For many years, active remote probing by means of high-frequency (HF) radio waves has been a standard technique for diagnosing the ionosphere. This is because the recording and analysis of the reflected or scattered part of the HF radiation constitutes a convenient method of determining a number of ionospheric parameters or of investigating various physical processes occurring in the ionospheric plasma. The ionosphere is also treated as a natural space plasma laboratory and modulated more actively using high-power HF pump waves, so as to study the interactions of the electromagnetic waves and plasma. Research attempts in this area began with the Platteville heating experiments conducted in Colorado, USA, in the 1960s (Utlaut, 1970; Utlaut and Cohen, 1971; Utlaut and Violette, 1972). In ionospheric modification experiments, a powerful HF electromagnetic wave incident on the ionosphere can produce nonlinear effects on time scales ranging from tens of microseconds to minutes, and on size scales ranging from meters to kilometers. Strong nonlinear processes including self-focusing, parametric and resonant instability, and accompanying phenomena such as enhanced airglow production, Langmuir turbulence (LT), and the generation of geomagnetic-field-aligned density irregularities (FAI) have been found to occur (Gondarenko et al. 2003; Stubbe et al. 1984).

A characteristic feature of the instabilities excited during ionospheric modulation is the existence of a finite threshold value of the HF pump electric field \( \overrightarrow{E} \) intensity, which must be exceeded for these instabilities to be excited (Fejer, 1979; Fejer, 1981; Fejer et al. 1983). In order to correctly interpret the observations made in these experiments, it is therefore essential to be able to accurately calculate the \( \overrightarrow{E} \) wave pattern in the whole reflection region.

In this regard, the spatial distributions of the pump-wave \( \overrightarrow{E} \) are derived using a wave equation, which is in turn derived within the WKB or geometrical optics approximation. However, it is impossible to obtain the \( \overrightarrow{E} \) value in the vicinity of the reflection (conversion) point, where the WKB approximation breaks down (Ginzburg, 1970). A purely numerical simulation method for obtaining the variation of the pump-wave \( \overrightarrow{E} \) in the whole reflection region is used by Gondarenko et al. 2004. In addition, the ray tracing method proposed by Field et al. 1990 and Hinkel et al. 1993 yields the spatial distribution of the pump-wave \( \overrightarrow{E} \) by accurately calculating the phase of the pump-wave propagation path. An ingenious analytical method devised by Lundborg and Thide 1985, 1986 is used to calculate the variation of the characteristic-wave \( \overrightarrow{E} \) near the reflection region, but only ordinary (O)-mode waves are considered. As regards Chinese scholars, the empirical model is mostly used directly, to estimate the spatial distributions of the pump-wave \( \overrightarrow{E} \); however, these estimates are accurate to within an order of magnitude only (Huang and Gu, 2003; Hao et al. 2013). Of course, purely numerical methods can obtain the variation of the pump-wave \( \overrightarrow{E} \); however, the numerical models are very complex and the computational requirements are high. In addition, analytical methods always provide more information about the solution than pure numbers, and analytic formulas are very quickly evaluated, even on a moderately sized desktop computer.

We have therefore adopted the “uniform approximation” analytical method, which is similar to the method used by Lundborg and Thide 1986, to derive accurate approximations for the variation of the \( \overrightarrow{E} \) of both the O and the extraordinary (X)-mode characteristic waves. In contrast to the similar WKB or phase integral approximations, these approximations do not break down in the reflection region (Langer, 1937; Miller and Good, 1953). In section 2, we provide the mathematical expressions of the \( \overrightarrow{E} \) components of both characteristic mode waves, which are derived using an approach that couples the equation describing a wave initially impinging vertically upon the ionosphere with the Forsterling equation. Then, analytic solutions of each component calculated using the “uniform approximation” method are presented in section 3. In section 4, we present the numerical results for the \( \overrightarrow{E} \) intensity variation of the standing wave pattern of the O- and X-mode characteristic waves, under a specific density profile and throughout the whole reflection region (including the upper-hybrid resonance altitude). These calculations are conducted for different latitudes and at different local times in one location. Along with the real parts of the effective refractive index functions for the O- and X-mode waves, the swelling factors for both characteristic mode waves are also calculated. The field strength obtained for an unmagnetized plasma is given last and compared with the previously obtained results. Finally, our conclusions are outlined in section 5.

