2. Formulation

Consider the stability of a homogeneous, unmagnetised, weakly collisional plasma containing a small amplitude, propagating monochromatic electrostatic plasma wave of wave frequency ${\omega}$ and $\boldsymbol {{z}}$ directed wavenumber ${k}_{\|}$ for the electric field

(2.1) \begin{align}\boldsymbol {{E}}=\boldsymbol {{z}}\mathrm{Im} [\tilde{{E}}{e}^{{i}{({k}_{\|}}{z}-{\omega} {t})}]\approx \boldsymbol {{z}}{E}_{\|}\sin ({k}_{\|}{z}-{\omega} {t}).\end{align}

For the linear regime of the next section the full complex amplitude $\tilde{{E}}$ enters, while for what will be referred to as the nonlinear case, the lowest-order approximation on the right of (2.1) is adequate. The phase velocity is assumed to satisfy ${\omega} /{k}_{\|}\gg {v}_{{e}}=(2{T}/{m})^{1/2}$ and it is convenient to view ${E}_{\|}\propto {e}^{{\gamma} {\tau} }$ , with ${E}_{\|}$ the positive amplitude of the lowest order electric field with a small positive growth or negative damping rate ${\gamma}$ . The real part of the frequency satisfies ${\omega} ^{2}={{\omega} }_{{p}}^{2}+3{{k}}_{\|}^{2}{{v}}_{{e}}^{2}/2$ , with ${\omega} _{{p}}=(4{\pi} {e}^{2}{n}/{m})^{1/2}\gg {\gamma}$ the plasma frequency. Here and elsewhere n, T, m and e are the electron density, temperature, mass and charge magnitude.

Assume the initial electron distribution function is ${f}_{0}={f}_{{M}}+{f}_{{b}}$ , where ${f}_{{M}}$ is the Maxwellian

(2.2) \begin{align}{f}_{{M}}={f}_{{M}}({v})={n}{\pi} ^{-3/2}{{v}}_{{e}}^{-3/2}{e}^{-{{v{2}}}/{{v}}_{{e}}^{2}}.\end{align}

The bump on the tail contribution to ${f}_{0}$ is assumed to be centred and localised about ${v}_{\|}={V}\approx {\omega} /{k}_{\|}\gt {v}_{{e}}$ with a form that does not perturb the density. It is convenient to assume

(2.3) \begin{align}{f}_{{b}}=\big(2{\Gamma} /{\pi} ^{3/2}{{v}}_{{b}}^{5}\big)({v}_{\|}-{V}){e}^{-{{v}}_{\bot }^{2}/{{v}}_{{b}}^{2}-({{v}_{\|}}-{V}{)^{2}}/{{v}}_{{b}}^{2}},\end{align}

with ${v}_{\|}=\boldsymbol {{z}}\boldsymbol{\cdot }\boldsymbol {{v}}$ and ${v}^{2}={{v}}_{\|}^{2}+{{v}}_{\bot }^{2}={{v}}_{\|}^{2}+{{v}}_{{x}}^{2}+{{v}}_{{y}}^{2}$ , and ${{v}}_{{b}}^{2}\ll {{v}}_{{e}}^{2}$ to provide the desired localisation for the bump. Then $\int \mathrm{d}^{3}{vf}_{{b}}=0$ , but $\int \mathrm{d}^{3}{v}{{v}_{\|}}{f}_{{b}}={\Gamma} \gt 0$ is a $\boldsymbol {{z}}$ directed flux that is assumed to always be small enough to keep ${f}_{0}\gt 0$ . Picking the bump of this form allows a positive slope region for ${f}_{{b}}$ about ${v}_{\|}={V}\gt 0$ , while if ${\Gamma}$ is small enough to keep the distribution function ${f}_{0}$ monotonic, then by Gardner’s theorem (1963) stability is assured. However, when ${\Gamma}$ is large enough to make the distribution function ${f}_{0}$ non-monotonic, then the Penrose criterion (1960) means a linearised treatment will lead to instability. Other, less convenient choices can be made for the bump as long as it is localised to the tail of the Maxwellian.

Here, the ultimate task is to solve the full electron kinetic equation

(2.4) \begin{align}\partial {f}/\partial {t}+{v}_{\|}\,\partial {f}/\partial {z}-\left({e}/{m}\right)\boldsymbol {{z}}\boldsymbol{\cdot }\boldsymbol {{E}}\partial {f}/\partial {v}_{\|}={C}\{{f}\},\end{align}

where the collision operator retains like and unlike collisions, with the full electron collision operator giving ${C}\{{f}_{{M}}\}=0$ , and $\partial {f}_{{M}}/\partial {t}=0=\partial {f}_{{M}}/\partial {z}$ also assumed. The background ions are assumed stationary so a Lorentz operator is adequate for them.

The non-Maxwellian bump will be on the tail if ${V}\approx {\omega} /{k}_{\|}\approx {v}_{\|}\gg {v}_{{e}}$ . This limit suggests expanding the electron–electron collision operator for ${v}^{2}\gg {{v}}_{{e}}^{2}$ . As a result, the following linearised collision operator may be employed (Catto Reference Catto2025b ):

(2.5) \begin{align}{C}\{{f}\}=\boldsymbol{\nabla} _{{v}}\boldsymbol{\cdot }\left\{{\nu}\left[\big({v}^{2}{\mathop{I}\limits^{\leftrightarrow}}-\boldsymbol {{v}}\boldsymbol {{v}}\big)\boldsymbol{\cdot }\boldsymbol{\nabla} _{{v}}{f}+\frac{2\textit{Tf}_{{\!M}}}{({Z}+1){m}}\boldsymbol{\nabla} _{{v}}\left(\frac{{f}}{{f}_{{M}}}\right)\right]\right\}\!,\end{align}

where ${x}={v}/{v}_{{e}}$ , ${\nu}_{{e}}=3\sqrt{\pi }(Z+1){\nu}_{{ee}}/4$ , ${\nu}_{{ee}}=4\sqrt{2{\pi} }{e}^{4}{n}\ell{n}{\Lambda} /3{m}^{1/2}{T}^{3/2}={\nu}_{{ei}}/{Z}$ and ${\nu}={\nu}_{{e}}/2{x}^{3}$ , with Z the ion charge number and $\ell{n}{\Lambda} \gg 1$ the Coulomb logarithm.

