document: pulsed high - frequency ( hf ) electromagnetic ( em ) waves from transmitters on the ground are regularly used for sounding the density profile and drift velocity of the overehead ionosphere [ _ hunsucker _ , 1991 ; _ reinisch et al . 
   _ , 1995 , _ reinisch _ , 1996 ] . 
   in 1971 , it was shown theoretically by _ perkins and kaw _ [ 1971 ] that if the injected hf radio beams are strong enough , weak - turbulence parametric instabilities in the ionospheric plasma of the type predicted by _ silin _ [ 1965 ] and _ dubois and goldman _ [ 1965 ] would be excited . 
   ionospheric modification experiments by a high - power hf radio wave at platteville in colorado [ _ utlaut _ , 1970 ] , using ionosonde recordings and photometric measurements of artificial airglow , demonstrated the heating of electrons , the deformation in the traces on ionosonde records , the excitation of spread @xmath0 , etc . 
   , after the hf transmitter was turned on . the triggering of weak - turbulence parametric instabilities in the ionosphere 
   was first observed in 1970 in experiments on the interaction between powerful hf radio beams and the ionospheric plasma , conducted at arecibo , puerto rico , using a scatter radar diagnostic technique [ _ wong and taylor _ , 1971 ; _ carlson et al . 
   _ , 1972 ] . 
   a decade later it was found experimentally in troms that , under similar experimental conditions as in arecibo , strong , systematic , structured , wide - band secondary hf radiation escapes from the interaction region [ _ thid et al . _ , 1982 ] . 
   this and other observations demonstrated that complex interactions , including weak and strong em turbulence , [ _ leyser _ , 2001 ; _ thid et al . _ , 2005 ] and harmonic generation [ _ derblom et al . _ , 1989 ; _ blagoveshchenskaya et al . _ , 
   1998 ] are excited in these experiments . 
   numerical simulations have become an important tool to understand the complex behavior of plasma turbulence . 
   examples include analytical and numerical studies of langmuir turbulence [ _ robinson _ , 1997 ] , and of upper - hybrid / lower - hybrid turbulence in magnetized plasmas [ _ goodman et al . _ , 1994 ; _ xi _ , 2004 ] . in this letter , we present a full - scale simulation study of the propagation of an hf em wave into the ionosphere , with ionospheric parameters typical for the high - latitude eiscat heating facility in troms , norway . to our knowledge , this is the first simulation involving realistic scale sizes of the ionosphere and the wavelength of the em waves . 
   our results suggest that such simulations , which are possible with today s computers , will become a powerful tool to study hf - induced ionospheric turbulence and secondary radiation on a quantitative level for direct comparison with experimental data . 
   we use the mks system ( si units ) in the mathematical expressions throughout the manuscript , unless otherwise stated . 
   we assume a vertically stratified ion number density profile @xmath1 with a constant geomagnetic field @xmath2 directed obliquely to the density gradient . 
   the em wave is injected vertically into the ionosphere , with spatial variations only in the @xmath3 direction . 
   our simple one - dimensional model neglects the em field @xmath4 falloff ( @xmath5 is the distance from the transmitter ) , the fresnel pattern created obliquely to the @xmath3 direction by the incident and reflected wave , and the the influence on the radio wave propagation due to field aligned irregularities in the ionosphere . 
   for the em wave , the maxwell equations give @xmath6 @xmath7 where the electron fluid velocity is obtained from the momentum equation @xmath8\ ] ] and the electron density is obtained from the poisson equation @xmath9 . 
   here , @xmath10 is the unit vector in the @xmath3 direction , @xmath11 is the speed of light in vacuum , @xmath12 is the magnitude of the electron charge , @xmath13 is the vacuum permittivity , and @xmath14 is the electron mass . 
   ms.,scaledwidth=48.0% ]    the number density profile of the immobile ions , @xmath15 $ ] ( @xmath3 in kilometers ) is shown in the leftmost panel of fig . 
