Fig.<ref> Comparison between the WKB $\alpha$ (blue line) and the wave-packet $\alpha$ (red line), for reduced (by a factor $10^2$) electron Landau damping, as functions of the antenna frequency in a neighborhood of the reference frequency $\omega_0$.

Fig.<ref> Dispersion relation about a "resonant cavity" mode: wave-packet solution (red), WKB solution (blue) and no-damping $b_k\ll1$ solution (green).

Fig.<ref> Normalized KAW radial electric field $\delta E_r$ (for reduced electron Landau damping).

Fig.<ref> Normalized KAW parallel electric field $\delta E_\parallel$ (for reduced electron Landau damping).

Fig.<ref> Modulus of KAW electric field ratio $\delta E_\parallel/\delta E_\perp$ (for reduced electron Landau damping).

§ SUMMARY

In this chapter, we have revisited the ideal MHD analysis of chapter 3 using gyrokinetic theory, highlighting the microscale phenomena that underlie the resonant absorption of the SAW at the radial location where the imposed antenna frequency matches the continuously varying SAW frequency spectrum; consistent with earlier analyses [40, 41], the resonant absorption consists in the mode conversion to the short wavelength kinetic Alfvén wave, which is typically absorbed within a short distance from the resonant layer and within a few spatial oscillation period of the mode converted wave (strong absorption or strong damping). Despite this case was and has been of interest for existing studies so far due to its direct relevance in plasma heating [40, 41, 44, 45], here we have rather focused on the opposite weak absorption case, where, even with a modest amplitude SAW launched by the antenna, large KAWs can be excited inside the resonant (mode conversion) layer and yield a number of interesting nonlinear behaviors. As new novel results of chapter 4, it has been demonstrated that the plasma region inside the mode conversion layer behaves as a "resonant cavity"; that is, when the antenna frequency matches a given discrete spectrum, the plasma response is stronger, exhibiting the clear behavior of a resonantly driven and weakly damped oscillator. This is a result of relevance for chapter 5, in which we shall investigate nonlinear behaviors induced by large amplitude kinetic Alfvén waves, fine-tuned in phase and amplitude by the external antenna.

CHAPTER: KINETIC ALFVÉN WAVES IN A CYLINDRICAL PLASMA:
NONLINEAR DESCRIPTION

Within the MHD description, the shear Alfvén wave launched by the antenna develops radial singular structures at the radial position where the antenna frequency matches the local shear Alfvén continuous spectrum frequency. Correspondingly, the incoming Poynting flux vanishes at the resonant surface, suggesting that the long wavelength fluctuation is completely absorbed at that very position. Within the linear gyrokinetic analysis, we have shown how the power flux actually transfers to the kinetic Alfvén wave by the mode conversion process. The new kinetic wave is damped and, for typical parameters, absorbed by thermal plasma electrons via Landau damping. This process was extensively studied in the literature from the discovery of kinetic Alfvén waves [40, 41] to the many application studies looking at plasma heating by Alfvén waves, e.g., in [44, 45]. The scope of this thesis work is to investigate the exactly opposite limit, where the mode converted kinetic Alfvén wave is weakly absorbed. While it was shown in chapter <ref> that this case is of little interest for Alfvén wave heating, we also noted that it is characterized by the plasma behavior as a "resonant cavity" inside the mode conversion layer. Thus, with a relatively small perturbation amplitude imposed at the antenna, a significantly large kinetic Alfvén wave can be excited with consequences on the nonlinear plasma behaviors. In the following, we adopt the nonlinear gyrokinetic description and show that the power that is mode converted to kinetic Alfvén waves is only partly absorbed by electrons through Landau damping, while the dominant fraction forms a nonlinear convective cell which is characterized by $k_\parallel \equiv 0$. Our study of the generation of convective cells by modulational instability, summarized in this chapter, generalizes a previous work in uniform magnetized plasmas [85], and a local analysis of magnetostatic convective cells in non-uniform plasmas [47]. This problem is of interest both for the physics of planetary magnetospheres and for the production of longitudinal electric fields in laboratory plasmas by external excitation of high-frequency waves (particle acceleration). The present theoretical analyses, yielding both analytical as well as numerical new original results, can be the basis for the proposal of a small table-top experiment as well as the test-bed for novel numerical simulation codes, such as TRIMEG [56, 57], STRUPHY [43] and GeFi [54], addressing the same fundamental nonlinear interactions.

Since the modulational instability that underlies the formation of convective cells has similarities with the parametric decay instability, we start introducing the qualitative features of this universal process. The concept of convective cells is then introduced, explaining their role in cross-field transport in magnetized plasmas as a reason that has attracted significant attention in plasma research. We also explain the excitation of convective cells by modulational instability and the special role played by kinetic Alfvén waves in this process. After this first general preamble, we carry out a systematic analysis of the nonlinear interplay of kinetic Alfvén waves and convective cells in cylindrical plasmas. In the more general, nonlinear regime, fluctuations are allowed to be of finite amplitude with, however, the constraint that the corresponding nonlinear frequencies, $\omega_{nl}$, be much less than the cyclotron frequency. In other words, consistent with the linear gyrokinetic orderings,
with $\delta\mathbf{u}_\perp$ representing the fluctuation-induced particle (guiding-center) jiggling velocity, for example $\delta\mathbf{u}_\perp\approx v_\parallel\delta\mathbf{B}_\perp/B_0$ or $\delta\mathbf{u}_\perp\approx (c/B_0^2)(\delta \mathbf{E}_\perp \times \mathbf{B}_0)$ [17], and with $\varepsilon= \rho_i/L \sim |\omega/\Omega_i|$ the small parameter introduced in subsection <ref>.

As a result of our nonlinear gyrokinetic analysis, we demonstrate that optimal conditions for convective cell generation in cylindrical plasmas are those of a "$\theta$-pinch" equilibrium, that is, with an azimuthal magnetic field only and no magnetic curvature. Earlier results of convective cell excitation by kinetic Alfvén waves in uniform plasmas [85] are recovered in the proper limiting case. Meanwhile, the new original results of this thesis work are presented. In particular, the effect of plasma nonuniformity is shown to importantly affect the convective cell stability both qualitatively and quantitatively via the diamagnetic response [13, 47]. In fact, the convective cell growth is significantly stronger (typically up to an order of magnitude) than in uniform plasmas, and the unstable parameter region is significantly broader. Another element of crucial importance is the subtle interplay between nonlinearity and plasma nonuniformity, which ultimately impacts the plasma self-organization that can be controlled by fine-tuning the amplitude of the antenna driven mode converted kinetic Alfvén wave.

§ PARAMETRIC DECAY INSTABILITIES

The parametric decay instability (PDI) is a fundamental nonlinear process [46, 55] involving three nonlinear coupled waves/oscillators: a large-amplitude pump ("mother") wave decays into two daughter waves once its amplitude exceeds a nonlinear threshold. Since the pump wave can be either spontaneously or externally excited, PDI is an important channel for wave energy transfer along with its associated consequences on plasma heating, acceleration and transports. Consider, as a prototype, a system of two coupled driven harmonic oscillators of equations
^2x_1t^2+ω_1^2x_1   =c_1x_2E_0,

^2x_2t^2+ω_2^2x_2   =c_2x_1E_0,
where $\mathbf{k}_{1,2}$ and $\omega_{1,2}$ are taken to be the resonant wave vectors and frequencies, $c_{1,2}$ are the constants representing the strength of the coupling and $E_0=\overline{E}_0\cos(\mathbf{k}_0\cdot\mathbf{x}-\omega_0t)$ is the amplitude of the driver wave, namely the pump. Assuming $x_1=\overline{x}_1\cos(\mathbf{k}_1\cdot\mathbf{x}-\omega t)$ and $x_2=\overline{x}_2\cos(\mathbf{k}_2\cdot\mathbf{x}-\omega't)$, the second of (<ref>) can be written as
(ω_2^2-ω'^2)x_2cos(𝐤'·𝐱-ω't)   =c_2E_0x_1cos(𝐤_0·𝐱-ω_0t)cos(𝐤·𝐱-ωt)=
so, in particular, the driving terms on the right can excite oscillators $x_2$ with frequencies $\omega'=\omega_0\pm\omega$, and oscillators $x_1$ and $x_2$ constitute a couple of decay waves. Equation (<ref>) suggests that nonlinear driving terms are still effective assuming a finite frequency shift; so $\omega'$ does not need to be exactly $\omega_2$, but only approximately equal to $\omega_2$. Furthermore, $\omega'$ can be complex because there can be damping or growth, so the oscillator $x_2$ can respond to a range of frequencies. In conclusion, the frequency and wave-vector matching conditions are

A parametric decay instability may occur when an incident electromagnetic wave of high frequency and large phase velocity excites two waves with $\mathbf{k}_1\approx-\mathbf{k}_2$. Keep in mind that even a small amount of damping (either collisional or collisionless) will prevent parametric instability unless the pump wave is strong enough. To calculate the threshold amplitude for wave excitation, damping rates $\gamma_1$ and $\gamma_2$ in the first and second of (<ref>) are introduced by means of damping terms $2\gamma_{1,2}\dot{x}_{1,2}$, ultimately obtaining

The PDI can be either resonant, if both decay waves are marginally stable or weakly damped normal modes, or non-resonant, if one of the decay waves is a heavily damped quasi mode. The conditions $\omega\approx\omega_1$ and $\omega_0-\omega\approx\omega_2$ give, from the above equations, an instability threshold as a function of the damping coefficients, that is, expressing $x_1$ and $x_2$ with their peak amplitudes,
thus, the threshold goes to zero for vanishing damping.

