SOME  MATHEMATICAL  ASPECTS  OF  THE  PLANET  EARTH    

José  Francisco  Rodrigues  (University  of  Lisbon)    Article  of  the  Special  Invited  Lecture,  6th  European  Congress  of  Mathematics  3  July  2012,  Krakow.       The   Planet   Earth   System   is   composed   of   several   sub-­‐systems:   the  atmosphere,  the  liquid  oceans  and  the  icecaps  and  the  biosphere.  In  all  of  them  Mathematics,  enhanced  by  the  supercomputers,  has  currently  a  key  role  through  the  “universal  method"  for  their  study,  which  consists  of  mathematical  modeling,  analysis,  simulation  and  control,  as  it  was  re-­‐stated  by  Jacques-­‐Louis  Lions  in  [L].  Much   before   the   advent   of   computers,   the   representation   of   the   Earth,   the  navigation   and   the   cartography   have   contributed   in   a   decisive   form   to   the  mathematical   sciences.   Nowadays   the   International   Geosphere-­‐Biosphere  Program,   sponsored   by   the   International   Council   of   Scientific   Unions,   may  contribute  to  stimulate  several  mathematical  research  topics.     In   this   article,  we   present   a   brief   historical   introduction   to   some   of   the  essential   mathematics   for   understanding   the   Planet   Earth,   stressing   the  importance   of   Mathematical   Geography   and   its   role   in   the   Scientific  Revolution(s),  the  modeling  efforts  of  Winds,  Heating,  Earthquakes,  Climate  and  their  influence  on  basic  aspects  of  the  theory  of  Partial  Differential  Equations.  As  a  special   topic   to   illustrate   the  wide  scope  of   these   (Geo)physical  problems  we  describe   briefly   some   examples   from   History   and   from   current   research   and  advances   in   Free   Boundary   Problems   arising   in   the   Planet   Earth.   Finally   we  conclude   by   referring   the   potential   impact   of   the   international   initiative  Mathematics  of  Planet  Earth  (www.mpe2013.org)  in  Raising  Public  Awareness  of  Mathematics,   in   Research   and   in   the   Communication   of   the   Mathematical  Sciences  to  the  new  generations.    Ancient  Mathematics  and  the  Earth       There   is   no   doubt   that   the   planet   Earth   is   one   a   main   ancient   root   of  mathematics.   Distancing,   constructing,   spacing,   surveying   or   angulating   led   to  Geometry,   that  means   literally  measurement  of   the   earth   (respectively,  metron  and  geo,   from  ancient  Greek).  The  Babylonian   tablets   and   the  Egyptian  papyri,  which   are   dated   back   about   4000   years,   are   the   first   known   records   of  elementary  geometry.     Even  if  it  may  be  controversial  to  attribute  to  Pythagoras  the  idea  that  the  shape  of  the  Earth  is  a  sphere,  this  was  clear  already  to  Aristotle  (384  –  322  BCE)  in   his   “On   the   Heavens”:   “Its   shape   must   be   spherical…   If   the   earth   were   not  spherical,  eclipses  of   the  moon  would  not  exhibit   segments  of   the  shape  they  do…  Observation  of  the  stars  also  shows  not  only  that  the  earth  is  spherical  but  that  it  is  not   no   great   size,   since   a   small   change   of   position   on   our   part   southward   or  northward  visibly  alters  the  circle  of  the  horizon,  so  that  the  stars  above  our  heads  change  their  position  considerably,  and  we  do  not  see  the  same  stars  as  we  move  to  the  North  or  South.”       But  if  the  Hellenistic  scientists  had  observed  the  sphericity  of  the  planet,  they   had   also   obtained   a   relatively   accurate   estimate   of   its   radius.   Indeed,  we  owe  one  of  the  first  estimates  of  the  circumference  of  the  earth  to  Erastosthenes  

(276-­‐194   BCE),   a   member   of   the   Alexandrine   school,   who   established   it   in  250,000   stadia.   He   measured   in   Alexandria   the   angle   elevation   of   the   sun   at  midday,  i.e.  the  angular  distance  from  the  zenith  at  the  summer  solstice,  and  he  found  1/50th  of   a   circle   (about  7°12’)  making   then   the  proportion,  by  knowing  that   Syene   (Aswan)   was   on   the   Tropic   of   Cancer   at   a   distance   of   about   5000  stadia.  If  the  stadion  meant  185  m,  he  obtained  46,620  km,  an  error  of  16.3%  too  great,   but   if   the   stadion  meant   157.5  m,   them   the   result   of   39,690   km   has   an  error  less  than  2%!  

     

  Erastosthenes  of  Cyrene,  as  Heath  wrote  [H], “was,  indeed,  recognised  by  his  contemporaries  as  a  man  of  great  distinction  in  all  branches  of  knowledge”.  He  is  remembered  for  his  prime  number  sieve,  still  a  useful  tool  in  number  theory,  and  was  the  first  to  use  the  word  geography  and  to  attempt  to  make  a  map  of  the  world  for  which  he  invented  a  line  system  of  latitude  and  longitude.     Another  old  trigonometric  technic,  the  basic  principle  of  triangulation  to  determine  distances  of  inaccessible  points  on  earth,  was  used  by  Aristarchus  of  Samos   (about  310-­‐230  BCE)   to  estimate   the   relative   sizes  and  distances  of   the  Sun   and   the  Moon.   Even   if   these   estimates  were   of   an   order   of  magnitude   too  small,   this   was   a   remarkable   intellectual   achievement   of   the   Hellenic  mathematician.   He   was   also   a   precursor   of   Copernicus,   as   one   of   the  philosophers  of  the  Antiquity  to  suggest  the  heliocentric  theory  in  Astronomy.  

http://en.wikipedia.org/wiki/File:PtolemyWorldMap.jpg (15th century redrawn of Ptolemy’s world map)

  Ptolemy   (about   100-­‐178),   the  most   influential   Hellenic   astronomer   and  geographer   of   his   time,   credited  Eratosthenes   to   have  measured   the   tilt   of   the  Earth's   axis  with   great   accuracy  obtaining   the   value  of   11/83  of   180°   (23°  51'  