To solve the kinetic equation (2.4) in the linear and nonlinear limits, it is convenient to let ${f}={f}_{0}+{f}_{1}$ with ${f}_{1}\ll {f}_{0}={f}_{{M}}+{f}_{{b}}$ and allow ${\Gamma} ={\Gamma} ({t})$ to satisfy $\partial {f}_{{b}}/\partial {t}={C}\{{f}_{{b}}\}$ , then

(2.6) \begin{align}\partial {f}_{1}/\partial {t}+{v}_{\|}\,\partial {f}_{1}/\partial {z}-\left({e}/{m}\right)\boldsymbol {{z}}\boldsymbol{\cdot }\boldsymbol {{E}}\partial \left({f}_{0}+{f}_{1}\right)/\partial {v}_{\|}={C}\left\{{f}_{1}\right\}.\end{align}

The drive for the bump on tail instability is $\partial {f}_{0}/\partial {v}_{\|}=\partial ({f}_{{M}}+{f}_{{b}})/\partial {v}_{\|}$ . The assumption ${f}_{1}\ll {f}_{0}$ is made to ensure the perturbations ${E}_{\|}$ and ${f}_{1}$ satisfy Poisson’s equation to lowest order. The nonlinear limit retains the product of the two linear terms $\boldsymbol {{z}}\boldsymbol{\cdot }\boldsymbol {{E}}$ and $\partial {f}_{1}/\partial {v}_{\|}$ .

Collisions are assumed weak, but can never be neglected for the resonant electrons (Zakharov & Karpman Reference Zakharov and Karpman1963; Catto Reference Catto2025a ). This means they always must be retained in a narrow collisional boundary layer about resonance. The width of this collisional layer can be estimated by letting ${\Delta} {v}_{\|}={v}_{\|}-{\omega} /{k}_{\|}$ and using ${C}\{{f}_{1}\}\sim {\nu}{{v}}_{{e}}^{2}{f}_{1}/({\Delta} {v}_{\|})^{2}$ , with ${\nu}={\nu}_{{e}}/2{x}^{3}$ . Balancing this collisional term with $\partial {f}_{1}/\partial {t}+{v}_{\|}\,\partial {f}_{1}/\partial {z}\sim {f}_{1}{k}_{\|}{\Delta} {v}_{\|}$ gives the narrow collisional boundary layer width to be of the order of

(2.7) \begin{align}({\Delta} {v}_{\|})_{{\nu}}\sim \big({\nu}{{v}}_{{e}}^{2}/{k}_{\|}\big)^{1/3}=({\nu}/{k}_{\|}{v}_{{e}})^{1/3}{v}_{{e}}.\end{align}

Assuming $({\Delta} {v}_{\|})_{{\nu}}\ll {v}_{{b}}\ll {v}_{{e}}$ allows the collision operator to be adequately approximated in the vicinity of the resonance by

(2.8) \begin{align}{C}\{{f}_{1}\}\approx {\nu}\big[{{v}}_{\bot }^{2}+{({Z}+1)^{-1}{v}}_{{e}}^{2}\big]\partial ^{2}{f}_{1}/\partial {{v}}_{\|}^{2}.\end{align}

The preceding assumes localisation of ${f}_{{b}}$ to a width ${v}_{{b}}$ with an effective collision frequency for the bump of ${\nu}{{v}}_{{e}}^{2}/{{v}}_{{b}}^{2}$ , which is much slower than the effective collision frequency ${\nu}{{v}}_{{e}}^{2}/({\Delta} {v}_{\|}{)}_{{\nu}}^{2}$ for ${f}_{1}$ so ${f}_{{b}}/{{v}}_{{b}}^{2}\ll {f}_{1}/({\Delta} {v}_{\|}{)}_{{\nu}}^{2}$ may be assumed. As a result, bump on tail growth/damping occurs on a faster time scale. Notice velocity space is two-dimensional, but ${v}_{\bot }$ only enters ${{v}}_{\bot {z}}^{2}={{v}}_{\bot }^{2}+{{v}}_{{e}}^{2}/({Z}+1)$ as a parameter.

When ${\gamma} \ll {\omega} \sim {\omega} _{{p}}$ , only the lowest order kinetic equation

(2.9) \begin{align}{k}_{\|}{u}\frac{\partial {f}_{1}}{\partial {\phi} }-\frac{{eE}_{\|}}{{m}}\,\sin{\phi}\! \left(\frac{\partial {f}_{1}}{\partial {u}}+\frac{\partial {f}_{0}}{\partial {u}}| _{{u}=0}\right)={{\nu}{v}}_{\bot {z}}^{2}\frac{\partial ^{2}{f}_{1}}{\partial {u}^{2}}\end{align}

need be solved. For a monochromatic wave ${f}_{1}={f}_{1}({\phi} ,{u})$ may be assumed, with ${u}={v}_{\|}-{\omega} /{k}_{\|}$ , as all spatial and fast time variation is in the phase ${\phi} ={k}_{\|}{z}-{\omega} {t}$ . The assumption $({\Delta} {v}_{\|})_{{\nu}}\ll {v}_{{b}}\ll {v}_{{e}}$ means the bump on the tail is slowly varying, implying

(2.10) \begin{align}\partial {f}_{{b}}/\partial {u}| _{{u}=0}=\big(2{\Gamma} /{\pi} ^{3/2}{{v}}_{{b}}^{5}\big){e}^{-{{v}}_{\bot }^{2}/{{v}}_{{b}}^{2}-({V}-{\omega} /{{k}_{\|}}{)^{2}}/{{v}}_{{b}}^{2}}\end{align}

and

(2.11) \begin{align}\partial {f}_{{M}}/\partial {u}| _{{u}=0}=-\big(2{n}{\omega} /{\pi} ^{3/2}{k}_{\|}{{v}}_{{e}}^{5}\big){e}^{-{{v}}_{\bot }^{2}/{{v}}_{{e}}^{2}-{{\omega} ^{2}}/{{k}}_{\|}^{2}{{v}}_{{e}}^{2}}\end{align}

are the useful expansions, and give

(2.12) \begin{align}\frac{\partial {f}_{0}}{\partial {u}}| _{{u}=0}=-\frac{2}{{\pi} ^{3/2}}\left[\frac{{n}{\omega} }{{k}_{\|}{{v}}_{{e}}^{5}}{e}^{-{{v}}_{\bot }^{2}/{{v}}_{{e}}^{2}-{{\omega} ^{2}}/{{k}}_{\|}^{2}{{v}}_{{e}}^{2}}-\frac{{\Gamma} }{{{v}}_{{b}}^{5}}{e}^{-{{v}}_{\bot }^{2}/{{v}}_{{b}}^{2}-({V}-{\omega} /{{k}_{\|}}{)^{2}}/{{v}}_{{b}}^{2}}\right].\end{align}

The next two sections solve this equation in the linear and nonlinear regimes.