Let now the three interacting waves be the pump KAW $\boldsymbol{\Omega}_0=(\omega_0,\mathbf{k}_0)$, the low-frequency daughter ion sound wave (ISW) $\boldsymbol{\Omega}_s=(\omega_s,\mathbf{k}_s)$ and the daughter KAW $\boldsymbol{\Omega}_-=(\omega_-,\mathbf{k}_-)$ with $\omega_-=\omega_s-\omega_0$ and $\mathbf{k}_-=\mathbf{k}_s-\mathbf{k}_0$, consistent with frequency and wave-vector matching conditions. Let the small but finite pump wave amplitude be denoted as $\Phi_0=e\delta\phi_0/T_e$. As $\Omega_s$ could be a quasi mode, we then need to retain terms of order $|\Phi_0|^2$ in view of a proper account of non-resonant PDI. The KAW PDI dispersion relation is [17, 14]
with the ISW dielectric constant linearly approximated as
while $\epsilon_{Ak-}$ is the linear dielectric constant of the KAW decay wave,
where $\Gamma_-$, $b_-$ and $\sigma_-$ are the usual functions introduced in section <ref> and as yet expressed with the generic subscript $k$; furthermore, $\chi_{A_{-}}^{(2)}$ is a function accounting for nonlinear ion Compton scattering [68, 14, 17] and $C_k$ is the nonlinear coupling coefficient between the two daughter waves via the pump mode [14, 17]. It is beyond the illustrative scope of PDI given in this section to provide detailed derivation of the $\chi_{A -}^{(2)}$ and $C_k$ terms. So, they will be omitted here. In the case of resonant decay, for a frequency $\omega_s=\omega_{sr}+\text{i}\gamma$, and denoted $\gamma_{dA_{-}}$ and $\gamma_{ds}$ the linear damping rates of the KAW and ISW daughter waves, respectively, relation (<ref>) reduces to
We see that the parametric decay instability growth rates increase with $|C_k|\Phi_0|^2|$, which scales with $\left|k_\perp\rho_i\right|^4\left|\delta\mathbf{B}_{\perp0}/B_0\right|^2$ for $\left|k_\perp\rho_i\right|^2\ll1$ and $\left|\delta\mathbf{B}_{\perp0}/B_0\right|^2/\left|k_\perp\rho_i\right|$ for $\left|k_\perp\rho_i\right|^2\gg1$ [17, 14]. The decay instabilities are, thus, strongest when $\left|k_\perp\rho_i\right|\sim1$, and it clearly demonstrates the necessity of keeping finite ion Larmor radius kinetic effects in dealing with the decay instabilities of KAWs.

Finally, it is illuminating to compare the decay instabilities of KAWs versus those of SAWs in the MHD regime. Essentially, by employing the ideal MHD fluid theory the PDI dispersion relation takes the form similar to the KAW PDI dispersion relation (<ref>), with KAW terms replaced by corresponding SAW terms. The more fundamental change lies in the nonlinear coupling term, that is, $C_k$ replaced by $C_I$, with [17]
C_I∝cos^2θ_0,    C_k∝sin^2θ_0,
where $\theta_0$ is the angle between $\mathbf{k}_{\perp0}$ and $\mathbf{k}_{\perp-}$; in particular,
for $\left|k_\perp\rho_i\right|^2\ll1$, and
for $\left|k_\perp\rho_i\right|^2\sim1$. The formula (<ref>) indicates that, for $1>\left|k_\perp\rho_i\right|^2 >\left|\omega_0/\Omega_i\right|$, nonlinear couplings via kinetic effects dominate. Noting that $\left|\omega_0/\Omega_i\right|\sim10^{-3}$ in typical laboratory plasmas, the validity regime of MHD fluid theory for the SAW nonlinear physics is rather limited. Furthermore, at the $\left|k_\perp\rho_i\right|\sim1$ regime where KAW nonlinear effects maximize, we have $\left|C_k\right|/\left|C_I\right|\sim\left|\Omega_i/\omega_0\right|^2$ for typical parameters. In summary, in addition to the significantly enhanced PDI growth rates, there is, perhaps, more significant qualitative difference between KAW and SAW PDI in terms of the wave vector of the scattered daughter wave with respect to that of the pump wave. Indeed, since $C_I\propto\cos^2\theta_0$ according to (<ref>), the SAW scattering maximizes around $\theta_0=0,\pi$, i.e., when $\mathbf{k}_{\perp-}$ is parallel or anti-parallel to $\mathbf{k}_{\perp0}$. In contrast, the behavior of $C_k$ implies that the KAW scattering maximizes around $\theta_0=\pm\pi/2$, i.e., $\mathbf{k}_{\perp0}$ and $\mathbf{k}_{\perp-}$ are orthogonal. This difference affects the nonlinear evolution of KAW turbulence and the charged particle transports induced by the KAW decay processes [17]. In fact, assuming the pump wave is a mode converted KAW and $\mathbf{k}_{\perp 0}$ is predominantly radial, the daughter KAW will also have $\mathbf{k}_{\perp -}$ predominantly radial in the ideal MHD analysis, with little effect on plasma transport. Meanwhile, $\mathbf{k}_{\perp -}$ of the daughter KAW will be predominantly azimuthal in the kinetic regime, with important consequences on plasma transport.

§ CONVECTIVE CELLS

When, along with the confining magnetic field, an electric field is present, the motion of particles is the sum of the usual circular Larmor gyration plus a drift of the guiding center, resulting in an additional transverse component of velocity [14],
𝐯_E=c 𝐄×𝐁/B^2.
For long wavelengths, electron and ions have similar responses; however, on shorter scales of the order of the ion Larmor radius, electron and ion responses become unbalanced and this, in turn, generates local charge separation and therefore strong radial electric fields. The ion flow forms two-dimensional (in the plane perpendicular to the ambient magnetic field) drift patterns named convective cells [75, 60, 18, 20, 53, 70, 71, 27], namely perturbations which are constant along the field line but change in the perpendicular direction; as a consequence, these perturbations play an important role in the vortex dynamics and in the diffusion and transport of plasma across the confining magnetic field. A classification scheme for convective cells (CCs) can be made between electrostatic convective cells (ESCCs), where $\delta E=\delta E_\perp$ [60, 75], and magnetostatic convective cells (MSCCs), with $\delta B=\delta B_\perp$ [20]. Two-dimensionality means, for convective cells, $k_\parallel=0$, with typically $\text{Re}(\omega)\approx0$. Convective-cell motion is therefore similar to that of two-dimensional vortexes in an incompressible fluid, both motions being divergence free. In the presence of collisions, the cells are damped because of ion viscosity, leading to a normal mode with purely imaginary frequency. Their slow motion can lead to anomalously rapid plasma transport across the magnetic field even in thermal equilibrium, mostly when they are connected to instability mechanisms such as a turbulence in an inhomogeneous plasma: initially given linearly unstable drift-waves, namely the waves or collective excitations driven by a pressure gradient destabilized by differences between ion and electron motion, can nonlinearly interact and excite them. Another nonlinear route leading to the formation of convective cells, which is of central importance for the present work, is through Alfvén waves and the modulational instability mechanism discussed in the following.

It is worthwhile recalling, here, the connection of the modulational instability with the parametric decay instability, which was discussed in Sec. 5.1 and is the first example that was historically studied, where the importance of wave scattering in plasmas was recognized and explained [45, 54]. This connection between modulational instability and parametric deday instability will be further elaborated in the following.

§.§ Modulational instability

In laboratory fusion plasmas, nonlinear excitations of convective cells usually occur via mode-mode couplings of ambient drift-wave and/or Alfvén-wave instabilities. Since convective cells have no parallel wavenumber, their nonlinear interactions involve couplings between co-propagating Alfvén waves with the same $k_\parallel$, which vanishes for SAWs in the ideal MHD limit. Thus, it has long been recognized that only non-ideal MHD fluctuations, such as KAWs, can nonlinearly excite convective cells. Furthermore, since convective cells are excited at $\omega=0$, it is also recognized that their nonlinear excitation takes the form of a modulational instability. In general, a modulational instability is a deviation from the wave periodic behavior further reinforced by nonlinearity: a weakly modulated continuous wave in a nonlinear medium grows and leads to spectral sidebands and possibly to the breaking of the periodic fluctuation into modulated pulses. In space, it transforms weakly modulated plane waves into spatially periodic patterns. So, modulational instability is formally quite similar to a PDI, where the carrier wave is the pump, and the two sidebands behave as analogs of the daughter waves. Previous theoretical studies about convective cells excited by KAWs via modulational instability suffer from being restricted to the two-fluid or drift-kinetic descriptions (ignoring finite Larmor radius effects), and/or the limiting assumption that ESCCs and MSCCs are decoupled. These hypotheses have been adopted in order to simplify the theoretical analysis but lead to erroneous conclusions on the nonlinear excitation mechanisms. Notably, spontaneous excitation of CCs can only occur when one keeps the crucial finite ion Larmor radius (FILR) effects, $|k_\perp\rho_i|\sim1$; and in this regime one finds that ESCCs and MSCCs are, in general, intrinsically coupled [85]. As new original element with respect to previous works [47, 85], we will adopt nonuniform cylindrical plasma geometry. Meanwhile, we will employ the nonlinear gyrokinetic equations and demonstrate that both the FILR as well as the finite coupling between ESCCs and MSCCs play qualitatively crucial roles in the dynamics of the modulational excitations of convective cells.

§ NONLINEAR GYROKINETIC EQUATIONS

In the cylindrical plasma equilibrium of interest here, which was introduced in chapter <ref>, the vorticity equation can be written as (see also [85])

here $\mathbf{k}=\mathbf{k}'+\mathbf{k}''$, where $\mathbf{k}'$ and $\mathbf{k}''$ are two
wave vectors involved in the three-wave interaction with $\mathbf{k}$ (similar to parametric decay instability), and $\omega_{{k}}=\omega_{{k}'}+\omega_{{k}''}$ from the frequency matching condition. Furthermore, $\Lambda_{k'}^{k''}=\left(\mathbf{k}'\times\mathbf{k}''\right)\cdot\hat{\mathbf{z}}$, while $\Gamma_k:=I_0(b_k)e^{-b_k}$ with $I_0$ the modified Bessel function. The KAW generated by the external antenna via mode conversion interacts with low-amplitude convective cells and generates sidebands. In particular, as we show in the following, an externally excited KAW pump of sufficiently high amplitude can spontaneously generate a convective cell and the two coupling sidebands which are due to the beat between convective cell and pump mode. As already mentioned in the previous section, we know [85] that spontaneous excitation of CCs can only occur when one keeps the crucial finite ion Larmor radius (FILR) effects, i.e., in the $|k_\perp\rho_i|\sim1$ regime, and that ESCCs and MSCCs are intrinsically coupled. Also note that (<ref>) readily reduces to the already discussed linear limit. In WKB form, (<ref>) can be cast as

The mode decomposition is made by writing explicitly the expressions for $\delta\phi$:
δϕ_0:=T_i/2e[δϕ̂_0e^i(n/Rz-mθ-ω_0t)]+c.c.       (KAW pump),

