.MCAD 310000000 [  docDocument LmcObjectZ  d2_graph_format graphData% axisFormat)L)Ltrace2D&&&&&&&&& & & & & &&& dim_formatSmasslengthtimecharge temperature luminosity substanceNumericalFormatPdii  shpRectUmcDocumentObjectState[ mcPageModelJ????mcHeaderFooterH@H CHeaderFooterI@{\rtf1\ansi\ansicpg1252\deff0\deflang1033{\fonttbl{\f0\fswiss\fprq15\fcharset0 Arial;}{\f1\fnil\fcharset0 Arial;}} \viewkind4\uc1\pard\f0\fs18\{d\}\f1\par }@{\rtf1\ansi\ansicpg1252\deff0\deflang1033{\fonttbl{\f0\fswiss\fprq15\fcharset0 Arial;}{\f1\fnil\fcharset0 Arial;}} \viewkind4\uc1\pard\qc\f0\fs18 Electrostatics\f1\par \par }@{\rtf1\ansi\ansicpg1252\deff0\deflang1033{\fonttbl{\f0\fswiss\fprq15\fcharset0 Arial;}{\f1\fnil\fcharset0 Arial;}} \viewkind4\uc1\pard\qr\f0\fs18 Separation of ions by gravity \{n\}\f1\par \par }@I@I@I MbP?MbP? TextState> TextStyle=@ ArialSerial_ParPropDefaultVNormal=@Arial@V Heading 1=@ Arial@V Heading 2 =@ Arial@V Heading 3 =@ Arial@V Paragraph =@ Arial@VList =@ Arial@VIndent =@Times New Roman@VTitle=@Times New Roman@VSubtitle font_style_listN font_styleO  VariablesTimes New Roman@O  ConstantsTimes New Roman@O TextArial@O Greek VariablesSymbol@O User 1Arial@O User 2 Courier New@O User 3Arial@O User 4Times New Roman@O User 5Times New Roman@O User 6Arial@O User 7Times New Roman@O SymbolsSymbol@O Current Selection FontArial@O Undefined Font@O HeaderArial@O FooterArial@O Rotated Math FontTimes New RomanN TextRegion* docRegionFshpBoxTdocRegionAttributeG@G" CharacterMap-RangeMap:-The ionosphere: Separation of ions by gravity ChrPropMap6- ParPropMap8- RangeElem;- ParPropData9 RangeData<@VEmbedMap0;LinkMap.-;-LinkData/@NormalArial  BitmapRegion PictureRegionE@T@(F@(@G@G" mc_dibitmap 8Q( Q*@T''8@G@G"-BfThe ionosphere has hydrogen ions H+ and oxygen ions O+. The proportion of H+ ions increases with altitude. This increase satisfies our intuition that hydrogen, being lighter, "must go up." This intuitive notion is given some mathematical underpinning if we define a gravitational scale height kT/mg, where kT is the thermal energy, m is the ion mass, and g is the acceleration of gravity. The density of the ion decreases with altitude as exp[-mgz/kT], where mgz is the gravitational potential energy. The proportion of H+ increases with altitude because the scale height of H+ is greater than that of other ions. 6A"f;" ChrPropData7 ;!7_";#7 $;%7_"&;'7$(;)7_&*;o+7(,;-7` Symbol*.;M/7,0;17_.2;63704;57_26;%7746 8f8;f99@V0:;.f;;f@G?@G"  -@RDensity of Oxygen ions (dotted) and hydrogen ions (solid) as a function of height.6R8R@@;R@A9@V0@B;.R@C;R@D/@NormalArial @E*@T3]@@F@G@G@G"UUxUx-BNow consider the electrons in the ionosphere. Their gravitational scale height is about 1836 times greater than that of H+ and the electrons would be higher than the ions if there were no other effects. The electrons being higher, however, suggests an electric field pointing upward that will tend to draw the electrons and ions together. Quasineutrality is a strong requirement, thus we can conclude that the electric field equalizes densities. The total potential energy for each particle is the sum of the gravitational potential energy mgh and the electrostatic potential energy +qF (depending on the sign of the charge). Setting the scale heights of electrons and H+ ions equal, we obtain: 6y@H;y@I7@E@J;@K7@E_@H@L;@M7@E@J@N;@O7@Ew@L@P;@Q7@E@N@R;@S7@ESymbol@P@T;S@U7@E@R@V;@W7@E_@T@X;@Y7@E@V@X@J8@Z;@[9@V,0@\;.