3. Linear regime

To solve in the more familiar linear limit, $\partial {f}_{0}/\partial {u}| _{{u}=0}\gg \partial {f}_{1}/\partial {u}$ must be assumed. This assumption implies the collisional boundary layer width $({\Delta} {v}_{\|})_{{\nu}}$ is being assumed much wider than the island width

(3.1) \begin{align}({\Delta} {v}_{\|})_{\textit{is}}\sim ({eE}_{\|}/{mk}_{\|})^{1/2}\end{align}

obtained by balancing the resonant and nonlinear terms

(3.2) \begin{align}{k}_{\|}({\Delta} {v}_{\|})_{\textit{is}}{f}_{1}\sim {k}_{\|}{u}\frac{\partial {f}_{1}}{\partial {\phi} }\sim \frac{{e}}{{m}}\boldsymbol {{z}}\boldsymbol{\cdot }\boldsymbol {{E}}\frac{\partial {f}_{1}}{\partial {u}}\sim \frac{{eE}_{\|}{f}_{1}}{{m}({\Delta} {v}_{\|})_{\textit{is}}}.\end{align}

The $({\Delta} {v}_{\|})_{{\nu}}\gg ({\Delta} {v}_{\|})_{\textit{is}}$ assumption means only the linear inhomogeneous Airy equation

(3.3) \begin{align}{k}_{\|}{u}\frac{\partial {f}_{1}}{\partial {\phi} }-{{\nu}{v}}_{\bot {z}}^{2}\frac{\partial ^{2}{f}_{1}}{\partial {u}^{2}}=\frac{{eE}_{\|}}{{m}}\sin {\phi} \frac{\partial {f}_{0}}{\partial {u}}| _{{u}=0}\end{align}

need be solved. The Su & Oberman (Reference Su and Oberman1968) solution in the vicinity of the resonance is

(3.4) \begin{align} \left.{f}_{1}\right| _{{res}} & =\frac{{e}/{m}}{\big({{k}}_{\|}^{2}{{v}}_{\bot {z}}^{2}{\nu}\big)^{1/3}}\frac{\partial {f}_{0}}{\partial {u}}| _{{u}=0}\mathrm{Im}\left[\tilde{{E}}{e}^{{i}{\phi} }{\int }_{\!\!\!0}^{{\infty }}{\rm d}{\tau} {e}^{-{isu}{\tau} -{{\tau} ^{3}}/3}\right]\nonumber\\[4pt]& \quad \times \underset{{su}\gt 1}{\longrightarrow }\frac{\textit{esE}_{\|}}{{mk}_{\|}}\frac{\partial {f}_{0}}{\partial {u}}| _{{u}=0}\sin {\phi} {\int }_{\!\!\!0}^{{\infty }}{\rm d}{\tau} {e}^{-{{\tau} ^{3}}/3}\cos (\textit{su}{\tau}), \end{align}

as can be verified by direct substitution, with ${s}=({k}_{\|}/{{v}}_{\bot {z}}^{2}{\nu})^{1/3}$ . Far from the resonance, $s| {u}| \gg 1,{\int }_{0}^{{\infty }}{\rm d}{\tau} {e}^{-{isu}{\tau} -{{\tau} ^{3}}/3}\rightarrow 1/{isu}$ giving ${f}_{1}| _{{non}}=-({eE}_{\|}\cos {\phi} /{mk}_{\|}{u})$ $\partial {f}_{0}/\partial {u}| _{{u}=0}$ . The delta function behaviour in the resonant contribution leads to

(3.5) \begin{align}\frac{{s}}{{\pi} }{\int }_{\!\!\!-{\beta} /{s}}^{{\beta} /{s}}{{\rm d}u}{\int }_{\!\!\!0}^{{\infty }}{\rm d}{\tau} {e}^{-{{\tau} ^{3}}/3}\cos (\textit{su}{\tau})=\frac{2}{{\pi} }{\int }_{\!\!\!0}^{{\infty }}\frac{{\rm d}{\tau} }{{\tau} }{e}^{-{{\tau} ^{3}}/3}\sin ({\beta} {\tau})\approx 1\end{align}

for ${\beta} \gg 1$ (see the Appendix). The solution for ${f}_{1}$ can be adequately estimated using

(3.6) \begin{align}{f}_{1}\sim [{eE}_{\|}/{mk}_{\|}({\Delta} {v}_{\|})]\partial {f}_{0}/\partial {u}| _{{u}=0},\end{align}

with $({\Delta} {v}_{\|})\rightarrow ({\Delta} {v}_{\|})_{{\nu}}$ to obtain the small island restriction

(3.7) \begin{align}\frac{\partial {f}_{1}/\partial {u}}{\partial {f}_{0}/\partial {u}}\sim \frac{{v}_{{e}}{f}_{1}}{({\Delta} {v}_{\|})_{{\nu}}{f}_{0}}\sim \frac{{eE}_{\|}}{{mk}_{\|}{{v}}_{{e}}^{2}}\left(\frac{{k}_{\|}{v}_{{e}}}{{\nu}}\right)^{2/3}\sim \frac{({\Delta} {v}_{\|}{)}_{\textit{is}}^{2}}{({\Delta} {v}_{\|}{)}_{{\nu}}^{2}}\ll 1.\end{align}

However, a comparison of the estimated quasilinear and collisional diffusivities reveals a subtle and often overlooked complication. The collisional diffusivity associated with ${f}_{{M}}$ is simply ${D}_{{\nu}}\sim {\nu}{{v}}_{{e}}^{2}$ . To estimate the many overlapping modes quasilinear diffusivity ${D}_{{q}}$ the velocity step ${eE}_{\|}/{m}{\nu}_{\textit{eff}}$ in an effective collision time of

