1. Introduction
Plasma heating and current drive by radio-frequency (RF) waves in magnetically confined thermonuclear fusion scenarios are always coupled with parametric instabilities (PI). The PI process is induced when the RF wave injected into the plasma couples with a low-frequency plasma fluctuation and scatters into a sideband decay mode. Consequently, anomalous power loss of the RF wave occurs, which may result in the loss of heating and current drive efficiency [1–3] or an intensive reflection of the RF energy that might damage the antennas [4–6].
Two sets of saturation mechanisms are mainly considered for PI. In the PI scenarios with a strong field pump wave, high-order nonlinearities including secondary wave-wave coupling [7] and detuning due to particle trapping [8] serve as the major saturation mechanisms. Whereas convective saturation due to plasma inhomogeneity [9, 10] and finite coupling region of PI [11, 12] are mainly considered when the pump electric field is assumed weak and cannot drive the high-order nonlinearities. The separatrix of the two sets of mechanisms is the threshold of the absolute instability, which means a well-behaved temporal eigenmode of the coupling equation of PI in an inhomogeneous plasma and finite pump region that can grow to infinity as t → ∞. White et al [13] derived the threshold of absolute parametric instability in a homogeneous plasma when both daughter waves are resonant
$\begin{eqnarray}{\gamma }_{0}\gt \frac{\sqrt{{v}_{1}{v}_{L}}}{2}\left(\frac{{\nu }_{1}}{{v}_{1}}+\frac{{\nu }_{L}}{{v}_{L}}\right).\end{eqnarray}$
Here, γ0 is the linear growth rate of PI, subscripts j = 1, L refer to the sideband daughter wave and the low-frequency daughter wave respectively, vj and νj represent the group velocities and damping rates of the daughter waves.In laser plasma interaction scenarios, the intensive lasers easily exceed the threshold of absolute instability regardless of inhomogeneous plasma [14], hence the high-order nonlinearities are of major concern. On the contrary, for the RF heating and current drive cases in magnetically confined plasma, previous studies usually assume the pump wave to be weaker than the threshold in equation (1 ) due to the lower electric field of the RF waves compared to the intensive lasers and the highly damped low-frequency modes involved in these PI scenarios [1, 12, 15]. However, as the power intensity of the RF waves (and consequently, the linear growth rate) increases with the development of fusion devices, the threshold introduced by equation (1 ) might be achieved in RF heating and current drive scenarios for contemporary and future fusion devices. For instance, in the case of lower hybrid current drive (LHCD), recent results [16] show that the linear growth rate of the PI induced in the scrape-off layer (SOL) plasma exceeds the level of ion-cyclotron frequency at high plasma density with the RF power of several megawatts, which is sufficient to induce the absolute parametric instability if we simply consider the threshold equation (1 ) of resonant decay. Note that the anomalous power loss of the lower hybrid wave in the SOL region due to PI has been the major challenge of applying LHCD to drive plasma current in high density plasma for a long time [2, 17–19]; it becomes necessary to re-investigate the problem of absolute instability to clarify the saturation mechanism of PI for such cases.
The problem of absolute instability becomes complicated when we consider PIs induced in RF heating and current drive scenarios, for they usually involve non-resonant daughter waves (i.e. low-frequency quasi-modes that are highly damped). Such quasi-mode PI leads to coupling equations with complex parameters [20], hence the problem of absolute instability turns to an eigenvalue problem of a Schrodinger equation with non-Hermitian Hamiltonian. The group velocities and damping rates of equation (1 ) also have imaginary parts. Although remarkable efforts have been made to solve the eigenvalue problem of absolute instability since the 1970s [21–26], the mathematical methods introduced in these studies are mainly based on ordinary Schrodinger equations with Hermitian Hamiltonian, thus their conclusions about absolute instability become inapplicable when non-resonant daughter waves are involved. Consequently, previous studies on the PI process in RF-plasma interaction scenarios usually neglect the possibility of stimulating an absolute instability [1, 15] without further clarification. However, as the clarification of the excitation of absolute instability gets essential for LHCD induced PI and other RF heating and current drive scenarios, it becomes imperative to revisit the non-Hermitian eigenvalue problem of the quasi-mode PI to present an applicable threshold of absolute instability of such cases.