@];@^/@NormalArial @_ COleDocItemDocItem @T@`@G@a@G"@b@G@c@G"@d@G@e@G"ࡱ> Root EntryFtS@Ole Equation Native uCompObjf B E Y2Q m H gz+q=m e gz"q FMicrosoft Equation 3.0 DS Equation Equation.39q{ E .  @`&  & MathType`OlePres000Ole Symbol ;@ww 0wf-2  Fy2  -y2 y=y2 Fy2 +yTimes New Romanww 0wf-2 3 qy2 c gz2 mz2 'qz2 Qgz2 :mz Times New Romanww 0wf-2  ez2 _Hz & "Systemf !\-NANI@f@T@g@G@h@G"@i@G@j@G"@k@G@l@G"ࡱ> Root EntryFSOle Equation Native CompObjf B E k q=" 12 m H "m e ()g FMicrosoft Equation 3.0 DS Equation Equation.39q{  .  @ &`  & MathType`OlePres000Ole -  tSymbol pww 0wfC-2 ~(ySymbolo \pww 0wfC-2 ~p )yTimes New Romanww 0wfC-2  gz2  mz2 'mz2 :qz Times New Romanww 0wfC-2  ez2 LHzSymbol @ww 0wfC-2 -z2 -z2 =z2 Fzz Times New Romanww 0wfC-2 2z2 1z & "SystemCfC !\-NANI@m*@T@n@G@o@G"-or 68@p;@q9@V0@r;.@s;@t/@NormalArial @u@T @v@G@w@G"@x@G@y@G"@z@G@{@G"ࡱ> Root EntryFS@Ole Equation Native yCompObjf B E ]3 E=g2qm H "m e ())g FMicrosoft Equation 3.0 DS Equation Equation.39qg ~  .   @ &  & MathTypeOlePres000Ole -Symbol 2pww 0wo fe-2 ^(ySymbol pww 0wo fe-2 ^ )y Times New Romanww 0wo fe-2  ey2 vHyTimes New Romanww 0wo fe-2 `my2 `Qmy2 qy2 nagy2 `FEySymbol 4@ww 0wo fe-2 `-y2 `=yzTimes New Romanww 0wo fe-2 2y & "Systemeo fe !\-NANINANI@|*@T` @}@G@~@G",,-7where mH is the mass of H+ and me is the electron mass.6!7@;@7@|@;@7@|_@@;@7@|@@;@7@|_@@;@7@|@@;@7@|_@@;@7@|@@@87@;7@9@V0@;.7@;7@/@NormalArial @*@T .@@G@@G" -@LDifferentiating F, we find the electric field that equalizes the net force:6L@;@7@@;@7@` Symbol@@;;@7@@@@8))#L@;)@9@V@;#@9@V@@@0@;.L@;L@/@NormalArial @*@T++(M@@G@@G"###-&An ionosphere of electrons, H+ and O+ 6%&@;@7@@;@7@_@@;@7@@@;@7@_@@;@7@@@@8&@;&@9@V0@;.&@;&@/@NormalArial @*@T/]@J@@G@@G"UUhUh-B=Suppose that there are electrons and two types of ions, H+ and O+. How do the densities of the three species vary with altitude? The ions are mostly O+ at the bottom of the ionosphere and mostly H+ at the top. Thus the mass of the ion in the expression above for the electric field is some varying "blend" of the two masses. There is no simple expression for E. E must be found by solving Poisson's equation. For the solution to this equation we will use the relaxation method reviewed on the next page. For the densities of the species, we will use the equilbrium values: 69=@;9@7@@;@7@_@@;@7@@@;@7@_@@;U@7@@@;@7@_@@;-@7@@@;@7@_@@;x@7@@@@8=@;=@9@V0@;.=@;=@/@NormalArial @@T@@G@@G"@@G@@G"@@G@@G"ࡱ> Root EntryFVSOle Equation Native CompObjf  MarkedRegions_mcObject`4 n e (z)=n e0 exp"(m e gz"q)T[] FMicrosoft Equation 3.0 DS Equation Equation.39q   .  &OlePres000ZOle `% & MathType-@ @Symbol $@ww 0wf-2 -y2 y2 y2 -B y2 B y2 B y2 Fy2 -y2  -y2 =yTimes New Romanww 0wf-2 LTy2 $qy2 T gz2  mz2 nz2 :zz2 :nz Times New Romanww 0wf-2  ez2 ez2 ezTimes New Romanww 0wf-2 )z2  (z 2 expe2 )x2 (x Times New Romanww 0wf-2 70x & "Systemf !\-NANIFMicrosoft Equa@@T@@G@@G"@@G@@G"@@G@@G"ࡱ> Root EntryF SOle Equation Native CompObjf  MarkedRegions_mcObject` n H (z)=n H0 exp"(m H gz+q)T[] FMicrosoft Equation 3.0 DS Equation Equation.39q   .  &OlePres000ZOle " & MathType-@ @Symbol dww 0w1 f-2 0y2 y2 y2 0 y2  y2  y2 Fy2 3+y2  -y2 +=yTimes New Romanww 0w1 f-2 KTy2 Yqy2 gz2 l mz2 cnz2 zz2 :nz Times New Romanww 0w1 f-2  Hz2 4Hz2  HzTimes New Romanww 0w1 f-2 K)z2  (z 2 expe2 G)x2 (x Times New Romanww 0w1 f-2 0x & "System1 f !-NANIFMicrosoft Equa@@T@@G@@G"@@G@@G"@@G@@G"ࡱ> Root EntryFP| SOle Equation Native CompObjf  MarkedRegions_mcObject`` n O (z)=n O0 exp"(m O gz+q)T[] FMicrosoft Equation 3.0 DS Equation Equation.39qv   .  `&OlePres000ZOle  % & MathType-@ @kSymbol Sdww 0w f-2 -y2 y2 y2 - y2  y2  y2 Fy2 +y2 \ -y2 =yTimes New Romanww 0w f-2 Ty2 qy2 gz2  mz2 =nz2 yzz2 :nz Times New Romanww 0w f-2 8 Oz2 Oz2 OzTimes New Romanww 0w f-2 )z2  (z 2 expe2 !)x2 (x Times New Romanww 0w f-2 0x & "SystemІ f !-NANIFMicrosoft Equa@*@T+(@@G@@G"-@[where T is the temperature in energy units (so that Boltzmann's constant does not appear). 6[8[@;[@9@V0@;.[@;[@/@NormalArial @@TPUhP@@G@@G"@@G@@G"@@G@@G"ࡱ> Root EntryF0'}XOle Equation Native CompObjf  MarkedRegions_mcObject`hQ ""P s "n s m s g"n s  FMicrosoft Equation 3.0 DS Equation Equation.39qoft Equation 3.0 DS Eq{2  .  @& & MathType`OlePres000tOle Times New Romanww 0w f-2 0ySymbol @ww 0w f-2 =y2  Fy2  y2 G -y2 -y2 Ny2 :-y Times New Romanww 0w f-2  sy2 5 sy2 sy2 sy2 %syTimes New Romanww 0w f-2  qy2 g ny2 !gy2 omy2 #ny2 bPy & "System f !q s "=0+\-NANI@*@TKpkX@@G@@G"hh h -@wTry it: Derive the expressions for the densities by noting that for species s the momentum equation for equilbrium is: 6Mw@;@7@}@;E@7@@@;@7@{@@;*@7@@@@8w@;w@9@V0@;.wA;wA/@NormalArial A@TxxA@GA@G"A@GA@G"A@GA@G"ࡱ> Root EntryF07*}XOle Equation Native fCompObjf  MarkedRegions_mcObject`Jt "P s =T s "n s=00 docRe FMicrosoft Equation 3.0 DS Equation Equation.39qoft Equation 3.0 DS Eq{J % .  @& & MathType` OlePres000Ole Times New Romanww 0w f-2 Csy2 sy2 syTimes New Romanww 0w f-2 uny2 Ty2 APySymbol[ @ww 0w f-2 by2 =y2 .y & "System f !\-NANI2 my2 bPy & A *@T{T A @GA @G"LLL-8For simplicity, we assume an isothermal ionosphere with:6888A ;8A 9@V0A;.8A;8A/@NormalArial A*@TA@GA@G"-(Relaxation method for Poisson's equation6(8(A;(A9@V0A;.(A;(A/@NormalArial A*@T`cA@GA@G"XXX-CIThe values of the potential F will be assigned on grid points equally spaced in z. For example, Fn is the value of F(z) at the location zn, the nth grid point. The first equation below is Poisson's equation written in the usual way. The second equation is the definition of the z derivative of Fn, using values from the grid. The third equation defines the second derivative of Fn. In the last equation, Poisson's equation is written with the finite-difference form of d2F/dz2. In the last equation, we use the present values of Fn-1 and Fn-1 to calculate the next iteration for Fn. The left side of the equation is the new guess based on the old guess that is on the right side. This is done repetitively to get better values for Fn. The iterations are stopped when the new guess is no longer significantly different from the old guess. 