2.1. Cold-plasma dispersion relations for ion cyclotron waves

The real frequency $\omega _{k{\mathrm{r}}}$ of an ICW with wavenumbers $k_\parallel$ and $k_\perp$ , parallel and perpendicular to the background magnetic field, respectively, can be approximated by the cold-plasma dispersion relation (equation (21) in chapter 2 of Stix (Reference Stix1992)):

(2.1a) \begin{align} \omega _{k{\mathrm{r}}}= \varOmega _{\mathrm{p}}\sqrt { \dfrac {1}{2}\big(b-\sqrt {b^2-4c}\big)}, \end{align}

where

(2.1b) \begin{align} b&=(k d_{\mathrm{p}})^2 + (k_\parallel d_{\mathrm{p}})^2 + c,\\[-8pt]\nonumber \end{align}

(2.1c) \begin{align} c &= k^2k_\parallel ^2d_{\mathrm{p}}^4. \end{align}

Here, $k=(k_\parallel ^2+k_\perp ^2)^{1/2}$ is the magnitude of the wavenumber. If we now set $k_\perp =0$ , we obtain the PICW dispersion relation (Hollweg & Isenberg Reference Hollweg and Isenberg2002):

(2.2) \begin{align} \omega _{k_\parallel {\mathrm{r}},{\rm P}} =\dfrac {\varOmega _{\mathrm{p}}(k_\parallel d_{\mathrm{p}})^2}{2}\Big[\sqrt {1+4(k_\parallel d_{\mathrm{p}})^{-2}}-1\Big]. \end{align}

2.2. Ion cyclotron wave stability

The linear stability of a plasma to a given wave mode is entirely due to the preferred direction of energy exchange between the wave mode and particle population, as mediated by resonant wave–particle interactions. Waves and particles, here cyclotron waves and protons, interact resonantly when $\omega _{k{\mathrm{r}}}$ , as measured in a frame moving along the background magnetic field at the parallel velocity of the proton $v_\parallel$ , is an integer ( $n$ ) multiple of the proton cyclotron frequency $\varOmega _{\mathrm{p}}$ (Stix Reference Stix1992), i.e.

(2.3) \begin{align} \omega _{k{\mathrm{r}}}-k_\parallel v_\parallel =n\varOmega _{\mathrm{p}}. \end{align}

Here, $v_\parallel$ is measured in the frame travelling at the solar-wind velocity $U$ . One can solve (2.3) either for $k_{\parallel \mathrm{res}}=k_{\parallel \mathrm{res}}(v_\parallel ,k_\perp ,n)$ , the $k_\parallel$ of a wave with given $k_\perp$ whose $n{\mathrm{th}}$ cyclotron harmonic is resonant with a particle travelling at $v_\parallel$ , or for

(2.4) \begin{align} v_{\parallel \mathrm{res}}(k_\parallel ,k_\perp ,n) = \dfrac {\omega _{k{\mathrm{r}}}-n\varOmega _{\mathrm{p}}}{k_\parallel }, \end{align}

the velocity of a particle that is resonant with the $n{\mathrm{th}}$ cyclotron harmonic of a wave with a given $k_\parallel$ and $k_\perp$ .

For $n=0$ , $v_{\parallel \mathrm{res}} = \omega _{k{\mathrm{r}}}/k_\parallel$ , i.e. the parallel resonant velocity for a given wave is its phase speed. The $n=0$ resonance is often referred to as the Landau resonance. Landau-resonant wave–particle interactions can be mediated by the component of the electric field parallel to the magnetic field, as is the case with Landau damping, or by the electric field components perpendicular to the magnetic field, as is the case with transit-time damping (Stix Reference Stix1992). Qualitatively, if the derivative of the proton velocity distribution function, $f_{\mathrm{p}}$ , with respect to $v_\parallel$ has the same sign as the wave phase velocity, $\omega _{k{\mathrm{r}}}/k_\parallel$ , in the immediate vicinity of the resonant velocity $v_\parallel =\omega _{k{\mathrm{r}}}/k_\parallel$ , more protons will give energy to the wave than take away from it. The net effect will be wave growth – instability. Conversely, wave energy will be absorbed (damped) if the opposite is true.

