3.1 Accounting for the plasma inhomogeneity

Intending to adapt the leading-order approximation (3.19) to the case of an inhomogeneous plasma, one discovers that this time all its constituents depend on $\boldsymbol{r}$ . Accordingly, it becomes necessary to consider the terms of (3.10), (3.11) with derivatives with respect to $\boldsymbol{r}$ and $\boldsymbol{k}$ . In view of this, the iterations of the two-time correlation function should be modified. We again use the leading order of the function in the form (3.19). Here the recipe for calculating the natural frequency $\unicode[STIX]{x1D714}_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}s}$ and the polarization tensor $\unicode[STIX]{x1D641}_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}s}$ does not need any modifications; the only thing is that uncertainties appear in the choice of the dependencies of the phase of the polarization tensor on $\boldsymbol{r}$ and $\boldsymbol{k}$ . (In homogeneous plasmas, this phase does not influence the final expressions. Further, we shall show that the same is true within the current physical content. However, this is hardly a common case for turbulent wave fields in inhomogeneous plasmas.)

As a matter of fact, the terms with the $\boldsymbol{r}$ and $\boldsymbol{k}$ -derivatives of the approximation ${\text{}^{0}\unicode[STIX]{x1D731}}_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}s}(\boldsymbol{r},t,t^{\prime })$ within (3.10), (3.11) result in adding some correction $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D731}_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}s}(\boldsymbol{r},t,t^{\prime })$ to this approximation. It is natural to regard it as a tensor of some ‘forced oscillations’. Their most important part is the one that resembles ${\text{}^{0}\unicode[STIX]{x1D731}}_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}s}(\boldsymbol{r},t,t^{\prime })$ in tensorial structure. It is the tensor $\text{}^{0\!}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D731}_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}s}(\boldsymbol{r},t,t^{\prime })=\unicode[STIX]{x1D641}_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}s}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6F7}_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}s}(\boldsymbol{r},t,t^{\prime })\otimes (\unicode[STIX]{x1D641}_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}s})^{\dotplus }$ , where $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6F7}_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}s}(\boldsymbol{r},t,t^{\prime })$ is the convolution of the tensor $\unicode[STIX]{x1D6FF}\pmb{\unicode[STIX]{x1D6F7}}_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}s}(\boldsymbol{r},t,t^{\prime })$ with the tensor $(\unicode[STIX]{x1D641}_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}s})^{\dotplus }$ at the entry and with the tensor $\unicode[STIX]{x1D641}_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}s}$ at the exit.Footnote ⁴

From the point of view of carrying out iterations, it is desirable that the tensor $\text{}^{0\!}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D731}_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}s}(\boldsymbol{r},t,t^{\prime })$ should not compete, in the sense of its norm, with the tensor $\text{}^{0}\unicode[STIX]{x1D731}_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}s}(\boldsymbol{r},t,t^{\prime })$ for all values of the input and output time arguments. One can hope to achieve this by properly choosing the time dependence of the wave spectral density $n_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}}(\boldsymbol{r},t)$ . To analyse this possibility, it is necessary to devise a recipe for calculating $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6F7}_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}s}(\boldsymbol{r},t,t^{\prime })$ .

Conceptually, the simultaneous equations (3.10), (3.11) constitute a tensorial equation for the evolution of the two-time correlation function $\unicode[STIX]{x1D731}_{\boldsymbol{k}}(\boldsymbol{r},t,t^{\prime })$ . We substitute the sum ${\text{}^{0}\unicode[STIX]{x1D731}}_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}s}(\boldsymbol{r},t,t^{\prime })+\unicode[STIX]{x1D641}_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}s}\otimes (\unicode[STIX]{x1D641}_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}s})^{\dotplus }\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6F7}_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}s}(\boldsymbol{r},t,t^{\prime })$ for the function $\unicode[STIX]{x1D731}_{\boldsymbol{k}}(\boldsymbol{r},t,t^{\prime })$ into this evolution equation and then convolute the equation entry with the tensor $(\unicode[STIX]{x1D641}_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}s})^{\dotplus }$ and the equation exit with the tensor $\unicode[STIX]{x1D641}_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}s}$ . This yields a scalar equation that defines the derivative of $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6F7}_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}s}(\boldsymbol{r},t,t^{\prime })$ over the entry time $t$ . We use the result of integration of this derivative from $t=t^{\prime }$ to the current time $t$ for the function $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6F7}_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}s}(\boldsymbol{r},t,t^{\prime })$ at $t>t^{\prime }$ . In the integration, the function ${\text{}^{0}\unicode[STIX]{x1D731}}_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}s}(\boldsymbol{r},t_{1},t^{\prime })$ can be well approximated by the expression

(3.20) $$\begin{eqnarray}{\text{}^{0}\unicode[STIX]{x1D731}}_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}s}(\boldsymbol{r},t_{1},t^{\prime })=\unicode[STIX]{x1D641}_{\boldsymbol{ k}}^{\unicode[STIX]{x1D70E}s}\otimes (\unicode[STIX]{x1D641}_{\boldsymbol{ k}}^{\unicode[STIX]{x1D70E}s})^{\ast }n_{s\boldsymbol{k}}^{\unicode[STIX]{x1D70E}}(t^{\prime })\exp \left(-\text{i}\int _{t^{\prime }}^{t}\unicode[STIX]{x1D714}_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}s}(\unicode[STIX]{x1D70F})\,\text{d}\unicode[STIX]{x1D70F}\right).\end{eqnarray}$$

The point is that the time advances of $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6F7}_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}s}(\boldsymbol{r},t,t^{\prime })$ at $t>t^{\prime }$ depend only on the leading-order approximation ${\text{}^{0}\unicode[STIX]{x1D731}}_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}s}(\boldsymbol{r},t_{1},t^{\prime })$ at time delays ${t_{1}-t}^{\prime }\gtrsim -(\unicode[STIX]{x0394}\unicode[STIX]{x1D714})^{-1}$ , due to the rapid decay of the conductivity tensor $\unicode[STIX]{x1D748}_{\boldsymbol{k}}(t,t_{1})$ with increase of $t-t_{1}$ . In the corresponding time domain, the difference between the leading-order approximation and the above approximate expression is insignificant.