In this study, the Wentzel–Kramers–Brillouin (WKB) analysis on the entire complex plane (instead of the WKB analysis near the real axis, which is applied in previous literature [27]) is applied to solve the eigenvalue problem of the coupling equations of quasi-mode PIs. In section 2 , we introduce the basic concept of WKB analysis on the complex plane and reconsider the eigenvalue problem of absolute instability. In section 3 , we solve the eigenvalue problem under different circumstances to obtain the threshold of absolute instability. Major conclusions and discussions are presented in section 4 .
2. Eigenvalue problem of absolute instability
2.1. Coupling equations
We start from the coupling equations of a quasi-mode PI in an inhomogeneous plasma with a finite pump profile, which is obtained under the electrostatic approximation and the WKB approximation. For the PI interactions in RF heating and current drive scenarios that are dominated by electrostatic effects [20, 28], the coupling equations for a quasi-mode PI have a similar form as the coupling equations of a resonant PI, except that some of the parameters might be complex. Equation (2 ) represents the coupling equations for a quasi-mode PI.
$\begin{eqnarray}\begin{array}{rcl}\left[{\epsilon }_{1}+\frac{\partial {\epsilon }_{1}}{\partial {\omega }_{1}}\cdot {\rm{i}}\frac{\partial }{\partial t}-\frac{\partial {\epsilon }_{1}}{\partial {k}_{1x}}\cdot {\rm{i}}\frac{\partial }{\partial x}\right]{\phi }_{1} & = & {\alpha }_{L\to 1}{\phi }_{0}^{* }(x){\phi }_{L},\\ \left[{\epsilon }_{L}+\frac{\partial {\epsilon }_{L}}{\partial {\omega }_{L}}\cdot {\rm{i}}\frac{\partial }{\partial t}-\frac{\partial {\epsilon }_{L}}{\partial {k}_{Lx}}\cdot {\rm{i}}\frac{\partial }{\partial x}\right]{\phi }_{L} & = & {\alpha }_{1\to L}{\phi }_{0}(x){\phi }_{1},\end{array}\end{eqnarray}$
in which εj refers to the dielectric function of the wave j, and αi→j the coupling coefficient from the daughter wave i to the daughter wave j. The profile of the pump wave is presented by ${\phi }_{0}\left(x\right)$. Considering the variation of εj due to plasma inhomogeneity, we have $\begin{eqnarray}\begin{array}{rcl}\left[{\rm{i}}{\rm{\Delta }}{k}_{1}^{\left(0\right)}-{\rm{i}}{\rm{\Delta }}{k}_{1}^{L}(x)+\frac{1}{{v}_{1gx}}\frac{\partial }{\partial t}+\frac{\partial }{\partial x}\right]{\phi }_{1} & = & \frac{{\alpha }_{L\to 1}{\phi }_{0}^{* }}{\partial {\epsilon }_{1}/\partial {k}_{1x}}{\rm{i}}{\phi }_{L},\\ \left[{\rm{i}}{\rm{\Delta }}{k}_{L}^{\left(0\right)}-{\rm{i}}{\rm{\Delta }}{k}_{L}^{L}(x)+\frac{1}{{v}_{Lgx}}\frac{\partial }{\partial t}+\frac{\partial }{\partial x}\right]{\phi }_{L} & = & \frac{{\alpha }_{1\to L}{\phi }_{0}}{\partial \epsilon /\partial {k}_{Lx}}{\rm{i}}{\phi }_{1},\end{array}\end{eqnarray}$
where $\begin{eqnarray}{\rm{\Delta }}{k}_{j}^{\left(0\right)}=\frac{{\epsilon }_{j}{| }_{x=0}}{\partial {\epsilon }_{j}/\partial {k}_{jx}},\quad {\rm{\Delta }}{k}_{j}^{L}=-\frac{\partial {\epsilon }_{j}/\partial x}{\partial {\epsilon }_{j}/\partial {k}_{jx}}\cdot x.\end{eqnarray}$
Since equation (2 ) is obtained from the Taylor expansion of the nonlinear dispersion relation of PI in a homogeneous plasma based on the WKB approximation [20], the ∂εj/∂kjx term and the ∂εj/∂ωj are considered constants. As a result, the definition of ${\rm{\Delta }}{k}_{j}^{L}$ presented in equation (4 ) ensures ${\epsilon }_{j}\left({\omega }_{j},{k}_{jx}+{\rm{\Delta }}{k}_{j}^{L}\right)$ stays constant for different x. Here, the ∂εj/∂x term is also treated as a constant, thus the effects of inhomogeneous plasma can be evaluated by the linear expansion of εj. When wave j is resonant, ${\rm{\Delta }}{k}_{j}^{L}$ degrades to the variation of the wavenumber calculated by the linear dispersion relation.
