Sagir_FC_OK:Layout 6 ANNALS OF GEOPHYSICS, 57, 2, 2014, A0216; doi:10.4401/ag-6469 A0216 The characterization of diffusion tensor for mid-latitude ionospheric plasma Selcuk Sagir1,*, Ali Yesil2, Gulay Sanac2, Ibrahim Unal3 1 Mus Alparslan University, Department of Physics, Mus, Turkey 2 Fırat University, Department of Physics, Elazig, Turkey 3 Inonu University, Department of Science Teaching, Malatya, Turkey ABSTRACT In this study, the relationship between diffusion and the electrical con- ductivity tensor is investigated by using the real geometry of the Earth’s magnetic field in the Northern Hemisphere for mid-latitudes. This rela- tionship is derived from equation. Only in the elements of the diagonal of the tensor is an equal relationship found between the electrical conduc- tivity called the longitudinal conductivity and the diffusion coefficient. The diagonal elements are equal to each other and do not depend on the Earth’s magnetic field; however, the other elements of the tensor strongly depend on the Earth’s magnetic field and they have characteristic of Bohm diffusion became a semi-empirical model (DB= (1/16)kT/eB) but they do not depend on the electrical conductivity or the classical dif- fusion coefficient. 1. Introduction The ionosphere is the layer of the Earth’s atmos- phere that extends from 50 to 1000 km. The ionosphere significantly affects the propagation of radio waves so the variability of the ionosphere is important for ionospheric physics and radio communications. The ionosphere can be characterized by its electron density distribution, which is a complex function of spatial and temporal vari- ations and geomagnetic and solar activity [Rishbeth 1965, Rishbeth and Garriott 1969, Whitten and Poppoff 1971, Banks and Kockarts 1973]. Usually, the ionosphere with respect to its electron density distribution is separated into five independent regions, sometimes called layers. The bottom one, from 50 to 80 km, is called the D layer, from 80 to 130 km is the E layer, and above 150 km is the F layer. The F region is usually separated at the F1 layer, from 130 to 200-250 km, and the F2 layer is above 250 km [Rishbeth and Garriott 1969, Whitten and Poppoff 1971, Banks and Kockarts 1973, Rishbeth 1977]. The iono- sphere is a natural plasma. Many plasma phenomena can be analyzed using macroscopic transport equations, ei- ther by considering the plasma as a multi-constituent fluid or by treating the plasma as a single conducting fluid [Rishbeth 1975, St.-Maurice and Schunk 1977]. In some cases, however, a satisfactory description can be obtained only through the use of kinetic theory. The presence of a pressure gradient term in the momentum transport equation provides a force that tends to smooth out any in- homogeneities in the plasma density [Zhilinskiĭ and Tsendin 1980]. Ion electrons diffuse under the influence of partial pressure gradients and gravity. The diffusion of the particles in a plasma results from this pressure gradi- ent force. To deduce the expression for the electron dif- fusion coefficient for a warm weakly ionized element, the momentum transport equations for the electrons are used with a constant electron-neutral collision frequency in the ionospheric plasma [Banks and Kockarts 1973, St.- Maurice and Schunk 1977, Abbas et al. 2011]. The elec- trons have far lower masses than ions, and they have far higher typical speeds at fixed temperature and are much more easily accelerated (i.e., they are much more mo- bile). As a result, it is the motion of the electrons, not the ions, that is responsible for the transport of heat and charge through plasma. Transport properties can now be computed, including the electric conductivity and ther- mal conductivity, on the presumption that it is Coulomb collisions that determine the electron mean free paths and that magnetic fields are unimportant in providing a more serious impediment to charge and heat flow than