Calculation in accordance with the above ideas yields the following formula for function $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6F7}_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}s}(\boldsymbol{r},t,t^{\prime })$ at $t>t^{\prime }$ :

(3.21) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6F7}_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}s}(\boldsymbol{r},t,t^{\prime })=\unicode[STIX]{x1D6FF}\tilde{\unicode[STIX]{x1D6F7}}_{\boldsymbol{ k}}^{\unicode[STIX]{x1D70E}s}(\boldsymbol{r},t,t^{\prime })n_{s\boldsymbol{k}}^{\unicode[STIX]{x1D70E}}(\boldsymbol{r},t^{\prime })\exp \left(-\text{i}\int _{t^{\prime }}^{t}\unicode[STIX]{x1D714}_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}s}(\boldsymbol{r},\unicode[STIX]{x1D70F})\,\text{d}\unicode[STIX]{x1D70F}\right),\end{eqnarray}$$

(3.22)

Here

(3.23) $$\begin{eqnarray}\unicode[STIX]{x1D70E}_{\boldsymbol{k}\unicode[STIX]{x1D714}}^{\unicode[STIX]{x1D70E}s}(t)=(F_{\boldsymbol{ k}\;\;\unicode[STIX]{x1D6FD}\,\cdot }^{\unicode[STIX]{x1D70E}s\,\cdot \;0})^{\ast }\unicode[STIX]{x1D70E}_{\boldsymbol{k}\unicode[STIX]{x1D714}\,\cdot \cdot \;\unicode[STIX]{x1D6FE}}^{\unicode[STIX]{x1D6FD}m\,\cdot }(t)F_{\boldsymbol{ k}\;\;\;m\cdot }^{\unicode[STIX]{x1D70E}s\,\cdot \;\unicode[STIX]{x1D6FE}}.\end{eqnarray}$$

In (3.22), we have replaced the natural frequency $\unicode[STIX]{x1D714}_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}s}$ under the signs of the $\boldsymbol{k}$ and $\boldsymbol{r}$ -derivatives by its real part $s\unicode[STIX]{x1D714}_{s\boldsymbol{k}}^{\unicode[STIX]{x1D70E}}$ since accounting for the corresponding wave damping rates is inessential.

It is worth noting that, according to the logic of its formation, the sum ${\text{}^{0}\unicode[STIX]{x1D731}}_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}s}(\boldsymbol{r},t,t^{\prime })+\unicode[STIX]{x1D641}_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}s}\otimes (\unicode[STIX]{x1D641}_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}s})^{\dotplus }\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6F7}_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}s}(\boldsymbol{r},t,t^{\prime })$ surely satisfies (3.10), (3.11) only in the time domain $t-t^{\prime }\gg (\unicode[STIX]{x0394}\unicode[STIX]{x1D714})^{-1}$ . The property (3.9) permits one to construct some approximation of the two-time correlation function with the reverse sequence of the time arguments: $\unicode[STIX]{x1D731}_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}s}(\boldsymbol{r},t,t^{\prime })=(\unicode[STIX]{x1D731}_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}s}(\boldsymbol{r},t^{\prime },t))^{\ast }$ . One may hope that the wave spectral density $n_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}}$ evolves in such a way that the corresponding approximation is consistent with the bulk of the real two-time correlation function in the corresponding time domain. This provides the key idea for the derivation of the rate of change of $n_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}}$ . Consider (3.10), (3.11) in the asymptotic time domain $t-t^{\prime }\ll -(\unicode[STIX]{x0394}\unicode[STIX]{x1D714})^{-1}$ . Substitute the two-time correlation function on the right-hand sides of the equations by the tensor

(3.24) $$\begin{eqnarray}\unicode[STIX]{x1D641}_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}s}\otimes (\unicode[STIX]{x1D641}_{\boldsymbol{ k}}^{\unicode[STIX]{x1D70E}s})^{\dotplus }\left[n_{s\boldsymbol{k}}^{\unicode[STIX]{x1D70E}}(t)\exp \left(-\text{i}\int _{t^{\prime }}^{t}(\unicode[STIX]{x1D714}_{\boldsymbol{ k}}^{\unicode[STIX]{x1D70E}s}(\unicode[STIX]{x1D70F}))^{\ast }\,\text{d}\unicode[STIX]{x1D70F}\right)+(\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6F7}_{\boldsymbol{ k}}^{\unicode[STIX]{x1D70E}s}(\boldsymbol{r},t^{\prime },t))^{\ast }\right].\end{eqnarray}$$

This yields expressions that contain time derivative of the wave spectral density $n_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}}$ . One can try to select a value of this derivative for which the ‘motive force’ of the extra correction to structure (3.24) is orthogonal to the entry polarization tensor $\unicode[STIX]{x1D641}_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}s}$ throughout the domain.Footnote ⁵ However, the kernel of the correction (3.22) contains some integrals that might not permit such a selection. We mean the sum

