Effects of cold electron number density variation on whistler-mode wave growth

We examine how the growth of magnetospheric whistler-mode waves depends on the cold (background) electron number density N0. The analysis is carried out by varying the cold-plasma parameter a = (electron gyrofrequency )2/(electron plasma frequency )2 which is proportional to 1/N0. For given values of the thermal anisotropy AT and the ratioNh/N0, whereNh is the hot (energetic) electron number density, we find that, as N0 decreases, the maximum values of the linear and nonlinear growth rates decrease and the threshold wave amplitude for nonlinear growth increases. Generally, as N0 decreases, the region of (Nh/N0,AT)-parameter space in which nonlinear wave growth can occur becomes more limited; that is, as N0 decreases, the parameter region permitting nonlinear wave growth shifts to the top right of (Nh/N0,AT) space characterized by largerNh/N0 values and larger AT values. The results have implications for choosing input parameters for full-scale particle simulations and also in the analysis of whistler-mode chorus data.

Chorus waves typically comprise discrete "rising-tone" elements with a time-increasing frequency.The initial stage of the generation process of magnetospheric whistler-mode chorus is considered to be linear, that is, classical linear wave growth produced by an injection at the magnetic equator of an anisotropic distribution of energetic (hot) electrons.The generation of the rising-tone elements, however, requires fully nonlinear theory.Considerable progress has been made recently in developing a nonlinear cyclotron theory of the generation of whistler-mode chorus by Omura et al. (2008Omura et al. ( , 2009Omura et al. ( , 2012)), with further extensions by Summers et al. (2012aSummers et al. ( , 2013)).Omura et al. (2008Omura et al. ( , 2009) ) describe the detailed nonlinear dynamics of cyclotron-resonant electrons and postulate the formation of electromagnetic electron "holes" that result in resonant currents generating rising-tone emissions.Various aspects of this nonlinear theory have been verified by sophisticated full-scale simulations of the generation and growth of whistler-mode chorus elements by Omura et al. (2008Omura et al. ( , 2009)), Hikishima et al. (2009); Hikishima et al. (2010) and Katoh and Omura (2011).Summers et al. (2012b) Published by Copernicus Publications on behalf of the European Geosciences Union.

R. Tang et al.: Effects of cold electron number density variation on whistler-mode wave growth
analyze the nonlinear spatiotemporal evolution of whistlermode chorus waves propagating along a magnetic field line from their equatorial source.The wave profiles exhibit convective growth, due to nonlinear wave trapping, followed by saturation, due partly to a decreasing resonant current with latitude.
The present study concentrates on the conditions required for effective nonlinear growth of whistler-mode waves at the magnetic equator, and in particular extends the work of Summers et al. (2012aSummers et al. ( , 2013)).These authors treated the growth of magnetospheric whistler-mode waves in terms of a linear growth phase followed by a nonlinear growth phase.They then constructed complete time profiles of the wave amplitude by smoothly matching the solutions in the linear and nonlinear regimes.It was found that this matching procedure could only take place over a restricted "matching region" in (N h /N 0 , A T ) space where A T is the electron thermal anisotropy, N h is the hot (energetic) electron number density, and N 0 is the cold (background) electron number density.In a complementary analysis, using a condition based on the maximum linear growth rate, Summers et al. (2013) determined a boundary in (N h /N 0 , A T ) space separating a region in which only linear whistler-mode wave growth can occur from a region in which whistler-mode waves can achieve fully nonlinear growth.The whole analysis of Summers et al. (2012aSummers et al. ( , 2013) ) was carried out at a fixed L shell and a fixed cold electron number density N 0 .That is, in terms of the cold-plasma parameter a = | e | 2 /ω 2 pe where | e | is the electron gyrofrequency and ω pe is the electron plasma frequency, Summers et al. (2012aSummers et al. ( , 2013) ) assumed that a is constant (the value a = 1/16 was assumed to characterize the region outside the plasmasphere at or near L = 4).The particular objective of the present paper is to investigate how the conditions for nonlinear growth of whistler-mode waves in the magnetosphere, as determined by Summers et al. (2012aSummers et al. ( , 2013)), depend on the parameter a.Thus, herein we adopt a selection of a values to represent the differing conditions in the inner magnetosphere experienced during geomagnetic disturbances including storms and substorms.
The plan of our paper is as follows.In Sect.2, for a chosen loss cone distribution, we examine the (relativistic) linear growth rate for whistler-mode waves and its dependence on the cold-plasma parameter a and the thermal anisotropy A T .In Sect. 3 we give a brief account of the nonlinear cyclotron resonance theory for whistler-mode waves required in the present study.This involves introducing the "total" nonlinear growth rate N (t) and the local nonlinear growth rate γ N (t).As well, we describe the "chorus equations" (Eqs.16 and 17) used to model the nonlinear growth of a chorus element, and derive the threshold wave amplitude Bth required for nonlinear growth.In Sect.4, we show how to construct the matching region in (N h /N 0 , A T ) space in which smooth matching of linear and nonlinear solutions is possible, and we also construct the (generally different) parameter region  (with boundary 31) in which fully nonlinear wave growth can occur.Finally, in Sect. 5 we state our conclusions.