δϕ_±:=T_i/2e[δϕ̂_±e^i(±n/Rz-(m_z±m)θ-(ω_z±ω_0)t)]+c.c.       (KAW sidebands),

δϕ_z:=T_i/2e[δϕ̂_z e^-i(m_zθ+ω_zt)]+c.c.       (CC);
so the subscript $0$ is for the pump, the subscripts $\pm$ are for the two sidebands and the subscript $z$ is for the convective cell. The expressions for $\delta\psi$ follow consistently. Note that the normalization used here is fully consistent with [12] and slightly different from [85]. The fields $\delta\hat{\phi}_0$, $\delta\hat{\phi}_\pm$, $\delta\hat{\phi}_z$ are dimensionless: the corresponding magnetic stream functions are
δψ̂_0=ω_0/k_∥cδÂ_∥0,    δψ̂_±=±ω_z±ω_0/k_∥cδÂ_∥±,    δψ̂_z=ω_0/k_∥cδÂ_∥z,
where $k_\parallel = k_{\parallel 0}$ and $\psi_{0,\pm,z}$ are the components of the effective induced parallel potential (they are
proportional to the vector potential). Direct substitution of (<ref>), (<ref>), (<ref>), and (<ref>) into (<ref>) gives the following coupled nonlinear vorticity equations for KAW pump, sidebands and CC:

Note that, in the ideal MHD limit, the right-hand sides of these equations go to 0 since the differences involving $\Gamma_k$ functions and the terms with $b_k$ vanish. On the other hand, the measure of nonlinear behavior is given by the coupling coefficients $\Lambda_{k'}^{k''}$, which are maximized when the pump wavenumber is orthogonal to the sidebands wavenumbers. These equations must be closed by the quasineutrality conditions
where $\delta g_k$ satisfies the Frieman-Chen nonlinear gyrokinetic equation for electrons and ions [35, 85] which, in uniform plasma, reads
Solving (<ref>) for $k_\parallel\neq0$ we obtain [85], at the lowest (linear) order,
δg_ki^(1)≈e F_MiJ_k/T_iδL_k,    δg_ke^(1)≈-e F_Me/T_eδψ_k,
with $\delta L_k=\delta\phi_k-(v_\parallel/c)\delta A_\parallel$. Substitution back into (<ref>) yields
with $\sigma_k$ as defined in <ref>; however, including Landau damping gives $\sigma_k$ a negative complex imaginary part, $\sigma_k=1+\tau(1-\Gamma_k)(1-\text{i}\sqrt{\pi}\xi_e)$. The anti-Hermitian response is predominantly due to electron-Landau damping. At the next order,
δg_ki^(2)≈0,    k_∥δg_ke^(2)≈-ic/B_0Λ_k'^k”(δg_k'eδA_∥k”/c-δg_k”eδA_∥k'/c).
Noting (keep in mind (<ref>))
and substituting back into (<ref>), we obtain
σ_0δϕ̂_0-δψ̂_0   =-i/2ω_ciρ_i^2[Λ_k_+^k_z^*(δψ̂_z^*/ω_0-δϕ̂_z^*/ω_0+ω_z)δψ̂_+
In order to close the system, the equation for $\delta\hat{\psi}_z$ must be determined independently of (<ref>), which is degenerate with (<ref>) for the CC. For wavelength longer than the collisionless skin depth, the equation for $\delta\hat{\psi}_z$ is obtained imposing the vanishing of $\delta J_{\parallel z}$ for electrons:
k_⊥^2c^2/ω_pe^2≪1,    δJ_∥ze=-e∫v_∥δf_ze d𝐯≈0.
For $k_\parallel=0$, the nonlinear gyrokinetic equation becomes [85]
from which (<ref>) readily follows. By substitution of (<ref>) into (<ref>) we finally obtain
that is,
The nonlinear gyrokinetic vorticity equations (<ref>), (<ref>) and (<ref>) and the electron parallel force balance/quasineutrality equations (<ref>), (<ref>) and (<ref>) provide a complete set of nonlinear equations that describe the formation of CCs by KAW decay in uniform plasmas. They extend the results of [85] since they include, with (<ref>) and (<ref>), the feedback of CC and sideband onto the KAW pump. These equations will be further extended to nonuniform plasmas in the next section. So we have six equations for 6 unknowns, to be resolved in time-dependent or time-independent way: for assigned $\delta\psi_0$ and $\delta\phi_0$, that is for an assigned pump, we reduce to 4 algebraic equations. Parametric decay instability can be investigated by (<ref>) and (<ref>) coupled with (<ref>) and (<ref>), assuming a fixed amplitude KAW pump. After determining the nonlinear threshold, the further nonlinear evolution of the system can be studied by exploring the whole set of equations.

Varying the NL coupling coefficients $\Lambda_{k'}^{k''}$, as a function of $\mathbf{k}_0$ and $\mathbf{k}_z$ (with $\mathbf{k}_\pm=\mathbf{k}_z\pm \mathbf{k}_0$), we can study the local growth rate of modulational instability as a function of wavenumber and amplitude. Since for a CC $k_\parallel=0$, with $k_\parallel$ given by (<ref>), we see that, in general screw pinch cylindrical plasma equilibria, CC existence implies a strong constraint: $n=m=0$. Our interest, meanwhile, is to maximize nonlinear coupling. Thus, it seems reasonable to consider a $\theta$-pinch equilibrium as a special case of the general screw pinch equilibrium with vanishing azimuthal magnetic field and $q \rightarrow \infty$. In this way, the CC existence imposes $n=0$ and we are allowed to freely choose $m_z$ to maximize nonlinear coupling. Meanwhile, due to the intrinsic stability of the screw pinch configuration, we can properly choose electron and ion temperature profiles to minimize Landau damping. In other words, this special case of cylindrical plasma equilibrium is particularly suitable for our studies intended at investigating CC generation by high amplitude mode converted KAW excited by an external antenna.

§.§ Nonuniform plasma in $\theta$-pinch configuration

Considering that $|k_\parallel v_{Ti}|\ll\omega\ll|k_\parallel v_{Te}|$, drifts effects are possibly changing ion gyrokinetic response in the short-wavelength limit. However, from equilibrium condition (<ref>) in a $\theta$-pinch magnetic field configuration, that is,
one readily shows that $v_d\sim(\beta/2)v_*$ for thermal ions. Thus, the characteristic ion magnetic drift velocity, $v_d$, is typically much smaller that the corresponding ion magnetic drift, $v_*$. Therefore, we may include plasma nonuniformity only via ions and electrons diamagnetic responses. In particular, for the sake of simplicity, let us consider the effect of density gradient only, while ion and temperature profiles are taken as constants; see also [13]. The former equation can be solved, with $k_\parallel\neq0$, at the lowest linear order given by
δg_ki^(1)≈e F_MiJ_k/T_i(1-ω_*i/ω)_kδL_k,    δg_ke^(1)≈-e F_Me/T_e(1-ω_*e/ω)_kδψ_k,
with the diamagnetic frequencies defined as
ω_*i   :=cT_i/eB_0k_θ∂_r n_0/n_0,

ω_*e   :=-cT_e/eB_0k_θ∂_r n_0/n_0=-τω_*i.
This modifies the quasineutrality condition (see (<ref>)) of $k_\parallel\neq0$ modes as
for electrons, this equation follows the solution of (<ref>), and yields
Meanwhile, $\delta g_{ki}^{(2)}$ contribution does not vanish (it vanishes for $\omega_{*i}=0$) and becomes
where $F_{1+}=\langle J_zJ_{k0}J_{k+}F_{Mi}/n_0\rangle$, $F_{1-}=\left\langle J_zJ_{k0^*}J_{k-}F_{Mi}/n_0\right\rangle$ and where terms proportional to $\delta A_\parallel$ have been neglected as they are $\mathcal{O}(\beta_i)$. Equation (<ref>) is modified into the following sidebands quasineutrality condition
σ̂_±δϕ̂_±-δψ̂_±   =±iω_ciρ_i^2/2ω_0(1-ω_*e/ω)_±(Λ_k_0^k_zδψ̂_0

with $\hat{\sigma}_\pm=\left[1+\tau-\tau\Gamma_\pm(1-\omega_{*i}/\omega)_\pm\right]/(1-\omega_{*e}/\omega)_\pm$, as shown by [13]. A similar procedure can be adopted for computing the parallel force balance for electrons, keeping in mind, in massless approximation for electrons, (<ref>),
Following the same procedure delineated after (<ref>), and looking for a solution to $\delta J_{\parallel ze}=0$, one can verify that (<ref>) and (<ref>) yield to a simplification of electron diamagnetic effects and that (<ref>) still applies,
So, closing equations by means of vorticity equations, we get

which can be specialized to both KAW sideband,

Λ_k_0^*^k_zδϕ̂_0^*)δϕ̂_z-k_∥^2v_A^2/ω_0^2(b_0-b_z) (Λ_k_0^k_zδψ̂_0
and convective cells,

In summary, (<ref>), (<ref>), (<ref>), and (<ref>) are the complete set of equations to solve for computing CC modulational instability growth. They include diamagnetic effects (symmetry breaking), reducing to the previous ones for $\omega_{*i,e}\rightarrow0$, and are consistent with the analysis in [13]. These theoretical analyses are an original novel contribution of the present thesis work together with the corresponding numerical solutions that will be presented in the next sections.