(3.25) $$\begin{eqnarray}\displaystyle & & \displaystyle -\text{i}sc\text{Re}((F_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}s\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6FE}})^{\ast }F_{\boldsymbol{k}0\unicode[STIX]{x1D6FE}}^{\unicode[STIX]{x1D70E}s})\int _{t^{\prime }}^{\tilde{t}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}r^{\unicode[STIX]{x1D6FD}}}\unicode[STIX]{x1D714}_{s\boldsymbol{k}}^{\unicode[STIX]{x1D70E}}(\unicode[STIX]{x1D70F})\,\text{d}\unicode[STIX]{x1D70F}\nonumber\\ \displaystyle & & \displaystyle \quad +\,2\unicode[STIX]{x03C0}s(F_{\boldsymbol{k}0\unicode[STIX]{x1D6FD}}^{\unicode[STIX]{x1D70E}s})^{\ast }\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}k_{\unicode[STIX]{x1D6FF}}}\unicode[STIX]{x1D70E}_{\boldsymbol{k}\unicode[STIX]{x1D714}}^{\;\;\unicode[STIX]{x1D6FD}m\unicode[STIX]{x1D6FE}}(\boldsymbol{r},\tilde{t})F_{\boldsymbol{ k}m\unicode[STIX]{x1D6FE}}^{\unicode[STIX]{x1D70E}s}\int _{t^{\prime }}^{\tilde{t}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}r^{\unicode[STIX]{x1D6FF}}}\unicode[STIX]{x1D714}_{s\boldsymbol{k}}^{\unicode[STIX]{x1D70E}}(\boldsymbol{r},\unicode[STIX]{x1D70F})\,\text{d}\unicode[STIX]{x1D70F}\nonumber\\ \displaystyle & & \displaystyle \quad \left.-\,2\unicode[STIX]{x03C0}s(F_{\boldsymbol{k}0\unicode[STIX]{x1D6FD}}^{\unicode[STIX]{x1D70E}s})^{\ast }\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}r^{\unicode[STIX]{x1D6FF}}}\unicode[STIX]{x1D70E}_{\boldsymbol{k}\unicode[STIX]{x1D714}}^{\;\;\unicode[STIX]{x1D6FD}m\unicode[STIX]{x1D6FE}}(\boldsymbol{r},\tilde{t})F_{\boldsymbol{ k}m\unicode[STIX]{x1D6FE}}^{\unicode[STIX]{x1D70E}s}\int _{t^{\prime }}^{\tilde{t}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}k_{\unicode[STIX]{x1D6FF}}}\unicode[STIX]{x1D714}_{s\boldsymbol{k}}^{\unicode[STIX]{x1D70E}}(\unicode[STIX]{x1D70F})\,\text{d}\unicode[STIX]{x1D70F}\right|_{\unicode[STIX]{x1D714}=\unicode[STIX]{x1D714}_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}s}(\boldsymbol{r},\tilde{t})}.\end{eqnarray}$$

If it does not turn to zero, none of the choices of the time dependence of wave spectral density will provide a good approximation of the two-time correlation function in the whole domain $t^{\prime }-t\gg (\unicode[STIX]{x0394}\unicode[STIX]{x1D714})^{-1}$ . Previously, we have tried to formulate a recipe for circumventing the respective difficulty (Erofeev Reference Erofeev2015b ). It consisted in adding an extra phase shift to the argument of the exponent in the leading-order approximation ${\text{}^{0}\unicode[STIX]{x1D731}}_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}s}(\boldsymbol{r},t,t^{\prime })$ at $t>t^{\prime }$ . However, this recipe turned out to be inconsistent. It is more likely that the above sum always falls out, at least in its leading order.Footnote ⁶ Note that just this took place in our previous study (Erofeev Reference Erofeev2015b ). The same takes place in the current problem of drift of Langmuir waves in a cold ionized magnetized plasma, which can be checked on the basis of further data on these oscillations. However, currently, we do not have a full-scale proof of the hypothesis that the nullification of the above problematic sum always takes place. We suspect that such a proof can be developed on the basis of the following property of natural oscillations (Erofeev Reference Erofeev2014, Reference Erofeev2015b ),

(3.26) $$\begin{eqnarray}\unicode[STIX]{x1D714}_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}s}(t)-2k_{\unicode[STIX]{x1D6FE}}c\text{Re}[(F_{\boldsymbol{ k}}^{\unicode[STIX]{x1D70E}s\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6FE}})^{\ast }F_{\boldsymbol{ k}\;\;0\unicode[STIX]{x1D6FD}}^{\unicode[STIX]{x1D70E}s}]=-4\unicode[STIX]{x03C0}\text{i}\unicode[STIX]{x1D70E}_{\boldsymbol{ k}\unicode[STIX]{x1D714}}^{\unicode[STIX]{x1D70E}s}(t)|_{\unicode[STIX]{x1D714}=\unicode[STIX]{x1D714}_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}s}(t)}.\end{eqnarray}$$

It arises from the convolution of the entries of the lowest-order approximations of the simultaneous equations (3.15), (3.16) with the tensor $(\unicode[STIX]{x1D641}_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}s})^{\dotplus }$ .

If the above hypothesis is valid, it is easy to calculate the value of $(\unicode[STIX]{x2202}n_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}})/(\unicode[STIX]{x2202}t)$ for which the driving force for the additional correction to the solution $({\text{}^{0}\unicode[STIX]{x1D731}}_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}s}(\boldsymbol{r},t^{\prime },t)+{\text{}^{0}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D731}}_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}s}(\boldsymbol{r},t^{\prime },t))^{\dotplus }$ is orthogonal to the polarization tensor of the natural oscillation. Due to this, the above correction is automatically small compared to $({\text{}^{0}\unicode[STIX]{x1D731}}_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}s}(\boldsymbol{r},t^{\prime },t)+{\text{}^{0}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D731}}_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}s}(\boldsymbol{r},t^{\prime },t))^{\dotplus }$ at all times from the interval $t-t^{\prime }\ll -(\unicode[STIX]{x0394}\unicode[STIX]{x1D714})^{-1}$ . This is the general recipe for developing the wave kinetic equation in inhomogeneous plasmas. It yields the following rate of wave drift in phase space:

(3.27)