Relativistic linear growth rate
We assume that field-aligned electromagnetic R-mode waves are generated by a hot anisotropic electron population in the presence of a dominant cold electron population.We choose the AAK (Ashour-Abdalla and Kennel, 1978) loss cone particle distribution as the hot electron distribution function, i.e., where p = γ m e v and p ⊥ = γ m e v ⊥ are the components of relativistic momentum p = γ m e v, m e is the electron rest mass, v is the electron velocity with components v and v ⊥ , parallel and perpendicular, respectively, to the ambient magnetic field, and and c is the speed of light; θ and θ ⊥ are the thermal momenta of the energetic electrons parallel and perpendicular to the background magnetic field; and N h is the hot electron number density.We require that the distribution satisfies f (p , p ⊥ )d 3 p = N h , with d 3 p = 2πp ⊥ dp ⊥ dp .The parameter β (where 0 < β < 1) is a measure of the angular size of the loss cone.When β → 1, distribution in Eq. (1) reduces to a particular form of the Dory-Guest-Harris loss cone distribution (Dory et al., 1965).When β → 0, distribution in Eq. ( 1) reduces to a bi-Maxwellian distribution.The thermal anisotropy, defined by A T = T ⊥ /T − 1, where T and T ⊥ are the parallel and perpendicular temperatures, is given by A T = (1+β)θ 2 ⊥ /θ 2 − 1 for distribution in Eq. (1).We write the cold-plasma dispersion relation for electromagnetic R-mode waves propagating parallel to an assumed uniform magnetic field in the form where and a is the cold-plasma parameter defined by where ω is the (real) wave frequency, k is the (real) wave number, | e | = eB 0 /(m e c) is the electron gyrofrequency, ω pe = (4π N 0 e 2 /m e ) 1/2 is the plasma frequency, −e is the electron charge, N 0 is the cold electron number density, and B 0 is the magnitude of the zeroth-order magnetic field.In Fig. 1, we show typical contours of constant phasespace density for the AAK loss cone distribution in Eq. ( 1) In Fig. 2, we plot the relativistic linear growth rates for the AAK loss cone distribution for θ /(m e c) = 0.49, A T = 1.5, the indicated values of the cold-plasma parameter a, and the various values of the electron number density ratio N h /N 0 .We find that the maximum linear growth rate increases as the number density ratio N h /N 0 increases as expected, but it decreases as the cold-plasma parameter a increases.As a increases (or N 0 decreases), the bandwidth for wave growth (ω i > 0) slightly decreases, and the frequency at which the maximum growth rate occurs increases.Thus, the variation of a does not significantly change the frequency band for wave growth, but does affect the frequency at which the wave growth maximizes.
For the rest of the calculations and figures in this paper, we have fixed θ /(m e c) = 0.49 as used in previous work (e.g., Summers et al., 2013).
In Fig. 3, we present the contours of constant wave frequency ωm = ω m /| e | at which the relativistic linear growth rate maximizes, as a function of the cold-plasma parameter a and the thermal anisotropy A T .For smaller values of a, say a < 0.1, ωm is only moderately dependent on the values of a and A T .
In Fig. 4, we show two-dimensional plots of the maximum relativistic linear growth rate max( ωi ) as a function of the cold-plasma parameter a and the thermal anisotropy A T , for N h /N 0 = 10 −2 , 10 −3 , and 10 −4 .The white regions are regions of parameter space in which max( ωi ) values are outside the typical practical range.The panels indicate that, as N h /N 0 decreases, useful (sufficiently large) values of max( ωi ) can only be obtained for larger values of A T and smaller values of a. Figures 3 and 4 in combination are useful as an aid for selecting input parameters in future computer simulations.3 Nonlinear wave growth Omura et al. (2008Omura et al. ( , 2009Omura et al. ( , 2012) ) have developed a nonlinear cyclotron resonance theory which describes the generation and growth of whistler-mode chorus waves.Summers et al. (2012aSummers et al. ( , 2013) ) generalized the so-called "chorus equations" (Omura et al., 2009) to an arbitrary energetic electron distribution and calculated conditions for sustained nonlinear growth.Following their work, we express the nonlinear growth rate N for field-aligned whistler-mode (R-mode) waves of frequency ω(t) and wave magnetic field amplitude B w (t) by the equation and where V g (t) is the wave group speed, γR (t) is the resonant Lorentz factor, ṼR (t) is the resonant parallel particle velocity, V ⊥0 (= constant) is the average perpendicular particle velocity, a is the cold-plasma parameter defined by Eq. ( 4), and Q is the dimensionless factor that represents the depth of the electromagnetic electron hole within which nonlinear particle trapping takes place.The quantity G is a measure of the average value of the hot electron distribution F T trapped by the wave.We express the trapped distribution F T as the electron ring distribution, with where δ is the Dirac delta function, and is a function of parallel particle momentum p only; G is given by where pR = γR m e ṼR .Following Omura et al. (2008Omura et al. ( , 2009) ) and Summers et al. (2012aSummers et al. ( , 2013)), we approximate the AAK distribution in Eq. ( 1) by the ring distribution where Hence from Eqs. ( 10)-( 12) it follows that The normalized wave amplitude Bw = B w (t)/B 0 and normalized frequency ω = ω(t)/| e | are found to satisfy the "chorus equations" (Omura et al., 2009;Summers et al., 2012aSummers et al., , 2013) ) given by with t = | e |t, and s where V P is the wave phase speed.Equations ( 16) and ( 17) hold at the magnetic equator of an assumed dipole field, and in general for wave frequencies in the range 0.1 ≤ ω ≤ 0.5.The parameter ã in Eq. ( 21) arises from a Taylor expansion of the Earth's dipole magnetic field about the equator; L denotes magnetic shell and R E is the Earth's radius.For self-sustaining emissions to exist, the wave amplitude must satisfy a threshold condition.By setting ∂ Bw /∂ t = 0 in Eq. ( 16) and then solving for B w , we determine that the normalized threshold wave amplitude ( Bth = B th /B 0 ) is given by In Fig. 5, we present two-dimensional plots of the normalized threshold wave amplitude Bth as a function of the cold-plasma parameter a and the thermal anisotropy A T , for N h /N 0 = 10 −2 , 10 −3 , and 10 −4 .The white regions represent regions of parameter space corresponding to unreasonably large values of Bth .It is clear that Bth can be strongly dependent on a, subject to the particular values of N h /N 0 and A T considered.When N h /N 0 decreases, reasonable values of Bth are obtained in the region typified by smaller a values and larger A T values.
In general, Eqs. ( 16) and ( 17) are solved as an initial-value problem for Bw (t) and ω(t), subject to the initial condition ω(0) = ω0 and Bw (0) = B w (0)/B 0 , where ω0 is the initial wave frequency and Bw (0) is chosen to be greater than the value of Bth calculated from Eq. ( 22) corresponding to ω0 .
The nonlinear growth rate N specified by Eqs. ( 5) and ( 6) can be regarded as a "total" growth rate since d/dt ≡ ∂/∂t + V g ∂/∂h, where h is the distance measured along the magnetic field line from the magnetic equator.Following Summers et al. (2012aSummers et al. ( , 2013)), we define a "temporal" nonlinear growth rate γ N given by where Since the temporal linear growth rate ω i satisfies then Eqs. ( 25) and ( 23) are analogous and respectively describe wave growth during the linear and nonlinear phases.We can construct complete time profiles for B w by a suitable matching of the linear and nonlinear solutions at the interface of the linear and nonlinear regimes.We describe this procedure in the following section.