§ SOLUTION OF THE MODULATIONAL INSTABILITY

For a more focused study, let us introduce the following simplifications that are aimed at maximizing the non-linear interactions, given that the pump KAW is the mode converted wave generated by the external antenna drive with prescribed frequency $\omega_0$ as well as (low) poloidal and azimuthal mode numbers $(n,m)$:
𝐤_⊥0≈𝐤_0r𝐫̂,    𝐤_∥0=𝐤_∥𝐳̂=n/R𝐳̂,

𝐤_⊥±≈-m_z/rθ̂±k_0r𝐫̂,    𝐤_∥±=±n/R𝐳̂,
in addition to the general request $k_{\parallel z}=0$. Note that the underlying assumption here is that the pump KAW has $\mathbf{k}_\perp$ predominantly in the radial direction (as we expect, since we construct the pump/mother KAW by mode conversion at the SAW resonance) and that its behavior is predominantly oscillatory with a modulation in amplitude occurring on a scale longer than $\rho_i$. Meanwhile, the CC is characterized by a macroscale radial structure with a predominant short poloidal scale that is connected with $|m_z|\gg |m|$. This feature clearly maximizes the nonlinear coupling coefficient $\Lambda_{k_0}^{k_z}$. It will be shown in the following that this condition is also consistent with the maximization of the CC excitation rate. Thanks to these assumptions, we have
b_±≈b_0+b_z,    b_z≈m_z^2ρ_i^2/r^2,    b_0≈k_0r^2ρ_i^2,
and so
Γ_+≈Γ_-,    F_1+≈F_1-=∫_0^∞e^-tJ_0(√(2b_zt))J_0(√(2b_0t))J_0(√(2b_+t)) dt.
Finally, in order to more easily manipulate equations, let us introduce the symbolic notations $R^\pm_1$ and $R^\pm_2$ to denote, respectively, the right-hand side of (<ref>) and of (<ref>) for the $\pm$ sign. So, from (<ref>) and (<ref>) we obtain

[(1-ω_*i/ω)_±(1-Γ_±)-b_±k_∥^2v_A^2/(ω_z±ω_0)^2σ̂_±]δψ̂_±   =-R^±_1(1-ω_*i/ω)_±(1-Γ_±)

where, considering (<ref>), and noting that $\Lambda_{k_0^*}^{k_z}=-\Lambda_{k_0}^{k_z}$,
R_1^±   =iω_ciρ_i^2/2ω_0Λ_k_0^k_zσ_0/(1-ω_*e/ω)_±{[1+(ω_*i/ω)_zτ-(ω_*i/ω)_zτF_1/σ_0(1±ω_z/ω_0)](δϕ̂_z-δψ̂_z)

±R_2^±   =iω_ciρ_i^2/2Λ_k_0^k_z{[Γ_0-Γ_z-(ω_*i/ω)_z(F_1-Γ_z)-1-Γ_0/b_0(b_z-b_0)]δψ̂_z

Introducing the KAW dielectric constants for sidebands
ϵ_A±:=(1-ω_*i/ω)_±1-Γ_±/b_±-k_∥^2v_A^2/(ω_z±ω_0)^2σ̂_± and collecting terms, equations (<ref>) and (<ref>) can be cast as
(ω_0±ω_z)b_±ϵ̂_A±δϕ̂_±   =iω_ciρ_i^2/2Λ_k_0^k_z[c_ϕ^±(δϕ̂_z-δψ̂_z)+d_ϕ^±δψ̂_z](δϕ̂_0

(ω_0±ω_z)b_±ϵ̂_A±δψ̂_±   =iω_ciρ_i^2/2Λ_k_0^k_z[c_ψ^±(δϕ̂_z-δψ̂_z)+d_ψ^±δψ̂_z](δϕ̂_0
c_ϕ^±   :=Γ_0-Γ_z-(ω_*i/ω)_z(F_1-Γ_z)

d_ϕ^±   :=Γ_0-Γ_z-(ω_*i/ω)_z(F_1-Γ_z)-(b_z-b_0)(1-Γ_0)/b_0

c_ψ^±   :=σ̂_±[Γ_0-Γ_z-(ω_*i/ω)_z(F_1-Γ_z)]

d_ψ^±   :=σ̂_±[Γ_0-Γ_z-(ω_*i/ω)_z(F_1-Γ_z)-(b_z-b_0)(1-Γ_0)/b_0]

With (<ref>), (<ref>), (<ref>), (<ref>), (<ref>) we can compute the modulational instability dispersion relation. First note that
$\omega_{*i}$ being the CC diamagnetic frequency (<ref>). Meanwhile, using the notation $\sigma_\pm$ for $\hat{\sigma}_\pm$ with no $\omega_{*i}$, we can write
b_±ϵ̂_A±   =(1-Γ_±)(1±ω_z/ω_0)(1±ω_z/ω_0±τω_*i/ω_0)(1±ω_z/ω_0∓ω_*i/ω_0)/(1±ω_z/ω_0)^2(1±ω_z/ω_0±τω_*i/ω_0)

from which we obtain

being the leading-order frequency mismatch between the sideband frequency and the expected normal mode frequency. In fact, (<ref>) and (<ref>) express that KAW sidebands are not normal mode of the system and, thus, $\hat \epsilon_{A\pm} \neq 0$.

In order to simplify notation, let us adopt from now on the subscript + for denoting both upper and lower KAW sidebands. Collecting terms and noting that $\Delta_\pm=\Delta_+$, $b_\pm=b_+$, $\Gamma_\pm=\Gamma_+$ and $\sigma_\pm=\sigma_+$, we can write

Meanwhile, equations (<ref>) become
(±ϵ̃_A1-ϵ̃_A2)δϕ̂_±   =iω_ciρ_i^2

(±ϵ̃_A1-ϵ̃_A2)δψ̂_±   =iω_ciρ_i^2
where, from (<ref>), (<ref>), (<ref>), (<ref>), we have

d_ϕ_1   =(ω_z/ω_0+τω_*i/ω_0)[Γ_0-Γ_z-ω_*i/ω_z(F_1-Γ_z)-(1-Γ_0)(b_z-b_0)/b_0]

d_ϕ_2   =Γ_0-Γ_z-ω_*i/ω_z(F_1-Γ_z)

c_ψ_1   =σ_+(ω_z/ω_0+τω_*iΓ_+/ω_0σ_+)[Γ_0-Γ_z-(ω_*i/ω)_z(F_1-Γ_z)]

c_ψ_2   =σ_+[Γ_0-Γ_z-(ω_*i/ω)_z(F_1-Γ_z)]

d_ψ_1   =σ_+(ω_z/ω_0+τω_*iΓ_+/ω_0σ_+)[Γ_0-Γ_z-(ω_*i/ω)_z(F_1-Γ_z)-(1-Γ_0)(b_z-b_0)/b_0]

d_ψ_2   =σ_+[Γ_0-Γ_z-(ω_*i/ω)_z(F_1-Γ_z)-(1-Γ_0)(b_z-b_0)/b_0]

Considering now (<ref>), we write
δψ̂_z   =iω_ciρ_i^2/2ω_0Λ_k_0^k_zσ_0(δϕ̂_0δψ̂_-/1-ω_z/ω_0+δϕ̂_0^*δψ̂_+/1+ω_z/ω_0)=

then, using the second of the (<ref>),

This equation reduces to those in [85] and [17] in the uniform plasma limit. The same procedure can be adopted for the CC vorticity equation, which, from (<ref>), can be rewritten as
ω_z/ω_0(1-ω_*i/ω_z)(1-Γ_z)δϕ̂_z   =-iω_ciρ_i^2/2ω_0Λ_k_0^k_z{[Γ_+-Γ_0-(F_1-Γ_+)ω_*iω_z/ω_0^2(1-ω_z^2/ω_0^2)

-[(F_1-Γ_+)ω_*i/ω_0(1-ω_z^2/ω_0^2)-b_z(1-Γ_0)ω_z/b_0ω_0(1-ω_z^2/ω_0^2)](δϕ̂_0δϕ̂_-+δϕ̂_0^*δϕ̂_+)}. Again, using the first of the (<ref>), this equation becomes

Also this equation reduces to those in [85] and [17] in the uniform plasma limit. Equations (<ref>) and (<ref>) are two coupled equations that, for a given power spectrum and $\mathbf{k}_{\perp z}$ give, as a solution, $\omega_z$ and the ratio $\delta\hat{\psi}_z/\delta\hat{\phi}_z$. That is, they fully solve the modulational instability problem and CC polarization, which is generally mixed. In fact, it is not possible to postulate a priori a magnetostatic and/or electrostatic CC, and the polarization has to be computed self-consistently (the polarization issue is discussed in [85] and [17]). Given this, the present analysis has some common features with [47], in particular the consideration of the important role of plasma nonuniformity, which will be further discussed in the next section. Compared with [47], the present work is not limited to magnetostatic CCs and is extended to cylindrical plasmas. The standard form of CC equations, by analogy with the analyses of [85, 17], can be cast as
ϵ̃_A2^2-ϵ̃_A1^2/4(1-Γ_+)^2(1+Δ_+/ω_0)δϕ̂_z   =-α_ϕ(δϕ̂_z-δψ̂_z)+β_ϕδψ̂_z,

ϵ̃_A2^2-ϵ̃_A1^2/4(1-Γ_+)^2(1+Δ_+/ω_0)δψ̂_z   =-α_ψ(δϕ̂_z-δψ̂_z)+β_ψδψ̂_z,
with $\alpha_{\phi,\psi}$ and $\beta_{\phi,\psi}$ defined by inspection comparing (<ref>) with (<ref>) and (<ref>). In the uniform plasma limit, equations (<ref>) represent the counterpart of equation (16) in [85] and equation (120) in [17]. In nonuniform plasmas, meanwhile, the symmetry breaking is evident by transforming $\omega_{*i} \mapsto - \omega_{*i}$. This clearly impacts the modulational instability process, which is no longer characterized by $\omega_z = 0$ at the critical threshold of pump KAW amplitude for CC excitation [85, 17]. In fact, marginally unstable CCs typically rotates in the electron diamagnetic direction, opposite to $\omega_{*i}$, consistent with [47], and this fact is expected to influence the CC effect on plasma transport. However, the symmetry breaking effect connected with finite $\omega_{*i}$ does not violate the overall cylindrical symmetry property of the system, which is reflected by the invariance of the CC dispersion relation under the transformation $\omega_{*i} \mapsto - \omega_{*i}$ and $\omega_z \mapsto - \omega_z$.

