D Gubbins
School of Earth Sciences, University of Leeds, UK.
\message {REFERENCE FORMATTING FILES:} % Basic control strings and macros for tib bibliography formatting, % and default definitions. % % tib and the accompaning TeX macros assume the \catcode of plain TeX % e.g. { demarks the beginning of a group, % demarks a comment, etc. % See D.E. Knuth, The TeXbook, p. 37. % So far as possible, definitions have been reduced to primitive TeX % strings, to make tib as compatible with any TeX as possible. % Two possible exceptions: the fonts \rm, \sl, \sevenrm are assumed % to have been loaded by TeX. If not, this file should be modified below. % Also macros for footnotes are tailored to the particular TeX, since % the footnote macros of different TeXs differ. Footnote macro files % ending in .p are for plain TeX; those ending in .l are for LaTeX. % % \Resetstrings and \Refformat are called within a local group--- % the string names will not conflict with uses elsewhere in the document. % Other control strings are defined globally. The following is % a complete list of such globally defined control strings (except those % the user might define in a private tib formatting file). % % \Resetstrings \Refformat \Rpunct \Lpunct % \Lspace \Lperiod \Lcomma \Lquest % \Lcolon \Lscolon \Lbang \Lquote % \Lqquote \Lrquote \Rspace \Rperiod % \Rcomma \Rquest \Rcolon \Rscolon % \Rbang \Rquote \Rqquote \Rrquote % \Refstd \Refstda \Smallcapsaand \Smallcapseand % \Acomma \Aand \Aandd \Ecomma % \Eand \Eandd \acomma \aand % \aandd \ecomma \eand \eandd % \Namecomma \Nameand \Nameandd \Revcomma % \Initper \Initgap \Citefont \ACitefont % \Authfont \Titlefont \Tomefont \Volfont % \Flagfont \Reffont \Smallcapsfont \Underlinemark % \Citebrackets \Citeparen \Citesuper \Citenamedate % \Lcitemark \Rcitemark \LAcitemark \RAcitemark % \LIcitemark \RIcitemark \Citehyphen \Citecomma % \Citebreak \Resetstrings \ztest \zstr % \Ztest \Zstr % \journalarticleformat \conferencereportformat \bookformat % \technicalreportformat \bookarticleformat \otherformat % % This file is \input first in a document processed by tib. % Secondly any tib format files are processed. Then the input % document file is read and processed. Thus the default definitions % of this file can be overridden either in the tib format files % or the input document. % % \Refformat calls macros for creating the actual bibliography listings. % Such macros are generally kept in tibtex files and \input by a statement % in the tib format file. \def\Resetstrings{%Clears all strings before processing reference listing. % The strings (\Astr, etc.) are fields taken from the database. % If the string is present, the appropriate test (\Atest, etc) is set % equal to \present---thus allowing the macros to test for the presence % or absence of a field. All reference processing is done in a local % group--the control string names will not conflict with uses % elsewhere in the document. \def\present{ }\let\bgroup={\let\egroup=}%primitive TeX \def\Astr{}\def\astr{}\def\Atest{}\def\atest{}% \def\Bstr{}\def\bstr{}\def\Btest{}\def\btest{}% \def\Cstr{}\def\cstr{}\def\Ctest{}\def\ctest{}% \def\Dstr{}\def\dstr{}\def\Dtest{}\def\dtest{}% \def\Estr{}\def\estr{}\def\Etest{}\def\etest{}% \def\Fstr{}\def\fstr{}\def\Ftest{}\def\ftest{}% \def\Gstr{}\def\gstr{}\def\Gtest{}\def\gtest{}% \def\Hstr{}\def\hstr{}\def\Htest{}\def\htest{}% \def\Istr{}\def\istr{}\def\Itest{}\def\itest{}% \def\Jstr{}\def\jstr{}\def\Jtest{}\def\jtest{}% \def\Kstr{}\def\kstr{}\def\Ktest{}\def\ktest{}% \def\Lstr{}\def\lstr{}\def\Ltest{}\def\ltest{}% \def\Mstr{}\def\mstr{}\def\Mtest{}\def\mtest{}% \def\Nstr{}\def\nstr{}\def\Ntest{}\def\ntest{}% \def\Ostr{}\def\ostr{}\def\Otest{}\def\otest{}% \def\Pstr{}\def\pstr{}\def\Ptest{}\def\ptest{}% \def\Qstr{}\def\qstr{}\def\Qtest{}\def\qtest{}% \def\Rstr{}\def\rstr{}\def\Rtest{}\def\rtest{}% \def\Sstr{}\def\sstr{}\def\Stest{}\def\stest{}% \def\Tstr{}\def\tstr{}\def\Ttest{}\def\ttest{}% \def\Ustr{}\def\ustr{}\def\Utest{}\def\utest{}% \def\Vstr{}\def\vstr{}\def\Vtest{}\def\vtest{}% \def\Wstr{}\def\wstr{}\def\Wtest{}\def\wtest{}% \def\Xstr{}\def\xstr{}\def\Xtest{}\def\xtest{}% \def\Ystr{}\def\ystr{}\def\Ytest{}\def\ytest{}% } \Resetstrings\def\Ztest{}\def\ztest{} \def\Refformat{%Determines the kind of reference by the presence or % absence of certain fields in the database listing, and calls the % appropriate macro. \if\Jtest\present {\if\Vtest\present\journalarticleformat \else\conferencereportformat\fi} \else\if\Btest\present\bookarticleformat \else\if\Rtest\present\technicalreportformat \else\if\Itest\present\bookformat \else\otherformat\fi\fi\fi\fi} \def\Rpunct{%Default punctuation control strings if the punctuation % is to appear after the citation. (tib looks for punctuation to % precede the incomplete citation in the input document; the TeX % output puts it to the left or right depending on the style of citation.) \def\Lspace{ }% \def\Lperiod{ }% . \def\Lcomma{ }% , \def\Lquest{ }% ? \def\Lcolon{ }% : \def\Lscolon{ }% ; \def\Lbang{ }% ! \def\Lquote{ }% ' \def\Lqquote{ }% " \def\Lrquote{ }% ` \def\Rspace{}% \def\Rperiod{.}% . \def\Rcomma{,}% , \def\Rquest{?