Linear-nonlinear matching region
In order to construct a complete time profile of B w , we assume that linear wave growth smoothly merges into nonlinear wave growth at a particular wave amplitude Bm = B m /B 0 which we call the matching wave amplitude.We assume that the smooth matching occurs when where γ N is the local nonlinear growth rate given by Eq. ( 24) and (ω i ) max is the maximum value of the relativistic linear growth rate.It then follows from Eqs. ( 6), ( 24) and ( 26) that the matching wave amplitude is given by where X = Bm , and The parameters k 1 and k 2 are evaluated at ω = ωm = ω m /| e | where ω m is the frequency at which the linear growth rate maximizes; ωm , k 1 , k 2 and k 3 can each be regarded as a function of A T , θ /(m e c), and the cold-plasma parameter a.The detailed algebraic expressions for k 1 , k 2 and k 3 are given in Appendix A. From Eq. ( 27) we find where we select the lower (−) root sign.From Eq. ( 29) we see that the matching process is 1 , and hence the linear-nonlinear matching region in the (N h /N 0 , A T ) plane is straightforward since the parameters k 1 , k 2 and k 3 are functions of A T only, for fixed values of a and θ /(m e c).
In Fig 6, top panels, we show the matching boundary N h /N 0 = 4k 2 k 3 /k 2 1 (black curve) and the (blue) region over which linear-nonlinear matching is possible, for the specified values of a. Corresponding to each case in the top panel, we show in the bottom panel the dependence on the thermal anisotropy A T of the wave frequency ωm = ω m /| e | at which the relativistic linear growth rate maximizes.When a increases (or N 0 decreases) the matching region is seen to shift toward the top-right region of (N h /N 0 , A T ) space corresponding to larger N h /N 0 values and larger A T values.Thus for a lower cold electron number density, more energetic particles are needed to enable the linear-nonlinear matching process.
Motivated by a series of electron hybrid simulations of whistler-mode chorus by Katoh and Omura (2011), Summers et al. (2013) proposed that the boundary separating linear wave growth and nonlinear wave growth can be given by the relation where M is a fixed bound.The criterion in Eq. ( 30) implies that linear wave growth only occurs when initial linear growth rates satisfy max(ω i /| e |) < M, while nonlinear wave growth occurs when linear growth rates satisfy max(ω i /| e |) > M. Summers et al. (2013) showed that Eq. ( 30) can be written as where the parameters c 1 and c 2 are relatively weak functions of the thermal anisotropy A T .For given values of a and θ /(m e c), we can readily construct the boundary curve in Eq. ( 31) in the (N h /N 0 , A T ) plane and hence determine the region ] in which nonlinear chorus occurs.
In Fig. 7, for the indicated values of a, we plot Eq. ( 31) with M = 10 −3 as a red curve, which acts as the boundary of the pink region representing max(ω i /| e |) > 10 −3 in which nonlinear wave growth takes place.In each panel we superimpose on top of the pink region the relevant blue (linearnonlinear matching) region depicted in Fig. 6.Evidently the linear-nonlinear matching procedure does not in general apply throughout the nonlinear chorus growth region.As in Fig. 6, when a increases, the nonlinear chorus growth region    31) with M = 10 −3 .In the pink region above the red boundary, nonlinear wave growth occurs, while in the white region only linear growth occurs.Superimposed on the pink region of nonlinear growth are the blue linear-nonlinear matching regions shown in Fig. 6. in (N h /N 0 , A T ) space shifts to the top-right region of larger N h /N 0 values and larger A T values.
In Fig. 8, we present four sets of three panels, for the respective parameter values a = 0.04, 0.06, 0.08, and 0.1, showing linear-nonlinear matching wave amplitude profiles.Also shown are corresponding profiles of the wave frequency ω(t) during the nonlinear phase and the local nonlinear wave growth rate γ N (t).Values of the electron number density ratio N h /N 0 are selected in the range 7.5 × 10 −4 to 2.5 × 10 −3 .Construction of the matching profiles involves matching nonlinear solutions of the chorus Eqs. ( 16)-( 17) to appropriate linear solutions of Eq. ( 25).In the solution of Eqs. ( 16)-( 17), we apply the initial conditions ω(0) = ω0 = ωm , Bw (0) = Bm , where ωm is the wave frequency at which the linear growth rate maximizes and Bm is the matching wave amplitude given by Eq. ( 29).We see from Fig. 8    sufficiently large values of N h /N 0 .As well, as a increases, the maximum values of the nonlinear growth rate γ N decline, and the profiles of γ N become flatter.The results in Fig. 8 reinforce the conclusion from Fig. 7 that the conditions for nonlinear wave growth generally become less favorable as the parameter a increases.

