HF Doppler oscillations due to mixed ULF wave modes

C.L. Waters, M. Sciffer, I.S. Dunlop, F.W. Menk
Department of Physics and CRC for Satellite Systems,
The University of Newcastle, Callaghan, 2308
New South Wales, Australia
Phone: (02) 4921 5421 Email: physpuls8@cc.newcastle.edu.au

Abstract

HF signals that propagate via the ionosphere may exhibit Doppler shifts due to a number of processes that give rise to a time dependent refractive index. This paper examines the role of ULF (1-100 mHz) wave energy, incident from the magnetosphere, in causing Doppler shifts in HF signals. We describe a model of ULF wave propagation through the ionosphere that allows the ULF wave mode mix to be specified. It is found that the shear Alfven mode wave gives the larger Doppler velocities of order 2 ms-1 in the F region, caused mostly by a mechanism that involves the divergence of the electron velocity from the passage of the ULF wave. Pure fast mode perturbations of similar amplitude gives Doppler velocities around ten times smaller. In both cases the role of vertical bulk motion of the electrons is small. An experiment that recorded data from a Doppler sounder and magnetometer is described and the amplitude ratio and cross phase of data recorded by the north-south magnetometer sensor and the Doppler sounder are presented. The amplitude ratio data show a distinctive maximum and the cross phase a large phase shift around a ULF field line resonance. The paper presents the relevant theory and modelling of this feature and illustrates the dependence on the horizontal spatial structure of the ULF energy.

1. Introduction

The role of the ionosphere in determining propagation properties of HF (3-30 MHz) signals has been studied since the advent of radio. At appropriate frequencies, the ionospheric plasma causes HF signals to be continually refracted. This makes long distance communications possible yet very dependent on ionospheric parameters. The ionosphere is birefringent due to the imposed geomagnetic field which also supports ultra-low frequency (ULF, 1-100 mHz) wave energy incident from the magnetosphere. Near Earth space is a magnetised plasma environment where ULF energy propagates as magnetohydrodynamic (MHD) waves which can be measured using sensitive magnetometers. When ULF waves interact with the ionosphere, they cause frequency shifts in any HF signals present.

The relationship between ULF waves recorded by ground based magnetometers and Doppler variations in ionosphere propagating HF waves was first noticed by Harang [1939]. Since then, the effect has been extensively examined experimentally [Watermann, 1987 and references therein]. A mathematical description was formulated by Rishbeth and Garriott [1964] who proposed two mechanisms for the observed Doppler effects. The first involved a polarisation electric field generated in the E region, influencing the F region as an E×B drift. The second mechanism describes bulk motion of the F region due to the MHD wave. Jacobs and Watanabe [1966] improved the model by including changes in the refractive index due to variations in the ionospheric electron distribution. A more complete and rigorous theory has been developed in a series of papers by Poole and Sutcliffe [1987], Poole et al., [1988] and Sutcliffe and Poole [1989, 1990] which we will denote as the SP model. This model identifies three mechanisms that might alter the refractive index and lead to Doppler shifts in HF signals. In this paper we use the latest modifications of the SP model described by Sutcliffe and Poole [1989, 1990]. A number of issues that involve the electron collision terms, complex refractive index and electron velocity terms are discussed.

The HF signal propagation path, s, in the ionosphere is determined by the signal frequency, f, and the medium refractive index, n. Any time variation in the refractive index will also vary the propagation path, giving rise to a Doppler frequency shift, f, of the HF signal. For , the angle between the direction of HF signal energy transport and the wave normal, the frequency shift is given by:

(1)

In this paper, Doppler shifts in HF signals arising from changes in refractive index associated with ULF (1-100 mHz) waves incident from the magnetosphere are discussed. The main result is the extension of previous theoretical work to include effects from a wider range of incident ULF wave modes. The paper also highlights how the combination of HF and ULF data can provide considerable insight into ionospheric processes and the effects of incident ULF energy on HF signal propagation.

2. ULF Waves through the ionosphere

ULF wave energy propagates through the magnetised plasma environment of the Earth's magnetosphere where the ionosphere represents the inner boundary. The energy source for these waves can be traced ultimately to solar activity while closer to Earth, the bow shock and magnetopause are ULF wave generation regions. ULF energy in the cold plasma of the magnetosphere exists in two wave modes known as the fast MHD and shear Alfven wave modes. The presently accepted, although debated, scenario is for fast mode energy, that can propagate across the magnetic field, to mode convert and excite the shear Alfven mode as a resonant oscillation between conjugate ionospheres [Chen and Hasegawa, 1974; Southwood, 1974; Yumoto et al., 1985]. An important property of the shear Alfven mode is that the wave energy is magnetic field line guided even if the propagation vector, k, has a component across the field. The shear Alfven mode energy can reflect at conjugate ionospheres to form field line resonances (FLRs) where the frequency depends on the plasma density and magnetic field topology and hence latitude. This means that in the 1-100 mHz band for a detector at a given latitude, FLRs occupy a small section of the spectrum so that most of the energy is in the fast mode or some mixture of the two modes.