}% ? \def\Rcolon{:}% : \def\Rscolon{;}% ; \def\Rbang{!}% ! \def\Rquote{'}% ' \def\Rqquote{"}% " \def\Rrquote{`}% ` } \def\Lpunct{%Default punctuation control strings if the punctuation % is to appear before the citation. (tib looks for punctuation to % precede the incomplete citation in the input document; the TeX % output puts it to the left or right depending on the style of citation.) \def\Lspace{}% \def\Lperiod{\unskip.}% . \def\Lcomma{\unskip,}% , \def\Lquest{\unskip?}% ? \def\Lcolon{\unskip:}% : \def\Lscolon{\unskip;}% ; \def\Lbang{\unskip!}% ! \def\Lquote{\unskip'}% ' \def\Lqquote{\unskip"}% " \def\Lrquote{\unskip`}% ` \def\Rspace{\spacefactor=1000}% \def\Rperiod{\spacefactor=3000}% . \def\Rcomma{\spacefactor=1250}% , \def\Rquest{\spacefactor=3000}% ? \def\Rcolon{\spacefactor=2000}% : \def\Rscolon{\spacefactor=1250}% ; \def\Rbang{\spacefactor=3000}% ! \def\Rquote{\spacefactor=1000}% ' \def\Rqquote{\spacefactor=1000}% " \def\Rrquote{\spacefactor=1000}% ` } \def\Refstd{%Standard control strings for formatting bibliography listings. \def\Acomma{\unskip, }%between multiple author names \def\Aand{\unskip\ and }%between two author names \def\Aandd{\unskip\ and }%between last two of multiple author names \def\Ecomma{\unskip, }%between multiple editor names \def\Eand{\unskip\ and }%between two editor names \def\Eandd{\unskip\ and }%between last two of multiple author names \def\acomma{\unskip, }%same for authors of reviewed material \def\aand{\unskip\ and }%same for authors of reviewed material \def\aandd{\unskip\ and }%same for authors of reviewed material \def\ecomma{\unskip, }%same for translators \def\eand{\unskip\ and }%same for translators \def\eandd{\unskip\ and }%same for translators \def\Namecomma{\unskip, }%same for citations using authors' names \def\Nameand{\unskip\ and }%same for citations using authors' names \def\Nameandd{\unskip\ and }%same for citations using authors' names \def\Revcomma{\unskip, }%between last and first name of reversed name \def\Initper{.\ }%punctuation after initial \def\Initgap{\dimen0=\spaceskip\divide\dimen0 by 2\hskip-\dimen0}% %gap between initials of abbreviated first name } \def\Smallcapsaand{%Smallcaps redefinition of \Aand and \Aandd for \Refstd \def\Aand{\unskip\bgroup{\Smallcapsfont\ AND }\egroup}% \def\Aandd{\unskip\bgroup{\Smallcapsfont\ AND }\egroup}% \def\eand{\unskip\bgroup\Smallcapsfont\ AND \egroup}% \def\eandd{\unskip\bgroup\Smallcapsfont\ AND \egroup}% } \def\Smallcapseand{%Smallcaps redefinition of \Eand, \Eeand, etc for Refstd \def\Eand{\unskip\bgroup\Smallcapsfont\ AND \egroup}% \def\Eandd{\unskip\bgroup\Smallcapsfont\ AND \egroup}% \def\aand{\unskip\bgroup\Smallcapsfont\ AND \egroup}% \def\aandd{\unskip\bgroup\Smallcapsfont\ AND \egroup}% } \def\Refstda{%Standard control strings for formatting bibliography listings. % \Refstda sets an Ampersand instead of the word "and". \chardef\Ampersand=`\&%primitive TeX \def\Acomma{\unskip, }%between multiple author names \def\Aand{\unskip\ \Ampersand\ }%between two author names \def\Aandd{\unskip\ \Ampersand\ }%between last two of multiple author names \def\Ecomma{\unskip, }%between multiple editor names \def\Eand{\unskip\ \Ampersand\ }%between two editor names \def\Eandd{\unskip\ \Ampersand\ }%between last two of multiple author names \def\acomma{\unskip, }%same for authors of reviewed material \def\aand{\unskip\ \Ampersand\ }%same for authors of reviewed material \def\aandd{\unskip\ \Ampersand\ }%same for authors of reviewed material \def\ecomma{\unskip, }%same for translators \def\eand{\unskip\ \Ampersand\ }%same for translators \def\eandd{\unskip\ \Ampersand\ }%same for translators \def\Namecomma{\unskip, }%same for citations using authors' names \def\Nameand{\unskip\ \Ampersand\ }%same for citations using authors' names \def\Nameandd{\unskip\ \Ampersand\ }%same for citations using authors' names \def\Revcomma{\unskip, }%between last and first name of reversed name \def\Initper{.\ }%punctuation after initial \def\Initgap{\dimen0=\spaceskip\divide\dimen0 by 2\hskip-\dimen0}% %gap between initials of abbreviated first name } %default fonts \def\Citefont{}%citations \def\ACitefont{}%alternate citations \def\Authfont{}%authors \def\Titlefont{}%titles \def\Tomefont{\sl}%journals or books \def\Volfont{}%volume number of journal \def\Flagfont{}%citation flag \def\Reffont{\rm}%set at beginning of reference listing \def\Smallcapsfont{\sevenrm}%small caps \def\Flagstyle#1{\hangindent\parindent\indent\hbox to0pt%flag style {\hss[{\Flagfont#1}]\kern.5em}\ignorespaces}% for references \def\Underlinemark{\vrule height .7pt depth 0pt width 3pc}%for replacing % successive listings of identical author(s) by underline (U option % in tib format file or -u flag on call). \def\Citebrackets{\Rpunct%defaults for putting citations in brackets []. \def\Lcitemark{\def\Cfont{\Citefont}[\bgroup\Cfont}%mark at left of citation \def\Rcitemark{\egroup]}%mark at right of citation \def\LAcitemark{\def\Cfont{\ACitefont}\bgroup\ACitefont}% %mark at left of alternate citation \def\RAcitemark{\egroup}%mark at right of alternate citation \def\LIcitemark{\egroup}%mark at left of insertion in citation \def\RIcitemark{\bgroup\Cfont}%mark at right of insertion