When absolute instability is induced, the amplitude of the daughter wave φj can be treated as a superposition of a series of temporal eigenmodes 3 ) become 6 ) to the following ordinary differential equation 7 ) comes from the spatial variation of the pump wave, in which both the wavenumber variation and the finite pump profile are included. The wavenumber variation of the pump wave ${\rm{\Delta }}{k}_{0}^{L}$ can be directly calculated by the linear dispersion relation, thus we can separate the $\frac{{\rm{d}}{\phi }_{0}^{* }}{{\phi }_{0}^{* }{\rm{d}}x}$ term into a $-{\rm{i}}{\rm{\Delta }}{k}_{0}^{L}$ term representing the variation of wavenumber and a $\frac{{\rm{d}}{\phi }_{0}^{* }}{{\phi }_{0}^{* }{\rm{d}}x}+{\rm{i}}{\rm{\Delta }}{k}_{0}^{L}$ term representing the finite profile of the pump wave. For further simplification we set $\kappa ={\rm{\Delta }}{k}_{L}^{L}-{\rm{\Delta }}{k}_{0}^{L}-{\rm{\Delta }}{k}_{1}^{L}={\kappa }^{{\prime} }x$ to represent the effect of inhomogeneous plasma, and consider the following transformation 7 ) to a Schrodinger equation
$\begin{eqnarray}{\phi }_{j}\left(x,t\right)=\displaystyle \sum _{m}{\phi }_{jm}\left(x\right)\exp \left({p}_{m}t\right).\end{eqnarray}$
For each eigenvalue, the coupling equation ( $\begin{eqnarray}\begin{array}{rcl}\left[{\rm{i}}{\rm{\Delta }}{k}_{1}^{\left(0\right)}-{\rm{i}}{\rm{\Delta }}{k}_{1}^{L}(x)+\frac{p}{{v}_{1gx}}+\frac{\partial }{\partial x}\right]{\phi }_{1} & = & \frac{{\alpha }_{L\to 1}{\phi }_{0}^{* }}{\partial {\epsilon }_{1}/\partial {k}_{1x}}{\rm{i}}{\phi }_{L}.\\ \left[{\rm{i}}{\rm{\Delta }}{k}_{L}^{\left(0\right)}-{\rm{i}}{\rm{\Delta }}{k}_{L}^{L}(x)+\frac{p}{{v}_{Lgx}}+\frac{\partial }{\partial x}\right]{\phi }_{L} & = & \frac{{\alpha }_{1\to L}{\phi }_{0}}{\partial \epsilon /\partial {k}_{Lx}}{\rm{i}}{\phi }_{1}.\end{array}\end{eqnarray}$
Here we apply a formal derivation similar to White [13], eliminating the φL terms to transform equation ( $\begin{eqnarray}\begin{array}{l}\left[{\rm{i}}{\rm{\Delta }}{k}_{L}^{\left(0\right)}-{\rm{i}}{\rm{\Delta }}{k}_{L}^{L}(x)-\frac{{\rm{d}}{\phi }_{0}^{* }}{{\phi }_{0}^{* }{\rm{d}}x}+\frac{p}{{v}_{Lgx}}+\frac{\partial }{\partial x}\right]\\ \quad \times \left[{\rm{i}}{\rm{\Delta }}{k}_{1}^{\left(0\right)}-{\rm{i}}{\rm{\Delta }}{k}_{1}^{L}(x)+\frac{p}{{v}_{1gx}}+\frac{\partial }{\partial x}\right]\\ \quad \times {\phi }_{1}=-\frac{{\alpha }_{1\to L}{\alpha }_{L\to 1}{\left|{\phi }_{0}(x)\right|}^{2}}{(\partial {\epsilon }_{1}/\partial {k}_{1x})(\partial {\epsilon }_{L}/\partial {k}_{Lx})}{\phi }_{1}.\end{array}\end{eqnarray}$
The $\frac{{\rm{d}}{\phi }_{0}^{* }}{{\phi }_{0}^{* }{\rm{d}}x}$ term in equation ( $\begin{eqnarray}{A}_{1}={\phi }_{1}\exp \left[\frac{i}{4}{\kappa }^{{\prime} }{x}^{2}-\frac{\mu }{2}x\right],\end{eqnarray}$
where $\mu =\frac{p}{{v}_{1gx}}+\frac{p}{{v}_{Lgx}}+{\rm{i}}\left({\rm{\Delta }}{k}_{1}^{(0)}+{\rm{\Delta }}{k}_{L}^{(0)}\right)$. We then transform equation ( $\begin{eqnarray}\begin{array}{l}\frac{{\partial }^{2}}{\partial {x}^{2}}{A}_{1}(x,p)-{V}_{A}(x){A}_{1}(x,p)=0,\\ {V}_{A}(x)=\frac{1}{4}{\left(i{\kappa }^{{\prime} }x+\gamma (x)+\sigma \right)}^{2}-\lambda -\frac{{\rm{i}}{\kappa }^{{\prime} }+{\gamma }^{{\prime} }(x)}{2},\end{array}\end{eqnarray}$
in which $\begin{eqnarray}\begin{array}{rcl}\gamma & = & \frac{1}{{\phi }_{0}^{* }}\frac{{\rm{d}}{\phi }_{0}^{* }}{{\rm{d}}x}+{\rm{i}}{\rm{\Delta }}{k}_{0}^{L},\\ \lambda & = & \frac{{\alpha }_{1\to L}{\alpha }_{L\to 1}{\left|{\phi }_{0}(x)\right|}^{2}}{(\partial {\epsilon }_{1}/\partial {k}_{1x})(\partial {\epsilon }_{L}/\partial {k}_{Lx})},\\ \sigma & = & {\rm{i}}\left({\rm{\Delta }}{k}_{1}^{(0)}-{\rm{\Delta }}{k}_{L}^{(0)}\right)+\frac{p}{{v}_{1gx}}-\frac{p}{{v}_{Lgx}},\end{array}\end{eqnarray}$
Here, γ refers to the profile of the pump wave, λ refers to the coupling coefficient of PI, and σ is equivalent to the summation of damping rates.For simplicity, we consider a Gaussian profile for the pump wave $\gamma (x)=-2x/{l}_{0}^{2}$ where l0 the length of the pump region, thus the Hamiltonian for Schrodinger equation (9 ) takes the following form 9 ) with x1, x2 two adjacent zeros of VA. However, for the quasi-mode PI scenarios, Im(σ) is not negligible, and equation (12 ) becomes invalid. We must turn to the WKB analysis on the full complex plane to solve such eigenvalue problem.