15").  In  his  Guide  to  Geography  he  gave  information  on  the  construction  of  maps  of  the  known  world  in  Europe,  Africa  and  Asia.  However,  as  we  may  see  from  a  world   map   redrawn   in   the   15th   century,   from   the   present   point   of   view   his  representation   of   the   earth   is   not   accurate   at   all,   in   particular   showing   the  Atlantic  and   the   Indian  Oceans  as   closed  seas. Ptolemy  used  Strabo’s  value   for  the  circumference  of  the  Earth,  which  was  too  small  with  an  error  of  27.7%.  This  crude  estimate  has  been  used  to  explain  the  Columbus’  error  of  looking  Cipango  (Japan)  going  West  more  than  thirteen  centuries  later  [RW],  but  historians  have  recently  discovered  other  reasons  for  this  fact.     In   its   great   astronomical   treatise   of   the   second   century,   the   Almagest,  which   geocentric   theory  was   not   superseded   until   a   century   after   Copernicus’  book   De   Revolutionibus   Orbium   Coelestium   of   1543,   Ptolemy   describes,   in  particular,  a  kind  of  ‘astrolabe’,  which  is  a  combination  of  graduated  circles  that  later  became  a  more   sophisticated   chief   astronomical   instrument   reintroduced  into  Europe  from  the  Islamic  world.  The  nautical  adaptation  of  the  planispheric  astrolabe  was  one  of  the  tools  used  by  the  Portuguese  navigator  Bartolomeu  Dias  in  his  ocean  expedition  rounding  Africa  and  crossing  the  Cape  of  “Boa  Esperança”  in   1488.   This   has   shown   the   connection   between   the   Atlantic   and   the   Indian  Oceans,   a   discovery   that  would   change   dramatically   the   geographical   vision   of  the   world,   and   has   happened   four   years   before   Columbus   first   travel   to   the  Antilles.    

 http://en.wikipedia.org/wiki/File:Martellus_world_map.jpg  (Martellus  world  map  of  1489  or  1490)    

    This   fact  was   immediately   reflected   in   the  world  map  made   in   1489   or  1490  by  Henricus  Martellus   and   in   the  Nuremberg  Globe  of  Martin  Behaim,   of  1492   [Da].   Atlas   and   globes   are   treasures   of   the  Renaissance   cartography   that  illustrate  how  useful  mathematical  techniques  were  necessary  for  map  making  in  the  late  15th  century,  for  practical  navigations  or  for  helping  the  European  minds  to   change   their   concept   of   the  world,   as   did   the   famous  Globus   Jagellonicus   of  1510  that  is  considered  as  being  the  oldest  existing  globe  showing  the  Americas.       The  strategic  importance  of  the  new  terrestrial  representations  and  of  the  ocean  navigations,  as  new  key  technologies,  goes  beyond  their  scientific  meaning  and   consequences.   They   represented   technological   breakthroughs   and   were  decisive   tools   for   the   European   expansion   in   the   period   1400-­‐1700,   as   “the  

conquest   of   the   high   seas   gave   Europe   a   world   supremacy   that   lasted   for  centuries”  [B].       If   shape,  measure  and  representation  of   the  Earth  were  key  elements   in  ancient   mathematics,   the   novel   problems   and   concepts   of   Renaissance  mathematics,   in  particular,   those  associated  with  a  new  geometric  approach   to  the   theory   and   practice   of   navigation,   as  well   as   to  mapping   techniques,  were  instrumental   in  the  rise  of  modern  science.  As  the  Dutch  historian  of  science  R.  Hooykaas  has  stressed,  “the  great  change  (not  only  in  astronomy  or  physics,  but  in  all   scientific   disciplines)   occurred   when,   not   incidentally   but   in   principle   and   in  practice,   the   scientists   definitively   recognized   the   priority   of   Experience.   The  change  of  attitude  caused  by  the  voyages  of  discovery  is  a  landmark  affecting  not  only  geography  and  cartography,  but  the  whole  of  'natural  history'.”  [Ho]  

Mathematical  Geography  and  the  Scientific  Revolution(s)       Recently  historians  of  Mathematics  have  been  recognizing  the  importance  of   Renaissance   methods   [K],   often   invoking   the   significant   and   countlessly  repeated  phrase  of  Galileo  in  “Il  Saggiatore”  (1623):  “Philosophy  is  written  in  this  grand   book,   the   universe,   (…)  written   in   the   language   of  mathematics,   and   its  characters  are  triangles,  circles,  and  other  geometric  figures  without  which  it  is  humanly   impossible   to   understand   a   single   word   of   it;   without   these,   one  wonders  about  in  a  dark  labyrinth.”       The  Elements   of   Euclides  were   first   printed,   in   Latin,   in  Venice   in  1482,  and   had   several   vernacular   translations   in   the   following   century   in   Italian,  German,  French,  English  and  Spanish.  The  English’s  edition,  printed  in  London  in  1570,   contains   a   “very   fruitfull   Preface   made   by   M.I.Dee,   specifying   the   chiefe  Mathematicall  Sciences”  [AG].  In  this  influential  text  of  the  English  scientist  John  Dee  (1527-­‐1608),  after  stating  that  “Of  Mathematicall  thinges,  are  two  principall  kindes:  namely,  Number,  and  Magnitude”,  he  describes  among  the  branches  of  his  remarkable  “Mathematicall  Tree”,  the  “Arte  of  Nauigation,  demonstrateth  how,  by  the   shortest   good  way,   by   the   aptest   Directiõ,   &   in   the   shortest   time,   a  sufficient  Ship,  betwene  any  two  places  (in  passage  Nauigable,)  assigned:  may  be  cõducted:  and  in  all  stormes,  &  naturall  disturbances  chauncyng,  how,  to  vse  the  best  possible  meanes,  whereby  to  recouer  the  place  first  assigned”  [D].       Geography   and   navigation  were   in   fact   extremely   important   in   the   16th    century  [LA]  and  it  became  now  clear  that  Dee’s  mathematical  program  has  roots  in   the  works  of   the  Portuguese  mathematician  and  cosmographer  Pedro  Nunes  (1502-­‐1578).  In  a  letter  of  1558  to  Mercator,  Dee  considered  Nunes  as  the  “most  learned   and   grave   man   who   is   the   sole   relic   and   ornament   and   prop   of   the  mathematical  arts  among  us”   [A].  Gerardus  Mercator   (1512-­‐1594),   the  German  cartographer   and  mathematician,   in  his  Mapamundi  of  1569  had   constructed  a  new  projection  to  represent  the  rhumb  lines,  i.e.  the  curves  with  a  constant  angle  V  (0<V<π/2)  with  all  meridians,  as  straight  lines  on  a  flat  map  [C].  Those  spiral  curves  on  the  sphere,  later  also  called  loxodromes,  as  it  is  confirmed  with  recent  new  evidence  by  historians  [A],  were  brought  to  Mercator’s  attention  by  Nunes’  works,  who   a   few   years   earlier   had   imagined   and   had   discussed   a  method   for  their  representation  for  nautical  purposes.       Although   this  major   advance   in  mathematical   cartography  was   of   great  importance   for   navigation   [D’H],   it   took   one   and   half   century   to   understand  