When $n\ne 0$ , the wave–particle interactions are said to be cyclotron resonant. Consider a single plane wave. In the ‘wave frame’, which moves at the wave phase velocity along the background magnetic field $\hat{\boldsymbol{b}} \omega /k_\parallel$ (here $\hat {\boldsymbol{b}}=\boldsymbol{B}/B$ is the unit vector along the magnetic field), the partial time derivative of the wave fields vanishes, and the total particle energy $H = ({1}/{2}) mv^2 + q_{\mathrm{p}} \varPhi$ is constant, where $\varPhi$ is the electrostatic potential. For small-amplitude waves, $\varPhi$ is a constant plus a small sinusoidal oscillation. Therefore, the particle kinetic energy $({1}/{2}) m_{\mathrm{p}} v^2$ undergoes only small oscillations, cannot change secularly and is effectively conserved. Cyclotron-resonant wave–particle interactions can, however, cause particles to diffuse in the $v_\parallel {-} v_\perp$ plane along contours of constant energy in the wave frame, which are defined by the equation (Kennel & Engelmann Reference Kennel and Engelmann1966)

(2.5) \begin{align} \bigg(v_\parallel -\dfrac {\omega _{k{\mathrm{r}}}}{k_\parallel }\Big|_{k_{\parallel \mathrm{res}}} \bigg)^2 + v_\perp ^2 = {\mathrm{const}}. \end{align}

The solutions to (2.5) are called quasi-linear contours. The intuition behind the stability of cyclotron-resonant waves (like ion cyclotron and whistler waves) is much the same as waves mediated by the Landau resonance. However, the relevant velocity-space derivative of $f_{\mathrm{p}}$ is given by the differential operator (Kennel & Wong Reference Kennel and Wong1967)

(2.6) \begin{align} G=\left (\left .\dfrac {\omega _{k{\mathrm{r}}}}{k_\parallel }\right . -v_\parallel \right )\frac {\partial }{\partial v_\perp }+v_\perp \frac {\partial }{\partial v_\parallel }, \end{align}

which calculates the derivative of its operand along the quasi-linear contours (2.5). To determine the stability of a wave mode, one must consider the net effect of all the resonant particles. Indeed, this consideration is taken into account in the full calculation of the linear growth rate in Kennel & Wong (Reference Kennel and Wong1967). For PICWs, one can show that the growth rate $\gamma _{\mathrm{P}}(k_\parallel )$ is positive when

(2.7) \begin{align} \gamma _{\mathrm{P}}(k_\parallel )\propto {\mathrm{sign}}\left (\left .\dfrac {\omega _{k_\parallel {\mathrm{r}},P}}{k_\parallel }\right . \right )\int _0^\infty {\mathrm{d}}v_\perp \,v_\perp ^2\left .\left [G_{\mathrm{P}}f_{\mathrm{p}}(v_\parallel ,v_\perp )\right ]\right |_{v_{\parallel {\mathrm{P}}}} \gt 0. \end{align}

Here, $G_{\mathrm{P}}$ is the differential operator $G$ of (2.6) evaluated for $\omega _{k{\mathrm{r}}}=\omega _{k_\parallel {\mathrm{r}},{\rm P}}$ and $v_{\parallel {\mathrm{P}}}=v_{\parallel {\mathrm{res}}}(k_\parallel ,0,1)$ is the resonant parallel velocity for a PICW with parallel wavenumber $k_\parallel$ .

The PICWs are left circularly polarised and therefore only able to interact with protons via the $n=1$ cyclotron resonance (Stix Reference Stix1992). It then follows that each $v_\parallel$ can resonate with only a single $k_\parallel$ , and there is a unique quasi-linear contour passing through each point in velocity space (Isenberg, Lee & Hollweg Reference Isenberg, Lee and Hollweg2001). This can be seen graphically with the aid of figure 3(c). Technically, protons with a given $v_\parallel$ whose magnitude is ${\lesssim}v_{\mathrm{A}}$ could interact with oblique ICWs at multiple values of $k_\parallel$ via different resonances (i.e. $n\neq 1$ ). However, the values of $k_\parallel$ for $n\neq 1$ are far outside the inertial range, and so we neglect them. Thus, effectively, when protons interact with oblique ICWs, the protons at each $v_\parallel$ interact with only a single $k_\parallel$ , and there is again a unique quasi-linear contour passing through each point in the $v_\parallel {-} v_\perp$ plane.