$\begin{eqnarray}{V}_{A}=\frac{1}{4}{\left(\left(2-{\rm{i}}\theta \right)\frac{x}{{l}_{0}^{2}}-\sigma \right)}^{2}+\frac{2-{\rm{i}}\theta }{2{l}_{0}^{2}}-{\lambda }_{0}{{\rm{e}}}^{-{x}^{2}/{l}_{0}^{2}},\end{eqnarray}$
where $\theta ={\kappa }^{{\prime} }{l}_{0}^{2}$ indicates the phase mismatch due to inhomogeneous plasma in the pump region, λ0 refers to the coupling coefficient λ at the center of the pump region x = 0. VA becomes real if we consider θ ≪ 1 and neglect the imaginary part of σ, thus the Bohr–Sommerfeld approximation $\begin{eqnarray}{\int }_{{x}_{1}}^{{x}_{2}}\sqrt{{V}_{A}}{\rm{d}}x=\left(n+\frac{1}{2}\right){\rm{i}}\pi ,\quad n\in {\mathbb{N}},\end{eqnarray}$
becomes applicable for the eigenvalue problem equation (2.2. Boundary conditions and WKB analysis
Applying the WKB method to equation (9 ) to get the WKB solutions A1± and corresponding φ1±, as x approaching infinity, we get 6 ) and get WKB solutions φL±. Take φL+ corresponding to linear propagation and φL− driven by the sideband, then the asymptotic behavior of φL satisfies
$\begin{eqnarray}\begin{array}{rcl}{\phi }_{1+} & \sim & \exp \left[-\left({\rm{i}}{\rm{\Delta }}{k}_{1}^{(0)}+\frac{p}{{v}_{1gx}}\right)x\right].\\ {\phi }_{1-} & \sim & \exp \left[-\left({\rm{i}}{\rm{\Delta }}{k}_{L}^{(0)}+\frac{p}{{v}_{Lgx}}\right)x+\displaystyle \int \left({\rm{i}}{\kappa }^{{\prime} }x+\gamma \right){\rm{d}}x\right].\end{array}\end{eqnarray}$
From the exponential terms we find that the WKB solution φ1+ corresponds to a linearly propagating sideband wave, whereas φ1− corresponds to a sideband wave driven by the coupling of the pump wave and the low frequency daughter wave. We express the boundary conditions through the asymptotic behavior of φ1 as x → ∞ $\begin{eqnarray}{\phi }_{1}\sim {c}_{1}{\phi }_{1+}+{c}_{2}{\phi }_{1-},\end{eqnarray}$
in which the coefficients c1, c2 might differ by directions. Note that we can apply similar analysis to obtain an eigenvalue problem of φL by eliminating φ1 in equation ( $\begin{eqnarray}{\phi }_{L}\sim {c}_{1}{\phi }_{L-}+{c}_{2}{\phi }_{L+}.\end{eqnarray}$
Since φ1+ and φL+ are convergent only in one direction, the trivial solution would be the only solution to let φ1,L behave well as x → ∞ if φ1+, φL+ were convergent in the same direction. Therefore, we take φL+ convergent as x → + ∞ and φ1+ convergent as x → − ∞, then the following boundary conditions are acquired $\begin{eqnarray}\left\{\begin{array}{ll}{c}_{1}=0,\quad {c}_{2}\ne 0\quad & x\to -\infty ,\\ {c}_{1}\ne 0,\quad {c}_{2}=0\quad & x\to +\infty ,\end{array}\right.\end{eqnarray}$
which means the asymptotic behavior of φ1,L must be different as x → ± ∞.If VA were real, the asymptotic behavior would change at the zeros of VA and the boundary conditions simply lead us to equation (12 ). However, the asymptotic behavior of Schrodinger equations gets extremely complicated with a non-Hermitian Hamiltonian. At such circumstances, the WKB analysis is introduced by the Liouville–Green transform as follows [29, 30]18 ) has the form of the Schrodinger equation of a harmonic oscillator with perturbation term $h\left(\zeta \right)$ and we can find the WKB solutions w±(ζ) by setting $h\left(\zeta \right)=0$. The formal theory of asymptotic analysis provides us with the following theorem to express the asymptotic behavior of the solutions.