in citation \def\Citehyphen{\egroup--\bgroup\Cfont}%separater for string of citations \def\Citecomma{\egroup,\hskip0pt\bgroup\Cfont}% %separater for multiple citations \def\Citebreak{}%mark between parts of citation (e.g. author\Citebreak date) } \def\Citeparen{\Rpunct%defaults for putting citations in parenthesis (). \def\Lcitemark{\def\Cfont{\Citefont}(\bgroup\Cfont}%mark at left of citation \def\Rcitemark{\egroup)}%mark at right of citation \def\LAcitemark{\def\Cfont{\ACitefont}\bgroup\ACitefont}% %mark at left of alternate citation \def\RAcitemark{\egroup}%mark at right of alternate citation \def\LIcitemark{\egroup}%mark at left of insertion in citation \def\RIcitemark{\bgroup\Cfont}%mark at right of insertion in citation \def\Citehyphen{\egroup--\bgroup\Cfont}%separater for string of citations \def\Citecomma{\egroup,\hskip0pt\bgroup\Cfont}% %separater for multiple citations \def\Citebreak{}%mark between parts of citation (e.g. author\Citebreak date) } \def\Citesuper{\Lpunct%defaults for making superscript citations \def\Lcitemark{\def\Cfont{\Citefont}\raise1ex\hbox\bgroup\bgroup\Cfont}% %mark at left of citation \def\Rcitemark{\egroup\egroup}%mark at right of citation \def\LAcitemark{\def\Cfont{\ACitefont}\bgroup\ACitefont}% %mark at left of alternate citation \def\RAcitemark{\egroup}%mark at right of alternate citation \def\LIcitemark{\egroup\egroup}%mark at left of insertion in citation \def\RIcitemark{\raise1ex\hbox\bgroup\bgroup\Cfont}% %mark at right of insertion in citation \def\Citehyphen{\egroup--\bgroup\Cfont}%separater for string of citations \def\Citecomma{\egroup,\hskip0pt\bgroup% \Cfont}%separater for multiple citations \def\Citebreak{}%mark between parts of citation (e.g. author\Citebreak date) } \def\Citenamedate{\Rpunct%defaults for making name-date citations \def\Lcitemark{%mark at left of citation--also sets internal punctuation \def\Citebreak{\egroup\ [\bgroup\Citefont}%separater in citation \def\Citecomma{\egroup]; %between multiple citations \bgroup\let\uchyph=1\Citefont}(\bgroup\let\uchyph=1\Citefont}% \def\Rcitemark{\egroup])}%mark at right of citation \def\LAcitemark{%mark at left of alternate citation \def\Citebreak{\egroup\ [\bgroup\Citefont}\def\Citecomma{\egroup], % \bgroup\ACitefont }\bgroup\let\uchyph=1\ACitefont}% \def\RAcitemark{\egroup]}%mark at right of alternate citation \def\Citehyphen{\egroup--\bgroup\Citefont}%separater for string of citations \def\LIcitemark{\egroup}%mark at left of insertion in citation \def\RIcitemark{\bgroup\Citefont}%mark at right of insertion in citation } \message {(../journals.ttz} \message{)}\message {(gji.tib} \message {(/usr/local/lib/tex/inputs/amsabb.ttz} \message {(/usr/local/lib/tex/inputs/pub.ttz} \message{)}\message {(/usr/local/lib/tex/inputs/mscabb.ttz} \message{)}\message{)} %\input\TMACLIB ijrr.ttx \input gji.ttx \message{*** Note: multiple citations should be entered in chronological order. ***} \Refstd\Rpunct%set general formats for reference list and citations \def\Titlefont{\Reffont}\def\Volfont{\bf} \def\Tomefont{\it}%redefine some fonts \def\Lcitemark{%mark at left of citation--also sets internal punctuation \let\uchyph=1\def\Cfont{}% \def\Citebreak{\egroup\ \bgroup\Cfont}% \def\Citecomma{\egroup; \bgroup\Cfont }(\bgroup\Cfont}% \def\Rcitemark{\egroup)}%mark at right of citation \def\LAcitemark{%mark at left of alternate citation \def\Cfont{}% \let\uchyph=1\def\Citebreak{\egroup\ (\bgroup\Cfont}% \def\Citecomma{\egroup, \bgroup\Cfont }\bgroup\Cfont}% \def\RAcitemark{\egroup)}%mark at right of alternate citation \def\Citehyphen{\egroup--\bgroup\Cfont}%separater for string of citations \def\LIcitemark{\egroup}%mark at left of insertion in citation \def\RIcitemark{\bgroup\Cfont}%mark at right of insertion in citation \def\Aand{\unskip\ \& } \def\Aandd{\unskip\ \& } \def\Nameand{\unskip\ \& } \def\Nameandd{\unskip\ \& } \message {)}\message {DOCUMENT TEXT} \documentclass{article} \oddsidemargin=-0.5pt \topmargin -0.25in \setlength{\textwidth}{6.0in} \setlength{\textheight}{9.0in} % % newcommands for tex: includes bold face math, vectors etc for % geomagnetic stuff, derivative operators, labelling, journals % \newcommand {\labeq}[1]{\label {eq:#1}} \newcommand {\eqn}[1]{(\ref {eq:#1})} \newcommand{\beq}{\begin{equation}} \newcommand{\eeq}{\end{equation}} % \newcommand{\E}[1]{\mbox {$\times 10^{#1}$}} \newcommand{\implies}{\Longrightarrow} \newcommand{\less}{\stackrel{\textstyle <}{\sim}} \newcommand{\more}{\stackrel{\textstyle >}{\sim}} \newcommand {\degree}{$^\circ$} \newcommand {\kmyr}{km yr$^{-1}$} % \newcommand{\dint}{\int\!\!