Coulomb collisions and thus dominate the conductivities [Rishbeth and Garriott 1969, Whitten and Poppoff 1971, Banks and Kockarts 1973]. Article history Received November 22, 2013; accepted March 6, 2014. Subject classification: Ionosphere, Diffusion tensor, Electrical conductivity. This motion is hindered by collisions with the neu- tral air. The equation for the diffusion of plasma is ob- tained by adding the ion and electron force equations. The gravity term for electrons is omitted as it is quite negligible [Banks and Kockarts 1973]. The diffusion co- efficient, called diffusivity, is an important parameter in- dicative of the diffusion mobility. The diffusion coefficient is not only encountered in Fick’s law, but is found in nu- merous other equations of physics and chemistry. The diffusion coefficient is generally prescribed for a given pair of species. For a multi-component system, it is pre- scribed for each species in the ionospheric plasma. The higher the diffusivity, the faster the diffusion. The diffu- sion coefficient is proportionally constant between the diffusion flux and gradient in the concentration of the diffusing species, and it depends on both temperature and pressure [Rishbeth and Garriott 1969, Whitten and Pop- poff 1971, Banks and Kockarts 1973]. 2. The relationship between diffusion and electrical conductivity tensor The behavior of the ionospheric plasma and the neutral thermosphere is subject to the equation of state (perfect gas law) and the general conservation equa- tions for mass, momentum and energy [Rishbeth 1965, Bittencourt 1995]: a) Continuity equation: Density change=Production–Loss–Transport b) Equation of motion: Acceleration=Force–Drag–Transport c)Temperature change=Heating–Cooling–Conduction. If it does not have an impact outside, the transport of particles in the ionosphere plasma from place to place results from the pressure-gradient (∇P). This force occurs in any part of the plasma density to elim- inate inhomogeneity. If B≠0, the medium is called the anisotropic [Banks and Kockarts 1973]. Due to this, the ionospheric plasma is anisotropic. The real geometry of the Earth’s magnetic field for the Northern Hemi- sphere is shown in Figure 1, where Bx = BCosISind, By = BCosICosd, and Bz = −BSinI. I is the magnetic dip angle and d is the magnetic declination angle. If U and m are the velocity and mass of the electron, respec- tively, then the force acting on the electron is as follows: where and, are the electron-ion and electron-neutral collision fre- quencies, respectively. It is assumed that the velocity and fields vary as ei(k · r−~t), where ~ is angular wave fre- quency, and ~c is electron angular gyro frequency. Its vector components are: The current density can be obtained as follows: in which is the longitudinal conductivity. We assumed that the z- axis of the coordinate system with its origin located on the ground is vertical upwards. The x-and y-axes are ge- ographic eastward and northward in the Northern Hemisphere [Aydoğdu et al. 2004]. When the real geom- etry of the Earth’s magnetic field is used for the steady- state (∂/∂t=0, that is, ~=0), the conductivity tensor of the solution of Equation (2) is obtained as follows: with All of the tensor elements of Equation (4) are real if Equation (1) is used for the diffusion coefficient, de- pending on the real geometry of the Earth: where, m dt d e mU U B E U# o=- + -^ h ei eno o o= + N . N T T 5.4 10 N T m.k.s 10 m.k.s andlog59 4 18ei e en n e e 3 16 21 6 3 2 # #o o = + = - - -; 6 6 E @ @ c m eB eB- - m , m eB and m .cx x cy c y z z ~ ~ ~= = - = m i eJ E J B0 #v o ~ = - -^ h m i Ne 0 2 v o ~ = -^ h J Ev= c cx cx cy cz cx cz cy cx cy cz cy cy cz cx cx cz cy cy cz cx cz 2 2 0 2 2 2 2 2 2 v o ~ v o ~ ~ ~ ~ o ~ ~ ~ o ~ ~ ~ o o ~ ~ ~ ~ o ~ ~ ~ o ~ ~ ~ o o ~ = + + - + + + - + + -> H n nB E D# dnC C= + -^ h m e , D m k T and nUbn o o C= - = =^ h SAGIR ET AL. 2 Figure 1. The geometry of