$\begin{eqnarray}\begin{array}{rcl}\zeta & = & {\displaystyle \int }_{{x}_{0}}^{x}\sqrt{{V}_{A}\left({x}^{{\prime} }\right)}{\rm{d}}{x}^{{\prime} },\\ w\left(\zeta \right) & = & {A}_{1}\left(x\right)\cdot {\left[{V}_{A}\left(x\right)\right]}^{1/4},\end{array}\end{eqnarray}$
where x0 is a zero of VA. Then we get an equivalent equation of $w\left(\zeta \right)$ $\begin{eqnarray}\begin{array}{l}\frac{{{\rm{d}}}^{2}}{{\rm{d}}{\zeta }^{2}}w\left(\zeta \right)-\left[1-h\left(\zeta \right)\right]w\left(\zeta \right)=0,\\ h\left(\zeta \right)=-{V}_{A}^{-1/4}\cdot \frac{{{\rm{d}}}^{2}}{{\rm{d}}{\zeta }^{2}}\left({V}_{A}^{-1/4}\right).\end{array}\end{eqnarray}$
Equation ( The Liouville–Green transformation z → ζ turns the complex plane z into a multi-sheet Riemann surface ζ. For each sheet of the Riemann surface ζ cut along the positive or the negative part of the imaginary axis, the WKB solutions $\left\{{w}_{+}(\zeta ),{w}_{-}(\zeta )\right\}$ are asymptotic to a fundamental set of solutions $\left\{{w}^{(1)}(\zeta ),{w}^{(2)}(\zeta )\right\}$ for the Schrodinger equation (
Combining Theorem. 1 and the required asymptotic behavior equation (16 ), we get the following corollary.
If there exists a zero x0 of VA to transfer the real axis of the x plane to a curve lζ that cannot pass through both the positive and negative parts of the imaginary ζ axis, then no solution of equation (
When VA is real, Corollary. 1 degrades to the familiar expression: an eigenmode cannot without zeros of VA. However, due to the complicated behavior of the zeroes of VA on the complex plane (since VA = 0 is a transcendental equation with undetermined parameters λ0, θ, σ), it is nearly impossible to directly obtain the threshold of absolute instability through Corollary 1 . Therefore, we present a weaker, and consequently, less complicated statement based on Corollary 1 that is applicable for providing a threshold for absolute instability.
If there exists ${\rm{\Delta }}x\in {\mathbb{C}}$ letting the curve correspond to the function ${V}_{A}\left(x+{\rm{\Delta }}x\right),x\in {\mathbb{R}}$ has no cross point with the negative part of the real axis on the complex plane, then no solution of equation (
It is obvious that we only need to prove Corollary.
We then turn to the behavior of VA. Choosing the negative real axis as the cut line of the multi-valued function $\sqrt{{V}_{A}}$ to find $\,\rm{Re}\,\left[\sqrt{{V}_{A}\left(x\right)}\right],x\in {\mathbb{R}}$ is always positive if ${V}_{A}\left(x\right)$ has no cross point with such cut line. Therefore, for any zero x0 of VA, $\,\rm{Re}\,\left(\zeta \right)$ gets monotone on the course lζ and lζ has at most one cross point with the imaginary axis on ζ Riemann surface. As a result, according to Corollary.
Although explicit eigenmodes cannot be obtained directly from Corollary 2 , we may still find an effective threshold for absolute instability by choosing a suitable Δx and applying the Corollary.
As an application, we consider the simplest case where a PI is induced in an infinite pump wave and inhomogeneous plasma, thus the Hamiltonian takes the form of
$\begin{eqnarray}{V}_{A}=\frac{1}{4}{\left({\rm{i}}{\kappa }^{{\prime} }x+\sigma \right)}^{2}-\frac{{\rm{i}}}{2}{\kappa }^{{\prime} }-{\lambda }_{0},\end{eqnarray}$
in which any translation operator x → x + Δx only affects the ${\rm{i}}{\kappa }^{{\prime} }x+\sigma $ term. Therefore, we may always choose a Δx to make $\,\rm{Re}\,\left({\left({\kappa }^{{\prime} }x+\sigma \right)}^{2}\right)$ sufficiently large at the zero of $\,\rm{Im}\,\left({V}_{A}\left(x+{\rm{\Delta }}x\right)\right)$, which means no absolute instability exists under such conditions. The conclusion we obtained through the analysis above agrees with the well-known analysis of Rosenbluth [9].Note that Corollary 2 provides us only a necessary condition rather than a sufficient condition for the absolute instability, we might still need to introduce other mathematical or physical conditions to acquire an effective threshold. In the next section, we present several thresholds for absolute parametric instability at different PI scenarios.