2.3. Quasi-linear focusing

Quasi-linear focusing (Chandran et al. Reference Chandran, Pongkitiwanichakul, Isenberg, Lee, Markovskii, Hollweg and Vasquez2010b ; Isenberg & Vasquez Reference Isenberg and Vasquez2011) is a process in which oblique ICWs from the turbulent cascade are ‘focused’ into PICWs via wave–particle interactions. To explain this process, we assume for simplicity that all the oblique ICWs propagate with a $k_\perp$ that is an a priori known function of $k_\parallel$ , $k_{\perp ,\mathrm{CB}}(k_\parallel )$ , given by critical balance (Goldreich & Sridhar Reference Goldreich and Sridhar1995; Maron & Goldreich Reference Maron and Goldreich2001; Sioulas et al. Reference Sioulas2025) and are present at all $k_\parallel d_{\mathrm{p}}\lesssim 1$ . Function $k_{\perp ,\mathrm{CB}}(k_\parallel )$ is further discussed in § 3.1; however, it should be stated here that the results of this section – and this work in general – are insensitive to the exact functional dependence of $k_{\perp ,\mathrm{CB}}$ on $k_\parallel$ . Let us define the real frequency of the oblique ICW with this choice of $k_\perp$ to be $\omega _{k_\parallel {\mathrm{r}},{\rm O}}\equiv \omega _{k{\mathrm{r}}}(k_\parallel ,k_{\perp ,\mathrm{CB}}(k_\parallel ))$ . As these ICWs heat an initially Maxwellian plasma, the plasma will eventually reach a state, denoted $f_{\mathrm{p,O}}$ , of marginal stability to the ICWs, in which (Kennel & Engelmann Reference Kennel and Engelmann1966; Isenberg et al. Reference Isenberg, Lee and Hollweg2001; Isenberg & Vasquez Reference Isenberg and Vasquez2011)

(2.8) \begin{align} G_{\mathrm{O}}|_{k_{\parallel \mathrm{O}}} f_{\mathrm{p,O}}(v_\parallel ,v_\perp )=0. \end{align}

Here, $G_{\mathrm{O}}$ is the differential operator $G$ of (2.6), evaluated for $\omega _{k{\mathrm{r}}}=\omega _{k_\parallel {\mathrm{r}},{\rm O}}$ . So that (2.8) is purely a function of velocity, $G_{\mathrm{O}}$ is evaluated at the resonant parallel wavenumber for oblique ICWs, $k_{\parallel \mathrm{O}}\equiv k_{\parallel {\mathrm{res,O}}}(v_\parallel )$ . After $f_{\mathrm{p}}$ reaches the state $f_{\mathrm{p,O}}$ , plasma heating by oblique ICWs ceases, and the oblique ICWs no longer damp. Moreover, $f_{\mathrm{p,O}}$ , which is marginally stable to oblique ICWs, is unstable to PICWs, as we now show. Henceforth, we assume that both oblique ICWs and PICWs propagate anti-sunward, so ${{\mathrm{sign}}\ (\omega _{k_\parallel {\mathrm{r}},{\rm O}}/k_\parallel )={\mathrm{sign}}(\omega _{k_\parallel {\mathrm{r}},{\rm P}}/k_\parallel )}=1$ . The condition (2.7) can be simplified by adding and subtracting the phase speed of the co-resonant oblique ICW, $(\omega _{k_\parallel {\mathrm{r}},{\rm O}}/k_\parallel )|_{k_{\parallel \mathrm{O}}}$ :

(2.9) \begin{align} G_{\mathrm{P}}f_{\mathrm{p,O}}(v_\parallel ,v_\perp )&=\left [\left (\dfrac {\omega _{k_\parallel {\mathrm{r}},{\rm P}}}{k_\parallel }-\left .\dfrac {\omega _{k_\parallel {\mathrm{r}},{\rm O}}}{k_\parallel }\right |_{k_{\parallel \mathrm{O}}}\right )\frac {\partial }{\partial v_\perp }+G_{\mathrm{O}}|_{k_{\parallel \mathrm{O}}}\right ] f_{\mathrm{p,O}}(v_\parallel ,v_\perp )\nonumber\\[5pt] &=\left (\dfrac {\omega _{k_\parallel {\mathrm{r}},{\rm P}}}{k_\parallel }-\left .\dfrac {\omega _{k_\parallel {\mathrm{r}},{\rm O}}}{k_\parallel }\right |_{k_{\parallel \mathrm{O}}}\right )\frac {\partial }{\partial v_\perp }f_{\mathrm{p,O}}(v_\parallel ,v_\perp ), \end{align}