Equation (3.27) includes a term that contains the convolution of the tensor $\unicode[STIX]{x1D641}_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}s}(t^{\prime })$ with the spatial derivative of the exit polarization tensor $(\unicode[STIX]{x1D641}_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}s})^{\dotplus }(t^{\prime })$ of the leading order ${\text{}^{0}\unicode[STIX]{x1D731}}_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}s}(\boldsymbol{r},t,t^{\prime })$ . For our case of Langmuir waves, the dependence of the phase of the polarization tensor on $\boldsymbol{r}$ can be adjusted so that this convolution vanishes:

(3.28) $$\begin{eqnarray}F_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}s\unicode[STIX]{x1D6FD}0}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}r^{\unicode[STIX]{x1D700}}}(F_{\boldsymbol{k}\;\;\;\unicode[STIX]{x1D6FD}\;\cdot }^{\unicode[STIX]{x1D70E}s\,\cdot \;0})^{\ast }+\frac{1}{2}F_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}s\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6FE}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}r^{\unicode[STIX]{x1D700}}}(F_{\boldsymbol{k}\;\;\;\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6FE}}^{\unicode[STIX]{x1D70E}s})^{\ast }=0.\end{eqnarray}$$

Conceptually, the latter restriction implies that the phase of the polarization tensor does not depend on spatial coordinates.

The situation is different for the dependence of the phase on the wave vector. Unlike in our previous study of electromagnetic waves (Erofeev Reference Erofeev2015b ), no choice of this dependence yields the vanishing of the corresponding convolution:

(3.29) $$\begin{eqnarray}F_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}s\unicode[STIX]{x1D6FD}0}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}k^{\unicode[STIX]{x1D700}}}(F_{\boldsymbol{k}\;\;\;\unicode[STIX]{x1D6FD}\;\cdot }^{\unicode[STIX]{x1D70E}s\,\cdot \;0})^{\ast }+\frac{1}{2}F_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}s\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6FE}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}k^{\unicode[STIX]{x1D700}}}(F_{\boldsymbol{k}\;\;\;\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6FE}}^{\unicode[STIX]{x1D70E}s})^{\ast }\neq 0.\end{eqnarray}$$

That is, the phase of the polarization tensor cannot be regarded as independent of $\boldsymbol{k}$ . Still, terms with the derivative of the wave phase with respect to the wave vector fall out of the wave drift velocity in our physical situation. This fact will be commented in the next section.

We conclude this section by the following remark. We have mentioned above that it is desirable that the tensor of forced oscillations $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D731}_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}s}(\boldsymbol{r},t^{\prime },t)$ be small in comparison with the tensor ${\text{}^{0}\unicode[STIX]{x1D731}}_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}s}(\boldsymbol{r},t,t^{\prime })$ . Meanwhile, with increase of time delay $t-t^{\prime }$ in our calculations, the forced oscillation tensor will ultimately override the tensor of the leading-order approximation ${\text{}^{0}\unicode[STIX]{x1D731}}_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}s}(\boldsymbol{r},t,t^{\prime })$ . Still, our leading-order approximation ${\text{}^{0}\unicode[STIX]{x1D731}}_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}s}(\boldsymbol{r},t,t^{\prime })$ represents the bulk of the two-time correlation function in a rather wide piece of the domain $t-t^{\prime }\gg (\unicode[STIX]{x0394}\unicode[STIX]{x1D714})^{-1}$ provided that the plasma inhomogeneity parameter $\unicode[STIX]{x1D706}/l$ is adequately small. Then the obtained picture of wave drift adequately reflects the objective realities of this phenomenon: our calculation is meaningful.

4.1 Conductivity tensor in a magnetized plasma

Following formulae (3.7) and (3.2),

(4.1) $$\begin{eqnarray}\unicode[STIX]{x1D70E}_{\cdot \;\cdot \;\unicode[STIX]{x1D6FE}}^{\unicode[STIX]{x1D6FD}m\cdot }(\boldsymbol{r},t,\boldsymbol{r}_{1},t_{1})=-\mathop{\sum }_{\unicode[STIX]{x1D6FC}}\frac{e_{\unicode[STIX]{x1D6FC}}^{2}}{c}\int \,\text{d}^{3}\boldsymbol{p}v^{\unicode[STIX]{x1D6FD}}\int \,\text{d}^{3}\boldsymbol{p}_{1}{\text{}^{0}\!G}_{\unicode[STIX]{x1D6FC}}(\boldsymbol{r},\boldsymbol{p},t,\boldsymbol{r}_{1},\boldsymbol{p}_{1},t_{1})v_{1}^{m}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}p_{1}^{\unicode[STIX]{x1D6FE}}}f_{\unicode[STIX]{x1D6FC}}(\boldsymbol{r}_{1},\boldsymbol{p}_{1},t_{1}).\end{eqnarray}$$

Bearing in mind that wave drift is due to the plasma inhomogeneity, we analyse the effects of the inhomogeneity on this tensor. Let us assume that the magnetic pressure in the plasma is great compared to the hydrodynamic plasma pressure, i.e.