### Wave formulation

The general equation for wave propagation in an isotropic inhomogeneous plasma medium can be derived from the Maxwell equations, which are expressed in the Cartesian coordinate system in the form (Gurevich, 1978):

$$ \Delta\!\! \overrightarrow{E}-\nabla \left(\nabla \bullet \overrightarrow{E}\right)+\frac{\omega^2}{c^2}{\varepsilon}^{\hbox{'}}\left(\omega, r\right)\overrightarrow{E}=0, $$

(1)

where \( {\varepsilon}^{\hbox{'}}\left(\omega, r\right)=\varepsilon \left(\omega \right)-i\frac{4\pi }{\omega}\sigma \left(\omega \right) \), *σ* is the conductivity tensor and *ω* is the wave angular frequency. When considering a plane electromagnetic wave impinging vertically upon the ionosphere, the changes in \( \overrightarrow{E} \) depend on the *z*-coordinate only. Then, Eq. (1) can be rewritten as:

$$ \frac{d^2E}{d{z}^2}+\frac{\omega^2}{c^2}{\varepsilon}^{\hbox{'}}\left(\omega, z\right)E=0. $$

(2)

This equation is applicable to both of the horizontal \( \overrightarrow{E} \) components, *E*
_{
x
} and *E*
_{
y
}. When the background ionosphere is considered to be a linear plasma layer without absorption, Eq. (2) becomes:

$$ {\varepsilon}^{\hbox{'}}\left(\omega, z\right)=\varepsilon (z)={n}^2(z),\;\sigma =0. $$

(3)

where *n*
^{2} is the square of the complex refractive index. *E*
_{
x
} and *E*
_{
y
} are determined from the one-dimensional time-independent wave equation:

$$ \frac{d^2E}{d{z}^2}+{k}^2{n}^2(z)E=0, $$

(4)

where *k* = *ω*/*c*.

In this paper, we are interested in analyzing the variation of the \( \overrightarrow{E} \) intensity near the reflection point caused by the initial vertical propagation of the HF radio waves in the ionosphere. We therefore model the ionosphere as a cold, magnetoactive, collisional plasma. In the right-handed Cartesian coordinate system we have chosen, the *x*-, *y*-, and *z*-axes respectively point toward magnetic east, magnetic north, and vertically upward, that is, the *z*-axis parallel to the *k* vector of the radio wave launched from a transmitter on the ground. The geomagnetic field *B*
_{0} is considered to lie in the *yoz*-plane and to create an angle *θ* to the negative *z*-axis, as shown in Fig. 1. We assume that the standard magnetoionic notations are suitable for our model. Then, we have (Rishbeth and Garriott, 1969):

$$ X={\omega}_{pe}^2/{\omega}^2,\;Y={\omega}_{ce}/\omega,\;Z=\nu /\omega, $$

(5)

where *ω*
_{
pe
} and *ω*
_{
ce
} are the electron plasma (angular) frequency and the electron cyclotron frequency, respectively. For the specific expressions of these two terms, refer to Rishbeth and Garriott 1969. The electron collision frequency *ν*
_{
e
} = *ν*
_{
em
} + *ν*
_{
ei
}, where *ν*
_{
em
} and *ν*
_{
ei
} are the collision frequencies of electrons with neutral particles and with ions, respectively. The specific expressions of these collision frequencies are given in detail in Banks and Kocharts 1973 and Schunk and Walker 1980.

The wave equations for the components of \( \overrightarrow{E} \) can be derived from Maxwell’s equations in the usual, well-known way, as we have discussed above. Assuming a time variation of exp[−*iωt*], we obtain the following wave equations (Ginzburg, 1970):

$$ \frac{d^2{E}_x}{d{z}^2}+{k}^2{Q}_{11}(z){E}_x+{k}^2{Q}_{12}(z){E}_y=0, $$

(6a)

$$ \frac{d^2{E}_y}{d{z}^2}+{k}^2{Q}_{21}(z){E}_x+{k}^2{Q}_{22}(z){E}_y=0, $$

(6b)

$$ {E}_z+{Q}_{31}(z){E}_x+{Q}_{32}(z){E}_y=0, $$

(6c)

where the functions *Q*
_{
ij
}(*z*) are given by

$$ {Q}_{11}(z)=1-\frac{X(z)\left[1+iZ\right]\left[1+iZ-X(z)\right]}{D(z)}, $$

$$ {Q}_{12}(z)=-{Q}_{21}(z)=-i\frac{X(z)\left[1+iZ-X(z)\right]Y \cos \theta }{D(z)}, $$

$$ {Q}_{22}(z)=1-\frac{X(z)\left[1+iZ\right]\left[1+iZ-X(z)\right]}{D(z)}+\frac{X(z){Y}^2{ \sin}^2\theta }{D(z)}, $$

$$ {Q}_{31}(z)=i\frac{X(z)\left[1+iZ\right]Y \sin \theta }{D(z)}, $$

$$ {Q}_{32}(z)=-\frac{X(z){Y}^2 \sin \theta \cos \theta }{D(z)}, $$

with the common denominator

$$ D(z)=\left[1+iZ-X(z)\right]\left[{\left(1+iZ\right)}^2-{Y}^2\right]-X(z){Y}^2{ \sin}^2\theta . $$

(7)