The procedure for determining HF Doppler shifts due to ULF wave energy in the ionosphere requires a knowledge of the ULF wave fields as a function of height from the ground up to the HF reflection point. The SP papers used the model of Hughes [1974] and Hughes and Southwood [1976] to determine the ULF wave fields through the ionosphere. This model is formulated as an initial value problem but has difficulties with numerical swamping [eg. Pitteway, 1965]. The SP papers also limited the application to shear Alfven mode waves. A more recent and flexible formulation of the problem of ULF wave propagation through the ionosphere has been given by Zhang and Cole [1994, 1995]. This recasts the equations as a boundary value problem and allows greater flexibility for tailoring the boundary conditions, particularly the incident ULF wave properties. A variant of this scheme is described and used in this paper.

Assuming the medium changes only in the vertical direction (z), the relevant Maxwell equations are:

(2)
(3)

where:

(4)

The formulation of the problem for any coordinate system is achieved through the terms in the conductivity tensor, . By taking the curl of (2) and substituting in (3), Zhang and Cole [1994] derived equations for the horizontal electric field quantities in terms of two second order differential equations with eight rather involved coefficients. Keeping equations (2) and (3) separate and formulating the problem as a system of first order equations appeared to be a much simpler route. For the permittivity tensor related to the conductivity by:

(5)

and a vertical ambient magnetic field, the relevant equations for the ULF perturbation electric and magnetic fields are:

(6)
(7)
(8)
(9)
(10)
(11)

where 1, 2 and 3 are associated with the Pedersen, Hall and direct conductivities respectively and k0=(μ0e0)½.

2.1 Boundary Conditions

To obtain a solution to the differential equations (6)-(9) we require four boundary conditions. The bottom boundary conditions allow for finite ground conductivity where the ULF wave decays in a uniform, isotropic medium. The bottom boundary conditions are therefore:

(12)
(13)

where alpha represents the skin depth.

The upper boundary assumes an interface between the ionospheric plasma and uniform, ideal MHD conditions. It is then possible to specify various mixtures of fast and shear Alfven mode incident energy. For the shear Alfven mode (in ideal MHD) there is no magnetic perturbation in the direction of the background magnetic field. Therefore, from (2):

(14)

A pure fast mode wave must satisfy Div E=0 [Cross, 1988]. For ideal MHD, the field aligned electric field is zero (Ez=0) so that:

(15)

If the energy is a mixture of these two distinct MHD modes then we can resolve the perturbation electric field of the wave into fast and shear Alfven mode components as:

(16)

where alpha and beta are complex constants that represent the amount each distinct mode contributes to the total incident ULF energy.

To set the upper boundary condition for a given mixture of fast and shear Alfven mode waves and a specified horizontal structure (kx and ky), we choose alpha and beta and one of the perturbed electric field components, say Ex. The other component, Ey, is then fixed by:

(17)

The system can now be solved for the ULF wave fields provided the medium is specified through the conductivity tensor as described in Zang and Cole [1994]. The atmospheric and ionospheric parameters were obtained from the MSIS-86 and IRI-90 models for the day 12 January, 1994.

3. HF Waves in the ionosphere

Considering the properties of the ionosphere and the frequency range of interest (HF), Appleton (1927) developed an equation that describes how the refractive index, n, varies with frequency, omegaHF. For vertical incidence:

(18)

where:

(19)
(20)
(21)

and me is the electron mass, e is the electron charge (negative), Z=nu/omega HF for nu the electron collision frequency, Ne is the electron number density, and BL,T is the longitudinal and transverse components of the magnetic field relative to the HF wave propagation vector. The electron collision term gives a complex refractive index where n=μ+i*xi [Budden, 1985]. For frequencies used for sounding the ionosphere, Z is usually very small and the refractive index is real so that (18) becomes:

(22)

The ± sign identifies the ordinary and extraordinary propagation modes.

4. Doppler shifts in HF signals due to ULF waves

The SP model identifies three mechanisms that contribute to Doppler shifts in HF signals due to ULF wave energy. For a coordinate system where x is positive northward, y is positive eastward and z is positive down, the Doppler velocity from the magnetic mechanism is:

(23)

where the magnetic field is B=BL+BT, the longitudinal and transverse components respectively and the ULF magnetic field varies as B=B0+b0e-i*omega*t. Equation (23) describes the change in refractive index due to magnetic field variations from the ULF wave and shows that the refractive index can change from mechanisms other than electron motion.