\int} \newcommand{\delh}{\nabla_{\rm h}} \newcommand{\del}{\nabla} \newcommand{\Lap}{\nabla^2} \newcommand{\bm}[1]{\mbox{\boldmath $ #1 $}} \newcommand{\ds}{\displaystyle} \newcommand{\pdiff}[2]{\ds \frac {\partial #1}{\partial #2}} \newcommand{\pdline}[2]{\partial #1 / \partial #2} \newcommand{\ddx}[1]{\left( \frac {d}{d #1} \right)} \newcommand{\recip}[1]{\frac {1}{#1}} \newcommand{\half}{\frac {1}{2}} % \renewcommand {\u}{\bm{u}} %\renewcommand {\v}{\bm {v}} %\newcommand {\V}{\bm {v}} \newcommand {\B}{\bm {B}} \newcommand {\J}{\bm {J}} \newcommand {\Bdot}{\dot {\bm {B}}} \newcommand {\e}{\bm {e}} \newcommand {\g}{\bm {g}} \newcommand{\A}{\bm{A}} \newcommand{\G}{\bm{G}} \newcommand{\Q}{\bm{Q}} \newcommand{\V}{\bm{V}} \newcommand{\Svec}{\bm{S}} \renewcommand{\P}{\bm{S}} \newcommand{\T}{\bm{S}} \newcommand{\rhat}{\hat{\bm r}} \newcommand{\om}{\bm{\omega}} \newcommand{\Ombold}{\bf \Omega} % \begin{document} % \noindent {\bf Thermal History of the Geodynamo} \\ {\it David Gubbins, School of Earth Sciences, University of Leeds, UK} \\ \vspace{0.1in} \noindent The geodynamo is driven ultimately by heat which must pass across the core-mantle boundary. The permanent existence of a magnetic field provides strong and useful contraints on both the Earth's thermal evolution and the lower boundary condition for mantle convection. Heat flux into the mantle is possibly more than 25\%\ of the total heat flux at the Earth's surface, 12~TW. If we impose the additional constraint of an ancient inner core, that the Earth was never so hot in the past 3--4~Ga to melt the entire core, we bracket the allowable parameters into a rather narrow range\Lspace \Lcitemark Labrosse\Namecomma Poirier\Nameandd LeMou\"{e}l\Citebreak 1997\Rcitemark \Rspace{}. These constraints and the present consensus on core properties make it impossible to power the geodynamo without some radioactive heating in the core. The usual model for the core assumes near-hydrostatic pressure, uniform composition, and near-adiabatic temperature. Core cooling is determined by the drop in temperature of the base of the mantle. Convection involves some essential departures from this basic state; they are an exceedingly small fraction of the total pressure, temperature, and composition, but are nevertheless vital for the dynamo. Thermal convection is driven by internal radioactive heat sources, latent heat of freezing of the liquid core, and specific heat. The difference in composition between inner and outer cores required to explain the seismologically-determined density difference means that freezing will be accompanied by expulsion of some light components into the fluid. This drives compositional convection. Estimates of the heat required to drive the dynamo are proportional to the cooling rate and concentration of radiogenic elements within the core, and on the material properties of liquid and solid iron and the lighter constituents of the core. Uncertainties in these properties frustrated early efforts to make meaningful estimates of core heat, but recent {\it first principles} quantum mechanical calculations have reduced the uncertainties of relevant properties considerably \Lcitemark Alf\`{e}\Namecomma Gillan\Nameandd Price\Citebreak 1999\Citecomma Alf\`{e}\Namecomma Kresse\Nameandd Gillan\Citebreak 2000\Rcitemark \Rspace{}. Relevant parameters are listed in Table~1. The temperature in the core is obtained by integrating the adiabatic gradient from the melting temperature at the inner core boundary upwards to the core-mantle boundary using the formula \beq \frac{1}{T_a(r)}\frac{dT_{\rm a}(r)}{dr} = \frac{g\gamma}{\phi} \labeq{adtemp} \eeq where $g$ is acceleration due to gravity, $\phi$ the seismic parameter, and $\gamma$ Grun\"{e}issen's constant. Estimates of $\gamma$ have now settled around 1.5 \Lcitemark Poirier\Citebreak 2000\Rcitemark \Rspace{}, so the main uncertainty in determining core temperature arises from the estimate of inner core boundary temperature. This is found from the melting point of iron reduced by the presence of impurities. First principles calculations show ideal solution theory to be a good predictor of the lowering of the melting point, but there is some disagreement between the various estimates of the melting point of iron at core pressures \Lcitemark Boehler\Citebreak 1996\Citecomma Alf\`{e}\Namecomma Gillan\Nameandd Price\Citebreak 1999\Rcitemark \Rspace{}. The temperature gradient is better known than the temperature itself. Equation \eqn{adtemp} yields a useful approximation for the cooling rate anywhere in the core. Assuming the core properties on the right hand side change little with time we get \beq \frac{1}{T}\frac{DT}{Dt} = \frac{1}{T_{\rm cmb}}\frac{DT_{\rm cmb}}{Dt} \labeq{adtemp1} \eeq with error less than 1\% . The logarithmic cooling rate is independent of radius, and therefore the cooling rate anywhere can be simply related to the cooling rate at the core-mantle boundary. The core cools more rapidly at depth than at the surface. The simplest estimate of heat flux required to drive the dynamo is obtained from the heat flux down the adiabat at the core-mantle boundary, 4.5~TW. The heat capacity of the present inner core is $7.3\times 10^{28}$~J, so this rate of heat loss could only be sustained without further sources for 520~Myr. This result defines the heart of the difficulty; it depends only on seismologically determined parameters, the latent heat, and Grun\"{e}issen's constant. More sophisticated calculations involve models for core cooling and the needs of the geodynamo. The importance of the entropy balance was recognised in the 1970's\Lspace \Lcitemark Hewitt\Namecomma McKenzie\Nameandd Weiss\Citebreak 1975\Citecomma Backus\Citebreak 1975\Citecomma Gubbins\Citebreak 