completely  the  mathematical  equation  of  the  loxodrome  and  to  establish  that  its  stereographic   projection   in   the   plane   is   the   logarithmic   spiral,   as   it   was  published  by  the  English  astronomer  Edmund  Halley  in  1696  [Ha].     Directly   questioned   by   the   seamen   returning   from   the   oceanic  navigations,  Pedro  Nunes  was  the   first   to  distinguished  on  the  terrestrial  globe  between  the  loxodromic  course,  then  called  rhumb  lines,  consisting  in  navigating  with  a  constant  angle  and  the  orthodromic  course,  which  is  the  shortest  distance  on   the   arc   of   a   great   circle,   i.e.   the   geodesic.   In   two   small   original   treatises  published  in  Portuguese  in  1537,  in  particular  in  “Tratado  em  defensam  da  carta  de  marear”   (‘Treatise   defending   the   nautical   chart’)   [LA],   Nunes   described   the  spiral  nature  of  the  rhumb  lines  and  represented  them  in  a  symmetrical  picture  inside  an  equinoctial  circle.  However,  only  in  is  Opera,  a  Latin  Collectanea  of  his  extended  works  published  in  Basel  in  1566,  Nunes  clearly  stated  that  the    

       

The  1537’s  original  representation  of  the  loxodromes  by  Pedro  Nunes  [N].    

loxodrome  behaves  similarly  to  a  helix  never  reaching  the  pole,  a  concept  he  also  described  in  a  manuscript  of  the  1540’s  found  in  Florence.  Later  in  a  manuscript  of   1595   Thomas   Harriot   suggested   a   relation  with   the   logarithm   spiral   in   the  plane.   In   the  book   “Certain  errors   in  navigation”  published   in  London   in  1599,  another  English  mathematician  and  cartographer,  Edward  Wright   (1561-­‐1615)  that   studied   carefully   Nunes’   works,   described   precisely   the   process   of  representing  rumbs  lines  as  straight  lines  in  Mercator  charts.     In   modern   notations,   in   a   sphere   with   unit   radius   the   equation   of   the  loxodrome  with  angle  V  may  be  given  by      

φ  =  −  τ  log  tan  (θ/2)  ,    where  τ =  tan  V,  φ is   the   longitude   (φ  =  0  at   its   intersection  with   the  equator)  and  θ  the  colatitude.  Hence,  in  Cartesian  coordinates,  z  =  cos  θ  and  in  the  xy-­‐plan  of  the  equator  x  =  sin  θ  cos  φ    and  y  =  sin  θ  sin  φ.    By  eliminating  θ  in  the  equation  ρ  =  sin  θ  and  using  the  loxodrome  equation,  we  obtain      

ρ  =  sech  (φ/τ )  =  2  (eφ /τ  +    e−φ /τ )−1  .  

    This   equation   is   the   orthogonal   projection   of   the   loxodrome   in   polar  coordinates  on   the  equator  plan  and   represents  a  Poinsot   spiral   [GT],   a  planar  curve   considered   by   the   French   mathematician   Louis   Poinsot   in   his   1834  geometrical  mechanics  theory  [P].       Another   remarkable   work   of   Pedro   Nunes   is   the   book   on   twilights,  De  Crepusculis   published   for   the   first   time   in   Lisbon   in   1542.   This   geophysical  problem   had   been   considered   by   the   Islamic   Al-­‐Andalus   mathematician   Ibn  Mu’adh  in  the  11th  century  and  by  the  Polish  scientist  Witelo  in  the  13th  century.  In  his  book,  Nunes  treated  the  twilight  variation  produced  by  the  sun  during  the  annual   course   through   the   ecliptic.   Using   spherical   trigonometry   only,   he  was  able,  in  particular,  to  answer  completely  the  question  about  the  shortest  twilight,  while   Jakob   Bernouilli,   l’Hospital   and   d’Alembert   gave   only   an   indirect,  incomplete   solution   of   the   problem,   as   recognized   by   Delambre   in   1815   and  refereed  by  Gauss  in  1817  [Kno].       No   less   important   is   the   method   invented   by   Nunes   to   improve   the  measurements  of  angles  in  the  division  of  the  scale  of  a  quadrant  or  of  a  nautical  astrolabe.   The   concept   of   the   “nonius”   arise   in   the   second   part   of   the   “De  Crepusculis”,   after   proposition   III:   “to   construct   an   instrument   well   suited   to  observation   of   the   heavenly   bodies,   with   which   one   can   accurately   determine  their  altitudes”  [ER].  The  method  was  used  by  Tycho  Brahe  (1546-­‐1601)  in  two  quadrants   he   constructed   for   his   observatory   in   Uraniborg,   in   Denmark,   that  were   equipped   with   the   nonius   and   he   described   in   the   book   “Astronomiae  Insaturatae  Mechanica”   [Br],   first  published   in  1598,   referring  explicitly   “inside  this  division  there   is  yet  another  according  to  the  principles  set   forth  by  …  Petrus  Nonnius   in   his   learned   little   book   on   the   Twilight”.   The   only   known   existing  instrument  of  this  period  reproducing  the  Nunes’  invention  is  a  quadrant  used  to  measure   altitudes   dated   1595   and   belonging   to   the  Museum   of   the   History   of  Science  in  Florence  [ER].  