§ EQUILIBRIUM CONFIGURATION AND NUMERICAL SOLUTION

The plasma equilibrium profiles and parameters introduced in section <ref> define a generic screw pinch configuration, with the axial component of the magnetic field dominant with respect to the azimuthal one. Meanwhile, in section <ref> we have demonstrated that, in order to maximize nonlinear coupling and the generation of CCs by modulational instability, the preferred cylindrical plasma equilibrium is a $\theta$-pinch, which can be seen as a limiting case of a screw pinch with vanishing azimuthal magnetic field and/or $q \rightarrow \infty$. This situation can be achieved by lowering the value of $J_{0z}$ by several orders of magnitude so that, according to (<ref>), the azimuthal component becomes zero in any practical respect. We also adjust the value of $B_{0z}(0)$ and $n_0(0)$ for further convenience. The novel equilibrium considered hereafter is characterized by significant diamagnetic plasma response (smaller $B_{0z}$ inside the plasma) and, at the center, $\beta$ reaches up to nearly 0.4 (higher than in the previous case) to keep KAW electron Landau damping small. The antenna frequency is $\omega_0=1.53$ MHz, and, for the sake of simplicity, we assume $n_0 = m_0 = 2$. These choices move the resonant layer to $x_0=0.746$. In the following, we also assume

$n_0(0)=5\cdot10^{14}$ cm$^{-3}$

$J_{0z}(0)=0.7\cdot10^{4}$ statA

$B_{0z}(0)=1.6\cdot10^4$ G

while other parameters remain the same as in section <ref>. As in chapter <ref>, with the present launched antenna spectrum, diamagnetic effects can be neglected for the mode converted KAW pump. However, they are generally important for both CC and KAW sidebands. For the present application, we arbitrarily choose $\delta\hat B_{rh}/B_0=0.01$, corresponding, for the KAW pump (on the slow radial scale), to $\delta B_{\theta 0}/B_0\approx0.003$. The radial behavior of CC frequency and growth rate and the polarization are obtained with these parameters.

Fig.<ref> Radial displacement, normalized to $(-\delta \hat B_{rh}/B_0)$, and normalized radial electric field, for the $\theta$-pinch case (nominal damping).

Fig.<ref> Radial displacement, normalized to $(-\delta \hat B_{rh}/B_0)$, and normalized radial electric field, for the $\theta$-pinch case ($10\%$ of nominal damping).

Fig.<ref>  Radial displacement, normalized to $(-\delta \hat B_{rh}/B_0)$, and normalized radial electric field, for the $\theta$-pinch case ($1\%$ of nominal damping).

The normalized radial displacement and radial electric field are computed for a nominal value of damping (fig.<ref>), for $10\%$ of nominal value (fig.<ref>), and for $1\%$ of nominal value (fig.<ref>), all at the same frequency $\omega_0=1.53$ MHz: the second and third couple of figures are qualitatively identical, showing that the mode structure becomes essentially independent from damping (except for amplitude and phase effects) at about $10\%$ of the nominal value. In the following, we will adopt the aforementioned plasma parameters, corresponding to a $\theta$-pinch equilibrium, assuming the reduced damping case with $10^{-2}$ of the nominal damping to better illustrate the effects of nonlinear excitations of CCs.

Hereafter, we study the problem of the generation of a convective cell by modulational instability, due to nonlinear excitation by a mode converted KAW pump mode. As regards the dispersion relation, the equation that is solved is (<ref>), with (<ref>) and (<ref>) that only provide the definition of the coefficients entering in (<ref>). Correspondingly, the polarization vector is computed as solution of (<ref>) at the obtained eigenvalue $\omega_z$[Bessel functions are computed by means of the scipy.special python library, while the function $F_1$ from (<ref>) is computed by quadrature using the integrate package from scipy. Finally, the roots of the dispersion function are obtained by the fmin routine from scipy.optimize, minimizing the absolute value of the dispersion function nearby a guess in the complex plane.]. We can restrict our analysis to a conventionally positive $\omega_0$ and a positive $\omega_{*i}$ (the ion diamagnetic frequency (<ref>)) due to the fact that the CC dispersion relation is symmetric under the transformation $\omega_{*i}\mapsto-\omega_{*i}$ and $\omega_z\mapsto-\omega_z$. As a first test for our theoretical approach, we take the uniform plasma limit, that is $\omega_{*i}=0$, and recover the marginal stability curves obtained in [85] (here in fig.<ref>). Direct connection with the uniform plasma slab of [85] is obtained, identifying the radial direction with $x$ and the azimuthal direction with $y$. Remember that $\omega_{*i}$, according to its definition (<ref>), depends on the CC azimuthal wave vector but also on the density gradient: so, for finite $k_y\rho_i$, $\omega_{*i}=0$ if we take the uniform plasma limit. Fig.<ref> assumes $k_{\parallel 0} \rho_i = k_\parallel \rho_i = 0.02$, consistent with [85], and previous results are exactly reproduced as expected. Fig.<ref>, meanwhile, assumes $k_{\parallel 0} \rho_i = k_\parallel \rho_i = 0.003$, consistent with typical values of the $\theta$-pinch equilibrium considered in the present analysis and with the aforementioned wave spectrum launched by the external antenna. It is evident that the behavior of the modulational instability is qualitatively the same as in [85] in the uniform plasma limit, upon replacing  $k_x \rho_i \mapsto k_0 \rho_i$ and $k_y \rho_i \mapsto k_z \rho_i$, the main quantitative difference being due to the different values of $k_\parallel \rho_i$. The plots in figs.<ref>-<ref> show the upper and low limits of instability domain: the upper limit is given by $|k_0|\approx |k_z|$ (or, in the first figure, $|k_x|\approx |k_y|$), namely, in the uniform limit, modulational instability can't generate a CC whose wavelength is shorter than the pump wavelength (see [85] for further discussion about this point). This result is the main motivation for the conditions maximizing the modulational instability given by (<ref>), (<ref>) and (<ref>). In fact, $\mathbf{k}_z$ must simultaneously satisfy the condition $|k_z| <  |k_0|$ as well as of being perpendicular to $\mathbf{k}_0$ (predominantly radial, consistent with the mode converted KAW pump) and to the ambient magnetic field. In what follows, we demonstrate that this condition is not qualitatively modified by the inclusion of diamagnetic effects.

Fig.<ref> Marginal stability curves in the $(k_x\rho_i,k_y\rho_i)$-plane obtained from the present analysis in the uniform plasma limit and for $k_\parallel \rho_i = 0.02$, consistently reproducing the original results for different values of $\delta B_y/B_0$ studied in [85, 17].

Fig.<ref> Plot of marginal stability for the uniform case. Here $k_z$ is the wave number (predominantly in the $\theta$-direction) of the CC, while $k_0$ is that of the pump KAW (predominantly in the radial direction). Results are consistent with [85], while quantitative differences are due to the value of $k_\parallel \rho_i = 0.003$. Note that the azimuthal KAW pump magnetic field fluctuation, $\delta B_{\theta 0}/B_0$, replaces $\delta B_y/B_0$ in fig.<ref>.

Let us now focus on the more general non-uniform plasma case, looking at the diamagnetic effect on marginal stability curves, CC real frequency and modulational instability growth rate, and at the related symmetry breaking effects. In fig.<ref> we see the result for different values of $\omega_{*i}$, showing the crucial quantitative effect of equilibrium plasma non-uniformity. Note that, similar to previous considerations made for taking the proper physically meaningful uniform plasma limit, we can vary independently $k_{ z} \rho_i$ and $\omega_{*i}$ by acting on the density gradient. For the present $\theta$-pinch cylindrical plasma equilibrium, the range of $\omega_{*i}$ values corresponds to $|m_z| = {\cal O}(10) \gg |m_0|$. Thus, as anticipated above, diamagnetic effects are important for CC and KAW sidebands but not for the KAW pump, consistent with equations (<ref>), (<ref>) and (<ref>). Looking at the upper part of the diagram, we see that, differently from the uniform case, CC modulational instability can occur at wavelengths shorter than that of the pump KAW: in fact, part of the region $|k_z|>|k_0|$ gives access to instability. Actually, non-uniformity also affects the lower bound; thus, in general, it extends the region of modulational instability both for lower and for higher CC wavelengths.

Next, we discuss the dependence of the CC real frequency, growth rate and polarization on the amplitude of the KAW pump, $\delta B_{\theta 0}/B_0$, which is connected with the radial electric field as $\delta B_{\theta 0}/B_0 =  (k_{\parallel 0}/\omega_0) \sigma_0 (c/B_0) \delta E_{r 0}$. The input quantities for the nonlinear local dispersion relation are $k_0\rho_i$ and $k_z\rho_i$, the amplitude of the pump wave $\delta B_{\theta 0}$, the CC diamagnetic frequency $\omega_{*i}/\omega_0$, and $k_{\parallel 0} \rho_i = k_\parallel$. The solution of the dispersion relation is the CC frequency $\omega_z/\omega_0$ and polarization $\delta \hat \psi_z/\delta \hat \phi_z$. In the very low dissipation limit that we are studying the dispersion relation is polynomial with real coefficients, so with real or complex conjugate roots: a conjugate couple is made by a stable solution and an unstable solution which is the one of our interest. Notably, from fig.<ref>, representing the growth rate $\text{Im}(\omega_z/\omega_0)$, we see that for low diamagnetic frequency we actually have two thresholds, represented by two "elbows" of the curves, due to different branches of the dispersion relation and to the complex interplay of nonlinearity and nonuniformity; these elbows fade into flexes for higher diamagnetic frequencies (orange and red lines). For $\omega_{*i}/\omega_0=0.3$ the change in slope of the growth rate vs. pump amplitude is almost inappreciable. Fig.<ref>, meanwhile, shows that the CC near marginal stability rotates in the electron diamagnetic direction (whose sign is opposite with respect to the ion diamagnetic frequency, taken here as positive), consistently with the results in [47]. However, for increasing instability the CC tends to rotate in the ion diamagnetic direction. This confirms the complex interplay of nonlinearity and nonuniformity, and that the KAW and drift-wave branches mix and couple in the nonlinear regime.

Fig.<ref> Marginal stability curves for the CC modulational instability as a function of plasma nonuniformity, accounted for by the local value of $\omega_{*i}/\omega_0$; in particular, $\omega_{*i}/\omega_0=0$ (black line) represents the uniform plasma limit. $\delta B_{\theta 0}/B_0 = 0.005$ is fixed and the black curve is consistent with the iso-lines displayed in fig.<ref>.

Fig.<ref> Imaginary CC frequency vs. pump KAW amplitude.

Fig.<ref> Real CC frequency vs. pump KAW amplitude.

