We change the gradient of the background magnetic field in both theoretical analyses and simulations to check the correlation between them. From the nonlinear wave growth theory (Omura 2021), the ideal nonlinear process happens at the inhomogeneity factor \(S \approx -0.4\), as shown in (3),

$$\begin{aligned} S= & {} -\frac{1}{s_0\omega \Omega _w}\left( s_1\frac{\partial \omega }{\partial t}+cs_2\frac{\partial \Omega _e}{\partial h}\right) \approx -0.4, \end{aligned}$$

(3)

where \(s_0\), \(s_1\), and \(s_2\) are three parameters defined in (38) to (40) in Omura (2021), and \(\omega\), \(\Omega _w\), and \(\Omega _e\) are the frequency, amplitude of waves, and the gyrofrequency of electrons, respectively. From (3), the whole nonlinear wave growth process can be divided into two stages. The first stage is due to the frequency variation \(\partial \omega / \partial t\), and the second stage is due to the gradient \(\partial \Omega _e / \partial h\). Specifically, the first stage is dominant near the equator where the gradient is 0, and wave packets are formed and growing at this stage. After the formation of wave packets, their frequencies will not change through the propagation. When the wave packets propagate from the equator to higher latitudes, the second stage plays a more important role in keeping \(S \approx -0.4\). The second stage is also called the convective growth stage.

Assuming the gradient term to be 0 and letting \(S=-0.4\), we derive the optimum amplitude that represents an amplitude that gives the maximum nonlinear wave growth condition. In the first stage of the nonlinear process, waves cannot grow much greater than the optimum amplitude. Similarly, if the frequency variation term is omitted and letting \(S=-0.4\), we obtain the threshold amplitude, meaning the minimum amplitude that a wave packet can grow locally as an absolute instability and enter an ideal convective growth stage. We refer to the equations of the normalized optimum and threshold amplitudes in Omura (2021) and write them in (4) and (5):

$${\tilde{\Omega }}_{op} = 0.8\pi ^{-5/2}\frac{|Q|{\tilde{V}}_p{\tilde{V}}_g}{\tau {\tilde{\omega }}}\frac{{\tilde{U}}_{\perp 0}}{{\tilde{U}}_{t\parallel }}{\tilde{\omega }}_{ph}^2 \left( 1 - \frac{{\tilde{V}}_R}{{\tilde{V}}_g}\right) ^2 \text {exp}\left( -\frac{\gamma ^2{\tilde{V}}_R^2}{2{\tilde{U}}_{t\parallel }^2}\right) ,$$

(4)

$${\tilde{\Omega }}_{th} = \frac{100\pi ^3\gamma ^4\xi }{{\tilde{\omega }}{\tilde{\omega }}_{ph}^4(\chi {\tilde{U}}_{\perp 0})^5}\left( \frac{{\tilde{a}}s_2{\tilde{U}}_{t\parallel }}{Q}\right) ^2 \text {exp}\left( \frac{\gamma ^2{\tilde{V}}_R^2}{{\tilde{U}}_{t\parallel }^2}\right) .$$

(5)

Here, \({\tilde{\Omega }}_{op}=\Omega _{op}/\Omega _{e0}\), \({\tilde{\Omega }}_{th}=\Omega _{th}/\Omega _{e0}\), \({\tilde{V}}_p=V_p/c\), \({\tilde{V}}_g=V_g/c\), \({\tilde{U}}_{\perp 0}=U_{\perp 0}/c\), \({\tilde{U}}_{t\parallel }=U_{t\parallel }/c\), \({{\tilde{\omega }}} = \omega / \Omega _{e0}\), \({{\tilde{\omega }}}_{ph} = \omega _{ph} / \Omega _{e0}\), \({\tilde{V}}_R=V_R/c\), and \({{\tilde{a}}} = ac^2 / \Omega _{e0}^2\) are normalized optimum amplitude, threshold amplitude, phase velocity, group velocity, average perpendicular momentum, parallel thermal momentum, wave frequency, hot plasma frequency, resonant velocity and gradient coefficient of background field, respectively. Parameters \(\xi\) and \(\chi\) are defined by \(\xi ^2 = \omega (\Omega _{e0}-\omega ) / \omega _{pe}^2\) and \(\chi ^2 = 1/(1+\xi ^2)\), respectively, specifying the dispersion relation. The Lorentz factor is given by \(\gamma =[1 - (v_\parallel ^2 + v_\perp ^2)/c^2]^{-1/2}\), where \(v_\parallel\) and \(v_\perp\) are the parallel and perpendicular velocity of particles. In addition, *Q* is the depth of an electron hole due to the depletion of trapped resonant electrons in the velocity phase space, and \(\tau\) represents the ratio between the nonlinear transition time and the nonlinear trapping time. In this study, we assume \(Q=0.1\) and \(\tau =0.5\) to keep the consistency with the previous research.