6GIA;A7AA;A7A` SymbolAA ;CA!7A` ArialAA";A#7A` SymbolA A$;A%7A@ ArialA"A&;A'7A` ArialA$A(;A)7A` SymbolA&A*;A+7A` ArialA(A,;A-7A_A*A.;A/7A` ArialA,A0;A17A` SymbolA.A2;A37A_A0A4;RA57A` ArialA2A6;A77A` SymbolA4A8;A97A_A6A:;A;7A` ArialA8A<;A=7A` ArialA:A>;CA?7A` ArialAAB;AC7A` SymbolA@AD;AE7A` ArialABAF;AG7A_ADAH;5AI7A` ArialAFAJ;AK7ASymbolAHAL;AM7A_AJAN;AO7A` ArialALAP;AQ7ASymbolANAR;AS7A_APAT;AU7A` ArialARAV;$AW7A` ArialATAX;AY7A` SymbolAVAZ;A[7A_AXA\;A]7A` ArialAZA^;A_7ASymbolA\A`;Aa7A_A^Ab;jAc7A` ArialA`Ad;Ae7AAbAdA8IAf;IAg9@V0Ah;.IAi;IAj/@NormalArial Ak@Tp p  Al@GAm@G"An@GAo@G"Ap@GAq@G"Ar@GAs@G"At@GAu@G"   ࡱ>  Root EntryF07*}XOle Equation Native CompObjf&  !"#$%-'()*+,H./0123456789:;<=>?@ABCDEFGIJMKLNOPQRtry\Machine\Software\C "d 2 dz 2 (z)=n i (z)"n e (z)()q/ 0 ,                n i (z)=n H (z)+n O (z) FMicrosoft Equation 3.0 DS Equation Equation.39q4959}+0@  .  '&') & MathType-@`@Symbol pww 0w[ f-2 y(ySymbol }pww 0w[ f-2 )yQ@QZQ QOlePres000 Ole Symbol Hww 0w[ f-2 [ySymbol ~Hww 0w[ f-2  ]y @ (  TSymbol Hww 0w[ f-2 [ySymbol Hww 0w[ f-2 ]yoWSymbol pww 0w[ f-2 [ySymbol pww 0w[ f-2 ]yM5Symbol pww 0w[ f-2 p(ySymbol pww 0w[ f-2 )ySymbol pww 0w[ f-2 [ySymbol pww 0w[ f-2  ]y Times New Romanww 0w[ f-2 ]"0y2 52y2 ]1y2 ] 1y2 L 1y2 L ) 1y2 s 2y2 s (2y2 N _2y2 1y2 y0y2 '2y2 2yTimes New Romanww 0w[ f-2  /y2 )y2 (y2 C)y2 N(y2 )g2y2 a1yddz=1z n+1 " n []d 2 dz 2 =1z 2  n+1 "2 n + n"1 [], new  n =12, old  n+1 +, old 2 )2y2 1y2 ' 2y2 1y2 1y2 &)y2 %(y2 ")y2  (y2 )y2 (y 2 ( 2 h 2 , 2 / 2  ) 2 ( 2 v ) 2 2 ( 2 d) 2  ( Symbol @ww 0w[ f-2 :!e 2 e  Times New Romanww 0w[ f-2 ]n 2 ]e 2 ]n 2 ]i 2 ]n 2 QV old 2 ] nl 2 Qold 2 ]&nl 2 Q=new 2 L ne2 L 4ne2 L 8 ne2  ne2 ne2 $Oe2 He2 &ie2 )ee2  ieTimes New Romanww 0w[ f-2 /ze2 /ne2 ze2  ne2 ze2  ze2  dz2 Zdz2 zz2 dz2 Zdz2 A&zz2 $nz2 h!zz2 nz2 ?zz2 bnz2 qz2 bzz2 b nz2  zz2 nz2 zz2 zdz2 dzSymbol @ww 0w[ f-2 -z2 Dz2 !+z2  Fz2 a +z2 Fz2 1=z2 Fz2 GFz2 '+z2 Fz2  -z2 Fz2  Dz2 =z2 Fz2  Fz2 w -z2 [Fz2 $Dz2 =z n"1 []+122 Fz2 "+z2 =z2 B -zz() 2 n i (z n )"n e (z n )[]/ 0 f!-2 ]2 H=z2 Fz2 :-z Symbol @ww 0w[ f-2 ]-z2 ]R +z2 L -z2 L +z2 &+z & "System[ f !\-NANIAv*@TASPAw@GAx@G"- Equations in dimensionless units6 8 Ay; Az9@V0A{;. A|; A}/@NormalArial A~*@T[M hA@GA@G"EE@E@-AmIn the second line below, the terms in Poisson's equation have been multiplied by constants so that the last two bracketed terms are dimensionless. We recognize that the first bracketed term is the Debye length squared and that this can be used to make the z derivatives dimensionless as well. The last three lines show the dimensionless versions of the variables. 6m8mA;mA9@V0A;.mA;mA/@NormalArial A@Tck A@GA@G"A@GA@G"A@GA@G"A@GA@G"A@GA@G" ࡱ>  Root EntryFaSOle Equation Native CompObjf & !"