(3.8) \begin{align}{\nu}_{\textit{eff}}\sim {\nu}{{v}}_{{e}}^{2}/({\Delta} {v}_{\|})^{2}\end{align}

is employed with $({\Delta} {v}_{\|})\rightarrow ({\Delta} {v}_{\|})_{{\nu}}\ll {eE}_{\|}/{m}{\nu}_{\textit{eff}}$ to ignore the nonlinear term and thus

(3.9) \begin{align}{D}_{{q}}\sim {\nu}_{\textit{eff}}\big({eE}_{\|}/{m}{\nu}_{\textit{eff}}\big)^{2}\sim {\nu}{{v}}_{{e}}^{2}\left(\frac{{eE}_{\|}}{{mk}_{\|}{{v}}_{{e}}^{2}}\right)^{2}\left(\frac{{k}_{\|}{v}_{{e}}}{{\nu}}\right)^{4/3}.\end{align}

Then the ratio of the diffusivities is

(3.10) \begin{align}\frac{{D}_{{q}}}{{D}_{{\nu}}}\sim \left(\frac{{eE}_{\|}}{{mk}_{\|}{{v}}_{{e}}^{2}}\right)^{2}\left(\frac{{k}_{\|}{v}_{{e}}}{{\nu}}\right)^{4/3}\sim \frac{({\Delta} {v}_{\|}{)}_{\textit{is}}^{4}}{({\Delta} {v}_{\|}{)}_{{\nu}}^{4}}\ll 1.\end{align}

As a result, if the nonlinear term is ignored as small by assuming $\partial {f}_{1}/\partial {u}\ll \partial {f}_{0}/\partial {u}$ , as in standard quasilinear theory (QLT), then quasilinear flattening must be ignored as small compared with collisional relaxation (Catto & Tolman Reference Catto and Tolman2021; Catto Reference Catto2025b ). Based on the preceding estimates, the bump on tail quasilinear operator is expected to be valid only if $({\Delta} {v}_{\|})_{\textit{is}}\ll ({\Delta} {v}_{\|})_{{\nu}}$ , and it should only be trusted when it gives small corrections to ${f}_{0}$ .

Normally, collisions are ignored in QLT and a statistical treatment of a bumpy tailed Maxwellian invoked, as discussed in mathematical detail in Besse et al. (Reference Besse, Elskens, Escande and Bertrand2011). When weak like collisions are retained for plasmas immersed in a bath of imposed stochastic fields (Nastac et al. Reference Nastac, Ewart, Sengupta, Schekochihin, Barnes and Dorland2024, Reference Nastac, Ewart, Juno, Barnes and Schekochihin2025), some features discussed here are recovered even in the presence of the artificial dissipation introduced by a statistical random field approach. It is unclear how to completely reconcile these many-wave, phase mixing treatments to the single wave, collisional model presented here, but the key roles of collisions retained here and in Nastac et al. (Reference Nastac, Ewart, Sengupta, Schekochihin, Barnes and Dorland2024, Reference Nastac, Ewart, Juno, Barnes and Schekochihin2025) leave no doubt they must be retained in standard QLT.

When $\partial {f}_{0}/\partial {u}| _{{u}=0}\gt 0$ the preceding collisional solution is consistent with growth of a bump on tail instability as in Penrose (Reference Penrose1960) treatment without collisions. Moreover, for $\partial {f}_{0}/\partial {u}| _{{u}=0}\lt 0$ no bump occurs, implying collisional Landau damping dominates and drives ${f}_{0}$ toward the Maxwellian ${f}_{{M}}$ consistent with Gardner (Reference Gardner1963). It is tempting to argue that there is no need to retain collisions to obtain the Penrose criterion (1960). However, the limit of weak collisions will be considered shortly, and it will be shown that collisions can never be neglected, as is the case for Landau damping (Zakharov & Karpman Reference Zakharov and Karpman1963; Catto Reference Catto2025a ), for which a plasma wave transfers energy to the background electrons.

To evaluate the small wave growth or damping rate, it is convenient to employ the power absorbed (lost) by the growing (damped) plasma wave by using the usual relation

(3.11) \begin{align}{P}=-{eE}_{\|}\left\langle \sin {\phi} \int {d}^{3}{vv}_{\|}{f}_{1}\right\rangle _{{\phi} }=-2{\gamma} {{E}}_{\|}^{2}/8{\pi},\end{align}

with $\langle \ldots \rangle _{{\phi} }=\oint {\rm d}{\phi} (\ldots )/2{\pi}$ . In the conventional linear limit, using ${v}_{\|}\approx {\omega} /{k}_{\|}$ gives

(3.12) \begin{align}{P}_{0}=-\frac{{e}^{2}{{E}}_{\|}^{2}}{2{m}{{k}}_{\|}^{2}}\int \mathrm{d}^{3}{v}\frac{\partial {f}_{0}}{\partial {u}}| _{{u}=0}{\int }_{\!\!\!0}^{{\infty }}{\rm d}{\tau} {e}^{-{{\tau} ^{3}}/3}\cos (\textit{su}{\tau}),\end{align}

since only the $\sin {\phi}$ term in $\left.{f}_{1}\right| _{{res}}$ contributes (the non-resonant term vanishes). As a result,

(3.13) \begin{align}{P}_{0}=\frac{{{\pi} ^{1/2}}{n}{\omega} {e}^{2}{{E}}_{\|}^{2}}{{m}{{k}}_{\|}^{2}{{v}}_{{e}}^{2}}\left[\frac{{\omega} }{{k}_{\|}v_{e}}{e}^{-{{\omega} ^{2}}/{{k}}_{\|}^{2}{{v}}_{{e}}^{2}}-\frac{{\Gamma} {{v}}_{{e}}^{2}}{{n}{{v}}_{{b}}^{3}}{e}^{-({V}-{\omega} /{{k}_{\|}}{)^{2}}/{{v}}_{{b}}^{2}}\right]\end{align}

leads the usual linear growth/damping rate ${\gamma}$ to be

(3.14) \begin{align}{\gamma} =\frac{{\pi} ^{1/2}{{\omega} }_{{p}}^{2}{\omega} }{{{k}}_{\|}^{2}{{v}}_{{e}}^{2}}\left[\frac{{\Gamma} {{v}}_{{e}}^{2}}{{n}{{v}}_{{b}}^{3}}{e}^{-({V}-{\omega} /{{k}_{\|}}{)^{2}}/{{v}}_{{b}}^{2}}-\frac{{\omega} }{{k}_{\|}{v}_{{e}}}{e}^{-{{\omega} ^{2}}/{{k}}_{\|}^{2}{{v}}_{{e}}^{2}}\right]\!,\end{align}