   [ fig1 ] . instead of modeling a transmitting antenna via a time - dependent boundary condition at @xmath16 km 
   , we assume that the em pulse has reached the altitude @xmath17 km when we start our simulation , and we give the pulse as an initial condition at time @xmath18 s. in the initial condition , we use a linearly polarized em pulse where the carrier wave has the wavelength @xmath19 ( wavenumber @xmath20 ) corresponding to a carrier frequency of @xmath21 ( @xmath22 ) . 
   the em pulse is amplitude modulated in the form of a gaussian pulse with a maximum amplitude of @xmath23 v / m , with the @xmath24-component of the electric field set to @xmath25\sin(0.1047\times 10^{3 } z)$ ] ( @xmath3 in kilometers ) and the @xmath26 component of the magnetic field set to @xmath27 at @xmath18 . 
   the other electric and magnetic field components are set to zero ; see fig .  [ 
   the spatial width of the pulse is approximately 30 km , corresponding to a temporal width of 0.1 milliseconds as the pulse propagates with the speed of light in the neutral atmosphere . 
   it follows from eq . 
   ( 1 ) that @xmath28 is time - independent ; hence we do not show @xmath28 in the figures . 
   the geomagnetic field is set to @xmath29 tesla , corresponding to an electron cyclotron frequency of 1.4 mhz , directed downward and tilted in the @xmath30-plane with an angle of @xmath31 degrees ( @xmath32 rad ) to the @xmath3-axis , i.e. , @xmath33 . in our numerical simulation , we use @xmath34 spatial grid points to resolve the plasma for @xmath35 km . 
   the spatial derivatives are approximated with centered second - order difference approximations , and the time - stepping is performed with a leap - frog scheme with a time step of @xmath36  s. 
   the splitting of the wave is due to faraday rotation.,scaledwidth=48.0% ]     ms . 
   b ) a closeup of the region of the turning points of the r - x and l - o modes . 
   we see that the wave - energy of the l - o mode is concentrated into one single half - wave envelop at @xmath37 km , while the turning point of the less localized r - x mode is at @xmath38 km.,scaledwidth=48.0% ]     ms . 
   b ) a closeup of the region of the turning points of the r - x and l - o modes . here , the l - o mode oscillations at @xmath37 km are radiating em waves with perpendicular ( to the @xmath3 axis ) electric field components.,scaledwidth=48.0% ]    in the simulation , the em pulse propagates without changing shape through the neutral atmosphere , until it reaches the ionospheric layer . at time @xmath39 
   ms , shown in fig . 
   [ fig2 ] , the em pulse has reached the lower part of the ionosphere . the initially linearly polarized em wave undergoes faraday rotation due to the different dispersion properties of the l - o and r - x modes ( we have adopted the notation `` l - o mode '' and `` r - x mode '' for the two high - frequency em modes , similarly as , e.g. , _ goertz and strangeway _ [ 1995 ] ) in the magnetized plasma , and the @xmath40 and @xmath41 components are excited . at @xmath42 ms , shown in fig . 
   [ fig3 - 4 ] , the l - o and r - x mode pulses are in the vicinity of their respective turning points , the turning point of the l - o mode being at a higher altitude than that of the r - x mode ; see panel a ) of fig . 
   [ fig3 - 4 ] . 
   a closeup of this region , displayed in panel b ) , shows that the first maximum of the r - x mode is at @xmath38 km , and the one of the l - o mode is at @xmath37 km . 
   the maximum amplitude of the r - x mode is @xmath43 
   v / m while that of the l - o mode is @xmath44 v / m ; the latter amplitude maximum is in agreement with those obtained by _ thid and lundborg _ , [ 1986 ] , for a similar set of parameters as used here . 
   the electric field components of the l - o mode , which at this stage are concentrated into a pulse with a single maximum with a width of @xmath45 m , are primarily directed along the geomagnetic field lines , and hence only the @xmath46 and @xmath47 components are excited , while the magnetic field components of the l - o mode are very small . at @xmath48 ms , 
   shown in panel a ) of fig . 