#$%F'()*+,-./0123456789:;<=>?@ABCDEHGIJtry\Machine\Software\CXd "d 2 dz 2 (z)=n i (z)"n e (z)()q/ 0 " 0 T FMicrosoft Equation 3.0 DS Equation Equation.39qdx 2n, 0 q 2 ()d 2 dz 2 q(z)#   .  @ &@  & MathType`-B`BSymbol pww 0w f-2 y(ySymbolG /pww 0w f-2 )yHD t OlePres000 Ole =p @p  "1 -_-g>  Times New Romanww 0w f- 2 mDebye2 0 ee2 0 rie2 ree2 rjie2 )ee2  ieTimes New Romanww 0w f-2 <ze2 dYqe2 dne2 dTe2 %ze2 Rze2  ne2  ne2  ne2 Te2 qe2 Fze2 F ne2 ze2 ne2 ze2  ze2  Zde2 de2 =+ne2 Wze2 Wne2 =hne2 ze2 neT()=n i (z)n, 0 "n e (z)n, 0 ()d 2 d2z 2 "(2z)=2n i (2z)"2n e (2z)"=q/T,        2n=nn 0 ,       2z=z  0 T/2 = Te2 : ze2  qe2 D^dz2 dz2 Dqz2 D.nz2 Tz2 qz2 bzz2 b nz2  zz2 nz2 zz2 zdz2 dzSymbol @ww 0w f-2 lz2 dez2 ez2 ezSymbolG 1@ww 0w f-2 =z2 =z2 N =z2 Fz2 =z2 4Fz2 & -z2 v=z2 Fz2 z2 z2 z2 z2 z2 z2 z2 z2 -z2 =z2 1 z2 k z2 ; z2 1z2 kz2 ;z2 l Fz2 xz2 xz2 xz2 xz2 Nz2 Nz2 Nz2 Nz2 :-z2 B -z2 H=z2 Fz2 :-z Times New Romanww 0w f-2 @2z2 0z2 0z2 mm 0z2 W 2z2 2 2z2 0z2 E0z2 2z2 s2z2 2z2  0z2 rW0z2 y0z2 )2z2 2zTimes New Romanww 0w f-2 dW/z2 QC~z 2  2 X, 2 K ~ 2   2 , 2 &/ 2 a~ 2 ) 2 @ 7~ 2 ( 2 : P ~ 2 Z ) 2 @ | ~ 2 ( 2 : ~ 2 ) 2 @ ~ 2 '( 2 "~ 2 s ~ 2 ) 2 ( 2 +) 2 ( 2  ) 2  ( 2 / 2  ) 2 ( 2 v ) 2 2 ( 2 d) 2 n, 0 q 2 =z Debye ( & "System f !\-NANIA*@T< A@GA@G"444-@XIn the pages below, we are using the dimensionless variables without the tildes on top. 6X8XA;XA9@V0A;.XA;XA/@NormalArial A*@T_A@GA@G"h-The particle densities 68A;A9@V0A;.A;A/@NormalArial A*@T`A@GA@G"h-@WThe expressions for the particle densities must be written in the dimensionless units. 6W8WA;WA9@V0A;.WA;WA/@NormalArial A*@T'A@GA@G"h-The paticle masses in SI units:68A;A9@V0A;.A;A/@NormalArial AeqRegionA@TD?A@GA@G"Atree? pA? AA?dAm.eA?AA?tA9.11A?AA?tA10A?KAA?A31A@A@TXr@A@GA@G"A? pA? AA?dAm.HA?AA?tA1.67A?AA?tA10A?KAA?A27A@A@TAA@GA@G"A? pA? AA?dAm.OA?AA?tA16A?Am.HA*@TRSA@GA@G"JJPJP-AThe dimensionless value of F was found by dividing qF by T. Similarly, we can find the dimensionless value of g by dividing mgz by T. However, this will make the atmosphere so many Debye lengths tall that too many grid points are required. In order to decrease the number of grid points, we will increase g so that the density falls by a factor of 100 in only 200 Debye lengths. This means 200 mg is assigned the value -ln(0.01). Then g is: 65A;A7AA;A7A` SymbolAA;A7AAA;A7A` SymbolAA;A7AAAA8A;A9@V?0A;.A;A/@NormalArial A@A@TcUxA@GA@G"A? pA? AA?dAgA?AA?K@AA?AA?dAlnA?pAA?A0.01A?AA?tA200A?Am.HA@A@Td}xA@GA@G"A? pA?AA?dAgA?AA?+@ASerial_DisplayNodeWA?AA*@Tk7xA@GA@G"`''-3This large value of g makes mHgz near order unity. 63A;B7AB;B7A_AB;B7ABBB83B;3B9@V0B;.3B;3B /@NormalArial B *@T&B @GB @G"h-@]The proportion of ions that are hydrogen will be made 1% at the lower boundary of the domain:6]8]B ;]B9@V0B;.]B;]B/@NormalArial B@A@TE&B@GB@G"B? pB? BB?dBn.H0B?B0.01B@A@TB@GB@G"B? pB? BB?dBn.O0B?B0.99B @A@TJ+B!@GB"@G"B#? pB$? B#B%?dB$n.e0B&?B$1.00B'*@TB(@GB)@G"h-@PWith the above value of g, the equilibrium densities in dimensionless units are:6P8PB*;PB+9@V0B,;.PB-;PB./@NormalArial B/@A@T; B0@GB1@G"B2? pB3? B2B4?