           The  quadrant  of  the  16th  century  of  the  Museum  of  the  History  of  Science  in  Florence  and  a  

picture  from  the  Brahe’s  “Astronomiae  Insaturatae  Mechanica”  of  1598.    

  A   pioneer   and   most   important   feature   of   this   Iberian   Renaissance  mathematician  was  his   professional   activity   as   royal   cosmographer   from  1529  on  and,  in  addition  of  being  mathematics  professor  at  the  University  of  Coimbra  since  1544,  his  appointment  as  Cosmographer-­‐in-­‐Chief  of  Portugal  in  1547,  who  was   in   charge   of   the   examination   the   chart   masters   and   nautical   instruments  manufacturers   and   of   the   certification   of   its   quality   [ER].   These   facts   placed  Pedro  Nunes  as  one  of  the  “few  learned  authors  [who]  began  to  be  interested  in  the   mechanical   arts,   which   had   become   economically   so   important”   in   the  sixteenth  century,  since  “natural  science  needs  theory  and  mathematics  as  well  as   experiments   and   observations”   and   “only   theoretically   educated   men   with  rationally  trained  intellects  were  able  to  supply  that  other  half  of  its  methods  to  science”,   so   that   paraphrasing   Edgar   Zilsel   [Z],   “eventually   the   social   barrier  between   the   two   components   of   the   scientific   method   broke   down,   and   the  methods   of   the   superior   craftsmen   were   adopted   by   academically   trained  scholars:  the  real  science  was  born”.     These   original   contributions   in   mathematical   navigation,   together   with  other  advances  in  geography,  astronomy,  architecture,  mechanics  and  music,  are  significant   examples   of   a   paradigm   of   Renaissance   mathematics   on   the  establishment  of  a  “program  for  the  mathematization  of  the  real  world”  [Le]  and  have   contributed   to   the   creation   of   the   first   “Academia   Real   Mathematica”   de  Madrid,  by  Filipe  II,  while  he  was  in  Lisbon  in  1582.  In  a  remarkable  text  on  the  institutional   foundations  of   this  pioneer  Iberian  academy,   Juan  de  Herrera  [He]  refers   the   importance   of   “las   disciplinas   Mathematicas   que   abren   la   entrada   y  puerta   a   todas   las   demas   sciencias   por   su   grande   certitude   y   mucha   euidencia,  donde  tomaron  el  nombre  de  Mathematicas  o  disciplinas  que  todo  es  vno,  porque  manifestan  el  methodo  verdadero  y  orden  de  saber”  (the  Mathematical  disciplines  that  open  the  door  and  entrance  to  all  other  sciences  for  his  great  certitude  and  much   evidence,   which   took   the   name   of   Mathematics   or   disciplines   that   the  whole   is  one,  because   they  express   the   true  method  and  order  of  knowledge").  Indeed,  almost  one  century  before  Galileo’s  famous  statement,  Nunes  in  his  1537  work  on  mathematical  navigation  had  already  written  “(…)  because  no  rule  that  is   based   on   speculative   or   theoretical   knowledge   can   be   well   practiced   and  understood   if   one   doesn’t   know   this   same   principles”   since   “nothing   is   most  evident  than  mathematical  demonstration,  which  by  no  means,  is  possible  to  be  contested”  [AL].    Winds,  Heating,  Earthquakes  and  Climate         The  atmosphere,  the  liquid  oceans,  the  icecaps,  the  internal  structure  and  the   biosphere   are   the   sub-­‐systems   of   the   Planet   Earth   whose   mathematical  models   and   simulations,   nowadays   enhanced   by   the   supercomputers,   play   an  essential  role,  as  observed  by  J-­‐L  Lions  (1928-­‐2001)  in  [L].       Much  before  the  advent  of  computers,  while  navigation  and  cartography  have   contributed   to   advance   the  mathematical   knowledge   in   the   Renaissance,  after  the  invention  of  the  Calculus  in  the  following  century,  the  planet  Earth  has  continued  to  contribute  in  several  ways  to  advance  Mathematical  Analysis  in  the  eighteen  and  nineteen  centuries  as  well.       The   partial   calculus   appeared   in   the   end   of   the   17th   century   with  trajectory   and   isoperimetric   problems   with   one   parameter   families   of   curves.  

However  the  creation  of  the  theory  of  partial  differential  equation  took  a  decisive  step   in   the   works   of   Jean   D’Alembert   (1717-­‐1783)   on   the   “vibrating   string  problem”  and  on  a  problem  of  the  Planet  Earth  [De].  The  introduction  in  1743  in  his  “Traité  de  dynamique”  of  the  first  wave  equation  of  the  type  

ddy = dyds

− (l − s) ddyds2

⎡⎣⎢

⎤⎦⎥dt 2    

was   followed   by   the   more   developed   work   published   in   Paris   in   1747   on  “Réflexions  sur  la  cause  générale  des  vents”.    In  this  book,  that  won  the  1746  prize  of   the   Academy   of   Sciences   of   Berlin,   d’Alembert   considered   the   geophysical  problem  of  the  vibrations  of  a  layer  of  air  (winds)  under  the  action  of  the  Earth  rotation  and  he  derives  for  the  first  time  the  general  formula  for  the  solution  of      

 the  wave   equation   that   has   now  his   name.  He   developed   the  method   of   Euler,  introducing  first  order  systems  of  the  type    

∂α∂t

= ∂β∂s      and      ν ∂β

∂t= ρ ∂α

∂s+ϕ(t, s)

   

