### Distribution functions of energetic electrons

As the velocity distribution function of energetic electrons, we often assume a bi-Maxwellian with Gaussian functions of velocity components \(v_{\parallel }\) and \(v_{\perp }\) which go to \(\pm \infty\). When we consider the relativistic energy range, the definition of the velocity distribution as the bi-Maxwellian becomes inconvenient because of the limitation by the speed of light *c*. To avoid the inconvenience, we use a momentum defined by \(u = \gamma v\) in place of a velocity \(v\). The range of the momentum is not limited, while the dimension is the same as the velocity \(v\) because the Lorentz factor \(\gamma = (1 - c^2 / v^2 )^{-1/2}\) is dimensionless. To implement the loss cone distribution of energetic electrons trapped by the Earth’s dipole magnetic field, we assume a subtracted Maxwellian distribution function at the equator given by

$$\begin{aligned} f ( u_{\parallel }, u_{\perp } ) =& \frac{N_0}{ (2 \pi )^{3/2} U_{t \parallel } U_{t \perp }^{2} ( 1 - \rho \beta )} \exp { \left( - \frac{u_{\parallel }^{2}}{2 U_{t \parallel }^{2}} \right) }\\ & \cdot \left[ \exp { \left( -\frac{ u_{\perp }^{2} }{ 2 U_{t \perp }^{2}} \right) } - \rho \exp { \left( -\frac{ u_{\perp }^{2} }{ 2 \beta U_{t \perp }^{2}} \right) } \right], \end{aligned}$$

(49)

where \(\rho\) (\(0 \le \rho \le 1\)) and \(\beta\) (\(0< \beta < 1\)) specify relative height and width of a momentum distribution subtracted from a Maxwellian distribution, respectively. With \(\rho = 1\), a complete loss cone is realized. Summers et al. (2012) showed that a bi-Maxwellian distribution function at the equator can keep the shape of the bi-Maxwellian distribution function at a distance away from the equator. Therefore, we assume a subtracted Maxwellian distribution function as a distance *h* from the equator as

$$\begin{aligned} f_h ( u_{\parallel h}, u_{\perp h} ) =& \frac{N_h }{ (2 \pi )^{3/2} U_{t \parallel h} U_{t \perp h}^{2} ( 1 - \rho \beta _h) } \exp { \left( - \frac{u_{\parallel h}^{2}}{2 U_{t \parallel h}^2} \right) }\\ & \cdot \left[ \exp { \left( -\frac{ u_{\perp h}^{2} }{ 2 U_{t \perp h}^{2}} \right) } - \rho \exp { \left( -\frac{ u_{\perp h}^{2} }{ 2 \beta _h U_{t \perp h}^{2}} \right) } \right]. \end{aligned}$$

(50)

From Liouville’s theorem, we have

$$f_h ( u_{\parallel h}, u_{\perp h}) = f [ u_{\parallel } (u_{\parallel h}, u_{\perp h}, h), u_{\perp }(u_{\parallel h}, u_{\perp h}, h) ].$$

(51)

Preservation of the first adiabatic invariant and energy conservation of an electron give

$$u_{\perp }^{2}= \frac{B_{0}}{B_h} u_{\perp h}^{2},$$

(52)

$$u_{\parallel }^{2}= u_{\parallel h}^{2} + (1 - \frac{B_{0}}{B_h} ) u_{\perp h}^{2}.$$

(53)

Substituting (52) and (53) into (51), we obtain

$$\begin{aligned} &f_h (u_{\parallel h}, u_{\perp h}) = \frac{N_0}{ (2 \pi )^{3/2} U_{t \parallel } U_{t \perp }^{2} ( 1 - \rho \beta ) } \exp { \left( - \frac{u_{\parallel h}^{2}}{2 U_{t \parallel }^{2} } \right) } \\ & \quad \cdot \left\{ \exp { \left[ - \left( \frac{1 - B_{0}/B_h}{2 B_h U_{t \parallel }^{2}} + \frac{B_{0}}{2B_h U_{t \perp }^{2}} \right) u_{\perp h}^{2} \right] } \right. \\ & \left.\quad- \exp { \left[ - \left( \frac{1 - B_{0}/B_h}{2 B_h U_{t \parallel }^{2}} + \frac{B_{0}}{2 \beta B_h U_{t \perp }^{2}} \right) u_{\perp h}^{2} \right] } \right\}. \end{aligned} $$

(54)

Comparing (50) and (54), and assuming \(B_h = B_{0} ( 1 + a h^{2} )\), we have

$$U_{t \parallel h}= U_{t \parallel } ,$$

(55)

$$U_{t \perp h}= W_h U_{t \perp } ,$$

(56)

$$\beta _h= \left[ 1 + \frac{W_h^{2}}{ 1 + a h^{2}} \left( \frac{1}{\beta } -1 \right) \right] ^{-1} ,$$

(57)

$$N_h= W_h^{2} \frac{1 - \rho \beta _h }{1 - \rho \beta } N_{0},$$

(58)

where

$$W_h = \left( 1 + \frac{ah^{2}}{1+a h^{2}} A_{0} \right) ^{-1/2} ,$$

(59)

and

$$A_{0} = \frac{ U_{t \perp }^{2} }{ U_{t \parallel }^{2} } -1.$$

(60)

For the analysis of the nonlinear trapping and associated wave growth at a distance *h*, a simplified distribution function \(f_{\text{t}} (u_{\parallel }, u_{\perp })\) with the following form is assumed.

$$f_{\text{t}} (u_{\parallel }, u_{\perp }) = K \exp {\left( - \frac{u_{\parallel }^{2}}{2 U_{t \parallel h}^2} \right) } \delta ( u_{\perp } - U_{\perp h} ).$$

(61)

Assuming the total density of the energetic electrons is the same with the two distribution functions, we have

$$\int f_{\text{t}} (u_{\parallel }, u_{\perp }) 2 \pi u_{\perp } \text{d} {u_{\perp }} \text{d} u_{\parallel } = \int f (u_{\parallel }, u_{\perp }) 2 \pi u_{\perp } \text{d} {u_{\perp }} \text{d} u_{\parallel } = N_h.$$

(62)

Substituting (61) into (62), we obtain

$$K = \frac{N_h }{ (2 \pi )^{3/2} U_{t \parallel h} U_{\perp h}}.$$

(63)

We also assume the equality of the perpendicular momentum

$$\int u_{\perp } f_{\text{t}} (u_{\parallel }, u_{\perp }) 2 \pi u_{\perp } \text{d} {u_{\perp }} \text{d} u_{\parallel } = \int u_{\perp } f (u_{\parallel }, u_{\perp }) 2 \pi u_{\perp } \text{d} {u_{\perp }} \text{d} u_{\parallel }.$$

(64)

Substituting (50) and (61) into (64), we obtain

$$U_{\perp h} = \left( \frac{\pi }{2} \right) ^{1/2} \frac{1 - \rho \beta _h^{3/2}}{1 - \rho \beta _h} U_{t \perp }.$$

(65)