@B3B5?dB4n.HB6?pB4B7? B6B8?dB7\fB9?B7zB:?B3B;?dB:n.H0B?pB<B??B>B@?K@B?BA?B@\fBB?B?BC?tBB1.0BD?BBBE?@BDBF?dBEm.HBG?BEgBH?BDzBI@A@TP  BJ@GBK@G"BL? pBM? BLBN?@BMBO?dBNn.eBP?pBNBQ?BP\fBR?BMBS?dBRn.e0BT?BRBU?dBTexpBV?pBTBW?BV\fBX@A@Th BY@GBZ@G"B[? pB\? B[B]?@B\B^?dB]n.OB_?pB]B`? B_Ba?dB`\fBb?B`zBc?B\Bd?dBcn.O0Be?BcBf?dBeexpBg?pBeBh?BgBi?K@BhBj?Bi\fBk?BhBl?@BkBm?dBlm.OBn?BlgBo?BkzBp*@T)P;8 Bq@GBr@G"HHH- The z grid6 8 Bs; Bt9@V0Bu;. Bv; Bw/@NormalArial Bx*@TKL{X By@GBz@G"DD0D0-@The grid spacing will be Dz= 0.5 Debye lengths. A vector (a matrix with one row) will contain the values of F. We will use 400 grid points so that the domain is 200 Debye lengths. The variable Phi will be the electrostatic potential that we are finding. 6B{;B|7Bx` ArialB};B~7Bx` SymbolB{B;B7Bx` ArialB}B;<B7Bx` ArialBB;B7Bx` SymbolBB;B7Bx` ArialBB;EB7Bx` ArialBB;B7BxBB;JB7Bx` ArialBBB{8JB;B9@VB;JB9@VBBB0B;.B;B/@NormalArial B@A@T6 B@GB@G"B? pB? BB?dB\DzB?B0.5B*@Th,r B@GB@G"-!The grid spacing in Debye lengths6!8!B;!B9@V0B;.!B;!B/@NormalArial B@A@TD( B@GB@G"B? pB? BB?dBjmaxB?B400B@A@Tht B@GB@G"B? pB? BB?dBjB?BB? @BB?tB0B?B1B?BjmaxB*@T B@GB@G"-!There will be jmax+1 grid points.6!8!B;!B9@V0B;.!B;!B/@NormalArial B@A@T8 B@GB@G"B? pB? BB?@BB?dBzB?BjB?BB?dBjB?B\DzB*@Thy B@GB@G"hhh-Define grid points.68B;B9@V0B;.B;B/@NormalArial B*@T  B@GB@G"h-The program loop68B;B9@V0B;.B;B/@NormalArial B*@T WG  & B@GB@G"OO4O4-A6The program loop is similar to the one used earlier for the Debye length. The successive iterations for the values of the potential Fk are saved in a matrix Phi. The first line of the loop initializes the matrix Phi to zero. The first "for loop" puts the initial guess, PhiAnalytic, into the first row of Phi. 6N6B;NB7BB;B7BArialBB;B7BBB;B7B` SymbolBB;B7B_BB;B7BBBB86B;6B9@V0B;.6B;6B/@NormalArial B*@T[ k h > B@GB@G"h-@_The initial guess to start the iteration process will be the analytic solution for F on page 1:6TS_B;SB7BB;B7B` SymbolBB; B7BBBB8_B;_B9@V0B;._B;_B/@NormalArial B@A@Tv S ? B@GB@G"B? pB? BB?@BC?dB PhiAnalyticC?BjC?BC?@CC?@CC?K@CC?Cm.HC?CgC?C2C ?CC ?dC zC ?C jC *@T w K C @GC@G"-'This is the analytic solution for Phi. 6'8'C;'C9@V0C;.'C;'C/@NormalArial C*@T " T C@GC@G"h-Boundary conditions68C;C9@V0C;.C;C/@NormalArial C*@T \{  V C@GC@G"TTT-CAt the left boundary, the potential is specified to be zero. The "for j" loop in the program omits j = 0 so that the boundary value is preserved. We do not know the potential at the right boundary where j = jmax. A way to allow this value to relax toward an equilibrium is to use periodic boundary conditions. This is implemented by creating a fictitious point at jmax+1 that has the same value of potential as the point at jmax-1. The potential at jmax is figured by the usual formula, except that the potential value at the next (nonexistent) grid point jmax+1 is assigned the same value as the potential at the previous point. In the "for j" loop, the point at jmax is omitted and several lines are added after this loop that assign Phi at jmax using the rule for the periodic boundary. Note that the new (temporary) potential value Temp is averaged with the previous potential value to prevent instability at the shortest wavelength. 