Finally, we compute the radial dependence of modulational instability, that is, the growth rate and frequency vs. the radial position, along with the polarization of the CC. From fig.<ref> it is clear that the pump KAW is dominated by its short radial scale, typical of the mode converted fluctuation structure. However, it is also clear that the most "effective" CC excitation is characterized by relatively short azimuthal wavelength and relatively long radial scale, as argued above. Thus, the "effective" KAW amplitude at long radial scale is that shown in fig.<ref>, obtained from fig.<ref> for reduced dissipation ($1\%$ of the nominal value). For the KAW pump, $b_k$ is almost purely real, as we expected, and in the range of interest for the CC modulational excitation, in agreement with previous results. In fig.<ref> we can recognize the location of the resonant surface, where the value of $b_0$ becomes negative. The behavior of $b_0$ is essentially linear until close to the magnetic axis, where the effect of cylindrical geometry becomes important. In fig.<ref>, $b_z$ goes as $1/r^2$, as it reflects the dominant behavior of the azimuthal wave number. Meanwhile, the normalized parallel wave number, $b_\parallel = k_\parallel^2 \rho_i^2$, is shown in fig.<ref>. As a result, going from the resonant layer to the magnetic axis, we have a first zone where $b_{0}>b_{z}$, namely the pump KAW radial wavelength is shorter than the CC azimuthal wavelength. Then, starting at $r/a\approx 0.2$, the situation is already reversed and $b_{z}$ dominates. In the light of the stability diagram of fig.<ref>, we expect this fact to have important consequences on the radial dependence of the CC growth rate. It is essentially in the region $b_0\gtrsim b_z$ that we expect the strongest growth rate, as we see from fig.<ref> (between $r/a=0.2$ and $r/a=0.4$, approximately). Ultimately, fig.<ref> suggests that CC growth rate should be suppressed near axis, because of increasing $b_z$ and decreasing $\omega_{*i}$. Growth rate should also be reduced while approaching the resonant layer since $b_k$ goes to zero (even though $\omega_{*i}$ is increasing) and we need FLR to excite CCs. The polarization of the CC is shown in fig.<ref> and confirms the finding of [85, 17] that, in general, electrostatic and/or magnetostatic CCs are not self-consistent nonlinear solutions produced by modulational instability and the polarization state must be properly accounted for and determined case by case.

In order to understand how the CC radial structure can be controlled externally, let us analyze the behavior of the local CC frequency, growth rate and polarization vs. the amplitude of the KAW pump at different radial locations. Figs.<ref> and <ref> illustrate these behaviors at $r/a = 0.2$, thus, in a region where $b_0 \lesssim b_z$. Consistent with previous findings in figs.<ref>-<ref>, the CC is effectively excited only in a limited range of KAW pump strength. Meanwhile, figs.<ref>-<ref> show the same behaviors at $r/a = 0.4$, where the CC growth vs. KAW pump strength monotonically increases as expected for $b_0 > b_z$. Again, these results demonstrate the complex interplay of nonlinearity and plasma nonuniformity and suggest that controlling the external antenna perturbation amplitude is a very efficient way to control the CC radial structure.

The excitation of CCs is accompanied by the formation of inductive parallel electric field structures, which are readily derived in the form
For the present case, the radial structure of $\delta E_{\parallel z}$ in units of kV/m and normalized to $\delta \hat \psi_z$ is given in fig.<ref>. The role of parallel electric field is very important in space physics, viz. in Earth's magnetosphere, where it can significantly impact particle acceleration [85, 17]. The present study suggests that the parallel electric field values that are typically produced in laboratory plasmas for present parameters are rather small, in the order of a few tens of V/m, consistent with the typically low values observed in high-temperature magnetized plasmas of fusion interest; nonetheless, our study also suggests typical scalings with plasma parameters, which could be used for generating stronger electric fields; e.g., higher electron temperature and shorter parallel wavelength of the launched antenna wave spectrum.

§ SUMMARY

In this chapter we have performed a theoretical nonlinear analysis of KAWs in a cylindrical plasma equilibrium, focusing on the spontaneous generation of convective cells by a large KAW and examining a configuration which maximizes nonlinear coupling and minimizes Landau damping; then, we have computed the numerical solutions. Earlier results of convective cell excitation by KAWs in uniform plasmas [85] have been recovered in the proper limiting case. The effect of plasma nonuniformity has been shown to importantly affect the convective cell stability both qualitatively and quantitatively via the diamagnetic response [13, 47]. As new novel results, we have shown that the convective cell rotates in the electron diamagnetic direction near the critical excitation threshold, that the convective cell growth is significantly stronger (typically up to an order of magnitude) than in uniform plasmas, and that the unstable parameter region is significantly broader. Furthermore, we have shown the subtle interplay between nonlinearity and plasma nonuniformity. As a result, the plasma self-organization can be controlled by fine-tuning the amplitude of the antenna driven mode converted KAW. Finally, we have computed self-consistently the convective cell (generally mixed) polarization and radial structure of the generated inductive parallel electric field.

Fig.<ref> Normalized effective KAW azimuthal magnetic field vs. the radial coordinate $r/a$.

Fig.<ref> Normalized pump KAW radial wavenumber as a function of radial position.

Fig.<ref> Normalized CC azimuthal wavenumber and diamagnetic frequency as a function of radial position.

Fig.<ref> Normalized KAW parallel wavenumber as a function of radial position.

Fig.<ref> Normalized CC frequency and growth rate as a function or radial position.

Fig.<ref> Normalized CC polarization as a function or radial position.

Fig.<ref> Normalized CC frequency and growth rate vs. KAW pump strength at $r/a = 0.2$.

Fig.<ref> Convective cell polarization vs. KAW pump strength at $r/a = 0.2$.

Fig.<ref> Normalized CC frequency and growth rate vs. KAW pump strength at $r/a = 0.4$.

Fig.<ref> Convective cell polarization vs. KAW pump strength at $r/a = 0.4$.

Fig.<ref> Normalized CC parallel electric field as a function of $r/a$ expressed in units of kV/m.

CHAPTER: CONCLUSIONS AND OUTLOOK

In this Ph.D. thesis work we have studied the physics of shear and kinetic Alfvén waves (SAWs and KAWs) in a cylindrical non-uniform magnetized plasma. This is a well-known classical problem of fusion plasma physics, which we have revisited and proposed under new light and with new results.

In the first part, we have presented an in-depth review of the linear ideal magneto-hydro-dynamic (MHD) and gyrokinetic (GK) analyses of propagation and absorption of waves launched by an external antenna. The process of resonant wave absorption, at the radial position of the nonuniform plasma where the launched wave frequency matches that of the local SAW continuous spectrum, has been illustrated as the time asymptotic generation of increasingly shorter wavelengths (consistent with the classical approach of existing literature [37, 39]), eventually breaking the MHD long wavelength assumption and requiring a GK analysis. This natural tendency of a SAW to generate localized radial singular structures in the ideal MHD model corresponds, in the GK description, to a complete transformation of the wave nature from a macroscopic (SAW) to a microscopic (KAW) fluctuation, which exists on the typical spatial scale of the thermal ion Larmor radius [40, 41, 17]. In particular, we have shown that the SAW Poynting flux is progressively and wholly converted to the KAW Poynting flux on the Airy scale, intermediate between the system macro-scale and the ion Larmor radius [41]. This explains why, when the mode converted KAW is strongly absorbed within a few radial wavelengths from the mode conversion layer, the power absorbed by the plasma in the GK description is essentially the same as that calculated using ideal MHD. In other words, the antenna impedance remains the same,  as noted in [41, 44, 45], regardless of the model adopted for computing it.

The second part of this work focused on the linear and nonlinear behaviors of weakly damped mode converted KAW, that is on the case where the micro-scale fluctuation may propagate up to the magnetic axis without significant absorption. This case has little relevance, if any, for the classical problem of plasma heating, but it has great interest for analyzing the nonlinear behaviors of KAWs in non-uniform cylindrical plasmas. As such, this problem is the core of the novel contribution of the present Ph.D. thesis research. In the linear limit, we have demonstrated that the plasma region inside the mode conversion layer behaves as a "resonant cavity"; that is, when the antenna frequency matches a given discrete spectrum, the plasma response is stronger, exhibiting the clear behavior of a resonantly driven and weakly damped oscillator. Not surprisingly, this behavior, demonstrated here for the first time for the KAW spectrum, is qualitatively similar to that of mode converted electron Bernstein waves, described in [6]. This behavior also is of particular interest for our scope of investigating nonlinear behaviors induced by large amplitude KAWs. In fact, by proper tuning of the antenna frequency, mode converted KAW can be pumped at significantly high amplitude, time asymptotically, even with a relatively small antenna power. Thus, after a brief review of the parametric decay process underlying nonlinear wave excitation, we have focused on the spontaneous generation of convective cells (CCs) by modulational instability of a weakly damped large amplitude mode converted KAW in a non-uniform cylindrical plasma equilibrium; in particular, we have focused on a $\theta$-pinch equilibrium, with purely axial ambient magnetic field, in order to maximize nonlinear wave-wave couplings. Earlier results have been recovered in the limiting case of uniform plasmas [85]. Meanwhile, as new novel results, we have discussed and demonstrated qualitative and quantitative effects of equilibrium non-uniformity, accounted for by the diamagnetic plasma response. First of all, the convective cell preferentially rotates in the electron diamagnetic direction near the critical excitation threshold, consistent with the prediction for magnetostatic CCs [47]: this means that the parity is broken for rotations about the symmetry axis of the cylindrical equilibrium, and that plasma rotation of the self-organized nonlinear plasma equilibrium can be controlled by the choice of the antenna launched wave spectrum. Furthermore, the modulational instability growth rate is significantly stronger (typically up to an order of magnitude) than in uniform plasmas, and the unstable parameter region is significantly broader: this is analogous to the similar effect discussed recently in [13] for KAW parametric decay instability and, thus, confirms the crucial role of symmetry breaking wave-wave couplings in reactor-relevant fusion plasmas. As a novel result of this work, we have also shown that, due to the subtle interplay between nonlinearity and plasma non-uniformity, the self-organized nonlinear plasma equilibrium can be controlled by fine-tuning the amplitude of the mode converted KAW; in fact, local CC rotation frequency, growth rate and polarization crucially depend on diamagnetic plasma response as well as on the spectrum of both pump KAW and CC. This is a consequence of cylindrical geometry and causes, e.g., the CC growth rate not to generally have a monotonic behavior vs. the mode converted KAW amplitude. Finally, we have self-consistently computed the convective cell polarization, which is generally mixed, and the radial structure of the generated inductive parallel electric field.

