av Uppsala läns naturreservat, Emelie Nilsson

Nr 95 Water resources and water management in Mauritius, Per Berg

Nr 96 Past and future of Nordenskiöldbreen, Svalbard, Peter Kuipers Munneke

Nr 97 Micropaleontology of the Upper Bajocian Ostrea acuminata marls of Champfromier (Ain, France) and paleoenvironmental implications, Petrus Lindh Nr 98 Calymenid trilobites (Arthropoda) from the Silurian of Gotland, Lena Söderkvist

Nr 99 Development and validation of a new mass-consistent model using terrain-influenced coordinates, Linus Magnusson

Nr 100 The Formation of Stratus in Rain, Wiebke Frey

Nr 101 Estimation of gusty winds in RCA, Maria Nordström

Nr 102 Vädermärken och andra påståenden om vädret - sant eller falskt?, Erica Thiderström

Nr 103 A comparison between Sharp Boundary inversion and Reduced Basis OCCAM inversion for a 2-D RMT+CSTMT survey at Skediga, Sweden, Adriana Berbesi

Nr 104 Space and time evolution of crustal stress in the South Iceland Seismic Zone using microearthquake focal mechanics, Mimmi Arvidsson

Nr 105 Carbon dioxide in the atmosphere: A study of mean levels and air-sea fluxes over the Baltic Sea, Cristoffer Wittskog

Nr 106 Polarized Raman Spectroscopy on Minerals, María Ángeles Benito Saz

Nr 107 Faunal changes across the Ordovician – Silurian boundary beds, Osmundsberget Quarry, Siljan District, Dalarna, Cecilia Larsson

Nr 108 Shrews (Soricidae: Mammalia) from the Pliocene of Hambach, NW Germany, Sandra Pettersson

Nr 109 Waveform Tomography in Small Scale Near Surface Investigations, Joseph Doetsch

Nr 110 Vegetation Classification and Mapping of Glacial Erosional and Depositional Features Northeastern part of Isla Santa Inés, 530S and 720W, Chile, Jenny Ampiala

Nr 111 Recent berm ridge development inside a mesotidal estuary The Guadalquivir River mouth case, Ulrika Åberg

Nr 112 Metodutveckling för extrahering av jod ur fasta material, Staffan Petré

Nr 113 Släntstabilitet längs Ångermanälvens dalgång, Mia Eriksson

Nr 114 Validation of remote sensing snow cover analysis, Anna Geidne

Nr 115 The Silver Mineralogy of the Garpenberg Volcanogenic Sulphide Deposit, Bergslagen, Central Sweden, Camilla Berggren

Nr 116 Satellite interferometry (InSAR) as a tool for detection of strain along End- Glacial faults in Sweden, Anders Högrelius

Nr 117 Landscape Evolution in the Po-Delta, Italy, Frida Andersson Nr 118 Metamorphism in the Hornslandet Area, South - East Central Sweden, Karl-Johan Mattsson Nr 119 Contaminated Land database - GIS as a tool for Contaminated Land Investigations, Robert Elfving

Nr 120 Geofysik vid miljöteknisk markundersökning, Andreas Leander

Nr 121 Precipitation of Metal Ions in a Reactive Barrier with the Help of Sulphate - Reducing Bacteria, Andreas Karlhager

Nr 122 Sensitivity Analysis of the Mesoscale Air Pollution Model TAPM, David Hirdman

Nr 123 Effects of Upwelling Events on the Atmosphere, Susanna Hagelin

Nr 124 The Accuracy of the Wind Stress over Ocean of the Rossby Centre Atmospheric Model (RCA), Alexandra Ohlsson

Nr 125 Statistical Characteristics of Convective Storms in Darwin, Northern Australia, Andreas Vallgren

Nr 126 An Extrapolation Technique of Cloud Characteristics Using Tropical Cloud Regimes, Salomon Eliasson

Nr 127 Downscaling of Wind Fields Using NCEP-NCAR-Reanalysis Data and the Mesoscale MIUU-Model, Mattias Larsson

Nr 128 Utveckling och Utvärdering av en Stokastisk Vädergenerator för Simulering av Korrelerade Temperatur- och Nederbördsserier, för Tillämpningar på den Nordiska Elmarknaden, Johanna Svensson

Nr 129 Reprocessing of Reflection Seismic Data from the Skåne Area, Southern Sweden, Pedro Alfonzo Roque

Nr 130 Validation of the dynamical core of the Portable University Model of the Atmosphere (PUMA), Johan Liakka

Nr 131 Links between ENSO and particulate matter pollution for the city of Christchurch, Anna Derneryd

Nr 132 Testing of a new geomorphologic legend in the Vattholma area, Uppland, Sweden, Niels Nygaard

Nr 133 Återställandet av en utdikad våtmark, förstudie Skävresjön, Lena Eriksson, Mattias Karlsson

Nr 134 A laboratory study on the diffusion rates of stable isotopes of water in unventilated firn, Vasileios Gkinis

Nr 135 Reprocessing of Reflection Seismic Data from the Skåne Area, Southern Sweden Wedissa Abdelrahman