where the last equality follows from the definition of $f_{\mathrm{p,O}}$ (2.8). With (2.9) in hand, we can integrate (2.7) by parts. The remaining integral is positive-definite, so the inequality reduces to $\gamma _{\mathrm{P}}(k_\parallel )\propto \left .(\omega _{k_\parallel {\mathrm{r}},{\rm O}}/k_\parallel )\right |_{k_{\parallel \mathrm{O}}(v_{\parallel {\mathrm{P}}})}-\omega _{k_\parallel {\mathrm{r}},{\rm P}}/k_\parallel \gt 0$ . Thus, a PICW will be unstable if the oblique ICW that is co-resonant, or resonates with protons of the same $v_\parallel$ , has a faster phase velocity.

Figure 1.

The frequency of an oblique ICW, $\omega _{k_\parallel {\mathrm{r}},{\rm O}}$ , here with (a very exaggerated value of) $k_\perp d_{\mathrm{p}}=10$ , is always larger than $\omega _{k_\parallel {\mathrm{r}},{\rm P}}$ for a given $k_\parallel$ . As a result, $k_{\parallel {\mathrm{P}}}\gt k_{\parallel {\mathrm{O}}}$ , as shown by the intersection of $\omega _{k{\mathrm{r}},{\rm O}}$ and $\omega _{k_\parallel {\mathrm{r}},{\rm P}}$ with $\varOmega _{\mathrm{p}}+k_\parallel v_\parallel$ in (a). One can then show that $(\omega _{k_\parallel {\mathrm{r}},{\rm O}}/k_\parallel )|_{k_{\parallel \mathrm{O}}}-(\omega _{k_\parallel {\mathrm{r}},{\rm P}}/k_\parallel )|_{k_{\parallel {\mathrm{P}}}}\gt 0$ (see text), a necessary condition for quasi-linear focusing. For this condition to be sufficient, the proton velocity distribution must be nearly constant along oblique ICW contours. In (b) (after figure 4 in Chandran et al. Reference Chandran, Pongkitiwanichakul, Isenberg, Lee, Markovskii, Hollweg and Vasquez(2010b )), we show the proton velocity distribution function (darker purple denotes contours of higher particle density) only for $v_\parallel \lt v_\parallel ^*$ , where $v_\parallel ^*$ is the resonant parallel velocity associated with the start of the turbulent transition region (see § 3.2). Physically, PICWs provide alternative quasi-linear contours (black semicircles) along which protons can diffuse. Because $(\omega _{k_\parallel {\mathrm{r}},{\rm O}}/k_\parallel )|_{k_{\parallel \mathrm{O}}}-(\omega _{k_\parallel {\mathrm{r}},{\rm P}}/k_\parallel )|_{k_{\parallel {\mathrm{P}}}}\gt 0$ (the difference in (b) is exaggerated for clarity), protons will on average diffuse towards regions of velocity space with lower proton density, i.e. in the direction given by the black arrow. These regions of low proton density also have lower energy (contours of larger constant energy are given by darker orange semicircles); the energy lost by protons is given to the PICWs, causing their amplitude to grow (Chandran et al. Reference Chandran, Pongkitiwanichakul, Isenberg, Lee, Markovskii, Hollweg and Vasquez2010b ). Purple, black and orange dashed lines, corresponding to the centre of the semicircular quasi-linear contours for oblique ICWs, PICWs and contour of constant energy, respectively, are included to aid the eye.

One can equivalently view $\gamma _{\mathrm{P}}(k_\parallel )$ as a function of $v_\parallel$ by evaluating at $k_{\parallel {\mathrm{P}}}=k_{\parallel {\mathrm{res,P}}}(v_\parallel )$ :

(2.10) \begin{align} \gamma _{\mathrm{P}}(k_{\parallel {\mathrm{P}}}(v_\parallel ))\propto \left .\dfrac {\omega _{k_\parallel {\mathrm{r}},{\rm O}}}{k_\parallel }\right |_{k_{\parallel \mathrm{O}}}-\left .\dfrac {\omega _{k_\parallel {\mathrm{r}},{\rm P}}}{k_\parallel }\right |_{k_{\parallel {\mathrm{P}}}}\gt 0. \end{align}