1977\Rcitemark \Rspace{}; energy considerations are not enough because the Ohmic heating does not appear in the gross energy balance. We must balance dissipative losses due to thermal, electrical, and molecular conduction, with sources of buoyancy that ultimately owe their origin to heat, chemical energy (through the heat of reaction), and the Earth's gravitational energy (through thermal contraction and differentiation of light material into the outer core). Source terms are proportional to the cooling rate at the CMB or concentration of radioactive heat sources, the constants of proportionality being integrals of core properties. Conservation of energy simply relates heat through the CMB to cooling rate and heat sources. The entropy of thermal dissipation is the integral of the local term, $k(\nabla T/T)^2$, where $k$ is thermal conductivity. Using this quantity defines exactly the needs of maintaining the adiabatic temperature throughout the core. Merely estimating conducted heat at the surface fails to take account of the internal requirements. Ohmic dissipation is the integral of $\bm{J}^2/\sigma=(\nabla\times\B )^2/\sigma$, where $\sigma$ is electrical conductivity, $\J$ the electrical conductivity, and $\B$ the magnetic field. According to the Wiedemann-Franz law $\sigma$ is proportional to $k$, which is usually estimated from laboratory estimates of $\sigma$. A reduction in $k$ reduces thermal dissipation but increases electrical dissipation, and errors in $k$ therefore cancel, to some extent, in the dissipation estimates. Molecular diffusion has a detrimental effect on compositional convection by leaking away some of the buoyancy that would otherwise drive convection. The largest contribution comes from conduction down the pressure gradient\Lspace \Lcitemark Braginsky\Citebreak 1963\Rcitemark \Rspace{}: the light material is literally ``squeezed'' out of the core. This effect is quantifiable and appears to be rather small. \begin{table} \caption{Properties of pure iron and iron alloys used in the calculations} \begin{center} \begin{tabular}{|lcr|r|r|r|r|} \hline & & units & Fe & FeO & FeS & FeSi \\ \hline thermal expansion & $\alpha$ & $10^{-5}$~K$^{-1}$& 1.02--1.95 & & & \\ specific heat & $C_p$ & Jkg$^{-1}$K$^{-1}$& 715 & & & \\ Gr\"{u}neissen parameter & $\gamma$ & -- & 1.5 & & & \\ latent heat & $L$ & $10^6$~Jkg$^{-1}$ & 0.75 & & & \\ thermal conductivity & $k$ & Wm$^{-1}$K$^{-1}$ & 50 & & & \\ ICB temperature & $T_{\rm i}$ & K & 5500 & & & \\ CMB temperature & $T_{\rm c}$ & K & 4123 & & & \\ melting gradient & $dT_{\rm m}/dP$ & KGPa$^{-1}$ & 9.0 & & & \\ ICB gradient & $\tau$ & Kkm$^{-1}$ & 0.14 & & & \\ % compositional expansion & $\alpha_c$ & -- & & -1.10 & -0.64 & -0.87 \\ diffusion constant & $D$ & $10^{-9}$ m$^2$s$^{-1}$ && $10$ & $5$ & $5$ \\ heat of reaction & $R_H$ & $10^6$~Jkg$^{-1}$ & & -27.7 & -- & -- \\ barodiffusion constant & $K_P$ & -- & & 0.17 & 0.34 & 0.45 \\ molecular flux & $\imath$ & $10^{-12}$ kgm$^{-2}$s$^{-1}$ & & 3.4 & 3.0 & 4.0 \\ \hline \end{tabular} \end{center} \end{table} % Estimates of the ohmic dissipation require a model for the magnetic field within the core. Previous estimates assumed simple fields that produced lower bounds, kinematic dynamo models, and simple large-scale fields. The main question was whether the internal toroidal field was comparable with, or much larger than, the observed poloidal field. The new generation of dynamical geodynamo models produce a complete magnetic spectrum and can therefore be used to compute an exact Ohmic heating. Short-wavelength fields can dominate the ohmic heating. Consider the simple case of two length scales: $l$ and $L\approx 1000$~km. The small scale flow and field are denoted by primes. We assume the small scale quantities participate in maintaining the large scale field: $\B '$ is generated by $\bm{v}'$ acting on $\B$ and the large-scale average of $\bm{v}'\times \B '$ helps regenerate $\B$. Balancing orders of magnitude leads to \beq \left(\frac{\nabla\times\B '}{\nabla\times\B}\right)^2\sim \left(\frac{L}{l}\right)^\frac{3}{2} \eeq showing that the small scales do dominate the ohmic heating. A large scale field with ohmic dissipation $2\times 10^6$~W/K and an acceptable ohmic contribution roughly equal to that of thermal conduction gives $L/l\approx 50$. This is a promising result: we should be able to represent that part of the geomagnetic field responsible for maintaining its observed part with a numerical resolution of about 20~km, which is achievable with existing computers. Heat is an essential source term. It is needed to maintain the adiabat, but it may not be the main contributor to driving the dynamo. The entropy contribution of any heat source $h$ is $h/T$, and since any heat gained by the core is lost through the CMB at the lowest temperature, the entropy contribution will contain a thermodynamic ``efficiency'' factor of the form $\Delta T/T$. Latent heat has highest efficiency because it is released at the hottest temperature; specific heat is next highest because the deep core cools fastest; radioactive heating is least efficient because it is uniformly distributed and most heat