Conclusions
We have considered the linear and nonlinear growth of magnetospheric whistler-mode waves as the parameter a = | e | 2 /ω 2 pe varies.Since a ∝ B 2 0 /N 0 , where N 0 is the cold electron number density and B 0 is the background magnetic field strength, then at a given L shell or fixed magnetic field, an increasing (decreasing) value of a implies a decreasing (increasing) value of the cold electron number density.Our general conclusions are as follows: 1.As the parameter a increases, the maximum linear growth rate decreases, for given values of the thermal anisotropy A T and the ratio N h /N 0 , where N h is the hot (energetic) electron number density.
2. As a increases, the threshold wave amplitude Bth (given by Eq. 22) for nonlinear wave growth increases, for given values of A T and N h /N 0 .
3. As a increases, the matching wave amplitude Bm (given by Eq. 29) occurring at the transition from linear to nonlinear growth increases, for given values of A T and N h /N 0 .
4. As a increases, the maximum value of the nonlinear growth rate γ N (given by Eq. 24) decreases, for given values of A T and N h /N 0 .
5. The region of (N h /N 0 , A T ) space that is favorable to nonlinear whistler-mode wave growth becomes more restricted as the parameter a increases.Specifically, as a increases, the region permitting nonlinear wave growth shifts toward the top right of (N h /N 0 , A T ) space characterized by larger N h /N 0 values and larger A T values.
6.This study sharpens the realization that the nonlinear growth of whistler-mode waves is only possible over a relatively restricted region of the three-dimensional (N h /N 0 , A T , a)-parameter space.The results reported here serve as an aid in choosing input parameters for computationally intensive particle simulations, and also as a practical tool to assist in the analysis of experimental whistler-mode wave data.
Contours of constant wave frequency ωm = ωm/|Ωe| at which the relativistic linear growth rate maxizes, as a function of the cold-plasma parameter a = |Ωe| 2 /ω 2 pe and thermal anisotropy AT .