In (2.10), as before, the growth rate of the resonant PICW is positive if the phase speed of the co-resonant oblique ICW is larger than that of the PICW. This condition was proven to be satisfied up to first order in an asymptotic expansion in $|v_\parallel |/v_{\mathrm{A}}$ in Chandran et al. (Reference Chandran, Pongkitiwanichakul, Isenberg, Lee, Markovskii, Hollweg and Vasquez2010b ). In what follows, we show that the condition is satisfied in general for the cold-plasma dispersion relation. In Appendix A, we show algebraically that $\omega _{k{\mathrm{r}}}(k_\parallel ,k_\perp )\gt \omega _{k_\parallel {\mathrm{r}}}$ for all $(k_\parallel ,k_\perp )\gt (0,0)$ . This fact is also illustrated by graphs of $\omega _{k_\parallel {\mathrm{r}},{\rm P}}$ and $\omega _{k_\parallel {\mathrm{r}},{\rm O}}$ with a chosen value $k_\perp$ , as shown in figure 1(a). It is therefore true that for any $k_\parallel$ , $\omega _{k_\parallel {\mathrm{r}},{\rm O}}/k_\parallel \gt \omega _{k_\parallel {\mathrm{r}},{\rm P}}/k_\parallel$ , and that for any $v_\parallel$ , $k_{\parallel {\mathrm{P}}}\gt k_{\parallel \mathrm{O}}$ (as shown in figure 1 a). Because $\omega _{k_\parallel {\mathrm{r}},{\rm P}}/k_\parallel$ is a monotonically decreasing function of $k_\parallel$ , $(\omega _{k_\parallel {\mathrm{r}},{\rm O}}/k_\parallel )|_{k_{\parallel \mathrm{O}}}\gt (\omega _{k_\parallel {\mathrm{r}},{\rm P}}/k_\parallel )|_{k_{\parallel \mathrm{O}}}\gt (\omega _{k_\parallel {\mathrm{r}},{\rm P}}/k_\parallel )|_{k_{\parallel {\mathrm{P}}}}$ and $\gamma _{\mathrm{P}}(k_{\parallel {\mathrm{P}}}(v_\parallel ))\gt 0$ .

Chandran et al. (Reference Chandran, Pongkitiwanichakul, Isenberg, Lee, Markovskii, Hollweg and Vasquez2010b ) also showed that, physically, PICWs can be driven unstable by strong oblique ICW heating because co-resonant PICWs provide alternative contours for protons to diffuse along. As the condition (2.10) is generally satisfied, PICW contours will be shallower than oblique ICW contours at a given $v_\parallel$ . There will therefore be a proton velocity-space density gradient along PICW contours in the direction of smaller parallel speed. Protons will then naturally diffuse in the direction of decreasing particle density along the PICW contours in the direction of increasing parallel speed. However, PICW contours are steeper than contours of constant energy, so the net diffusion of protons is towards velocity-space regions of lower kinetic energy. Total energy is conserved, so the kinetic energy lost by protons is resonantly transferred to the electromagnetic field as PICWs, driving the waves unstable. We show a highly simplified version of this process in figure 1(b).

3.1. The turbulent transition range and its associated scales

The turbulent transition range is an interval of the turbulent spectrum in which the spectral scaling is observed to be very steep. While reports of the spectral index of the transition range vary, it is usually described as having a mean spectral index ${\sim }k_\perp ^{-4}$ (Leamon et al. Reference Leamon, Smith, Ness, Matthaeus and Wong1998; Bowen et al. Reference Bowen2022; McIntyre et al. Reference McIntyre, Chen, Squire, Meyrand and Simon2025). This steep scaling is observed near $k_\perp \rho _{\mathrm{p}}\sim 1$ , where $\rho _{\mathrm{p}}=v_{\mathrm{thp}}/\varOmega _{\mathrm{p}}$ is the thermal proton gyroradius and $v_{\mathrm{thp}}$ is the proton thermal speed. The transition range is sandwiched by the ${\sim }k_\perp ^{-3/2}$ or ${\sim }k_\perp ^{-5/3}$ inertial range at larger scales and the ${\sim }k_\perp ^{-2.8}$ kinetic-Alfvén-wave range at smaller scales (Alexandrova et al. Reference Alexandrova, Saur, Lacombe, Mangeney, Mitchell, Schwartz and Robert2009; Podesta Reference Podesta2013). In keeping with recent work (Meyrand et al. Reference Meyrand, Squire, Schekochihin and Dorland2021), we denote the perpendicular wavenumber at the boundary between the inertial range and transition range $k_\perp ^*$ , as illustrated in figure 2.