is released high up where the core is coolest. The efficiency factors are of order 0.1--0.2, so typically 10 units of heat are needed to generate about 1 unit of electrical heating. Compositional convection arises from release of light material at the ICB, which either becomes distributed uniformly throughout the outer core by convection or rises to the CMB\Lspace \Lcitemark Braginsky\Nameand Roberts\Citebreak 1995\Rcitemark \Rspace{}. There is a change in the Earth's gravitational energy, which is the ultimate source for this mechanism. There is no thermodynamic efficiency factor, so the contribution to the entropy of dissipation is simply the gravitational energy released divided by the temperature. The rate of gravitational energy released depends on the rate of growth of the inner core and the density jump associated with the light material, or seismological value minus that due to freezing of pure iron. Ideal solution theory allows us to compute the gravitational energy from the density jump alone; we do not need to know the chemical composition of the light elutriant. The most difficult part of this calculation, and the one that is causing greatest controversy, is deciding what happens to the gravitational energy change consequent on thermal contraction and (a similar process) volume change on freezing. These energies are large and have been promoted as sources for powering the geodynamo\Lspace \Lcitemark H\"{a}ge\Nameand M\"{u}ller\Citebreak 1979\Citecomma Buffett et al.\Citebreak 1996\Rcitemark \Rspace{}. Much of this energy is taken up in work done by forces during compression of the material. There have been several attempts to compute the residual by subtracting the work done, computed using an equation of state or stress model\Lspace \Lcitemark Loper\Citebreak 1978\Citecomma Stacey\Nameand Stacey\Citebreak 1999\Citecomma H\"{a}ge\Nameand M\"{u}ller\Citebreak 1979\Rcitemark \Rspace{}, from the change in gravitational energy. This calculation is prone to error because it involves subtraction of two large but roughly similar quantities. It is better to isolate the thermal contributions. If the pressure remains hydrostatic, all of the gravitational energy arising from the volume change at constant pressure is taken up in compression. This leaves only the heat released by the pressure change available to drive convection, which is a small fraction, $\alpha\gamma T\approx 0.05$, of the total. Furthermore, the gravitational energy released by the effect of pressure changes on freezing may be shown to be, through the Clausius-Clapeyron equation, equal to the latent heat released by freezing associated with the change in melting point with pressure. There is therefore no additional heat supply to the geodynamo from this mechanism. There remains the difficult question of whether departures from hydrostatic pressure and adiabatic temperature associated with the convection can release further gravitational energy. I contend that it cannot because gravitational force is conservative and the Earth's gravitational energy depends only on the initial and final states, not on the details of the convective history.\Lspace \Lcitemark Buffett et al.\Citebreak 1996\Rcitemark \Rspace{} claim otherwise. \begin{table} \caption{Results for 3 choices of total dissipation entropy with and without radioactive heating.} \begin{center} \begin{tabular}{lrrrrrrr} \hline Model & Units & (i) & (ii) & (iii) & (iv) & (v) & (vi) \\ \hline $E$ & MW/K & 1000 & 546 & 262 & 1000 & 599 & 262 \\ $dT_{\rm c}/Dt$ & K/Gyr & 126 & 69 & 33 & 14 & 14 & 14 \\ IC age & Ga & 398 & 727 & 1521 & 3500 & 3500 & 3500 \\ $h$ & pW/kg & 0 & 0 & 0 & 14 & 7 & 3 \\ $Q$ & TW & 16 & 9 & 4 & 30 & 16 & 7 \\ \hline \end{tabular} \end{center} \end{table} % In Table~2 I present thermal core models based on core composition and properties computed by first principles calculations\Lspace \Lcitemark Alf\`{e}\Namecomma Price\Nameandd Gillan\Citebreak 2000\Rcitemark \Rspace{}. A mixture of Fe (84\% molar)+S/Si(8\% )+O(8\% ) has the right density for the liquid outer core and freezes to form an Fe-S/Si solid with the right density for the inner core. I assume 3 values of the total dissipation entropy: (a) $1000$~MW/K, a comfortable guess (b) 599~MW/K, the point at which compositional convection becomes essential to maintain the adiabat (c) 262~MW/K, when entropy sources fail to maintain convection. The two most critical properties are the density jump and temperature gradients at the ICB. Both involve differences of poorly-known quantities. The density jump determines the relative contribution of compositional to thermal convection; the temperature gradients determine the rate of freezing of the inner core. Other parameters play smaller roles in the final solution. If there is no inner core there is no latent heat or compositional convection: it is virtually impossible to sustain the geodynamo without these sources of buoyancy. We must therefore retain the inner core for at least 3~Gyr. This is only possible with substantial radioactive heating in the core. Model (v) is plausible if some way can be found for that much K to go into the core. The constraints