The advection mechanism involves the electron density, N and is given as:

(24)

This describes the vertical bulk motion of electrons driven by the electric field of the ULF wave and is essentially the same as the first mechanism described by Rishbeth and Garriott [1964].

The compression mechanism is:

(25)

which describes the Doppler shift due to the compression and rarefaction of the plasma due to the ULF wave energy. A fourth mechanism identified by Poole et al., [1988] is zero since electron production and loss terms are assumed to be equal. The Doppler velocity was calculated from equations (23)-(25) as V*=V1+V2+V3 .

5. Experimental data

The Doppler sounder technique monitors the frequency shift of a HF carrier reflected from ionospheric heights. During the southern hemisphere summer of 1994, the VNG signal from Llandillo (near Sydney) was received at Clarencetown, 30km north of Newcastle using modified commercial Kenwood R-5000 transceivers capable of detecting frequency variations of <0.1 Hz [Menk et al., 1995, Marshall, 1996]. The ULF data were obtained using induction magnetometers [Fraser et al., 1991] comprising two perpendicular sensors oriented geomagnetic north-south and east-west. Data were recorded every 2 seconds using a PC controlled 12 bit A/D card and stored on disk. Time accuracy was typically <10 ms. Extensive descriptions of the data and analysis details are given in Marshall [1996] and Dunlop [1998].

In this paper, we concentrate on modelling the Doppler shifts in HF due to ULF wave energy around a FLR frequency. Typical results for the amplitude ratio and cross phase between data obtained from the magnetometer north-south sensor and the Doppler receiver are shown in Figure 1a and Figure 1b. The FLR frequencies were identified by an independent technique using two magnetometers [e.g. Waters et al., 1991] and correspond with the maxima in the amplitude ratio and dips in cross phase. The SP models only model the shear Alfven ULF mode which is dominant at the FLR frequencies. Figure 1a and Figure 1b shows this mode has a narrow bandwidth at the two harmonics identified by the arrows while most of the spectrum exhibits a different and almost constant cross phase value. The modelling will concentrate on reproducing the observed amplitude and phase across the FLR at ~50mHz.

6. Data from the model

The first step for the modelling is to obtain the variation of the ULF electric and magnetic perturbation fields with height through the ionosphere/atmosphere system. The excitation of FLRs arises from inward propagating fast mode energy from the magnetosphere. At the location (magnetic field shell) where the frequencies match, the FLR grows and narrows in spatial extent [e.g. Tamao, 1966]. Choosing a frequency range of 40 to 60 mHz, both the ULF wave mode mixture and spatial structure change as the frequency passes through the FLR at 50 mHz. A number of measurements have been made of ULF azimuthal wavenumbers [e.g. Ansari and Fraser, 1986]. Based on these, we set a pure fast mode at both 40 and 60 mHz with kx=ky=8x10-8 m-1. At 50 mHz we set 90% shear Alfven mode and increase kx to 1x10-7 m-1. The values for mode mix and kx at the other frequencies were altered using a suitable gaussian fit joining these three points.

The amplitude and phases for the ULF perturbation magnetic fields for a fast mode at 40 mHz and an Alfven mode at 50 mHz are shown in Figure 2a and Figure 2b. It is important to keep in mind that the ULF energy obeys ideal MHD only at the topside boundary. Through the ionosphere, the Maxwell equations are solved and the medium exhibits finite, anisotropic conductivity. For the given horizontal spatial structure, the 40 mHz fast mode horizontal magnetic perturbations, bx and by, pass through the ionosphere without much change in amplitude. The shear Alfven case at 50 mHz shows a decrease in amplitude and large shift in phase in both bx and by around 120 km altitude. This is consistent with "ionospheric shielding" described by Hughes [1974]. This may be explained using the Maxwell equation where no currents flow in the neutral atmosphere so that:

(26)

This means that the horizontal components of the perturbation magnetic field must be either zero or parallel to k[perp] which gives rise to a rotation of the b[perp] components by 90°. This effect is usually shown with by decreasing to zero at ground level [e.g. Hughes and Southwood, 1976]. However, this only occurs when the shear Alfven wave at the top boundary has the magnetic perturbation mostly in the y component i.e. for kx >> ky. The wave polarisation is determined by the MHD conditions for the shear Alfven mode (k.E ne 0) and the horizontal spatial structure. The shielding still occurs for different wave polarisations while the magnitude of by on the ground is determined by the projection of the polarisation ellipse on the coordinate axes.