Using (65), (63), and (58), we can rewrite (61) as

$$f_{\text{t}} (u_{\parallel }, u_{\perp }) = \frac{N_h }{(2 \pi )^{3/2} U_{t \parallel h} U_{\perp h}} \exp { \left( - \frac{u_{\parallel }^2}{2 U_{t \parallel h}} \right) } \delta (u_{\perp } - U_{\perp h} ).$$

(66)

### Resonant currents and wave evolution

Resonant electrons \(v_{\parallel } \sim V_{\text{R}}\) are divided into two groups. One is trapped resonant electrons inside the nonlinear wave potential described above. The other is those outside the nonlinear potential. The shape of the trapping potential in \(\zeta - \theta\) phase space changes as a function of *S* as we have seen above. Trajectories of trapped and untrapped resonant electrons become very different because of the variation of the resonance velocity, by which the trapped electrons are guided, while the untrapped electrons follow adiabatic motion except for the moment crossing the resonance velocity. The difference in the number densities of the trapped and untrapped resonant electrons give rise to a resonant current \(\varvec{J}_{\text{R}}\), which is decomposed into \(J_{\text{B}}\) and \(J_{\text{E}}\) parallel to the wave magnetic field and electric field, respectively. These currents are calculated by

$$J_{\text{B}}= \int _{0}^{\infty } \int _{0}^{2 \pi } \int _{- \infty }^{\infty } [-e v_{\perp } \cos {\zeta } ] f( u_{\parallel }, \zeta , u_{\perp }) u_{\perp } \text{d} u_{\parallel } \text{d} \zeta \text{d} u_{\perp },$$

(67)

$$J_{\text{E}}= \int _{0}^{\infty } \int _{0}^{2 \pi } \int _{- \infty }^{\infty } [e v_{\perp } \sin {\zeta } ] f( u_{\parallel }, \zeta , u_{\perp }) u_{\perp } \text{d} u_{\parallel } \text{d} \zeta \text{d} u_{\perp },$$

(68)

where \(f(u_{\parallel }, \zeta , u_{\perp } )\) is the momentum distribution function of energetic electrons representing the phase space density in the three-dimensional momentum space. The cold electrons supporting the wave propagation is not included in the distribution.

From Maxwell’s equations and the equations of motion of cold and energetic electrons, we can obtain a set of equations describing the evolution of electromagnetic wave field (Omura et al. 2008).

$$\frac{\partial B_{\text{w}}}{\partial t} + V_{\text{g}} \frac{\partial B_{\text{w}}}{\partial h} = - \frac{\mu _{0} V_{\text{g}} }{2} J_{\text{E}},$$

(69)

$$c^{2} k^{2} - \omega ^{2} - \frac{\omega \omega _{\text{pe}}^2}{\Omega _e - \omega } = \mu _{0} c^2 k \frac{J_{\text{B}}}{B_{\text{w}}},$$

(70)

where \(\mu _{0}\) is the magnetic permeability in vacuum. The resonant current \(J_{\text{E}}\) contributes to the variation of the wave amplitude, i.e., wave growth or damping, while \(J_{\text{B}}\) changes the dispersion relation of the wave as a nonlinear term that changes the wave frequency. These resonant currents are initially formed by a triggering wave packet with the frequency \(\omega _{0}\) and the wave number *k* which satisfy (3). Namely, we have

$$c^{2} k^{2} = \omega _{0}^{2} + \frac{\omega _{0} \omega _{\text{pe}}^{2} }{\Omega _{\text{e}} - \omega _{0}}.$$

(71)

Since the spatial structure of the wave phase is imposed by the wave packet of the triggering wave, the wave number *k* or the wavelength does not change in a short time scale, while the rate of the wave phase variation in time or the wave frequency changes in the presence of \(J_{\text{B}}\). Denoting the frequency deviation from \(\omega _{0}\) as \(\delta \omega\), i.e., \(\omega = \omega _{0} + \delta \omega\) and assuming \(\delta \omega \ll \omega _{0}\), we expand (70) around \(\omega _{0}\) to obtain

$$\left\{ 2 \omega _{0} + \frac{\Omega _{\text{e}} \omega _{\text{pe}}^{2}}{(\Omega _{\text{e}} - \omega _{0} )^{2} } \right\} \delta \omega = - {\mu _{0}} c^{2} k \frac{J_{\text{B}}}{B_{\text{w}}}.$$

(72)

Differentiating (71) with respect to \(\omega _{0}\), we have

$$2 c^{2} k \frac{\partial k}{\partial \omega _{0}} = 2\omega _{0} + \frac{\Omega _{\text{e}} \omega _{\text{pe}}^{2} }{(\Omega _{\text{e}} - \omega _{0})^{2}}.$$

(73)

From (72) and (73), we obtain

$$\delta \omega = - \frac{\mu _{0} V_{\text{g}}}{2} \frac{J_{\text{B}}}{B_{\text{w}}}.$$

(74)

As we have analyzed the nonlinear motion of resonant electrons, the magnitude of the perpendicular velocity controls the width of the trapping potential and the period of the trapping motion. For simplicity of the analysis, we integrate the distribution function in \(u_{\perp }\) in the calculation of the resonant currents by assuming the distribution is expressed by the following form:

$$f(u_{\parallel }, \zeta , u_{\perp } ) = g(u_{\parallel }, \zeta ) p(u_{\perp }).$$

(75)

The average perpendicular momentum \(U_{\perp 0}\) is calculated by

$$U_{\perp 0} = \frac{ \int _{0}^{\infty } u_{\perp } p(u_{\perp }) 2 \pi u_{\perp } \text{d} u_{\perp } }{ \int _{0}^{\infty } p(u_{\perp }) 2 \pi u_{\perp } \text{d} u_{\perp }}.$$

(76)

Substituting (75) into (68), and replacing \(p(u_{\perp })\) with a Dirac delta function \(\delta (u_{\perp } - U_{\perp 0})\), we obtain

$$J_{\text{E}} = e \gamma ^{-1} U_{\perp 0}^{2} \int _{0}^{2 \pi } \int _{- \infty }^{\infty } g(u_{\parallel }, \zeta ) \sin {\zeta } \text{d} u_{\parallel } \text{d} \zeta.$$

(77)

To realize a loss cone distribution function, we often assume a subtracted Maxwellian distribution function given by

$$p(u_{\perp }) =\frac{1}{1 - \rho \beta } \left[ \exp { \left( - \frac{u_{\perp }^{2}}{2 U_{t \perp }^{2}} \right) } - \rho \exp { \left( - \frac{u_{\perp }^{2}}{2 \beta U_{t \perp }^{2}} \right) } \right].$$

(78)

The average perpendicular momentum \(U_{\perp 0}\) is obtained from (76) as

$$U_{\perp 0} = \sqrt{ \frac{\pi }{2} } \left( \frac{ 1 - \rho \beta ^{3/2}}{1 - \rho \beta } \right) U_{t \perp }.$$

(79)

Under the assumption that \(u_{\perp } \sim U_{\perp 0}\), formation of the resonance current \(J_{\text{E}}\) and \(J_{\text{B}}\) is described by the structure of \(g(u_{\parallel }, \zeta )\). Since the dynamics of trapped resonant electrons is much different from that of untrapped electrons, there occurs a distinct difference in the distribution of trapped electrons. Representing the initial distribution of trapped electrons by \(g_{t}(u_{\parallel }, \zeta )\), we express the total distribution function of resonant electrons by

$$g(u_{\parallel }, \zeta ) = g_{0}(u_{\parallel }) - Q g_{t} (u_{\parallel }, \zeta ),$$