Looking   for   total   differentials   and   using   the   usual   change   of   coordinates,  d’Alembert  arrived  to  the  general  solution  of  the  wave  equation  in  terms  of  two  arbitrary  functions  defined  on  the  characteristics  equations  t  +  λ s  = C1  and  t  − λ  s = C2   ,  where  λ2 = ν/ρ.   The   third   part   of   his  Mémoire   actually   initiated   this   new  branch  of  mathematical  analysis   that  was   immediately   followed  and  developed  by  Euler  and  Lagrange.     Also   related   to   the   Planet   Earth,   the   Academy   of   Sciences   of   Paris   had  proposed   in  1738   in   the   class  of  mathematics   the   theme   “the   cause  of   the   flux  

and  reflux  of  the  sea”  and  among  the  prizewinners  were  Daniel  Bernoulli,  Euler  and  MacLaurin.  But,  as   J-­‐L  Lions  also  observed  in  his   interesting   little  book  [L],  Joseph   Fourier   (1768-­‐1830)   became   a   forerunner   by   calling   the   attention   in  1824  on  the  possible  effect  of  anthropogenic  factors  on  the  surface  temperature  of  the  Earth.       In   fact,   Fourier,   recognized   in   [F2]:   “La   question   des   températures  terrestres  m'a  toujours  paru  un  des  plus  grands  objets  des  études  cosmologiques,  et  je   l'avais   principalement   en   vue   en   établissant   la   théorie   mathématique   de   la  chaleur”  (The  question  of  the  terrestrial  temperatures  has  always  seemed  to  me  one  of  greatest  subjects  of  the  cosmological  studies,  and  I  had  it  mainly  in  mind  in  establishing  the  mathematical  theory  of  heat);  and  in  [F2]  he  also  conjectured  explicitly   that   “L’établissement   et   le   progress   des   sociétés   humaines,   l’action   des  forces  naturelles  peuvent  changer  notablement,  et  dans  de  vastes  contrées,  l’état  de  la  surface  du  sol,  la  distribuition  des  eaux  et  les  grands  mouvements  de  l’air”  (The  establishment  and  progress  of   the  human  societies,   the  action  of  natural   forces  can  change  significantly,  and  in  large  parts,  the  surface  condition  of  the  soil,  the  distribution   of   water   and   the   large   air   movements),   as   possible   causes   of   the  variation  of  the  average  temperatures  in  the  course  of  several  centuries.     Indeed  the  interest  of  Fourier  on  the  temperature  of  the  Earth  dated  from  the   first   manuscripts   of   1807   of   his   “Théorie   Analytique   de   la   Chaleur”,   which  final  version  was  published  in  Paris  only  in  1822.  In  his  Mémoire  [F1]  of  1820  he  referred   three   factors   acting   upon   the   heat   of   the   planet:   the   heating   by   the  sunrays,   the   internal   heat   of   the   globe   and   the   secular   dissipation   due   to   the  cooling   of   the   Earth.   Then   he   described   the   model   of   the   “refroidissement  séculaire”   of   the   sphere   using   the   heat   equation   in   polar   coordinates   for   the  temperature  υ  at  time  t  and  at  the  layer  of  radius  x:

∂ν∂t

= KCD

∂2ν∂x2

+ 2x∂ν∂x

⎛⎝⎜

⎞⎠⎟

where  K,  C  and  D  are  physical  constants.  To  model  the  heat  change  at  the  surface  of   the   Earth   Fourier   was   the   first   to   write   the   differential   equation   of   the  qualitative  law  observed  experimentally  by  Newton  in  1701:  

K ∂ν∂x

+ hν = 0 ,  

which  means  that  the  heat  flux   is  proportional  to  the  surface  temperature  with  constant   h/K.   He   gave   then   the   exact   general   solution   in   the   form   of   a  trigonometric   series   in   x   with   an   exponential   in   t   with   precise   coefficients   in  terms  of  the  initial  temperature,  by  referring  to  his  earlier  work  of  1807.  Finally  he   concluded   that   “on   peut   connaître,   au   moyen   de   cette   formule,   toutes   les  circunstances   du   refroidissement   d’un   globe   solide   dont   le   diamètre   n’est   pas  extrêmement  grand”  (By  means  of  this  formula,  we  may  know  all  circunstances  of  the  cooling  of  a  solid  globe  whose  diameter  is  not  extremely  large).     More   than   a   century   later,   after   graduating   at   the   Physics   and  Mathematics  Faculty  of  the  Leningrad  (St  Petersburg)  University  in  1929,  Sergey  Sobolev  (1908-­‐1989)  worked  at  the  Theoretical  Department  of  the  Seismological  Institute   and   at   the   Steklov   Institute   of   Physics   and  Mathematics   of   the   USSR  Academy   of   Sciences   in   his   city.   In   the   first   decade   of   his   mathematical  production,   he   introduced   and  developed  new   theories   that  were   fundamental  for   the  development   of   partial   differential   equations   and   functional   analysis   in  

the   20th   century   [Ba].   Besides   his   deep   work   in   hyperbolic   partial   differential  equations   of   second   order   and   the   proof   of   the   seminal   Sobolev   inequalities,  which   are   at   the   basis   of   the   theory   of   Sobolev   spaces,   he   introduced   in   1934  generalized   solutions   [So]   for   the   wave   equation,   a   landmark   that   became  adaptable   to   an   enormous   number   of   problems   of   mathematical   physics   and  other  areas  of  mathematics.  The  following  year  he  introduces  the  novel  concept  “solution  in  functionals”   for  the  Cauchy  problem  for  hyperbolic  linear  equations,  which  corresponds  to  what  is  now  called  a  distribution.    

 The  young  S  L  Sobolev  with  A  N  Krilov  in  the  1930’s.  