3. Threshold for absolute parametric instability with non-resonant daughter waves
3.1. Absolute instability in an homogeneous plasma
We first consider the absolute instability of a quasi-mode PI with a finite pump and homogeneous plasma. The Hamiltonian of equation (9 ) takes the following form 2 , we obtain that absolute instability exists only when $\,\rm{Re}\,\left[{V}_{A}\left({x}_{1}\right)\right]\lt 0$. Consequently, we have the following threshold for absolute instability 21 ) gets invalid at large $\,\rm{Im}\,\left({\sigma }_{0}\right)$, thus we need to consider additional conditions. Noticing that the real part of σ0 refers to the damping rate of daughter waves and the imaginary part of σ0 refers to the mismatch of wavevectors, hence both parts of σ0 suppress absolute parametric instability. As a result, the threshold of absolute instability for $\,\rm{Im}\,\left({\sigma }_{0}\right)\gt 1/{l}_{0}$ must be larger than that for $\,\rm{Im}\,\left({\sigma }_{0}\right)\approx 0$. Therefore, we obtain an effective threshold which is stronger than equation (21 ) 22 ) to find the threshold degrades to the form of equation (1 ).
$\begin{eqnarray}{V}_{A}=\frac{1}{4}{\left(\frac{2x}{{l}_{0}^{2}}-\sigma \right)}^{2}+\frac{1}{{l}_{0}^{2}}-{\lambda }_{0}{{\rm{e}}}^{-{x}^{2}/{l}_{0}^{2}}.\end{eqnarray}$
For a typical quasi-mode PI, the equivalent damping rate σ has a considerable imaginary part, whereas the coupling coefficient λ0 is usually real. Therefore, there is only one zero for $\,\rm{Im}\,\left({V}_{A}\right)$ on the real axis, namely ${x}_{1}=\,\rm{Re}\,\left(\sigma \right){l}_{0}^{2}/2$. From Corollary $\begin{eqnarray}{\lambda }_{0}\gt \left(\frac{1}{{l}_{0}^{2}}-\frac{1}{4}\,\rm{Im}\,{\left({\sigma }_{0}\right)}^{2}\right)\exp \left(\frac{{l}_{0}^{2}}{4}\,\rm{Re}\,{\left({\sigma }_{0}\right)}^{2}\right),\end{eqnarray}$
where ${\sigma }_{0}={\rm{i}}\left({\rm{\Delta }}{k}_{1}^{(0)}-{\rm{\Delta }}{k}_{L}^{(0)}\right)$. Threshold equation ( $\begin{eqnarray}{\lambda }_{0}\gt \frac{1}{{l}_{0}^{2}}\exp \left(\frac{{l}_{0}^{2}}{4}\,\rm{Re}\,{\left({\sigma }_{0}\right)}^{2}\right).\end{eqnarray}$
When both daughter waves are resonant, the damping effect of the daughter waves within the pump region is weak, namely $\,\rm{Re}\,\left({\sigma }_{0}\right)\ll {l}_{0}$. Thus, we can expand the exponential term in equation (For typical quasi-mode PI, the quasi-mode is severely damped such that ${\rm{\Delta }}{k}_{L}^{(0)}$ shares the same order as kL. Therefore, we find $\,\rm{Re}\,\left({\sigma }_{0}{l}_{0}\right)\gg 1$, no absolute instability can be induced for this case.