We can easily see that wave Eqs. (6a) and (6b) are coupled and, therefore, it is a formidable task to obtain exact solutions in their current form. However, for a homogeneous medium, the exact solution of (6a) and (6b) can be obtained trivially by solving the eigenvalue problem of the corresponding matrix, which is then constant. This yields the eigenvalues (Lundborg and Thide, 1986):

$$ {n}_{O/X}^2=1-\frac{X}{2D}\left\{2\left(1+iZ\right)\left(1+iZ-X\right)-{Y}^2{ \sin}^2\theta \mp {\left[{Y}^4{ \sin}^4\theta +4{\left(1+iZ-X\right)}^2{Y}^2{ \cos}^2\theta \right]}^{\frac{1}{2}}\right\}, $$

(8)

and the corresponding eigenvectors described by the transverse polarization

$$ {\rho}_{O/X}=\frac{i}{2\left(1+iZ-X\right)Y \cos \theta}\left\{{Y}^2{ \sin}^2\theta \pm {\left[{Y}^4{ \sin}^4\theta +4{\left(1+iZ-X\right)}^2{Y}^2{ \cos}^2\theta \right]}^{\frac{1}{2}}\right\}. $$

(9)

The lower subscripts and signs ∓ (or ±) in Eqs. (8) and (9) correspond to the O and X modes, respectively. From Eq. (9), we easily find that the two polarizations satisfy *ρ*
_{
O
}
*ρ*
_{
X
} = 1. For convenience in what follows, we primarily use the quantity *ρ*
_{
X
} in our equations, as *ρ*
_{
O
} becomes very large in the O-mode reflection region.

The characteristic waves in the homogeneous medium are thus the well-known O- and X-mode wave with complex wave numbers *kn*
_{
O
} or *kn*
_{
X
}, according to (8), and with polarizations given by (9). However, it must be remembered that, in an inhomogeneous medium, these waves are no longer exact solutions of the wave equations. Obviously, the pump-wave reflection area in our calculation no longer satisfies this condition. However, if the medium is slowly varying, one might hope that there exist approximate solutions under certain conditions corresponding to the characteristic modes that are at least less strongly coupled to each other than the Cartesian field components. It might therefore be a good idea to transform the dependent variables *E*
_{
x
} and *E*
_{
y
} in (6a) and (6b) to new variables corresponding to the characteristic wave modes for a homogeneous plasma. The specific conversion procedure is described in both Lundborg and Thide 1986 and Budden 1966. Hence, we express the transverse field in terms of the two Forsterling functions *F*
_{
O
} and *F*
_{
X
}, where:

$$ \begin{array}{l}{E}_x={\rho}_X{E}_{y,O}+{E}_{x,X},\;{E}_y={E}_{y,O}+{\rho}_X{E}_{x,X},\\ {}{E}_{y,O}={\left({\rho}_X^2-1\right)}^{-\frac{1}{2}}{F}_O,\;{E}_{x,X}={\left({\rho}_X^2-1\right)}^{-\frac{1}{2}}{F}_X.\end{array} $$

(10)

The new variables *F*
_{
O
} and *F*
_{
X
} in Eq. (10) must satisfy

$$ {F}_O^{\hbox{'}\hbox{'}}+\left({k}^2{n}_O^2+{q}^2\right){F}_O={q}^{\hbox{'}}{F}_X+2q{F}_X^{\hbox{'}}, $$

(11a)

$$ {F}_X^{\hbox{'}\hbox{'}}+\left({k}^2{n}_X^2+{q}^2\right){F}_X={q}^{\hbox{'}}{F}_O+2q{F}_O^{\hbox{'}}, $$

(11b)

where the coupling function *q* is defined as

$$ q=\frac{i\left(i{Z}^{\hbox{'}}-{X}^{\hbox{'}}\right)Y \cos \theta { \sin}^2\theta }{Y^2{ \sin}^4\theta +4{\left(1+iZ-X\right)}^2{ \cos}^2\theta }. $$

(12)

Equations (11a) and (11b) are the Forsterling equations, which contain no approximations and, hence, are equivalent to the original equations, (6a) and (6b). Assuming the solutions of Eqs. (11a) and (11b) are known, we obtain *E*
_{
x
} and *E*
_{
y
} from Eq. (10) and, finally, *E*
_{
z
} from Eq. (6c). We can then achieve a formal simplification of these results by introducing the longitudinal polarization

$$ {\rho}_L={E}_z/{E}_x. $$

(13)

Substituting (13) into Eqs. (6c) and (10) yields

$$ {\rho}_{L,O/X}=\frac{iY \sin \theta }{1+iZ-X}\left({n}_{O/X}^2-1\right). $$

(14)

We may now write the exact total field in the form

where

$$ \begin{array}{l}{E}_O=\left(1,{\rho}_O,{\rho}_{L,O}\right){\rho}_X{\left({\rho}_X^2-1\right)}^{-\frac{1}{2}}{F}_O\\ {}\kern1.7em =\left({\rho}_X,1,{\rho}_X{\rho}_{L,O}\right){\left({\rho}_X^2-1\right)}^{-\frac{1}{2}}{F}_O,\\ {}{E}_X=\left(1,{\rho}_X,{\rho}_{L,X}\right){\left({\rho}_X^2-1\right)}^{-\frac{1}{2}}{F}_X.\end{array} $$

(15)