where ${1\gg {k}}_{\|}^{2}{{v}}_{{e}}^{2}/{{\omega} }_{{p}}^{2}\gg ({\omega} /{{k}_{\|}}{{v}_{{e}}}){e}^{-{{\omega} ^{2}}/{{k}}_{\|}^{2}{{v}}_{{e}}^{2}}\sim {\Gamma} {{v}}_{{e}}^{2}{e}^{-({V}-{\omega} /{{k}_{\|}}{)^{2}}/{{v}}_{{b}}^{2}}/{n}{{v}}_{{b}}^{3}$ allows consideration of the small, localised bump on the tail of the Maxwellian. The exponential ${e}^{-({V}-{\omega} /{{k}_{\|}}{)^{2}}/{{v}}_{{b}}^{2}}$ with ${V}\approx {\omega} /{k}_{\|}$ accounts for the fact that if the resonance is not in the bump, only wave damping will occur ( ${\gamma} \lt 0)$ . The plasma is bump on tail unstable when an adequate bump is present, $1\gg {\Gamma} {{v}}_{{e}}^{2}{{e}^{-({V}-{\omega} /{{k}_{\|}}{)^{2}}/{{v}}_{{b}}^{2}}}/{n}{{v}}_{{b}}^{3}\gt ({\omega} /{{k}_{\|}}{{v}_{{e}}}){e}^{-{{\omega} ^{2}}/{{k}}_{\|}^{2}{{v}}_{{e}}^{2}}$ (differing from the less precise condition $\partial {f}_{0}/\partial {u}| _{{u}=0}\gt 0$ because of the velocity space integration), and the plasma wave damps once the bump satisfies ${\Gamma} {e}^{-({V}-{\omega} /{k}_{\|})^{2}/{v}_{b}^{2}}{v}_{e}^{2}/{{nv}}_{{b}}^{3}\lt ({\omega} /{{k}_{\|}}{{v}_{{e}}}){e}^{-{{\omega} ^{2}}/{{k}}_{\|}^{2}{{v}}_{{e}}^{2}}\ll 1$ . Even though they are vital, collisions do not explicitly appear in the preceding expressions, implying the linear limit is a plateau regime. The absence of the collision frequency leads to the confusing belief collisions do not play a role. However, they play a key role as is confirmed next by solving the nonlinear or island problem in the weak collisionality, finite electric field limit.

4. Nonlinear or island regime

Recalling $\partial {f}_{{b}}/\partial {u}| _{{u}=0}$ is a slowly varying function of ${u}$ in the kinetic equation (2.9), and inserting

(4.1) \begin{align}{f}_{1}={g}({u},{\phi} )-({u}-{\phi} {\alpha} )\partial {f}_{0}/\partial {u}| _{{u}=0},\end{align}

where ${\sigma} ={u}/| {u}| =\pm 1$ (for unbound electrons) or 0 (for island bound electrons), and ${\alpha}$ is a constant speed to be determined, leads to

(4.2) \begin{align}{k}_{\|}{u}\partial {g}/\partial {\phi} -({eE}_{\|}/{m})\sin {\phi} \partial {g}/\partial {u}={{\nu}{v}}_{\bot {z}}^{2}\partial ^{2}{g}/\partial {u}^{2},\end{align}

where ${E}_{\|}\sin {\phi}$ is the initial applied plasma wave field satisfying the lowest-order Poisson equation $\boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol {{E}}=-4{\pi} {e}\int \mathrm{d}^{3}{vf}_{\!1}$ . The procedure follows Catto (Reference Catto2025a ) and starts by letting

(4.3) \begin{align}{j}=({m}/{eE}_{\|}{k}_{\|})^{1/2}{k}_{\|}{u}=({m}/{eE}_{\|}{k}_{\|})^{1/2}({k}_{\|}{v}_{\|}-{\omega} )\end{align}

and

(4.4) \begin{align}{\varDelta} ={\nu}{{k}}_{\|}^{2}{{v}}_{\bot {z}}^{2}({m}/{eE}_{\|}{k}_{\|})^{3/2}\propto {\nu}_{{ee}}/{{E}}_{\|}^{3/2}\end{align}

to find the same equation as the one solved numerically by Hamilton et al. (Reference Hamilton, Tolman, Arzamasskiy and Duarte2023)

(4.5) \begin{align}{j}\frac{\partial {g}}{\partial {\phi} }-\sin {\phi} \frac{\partial {g}}{\partial {j}}={\varDelta} \frac{\partial ^{2}{g}}{\partial {j}^{2}},\end{align}

where ${\varDelta} \gt 0$ since ${{v}}_{\bot }^{2}\approx {v}^{2}-{\omega} ^{2}/{{k}}_{\|}^{2}\gt 0$ in ${{v}}_{\bot {z}}^{2}$ . Incomplete analytic attempts to solve for g appear in Pao (Reference Pao1988) and Petviachvili (Reference Petviachvili1999), as well as Hamilton et al. (Reference Hamilton, Tolman, Arzamasskiy and Duarte2023) for ${\varDelta} \ll 1$ .

Equation (4.5) treats the island structure as well as collisions. Taking j ∼ 1 gives a velocity space island width of $({\Delta} {v}_{\|})_{\textit{is}}=({v}_{\|}-{\omega} /{k}_{\|})_{\textit{is}}\sim ({eE}_{\|}/{mk}_{\|})^{1/2}$ as before. A linearised treatment is only appropriate when the collisional boundary layer is wider than any island structure, requiring

(4.6) \begin{align}{\varDelta} =[({\Delta} {v}_{\|})_{{\nu}}/({\Delta} {v}_{\|})_{\textit{is}}]^{3}\gg 1.\end{align}

However, for a larger amplitude plasma wave in an extremely weak collisionality plasma, the

(4.7) \begin{align}{\varDelta} =[({\Delta} {v}_{\|})_{{\nu}}/({\Delta} {v}_{\|})_{\textit{is}}]^{3}\ll 1\end{align}

limit is of interest. In this case, using $({\Delta} {v}_{\|})_{\textit{is}}$ in (3.6) yields