(80)

where \(g_{0}(u_{\parallel })\) is a unperturbed distribution function, and *Q* is the depth of an electron hole due to depletion of trapped resonant electrons in the velocity phase space. Assuming that \(g_{t} ( u_{\parallel }, \zeta ) = G\) inside the trapping region and that \(g_{t} ( u_{\parallel }, \zeta ) = 0\) outside the trapping region, we rewrite (77) as

$$J_{\text{E}} = - J_{0} \int _{\zeta _{1}}^{\zeta _{2}} [\cos {\zeta _{1}} - \cos {\zeta } + S (\zeta - \zeta _{1}) ]^{1/2} \sin {\zeta } \text{d} \zeta.$$

(81)

Similarly we obtain

$$J_{\text{B}} = J_{0} \int _{\zeta _{1}}^{\zeta _{2}} [\cos {\zeta _{1}} - \cos {\zeta } + S (\zeta - \zeta _{1}) ]^{1/2} \cos {\zeta } \text{d} \zeta,$$

(82)

where

$$J_{0} = (2e)^{3/2} (m_{0} k )^{-1/2} \gamma ^{-1} \chi Q G U_{\perp 0}^{5/2} B_{\text{w}}^{1/2}.$$

(83)

The constants *e* and \(m_{0}\) are the absolute value of charge and the rest mass of an electron, respectively. The expression of \(J_{0}\) is slightly different from that in Omura et al. (2008). This is because we have assumed the distribution function (75) in momentum rather than in velocity. The value *G* in Omura et al. (2008) is in velocity, while the same *G* is used as in momentum in Omura et al. (2009) in deriving the nonlinear growth rate and the optimum and threshold wave amplitudes, which resulted in different powers of the Lorentz factor \(\gamma\) in these expressions. The nonlinear growth rates and the threshold wave amplitude are derived consistently based on the momentum distribution function in the followings.

In evaluating *G*, we assume the simplified momentum distribution function (66) at the magnetic equator as

$$f(u_{\parallel }, u_{\perp }) = \frac{N_{0}}{ (2 \pi )^{3/2} U_{t \parallel } U_{\perp 0}} \exp { \left( - \frac{u_{\parallel }^{2}}{2 U_{t \parallel }^{2}} \right) } \delta ( u_{\perp } - U_{\perp 0} ),$$

(84)

where \(U_{\perp 0} = \gamma V_{\perp 0}\), and \(U_{t \parallel }\) is the thermal momentum in the parallel direction. We have normalized the distribution to the density of hot electrons \(N_{0}\) at the magnetic equator. Integrating *f* over \(u_{\perp }\), we obtain *G* of the unperturbed distribution function \(g_{0}(u_{\parallel })\) at the resonance velocity \(V_{\text{R}}\) as

$$G = \frac{N_{0}}{ (2 \pi )^{3/2} U_{t \parallel } U_{\perp 0}} \exp { \left( - \frac{\gamma ^{2} V_{\text{R}}^{2}}{2 U_{t \parallel }^{2}} \right) }.$$

(85)

To evaluate *G* at a distance *h* from the equator, we replace \(N_{0}\) with \(N_{h}\) given by (58) and use \(U_{\perp h}\) given by (65) in place of \(U_{\perp 0}\), respectively. It should be noted that \(U_{t \perp }\) and \(\beta\) vary as functions of *h* as denoted by \(U_{t \perp h}\) and \(\beta _h\) in (56) and (57), respectively, while \(U_{t \parallel }\) is a constant as indicated by (55).

We evaluate the integrals in (81) and (82) numerically, and we plot the normalized currents \(- J_{\text{E}}/J_{0}\) and \(-J_{\text{B}}/J_{0}\) as functions of *S* for \(-1< S < 0\) in Fig. 3. The maximum value of \(- J_{\text{E}} / J_{0}\) is 0.975 at \(S = -0.413\), which gives \(J_{\text{B}} = - 1.3 J_{0}\). Since the negative \(J_{\text{E}}\) causes wave growth, we can expect the maximum wave growth at \(S = - 0.4\), which can be realized at the equator when we have the frequency increase as indicated in (37). Since the negative \(J_{\text{B}}\) causes a frequency increase, as shown in (74), we can assume an optimum condition for the nonlinear wave growth. Namely, when the frequency increase \(\delta \omega\) takes place because of gradual formation of \(J_{\text{B}}\) over a time \(T_{\text{N}}\), we have a frequency sweep rate on average specified by

$$\frac{\partial \omega }{\partial t} = \frac{\delta \omega }{T_{\text{N}}}.$$

(86)

We call the time \(T_{\text{N}}\) as the nonlinear transition time, and compare it with the nonlinear trapping time \(T_{\text{tr}}\) by introducing a parameter \(\tau = T_{\text{N}} / T_{\text{tr}}\), where the nonlinear trapping time is given by

$$T_{\text{tr}} = \frac{2 \pi }{\omega _{\text{tr}}} = \frac{2 \pi }{\chi } \left( \frac{m_{0} \gamma }{k V_{\perp 0} e B_{\text{w}}} \right) ^{1/2}.$$

(87)

Setting \(\partial \Omega _\text{e} / \partial h =0\) and \(S = - 0.4\) in (37), we have the optimum frequency sweep rate for the nonlinear wave growth at the equator as

$$\frac{\partial \omega }{\partial t} = \frac{0.4 s_{0} \omega }{s_{1}} \Omega _{\text{w}}.$$

(88)

The relation between the frequency sweep rate \(\partial \omega / \partial t\) and the wave amplitude \(\Omega _{\text{w}}\) (88) has been confirmed by the simulation shown in Fig. 5 and observations (Kurita et al. 2012; Foster et al. 2017).

### Role of linear growth rates

To initiate the nonlinear wave growth process, we need a triggering wave with a finite amplitude greater than the threshold amplitude for nonlinear wave growth. The triggering wave can be generated naturally from the thermal fluctuation of electromagnetic field if the linear growth rates of whistler mode waves are positive in the presence of energetic electrons. We assume a subtracted Maxwellian distribution function of the energetic electrons given by (49). With non-relativistic electrons, the parallel and perpendicular components \(U_{t \parallel }\) and \(U_{t \perp }\) of thermal momentum can be regarded as the parallel and perpendicular components of thermal velocity as defined in the dispersion solver KUPDAP (Sugiyama et al. 2015). With a momentum distribution function of subtracted Maxwellian distribution function including the bi-Maxwellian distribution function as a special case of \(\rho = 0\) with a temperature anisotropy \(U_{t \perp } > U_{t \parallel }\), we find the linear growth rate becomes positive over a range of frequency and corresponding wave number in the quasi-parallel direction with its maximum value with the wave number vector purely parallel to the background magnetic field, as shown in Fig. 4. We assumed a typical plasma frequency as \(\omega _{\text{pe}} = 4 \Omega _{\text{ce}}\). With energetic electrons higher than 30 keV, the linear growth rate \(\Gamma _{\text{L}}\) takes positive values only in the range below half the cyclotron frequency. In the presence of the temperature anisotropy, unstable wave modes grow from the thermal fluctuation level. Waves near the maximum linear growth rates grow with the linear growth rates initially. The mode with the maximum linear growth rate forms a coherent wave attaining a largest wave amplitude, and it suppresses the growth of adjacent wave modes. The coherent wave becomes a triggering wave for the nonlinear wave growth process.