    Also  in  1934  another  young  mathematician,  Jean  Leray  (1906-­‐1998)  after  completing  his  PhD  at  the  Faculty  of  Science  of  Paris  the  year  before,  introduced  the   fundamental   concept   of   weak   solutions   of   the   Navier-­‐Stokes   equations,  which   he   called   “solutions   turbulentes”,   based   on   the   new   definition   of   “quasi-­‐dérivées”   and   on   the   approximation   of   integrable   functions   by  mollification   in  [L1].   In   fact,   Leray   used   implicitly   the   Sobolev   space   H1( 3 )   in   his   existence  theory  and  raised  a  still  open  double  question.  Sixty  years  later  he  described  in  [L2]   this   double   question   that   became   known   as   one   of   “The   Millenium  Problems”:   “The  theoretical  study  of  a  fluid  motion  with  initial  condition  leads  in  various  cases  to  a  same  conclusion:  the  existence  of  at  least  one  weak  solution  that  is   regular   and  unique  near   the   initial   time,   and   exists   for   any   later   time   (…)  but  does  it  remain  regular  and  uniquely  determined?”       If   the   wave   equation   in   heterogeneous   media   is   an   essential   tool   in  theoretical  seismology,  the  Navier-­‐Stokes  equations  are  of  no  less  importance  in  meteorology,   in   particular,   in   developing   mathematical   models   of   the  atmosphere   or   of   the   ocean.   For   instance,  with   the   purpose   to   understand   the  long-­‐term  weather  prediction  and  climate  changes,   in  1992  J-­‐L  Lions  proposed,  in  collaboration  with  R  Temam  and  S  Wang  [LTW1],  a  long  range  project  based  on  the  mathematical  study  and  computational  simulations  of  an  important  class  in   the   hierarchy   of   models   for   in   geophysical   fluid   dynamics,   the   primitive  equations  of  the  dynamics  of  the  atmosphere  and  of  the  oceans,  as  well  as  their  coupling   effects,   by   analogy   to   the   mathematical   theory   of   Navier-­‐Stokes  equations  (see  [PTZ],  for  a  recent  survey).       From  a  different  point  of  view,  the  Serbian  mathematical-­‐physicist  Milutin  Milankovitch   (1879-­‐1958)   in   the   first   quarter   of   the   20th   century   studied   the  effect   of   solar   radiation   in   different   latitudes   and   its   consequence   in   the  

planetary  albedo  and  proposed  a  decisive  theory  on  “mathematical  climate”  [M]  to   explain   the  Earth’s   long   term  climate   changes   caused  by   the   variation  of   its  position,   including   obliquity   and   precession,   in   the   orbit   around   the   Sun   (the  Milankovitch  cycles).   In  a  certain  sense,   this  work  started  a  whole  hierarchy  of  climate  models,   paving   the  way   to   John  von  Neumann  optimistic   suggestion   in  1955,   by   referring   to   climate   control   through   managing   solar   radiation:  “Probably   intervention   in   atmospheric   and   climate   matters   will   come   in   a   few  decades,  and  will  unfold  on  a  scale  difficult  to  imagine  at  present”  [Neu]       In  recent  decades,   the  “rising  tide  of  scientific  data”  and  the  advances  of  mathematical  and  computational  tools  has  raised  climate  science  to  a  whole  new  level,   although  global   representations   and  predictions   are   still   very  difficult,   of  limited   range   and   often   quite   deficient.   Nevertheless,   applied   mathematics   is  bringing   new   contributions   among   the   whole   range   of  multiscale   problems   of  this   interdisciplinary   research,   by   blending   asymptotic,   qualitative,   numerical  simulations   with   rigorous   analysis.   For   instance,   the   modeling   climate  phenomena  in  the  tropics  in  the  range  of  scales  from  kilometers  to  ten  thousand  kilometers  and  shorter  time  scales  has  been  recently  surveyed  in  [KMS].  In  this  review  paper,  the  dynamics  of  precipitation  fronts  in  the  tropical  atmosphere  are  modeled  as  large-­‐scale  boundaries  between  moist  and  dry  regions  and  treated  as  a   new   hyperbolic   free   boundary   problem   for   which   rigorous   mathematical  analysis  is  possible  in  the  framework  of  weak  solutions.      Free  Boundary  Problems  (FBP)  in  the  Planet  Earth       Generally  speaking,  free  boundaries  are  interfaces,  for  instance  curves  or  surfaces,   which   are   a   priori   unknown   and   separate   different   regions   in   space  and/or   time   and   appear   typically   in   models   with   phase   changes.   Their  mathematical   treatment  has  a   long  history  and  a   large  scope  of  applications.   It  was   the   subject   of   a   five   years   scientific   program   of   the   European   Science  Foundation  in  the  1990’s  [R2]  and  interfaces  and  free  boundaries  appear  often  in  the   mathematics   of   models   for   climatology   and   environment   [Di].   As   typical  examples  of  FBPs  relevant  to  theses  models,  we  have  the  dynamic  of  glaciers  and  ice   sheets   that   lead   to   very   complex   and   rich  mathematical   questions,   like   for  instance,  the  motion  of  grounding  lines  which  solutions  with  zero  contact  angle  have   been   shown   to   exist   and  were   asymptotically   characterized   in   the   recent  work  [FM].     The   first   FBP   was   exactly   motivated   by   an   application   of   the   “Théorie  Analytique   de   la   Chaleur”   to   the   cooling   of   the   terrestrial   globe.   Lamé   and  Clapeyron  presented  at  the  session  of  10th  May  1830  of  the  Academy  of  Science  of   Paris,   the   elegant   solution   of   what   is   now   known   as   the   one   phase   Stefan  problem   in   one   space   dimension:   the   depth   of   the   solidification   front   of   the  Earth,   supposed   homogenous   and   initially   liquid   [LC].   Looking   for   solutions   of  the  heat  equation  of  the  single  variable   y = x / t ,  they  obtained  the  formulas  for  the  temperature  ν(x,t) in  the  solid  and  for  the  free  boundary   x =ϕ(t)  

ν(x,t) = A e− y2 /4 dy

0

x/ t

∫  and  ϕ(t) = β t ,  

where   the   constants   A   and β  are   determined   as   the   unique   roots   of   the  transcendental   equations   involving   the   normalized   latent   heat   λ  and   the  solidification  temperature  σ at  the  two  free  boundary  conditions    

σ = v(ϕ(t),t)    and    λ dϕdt(t) = ∂ν

∂x(ϕ(t),t) .  