6C;C 7CC!;C"7C` SymbolCC#;C$7CC!C%;gC&7CArialC#C';C(7CC%C'C!8C);C*9@V?C+;C,9@V,C)C+C+0C-;.C.;C//@NormalArial C0@A@T I ' a C1@GC2@G"C3? pC4? C3C5?dC4itersC6?C42048C7*@Tx  c C8@GC9@G"II I -@kThis is the number of iterations (found by trial and error) needed to converge to a final solution for Phi.6k8kC:;kC;9@V0C<;.kC=;kC>/@NormalArial C?@A@T \. ! C@@GCA@G"CB? pCC? CBCD?dCCPhiCE?CCCF?@CECG?@CFCH?@CGCI?@CHCJ?dCIPhiCK? CICL?dCKitersCM?CKjmaxCN?CH0CO?CGCP?@COCQ?@CPCR?dCQPhiCS? CQCT?tCS0CU?CSjCV?CPCW?dCV PhiAnalyticCX?CVjCY?COCZ?dCYjC[?CYC\?tC[0C]?C[jmaxC^?CFC_?@C^C`?@C_Ca?@C`Cb?@CaCc?@CbCd?@CcCe?dCdTempCf?6CdCg?@CfCh?@CgCi?@ChCj?dCiPhiCk? CiCl?@CkCm?dCliCn?Cl1Co?CkCp?dCojCq?Co1Cr?ChCs?dCrPhiCt? CrCu?@CtCv?dCuiCw?Cu1Cx?CtCy?dCxjCz?Cx1C{?Cg2C|?CfC}?@C|C~?@C}C?tC~1C?C~2C?C}C?dC\DzC?C2C?pC|C?CC?@CC?@CC?dCn.HC?pCC? CC?@CC?dCPhiC? CC?@CC?dCiC?C1C?CjC?CC?dCzC?CjC?CC?dCn.OC?pCC? CC?@CC?dCPhiC? CC?@CC?dCiC?C1C?CjC?CC?dCzC?CjC?CC?dCn.eC?pCC?CC?dCPhiC? CC?@CC?dCiC?C1C?CjC?CcC?@CC?dCPhiC? CC?dCiC?CjC?CC?tC0.5C?pCC?CC?dCTempC?CC?dCPhiC? CC?@CC?dCiC?C1C?CjC?CbC?dCjC?CC?tC1C?CC?dCjmaxC?C1C?CaC?dCjC?CjmaxC?C`C?dCTempC?CC?@CC?@CC?@CC?dCPhiC? CC?@CC?dCiC?C1C?CC?dCjC?C1C?CC?dCPhiC? CC?@CC?dCiC?C1C?CC?dCjC?C1C?C2C?CC?@CC?@CC?tC1C?C2C?CC?dC\DzC?C2C?pCC?CC?@CC?@CC?dCn.HC?pCC? CC?@CC?dCPhiC? CC?@CC?dCiC?C1C?CjC?CC?dCzC?CjC?CC?dCn.OC?pCC? CC?@CC?dCPhiD? CD?@DD?dDiD?D1D?DjD?CD?dDzD?DjD?CD ?dDn.eD ?pDD ?D D ?dD PhiD ? D D?@D D?dDiD?D1D?D jD?C_D?@DD?dDPhiD? DD?dDiD?DjmaxD?DD?tD0.5D?pDD?DD?dDTempD?DD?dDPhiD? DD ?@DD!?dD iD"?D 1D#?DjD$?C^D%?dD$iD&?D$D'?tD&1D(?D&itersD)?CEPhiD**@TS ^c ` D+@GD,@G"hVV-=A plot of the successive values for Phi is on the next page. 6=8=D-;=D.9@V0D/;.=D0;=D1/@NormalArial D2@A@T V8 D3@GD4@G"D5? pD6?D5D7?@D6D8?@D7D9?@D8D:?vD90D;?KD9D?@D=D??D=D@? D7DA? @D@DB? @DADC? @DBDD?@DCDE?dDDPhiDF? DDDG?tDF0DH?DFjDI?DCDJ?dDIPhiDK? DIDL?@DKDM?dDLitersDN?DL8DO?DKjDP?DBDQ?dDPPhiDR? DPDS?@DRDT?dDSitersDU?DS4DV?DRjDW?DADX?dDWPhiDY? DWDZ?@DYD[?dDZitersD\?DZ2D]?DYjD^?D@D_?dD^PhiD`? D^Da?dD`itersDb?D`jDc?D6Dd?@DcDe?@DdDf?vDe200Dg?De0Dh?DdDi?@DhDj?DhDk?DcDl?dDkzDm?DkjDn =! )L)LPotential as a function of z&&&&&&&&&& & & & & &&&Do*@TOM(`s Dp@GDq@G"EE<E<-A,The potential initially decreases rapidly because of the smaller scale height of the O+, then more slowly because of the H+. The initial guess for the first iteration was a decrease with the scale height of H+ (highest dashed line). At altitudes where O+ is depleted, the scale height of H+ applies. 6"V,Dr;VDs7DoDt;Du7Do_DrDv;#Dw7DoDtDx;Dy7Do_DvDz;UD{7DoDxD|;D}7Do_DzD~;,D7DoD|D;D7Do_D~D;#D7DoDD;D7Do_DD; D7DoDDDt8,D;,D9@V0D;.,D;,D/@NormalArial D*@TP"n D@GD@G"HHH-@SPlot of the H+ (solid line) and O+ (dottend line) densities for the two-ion plasma:6" SD; D7DD;D7D_DD;D7DDD;D7D_DD;1D7DDDD8SD;SD9@V 0D;.SD;SD/@NormalArial D@A@T;o D@GD@G"D? pD?DD?@DD?@DD?@DD?vD0.99D?DD?tD10D?KDD?D4D?DD?@DD?DD? DD?