### Nonlinear growth rate

The nonlinear wave growth is due to the formation of resonant currents through phase organization of resonant electrons in the presence of nonlinear trapping potential of a coherent triggering wave. The potential is formed by the Lorentz force \(-e {{\varvec{V}}}_{\perp 0} \times {{\varvec{B}}}_{\text{w}}\) acting on electrons with parallel velocities close to the cyclotron resonance velocity \(V_{\text{R}}\). Although a large \(U_{\perp 0}\) makes the trapping potential large, the temperature anisotropy of energetic electrons is not directly required for the nonlinear wave growth. This is very different from the linear growth rate which requires the temperature anisotropy \(T_{\perp } > T_{\parallel }\). Even with a condition of the negative linear growth rate, we can have nonlinear wave growth in the presence of large amplitude wave. The source of energy for the nonlinear growth comes from the perpendicular kinetic energy of resonant electrons as expressed by (68) for \(J_{\text{E}}\). Under a coherent triggering wave, resonant electrons are organized in gyrophase \(\zeta\), resulting in a negative \(J_{\text{E}}\). Because \({{\varvec{J}_{\text{R}}}} \cdot {{\varvec{E}}}_{\text{w}} < 0\), the transfer of energy from the resonant electrons to the wave field takes place.

We define the nonlinear growth rate based on the wave equation (69) describing the evolution of the wave amplitude. In a frame of reference moving with the group velocity \(V_{\text{g}}\), (69) is rewritten as

$$\frac{\text{d} B_{\text{w}}}{\text{d} t} = \Gamma _{\text{N}} B_{\text{w}},$$

(89)

where

$$\Gamma _{\text{N}} = - \frac{\mu _{0} V_{\text{g}}}{2} \frac{J_{\text{E}}}{B_{\text{w}}}.$$

(90)

Assuming an electron hole shown in Fig. 2, we can find the maximum value of \(- J_{\text{E}}\) with \(S = -0.4\). Since \(- J_{\text{E}} / J_{0} = 0.975 \sim 1\), we have from (83)

$$J_{\text{E}, \max} = - (2 e)^{3/2} ( m_{0} k ) ^{-1/2} \gamma ^{-1} \chi Q G U_{\perp 0}^{5/2} B_{\text{w}}^{1/2}.$$

(91)

Substituting \(J_{\text{E}, \max}\) with \(J_{\text{E}}\) in (90), and using (85) for the distribution function (84), we obtain the nonlinear growth rate

$$\Gamma _{\text{N}} = \frac{Q \omega _{\text{ph}}^{2} V_{\text{g}} }{2 \gamma U_{t \parallel }} \left( \frac{\xi }{\omega \Omega _{\text{w}}} \right) ^{1/2} \left( \frac{\chi U_{\perp h}}{\pi c } \right) ^{3/2} \exp { \left( - \frac{ \gamma ^{2} V_{\text{R}}^{2}}{2 U_{t \parallel }^{2}} \right) }.$$

(92)

The parameter \(\omega _{\text{ph}}\) is the plasma frequency of hot electrons given by \(\omega _{\text{ph}}^{2} = N_{h} e^{2} / (m_{0} \epsilon _{0} )\), where \(\epsilon _{0}\) is the vacuum permittivity, and \(N_{h}\) and \(U_{\perp h}\) are functions of *h* evaluated by (55) \(\sim\) (60) and (65). The nonlinear growth rate is evaluated at a distance *h* from the equator by (92). The Lorentz factor \(\gamma\) is calculated for the trapped resonant electrons from (12) with \(v_{\parallel } = V_{\text{R}}\) and \(v_{\perp } = U_{\perp h} / \gamma\).

The nonlinear growth rate is a function of the wave amplitude \(\Omega _{\text{w}} (= e B_{\text{w}} / m_{0})\), while the linear growth rate is a constant for a specific set of parameters regardless of the wave amplitude. In Fig. 6a, both linear and nonlinear growth rates are plotted for three different cold plasma densities as specified by the plasma frequencies \(\omega _{\text{pe}}/\Omega _{\text{e}} = 2, 4, 8\), while the density of energetic electrons is assumed to be constant as \(\omega _{\text{ph}} = 0.1789\) corresponding to \(n_{\text{h}} / n_{\text{c}} = 2 \times 10^{-3}\) in the case of \(\omega _{\text{pe}} = 4 \Omega _{\text{e}}\). The energetic electrons form a subtracted Maxwellian distribution functions given by (49) with \(\beta = 0.3\), \(\rho = 1.0\), \(U_{t \parallel } = 0.25 c\), and \(U_{t \perp } = 0.3 c\). The nonlinear growth rates are calculated for optimum amplitudes in solid lines and for threshold amplitudes in dashed lines, which are plotted in Fig. 6a. Derivations of threshold and optimum amplitudes are given in the following subsections. The linear growth rates plotted in dash-dot lines are much smaller than the nonlinear growth rates. Peaks in both linear and nonlinear growth rates shift to the lower frequency ranges with higher plasma densities.

### Absolute instability

As we have seen in (10), the frequency of the wave packet is constant in the frame of reference moving with the group velocity. The frequency only changes near the equator where we can have large \(-J_{\text{B}}/B_{\text{w}}\) inducing the frequency deviation \(\delta \omega\) given by (72). The wave amplitude \(B_{\text{w}}\) should increase to form a new wave packet. Expressing the derivative \(\text{d} B_{\text{w}} / \text{d} t\) in (89) in terms of temporal and spatial derivatives and normalizing the wave amplitude, we have

$$\frac{\partial \Omega _{\text{w}}}{\partial t} + V_{\text{g}} \frac{\partial \Omega _{\text{w}}}{\partial h} = \Gamma _{\text{N}} \Omega _{\text{w}}.$$

(93)

To have the wave growth locally, i.e., an absolute instability, we need \(\partial \Omega _{\text{w}} / \partial t > 0\). We obtain from (93)

$$\frac{\Gamma _{\text{N}}}{V_{\text{g}}} \Omega _{\text{w}} > \frac{\partial \Omega _{\text{w}}}{\partial h},$$

(94)

where we have assumed that the chorus wave packet propagates in the positive *h* direction, i.e., \(V_{\text{g}} > 0\).

Frequency variation is only possible at the time of localized wave generation before the wave number structure in space is formed over a distance much greater than a spatial scale of the nonlinear resonant current. Once the wave number structure is given it becomes difficult to change the frequency from the value determined by the cold plasma dispersion relation. Therefore, the chorus emission with substantial frequency variation is only possible by the localized absolute instability rather than the convective instability.