The implications of the novel findings of this Ph.D. thesis work are diverse [25, 26] and all connected with the fact that Alfvén waves, as fundamental electromagnetic oscillations in magnetized plasmas, can importantly contribute to heating and transport processes of charged particles. Having adopted a nonuniform magnetized plasma
equilibrium in cylindrical geometry is of fundamental importance since it addresses the essential physics elements that must be included into realistic models. In particular, our analytical and numerical results on the generation of convective cells can serve as test-bed for gyrokinetic and fully kinetic or hybrid codes. In fact, the present work shows the importance of accurately accounting for multiple spatiotemporal scales on the same footing for the correct description of the
underlying physics, which can serve as precious verification test. At present, comparisons are in progress within the CNPS[Center for Nonlinear Plasma Science, https://www.afs.enea.it/zonca/CNPS/] collaboration network of our findings against TRIMEG  [56, 57] and STRUPHY [43] numerical simulation results. As a further and future step, comparisons will be extended to the gyrokinetic electron and fully kinetic ion description of GeFi code [54]. This effort will imply the first step in the implementation of GeFi in realistic toroidal fusion plasmas equilibria as well as the verification of the validity limits of the GK reduced description, which is the foundation of TRIMEG and of other GK codes.

Despite the simplified cylindrical geometry, adopted in this work, the novel results discussed here bear important implications for toroidal fusion plasmas applications. We introduced the key assumption of weak KAW absorption to investigate the properties of wave structures propagating farther away from the mode conversion layer and to shed light into the corresponding nonlinear physics in nonuniform plasmas. In toroidal plasmas, similar conditions could be realized and externally controlled, viz. in the weak magnetic shear core regions of tokamaks and stellarators [21]. Controlling the finite parallel electric field by magnetic field configuration would allow KAWs to reach magnetic axis and efficiently drive plasma current in the near-axis region by electron Landau damping of non-symmetric fluctuation spectrum.

Furthermore, it is worth noting that KAW spectrum in the toroidal fusion plasma core is expected to be nearly isotropic due to parametric decay processes enhanced by plasma nonuniformity [13] and strongly excited by energetic particles [16, 82, 83]. Thus, zonal field structures predominantly depending on the radial flux coordinate could be readily excited, while poloidally varying CCs would be suppressed due to the incompatibility of $k_\parallel = 0$ with finite magnetic shear. In other words, the situation would be opposite than here, where CCs dominate in the simplified $\theta$-pinch configuration and nonlinear excitation of zonal field structures is weaker. However, other than these notable but not so crucial differences, similarities of the case discussed in this work with the near-axis region in the core of toroidal plasmas is evident, as evident is the relevance of symmetry-breaking effects, connected with plasma nonuniformity, on convective transport and current drive [13].

Looking further ahead, it is worthwhile making a further comment on possible applications of the present findings due to the remarkable applications of KAWs in laboratory, space, and astrophysical plasmas. Our research results suggest an ideal "table-top" plasma experiment where, by means of an external antenna, one can control the formation of self-organized plasma states resulting from the nonlinear interaction between KAW and CC. Given the relatively simple cylindrical geometry, a proper set of diagnostics could be conceived with the proper spatiotemporal resolutions to actually measure the underlying physical processes and the corresponding characteristic spatial and temporal scales. In particular, this approach may apply to the generation of parallel electric fields with applications to particle acceleration and transport. In other words, the study proposed in this Ph.D. thesis work could result into a simple and useful paradigm for verifying and validating nonlinear physics, supported by theory, by use of advanced kinetic (GeFi), gyrokinetic (TRIMEG), and hybrid (STRUPHY) codes as well as of experimental measurements.

[1] H. Alfvén, Existence of electromagnetic-hydrodynamic waves, Nature 150.3805, 405 (1942)

[2] H. Alfvén, R.D. Richtmyer, E. Teller, On the origin of cosmic rays, Physical Review 75.5, 892 (1949)

[3] T.M. Antonsen Jr., B. Lane, Kinetic equations for low frequency instabilities in inhomogeneous plasmas, The Physics of Fluids 23.6, 1205 (1980)

[4] C.K. Birdsall, A.B. Langdon, Plasma physics via computer simulation, CRC press (2004)

[5] A. Brizard, Nonlinear gyrokinetic Maxwell-Vlasov equations using magnetic co-ordinates, Journal of Plasma Physics 41.3, 541 (1989)

[6] S.J. Buchsbaum, A. Hasegawa, Longitudinal plasma oscillations
near electron cyclotron harmonics, Physical Review 143.1, 303 (1966)

[7] P.J. Catto, Linearized gyro-kinetics, Plasma Physics 20.7, 719 (1978)

[8] C.C. Chaston, J.W. Bonnell, J.P. McFadden, C.W. Carlson, C. Cully, O. Le Contel, A. Roux, H.U. Auster, K.H. Glassmeier, V. Angelopoulos, C.T. Russell, Turbulent heating and cross-field transport near the magnetopause from THEMIS, Geophysical Research Letters 35.17 (2008)

[9] C.C. Chaston, J.W. Bonnell, J.R. Wygant, F. Mozer, S.D. Bale, K. Kersten, A.W. Breneman, C.A. Kletzing, W.S. Kurth, G.B. Hospodarsky, C.W. Smith, E.A. MacDonald, Observations of kinetic scale field line resonances, Geophysical Research Letters 41.2, 209 (2014)

[10] F.F. Chen, An indispensable truth: how fusion power can save the planet, Springer (2011)

[11] F.F. Chen, Introduction to plasma physics and controlled fusion, Springer (2016)

[12] L. Chen, Theory of plasma transport induced by low‐frequency hydromagnetic waves, Journal of Geophysical Research: Space Physics 104.A2, 2421 (1999)

[13] L. Chen, Z. Qiu, F. Zonca, Parity-breaking parametric decay instability of kinetic Alfvén waves in a nonuniform plasma, Physics of Plasmas 29.5, 050701 (2022)

[14] L. Chen, F. Zonca, Gyrokinetic theory of parametric decays of kinetic Alfvén waves, Europhysics Letters 96.3, 35001 (2011)

[15] L. Chen, F. Zonca, On nonlinear physics of shear Alfvén waves, Physics of Plasmas 20.5, 055402 (2013)

[16] L. Chen, F. Zonca, Physics of Alfvén waves and energetic particles in burning plasmas, Reviews of Modern Physics 88.1, 015008 (2016)

[17] L. Chen, F. Zonca, Y. Lin, Physics of kinetic Alfvén waves: a gyrokinetic theory approach, Reviews of Modern Plasma Physics 5.1, 1 (2021)

[18] C.Z. Cheng, H. Okuda, Formation of convective cells, anomalous diffusion, and strong plasma turbulence due to drift instabilities, Physical Review Letters 38.13, 708 (1977)

[19] P.J. Chi, C.T. Russell, G. Le, Pc 3 and Pc 4 activity during a long period of low interplanetary magnetic field cone angle as detected across the Institute of Geological Sciences array, Journal of Geophysical Research: Space Physics 99.A6, 11127 (1994)

[20] C. Chu, M.S. Chu, T. Ohkawa, Magnetostatic mode and cross-field electron transport, Physical Review Letters 41.9, 653 (1978)

[21] M.S. Chu, V.S. Chan, P.A. Politzer, D.P. Brennan, M. Choi, L.L. Lao, H.E. St. John, A.D. Turnbull, Kinetic Alfvén wave and associated current drive at the center of tokamaks, Physics of Plasmas 13.11, 114501 (2006)

[22] L.B.N. Clausen, T.K. Yeoman, R.C. Fear, R. Behlke, E.A. Lucek, M.J. Engebretson, First simultaneous measurements of waves generated at the bow shock in the solar wind, the magnetosphere and on the ground, Annales Geophysicae Vol. 27 No. 1, Copernicus GmbH (2009)

[23] N.F. Cramer, The physics of Alfvén waves, John Wiley & Sons (2011)

[24] J.M. Dawson, Particle simulation of plasmas, Reviews of Modern Physics 55.2, 403 (1983)

[25] L. De Fabrizio, M.V. Falessi, Z. Qiu, F. Zonca, Linear and nonlinear kinetic Alfvén wave physics in cylindrical plasmas, AAPPS-DPP2022 Remote E-conference FP-9, October 10-14, 2022

[26] L. De Fabrizio et al., Linear and nonlinear kinetic Alfvén wave physics in cylindrical plasmas, submitted to Physics of Plasmas

[27] P.H. Diamond, S.-I. Itoh, K. Itoh, T.S. Hahm, Zonal flows in plasma—a review, Plasma Physics and Controlled Fusion 47.5, R35 (2005)

[28] M.J. Engebretson, L.J. Zanetti, T.A. Potemra, W. Baumjohann, H. Lühr, M.H. Acuna, Simultaneous observation of Pc 3–4 pulsations in the solar wind and in the Earth's magnetosphere, Journal of Geophysical Research: Space Physics 92.A9, 10053 (1987)

[29] M.J. Engebretson, N. Lin, W. Baumjohann, H. Luehr, B.J. Anderson, L.J. Zanetti, T.A. Potemra, R.L. McPherron, M.G. Kivelson, A comparison of ULF fluctuations in the solar wind, magnetosheath, and dayside magnetosphere, Journal of Geophysical Research: Space Physics 96.A3, 3441 (1991)

[30] D.H. Fairfield, W. Baumjohann, G. Paschmann, H. Lühr, D.G. Sibeck, Upstream pressure variations associated with the bow shock and their effects on the magnetosphere, Journal of Geophysical Research: Space Physics 95.A4, 3773 (1990)

[31] M.V. Falessi, F. Zonca, Transport theory of phase space zonal structures, Physics of Plasmas 26.2, 022305 (2019)

[32] A. Fasoli, J.B. Lister, S. Sharapov, D. Borba, N. Deliyanakis, C. Gormezano, J. Jacquinot, A. Jaun, H.A. Holties, G.T.A. Huysmans, W. Kerner, J.-M. Moret, L. Villard, Observation of multiple kinetic Alfvén eigenmodes, Physical Review Letters 76.7,
1067 (1996)

[33] E. Fermi, On the origin of the cosmic radiation, Physical Review 75.8, 1169 (1949)

[34] J.P. Freidberg, Ideal MHD, Cambridge University Press (2014)

[35] E.A. Frieman, L. Chen, Nonlinear gyrokinetic equations for low-frequency electromagnetic waves in general plasma equilibria, The Physics of Fluids 25.3, 502 (1982)