(4.8) \begin{align}\frac{\partial {f}_{1}/\partial {u}}{\partial {f}_{0}/\partial {u}}\sim \frac{{v}_{{e}}{f}_{1}}{({\Delta} {v}_{\|})_{\textit{is}}{f}_{0}}\sim 1.\end{align}

The weak collision limit is different than the collisional plateau limit because the island structure must be retained. In this ${\varDelta} \ll 1$ limit

(4.9) \begin{align}1\gg ({\Delta} {v}_{\|}{)}_{\textit{is}}^{3}/{{v}}_{{e}}^{3}\sim \big({eE}_{\|}/{mk}_{\|}{{v}}_{{e}}^{2}\big)^{3/2}\gg {\nu}/{{k}_{\|}}{v}_{{e}}\sim {\nu}_{{e}}{{k}}_{\|}^{2}{{v}}_{{e}}^{2}/{\omega} ^{3},\end{align}

for ${{v}}_{\bot {z}}^{2}\sim {{v}}_{{e}}^{2}$ . The collisional boundary layers are now much narrower than the islands. They enclose the separatrix between the bound (or librating) and unbound (or circulating) electron motion. All the collisional dissipation occurs in these narrow boundary layers.

Figure 1.

Contours of constant ${g}({j},{\varphi} )$ with the separatrix at h = 1. The ${g}({j},{\varphi} )=0$ bound region is inside the separatrix. The two unbound regions are above (red and yellow) and below (dark and light blue). The separatrix is surrounded by a narrow collisional boundary layer for this ${\varDelta} =0.001$ case. (Reprinted with permission from Hamilton et al. Reference Hamilton, Tolman, Arzamasskiy and Duarte2023.)

The equation for g with slow temporal evolution was solved numerically by Hamilton et al. (Reference Hamilton, Tolman, Arzamasskiy and Duarte2023) for an astrophysical application. The solution is skew symmetric, satisfying ${g}({j},{\phi} )=-{g}(-{j},-{\phi} )$ . By introducing the reduced constant of the motion

(4.10) \begin{align}{h}({j},{\phi} )={j}^{2}/2-\,{\cos}{\phi},\end{align}

an analytic treatment is possible for ${\varDelta} \ll 1$ . The separatrix at h = 1 is enclosed by a narrow collisional boundary layer separating the bound $(-1 \lt h \lt 1)$ and the unbound $(h \gt 1)$ electrons. The Hamilton et al. (Reference Hamilton, Tolman, Arzamasskiy and Duarte2023) ${\varDelta} =0.001$ solution shown in their figure 2(a) is reproduced here as figure 1 with their kind permission. Its nearly imperceptibly thin collisional boundary layer enclosing the separatrix is consistent with the g solution found analytically in Catto (Reference Catto2025a –Reference Catto c ) and obtained as follows. By using the reduced Hamiltonian, the kinetic equation is rewritten in terms of the new variables ${h},{\phi}$ as

(4.11) \begin{align}\left.\frac{\partial {g}}{\partial {\phi} }\right| _{{h}}={\varDelta} \left.\frac{\partial }{\partial {h}}\right| _{{\phi} }\left({j}\!\left.\frac{\partial {g}}{\partial {h}}\right| _{{\phi} }\right).\end{align}

A solution of the form ${g}={g}_{1}({h})+{g}_{2}({h},{\phi} )+\ldots$ exists in the ${\varDelta} \ll 1$ limit as lowest order requires $\partial {g}_{1}/\partial {\phi} | _{{h}}=0$ . To next order

(4.12) \begin{align}\left.\frac{\partial {g}_{2}}{\partial {\phi} }\right| _{{h}}={\varDelta} \left.\frac{\partial }{\partial {h}}\right| _{{\phi} }\left({j}\!\left.\frac{\partial {g}_{1}}{\partial {h}}\right| _{{\phi} }\right).\end{align}

The solution ${g}_{1}$ must satisfy the collisional solubility constraint

(4.13) \begin{align}\frac{\partial }{\partial {h}}\bigg| _{{\phi} }\bigg[\bigg(\!\oint _{{h}}{\rm d}{\phi} {j}\bigg)\frac{\partial {g}_{1}}{\partial {h}}\bigg| _{{\phi} }\bigg]=0.\end{align}

The solution ${g}_{1}({h})$ is independent of collision frequency, but its form is collisionally constrained by the form (4.13) of the collision operator. Hamilton et al. (Reference Hamilton, Tolman, Arzamasskiy and Duarte2023) found a partial solution for ${\varDelta} \ll 1$ that was completed by Catto (Reference Catto2025c ) for a stellarator transport calculation. The solution for ${g}_{1}$ vanishes for bound ( $\sigma =0$ ) orbits. For the unbound ( $\sigma =\pm 1$ ) the solution is found in terms of an integral over a complete elliptic integral E(k) of the second kind by using ${\int }_{-{\pi} }^{{\pi} }{\rm d}{\varphi} {j}=\sigma 8{k}^{-1}{E}({k})$ , with ${k}=\sqrt{2/({h}+1)}$ .

To match to the non-resonant circulating electrons far from the separatrix ( ${k}\rightarrow 0$ or ${h}\rightarrow {\infty }$ ) requires ${f}_{1}\rightarrow -({eE}_{\|}\cos {\phi} /{mk}_{\|}{u})\partial {f}_{0}/\partial {u}| _{{u}=0}\rightarrow 0$ , giving ${g}_{1}\rightarrow ({u}-{\sigma} {\alpha} )\partial {f}_{0}/\partial {u}| _{{u}=0}$ . Hence, $\partial {g}_{1}/\partial {h}| _{{\phi} }\rightarrow {\sigma} ({eE}_{\|}/2{hmk}_{\|})^{1/2}\partial {f}_{0}/\partial {u}| _{{u}=0}$ , and constant $(\oint _{{h}}{\rm d}{\phi} {j})\partial {g}_{1}/\partial {h}| _{{\phi} }$ gives

(4.14) \begin{align}\left.\frac{\partial {g}_{1}}{\partial {h}}\right| _{{\phi} }= \left(\frac{{eE}_{\|}}{{mk}_{\|}}\right)^{1/2}\frac{{\sigma} {\pi} {k}}{4{E}({k})}\frac{\partial {f}_{0}}{\partial {u}}| _{{u}=0}.\end{align}

Using ${\mathrm{d}h}=-4{\mathrm{d}k}/{k}^{3}$ to integrate from the separatrix at ${k}=1$ results in