the Earth’s Magnetic Field (for the North- ern Hemisphere). (1) (2) (3) (4) (5) 3 are respectively: the electron mobility, the electron dif- fusion coefficient and the flux of density. The flux of density in terms of the current density is as follows: From here, the electric field vector that depends on the flux of density could be written as follows: If this Equation (7) is used Equation (5), Equation (8) is obtained as follows: where I is the unit matrix and v is the conductivity ten- sor. From here, the diffusion tensor depends on both the classical diffusion coefficient [Abbas et al. 2011] and the electrical conductivity: where, With respect to Equation (9), the diffusion tensor is given by: The diagonal elements of tensor (Equation 10) are equal to each other and they only depend on the classi- cal diffusion coefficient (D) and the electrical conduc- tivity (v0). They do not depend on the Earth’s magnetic field. However, the tensor is symmetric and they have characteristic of Bohm diffusion became a semi-empir- ical model (DB= (1/16)kT/eB). This diffusion coeffi- cient does not depend on the density but it depends on magnetic field (that is, DBxy= −DByx, DBxz= −DBzx, and DByz= −DBzy). These results are interesting because the tensor (Equation 10) is a mixture of the electrical con- ductivity, the classical diffusion coefficient, and the dif- fusion depending on the magnetic field. 3. Numerical analysis and results The general system of transport equations is ap- plied to the mid-latitude topside ionosphere. The re- striction to this region of the ionosphere enables us to make several simplifying assumptions that significantly reduce the general system of transport equations. First is fully ionized plasma composed of two major ions, electrons, and a number of minors. Next, the steady state and the species temperature and flow velocity differ- ences are small in the mid-latitude topside ionosphere. This enables us to neglect stress and nonlinear accel- eration terms [Rishbeth 1975]. We wondered what the relationship between the electron diffusion and con- ductivity became in the ionospheric plasma by using the real geometry of the Earth’s magnetic field. This rela- tionship is given by Equation (10). According to Equa- tion (10), only in the elements of the diagonal of the tensor is the relationship found between the electrical conductivity called the longitudinal conductivity and the diffusion coefficient; the diagonal elements are equal and thus do not depend on the Earth’s magnetic field. How- ever, the other elements of the tensor strongly depend on the Earth’s magnetic field, but they do not depend on the electrical conductivity or the classical diffusion. In this study, the relationship between the diffusion tensor and the electrical conductivity is investigated by using the real geometry of the Earth’s magnetic field in the Northern Hemisphere for mid-latitudes iono- sphere plasma. The results obtained: I (dip angle)= 55.6°, d (Declination) = 3°, R=10, by using Equations (4)-(10). The ionospheric parameters used for calcula- tions are obtained using the IRI (International Refer- ence Ionosphere) model. For a given location, time, and date, the IRI model describes the electron density, elec- tron temperature, ion temperature, and ion composi- tion in the altitude range from about 50 to about 2000 km; it also describes the electron content. The model- provides monthly averages in the non-auroral iono- sphere for magnetically quiet conditions. The diagonal elements of Equation (10) are equal and only depend on the classic diffusion coefficient and longitudinal con- ductivity (v0). The seasonal diagram of the diagonal el- ement for mid-latitude in ionospheric plasma is given in Figure 1 for local time (12.00-24.00). According to Fig- ure 1, the diagonal element that depends on both the classical diffusion coefficient and electric conductivity decreases with altitude in