We study three-wave resonant interactions among kinetic-scale oblique sound waves in the low-frequency range below the ion cyclotron frequency. The nonlinear eigenmode equation is derived in the framework of a two-fluid plasma model. Because of dispersive modifications at small wavelengths perpendicular to the background magnetic field, these waves become a decay-type mode. We found two decay channels, one into co-propagating product waves (forward decay), and another into counter-propagating product waves (reverse decay). All wavenumbers in the forward decay are similar and hence this decay is local in wavenumber space. On the contrary, the reverse decay generates waves with wavenumbers that are much larger than in the original pump waves and is therefore intrinsically nonlocal. In general, the reverse decay is significantly faster than the forward one, suggesting a nonlocal spectral transport induced by oblique sound waves. Even with low-amplitude sound waves the nonlinear interaction rate is larger than the collisionless dissipation rate. Possible applications regarding acoustic waves observed in the solar corona, solar wind, and topside ionosphere are briefly discussed.

Kinetic sound waves (KSWs) are an extension of plasma sound waves (or
ion-acoustic waves) in the range of short (kinetic) perpendicular wavelengths

In situ and remote observations show that the sound waves (SWs) are a
widespread phenomenon in the solar corona, solar wind, and Earth's
magnetospheric and ionospheric plasma. For instance, the presence of
significant levels of electrostatic activity in the solar wind, identified as
ion-acoustic waves, has been confirmed by spacecraft observations

Identification of widespread slow (acoustic) modes in different regions of
the solar corona have revived interest in their application to coronal
seismology

There has also been much interest in naturally occurring (as opposed to
artificially stimulated) enhanced ion-acoustic spectra detected by incoherent
scatter radars (ISRs) in the auroral zone and cusp/cleft region

As most space plasmas are non-uniform, KSWs can be created by the phase
mixing of classic SWs, like in solar coronal loops

We consider a homogeneous hydrogen plasma in the ambient magnetic field
directed along

The total number density, velocity, electric field, and magnetic field are
presented as

We would like to note that, strictly speaking, kinetic ion sound waves and their nonlinear interaction can be rigorously described in the framework of kinetic theory. Instead, we use here the two-fluid MHD plasma model, as it gives a far more simple description and is still sufficiently accurate for the low-frequency waves we study.

In this section the nonlinear dispersion relation for kinetic sound waves is
derived. In the electrostatic approximation the electric field of the sound wave
can be presented as

In the ion-sound waves the electron number density follows the Boltzmann
distribution

In the nonlinear coupling of KSWs, the waves must satisfy the resonant
conditions

The restricting condition directly follows from the above equations:

The normalized growth rate

Using the linear responses due to the pump wave and the secondary KSWs, and
retaining the dominant terms in the approximation

From the expression (

Here we analyse characteristics of KSW decays in more detail. To this end, we
present the general expression of the normalized growth rate

The dependence of the normalized decay rate

We considered as well the higher value of the perpendicular wavenumber of the pump
wave, i.e.

The same as in Fig.

Importantly, from Figs.

The spectral transport induced by the nonlinear interaction among kinetic sound waves can be used for the explanation of wave phenomena in many solar and space plasmas. We consider here several possible examples.

First we analyse the nonlinear interaction of KSWs in the solar wind. The
presence of a significant level of electrostatic activity identified as
ion-acoustic waves has been recorded in the solar wind since the 1970s by the Helios
and Voyager spacecrafts

The nonlinear growth of the product waves may be balanced by their
collisionless dissipation. Similarly to the parallel-propagating
SWs, dissipation of oblique KSWs in weakly collisional plasmas is dominated
by the Landau damping. The Landau damping occurs on both electrons and ions,
with the relative importance depending on plasma parameters. As we consider
plasmas with

The threshold amplitude for the pump wave to excite waves 1 and 2 can be
obtained from the condition,

In Fig.

Here we consider decay instabilities of low-frequency kinetic sound waves
with

The dependence of threshold amplitude of pump wave on the frequency
of pump wave for the maximum value of growth rate

On the other hand, the low-frequency approximation that we used is directly
applicable to the slow-mode waves observed in the solar corona

The nonlinear interactions studied in our paper can also play a role in the
topside Earth's ionosphere. The Naturally Enhanced Ion Acoustic Lines (NEIALs)
are often observed at high latitudes in the incoherent scatter radar data.
NEIALs appear as ion-acoustic fluctuations enhanced by two or three orders
above the thermal level. The strength of one or both of the up- and down-shifted
ion lines is increased

A feasible explanation for two simultaneously observed spectral lines is
provided by the nonlinear KSW interactions. If the plasma instabilities
generate the ion-sound waves propagating in one direction, the reverse decay
of these waves will generate the waves propagating in the opposite direction.
As it is seen from Fig.

The nonlinear interaction of low-frequency kinetic ion-sound waves, with

We studied the three-wave resonant interaction among kinetic sound waves. The
nonlinear coupling equation describing both linear and nonlinear properties
of KSWs is derived in the framework of a two-fluid plasma model. Because of the
dispersive modification at small perpendicular wavelengths approaching the
ion gyroradius, KSW become a decay-type mode. We derived the nonlinear decay
rate of low-frequency KSWs with

There are two possible decay channels for KSWs: the forward decay into
two co-propagating product KSWs (

Contrary to

By accounting for the nonlinear terms

Both decay channels depend on the perpendicular wavenumber of the pump
wave. The reverse decay rate for

In the forward decay the perpendicular wavenumbers of product KSWs are
similar to the pump wavenumber,

The estimated thresholds of the KSW decays are very low in the solar wind and in the topside ionospheric conditions, which suggests their importance in these regions. The same may concern KSWs and their nonlinear dynamics in the solar corona and in laboratory plasmas.

The nonlinear interaction rate does not depend on the parallel wavenumbers, which reflects the dominant role of the perpendicular nonlinear dynamics where the parallel scales evolve kinematically.

These properties make KSW's nonlinear dynamics interesting in the context of acoustic-like wave activity observed in the solar corona, solar wind, and terrestrial magnetosphere.

This research was supported by the Belgian Federal Science Policy Office (via Solar-Terrestrial Centre of Excellence project “Fundamental science” and via IAP Programme project P7/08 CHARM) and by the European Commission (via FP7 Program project 313038 STORM). Topical Editor V. Fedun thanks two anonymous referees for their help in evaluating this paper.