(4.15) \begin{align} {g}_{1} & ={\sigma} {\pi} \left(\frac{{eE}_{\|}}{{mk}_{\|}}\right)^{1/2}\frac{\partial {f}_{0}}{\partial {u}}| _{{u}=0}{\int }_{{\!\!k}}^{1}\ \frac{{{\rm d}t}}{{t}^{2}{E}({t})} \underset{{k}\rightarrow 0}{\longrightarrow }{\sigma} \left(\frac{{eE}_{\|}}{{mk}_{\|}}\right)^{1/2}\frac{\partial {f}_{0}}{\partial {u}}| _{{u}=0}\nonumber\\[5pt]& \quad \times \left[\frac{2}{{k}}-1.379-\frac{{k}}{2}+{O}({k}^{3})\right]\approx \left({u}-{\sigma} {\alpha} -\frac{{eE}_{\|}\cos {\phi} }{{mk}_{\|}{u}}\right)\frac{\partial {f}_{0}}{\partial {u}}| _{{u}=0},\end{align}

with ${j}=\pm \sqrt{2({h}+\cos {\phi} )}=\pm (2/{k})\sqrt{1-{k}^{2}{\sin}^{2}({\phi} /2)}$ , and ${\alpha} =1.379| {eE}_{\|}/{m}{{k}_{\|}}| ^{1/2}$ now determined. The unbound solution satisfies ${g}_{1}\rightarrow 0$ at the separatrix (h = 1 = k), but $\partial {g}_{1}/\partial {h}| _{{\phi} }$ steps across it. The narrow, unresolved collisional boundary layer about the separatrix provides the smooth matching.

The power loss/absorption is once again evaluated using ${P}=-{eE}_{\|}\langle \sin {\phi} \int$ ${d}^{3}{vv}_{\|}{f}_{1}\rangle _{{\phi} }$ . The details differ only very slightly from a recent lower hybrid current drive (Catto Reference Catto2025b ) and plasma wave evaluation for an unmagnetised plasma (Catto Reference Catto2025a ). Skew symmetry gives $\langle {\int }_{\!\!\!-{\infty }}^{{\infty }}{{\rm d}uu\sin}{\phi} {f}_{1}({u},{\phi} )\rangle _{{\phi} }=0$ . Using $2\sin {\phi} =-{{\rm d}j}^{2}/{\rm d}{\phi} | _{{h}}$ at fixed h to integrate by parts, and inserting the kinetic equation leads to

(4.16) \begin{align}{P} & =\frac{{\omega} {eE}_{\|}}{2{k}_{\|}}\left\langle \int {\mathrm{d}}^{3}{vf}_{\!1}\!\left.\frac{{{\rm d}j}^{2}}{{\rm d}{\phi} }\right| _{{h}}\right\rangle _{{\phi} }=-\frac{{\omega} {eE}_{\|}}{2{k}_{\|}}\left\langle \int {\mathrm{d}}^{3}{vj}^{2}\!\left.\frac{\partial {g}_{2}}{\partial {\phi} }\right| _{{h}}\right\rangle _{{\phi} }\nonumber\\& =-\frac{{\omega} {eE}_{\|}}{2{k}_{\|}}\left\langle \int {\mathrm{d}}^{3}{v}{\Delta} {j}^{2}\!\left.\frac{\partial }{\partial {h}}\right| _{{\phi} }\left({j}\!\left.\frac{\partial {g}_{1}}{\partial {h}}\right| _{{\phi} }\right)\right\rangle _{{\phi} }.\end{align}

To perform the u or j velocity space integral use

(4.17) \begin{align}\left\langle {\int}^{\infty}_{\!\!\!-\infty}{{\rm d}\kern1pt\textit{jj}}^{2}\!\left.\frac{\partial }{\partial {h}}\right| _{{\phi} }{j}\!\left.\frac{\partial {g}_{1}}{\partial {h}}\right| _{{\phi} }\right\rangle _{{\phi} }\!=2\left\langle {\int }_{\!\!\!1}^{{\infty }}{\mathrm{d}hj}\!\left.\frac{\partial }{\partial {h}}\right| _{{\phi} }{j}\!\left.\frac{\partial {g}_{1}}{\partial {h}}\right| _{{\phi} }\right\rangle _{{\phi} }\!=0.384\left(\frac{{eE}_{\|}}{{mk}_{\|}}\right)^{1/2}\frac{\partial {f}_{0}}{\partial {u}}| _{{u}=0}.\end{align}

Using $\mathrm{d}^{3}{v}\rightarrow 2{\pi} {v}_{\bot }{{\rm d}v}_{\bot }{{\rm d}j}/({{\rm d}j}/{{\rm d}u})$ , recalling (2.12) and noticing this bump on tail case is just a very minor modification of the plasma wave result of Catto (Reference Catto2025a ) gives the power loss $(P \gt 0)$ or absorbed $(P \lt 0)$ by the decaying or growing plasma wave as

(4.18) \begin{align}{P}\approx 0.144({Z}+2){mn}{{v}}_{{e}}^{2}{\nu}_{{ee}}\left(\frac{{eE}_{\|}{k}_{\|}}{{m}{\omega} ^{2}}\right)^{1/2}\left[{e}^{-{{\omega} ^{2}}/{{k}}_{\|}^{2}{{v}}_{{e}}^{2}}-\frac{{\Gamma} {k}_{\|}{{v}}_{{e}}^{3}}{{n}{\omega} {{v}}_{{b}}^{3}}{e}^{-({V}-{\omega} /{{k}_{\|}}{)^{2}}/{{v}}_{{b}}^{2}}\right]\!,\end{align}

and the nonlinear or island limit growth/damping rate as

(4.19) \begin{align}{\gamma} =0.144\left({Z}+2\right){\nu}_{{ee}}\left(\frac{{m}{{v}}_{{e}}^{2}{k}_{\|}}{{eE}_{\|}}\right)^{3/2}\frac{{{\omega} }_{{p}}^{2}}{{\omega} ^{2}}\left[\frac{{\Gamma} {{v}}_{{e}}^{2}}{{n}{{v}}_{{b}}^{3}}{e}^{-({V}-{\omega} /{{k}_{\|}}{)^{2}}/{{v}}_{{b}}^{2}}-\frac{{\omega} }{{k}_{\|}{v}_{{e}}}{e}^{-{{\omega} ^{2}}/{{k}}_{\|}^{2}{{v}}_{{e}}^{2}}\right]\!,\end{align}

where ${\omega} ^{2}={{\omega} }_{{p}}^{2}+3{{k}}_{\|}^{2}{{v}}_{{e}}^{2}/2$ . The ratio of the island over the plateau power transfer is