   [ fig5 - 6 ] , both the r - x and l - o mode wave packets have widened in space , and the em wave has started turning back towards lower altitudes . in the closeup of the em wave in panel b ) of fig .  [ fig5 - 6 ] 
   , one sees that the l - o mode oscillations at @xmath37 km are now radiating em waves with significant magnetic field components . finally , shown in fig . 
   [ fig7 ] at @xmath49 , the em pulse has returned to the initial location at @xmath17 km . due to the different reflection heights of the l - o and r - x modes , the leading ( lower altitude ) part of the pulse is primarily r - x mode polarized while its trailing ( higher altitude ) part is l - o mode polarized . in the center of the pulse , where we have a superposition of the r - x and l - o mode , 
   the wave is almost linearly polarized with the electric field along the @xmath26 axis and the magnetic field along the @xmath24 axis . 
   the direction of the electric and magnetic fields here depends on the relative phase between the r - x and l - o mode . 
   ms.,scaledwidth=48.0% ]     at @xmath50 km , near the turning point of the r - x mode , and b ) the amplitude of the electric field component @xmath46 at @xmath51 km , near the turning point of the l - o mode . 
   c ) a snapshot of low - amplitude electrostatic waves of wavelength @xmath52 m ( wavenumber @xmath53 ) , observed at time @xmath54 ms , and d ) dispersion curves ( lower panel ) obtained from the appleton - hartree dispersion relation with parameters @xmath55 ( 5 mhz ) , @xmath56 ( 1.4 mhz ) and @xmath57 rad . 
   we identify the high - frequency r - x and l - o modes , as well as the z - mode which extends to the electrostatic langmuir / upper hybrid branch for large wavenumbers ; the circles indicate the approximate locations on the dispersion curve for the electrostatic oscillations shown in panel c ) . for completeness 
   we also show the low - frequency electron whistler branch in panel d).,scaledwidth=50.0% ]     at the altitude @xmath50 km , and b ) of @xmath46 at the altitude @xmath51 km.,scaledwidth=48.0% ]    in fig . 
   [ fig8 - 9 ] , panel a ) , we have plotted the electric field component @xmath47 at @xmath50 km , near the turning point of the r - x mode and in panel b ) we have plotted the @xmath46 component at @xmath51 km , near the turning point of the l - o mode . 
   we see that the maximum amplitude of @xmath47 reaches @xmath58 v / m at @xmath59 ms , and that of @xmath46 reaches @xmath60 v / m at @xmath61 ms . 
   the electric field amplitude at @xmath50 km has two maxima , due to the l - o mode part of the pulse , which is reflected at the higher altitude @xmath51 km and passes twice over the altitude @xmath50 km . 
   we also observe weakly damped oscillations of @xmath46 at @xmath51 km for times @xmath62 ms , which decrease exponentially in time between @xmath63 ms and @xmath64 ms as @xmath65 with @xmath66 s@xmath67 . 
   we found from the numerical values that @xmath68 , where @xmath69 is the inverse ion density scale length at @xmath70 km , but we are not certain how general this result is . 
   no detectable magnetic field fluctuations are associated with these weakly damped oscillations , and we interpret them as electrostatic waves that have been produced by mode conversion of the l - o mode . 
   the amplitudes of the @xmath47 and @xmath40 components are also much weaker than that of the @xmath46 component for these oscillations . 
   a closeup of these electrostatic oscillations at @xmath54 ms is displayed in panel c ) of fig . 
   [ fig8 - 9 ] , where we see that they have a wavelength of approximately 33 m ( wavenumber @xmath71 ) . in panel d ) of fig . 