would be eased considerably if the seismological value of the density jump at the ICB were raised to the upper limit consistent with present data. \noindent {\bf References} \small \message{REFERENCE LIST} \bgroup\Resetstrings% \def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{ }\def\Edcapsmallcapstest{}\def\Underlinetest{}% \def\NoArev{1000}\def\NoErev{1000}\def\Acnt{3}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}% \def\Ftest{ }\def\Ftrail{9}\def\Fstr{Alf\`{e}\Namecomma Gillan\Nameandd Price\Citebreak 1999}% \def\Atest{ }\def\Astr{Alf\`{e}\Revcomma D\Initper % \Acomma Gillan\Revcomma M\Initper \Initgap J\Initper % \Aandd Price\Revcomma G\Initper \Initgap D\Initper }% \def\Ttest{ }\def\Tstr{The melting curve of iron at the pressures of the Earth's core conditions}% \def\Jtest{ }\def\Jstr{Nature}% \def\Vtest{ }\def\Vstr{401}% \def\Ptest{ }\def\Pcnt{ }\def\Pstr{462--464}% \def\Dtest{ }\def\Dstr{1999}% \Refformat\egroup% \bgroup\Resetstrings% \def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{ }\def\Edcapsmallcapstest{}\def\Underlinetest{}% \def\NoArev{1000}\def\NoErev{1000}\def\Acnt{3}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}% \def\Ftest{ }\def\Ftrail{0}\def\Fstr{Alf\`{e}\Namecomma Price\Nameandd Gillan\Citebreak 2000}% \def\Atest{ }\def\Astr{Alf\`{e}\Revcomma D\Initper % \Acomma Price\Revcomma G\Initper \Initgap D\Initper % \Aandd Gillan\Revcomma M\Initper \Initgap J\Initper }% \def\Ttest{ }\def\Tstr{Constraints on the composition of the Earth's core from ab-initio calculations}% \def\Jtest{ }\def\Jstr{Nature}% \def\Vtest{ }\def\Vstr{405}% \def\Ptest{ }\def\Pcnt{ }\def\Pstr{172--175 }% \def\Dtest{ }\def\Dstr{2000}% \Refformat\egroup% \bgroup\Resetstrings% \def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{ }\def\Edcapsmallcapstest{}\def\Underlinetest{}% \def\NoArev{1000}\def\NoErev{1000}\def\Acnt{3}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}% \def\Ftest{ }\def\Ftrail{0}\def\Fstr{Alf\`{e}\Namecomma Kresse\Nameandd Gillan\Citebreak 2000}% \def\Atest{ }\def\Astr{Alf\`{e}\Revcomma D\Initper % \Acomma Kresse\Revcomma G\Initper % \Aandd Gillan\Revcomma M\Initper \Initgap J\Initper }% \def\Ttest{ }\def\Tstr{Structure and dynamics of liquid iron under Earth's core conditions}% \def\Jtest{ }\def\Jstr{Phys. Rev. B}% \def\Vtest{ }\def\Vstr{61}% \def\Ptest{ }\def\Pcnt{ }\def\Pstr{132--142}% \def\Dtest{ }\def\Dstr{2000}% \Refformat\egroup% \bgroup\Resetstrings% \def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{ }\def\Edcapsmallcapstest{}\def\Underlinetest{}% \def\NoArev{1000}\def\NoErev{1000}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}% \def\Ftest{ }\def\Ftrail{5}\def\Fstr{Backus\Citebreak 1975}% \def\Atest{ }\def\Astr{Backus\Revcomma G\Initper \Initgap E\Initper }% \def\Ttest{ }\def\Tstr{Gross thermodynamic heat engines in deep interior of Earth}% \def\Jtest{ }\def\Jstr{Proc. Natl. Acad. Sci. U.S.A.}% \def\Dtest{ }\def\Dstr{1975}% \def\Vtest{ }\def\Vstr{72}% \def\Ptest{ }\def\Pcnt{ }\def\Pstr{1555--1558}% \Refformat\egroup% \bgroup\Resetstrings% \def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{ }\def\Edcapsmallcapstest{}\def\Underlinetest{}% \def\NoArev{1000}\def\NoErev{1000}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}% \def\Ftest{ }\def\Ftrail{6}\def\Fstr{Boehler\Citebreak 1996}% \def\Atest{ }\def\Astr{Boehler\Revcomma R\Initper }% \def\Ttest{ }\def\Tstr{Melting temperature of the Earth's mantle and core: Earth's thermal structure}% \def\Dtest{ }\def\Dstr{1996}% \def\Jtest{ }\def\Jstr{Annu. Rev. Earth Planet. Sci.}% \def\Vtest{ }\def\Vstr{24}% \def\Ptest{ }\def\Pcnt{ }\def\Pstr{15--40}% \Refformat\egroup% \bgroup\Resetstrings% \def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{ }\def\Edcapsmallcapstest{}\def\Underlinetest{}% \def\NoArev{1000}\def\NoErev{1000}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}% \def\Ftest{ }\def\Ftrail{3}\def\Fstr{Braginsky\Citebreak 1963}% \def\Atest{ }\def\Astr{Braginsky\Revcomma S\Initper \Initgap I\Initper }% \def\Dtest{ }\def\Dstr{1963}% \def\Ttest{ }\def\Tstr{Structure of the F layer and reasons for convection in the Earth's core}% \def\Jtest{ }\def\Jstr{Dokl. Akad. Nauk. SSSR Engl. Trans.}% \def\Vtest{ }\def\Vstr{149}% \def\Ptest{ }\def\Pcnt{ }\def\Pstr{1311--1314}% \Refformat\egroup% \bgroup\Resetstrings% \def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{ }\def\Edcapsmallcapstest{}\def\Underlinetest{}% \def\NoArev{1000}\def\NoErev{1000}\def\Acnt{2}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}% \def\Ftest{ }\def\Ftrail{5}\def\Fstr{Braginsky\Nameand Roberts\Citebreak 1995}% \def\Atest{ }\def\Astr{Braginsky\Revcomma S\Initper \Initgap I\Initper % \Aand Roberts\Revcomma P\Initper \Initgap H\Initper }% \def\Ttest{ }\def\Tstr{Equations governing Earth's core and the geodynamo}% \def\Jtest{ }\def\Jstr{Geophys. Astrophys. Fluid Dyn.}% \def\Dtest{ }\def\Dstr{1995}% \def\Vtest{ }\def\Vstr{79}% \def\Ptest{ }\def\Pcnt{ }\def\Pstr{1--97}% \Refformat\egroup% \bgroup\Resetstrings% \def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{ }\def\Edcapsmallcapstest{}\def\Underlinetest{}% \def\NoArev{1000}\def\NoErev{1000}\def\Acnt{4}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}% \def\Ftest{ }\def\Ftrail{6}\def\Fstr{Buffett et al.