(4.20) \begin{align}\frac{{P}}{{P}_{0}}\approx 0.081\left({Z}+2\right)\frac{{{k}}_{\|}^{3}{{v}}_{{e}}^{3}}{{\omega} ^{3}}\frac{{\nu}_{{ee}}/{k}_{\|}{v}_{{e}}}{\big({eE}_{\|}/{mk}_{\|}{{v}}_{{e}}^{2}\big)^{3/2}}\sim \frac{({\Delta} {v}_{\|}{)}_{{\nu}}^{3}}{({\Delta} {v}_{\|}{)}_{\textit{is}}^{3}},\end{align}

and, therefore, the power loss/absorption is reduced as the collisional layer shrinks.

Unlike the earlier plateau result, there is an explicit dependence on collisions and electric field amplitude ${E}_{\|}$ , ${\gamma} \propto {\nu}_{{ee}}/{{E}}_{\|}^{3/2}$ , in the vanishingly small collision frequency limit for a finite amplitude plasma wave. Compared with the plateau limit, this weak collisionality, finite amplitude limit, requires a much longer time for bump on tail instability wave growth when ${\Gamma} {{v}}_{{e}}^{2}{{e}^{-({V}-{\omega} /{{k}_{\|}}{)^{2}}/{{v}}_{{b}}^{2}}}/{n}{{v}}_{{b}}^{3}\gt ({\omega} /{{k}_{\|}}{{v}_{{e}}}){e}^{-{{\omega} ^{2}}/{{k}}_{\|}^{2}{{v}}_{{e}}^{2}}$ and ${V}\approx {\omega} /{k}_{\|}$ . Similarly, collisional decay of a stable plasma wave occurs very slowly when ${\Gamma} {{v}}_{{e}}^{2}{{e}^{-({V}-{\omega} /{{k}_{\|}}{)^{2}}/{{v}}_{{b}}^{2}}}/{{nv}}_{{b}}^{3}\lt ({\omega} /{{k}_{\|}}{{v}_{{e}}}){e}^{-{{\omega} ^{2}}/{{k}}_{\|}^{2}{{v}}_{{e}}^{2}}\ll 1$ . It is important to realise there is no limit in which ${\nu}_{{ee}}=0$ may be assumed at the onset! As ${\nu}_{{ee}}\rightarrow 0$ the very small growth or damping rate in this limit might look somewhat like saturation for a single wave. For comparison, the long time, collisionless phase mixing limit of O’Neil (Reference O’neil1965) gives a vanishing damping rate in (34), rather than ${{\gamma} \propto {\nu}_{{ee}}{E}}_{\|}^{-3/2}$ .

In this large island width, weak collision limit ${f}_{1}\sim [{eE}_{\|}/{mk}_{\|}({\Delta} {v}_{\|})_{\textit{is}}]\partial {f}_{0}/\partial {u}| _{{u}=0}$ and

(4.21) \begin{align}\frac{\partial {f}_{1}/\partial {u}}{\partial {f}_{0}/\partial {u}}\sim \frac{{v}_{{e}}{f}_{1}}{({\Delta} {v}_{\|})_{\textit{is}}{f}_{0}}\sim \frac{{eE}_{\|}/{mk}_{\|}{{v}}_{{e}}^{2}}{({\Delta} {v}_{\|}{)}_{\textit{is}}^{2}}\sim 1\end{align}

as expected, and the effective collision frequency is ${\nu}_{\textit{eff}}\sim {\nu}{{v}}_{{e}}^{2}/({\Delta} {v}_{\|}{)}_{\textit{is}}^{2}\sim {\nu}{{v}}_{{e}}^{2}({mk}_{\|}/{eE}_{\|})$ . The step size ${eE}_{\|}/{m}{\nu}_{\textit{eff}}\sim ({\Delta} {v}_{\|})_{\textit{is}}/{\varDelta}$ is many island widths $({\Delta} {v}_{\|})_{\textit{is}}\sim ({eE}_{\|}/{mk}_{\|})^{1/2}$ as ${\varDelta} \ll 1$ . For many modes instead of a single mode, there will be large velocity space diffusion due to collisionally stochastic electron motion in the overlapping islands having narrow collisional dissipation regions enclosing their separatrices. The nonlinear island diffusivity is

(4.22) \begin{align}{D}_{\textit{is}}\sim {\nu}_{\textit{eff}}({eE}_{\|}/{m}{\nu}_{\textit{eff}})^{2}\sim ({eE}_{\|}/{m})^{3}/{\nu}{{v}}_{{e}}^{2}{k}_{\|},\end{align}

giving

(4.23) \begin{align}\frac{{D}_{\textit{is}}}{{D}_{{\nu}}}\sim \frac{({eE}_{\|}/{m})^{3}}{{\nu}^{2}{{v}}_{{e}}^{4}{k}_{\|}}\sim \left(\frac{{eE}_{\|}}{{mk}_{\|}{{v}}_{{e}}^{2}}\right)^{3}\left(\frac{{k}_{\|}{v}_{{e}}}{{\nu}}\right)^{2}\sim \frac{({\Delta} {v}_{\|}{)}_{\textit{is}}^{6}}{({\Delta} {v}_{\|}{)}_{{\nu}}^{6}}\gg 1,\end{align}

which is seen to be large. Consequently, it is only when the $\partial {f}_{1}/\partial {u}$ term as well as collisions are retained in the kinetic equation that islands with narrow dissipative separatrices lead to stochastic diffusivity competing with, $({\Delta} {v}_{\|})_{\textit{is}}\sim ({\Delta} {v}_{\|})_{{\nu}}$ , or dominating over, $({\Delta} {v}_{\|})_{\textit{is}}\gg ({\Delta} {v}_{\|})_{{\nu}}$ , Fokker–Planck diffusion ( ${D}_{{\nu}}\sim {\nu}{{v}}_{{e}}^{2}$ ), and perhaps even allowing the many plasma waves to locally modify ${f}_{0}$ away its initial form in velocity space.