mid-latitudes ionospheric plasma at 12.00 LT (Local Time) and for every season. It takes the minimum value and is negative at around n and enU J U EvC= =- = eE v C= - e ID n n B#d v n nC C= - - + ^ h` j n n n D 1 D D D D D D D D 1 1 1 1 1 1 1 1 Bz cz B c Bz cz B c By cy B c x x y z y y x x x x y z 0 0 0 $ d d d v o ~ o ~ o ~ v o ~ o ~ o ~ v C C C - + + + - - + - + + - = ^ ^ ^ h h h R T S S S S S S S > > V X W W W W W W W H H c c m m D eB k T , D eB k T and D eB k T .Bz z b B b B b y y x x = = = D 1 D D D D D D D D D 1 1 1 1 1 1 1 1 Bz cz B c Bz cz B c By cy B c B y y x x x x 0 0 0 v o ~ o ~ o ~ v o ~ o ~ o ~ v - + + + - - + - + + - = R T S S S S S S S V X W W W W W W Wc c m m DIFFUSION TENSOR IN THE IONOSPHERE (6) (7) (8) (9) (10) 225-400 km for March 21 and September 23 in general. However, for the same seasons, it suddenly decreases to north latitude (30°N) and is approximately −1.2 × 10-6 for March 21 and −7.0 × 10-6 at 300 km. For June 21 and December 21, it takes two times minimum values for south latitudes with an altitude (200-250 km; 275-400 km) except for the north latitudes. In addition, the diagonal elements of Equation (10) increased about 300 km after SAGIR ET AL. 4 Figure 2. The seasonal diagram of diagonal elements in Equation (10) for mid-latitude in ionospheric plasma. Figure 3. The seasonal diagram of DBxy (m 2/sn) in Equation (10) for mid-latitude in ionospheric plasma. Altitude (km) Altitude (km) 5 every season and latitude, but are negative. For 24.00 LT, the trend in change of altitude of the diagonal elements of Equation (10) is similar to the change of in 12.00 LT for every season. It is also the minimum altitude (about 300 km) but its magnitude has decreased with respect to 12.00 LT. The seasonal diagram of DBxy(m 2/sn) is given for 12.00-14.00 LT in Figure 2. DBxy(m 2/sn) is positive both for 12.00 LT and 24.00 LT in mid-latitude ionos- DIFFUSION TENSOR IN THE IONOSPHERE Figure 4. The seasonal diagram of DByz(m 2/sn) in Equation (10) for mid-latitude in ionospheric plasma. Figure 5. The seasonal diagram of DBxz(m 2/sn) in Equation (10) for mid-latitude in ionospheric plasma. Altitude (km) Altitude (km) pheric plasma for the Northern Hemisphere and all seasons, but is negative for the Southern Hemisphere. The magnitude of DBxy(m 2/sn) is bigger at 12.00 LT than 24.00 LT and seasonal. The change in altitude of DByz(m 2/sn) is the same as both the local time and the magnitude in Figure 2; however, the decrease in south lat- itudes and increase in north latitudes is exponential. The change in altitude of DBxz(m 2/sn) is shown in Figure 4. DBxz(m 2/sn) decreases with altitude for both seasonal and mid-latitudes (12.00-24.00LT) but the most notice- able results occur at 30° south latitude. DBxz(m 2/sn) quickly decreases with altitude for both season and lati- tude. DBxz(m 2/sn) is approximately maximum between 300 and 450 km depending on the season 30° south lat- itude. The relationship between the magnitudes is DBxz(m 2/sn) >DByz(m 2/sn)>DBxy(m 2/sn). 4. Conclusion This article has reviewed the diffusion tensor for electrons in the mid-latitude ionospheric plasma and in- vestigated whether any relationship between the diffu- sion and electrical conductivity exists. The findings indicate that the diffusion coefficient depends only on 0 (longitudinal conductivity) in Equation (10). However, the other conductivities, such as Pedersen and Hall con- ductivity, do not affect the diffusion in ionosopheric plasma. The diagonal elements of tensor (Equation 10) are equal to each other and they only depend on the clas- sical diffusion coefficient (D) and the electrical conduc- tivity (v0). They do not depend on the Earth’s magnetic field. However, the tensor is symmetric and they have the characteristic of Bohm diffusion became a semi-em- pirical model (DB=(1/16)kT/eB). This diffusion coeffi- cient does not depend on the density but it depends on magnetic field. (That