### Optimum wave amplitude

As the wave grows at a frequency of the largest linear growth rate, the wave becomes coherent suppressing the growth of other waves around the frequency. Once the wave amplitude exceeds a threshold amplitude for an absolute nonlinear instability, the wave amplitude grows with frequency increasing monotonically at the equator (Omura et al. 2009). The nonlinear wave growth stops near the optimum wave amplitude (Omura and Nunn 2011) and then decreases gradually to the level of the threshold amplitude, resulting in a short subpacket of a chorus wave element.

We evaluate \(J_{\text{B}}\) expressed by (82) with S = − 0.4 for the maximum \(J_{\text{E}}\), which gives \(J_{\text{B}} = -1.3 J_{0}\), as shown in Fig. 3. Namely, we have

$$J_{\text{B}} = - 1.3 (2 e)^{3/2} ( m_{0} k ) ^{-1/2} \gamma ^{-1} \chi Q G U_{\perp 0}^{5/2} B_{\text{w}}^{1/2}.$$

(95)

Substituting (95) into (74), and using (85) and (87), we calculate the frequency sweep rate \(\delta \omega / T_{\text{N}}\) due to formation of \(J_{\text{B}}\) over the nonlinear transition time given by (87).

$$\frac{\delta \omega }{T_{\text{N}}} = \frac{1.3 Q V_{\text{g}} }{4 \tau U_{t \parallel }} \pi ^{-5/2} \left( \frac{\omega _{\text{ph}} U_{\perp 0} \chi }{\gamma c} \right) ^{2} \exp { \left( -\frac{\gamma ^2 V_{\text{R}}^2 }{2 U_{t \parallel }^2 } \right) }.$$

(96)

Equating (86) and (88), we obtain an amplitude at which the optimum condition for nonlinear wave growth is satisfied. Solving for \(B_{\text{w}}\), we obtain the normalized optimum wave amplitude \({\tilde{\Omega }}_{\text{op}}\) as

$$\begin{aligned} {\tilde{\Omega }}_{\text{op}} = & 0.8 \pi ^{-5/2} \frac{|Q| {\tilde{V}}_{\text{p}} {\tilde{V}}_{\text{g}} }{\tau {\tilde{\omega }} } \frac{{\tilde{U}}_{\perp 0}}{{\tilde{U}}_{t \parallel } } {\tilde{\omega }}_{\text{ph}}^2 \\ & \cdot \left( 1 - \frac{{\tilde{V}}_{\text{R}}}{{\tilde{V}}_{\text{g}}} \right) ^{2} \exp { \left( - \frac{\gamma ^{2} {\tilde{V}}_{\text{R}}^2 }{2 {\tilde{U}}_{t \parallel }^{2} } \right) }, \end{aligned}$$

(97)

where \({\tilde{\Omega }}_{\text{op}} = \Omega _{\text{op}} / \Omega _{\text{e0}} = B_{\text{w}} / B_{0}\), \({\tilde{\omega }}_{\text{ph}} = \omega _{\text{ph}} / \Omega _{\text{e0}}\), \({\tilde{U}}_{t \parallel } = U_{t \parallel } / c\), and \({\tilde{V}}_{\text{p}} = V_{\text{p}}/c = \chi \xi\). We can apply the same logic to derive the optimum amplitude for the nonlinear wave growth due to an enhancement of trapped resonant electrons forming a positive \(J_{\text{B}}\) producing a falling tone emission (Omura et al. 2015a). We represent an electron enhancement forming an electron hill by a negative value of \(Q\). Therefore, we use the absolute value of \(Q\) in (97).

Using the optimum wave amplitude, we can rewrite the nonlinear transition time in a normalized form

$$T_{\text{N}} \Omega _{\text{e0}} = 2 \pi \gamma \tau \left( \frac{ \xi }{\chi {\tilde{U}}_{\perp 0} {\tilde{\omega }} {\tilde{\Omega }}_{\text{op}} } \right) ^{1/2}.$$

(98)

Over the period of \(T_{\text{N}}\) a subpacket grows and then damps out over nearly the same period of \(T_{\text{N}}\). The subpacket propagates away from the equator interacting with counter streaming resonant electrons in the downstream of the wave propagation. The gyro-phases of the resonant electrons are modulated by the wave with frequencies higher than that of the original triggering wave. The phase-modulated resonant electrons carry the information of the new wave packet by forming spatial structure with wavenumber *k* of the newly generated wave. The electrons move upstream, and generate a new triggering wave with the higher frequency in the upstream from the equator. The new wave triggers another cycle of the nonlinear wave growth, which is repeated to produce successive subpackets. Through the repetition of the subpacket formation, the wave frequency gradually increases, forming a rising-tone chorus element consisting of a series of subpackets.

An example of simulations generating chorus emission is shown in Fig. 5. This is a simulation by an electron hybrid code where cold electrons are treated as a fluid and hot energetic electrons are treated as particles undergoing cyclotron motion under a dipole magnetic field (Katoh and Omura 2006). Figure 5a shows the frequency spectra of the wave electric field and the theoretical sweep rate, in black solid line, obtained from (88) with instantaneous wave amplitude in the simulation.

### Critical distance

Near the magnetic equator, the inhomogeneity factor *S* is determined by the frequency sweep rate, which is nearly constant through propagation of the wave packet away from the equator. Since the dipole magnetic field is approximated by a parabolic function \(\Omega _{\text{e}} = \Omega _{\text{e}0} ( 1 + a h^{2})\) with \(a = 4.5 / (\text{LR}_{\text{E}})^2\), where \(\text{LR}_{\text{E}}\) represents the distance from the center of the Earth in the equatorial plane. The gradient of the magnetic field increases as a linear function of the distance *h*. We define the critical distance \(h_{\text{c}}\) at which the first term and the second term of *S* given by (37) become equal (Omura et al. 2009). Equating the two terms and using (88), we obtain

$$h_{\text{c}} = \frac{s_{0} \omega \Omega _{\text{w0}} }{ 5 c a s_{2} \Omega _{\text{e}0} }.$$

(99)

The black solid lines in Fig. 5b indicate the critical distances in the simulation by Katoh and Omura (2011). The critical distance varies as a function of the wave amplitude \(\Omega _{\text{w0}}\) at the generation region near the equator. Inside the critical distance, triggering of nonlinear wave growth due to frequency variation is possible, and the region within the critical distance can be regarded as the generation region of subpackets forming chorus emissions. The critical distance is used in identifying the dominant term of the inhomogeneity factor *S* as discussed in the following subsection.

### Convective wave growth

As a chorus sub-packet propagates away from the equator, it undergoes convective wave growth due to formation of an electron hole. At a distance much greater than \(h_{\text{c}}\) in the downstream, we can neglect the first term on the right-hand side of equation (37). Assuming the optimum wave growth condition \(S = -0.4\), we obtain

$$\Omega _{\text{w}} = \frac{c s_{2}}{0.4 s_{0} \omega } \frac{\partial \Omega _{\text{e}}}{\partial h} = \frac{5 c a s_{2} \Omega _{\text{e}0} }{s_{0} \omega } h.$$

(100)

The ideal condition for convective wave growth can be realized if the wave amplitude increases as a linear function of *h*. Assuming the optimum condition is maintained even at a shorter distance *h* ( \(h < h_{\text{c}}\) ), we find the gradient of the wave amplitude

$$\frac{\partial \Omega _{\text{w}}}{\partial h} =\frac{5 c a s_{2} \Omega _{\text{e}0} }{s_{0} \omega }.$$

(101)

The gradient of the wave amplitude is a condition for the optimum convective wave growth. The convective wave growth reaches a saturation when the flux of resonant electrons decreases as the absolute value of resonance velocity \(|V_{\text{R}}|\) increases.