3.2. Effect of an inhomogeneous plasma
We introduce the effects of an inhomogeneous plasma on the threshold of absolute instability through a simple case that both daughter waves are resonant. For small σ and θ (i.e. ∣σl0∣ ≈ 0 and θ ≪ 1), the imaginary part of VA is small enough that the asymptotic behavior of equation (9 ) can be dealt with through minimal adjustments of the WKB analysis on the real axis. White developed a numerical method to solve the eigenvalue problem equation (9 ) in such circumstances [27]. Subsequent studies [25, 26] provided the following threshold of absolute instability
$\begin{eqnarray}{\lambda }_{0}\gt 8{\rm{e}}\cdot \frac{{\theta }^{2}}{{l}_{0}^{2}}.\end{eqnarray}$
Similar conclusion can be extended to strong inhomogeneity and PI coupling scenarios (i.e. ∣θ∣ ≫ 1, ∣λ0l0∣ ≫ 1, ∣σl0∣ ≈ 0) through our model. Apply translation operator $x\to x+\frac{{x}_{0}}{{\rm{i}}\theta -2}$ on VA to find 27 ) provides us only an ambiguous understanding of the absolute parametric instability in an inhomogeneous plasma. However, we can combine equation (23 ) and equation (27 ) to find that plasma inhomogeneity always plays the role of suppressing absolute instability. Therefore, for quasi-mode PI, we can simply extend the threshold of absolute instability in a homogeneous plasma as a necessary threshold to induce absolute instability in an inhomogeneous plasma, and find that no absolute instability exists under such circumstances as well.
$\begin{eqnarray}\begin{array}{rcl}{V}_{A} & = & \frac{1}{4{l}_{0}^{2}}{\left(\left({\rm{i}}\theta -2\right)\frac{x}{{l}_{0}}+\frac{{x}_{0}}{{l}_{0}}+\sigma {l}_{0}\right)}^{2}-\frac{{\rm{i}}\theta -2}{2{l}_{0}^{2}}\\ & & -{\lambda }_{0}\exp \left[-\frac{1}{{l}_{0}^{2}}{\left(x+\frac{{x}_{0}}{{\rm{i}}\theta -2}\right)}^{2}\right].\end{array}\end{eqnarray}$
Choosing ${x}_{0}\in {\mathbb{R}}$ that satisfies l0 ≪ x0 ≪ θl0, thus the imaginary part of the exponential term can be neglected, we find two zeroes of Im(VA) as follows $\begin{eqnarray}{x}_{1}=\frac{{l}_{0}^{2}}{{x}_{0}},\quad {x}_{2}=\frac{1}{2}\left({x}_{0}+{x}_{1}\right),\end{eqnarray}$
where x1 ≪ l0 and x2 ≫ l0. Note that the eigenmode we desired must exist near the region of the pump wave, thus the behavior of x2 has no contribution to the absolute instability. For x1, we have $\begin{eqnarray}\,\rm{Re}\,\left[{V}_{A}({x}_{1})\right]=\frac{1}{4{l}_{0}^{2}}\left(\frac{{x}_{0}^{2}}{{l}_{0}^{2}}-{\theta }^{2}\frac{{l}_{0}^{2}}{{x}_{0}^{2}}\right)-{\lambda }_{0}\lt 0.\end{eqnarray}$
As θ getting larger, we can choose an x0 with an order slightly smaller than θl0, hence the ${\theta }^{2}\frac{{l}_{0}^{2}}{{x}_{0}^{2}}$ term can be neglected and the $\frac{{x}_{0}^{2}}{{l}_{0}^{2}}$ term approaches θ2 in magnitude. As a result, the threshold of absolute instability becomes $\begin{eqnarray}{\lambda }_{0}\gt \frac{{\theta }^{2}}{4{l}_{0}^{2}}.\end{eqnarray}$
Equation (4. Conclusion and discussion
We proved that no absolute instability exists for a quasi-mode PI which involves a non-resonant daughter wave by analyzing the asymptotic behavior of the coupling equations of PI. Due to the mathematical difficulties of solving a Schrodinger equation with a non-Hermitian Hamiltonian, the threshold we provided in this study is only a necessary condition to induce an absolute instability. But such a threshold equation (22 ) is large enough to put absolute instabilities out of consideration. Therefore, we only need to concentrate on convective saturation by inhomogeneous plasma and finite pump profile while evaluating the effects of a quasi-mode PI in RF heating and current drive scenarios, even if the linear growth rate of PI for such cases might be extremely large.