Figure 2.

A reduced magnetic fluctuation spectrum $\mathcal{E}_{\delta B}(k_\perp )$ , typical of the solar wind. The transition range (TR), characterised by a steep ${\sim }k_\perp ^{-4}$ spectral slope, is sandwiched between the inertial range (IR) and kinetic-Alfvén-wave range (KAW). We denote the perpendicular wavenumber at the boundary between the IR and TR as $k_\perp ^*$ .

Figure 3.

A diagrammatic description of quasi-linear focusing in the presence of the turbulent transition range. Our model turbulent spectrum for $\delta B$ is shown in (a), with darker green denoting larger values. The perpendicular and parallel start of the transition range, $k_\perp ^*$ and $k_\parallel ^*$ , and the critical balance anisotropy relationship $k_{\parallel ,\mathrm{CB}}(k_\perp )$ are given by the vertical, horizontal and slanted dashed black lines, respectively. Heating by the cascade (red circle in a) pushes the proton velocity distribution $f_{\mathrm{p}}$ , shown in (b), constant along oblique ICW quasi-linear contours for $v_\parallel \lt v_\parallel ^*$ , where $v_\parallel ^*$ is the resonant parallel velocity for an oblique ICW with $k_\parallel =k_\parallel ^*$ and $k_\perp =k_\perp ^*$ . Here, $f_{\mathrm{p}}$ is unstable to PICWs because oblique ICW quasi-linear contours are steeper than PICW contours (given by black lines in b). The PICWs will therefore be emitted, cooling $f_{\mathrm{p}}$ , in the wavenumber region given by the blue circle in (a). For $v_\parallel \gt v_\parallel ^*$ , $f_{\mathrm{p}}$ is much more isotropic because heating by oblique ICWs is much weaker there. Consequently, the contours of $f_{\mathrm{p}}$ are shallower than those of the PICWs, and the PICWs are damped. A given unstable PICW, resonating with protons with $v_\parallel$ given by the vertical blue line in (b), will necessarily have $k_\parallel \lt k_{\parallel {\mathrm{P}}}^*\simeq k_\parallel ^*$ . (c) The PICW resonance diagram shows $\omega _{k_\parallel {\mathrm{r,P}}}$ (black curved line) and the right-hand side of (3.2) (straight lines) – the intersection of which is the solution to the resonance condition. Wave–particle interactions with faster protons constitute steeper lines and therefore smaller resonant $k_\parallel$ . We approximate the extent of the excited spectrum of PICWs by the blue circle in (a).

We assume that the two-dimensional power spectrum of the fluctuating magnetic field $\mathcal{E}_{\delta B}(k_\perp , k_\parallel )$ , as reported in Squire et al. (Reference Squire, Meyrand, Kunz, Arzamasskiy, Schekochihin and Quataert2022), is a constant function of $k_\parallel$ between $k_\parallel =0$ and a non-zero parallel wavenumber given by the critical-balance cone, $k_{\parallel ,\mathrm{CB}}(k_\perp )$ . The inverse of this function, $k_{\perp ,\mathrm{CB}}(k_\parallel )$ , is used to compute $\omega _{k_\parallel {\mathrm{r}},{\rm O}}$ , as discussed in § 2.3. We take $k_{\parallel ,\mathrm{CB}}(k_\perp )$ to be an increasing function of $k_\perp$ . The bulk of the fluctuation energy at each $k_\perp$ then has a characteristic parallel wavenumber ${\sim }k_{\parallel ,\mathrm{CB}}(k_\perp )$ , even though smaller parallel wavenumbers are also present. Therefore, the characteristic parallel wavenumber at the start of the transition range, $k_\parallel ^* \equiv k_{\parallel ,\mathrm{CB}}(k_\perp ^*)$ , is the largest parallel wavenumber reached by the energy cascade within the inertial range. A depiction of $\mathcal{E}_{\delta B}(k_\perp , k_\parallel )$ that includes $k_\perp ^*$ , $k_\parallel ^*$ and the function $k_{\parallel ,\mathrm{CB}}(k_\perp )$ is shown in figure 3(a).