  This   type   of   phase   transition   problems   was   named   after   the   Slovene  mathematical-­‐physicist   Joseph   Stefan,   who   published   in   1889   a   series   of   four  papers   [St]   where   a   model   for   the   ice   formation   in   the   polar   oceans   was  discussed.  The  impressive  bibliography  [T],  containing  about  5900  references  up  to  the  year  2000,  reflects  the  huge  number  of  variants  and  applications  of  Stefan-­‐type  problems.   For   instance,   the   first   proof   of   existence   and  uniqueness  of   the  generalized  solution  of  the  multidimensional  (one  or  two-­‐phase)  Stefan  problem  was  given  by  Shoshana  Kamin   [Ka]   in  1958  her  doctoral   thesis   at   the  Moscow  University  under  the  supervision  of  Olga  Oleinik  [O].     From   the  mathematical   point   of   view,   the   one-­‐phase   Stefan   problem   is  closely  related  to  another  FBP  arising  in  filtration  through  porous  media.  In  the  “dam  problem”,   an  engineering  model  arising   in   the   control  of   a  particular  but  important   Earth   problem,   Claudio   Baiocchi   has   observed   in   1971   [Ba]   that   a  simple   transformation   allowed   to   reformulate   the   problem   as   a   variational  inequality,   opening   new   directions   of   research   [BC].   Similarly,   for   the   multi-­‐dimensional  one-­‐phase  Stefan  problem,   the  variational   inequality  approach  has  enabled  the  analysis  of  the  regularity  of  the  free  boundary,  in  particular,  by  Luis  Caffarelli  in  1976  [Ca].  As  a  consequence,  that  allowed,  under  certain  conditions,  to  show  that  the  generalized  solution  is  also  the  classical  one,  in  which  the  jump  conditions   are   satisfied   at   the   moving   interface   (see,   for   instance,   [R1]   for  references  and  details).     It  is  not  possible  to  give  a  complete  picture  of  the  FBPs  encountered  and  modeled  in  the  Planet  Earth,  but  in  order  to  illustrate  the  variety  of  mathematical  results  and  contributions  in  the  last  years  we  give  a  few  examples.  For  instance  the  design  of  freshwater  reservoirs  in  coastal  regions,  when  freshwater  overlays  salt  water  from  the  sea  may  lead,  in  a  special  case,  to  a  two-­‐phase  free  boundary  problem  for  the  stationary  flow  in  a  porous  medium.  In  [AD]  Alt  and  van  Duijn  using  a  weak  formulation  have  shown  rigorously  that  the  separation  of  fresh  and  salt  water,  in  two  or  three  dimensions,  is  a  continuous  interface,  which  ends  up      

     The  breakthrough  of  salt  water  creates  a  cusp  in  the  free  boundary    

separating  the  fresh  from  the  salted  water    in   the  well   and   has   an   asymptotic   behavior   that   can   be   precisely   described   in  function  of  real  parameters,   the  water  outflows  at   the  well.  This  special  case   is  related   to   another   relevant   but   different   model   for   the   joint   motion   of   two  immiscible   liquids,   the   Muskat   problem.   Starting   with   the   stationary   Stokes  system   for   an   incompressible   inhomogeneous   viscous   fluid,   occupying   a   pore  space,  coupled  with  the  stationary  Lamé  equations  for  an  incompressible  elastic  

solid  skeleton,   through  suitable  boundary  conditions  on  the  common  boundary  “solid  skeleton  –  pore  space”,  and  a   transport  equation   for   the  unknown   liquid  density,  Meirmanov  [Me]  has  recently  used  homogenization  techniques  to  derive  a  well-­‐posed  model  of  viscoelastic  filtration.       FBPs   for   the   Navier-­‐Stokes   and   related   equations   constitute   a   major  challenge  and  source  of  open  questions.  In  the  framework  of  coupled  models  for  the   dynamics   of   the   atmosphere   and   the   ocean,   a   free   surface   problem   is  considered   and   some   existence   and  uniqueness   results   are   shown   for   an   eddy  viscosity  model   in  [LTW2].  Another  classical   fundamental  problem  is  the  shape  and  stability  of  equilibrium  figures  of  a  uniformly  rotating  isolated  fluid,  at  least  since   the   famous   publication   in   1743   of   “Théorie   de   la   figure   de   la   Terre”,   by  Alexis   Clairaut,   supporting   the  Newton-­‐Huygens   conjecture   that   the   Earth  was  flattened  at  the  poles.  Studying  the  Navier-­‐Stokes  equations  with  surface  tension  and   kinematic   free   boundary   conditions,   Vsevolod   Solonnikov   in   [Sol]   has  recently   shown   that   the   positivity   of   the   second   variation   of   the   energy   in   an  appropriate   functional   space   is   a   sufficient   condition   for   the   stability   of   even  certain  nonsymmetrical  equilibrium  figures  of  rotating  viscous  fluids,  confirming  an  old  conjecture  of  Poincaré  and  Lyapunov  of  the  end  of  the  nineteenth  century.  

    A   very   interesting  mathematical  model   in  Aeolian   research   is   the   “sand  pile   problem”,   motivated   by   the   detachment,   transport   and   deposition   of  sediments   by   wind.   Among   different   approaches   [HK],   Prigozhin   in   [Pr]   has  observed   that   the   shape   of   a   growing   pile   z = u(x,t) ,   x ∈Ω⊂ 2,   t > 0 ,   of   a  cohesionless  granular  material,  being  characterized  by  its  angle  of  repose  α > 0 ,  is  constrained  through  its  surface  slope,  i.e.  

∇u(x,t) ≤ γ = tanα .  This  condition  is  complementary  to  a  general  conservation  of  mass  in  the  form  

∂u∂t

+∇⋅Φ(u,∇u) = f ,  

where   the   source   of   material   f  is   given.   The   horizontal   projection   of   the  material  flux  Φ in  the  case  without  convection  is  directed  to  the  steepest  descent  Φ = −µ∇u  and  subjected  to  the  unilateral  condition  

µ ≥ 0 ,     ∇u ≤ γ    and       ∇u < γ ⇒ µ = 0 .  This   model   corresponds   to   an   evolutionary   variational   inequality,   may   be  regarded   as   a   limit   case   of   the   p -­‐Laplacian   diffusion   when   p→∞  [AEW]   and  has   been   extended   in   several   directions.   The   case  γ = G(u)  corresponds   to   a  quasi-­‐variational  inequality  and  has  applications  to  other  critical-­‐state  problems,  such  as  magnetization  of  type-­‐II  superconductors,  formation  of  networks  of  lakes  and   rivers,   and  was   preceded   by   problems   in   elastic-­‐plastic   deformations   (see  [RS]  for  a  recent  mathematical  treatment  and  references).    