is, DBxy= –DByx,DBxz= –DBzx, and DByz= –DBzy). These results are interesting because the tensor (Equation 10) is a mixture of the electrical con- ductivity, the classical diffusion coefficient, and the dif- fusion depending on the magnetic field. Based on our results, the diagonal elements in Equation (10) take the minimum value around 200-400 km as both seasonal and latitudes are negative in gen- eral. This altitude (200-400 km) is the peak height of the region and has generally the maximum electron density in this region for every season. The tensorial el- ements of the diffusion expected for the diagonal ele- ment in Equation (10) do not depend on electrical conductivity, but the tensorial elements relate to the electron temperature and collision frequency due to a magnetic field constant in mid-latitudes ionospheric plasma. The diffusion coefficients with respect to all di- agrams change to linear with electron temperatures for mid-latitudes ionospheric plasma. Our procedure may be useful in assessing ionos- pheric variables such as conductivity and it can be used for ionospheric plasma density measurements. This study could become important in mid-latitude stations for ionosonde measurements because the electron con- centration of the medium determines the behavior of the medium against any external influence. References Abbas, Q.A., F.Y. Hadi and S.S. AL-Awadi (2011). Mod- ified Bohm Diffusione quation in Q-Machine, Bagh- dad Sci. J., 8, 339-344. Aydoğdu, M., A. Yeşil and E. Güzel (2004). The group refractive indicies of HF waves in the ionosphere and departure from magnitude without collisions, J. Atmos. Sol.-Terr. Phy., 66, 343-348. Baker, W.G., and D.F. Martyn (1953). Electric Currents in the Ionosphere. I. The Conductivity, Phil. Trans. R. Soc. Lond. A., 246, 281-294. Banks, P., and G. Kockarts (1973). Aeronomy, Part B, Academic Press, New York, 430 pp. Bittencourt, J.A. (1995). Fundamentals of Plasma Physics, Co-Edition FAPESP, 676 pp. Rishbeth, H. (1965). A review of ionospheric F region theory, Proceedings of the IEEE, 55, 16-35. Rishbeth, H., and O.K. Garriott (1969). Introduction to Ionospheric Physics, Academic Press, Amsterdam, 331 pp. Rishbeth, H. (1975). On the theory of diffusion in the ionosphere, Geophys. J. Roy. Astr. S., 41, 311-317. Rishbeth, H. (1997). The ionospheric E-layer and F- layer dynamos - a tutorial review, J. Atmos. Sol.-Terr. Phy., 59, 1873-1880. St.-Maurice, J-P., and R.W. Schunk (1977). Diffusion and heat flow equations for the mid-latitude topside ion- osphere, Planet. Space Sci., 25, 907-920. Whitten, R.C., and I.G. Poppoff (1971). Fundamentals of aeronomy, John Wiley & Sons, New York, 462 pp. Zhilinskiĭ, A.P., and L.D. Tsendin (1980). Collisional dif- fusion of a partly-ionized plasma in magnetic field, Sov. Phys. Uspekhi, 131, 343-385. *Corresponding author: Selcuk Sagir, Mus Alparslan University, Department of Physics, Mus, Turkey; email: s.sagir@alparslan.edu.tr or sagirselcuk@hotmail.com. © 2014 by the Istituto Nazionale di Geofisica e Vulcanologia. All rights reserved. SAGIR ET AL. 6 << /ASCII85EncodePages false /AllowTransparency false /AutoPositionEPSFiles false /AutoRotatePages /None /Binding /Left /CalGrayProfile (Dot Gain 20%) /CalRGBProfile (sRGB IEC61966-2.1) /CalCMYKProfile (U.S. Web Coated \050SWOP\051 v2) /sRGBProfile (sRGB IEC61966-2.1) /CannotEmbedFontPolicy /Warning /CompatibilityLevel 1.3 /CompressObjects /Tags /CompressPages true /ConvertImagesToIndexed true /PassThroughJPEGImages true /CreateJobTicket false /DefaultRenderingIntent /Default /DetectBlends true /DetectCurves 0.1000 /ColorConversionStrategy /LeaveColorUnchanged /DoThumbnails false /EmbedAllFonts true /EmbedOpenType false /ParseICCProfilesInComments true /EmbedJobOptions true /DSCReportingLevel 0 /EmitDSCWarnings false /EndPage -1 /ImageMemory 1048576 /LockDistillerParams true /MaxSubsetPct 100 /Optimize false /OPM 1 /ParseDSCComments true /ParseDSCCommentsForDocInfo true /PreserveCopyPage true /PreserveDICMYKValues true /PreserveEPSInfo true /PreserveFlatness true /PreserveHalftoneInfo false /PreserveOPIComments false /PreserveOverprintSettings true /StartPage 1 /SubsetFonts true /TransferFunctionInfo /Apply /UCRandBGInfo /Preserve /UsePrologue false /ColorSettingsFile (None) /AlwaysEmbed [ true /AndaleMono /Apple-Chancery /Arial-Black /Arial-BoldItalicMT /Arial-BoldMT /Arial-ItalicMT /ArialMT /CapitalsRegular /Charcoal /Chicago /ComicSansMS /ComicSansMS-Bold /Courier /Courier-Bold /CourierNewPS-BoldItalicMT /CourierNewPS-BoldMT /CourierNewPS-ItalicMT /CourierNewPSMT /GadgetRegular /Geneva /Georgia /Georgia-Bold /Georgia-BoldItalic /Georgia-Italic /Helvetica /Helvetica-Bold /HelveticaInserat-Roman /HoeflerText-Black /HoeflerText-BlackItalic /HoeflerText-Italic /HoeflerText-Ornaments /HoeflerText-Regular /Impact /Monaco /NewYork /Palatino-Bold /Palatino-BoldItalic /Palatino-Italic /Palatino-Roman /SandRegular /Skia-Regular /Symbol /TechnoRegular /TextileRegular /Times-Bold /Times-BoldItalic /Times-Italic /Times-Roman /TimesNewRomanPS-BoldItalicMT /TimesNewRomanPS-BoldMT /TimesNewRomanPS-ItalicMT /TimesNewRomanPSMT /Trebuchet-BoldItalic /TrebuchetMS /TrebuchetMS-Bold /TrebuchetMS-Italic /Verdana /Verdana-Bold /Verdana-BoldItalic /Verdana-Italic /Webdings ] /NeverEmbed [ true ] /AntiAliasColorImages false /CropColorImages true /ColorImageMinResolution 150 /ColorImageMinResolutionPolicy /OK /DownsampleColorImages true /ColorImageDownsampleType /Bicubic /ColorImageResolution 300 /ColorImageDepth -1 /ColorImageMinDownsampleDepth 1 /ColorImageDownsampleThreshold 1.10000 /EncodeColorImages true /ColorImageFilter /DCTEncode /AutoFilterColorImages true /ColorImageAutoFilterStrategy /JPEG /ColorACSImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /ColorImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /JPEG2000ColorACSImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /JPEG2000ColorImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /AntiAliasGrayImages false /CropGrayImages true /GrayImageMinResolution 150 /GrayImageMinResolutionPolicy /OK /DownsampleGrayImages true /GrayImageDownsampleType /Bicubic /GrayImageResolution 300 /GrayImageDepth -1 /GrayImageMinDownsampleDepth 2 /GrayImageDownsampleThreshold 1.10000 /EncodeGrayImages true /GrayImageFilter /DCTEncode /AutoFilterGrayImages true /GrayImageAutoFilterStrategy /JPEG /GrayACSImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /GrayImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /JPEG2000GrayACSImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /JPEG2000GrayImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /AntiAliasMonoImages false /CropMonoImages true /MonoImageMinResolution 1200 /MonoImageMinResolutionPolicy /OK /DownsampleMonoImages true /MonoImageDownsampleType /Bicubic /MonoImageResolution 1200 /MonoImageDepth -1 /MonoImageDownsampleThreshold 1.08250 /EncodeMonoImages true /MonoImageFilter /CCITTFaxEncode /MonoImageDict << /K -1 >> /AllowPSXObjects false /CheckCompliance [ /None ] /PDFX1aCheck false /PDFX3Check false /PDFXCompliantPDFOnly false /PDFXNoTrimBoxError true /PDFXTrimBoxToMediaBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXSetBleedBoxToMediaBox true /PDFXBleedBoxToTrimBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXOutputIntentProfile (None) /PDFXOutputConditionIdentifier () /PDFXOutputCondition () /PDFXRegistryName (http://www.color.org) /PDFXTrapped /Unknown /CreateJDFFile false /SyntheticBoldness 1.000000 /Description << /ENU (Use these settings to create PDF documents with higher image resolution for high quality pre-press printing. The PDF documents can be opened with Acrobat and Reader 5.0 and later. These settings require font embedding.) /JPN /FRA /DEU /PTB /DAN /NLD /ESP /SUO /NOR /SVE /KOR /CHS /CHT /ITA >> >> setdistillerparams << /HWResolution [2400 2400] /PageSize [595.000 842.000] >> setpagedevice