(4.2) $$\begin{eqnarray}\unicode[STIX]{x1D6FD}=\frac{nT_{e}}{B^{2}/(8\unicode[STIX]{x03C0})}\ll 1.\end{eqnarray}$$

Then spatial changes in the conductivity tensor are predominantly due to inhomogeneities of plasma density or (and) particle momentum distributions rather than due to the inhomogeneity of the confining magnetic field $\boldsymbol{B}(\boldsymbol{r})$ : either

(4.3) $$\begin{eqnarray}|\unicode[STIX]{x1D735}\ln (B)|\ll |\unicode[STIX]{x1D735}\ln (n_{e})|\end{eqnarray}$$

or (and)

(4.4) $$\begin{eqnarray}|\unicode[STIX]{x1D735}\ln (B)|\ll |\unicode[STIX]{x1D735}\ln (T_{e})|.\end{eqnarray}$$

Therefore, we shall neglect the influence of spatial variations of the leading magnetic field on the bare Green function ${\text{}^{0}\!G}_{\unicode[STIX]{x1D6FC}}(\boldsymbol{r},\boldsymbol{p},t,\boldsymbol{r}_{1},\boldsymbol{p}_{1},t_{1})$ . Let the $z$ axis of the Cartesian frame be directed along the external magnetic field $\boldsymbol{B}$ . Then integration of (3.3) yields the following formula for the Fourier–Laplace transform of the bare Green function:

(4.5) $$\begin{eqnarray}\displaystyle & & \displaystyle \text{}^{0}\!G_{\unicode[STIX]{x1D6FC}\boldsymbol{k}\unicode[STIX]{x1D714}}(\boldsymbol{p},\boldsymbol{p}_{1})=-\frac{1}{\unicode[STIX]{x1D714}_{B\unicode[STIX]{x1D6FC}}}\unicode[STIX]{x1D6FF}(p^{z}-p_{1}^{z})\frac{\unicode[STIX]{x1D6FF}(p^{\bot }-p_{1}^{\bot })}{p^{\bot }}\left[1-\exp \left(-2\unicode[STIX]{x03C0}\text{i}\frac{\unicode[STIX]{x1D714}-k^{z}v^{z}}{\unicode[STIX]{x1D714}_{B\unicode[STIX]{x1D6FC}}}\right)\right]^{-1}\nonumber\\ \displaystyle & & \displaystyle \quad \times \exp \left[-\text{i}\frac{\unicode[STIX]{x1D714}-k^{z}v^{z}}{\unicode[STIX]{x1D714}_{B\unicode[STIX]{x1D6FC}}}(\unicode[STIX]{x1D711}-\unicode[STIX]{x1D711}_{1})+\text{i}\frac{k^{\bot }v^{\bot }}{\unicode[STIX]{x1D714}_{B\unicode[STIX]{x1D6FC}}}(\sin (\unicode[STIX]{x1D711}-\unicode[STIX]{x1D711}^{\prime })-\sin (\unicode[STIX]{x1D711}_{1}-\unicode[STIX]{x1D711}^{\prime }))\right].\end{eqnarray}$$

(The transform is defined as $\text{}^{0}G_{\unicode[STIX]{x1D6FC}\boldsymbol{k}\unicode[STIX]{x1D714}}(\boldsymbol{p},\boldsymbol{p}^{\prime })=\int _{t^{\prime }}^{\infty }\,\text{d}t\int \,\text{d}^{3}\boldsymbol{r}{\text{}^{0}G}_{\unicode[STIX]{x1D6FC}}(\boldsymbol{r},\boldsymbol{p},t,\boldsymbol{r}^{\prime },\boldsymbol{p}^{\prime },t^{\prime })\exp (\text{i}\unicode[STIX]{x1D714}(t-t^{\prime })-\text{i}(\boldsymbol{k}\boldsymbol{\cdot }(\boldsymbol{r}-\boldsymbol{r}^{\prime })))$ .) Here we use cylindrical coordinates. The azimuthal angles $\unicode[STIX]{x1D711},\unicode[STIX]{x1D711}_{1}$ and $\unicode[STIX]{x1D711}^{\prime }$ are the angles between the projections of the vectors $\boldsymbol{p},\boldsymbol{p}_{1}$ and $\boldsymbol{k}$ onto the plane $xy$ and the axis $x$ , respectively. The sign $\bot$ is used to denote the components of the vectors perpendicular to the $z$ axis. It is assumed that the difference $\unicode[STIX]{x1D711}-\unicode[STIX]{x1D711}_{1}$ is within the interval $[0,2\unicode[STIX]{x03C0}[$ . The Larmor frequency of the particle is defined as $\unicode[STIX]{x1D714}_{B\unicode[STIX]{x1D6FC}}=e_{\unicode[STIX]{x1D6FC}}B/(m_{\unicode[STIX]{x1D6FC}}c)$ . (We assume that the particles are non-relativistic in our plasma.)

Generally, taking account of the Larmor rotation of plasma particles in the leading magnetic field, we assume that in our inhomogeneous plasma, the stationary particle distribution $f_{\unicode[STIX]{x1D6FC}}$ has the following structure:

(4.6) $$\begin{eqnarray}f_{\unicode[STIX]{x1D6FC}}(\boldsymbol{r},\boldsymbol{p})={\text{}^{0\!}f}_{\unicode[STIX]{x1D6FC}}(x+v_{y}/\unicode[STIX]{x1D714}_{B\unicode[STIX]{x1D6FC}},p_{z},p_{\bot }).\end{eqnarray}$$

Here the notation $p_{\bot }$ is used for the modulus of the component of the particle momentum that is orthogonal to the magnetic field. The function ${\text{}^{0\!}f}_{\unicode[STIX]{x1D6FC}}(x,p_{z},p_{\bot })$ characterizes the density of the particles with given $p_{\bot }$ and $p_{z}$ that rotate in the external magnetic field around centres located on the current level line of $x$ . This density does not depend on the phase of the rotation.