  The   Planet   Earth   if   full   of   very   complex   FBPs   and   challenges   to   the  mathematical  modeler,  as  the  mere  example  of  a  sand  pile  or  an  avalanche  may  illustrate.   As   a   simple   toy   problem   to   illustrate   the   variational   inequality  approach,   we   describe   the   free   boundary   evolution   for   the   sand   pile   with  constant   convection   in   one   space   dimension,   with   slope  γ = 1  and   constantly  increasing   source.   We  may   find   the   explicit   solution   that   solves   the   following  simple  problem:  Find  u = u(⋅,t)∈Κ = {v∈Η0

1 (0,1) : ′ν (x) ≤1} ,   t > 0 ,  such  that:  

0

1

∫∂u∂t(t)+ ∂u

∂x(t)− t⎛

⎝⎜⎞⎠⎟ (v − u(t))dx ≥ 0,     t > 0 ,    ∀v∈Κ ,  

with   initial   condition   u(x,0) = −x2 / 2, if 0 ≤ x ≤ ξ(0) , u(x,0) = x −1,  if ξ(0) ≤ x ≤1 .  Choosing   as   initial   free   boundary  ξ(0) = 3 −1 ,   a   second   free   boundary   point  ς (t) < ξ(t)  appears   for   t >1and   the   solution   attains   the   steady   state   exactly   at  t = 5 / 4when  ς (5 / 4) = ξ(5 / 4) = 1/ 2  (see  figure  at  times  t=0,  3/4,  8/9  and  5/4).    

   Satellite  image  and  simulated  rivers  and  lakes  of  the  Réunion  island,  by  Barrett  and  Prigozhin  (FBP2012).    http://www.uni-­‐regensburg.de/Fakultaeten/nat_Fak_I/fbp2012/FBP2012_files/Talks/Prigozhin_talk.pdf       It   is   interesting   to   notice   that   this   approach   is   applicable   to   other  geological  problems,  such  as  lakes  and  rivers,  by  taking  also  a  quasi-­‐variational  inequality   model   that   allows   the   solution   to   describe   “rivers”   running   out   of  “lakes”   and   flowing   in   the   steepest   descent   direction   until   reaching   the   next  horizontal  surface  of  a  lake.  By  approximating  the  continuous  problem,  the  zero-­‐repose-­‐angle  limit  of  a  growing  sand  pile  model,  with  a  network  of  fluxes  along  the   edges   of   a   triangulation,   in   2012   Barrett   and   Prigozhin   [BP]   were   able   to  provide   a   numerical   simulation   with   public   domain   (from   the   Shuttle   Radar  Topography  Mission)  real  data  of  the  Réunion  island  in  the  Indian  ocean  with  a  mesh  of  504  thousand  triangles  obtaining  a  very  realistic  picture.    

Raising  Awareness  of  Mathematics  and  MPE2103       The   “Mathematics   of   Planet   Earth"   initiative   (MPE2013)   aims   to   be   an  important  occasion  for  showing  the  essential  relevance  of  mathematical  sciences  in  planetary  issues  at  research  level  for  solving  some  of  the  greatest  challenges  of  our  century.  The  scope  is  broad  as  we  may  see  from  the  four  themes  compiled  by  C.  Rousseau  [Rou],   that  range  from  “a  planet  to  discover”  (oceans,  meteorology  and   climate,   mantle   processes,   natural   resources,   celestial   mechanics  

cartography)   to   “a   planet   at   risk”   (climate   change,   sustainable   development,  epidemics,   invasive   species,   natural   disasters),   nor   forgetting   that   “our   planet  supports  life”  (ecology,  biodiversity,  evolution)  and  is  “organized  by  humans  and  structured   by   civilization”   (political,   economic,   social   and   financial   systems,  transport  organization  and  communication  networks,  management  of  resources  and  energy).         Like   in   the   World   Mathematical   Year   2000,   it   should   also   be   an  opportunity  for  stimulating  a  collective  reflection  on  the  great  challenges  of  the  21st  Century,  on   the   role  of  mathematics  as  a  key   for  development  and  on   the  importance  of  the  image  of  mathematics  in  the  public  understanding.    

    As   observed   by   Jones   [J],   “since   the   Earth   is   not   available   for  experimentation,  climate  science  relies  on  mathematical  models   to  make  up   its  ‘laboratory’”,  the  mathematical  community  should  address  this  topic  and  be  also  aware  “that  the  greatest  challenges  as  well  as  the  greatest  promise  for  novel  and  innovative  mathematical  thinking  is  at  this  interface  between  data  and  models”.     On  another  hand,  for  the  non-­‐less  important  Raising  Public  Awareness  of  mathematics   at   the   educational   and   societal   levels,   mathematicians   should   be  encouraged   to  write   expository   versions   of   popular  mathematical   lectures,   for  instance,  on  the  global  change,  as  in  [Kl],  or  on  sand  piles  and  avalanches,  as  in  [CF]   and   [ADG].   But   any   other   well-­‐established   Public   Awareness   of   some  terrestrial   phenomenon   X   may   be   chosen   to   Raising   the   Public   Awareness   of  Mathematics,  as  the  example  of  X=wildfires,  illustrated  with  interesting  ideas  in  [Ma].   Among   several   initiatives   promoted   by   MPE2013,   the   international  competition  for  modules  for  a  virtual  global  exhibition,  which  launching  has  the  UNESCO   patronage,   and   the   research   initiatives   announced   for   2013,   in  particular,  in  Europe  by  the  ERCOM  centers,  deserve  special  reference.    Acknowledgements.   During   the   preparation   of   the   lecture,   that   was   sponsored   by   the   London  Mathematical  Society,  and  of  this  article,  the  author  has  profited  of  several  conversations  and  references  kindly  shared  with  B  Almeida,  J  I  Diaz,  K  Hutter,  H  Leitão,  L  Prigozhin,  L  Santos,  D  Tarzia  and  J  P  Xavier.    

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José  Francisco  Rodrigues     <rodrigue@fc.ul.pt>  Centro  de  Matemática  e  Aplicações  Fundamentais,    Universidade  de  Lisboa,  

Av  Prof  Gama  Pinto,  2.   1649-­‐003  Lisboa,     PORTUGAL