3.2. Quasi-linear focusing near the turbulent transition range

Let us ignore for a moment the expansion of the solar wind and focus on how the presence of the turbulent transition range bounds the regions of wave vector space and velocity space where wave–particle interactions lead to PICW emission via quasi-linear focusing. Relative to the inertial range, there is vanishingly little power in the transition range. Indeed, observations of imbalanced turbulence by PSP have shown that the spectral energy across the transition range drops by at least a factor of $100$ (Bowen et al. Reference Bowen2022, Reference Bowen, Bale, Chandran, Chasapis, Chen, Dudok de Wit, Mallet, Meyrand and Squire2024a ). Let us assume for the sake of simplicity that the power in the oblique ICW spectrum falls to zero for $k_\perp \gt k_\perp ^*$ and $k_\parallel \gt k_\parallel ^*$ . The highest-frequency oblique ICWs in the inertial range, depicted by the red circle in figure 3(a), dominate the oblique-ICW heating rate.

The negligible power in oblique ICWs at $k_\parallel \gtrsim k_\parallel ^*$ restricts strong cyclotron-resonant interactions between protons and oblique ICWs to $v_\parallel \lesssim v_\parallel ^*$ , where

(3.1) \begin{align} v_\parallel ^*=v_{\parallel \mathrm{res,O}}\big(k_\parallel ^*, 1\big)=\dfrac {\omega _{k_\parallel {\mathrm{r,O}}}(k_\parallel ^*)-\varOmega _{\mathrm{p}}}{k_\parallel ^*} \end{align}

is the solution to (2.4) for oblique ICWs at the start of the transition range. We note that $v_\parallel ^* \lt 0$ , so faster protons moving in the direction opposite the wave-propagation direction can resonate, but slower protons cannot. We plot $v_\parallel ^*$ with a vertical dashed black line over a model proton velocity distribution in figure 3(b). Let us assume the resonant heating by the oblique cascade is sufficiently strong to flatten the proton distribution constant along the quasi-linear contours of (2.5) for oblique ICWs at $v_\parallel \lt v_\parallel ^\ast$ . For $v_\parallel \gt v_\parallel ^*$ , we assume the proton distribution has not been extensively heated by oblique ICWs and is significantly more isotropic than for $v_\parallel \lt v_\parallel ^*$ . We depict the resulting distribution in figure 3(b). As discussed in § 2.3, the proton distribution for $v_\parallel \lt v_\parallel ^*$ is resonantly unstable to PICWs.

The resonance condition (2.3), now applied to the relevant $n=1$ cyclotron resonance for PICWs, is

(3.2) \begin{align} \omega _{k_\parallel {\mathrm{r,P}}}(k_\parallel )=k_\parallel v_\parallel +\varOmega _{\mathrm{p}}. \end{align}

Solutions to (3.2) are easily found pictorially in a resonance diagram, which we have included in figure 3(c). We plot the PICW frequency $\omega _{k_\parallel {\mathrm{r,P}}}$ , the left-hand side of (3.2), versus $k_\parallel$ with a solid black line. The right-hand side of (3.2) is a straight line originating from the top-left corner of the plot. As discussed in § 2.3, the point at which the two lines intersect is a solution of the resonance condition. We have included two lines for two different parallel velocities. The slanted dashed line is $k_\parallel v_\parallel ^*+\varOmega _{\mathrm{p}}$ , which intersects $\omega _{k_\parallel {\mathrm{r,P}}}$ at

(3.3) \begin{align} k_{\parallel {\mathrm{P}}}^*=k_{\parallel {\mathrm{P}}}\big(v_\parallel ^*\big), \end{align}

the parallel wavenumber for PICWs resonant with protons with $v_\parallel =v_\parallel ^*$ . We illustrate the value of $k_{\parallel {\mathrm{P}}}^*$ with a black dashed vertical line in figure 3(c). The blue solid line in figure 3(c) is $k_\parallel v_\parallel +\varOmega _{\mathrm{p}}$ for a parallel velocity $v_\parallel \lt v_\parallel ^*$ , which is itself shown by the vertical solid blue line in figure 3(b). The blue solid line in figure 3(c) intersects the wave frequency at a parallel wavenumber $k_\parallel \lt k_{\parallel {\mathrm{P}}}^*$ , where PICWs are unstable. This blue solid line is coloured this way to represent an unstable PICW actively cooling the plasma. Although our discussion so far has focused on a single wave mode, we expect PICW emission will occur in a band of wavenumbers at $k_\parallel \lesssim k_{\parallel {\mathrm{P}}}^*$ , as depicted by the blue circle in figure 3(a).