Taking into account the above structure of the particle distributions, we get

(4.7) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70E}_{\boldsymbol{k}\unicode[STIX]{x1D714}\;\cdot \;\cdot \,\unicode[STIX]{x1D6FE}}^{\;\;\unicode[STIX]{x1D6FD}m\,\cdot }(t) & = & \displaystyle -\mathop{\sum }_{\unicode[STIX]{x1D6FC}}\frac{e_{\unicode[STIX]{x1D6FC}}^{2}}{c}\int \,\text{d}^{3}\boldsymbol{p}\,\text{d}^{3}\boldsymbol{p}_{1}v^{\unicode[STIX]{x1D6FD}}v_{1}^{m}\nonumber\\ \displaystyle & & \displaystyle \times \left[\text{}^{0\!}G_{\unicode[STIX]{x1D6FC}\boldsymbol{k}\unicode[STIX]{x1D714}}(\boldsymbol{p},\boldsymbol{p}_{1})-\frac{\text{i}}{2}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}k_{x}}{\text{}^{0\!}G}_{\unicode[STIX]{x1D6FC}\boldsymbol{k}\unicode[STIX]{x1D714}}(\boldsymbol{p},\boldsymbol{p}_{1})\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x}\right]\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}p_{1}^{\unicode[STIX]{x1D6FE}}}\nonumber\\ \displaystyle & & \displaystyle \times \left[\left(1+\frac{p_{1\bot }\sin (\unicode[STIX]{x1D711}_{1})}{m_{\unicode[STIX]{x1D6FC}}\unicode[STIX]{x1D714}_{B\unicode[STIX]{x1D6FC}}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x}\right){\text{}^{0\!}f}_{\unicode[STIX]{x1D6FC}}(x,p_{1z},p_{1\bot })\right].\end{eqnarray}$$

The expression with the spatial gradient in the first square brackets accounts for the direct effect of the spatial inhomogeneity of the distribution ${\text{}^{0\!}f}_{\unicode[STIX]{x1D6FC}}$ (i.e. its dependence on $x$ ), and the term in the second square brackets accounts for the anisotropy of the genuine particle distribution $f_{\unicode[STIX]{x1D6FC}}$ in the $p_{x}$ – $p_{y}$ plane that stems from the above Larmor rotations of the particles.

The Larmor particle radii are presumably negligible compared to the spatial scale of the plasma inhomogeneity in our cold ionized plasma. Due to this, the difference between the expression in the second square brackets and the distribution ${\text{}^{0}f}_{\unicode[STIX]{x1D6FC}}$ can be ignored. Then we can write

(4.8) $$\begin{eqnarray}\unicode[STIX]{x1D70E}_{\boldsymbol{k}\unicode[STIX]{x1D714}\;\cdot \;\cdot \,\unicode[STIX]{x1D6FE}}^{\;\;\unicode[STIX]{x1D6FD}m\,\cdot }(\boldsymbol{r},t)=\left[1-\frac{\text{i}}{2}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}k_{x}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x}\right]{\text{}^{0}\unicode[STIX]{x1D70E}}_{\boldsymbol{k}\unicode[STIX]{x1D714}\;\cdot \;\cdot \,\unicode[STIX]{x1D6FE}}^{\;\;\unicode[STIX]{x1D6FD}m\,\cdot }(\boldsymbol{r},t),\end{eqnarray}$$

(4.9) $$\begin{eqnarray}\displaystyle & & \displaystyle {\text{}^{0}\!\unicode[STIX]{x1D70E}}_{\boldsymbol{k}\unicode[STIX]{x1D714}\;\cdot \;\cdot \,\unicode[STIX]{x1D6FE}}^{\;\;\unicode[STIX]{x1D6FD}m\,\cdot }(\boldsymbol{r},t)\nonumber\\ \displaystyle & & \displaystyle \quad =-\mathop{\sum }_{\unicode[STIX]{x1D6FC}}\frac{e_{\unicode[STIX]{x1D6FC}}^{2}}{c}\int \,\text{d}^{3}\boldsymbol{p}\,\text{d}^{3}\boldsymbol{p}_{1}v^{\unicode[STIX]{x1D6FD}}v_{1}^{m}\,{\text{}^{0}\!G}_{\unicode[STIX]{x1D6FC}\boldsymbol{ k}\unicode[STIX]{x1D714}}(\boldsymbol{p},\boldsymbol{p}_{1})\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}p_{1}^{\unicode[STIX]{x1D6FE}}}\,\text{}^{0\!}f_{\unicode[STIX]{x1D6FC}}(x,p_{1z},p_{1\bot })\nonumber\\ \displaystyle & & \displaystyle \quad =\mathop{\sum }_{\unicode[STIX]{x1D6FC}}\frac{e_{\unicode[STIX]{x1D6FC}}^{2}}{c\unicode[STIX]{x1D714}_{B\unicode[STIX]{x1D6FC}}}\int p_{\bot }\,\text{d}p_{\bot }\,\text{d}p_{z}p_{1\bot }\,\text{d}p_{1\bot }\,\text{d}p_{1z}\unicode[STIX]{x1D6FF}(p_{z}-p_{1z})\frac{\unicode[STIX]{x1D6FF}(p_{\bot }-p_{1\bot })}{p_{\bot }}\int _{0}^{2\unicode[STIX]{x03C0}}\,\text{d}\unicode[STIX]{x1D711}\int _{\unicode[STIX]{x1D711}-2\unicode[STIX]{x03C0}}^{\unicode[STIX]{x1D711}}\,\text{d}\unicode[STIX]{x1D711}_{1}\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,\frac{\exp [-\text{i}((\unicode[STIX]{x1D714}-k_{z}v_{z})/\unicode[STIX]{x1D714}_{B\unicode[STIX]{x1D6FC}})(\unicode[STIX]{x1D711}-\unicode[STIX]{x1D711}_{1})+\text{i}((k_{\bot }v_{\bot })/\unicode[STIX]{x1D714}_{B\unicode[STIX]{x1D6FC}})(\sin (\unicode[STIX]{x1D711}-\unicode[STIX]{x1D711}^{\prime })-\sin (\unicode[STIX]{x1D711}_{1}-\unicode[STIX]{x1D711}^{\prime }))]}{1-\exp (-2\unicode[STIX]{x03C0}\text{i}(\unicode[STIX]{x1D714}-k_{z}v_{z})/\unicode[STIX]{x1D714}_{B\unicode[STIX]{x1D6FC}})}\nonumber\\ \displaystyle & & \displaystyle \qquad \times \left(\begin{array}{@{}c@{}}-\\ v_{\bot }\cos (\unicode[STIX]{x1D711})\\ v_{\bot }\sin (\unicode[STIX]{x1D711})\\ v_{z}\end{array}\right)\otimes \left(\begin{array}{@{}c@{}}c\\ v_{1\bot }\cos (\unicode[STIX]{x1D711}_{1})\\ v_{1\bot }\sin (\unicode[STIX]{x1D711}_{1})\\ v_{1z}\end{array}\right)\otimes \left(\begin{array}{@{}c@{}}-\\ (\unicode[STIX]{x2202}\text{}^{0\!}f_{\unicode[STIX]{x1D6FC}})/(\unicode[STIX]{x2202}p_{1\bot })\cos (\unicode[STIX]{x1D711}_{1})\\ (\unicode[STIX]{x2202}\text{}^{0\!}f_{\unicode[STIX]{x1D6FC}})/(\unicode[STIX]{x2202}p_{1\bot })\sin (\unicode[STIX]{x1D711}_{1})\\ (\unicode[STIX]{x2202}\text{}^{0\!}f_{\unicode[STIX]{x1D6FC}})/(\unicode[STIX]{x2202}p_{1z})\end{array}\right).\end{eqnarray}$$