   [ fig8 - 9 ] , we have plotted the frequency @xmath72 as a function of the wavenumber @xmath73 , where @xmath74 is obtained from the appleton - hartree dispersion relation [ _ stix _ , 1992 ] @xmath75 here @xmath76^{1/2}$ ] , @xmath77 ( @xmath78 ) is the electron plasma ( cyclotron ) frequency , and @xmath79 is the angle between the geomagnetic field and the wave vector @xmath80 , which in our case is directed along the @xmath3-axis , @xmath81 . 
   we use @xmath82 ( corresponding to @xmath83 mhz ) , @xmath84 ( corresponding to @xmath85 mhz ) and @xmath57 rad . 
   the location of the electrostatic waves whose wavelength is approximately 33 m and frequency 5 mhz is indicated with circles in the diagram ; they are on the same dispersion surface as the langmuir waves and the upper hybrid waves / slow z mode waves with propagation parallel and perpendicular to the geomagnetic field lines , respectively . 
   the mode conversion of the l - o mode into electrostatic oscillations are relatively weak in our simulation of vertically incident em waves , and theory shows that the most efficient linear mode conversion of the l - o mode occurs at two angles of incidence in the magnetic meridian plane , given by , e.g. , eq . 
   ( 17 ) in [ _ mjlhus _ , 1990 ] . 
   the nonlinear effects at the turning point of the l - o and r - x modes are investigated in fig . 
   [ fig10 ] which displays the frequency spectrum of the electric field component @xmath47 at the altitude @xmath86 km and of @xmath46 at the altitude @xmath51 km . 
   the spectrum shows the large - amplitude pump wave at 5 mhz and the relatively weak second harmonics of the pump wave at 10 mhz at both altitudes ( the slight downshift is due to numerical errors produced by the difference approximations used in space and time ) . 
   visible are also low - frequency oscillations ( zeroth harmonic ) due to the nonlinear down - shifting / mixing of the high - frequency wave field . 
   in conclusion , we have presented a full - scale numerical study of the propagation of an em wave and its linear and nonlinear interactions with an ionospheric layer . 
   we observe the reflection of the l - o and r - x modes at different altitudes , the mode conversion of the l - o mode into electrostatic langmuir / upper hybrid waves as well as nonlinear harmonic generation of the high - frequency waves . 
   second harmonic generation have been observed in ionospheric heating experiments [ _ derblom et al . 
   _ , 1989 ; _ blagoveshchenskaya et al . 
   _ , 1998 ] and may be partially explained by the cold plasma model presented here . 
   blagoveshchenskaya , n. f. , v. a. kornienko , m. t. rietveld , b. thid , a. brekke , i. v. moskvin , and s. nozdrachev ( 1998 ) , stimulated emissions around second harmonic of troms  heater frequency observed by long - distance diagnostic hf tools . 
   _ geophys . 
   25_(6 ) , 863876 . 
   derblom , h. , b. thid , t. b. leyser , j. a. nordling ,  . 
   hedberg , p. stubbe , h. kopka , and m. rietveld ( 1989 ) , troms heating experiments : stimulated emission at hf pump harmonic and subharmonic frequencies , _ j. geophys . 
   94_(a8 ) , 1011110120 . 
   thid , b. , e. n. sergeev , s. m. grach , t. b. leyser , and t. d. carozzi ( 2005 ) , competition between langmuir and upper - hybrid turbulence in a high - frequency - pumped ionosphere , _ 
   _ 95 _ , 255002 .
summary: the time evolution of a large - amplitude electromagnetic ( em ) wave injected vertically into the overhead ionosphere is studied numerically . 
   the em wave has a carrier frequency of 5 mhz and is modulated as a gaussian pulse with a width of approximately 0.1 milliseconds and a vacuum amplitude of 1.5 v / m at 50 km . 
   this is a fair representation of a modulated radio wave transmitted from a typical high - power hf broadcast station on the ground . 
   the pulse is propagated through the neutral atmosphere to the critical points of the ionosphere , where the l - o and r - x modes are reflected , and back to the neutral atmosphere . 
   we observe mode conversion of the l - o mode to electrostatic waves , as well as harmonic generation at the turning points of both the r - x and l - o modes , where their amplitudes rise to several times the original ones . 
   the study has relevance for ionospheric interaction experiments in combination with ground - based and satellite or rocket observations .