More quantitative evaluation of the wave growth in space may be made by finding a steady state solution of (93). Assuming \(\partial \Omega _{\text{w}} / \partial t\) = 0, we have

$$\frac{\partial \Omega _{\text{w}} }{\partial h} = \frac{\Gamma _{\text{N}}}{V_{\text{g}}} \Omega _{\text{w}},$$

(102)

where we define the convective nonlinear growth rate \(\Gamma _{\text{N}} / V_{\text{g}}\). From (92), we have

$$\frac{\Gamma _{\text{N}}}{V_{\text{g}}} = \frac{Q \omega _{\text{ph}}^{2}}{2 \gamma U_{t \parallel }} \left( \frac{\xi }{\omega \Omega _{\text{w}}} \right) ^{1/2} \left( \frac{\chi U_{\perp h}}{ \pi c } \right) ^{3/2} \exp { \left( - \frac{ \gamma ^{2} V_{\text{R}}^{2}}{2 U_{t \parallel }^{2}} \right) },$$

(103)

where \(\omega_{\text{ph}}^{2} = N_{h} e^2 / (m_{0} \epsilon _{0})\) and other variables are functions of *h*. Since the group velocity decreases in the frequency rage above 0.25 \(\Omega _{\text{e}}\), as shown in Fig. 1, waves at higher frequencies undergo larger convective growth.

### Threshold wave amplitude

Substituting the gradient of the wave amplitude (101) to the condition for the absolute instability (94), we obtain the condition for the absolute instability, i.e., triggering of the nonlinear wave growth process as

$${\tilde{\Omega }}_{\text{w}0} > {\tilde{\Omega }}_{\text{th}},$$

(104)

where

$${\tilde{\Omega }}_{\text{th}} = \frac{100 \pi ^{3} \gamma ^{4} \xi }{{\tilde{\omega }} {\tilde{\omega }}_{\text{ph}}^{4} (\chi{\tilde{U}}_{\perp 0})^{5}} \left( \frac{ {\tilde{a}} s_{2} {\tilde{U}}_{t \parallel } }{Q} \right) ^{2} \exp { \left( \frac{\gamma ^{2} {\tilde{V}}_{\text{R}}^{2}}{ {\tilde{U}}_{t \parallel }^{2} } \right) },$$

(105)

where \(s_{2}\) is given by (40) with \(v_{\perp } / c = {\tilde{U}}_{\perp 0} / \gamma\). The parameters with tilde are normalized values as used in (97). The parameter of the parabolic magnetic field is normalized as \({\tilde{a}} = a c^{2} / \Omega _{\text{e}0}^{2}\). The wave amplitudes and frequencies are normalized by \(\Omega _{\text{e}0}\) as \({\tilde{\Omega }}_{\text{th}} = \Omega _{\text{th}} / \Omega _{\text{e}0}\) and \({\tilde{\omega }} = \omega / \Omega _{\text{e}0}\). The velocity and momentums are normalized by the speed of light *c* as \({\tilde{V}}_{\text{R}} = V_{\text{R}} / c\) and \({\tilde{U}}_{\perp 0} = U_{\perp 0} / c\). In Fig. 6b, we plot the optimum wave amplitude in solid lines and the threshold amplitude in dashed lines for different values of the plasma frequency \(\omega _{\text{pe}} / \Omega _{\text{e}} =\) 2 (blue), 4 (green), and 8 (red) with the same parameters of energetic electrons assumed in the linear and nonlinear growth rate calculation in Fig. 6a. The optimum wave amplitude becomes higher in the lower frequency range with higher plasma frequencies.

When a triggering wave with a constant frequency \(\omega _{0}\) and with an amplitude greater than the threshold amplitude (105) is present at the equator, there occurs an electron hole forming the resonant current \(J_\text{B}\) \((<0)\) causing an frequency increase by \(\delta \omega\) given by (72). The frequency increase makes the electron hole asymmetric with a finite *S*, resulting in the resonant current \(J_{\text{E}}\) \((<0)\) causing wave growth at a fixed position, i.e., an absolute instability. The wave amplitude grows locally with the increased frequency forming a new wave packet detached from the triggering wave. The amplitude reaches the optimum wave amplitude (97). The wave amplitude cannot grow much greater than the optimum value, because the nonlinear growth rate becomes smaller with a larger amplitude. The dynamics of the resonant electrons also causes saturation of the nonlinear wave growth because of entrapping of resonant electrons into the wave potential filling the electron hole. When the wave amplitude is growing locally there occurs efficient entrapping of resonant electrons because of enlargement of the trapping wave potential. The trapped electrons contribute to saturation of the wave amplitude by receiving energy from the wave. After the saturation, the wave amplitude gradually decreases, because the phase organized untrapped electrons move to an opposite phase resulting in a positive \(J_{\text{E}}\). The subpacket with an increased frequency \(\omega _{0} + \delta \omega\) propagates to the downstream undergoing the efficient convective wave growth.

### Chorus equations

The nonlinear growth process as an absolute instability can be described by the following set of equations obtained by normalizing the wave amplitude at the equator \(\Omega _{\text{w}0}\) in (88) and the frequency \(\omega\) in (93) as in (97).

$$\frac{\partial {\tilde{\omega }}}{\partial {\tilde{t}}} = 0.4\frac{s_{0}}{s_{1}} {\tilde{\omega }} {\tilde{\Omega }}_{\text{w}0}$$

(106)

and

$$\frac{\partial {\tilde{\Omega }}_{\text{w}0}}{\partial {\tilde{t}}} = {\tilde{V}}_g \left[ \frac{Q {\tilde{\omega }}_{\text{ph}}^2}{2 \gamma {\tilde{U}}_{t \parallel }} \left( \frac{\xi {\tilde{\Omega }}_{\text{w} 0}}{{\tilde{\omega }}} \right) ^{1/2} \left( \frac{\chi {\tilde{U}}_{\perp 0}}{\pi } \right) ^{3/2} \cdot \exp { \left( - \frac{\gamma ^2 {\tilde{V}}_{\text{R}}^{2}}{2 {\tilde{U}}_{t \parallel }^{2}} \right) } - \frac{5 s_2 {\tilde{a}}}{s_0 {\tilde{\omega }}} \right],$$