The dependence of the tensor on the superscript $\unicode[STIX]{x1D6FD}$ is represented in the left column at the bottom of the rightmost side of the given chain of equalities, the dependence on the superscript $m$ is represented in the middle column and the dependence on the subscript $\unicode[STIX]{x1D6FE}$ in the right column.

Within the convolution $\unicode[STIX]{x1D70E}_{\boldsymbol{k}\unicode[STIX]{x1D714}\;\cdot \;\cdot \,\unicode[STIX]{x1D6FE}}^{\;\;\unicode[STIX]{x1D6FD}m\,\cdot }F_{\boldsymbol{k}\;\;m\,\cdot }^{\unicode[STIX]{x1D70E}s\cdot \,\unicode[STIX]{x1D6FE}}$ , the summands with $m\neq 0$ are small compared to the other summands and these quantities are related as the ratio of the mean square electron velocity in the plasma (akin to the thermal velocity) to the speed of light. In view of this, we can ignore all the summands in this tensor convolution except for those with indices $m=0$ . Note also that plasma ions play only the role of macroscopically neutralizing background in Langmuir plasma oscillations: these oscillations depend exclusively on electrons.

Let us assume that the electron distribution ${\text{}^{0\!}f}_{e}$ is an isotropic Maxwellian distribution with temperature $T_{e}$ that does not depend on $x$ . Then the matrix of the conductivity tensor takes the form

(4.10) $$\begin{eqnarray}\displaystyle & & \displaystyle \text{}^{0\!}\unicode[STIX]{x1D70E}_{\boldsymbol{k}\unicode[STIX]{x1D714}\;\cdot \;\cdot \,\unicode[STIX]{x1D6FE}}^{\;\;\unicode[STIX]{x1D6FD}0\,\cdot }(x,t)=-\frac{e^{2}}{\unicode[STIX]{x1D714}_{Be}T_{e}}\int _{-\infty }^{\infty }\,\text{d}p_{z}\int _{0}^{\infty }p_{\bot }\,\text{d}p_{\bot }\text{}^{0\!}f_{e}(x,p_{z},p_{\bot })\nonumber\\ \displaystyle & & \displaystyle \quad \times \,\frac{-\text{i}\exp (\text{i}\unicode[STIX]{x03C0}a)}{2\sin (\unicode[STIX]{x03C0}a)}\!\int _{0}^{2\unicode[STIX]{x03C0}}\text{d}\unicode[STIX]{x1D711}\!\int _{\unicode[STIX]{x1D711}-2\unicode[STIX]{x03C0}}^{\unicode[STIX]{x1D711}}\text{d}\unicode[STIX]{x1D711}_{1}\exp [-\text{i}a(\unicode[STIX]{x1D711}-\unicode[STIX]{x1D711}_{1})+\text{i}z(\sin (\unicode[STIX]{x1D711}-\unicode[STIX]{x1D711}^{\prime })-\sin (\unicode[STIX]{x1D711}_{1}-\unicode[STIX]{x1D711}^{\prime }))]\nonumber\\ \displaystyle & & \displaystyle \quad \times \left(\begin{array}{@{}ccc@{}}v_{\bot }^{2}\cos (\unicode[STIX]{x1D711})\cos (\unicode[STIX]{x1D711}_{1}) & v_{\bot }^{2}\cos (\unicode[STIX]{x1D711})\sin (\unicode[STIX]{x1D711}_{1}) & v_{z}v_{\bot }\cos (\unicode[STIX]{x1D711})\\ v_{\bot }^{2}\sin (\unicode[STIX]{x1D711})\cos (\unicode[STIX]{x1D711}_{1}) & v_{\bot }^{2}\sin (\unicode[STIX]{x1D711})\sin (\unicode[STIX]{x1D711}_{1}) & v_{z}v_{\bot }\sin (\unicode[STIX]{x1D711})\\ v_{z}v_{\bot }\cos (\unicode[STIX]{x1D711}_{1}) & v_{z}v_{\bot }\sin (\unicode[STIX]{x1D711}_{1}) & v_{z}^{2}\end{array}\right).\end{eqnarray}$$

Here we have introduced the notation

(4.11a,b ) $$\begin{eqnarray}a=\frac{\unicode[STIX]{x1D714}-k_{z}v_{z}}{\unicode[STIX]{x1D714}_{Be}},\quad z=\frac{k_{x}v_{\bot }}{\unicode[STIX]{x1D714}_{Be}}.\end{eqnarray}$$