The Doppler shifts and phases as a function of height at 2km intervals for the shear Alfven mode case are shown in Figure 3. The major contribution to the Doppler shift comes from mechanism V3 which describes changes in the refractive index due to the divergence of the electron velocity at these altitudes. The velocities have been calculated from the perturbation electric fields of the ULF energy and the electron mobilities as described by Sutcliffe and Poole [1989]. The increase in Doppler shift from around 200 km and higher and the relatively constant phase with height is consistent with the results of Sutcliffe and Poole [1989]. A direct comparison is not possible as (i) the ionospheric parameters are different (ii) the ULF wave fields are for incident shear Alfven mode waves but the spatial structure parameters are different (iii) Sutcliffe and Poole [1989] used a main magnetic field inclined at 60° while ours is vertical.

The Doppler shifts associated with incident fast mode ULF energy at 40 mHz are shown in Figure 4. Mechanism V3 is still the largest contributor while the relative contribution of mechanism V2 [bulk vertical motion of electrons] has decreased. The fast and shear Alfven modes show similar contributions to the Doppler velocity from mechanism V1. This describes Doppler velocities due to changes in the magnetic field strength. Like the shear Alfven case, the phases are relatively constant in the F region. However, for HF reflected in the E region, the phases becomes quite sensitive to the medium properties. Finally, the relative contributions from V1 and V2 in Figure 4 are altitude dependent with V1 larger at higher compared with V2 larger at lower altitudes.

7. Discussion

Figure 1a and Figure 1b show experimental values for amplitude ratios and phase differences calculated from data recorded by the Doppler sounder and a magnetometer located beneath the reflection region. The Doppler receiver was tuned to 5 MHz. Assuming near vertical incidence and the ionospheric parameters for 12 January, 1994, this corresponded with a reflection height of around 200 km. The larger Doppler shift for the shear Alfven mode (FLR) compared with the fast mode incident energy is consistent with the ratio results in Figure 1a and Figure 1b. In order to compare the ratio and phase difference results with the model, various ULF mode mixtures at the top boundary were used and the associated Doppler shifts were calculated as described above. The data at 200 km were obtained from the model and the results are shown in Figure 5. The dip in phase difference is reproduced although not to the extent shown in Figure 1a and Figure 1b. The phase differences are sensitive to the choice of kx and ky. There are quite a few measurements of ky at low latitudes and we have used a typical value. However, the choice for kxis more guesswork. The difficulty lies in measuring the phase difference at latitudinally spaced magnetometer sites at frequencies which are resonant. We are investigating ways of obtaining accurate measurements of kx.

Doppler oscillations in HF signals may arise from other processes such as gravity waves, acoustic modes and other phenomena that alter the electron density. A further complication arises from the birefringent medium which gives the ordinary and extraordinary HF modes. These two modes at a given frequency reflect at different altitudes and may interfere (beat) at the receiver. All these effects make the role of the magnetometer vital to identifying Doppler shifts in HF due to ULF wave energy. The study by Marshall [1996] showed that for intervals of data that showed clear correspondence between the Doppler sounder and magnetometer data, the ordinary mode was around 20dB larger compared with the extraordinary mode. Therefore the modeling in this paper has only considered the ordinary HF mode.

The modeling for the ULF signal through the ionosphere assumed a vertical magnetic field. This allowed the definition of various ULF wave mode mixtures at the top side boundary. The difficulty with the vertical field configuration comes from equation (21) for vertical incidence of the HF signal where YT goes to zero. What actually happens is that we encounter a radio window [Budden, 1985]. The condition for wave reflection for HF propagation along the magnetic field only involves the extraordinary mode. If a wave propagates in the ordinary mode along the field through a medium where the refractive index decreases with distance then at the location where reflection would occur (the window point) if the field and HF propagation vector were not parallel an extraordinary wave appears. To avoid this problem, a small BT (2% of BL) was included in the calculations for the Doppler shifts. It would also be desirable to give the main field a realistic inclination to model the data recorded at Australian latitudes. We are presently working on formulating the boundary conditions required to retain the flexibility of being able to specify the ULF wave mode mixture for a non-vertical field.

8. Conclusion

HF signals that propagate via the ionosphere are known to exhibit Doppler oscillations caused by ULF wave energy incident from the magnetosphere. In this paper we have extended previous models of ULF wave propagation through the ionosphere so that the ULF wave mode mixture may be specified. This allows a greater range of the ULF spectrum and the effects on HF Doppler shifts to be investigated. We have found that the vertical bulk motion of the electrons is a minor contributor to the Doppler shifts with the major contribution arising from the divergence of the electron velocities. For fast mode energy, the role of the compression mechanism becomes larger. Future developments of the model include a formulation for non-vertical magnetic field, improving the estimate for kx and comparing with Doppler sounder and HF radar data.

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