(107)

where *s*_{0}, *s*_{1}, and *s*_{2} are calculated from (38), (39), and (40) with \(v_{\perp } /c = {\tilde{U}}_{\perp 0} / \gamma\), respectively. We call these equations as “chorus equations”, and we tried a simple numerical integration of the equations, as presented in Fig. 6 of Omura et al. (2009). We find a monotonic increase of the wave amplitude and frequency, in which the frequency increases rapidly like an exponential function contrary to observed chorus emissions. As it has been reported by Santolik et al. (2014), chorus wave packets consist of many subpackets. An example of a chorus rising-tone emissions is shown in Fig. 7. A rising-tone element of Fig. 7a is expanded in time, and instantaneous amplitudes and frequencies calculated from wave forms of the perpendicular wave magnetic field are plotted in Fig. 7b, c. We simulated the variation the wave amplitude and frequency using the chorus equations, as shown in Fig. 8. We integrated the equations using the parameters used in the calculation of the linear and nonlinear growth rates in Fig. 6. When the wave amplitude reached the optimum wave amplitude, we reversed the sign of the first term of (107) which corresponds to the resonant current \(J_{\text{E}}\). As the wave amplitude damps to a level below \(\Omega _{\text{th}} + 0.3 (\Omega _{\text{w}0} - \Omega _{\text{th}}) (rand)\), where (*rand*) is a uniform random number (\(0 \sim 1\)), we reversed the sign again, and the wave starts to grow. The process of wave growth and damping is repeated until the frequency reaches \(0.65 \Omega _{\text{e}}\). We have introduced some randomness assuming that there exist fluctuations of the electromagnetic fields which are radiated from counter-streaming energetic electrons, which are modulated in their wave phases through interaction with foregoing waves. The result is plotted in Fig. 8. The observed wave amplitudes of subpackets in Fig. 7a are greater than those of the modeled wave amplitudes shown in Fig. 8b. This is probably because of the convective wave growth from the source to the spacecraft.

### Formation of chorus element

In the model of nonlinear wave growth presented above, we assumed nonlinear wave growth takes place at the equator. As we find in Fig. 5b, formation of each subpacket takes place at different places around the equator. A wave packet produced by the triggering of the nonlinear wave growth is relatively short and the frequency increase is only by a small increment given by \(\delta \omega\), i.e., the frequency \(\omega _{1} = \omega _{0} +\delta \omega\). The wave packet propagates away from the triggering point with a wave number \(k_{1}\) undergoing the convective nonlinear wave growth due to the electron hole in the velocity phase space. The wave packet with \(k_{1}\) interacts with counter streaming resonant electrons going around the electron hole as untrapped resonant electrons. The amplitude of the wave packet reaches a substantially large amplitude. The counter streaming untrapped resonant electrons going though the electron hole are organized in phase with the wave number of the wave packet \(k_{1}\). It is noted that the frequency of a wave packet moving with the group velocity does not change in the absence of the resonant current as indicated by (10). The group of electrons in resonance with the wave packet are strongly modulated in gyro-phase with a wave number \(k_{1}\). The phase-modulated electrons move to the upstream region keeping the information of the new wave number \(k_{1}\). These electrons can work as an antenna which can radiate a helical wave with a new frequency \(\omega _{1}\) that satisfies the local dispersion relation with the wave number \(k_{1}\). The helical wave works as a new triggering wave for the next cycle of the nonlinear wave growth. The triggering process is repeated sequentially with slightly different frequencies. A model of the subpacket formation has been proposed based on the chorus equations integrated repeatedly at slightly different positions moving to the upstream region gradually. The model has reproduced the observed feature that the wave frequency drops between subpackets (Hanzelka et al. 2020). The tendency for points of the subpacket formation to shift to the upstream from the equator is often found in the simulation suggesting the sequential triggering as suggested by the model, but this is not always the case in the particle simulations, as presented in Fig. 3 of Hikishima et al. (2009).

In each process of the nonlinear wave growth, the wave amplitude saturates around the optimum wave amplitude. Therefore the spectrum of the chorus emission near the equator follows the profile of the optimum wave amplitude as a function of frequency. The optimum amplitude decreases at higher frequencies. When the optimum amplitude becomes less than the threshold amplitude, the nonlinear wave growth cannot take place. The frequency range of chorus emissions is determined from the relation of the optimum and threshold amplitudes. Since the threshold amplitude also decreases at higher frequency in most cases, the highest frequency of chorus elements is determined by another mechanism such as the cyclotron damping near the electron cyclotron frequency. The formation process suggested above is confirmed by an observation of chorus emissions by THEMIS spacecraft (Kurita et al. 2012). The process is also confirmed by simulation studies, as shown in Fig. 6 of Katoh and Omura (2013), and Fig. 3 of Katoh and Omura (2016).

### Plasmaspheric hiss

The threshold amplitude \(\Omega _{\text{th}}\) for the nonlinear wave growth strongly depends on the gradient of the magnetic field as we find \(a^{2}\) in (105). Katoh and Omura (2013) studied the effect of the gradient of the magnetic field on generation process of chorus and broadband hiss-like emissions. For the small gradient case, the threshold amplitude becomes very low, and there arises a big gap between the optimum and threshold amplitudes allowing the nonlinear wave growth process occurs in wide range of the amplitudes and frequencies. In the simulations by the electron hybrid code (Katoh and Omura 2007), we find broadband hiss-like emissions for the small gradient case, in which we find many rising-tone emissions and some falling-tone emissions with shorter duration periods being generated. With larger gradient, the threshold amplitude becomes greater than the optimum amplitude, and generation of rising-tone emissions are suppressed.

The generation of these emissions with frequency variation is due to a coherent wave that modify the velocity distribution function \(F(v_{\parallel })\) with its wave potential formed at the cyclotron resonance velocity \(V_{\text{R}}\) as we studied in the previous sections. Depending on the numbers of trapped and untrapped resonant electrons, we have either an electron hole or an electron hill as shown in Fig. 9a. When we have depletion of trapped electrons, an electron hole is generated, and we find more resonant electrons in the direction of the wave magnetic field vector \({{\varvec{B}}}_{\text{w}}\), which gives rise to a negative \(J_{\text{B}}\) inducing the frequency increase as indicated by (74). As the frequency increases, the absolute value of the resonance velocity decreases shifting to the higher density part of the velocity distribution function making the hole deeper. Because of the rising-tone frequency, the shape of the electron hole is distorted, as shown in Fig. 9b, which make the perpendicular velocities of the untrapped resonant electrons gathered in the direction of the wave electric field vector \({{\varvec{E}}}_{\text{w}}\), resulting in a negative \(J_{\text{E}}\) for the wave growth. On the other hand, when we have enhancement of trapped electrons, an electron hill is formed, and we find more resonant electrons in the opposite direction of \({{\varvec{B}}}_{\text{w}}\) giving rise to a positive \(J_{\text{B}}\) inducing the frequency decrease. As the frequency decreases, the absolute value of the resonance velocity increases shifting to the lower density part of the distribution function. The electron hill formed by the trapped resonant electrons is more enhanced with a less number of untrapped resonant electrons outside the trapping wave potential. Because of the distortion of the trapping potential due to the frequency decrease, the perpendicular velocities of trapped electrons are in the direction of \({{\varvec{E}}}_{\text{w}}\) forming a negative \(J_{\text{E}}\) for the wave growth. Therefore, the velocity distribution function \(F(v_{\parallel })\) is unstable in the presence of the coherent wave both for rising-tone and falling-tone triggered waves. As we have studied the convective wave growth, rising-tone emissions have a better chance of the wave growth because of the increasing gradient of the magnetic field in the downstream from the equator.