From (4) and (5), the gradient coefficient *a* only affects the threshold amplitude, which is also illustrated in Fig. 2 that the overlap between threshold and optimum amplitudes vanishes in large gradient cases. The theoretical result demonstrates that in Case 2 and Case 3, wave amplitudes after the first stage of the nonlinear wave growth are not large enough to enter an ideal convective growth stage, therefore, \(S=-0.4\) may not be satisfied at high latitudes and the nonlinear convective growth is suppressed there.

Figure 3 shows the spatial and temporal evolution of forward-propagating waves under different gradients of the magnetic field in the simulations. As the gradient increases, we find that the wave amplitudes at high latitudes are gradually decreasing especially in the first half period of the simulation run (\(0\sim 2\times 10^4[\Omega _{e0}^{-1}]\)), declaring a good agreement with the theory. It is worth noting that the group velocity of wave packets slightly increases for the large gradient case, leading to a curved form of wave path, as shown in Fig. 3c. This is because, in Case 3, we specified the background gradient that is 200 times greater than that in Fig. 3a (i.e., Case 1).

In the theory, we define the transition point of the two stages as a critical distance \(h_c\), at which the frequency variation term and the gradient term in (3) are equal. Inside the critical distance, the first stage is dominant, and we can assume wave packets are generated there. In Fig. 4, focusing on the typical frequency of hiss at around \(0.05\Omega _{e0}\), we find \(h_c\) in Case 1 is greater than half of the system length, meaning wave packets can be formed inside the whole simulation region. For Case 2 and Case 3, however, their critical distances are smaller, indicating that the generation of wave packets is limited to the equatorial region.

In Fig. 5, we obtain dynamic spectra of forward-propagating waves at various locations of the simulation region for all three gradient cases, and outline structures with wave amplitudes over \(10^{-4.5} [B_0]\). In Case 1, some typical wave packets are generated from \(h=-100[c\Omega _{e0}^{-1}]\). In Case 2, there are some typical structures at \(h=-50[c\Omega _{e0}^{-1}]\). However, we can only find obvious wave packets from \(h=0[c\Omega _{e0}^{-1}]\) in Case 3. The result is consistent with the theoretical critical distance.

From \(h=-100 [c\Omega _{e0}^{-1}]\) to \(-50 [c\Omega _{e0}^{-1}]\) in Case 2 and Case 3 of Fig. 5, wave amplitudes are growing. This is caused by the linear growth process. In Fig. 6, we examine the effectiveness of the linear and nonlinear process on the growth of wave amplitudes. In Fig. 6a, after rapid growth at the initial adjustment stage due to thermal fluctuations of hot electrons, the averaged wave amplitudes in the whole simulation region indicate that Case 2 and Case 3 have similar growth rates to the linear growth rate at the initial stage, while Case 1 undergoes a stronger nonlinear process, leading to a higher overall amplitude. We refer to the equation (92) in Omura (2021) and calculate the nonlinear growth rates at \(h=-100, -50\), and \(0 [c\Omega _{e0}^{-1}]\) in Case 1, as shown in Fig. 6b. As the wave amplitude becomes larger, the nonlinear growth rate will decrease correspondingly. This is illustrated in Fig. 6b that as wave packets approach the equator, the nonlinear growth rate continues decreasing and reaches the level of the linear growth rate.

In Fig. 7, we implement a band-pass filter of \(0.04\sim 0.06\Omega _{e0}\) to forward-propagating waves within the time range of \(0\sim 2\times 10^4[\Omega _{e0}^{-1}]\) and calculate their instantaneous frequencies and inhomogeneity factor *S*. From wave amplitudes and frequencies in the first and second row, the area where wave packets are generated becomes much closer to the equatorial region from Case 1 to Case 3, showing a good agreement with the theory. From Tobita and Omura (2018), the nonlinear process can exist even when the inhomogeneity factor *S* is down to \(-2\). To show the variation of S structures, we adjust the colorbar range to 0 to 5 in the bottom panels of Fig. 7. We notice that *S* values which correspond to the nonlinear process (i.e., \(|S|<2\)) widely locate in the whole simulation region in Case 1, yet are much limited to the equator in Case 2 and Case 3. This is consistent with the critical distance in each case, as indicated by the red dashed lines in the figure. The result implies that gradients in Case 2 and Case 3 are too large to satisfy the condition of nonlinear process at high latitudes, therefore not only their first stage of the nonlinear process has a very limited range, also their convective growth stage is strictly suppressed. Wave packets at high latitudes in Case 2 and Case 3 only undergo a weak linear growth process.