\Citebreak 1996}% \def\Atest{ }\def\Astr{Buffett\Revcomma B\Initper \Initgap A\Initper % \Acomma Huppert\Revcomma H\Initper \Initgap E\Initper % \Acomma Lister\Revcomma J\Initper \Initgap R\Initper % \Aandd Woods\Revcomma A\Initper \Initgap W\Initper }% \def\Ttest{ }\def\Tstr{On the thermal evolution of the Earth's core}% \def\Jtest{ }\def\Jstr{J. Geophys. Res.}% \def\Vtest{ }\def\Vstr{101}% \def\Ptest{ }\def\Pcnt{ }\def\Pstr{7989--8006}% \def\Dtest{ }\def\Dstr{1996}% \Refformat\egroup% \bgroup\Resetstrings% \def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{ }\def\Edcapsmallcapstest{}\def\Underlinetest{}% \def\NoArev{1000}\def\NoErev{1000}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}% \def\Ftest{ }\def\Ftrail{7}\def\Fstr{Gubbins\Citebreak 1977}% \def\Atest{ }\def\Astr{Gubbins\Revcomma D\Initper }% \def\Dtest{ }\def\Dstr{1977}% \def\Ttest{ }\def\Tstr{Energetics of the Earth's core}% \def\Jtest{ }\def\Jstr{J. Geophys.}% \def\Vtest{ }\def\Vstr{43}% \def\Ptest{ }\def\Pcnt{ }\def\Pstr{453--464}% \Refformat\egroup% \bgroup\Resetstrings% \def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{ }\def\Edcapsmallcapstest{}\def\Underlinetest{}% \def\NoArev{1000}\def\NoErev{1000}\def\Acnt{2}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}% \def\Ftest{ }\def\Ftrail{9}\def\Fstr{H\"{a}ge\Nameand M\"{u}ller\Citebreak 1979}% \def\Atest{ }\def\Astr{H\"{a}ge\Revcomma H\Initper % \Aand M\"{u}ller\Revcomma G\Initper }% \def\Ttest{ }\def\Tstr{Changes in dimensions, stresses and gravitational energy of the Earth due to crystallisation at the inner core boundary under isochemical conditions.}% \def\Jtest{ }\def\Jstr{Geophys. J. R. Astron. Soc.}% \def\Vtest{ }\def\Vstr{58}% \def\Ptest{ }\def\Pcnt{ }\def\Pstr{495--508}% \def\Dtest{ }\def\Dstr{1979}% \Refformat\egroup% \bgroup\Resetstrings% \def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{ }\def\Edcapsmallcapstest{}\def\Underlinetest{}% \def\NoArev{1000}\def\NoErev{1000}\def\Acnt{3}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}% \def\Ftest{ }\def\Ftrail{5}\def\Fstr{Hewitt\Namecomma McKenzie\Nameandd Weiss\Citebreak 1975}% \def\Atest{ }\def\Astr{Hewitt\Revcomma J\Initper % \Acomma McKenzie\Revcomma D\Initper \Initgap P\Initper % \Aandd Weiss\Revcomma N\Initper \Initgap O\Initper }% \def\Ttest{ }\def\Tstr{Dissipative heating in convective flows}% \def\Jtest{ }\def\Jstr{J. Fluid Mech.}% \def\Vtest{ }\def\Vstr{68}% \def\Ptest{ }\def\Pcnt{ }\def\Pstr{721--738}% \def\Dtest{ }\def\Dstr{1975}% \Refformat\egroup% \bgroup\Resetstrings% \def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{ }\def\Edcapsmallcapstest{}\def\Underlinetest{}% \def\NoArev{1000}\def\NoErev{1000}\def\Acnt{3}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}% \def\Ftest{ }\def\Ftrail{7}\def\Fstr{Labrosse\Namecomma Poirier\Nameandd LeMou\"{e}l\Citebreak 1997}% \def\Atest{ }\def\Astr{Labrosse\Revcomma S\Initper % \Acomma Poirier\Revcomma J-P\Initper % \Aandd LeMou\"{e}l\Revcomma J-L\Initper }% \def\Ttest{ }\def\Tstr{On cooling of the Earth's core}% \def\Jtest{ }\def\Jstr{Phys. Earth Planet. Int.}% \def\Vtest{ }\def\Vstr{99}% \def\Ptest{ }\def\Pcnt{ }\def\Pstr{1--17}% \def\Dtest{ }\def\Dstr{1997}% \Refformat\egroup% \bgroup\Resetstrings% \def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{ }\def\Edcapsmallcapstest{}\def\Underlinetest{}% \def\NoArev{1000}\def\NoErev{1000}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}% \def\Ftest{ }\def\Ftrail{8}\def\Fstr{Loper\Citebreak 1978}% \def\Atest{ }\def\Astr{Loper\Revcomma D\Initper \Initgap E\Initper }% \def\Ttest{ }\def\Tstr{The gravitationally powered dynamo}% \def\Jtest{ }\def\Jstr{Geophys. J. R. Astron. Soc.}% \def\Vtest{ }\def\Vstr{54}% \def\Ptest{ }\def\Pcnt{ }\def\Pstr{389--404}% \def\Dtest{ }\def\Dstr{1978}% \Refformat\egroup% \bgroup\Resetstrings% \def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{ }\def\Edcapsmallcapstest{}\def\Underlinetest{}% \def\NoArev{1000}\def\NoErev{1000}\def\Acnt{1}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}% \def\Ftest{ }\def\Ftrail{0}\def\Fstr{Poirier\Citebreak 2000}% \def\Atest{ }\def\Astr{Poirier\Revcomma J-P\Initper }% \def\Ttest{ }\def\Tstr{Introduction to the Physics of the Earth's Interior}% \def\Dtest{ }\def\Dstr{2000}% \def\Ptest{ }\def\Pcnt{}\def\Pstr{312pp}% \def\Itest{ }\def\Istr{Cambridge Univ. Press}% \Refformat\egroup% \bgroup\Resetstrings% \def\Loccittest{}\def\Abbtest{ }\def\Capssmallcapstest{}\def\Edabbtest{ }\def\Edcapsmallcapstest{}\def\Underlinetest{}% \def\NoArev{1000}\def\NoErev{1000}\def\Acnt{2}\def\Ecnt{0}\def\acnt{0}\def\ecnt{0}% \def\Ftest{ }\def\Ftrail{9}\def\Fstr{Stacey\Nameand Stacey\Citebreak 1999}% \def\Atest{ }\def\Astr{Stacey\Revcomma F\Initper \Initgap D\Initper % \Aand Stacey\Revcomma C\Initper \Initgap H\Initper \Initgap B\Initper }% \def\Ttest{ }\def\Tstr{Gravitational energy of core evolution: implications for thermal history and geodynamo power}% \def\Jtest{ }\def\Jstr{Phys. Earth Planet. Int.}% \def\Vtest{ }\def\Vstr{110}% \def\Ptest{ }\def\Pcnt{ }\def\Pstr{83--93}% \def\Dtest{ }\def\Dstr{1999}% \Refformat\egroup% \end{document}