In the plasmasphere, the ratio of the electron plasma frequency \(\omega _{\text{pe}}\) to the electron cyclotron frequency \(\Omega _{\text{e}}\) is much increased to 15–25, while the ratio is 2–5 outside the plasmasphere. The ratio controls the frequency range over which the nonlinear wave growth takes place, as shown in Fig. 10a. The frequency range over which the optimum amplitude (solid line) is greater than the threshold amplitude (dashed line) shifts to the lower frequency range as the plasma frequency increases (\(\omega _{\text{pe}} / \Omega _{\text{e}} = 5, 15, 25\)). We plot frequencies in Hz assuming the electron cyclotron frequency \(f_{\text{ce}} =\) 9 kHz. The large \(\omega _{\text{pe}} / \Omega_{\text{e}}\) makes the frequency ranges of the nonlinear wave growth much lower \((< 0.1 \Omega _{\text{e}})\). As we have seen in the case of hiss-like emissions, we can have formation of many sub-packets at the same time and the same position, when the threshold amplitude is much smaller than the optimum wave amplitude. The nonlinear transition time for formation of the resonant currents, which is calculated from the optimum wave amplitude, becomes shorter as the plasma frequency increases, as shown in Fig. 10b. This suggests the time scales of the generation of hiss waves are shorter than those of chorus emissions.

Initially we may need a seed wave that may grow from the linear instability or external sources such as whistlers or chorus coming from the outside of the plasmasphere. Once the nonlinear wave growth is initiated by a triggering wave, triggering of short wave packet expands over the entire frequency range over which the optimum amplitude is greater than the threshold amplitude. Because of the concurrent triggering, the wave frequency spectra look like noisy incoherent waves. The frequencies of these sub-packets are usually well separated each other so that their wave potentials do not overlap in the velocity phase space. Using the cyclotron resonance condition (21), we can calculate the minimum frequency separation \(\Delta \omega\) corresponding to twice of the trapping velocity \(V_{\text{tr}}\) as

$$\Delta \omega = \left( \frac{\text{d} V_{\text{R}}}{\text{d} \omega } \right) ^{-1} (2 V_{\text{tr}}).$$

(108)

From the resonance condition (21), we have

$$\frac{\text{d} V_{\text{R}}}{\text{d} \omega } = \frac{1}{k} + \frac{1}{k^{2}} \left( \frac{\Omega _{\text{e}}}{\gamma } - \omega \right) V_{\text{g}}^{-1}.$$

(109)

Using \(V_{\text{tr}} = 2 \omega _{\text{tr}} / k\) and (109), we obtain

$$\Delta \omega = 4 \omega _{\text{tr}} \left\{ 1 + \chi ^{2} \left( \frac{\Omega _{\text{e}}}{\gamma \omega } - 1 \right) \left[ \xi ^{2} + \frac{\Omega _{\text{e}}}{2 ( \Omega _{\text{e}} - \omega ) } \right] \right\} ^{-1}.$$

(110)

When the frequencies of two wave packets adjacent in frequency are separated much greater than \(\Delta \omega\), which we call the separability condition, the resonant interaction of each of the waves with energetic electrons is not affected by other waves. The interaction is the same as in the case of a single wave interacting with energetic electrons. Using the optimum wave amplitude, we calculated \(\Delta \omega\) for different \(\omega _{\text{pe}} / \Omega _{\text{e}}\), as plotted in Fig. 10c. The bandwidth in Fig. 3b of Omura et al. (2015a) was calculated with the threshold wave amplitude, showing that the separability condition is well satisfied at the moment of triggering of the nonlinear wave growth.

### Particle simulation of hiss emissions

A particle simulation has been conducted by using a particle code that was studied for chorus simulations (Hikishima et al. 2009) with the plasma frequency \(\omega _{\text{pe}} = 15 \Omega _{\text{e0}}\) (Hikishima et al. 2020). The simulation reproduced the generation process of hiss emissions, as shown in Fig. 11. The simulation was started with electromagnetic thermal noise due to limited number of super particles representing the dense cold electrons and energetic hot electrons with the density ratio \(n_\text{h} / n_\text{c} = 4 \times 10^{-4}\). Hot electrons have a temperature anisotropy given by thermal momentums \(U_{t \parallel } = 0.25 c\) and \(U_{t \perp } = 0.4c\) with \(\beta = 0.3\) for the subtracted-Maxwellian distribution at the equator. The transverse waves in the simulations are separated into forward and backward waves based on their spatial helicity as whistler-mode waves. Figure 11a shows the initial phase of the generation process where waves are gradually excited because of the positive linear growth rate and subsequent nonlinear wave growth process with frequency variations. Small scale structures with rising-tone and falling-tone frequencies grow concurrently at different frequencies. At a position \(h = 100 c \Omega _{\text{e0}}^{-1}\) away from the equator, the nonlinear convective growth makes the wave packet significantly larger than those near the equator. The generation of the wave packets, which we call hiss elements, continues for a long time, as shown in Fig. 11b. The spatial and temporal profile of hiss elements during a relatively short period of \(4.35 \sim 4.60 \times 10^{5} \Omega _{\text{e0}}^{-1}\). We find many discrete hiss elements propagating with different group velocities corresponding to different wave frequencies. We can understand that hiss emission consists of many discrete wave packets (hiss elements), which of them are undergoing the nonlinear wave-particle interaction as we find in the generation process of chorus emissions.

### Coherency and incoherency

Coherency is defined in different ways in different cases. In the case of a wave particle interaction, the interaction is called coherent when particles interact with a wave having a smooth variation of the amplitude and the wave phase. Even with two waves whose resonance velocities are well separated in the velocity phase space as we assumed in the separability condition of hiss elements, the interaction is coherent when the particles interact with one of the waves at a time. Particles interacting with a single wave undergo nonlinear trapping motion when the parallel velocity of a particle is within the range of trapping velocity from the resonance velocity

$$V_{\text{R}} - V_{\text{tr}}< v_{\parallel } < V_{\text{R}} + V_{\text{tr}}.$$

(111)

The velocity range shown above is the width of the trapping wave potential in the velocity phase space, which we call the trapping region. When \(|v_{\parallel } - V_{\text{R}}| \gg V_{\text{tr}}\), a particle hardly feels the effect of the wave, undergoing an adiabatic motion with very small perturbation. When resonance velocities of the two waves are close to each other, the trapping regions of the waves overlap. The particle motion becomes chaotic when it is under the direct influences of two waves with different frequencies. The particle motion becomes incoherent with the wave structures. In this case, we describe the wave-particle interaction as incoherent, when there occurs overlapping of trapping potentials in the velocity phase space. The trapping velocity \(V_{\text{tr}}\) of a whistler mode waves in the parallel propagation depends on both wave amplitude \(B_{\text{w}}\) and perpendicular velocity of a resonant electron \(v_{\perp }\), because the trapping potential is formed by the Lorentz force \(-e {{\varvec{v}}}_{\perp } \times {{\varvec{B}}}_{\text{w}}\). Therefore, the coherency of the cyclotron wave-particle interaction also depends on the particle property \(v_{\perp }\). In the quasi-linear diffusion theory, many waves forming a band of wave spectra are assumed, and the waves are incoherent for resonant electrons because of overlapping of trapping potentials of the waves.