What about PICWs with $k_\parallel \gt k_{\parallel {\mathrm{P}}}^*$ ? In our resonance diagram, these waves would be represented by lines that are shallower than the black dashed line. Shallower lines correspond to resonances with protons that have $v_\parallel \gt v_\parallel ^*$ . Thus, PICWs with $k_\parallel \gt k_{\parallel {\mathrm{P}}}^*$ are strongly damped by the proton distribution. The result of quasi-linear focusing is therefore very little wave energy for $k_\parallel \gt k_{\parallel {\mathrm{P}}}^*$ and a continuous buildup of PICW energy for $k_\parallel \lt k_{\parallel {\mathrm{P}}}^*$ . In a homogeneous plasma, there is no linear mechanism by which the wave energy at $k_\parallel \lt k_{\parallel {\mathrm{P}}}^*$ can be transported to $k_\parallel \gt k_{\parallel {\mathrm{P}}}^*$ , where the wave energy can damp and heat the plasma. As it turns out, such a transport mechanism is the natural consequence of the WKB propagation of PICWs in the inhomogeneous solar wind.

3.3. The WKB evolution of PICWs

We now detail how the propagation and advection of PICWs in the inhomogeneous solar wind change the properties of the waves and, inevitably, lead to their dissipation. As PICWs are driven unstable by the turbulent cascade, they will both propagate and be advected anti-sunward. Let us assume that the solar-wind background is in steady state. Then, the real part of the angular frequency of PICWs, measured in the heliocentric frame, will be constant with radius:

(3.4) \begin{align} \omega _{{\mathrm{r}}\odot }=k_\parallel U+\omega _{k_\parallel {\mathrm{r}},P}= {\mathrm{const}}. \end{align}

In the $U\gg v_{\mathrm{A}}$ super-Alvénic wind, $\omega _{k_\parallel {\mathrm{r,P}}}\leqslant k_\parallel v_{\mathrm{A}}\ll k_\parallel U$ , and so $\omega _{{\mathrm{r}}\odot }=\omega _{k_\parallel {\mathrm{r,P}}}+k_\parallel U\simeq k_\parallel U$ . For heliocentric distances much larger than the Alfvén critical point, $U\sim {\mathrm{const.}}$ , so $k_\parallel \sim {\mathrm{const}}$ . Thus, $k_\parallel d_{\mathrm{p}}\sim n^{-1/2}\sim x$ , where $x=r/R_{\odot }$ is the heliocentric distance normalised to the solar radius $R_{\odot }$ . In the sub-Alfvénic wind, the same analysis, again taking the non-dispersive limit, yields $k_\parallel v_{\mathrm{A}}\sim {\mathrm{const.}}$ , so $k_\parallel d_{\mathrm{p}}\sim B^{-1}\sim x^2$ at least for $x\gtrsim 3$ . While the radial scaling of $U$ near the Alfvén critical point somewhat complicates the preceding analysis, the general qualitative conclusion is clear: the normalised parallel wavenumber, $k_\parallel d_{\mathrm{p}}$ , of PICWs increases as the waves travel away from the Sun.

Depicted diagrammatically, the radial evolution of $\omega _{k_\parallel {\mathrm{r,P}}}/\varOmega _{\mathrm{p}}$ and $k_\parallel d_{\mathrm{p}}$ is very simple because the PICW cold-plasma dispersion relation (2.2) is invariant with heliocentric distance if $\omega _{k_\parallel {\mathrm{r,P}}}$ and $k_\parallel$ are normalised by $\varOmega _{\mathrm{p}}$ and $d_{\mathrm{p}}$ , respectively. As a result, the normalised frequency and parallel wavenumber of a given PICW simply increase along the normalised dispersion relation as the waves are transported to larger radii. We depict this evolution with a black arrow in figure 4(c) and vertical black arrow in figure 4(a). As $k_\parallel d_{\mathrm{p}}$ increases, the resonant parallel velocity decreases, as depicted by the shallower red resonance line in figure 4(c). If PICW emission and absorption occur continuously across a range of heliocentric distances, then at any given heliocentric distance one will observe PICWs cooling protons with $v_\parallel \lt v_\parallel ^*$ and heating protons with $v_\parallel ^*\lt v_\parallel \lt 0$ , as shown in figure 4(b). This distinct heating signature was recently produced by a quasi-linear analysis of PICWs observed by PSP during an encounter with a fast solar-wind stream (Bowen et al. Reference Bowen2024b ).