@DD?dDn.HD?pDD? DD?@DD?dDPhiD? DD?dDitersD?DjD?DD?dDzD?DjD?DD?dDn.OD?pDD? DD?@DD?dDPhiD? DD?dDitersD?DjD?DD?dDzD?DjD?DD?@DD?@DD?vD200D?D0D?DD?@DD?DD?DD?dDzD?DjD 6 )L)M&&&&&&&&& & & & & &&&D*@T># ~ D@GD@G"666-9This is the same plot with altitude on the vertical axis:6989D;9D9@V0D;.9D;9D/@NormalArial D@A@T@2A@ D@GD@G"D? pD?DD?@DD?@DD?@DD?vD200D?D0D?DD?@DD?DD? DD?@DD?dDzD?DjD?DD?dDzD?DjD?DD?@DD?@DD?vD0.99D?DD?tD10D?KDD?D4D?DD?@DD?DD? DD?@DD?dDn.HD?pDD? DE?@DE?dEPhiE? EE?dEitersE?EjE?DE?dEzE?EjE?DE ?dEn.OE ?pEE ? E E ?@E E ?dE PhiE? E E?dEitersE?EjE?E E?dEzE?EjE < )M)L&&&&&&&&& & & & & &&&E*@ToM( E@GE@G"EE\E\-AIt seems odd that the H+ density increases at low altitudes. Is this really what happens? The answer is yes. In real units, the H+ density increases from 300 to 600 km, according to the plot in the reference below. The equilibrium value of E is sufficiently large at low altitudes to reverse the sign of the gradient in the H+. In the ionosphere, the equilibrium density is also affected by ionization and recombination which have not been considered here. 6FE;E7EE;E7E_EE;iE7EEE;E7E_EE ;E!7EEE";E#7E_E E$;E%7EE"E$E8E&;E'9@V0E(;.E);E*/@NormalArial E+*@TX/ E,@GE-@G"PPTPT-ATry it: The periodic boundary conditions at jmax are equivalent to specifying E = 0 at the right boundary because these conditions impose mirror symmetry on F. Check that this periodic boundary condition has only a small effect on the solution by moving the boundary to 300 Debye lengths (jmax = 600). Observe that the converged solution at z = 200, Phiiters,400, is not significantly changed if the boundary is move further to the right.6jE.;E/7E+}E0;E17E+E.E2;E37E+` SymbolE0E4;E57E+E2E6;E77E+ArialE4E8;E97E+ArialE6E:; E;7E+_ArialE8E<;LE=7E+ArialE:E;E?9@V0E@;.EA;EB/@NormalArial EC*@TKpkX ED@GEE@G"hh h -@Try it: An alternate approach to finding the height dependence of the densities is to integrate the equilibrium momentum equations for the three species: 6EF;EG7EC}EH;EI7ECEFEHEF8EJ;EK9@V0EL;.EM;EN/@NormalArial EO@Txx EP@GEQ@G"ER@GES@G"ET@GEU@G"ࡱ> Root EntryFFSOle Equation Native CompObjf  MarkedRegions_mcObject`F "T"n s "n s m s g+n s FMicrosoft Equation 3.0 DS Equation Equation.39qoft Equation 3.0 DS EqJ{2  .  @&@ & MathType`OlePres000tOle Times New Romanww 0wGf-2 0ySymbol R@ww 0wGf-2 N=y2  +y2 -y2 y2 :-yTimes New Romanww 0wGf-2  Ey2  qy2 B ny2 gy2 Fmy2 ny2 .ny2 BTy Times New Romanww 0wGf-2 b sy2  sy2 hsy2 sy2 sy & "SystemGf ! q s E=0+\-NANIEV*@TV EW@GEX@G"NN@N@-A=The quasineutrality condition on the sum of the gradients provides a constraint that helps to find E in terms of the average ion mass. Find the expression for E and integrate the gradients in the three densities simultaneously using Runge-Kutta. Show that the result is the same as obtained using Poisson's equation. 6=8=EY;=EZ9@V0E[;.=E\;=E]/@NormalArial E^*@TT E_@GE`@G"hL L -@iReference: Asgeir Brekke, Physics of the Upper Polar Atmosphere, Wiley-Praxis, Chichester, 1997, p. 230.6? iEa; Eb7E^}Ec;Ed7E^EaEe;%Ef7E^{EcEg;*Eh7E^EeEgEa8iEi;iEj9@V,0Ek;.iEl;iEm/@NormalArial