Figure 3 .
Figure 3. Contours of constant wave frequency ωm = ω m /| e | at which the relativistic linear growth rate maximizes, as a function of the cold-plasma parameter a = | e | 2 /ω 2 pe and thermal anisotropy A T .

Fig. 4 .Figure 4 .
Fig. 4. Two-dimensional plots of the maximum relativistic linear growth rate max(ωi) as a function of the coldplasma parameter a = |Ωe| 2 /ω 2 pe and thermal anisotropy AT , for the indicated values of the electron number density ratio Nh/N0.

Fig. 6 .Figure 5 .
Fig. 6.(a) Top panels show the matching boundary Nh/N0 = 4k2k3/k 2 1 (black curve) and the (blue) region over which linear-nonlinear matching is possible, for the specified values of a = |Ωe| 2 /ω 2 pe .(b) Bottom panels show the dependence on the thermal anisotropy AT of the wave frequency ωm = ωm/|Ωe| at which the relativistic linear growth rate maximizes.

Fig. 6 .Figure 6 .
Fig. 6.(a) Top panels show the matching boundary N h /N 0 = 4k 2 k 3 /k 2 1 (black curve) and the (blue) region over which linear-nonlinear matching is possible, for the specified values of a = |Ω e | 2 /ω 2 pe .(b) Bottom panels show the dependence on the thermal anisotropy A T of the wave frequency ωm = ω m /|Ω e | at which the relativistic linear growth rate maximizes.

Fig. 7 .
Fig. 7. Variation of the parameter space for nonlinear whistler-mode wave growth with repsect to the coldplasma parameter a = |Ωe| 2 /ω 2 pe .The red boundary corresponds to condition (31) with M = 10 −3 .In the pink region above the red boundary, nonlinear wave growth occurs, while in the white region only linear growth occurs.Superimposed on the pink region of nonlinear growth are the blue linear-nonlinear matching regions shown in Figure6.

Figure 7 .
Figure 7. Variation of the parameter space for nonlinear whistler-mode wave growth with respect to the cold-plasma parameter a = | e | 2 /ω 2 pe .The red boundary corresponds to Eq. (31) with M = 10 −3 .In the pink region above the red boundary, nonlinear wave growth occurs, while in the white region only linear growth occurs.Superimposed on the pink region of nonlinear growth are the blue linear-nonlinear matching regions shown in Fig.6.