[36] X. Garbet, Y. Idomura, L. Villard, T.H. Watanabe, Gyrokinetic simulations of turbulent transport, Nuclear Fusion 50.4, 043002 (2010)

[37] H. Grad, Plasmas, Physics Today 22.12, 34 (1969)

[38] T.S. Hahm, W.W. Lee, A. Brizard, Nonlinear gyrokinetic theory for finite-beta plasmas, The Physics of Fluids 31.7, 1940 (1988)

[39] A. Hasegawa, L. Chen, Plasma heating by Alfvén-wave phase mixing, Physical Review Letters 32.9, 454 (1974)

[40] A. Hasegawa, L. Chen, Kinetic process of plasma heating due to Alfvén wave excitation, Physical Review Letters 35.6, 370 (1975)

[41] A. Hasegawa, L. Chen, Kinetic processes in plasma heating by resonant mode conversion of Alfvén wave, The Physics of Fluids 19.12, 1924 (1976)

[42] A. Hasegawa, C.G. Maclennan, Y. Kodama, Nonlinear behavior and turbulence spectra of drift waves and Rossby waves, The Physics of Fluids 22.11, 2122 (1979)

[43] F. Holderied, S. Possanner, X. Wang, MHD-kinetic hybrid code based on structure-preserving finite elements with particles-in-cell, Journal of Computational Physics 433, 110143 (2021)

[44] S.-I. Itoh, K. Itoh, K. Nishikawa, Kinetic Alfven wave propagation in cylindrical Tokamak, Plasma Physics 24.9, 1027 (1982)

[45] K. Itoh, S.-I. Itoh, Energy deposition profile of shear Alfven wave heating, Plasma Physics 25.9, 1037 (1983)

[46] E.A. Jackson, Parametric effects of radiation on a plasma, Physical Review 153.1, 235 (1967)

[47] P.K. Kaw, L. Chen, Nonlinear excitation of magnetostatic fluctuations by kinetic drift‐Alfvén waves, The Physics of Fluids 26.6, 1382 (1983)

[48] I. Langmuir, Scattering of electrons in ionized gases, Physical Review 26.5, 585 (1925)

[49] P. Lauber, Super-thermal particles in hot plasmas—Kinetic models, numerical solution strategies, and comparison to tokamak experiments, Physics Reports 533.2, 33 (2013)

[50] L.C. Lee, J.R. Johnson, Z.W. Ma, Kinetic Alfvén waves as a source of plasma transport at the dayside magnetopause, Journal of Geophysical Research: Space Physics 99.A9, 17405 (1994)

[51] W.W. Lee, Gyrokinetic approach in particle simulation, The Physics of Fluids 26.2, 556 (1983)

[52] W.W. Lee, Gyrokinetic particle simulation model, Journal of Computational Physics 72.1, 243 (1987)

[53] A.T. Lin, J.M. Dawson, H. Okuda, Thermal magnetic fluctuations and anomalous electron diffusion, Physical Review Letters 41.11, 753 (1978)

[54] Y. Lin, X.Y. Wang, Z. Lin, L. Chen, A gyrokinetic electron and fully kinetic ion plasma simulation model, Plasma Physics and Controlled Fusion 47.4, 657 (2005)

[55] C.S. Liu, M.N. Rosenbluth, Parametric decay of electromagnetic waves into two plasmons and its consequences, The Physics of Fluids 19.7, 967 (1976)

[56] Z.X. Lu, P. Lauber, T. Hayward-Schneider, A. Bottino, M. Hoelzl, Development and testing of an unstructured mesh method for whole plasma gyrokinetic simulations in realistic tokamak geometry, Physics of Plasmas 26.12, 122503 (2019)

[57] Z.X. Lu, G. Meng, M. Hoelzl, P. Lauber, The development of an implicit full f method for electromagnetic particle simulations of Alfvén waves and energetic particle physics, Journal of Computational Physics 440, 110384 (2021)

[58] R.R. Mett, S.M. Mahajan, Kinetic theory of toroidicity-induced Alfvén eigenmodes, Physics of Fluids B: Plasma Physics 4.9, 2885 (1992)

[59] A.B. Mikhailovskii, E.A. Kovalishen, M.S. Shirokov, A.I. Smolyakov, V.S. Tsypin, R.M.O. Galv$\mathrm{\tilde{a}}$o, Generation of zonal flows by kinetic Alfvén waves, Plasma Physics Reports 33.2, 117 (2007)

[60] H. Okuda, J.M. Dawson, Theory and numerical simulation on plasma diffusion across a magnetic field, The Physics of Fluids 16.3, 408 (1973)

[61] O.A. Onishchenko, O.A. Pokhotelov, M.A. Balikhin, L. Stenflo, R.Z. Sagdeev, Suppression of convective cell generation by Alfvén waves in the ionospheric auroral cavity, Physics of Plasmas 11.11, 4954 (2004)

[62] O.G. Onishchenko, O.A. Pokhotelov, R.Z. Sagdeev, L. Stenflo, R.A. Treumann, M.A. Balikhin, Generation of convective cells by kinetic Alfvén waves in the upper ionosphere, Journal of Geophysical Reearch: Space Physics 109.A3 (2004)

[63] O.A. Pokhotelov, O.G. Onishchenko, R.Z. Sagdeev, R.A. Treumann, Nonlinear dynamics of inertial Alfvén waves in the upper ionosphere: Parametric generation of electrostatic convective cells, Journal of Geophysical Research: Space Physics 108.A7 (2003)

[64] O.A. Pokhotelov, O.G. Onishchenko, R.Z. Sagdeev, M.A. Balikhin, L. Stenflo, Parametric interaction of kinetic Alfvén waves with convective cells, Journal of Geophysical Research: Space Physics 109.A3 (2004)

[65] Z. Qiu, Theories of geodesic acoustic modes in tokamak plasmas, Ph.D. Thesis, University of Science and Technology of China, Hefei, PRC (2010)

[66] Z. Qiu, F. Zonca, L. Chen, Kinetic theories of geodesic acoustic modes: Radial structure, linear excitation by energetic particles and nonlinear saturation, Plasma Science and Technology 13.3, 257 (2011)

[67] L. Rezeau, A. Morane, S. Perraut, A. Roux, R. Schmidt, Characterization of Alfvenic fluctuations in the magnetopause boundary layer, Journal of Geophysical Research: Space Physics 94.A1, 101 (1989)

[68] R.Z. Sagdeev, A.A. Galeev, Nonlinear plasma theory, Rev. and Edited by T.M. O'Neil and D.L. Book, W.A. Benjamin (1969)

[69] C.S. Salem, G.G. Howes, D. Sundkvist, S.D. Bale, C.C. Chaston, C.H.K. Chen, F.S. Mozer, Identification of kinetic Alfvén wave turbulence in the solar wind, The Astrophysical Journal Letters 745.1, L9 (2012)

[70] P.K. Shukla, M.Y. Yu, H.U. Rahman, K.H. Spatscheck, Excitation of convective cells by drift waves, Physical Review A 23.1, 321 (1981)

[71] P.K. Shukla, M.Y. Yu, H.U. Rahman, K.H. Spatscheck, Nonlinear convective motion in plasmas, Physics Reports 105.4-5, 227 (1984)

[72] P. Song, C. Russell, R. Strangeway, J. Wygant, C. Cattell, R. Fitzenreiter, R. Anderson, Wave properties near the subsolar magnetopause: Pc 3-4 energy coupling for northward interplanetary magnetic field, Journal of Geophysical Research: Space Physics 98.A1, 187 (1993)

[73] W.M. Stacey, Fusion plasma physics, John Wiley & Sons (2012)

[74] T.H. Stix, Waves in plasmas, Springer Science & Business Media (1992)

[75] J.B. Taylor, B. McNamara, Plasma diffusion in two dimensions, The Physics of Fluids 14.7, 1492 (1971)

[76] C. Walén, On the theory of sun-spots, Arkiv för matematik, astronomi och fysik 30A no.15, 1 (1944)

[77] B. Wang, Y. Nishimura, H. Zhang, X.-C. Shen, L. Lyons, V. Angelopoulos, Y. Ebihara, A. Weatherwax, A.J. Gerrard, H.U. Frey, The 2‐D structure of foreshock‐driven field line resonances observed by THEMIS satellite and ground‐based imager conjunctions, Journal of Geophysical Research: Space Physics 124.8, 6792 (2019)

[78] K.L. Wong, N. Bretz, G.Y. Fu, J. Machuzak, J.R. Wilson, Z. Chang, L. Chen, D.K. Owens, G. Schilling, Evidence of coupling between toroidal Alfvén eigenmodes and kinetic Alfvén waves, Physics Letters A 224.1-2, 99 (1996)

[79] D.-J. Wu, Kinetic Alfvén wave: Theory, experiment, and application, Science Press (2012)

[80] D.-J. Wu, L. Chen, Kinetic Alfvén waves in laboratory, space, and astrophysical plasmas, Springer (2020)

[81] Z. Zhang, Z. Yuan, S. Huang, X. Yu, Z. Xue, D. Deng, Z. Huang, Observations of kinetic Alfvén waves and associated electron acceleration in the plasma sheet boundary layer, Earth and Planetary Physics 6 (2022)

[82] J.S. Zhao, D.J. Wu, J.Y. Lu, L. Yang, M.Y. Yu, Generation of convective cells by kinetic Alfvén waves, New Journal of Physics 13.6, 063043 (2011)

[83] F. Zonca, L. Chen, S. Briguglio, G. Fogaccia, A.V. Milovanov, Z. Qiu, G. Vlad, X. Wang, Energetic particles and multi-scale dynamics in fusion plasmas, Plasma Physics and Controlled Fusion 57.1, 014024 (2014)

[84] F. Zonca, L. Chen, S. Briguglio, G. Fogaccia, G. Vlad, X. Wang, Nonlinear dynamics of phase space zonal structures and energetic particle physics in fusion plasmas, New Journal of Physics 17.1, 013052 (2015)

[85] F. Zonca, Y. Lin, L. Chen, Spontaneous excitation of convective cells by kinetic Alfvén waves, Europhysics Letters 112.6, 65001 (2015)

[86] F. Zonca, L. Chen, M.V. Falessi, Z. Qiu, Nonlinear radial envelope evolution equations and energetic particle transport in tokamak plasmas, Journal of Physics: Conference Series Vol. 1785 No. 1, IOP Publishing (2021)