The integration over azimuthal angles in formula (4.10) yields a somewhat more advanced form of the conductivity tensor. However, it remains rather cumbersome in the general case. We present it for the case $\unicode[STIX]{x1D711}^{\prime }=0$ (otherwise $k_{y}=0$ ). Then

(4.12) $$\begin{eqnarray}\displaystyle & & \displaystyle {\text{}^{0\!}\unicode[STIX]{x1D70E}}_{\boldsymbol{k}\unicode[STIX]{x1D714}\;\cdot \;\cdot \,\unicode[STIX]{x1D6FE}}^{\;\;\unicode[STIX]{x1D6FD}0\,\cdot }=\text{i}\frac{\text{e}^{2}}{T_{e}\unicode[STIX]{x1D714}_{Be}}\int \,\text{d}^{3}\boldsymbol{p}f_{e}(x,p_{z},p_{\bot })\nonumber\\ \displaystyle & & \displaystyle \quad \times \!\left(\begin{array}{@{}ccc@{}}v_{\bot }^{2}(-a(1+aG)/z^{2}) & \text{i}v_{\bot }^{2}aG^{\prime }/(2z) & -v_{\bot }v_{z}(1+aG)/z\\ -\text{i}v_{\bot }^{2}aG^{\prime }/(2z) & v_{\bot }^{2}(a(1+aG)/z^{2}-(zG^{\prime })^{\prime }/(2z)-G) & \text{i}v_{\bot }v_{z}G^{\prime }/2\\ -v_{\bot }v_{z}(1+aG)/z & -\text{i}v_{\bot }v_{z}G^{\prime }/2 & -v_{z}^{2}G\end{array}\right)\!.\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

Here

(4.13) $$\begin{eqnarray}\displaystyle G(z) & = & \displaystyle -\frac{1}{2\unicode[STIX]{x03C0}}\frac{\exp (\text{i}\unicode[STIX]{x03C0}a)}{2\sin (\unicode[STIX]{x03C0}a)}\int _{0}^{2\unicode[STIX]{x03C0}}\,\text{d}\unicode[STIX]{x1D711}\int _{\unicode[STIX]{x1D711}-2\unicode[STIX]{x03C0}}^{\unicode[STIX]{x1D711}}\,\text{d}\unicode[STIX]{x1D711}_{1}\exp [-\text{i}a(\unicode[STIX]{x1D711}-\unicode[STIX]{x1D711}_{1})+\text{i}z(\sin (\unicode[STIX]{x1D711})-\sin (\unicode[STIX]{x1D711}_{1}))]\nonumber\\ \displaystyle & = & \displaystyle -\mathop{\sum }_{n=-\infty }^{\infty }\frac{\text{J}_{n}^{2}(z)}{a-n}\equiv -\frac{\unicode[STIX]{x03C0}}{\sin (\unicode[STIX]{x03C0}a)}\text{J}_{a}(z)\text{J}_{-a}(z),\end{eqnarray}$$

where $\text{J}_{n}(z)$ is a first-kind Bessel function of integer order $n$ and $\text{J}_{\unicode[STIX]{x1D708}}(z)$ is a first-kind Bessel function of irrational (generally, complex) order $\unicode[STIX]{x1D708}$ .

Since the electron thermal velocity is small, our waves have lengths much larger than the typical electron Larmor radius $\unicode[STIX]{x1D70C}_{e}\approx v_{Te}/\unicode[STIX]{x1D714}_{Be}$ . Then argument $z$ takes small values, and the function $G$ can be expanded in a series in $z$ . For our case of cold plasma, we do not need more than two first orders of the expansion:

(4.14) $$\begin{eqnarray}\displaystyle G(z) & = & \displaystyle -\frac{1}{a}\left[1-\frac{1}{2}z^{2}\right]-\frac{1}{2}\frac{a}{a^{2}-1}z^{2}-\cdots \equiv -\frac{1}{a}\left[1+\frac{1}{4}\left(\frac{1}{a-1}-\frac{1}{a+1}\right)z^{2}+\cdots \,\right]\nonumber\\ \displaystyle & = & \displaystyle -\frac{1}{a}-\frac{z^{2}}{2a(a^{2}-1)}-\cdots \,.\end{eqnarray}$$

In this case, the tensor ${\text{}^{0\!}\unicode[STIX]{x1D70E}}_{\boldsymbol{k}\unicode[STIX]{x1D714}\;\cdot \;\cdot \,\unicode[STIX]{x1D6FE}}^{\;\;\unicode[STIX]{x1D6FD}0\,\cdot }$ becomes

(4.15) $$\begin{eqnarray}\text{i}\frac{\unicode[STIX]{x1D714}_{pe}^{2}}{4\unicode[STIX]{x03C0}}\left(\begin{array}{@{}ccc@{}}\unicode[STIX]{x1D714}/(\unicode[STIX]{x1D714}^{2}-\unicode[STIX]{x1D714}_{Be}^{2}) & -\text{i}\unicode[STIX]{x1D714}_{Be}/(\unicode[STIX]{x1D714}^{2}-\unicode[STIX]{x1D714}_{Be}^{2}) & 0\\ \text{i}\unicode[STIX]{x1D714}_{Be}/(\unicode[STIX]{x1D714}^{2}-\unicode[STIX]{x1D714}_{Be}^{2}) & \unicode[STIX]{x1D714}/(\unicode[STIX]{x1D714}^{2}-\unicode[STIX]{x1D714}_{Be}^{2}) & 0\\ 0 & 0 & 1/\unicode[STIX]{x1D714}\end{array}\right).\end{eqnarray}$$

We see tensor ${\text{}^{0\!}\unicode[STIX]{x1D70E}}_{\boldsymbol{k}\unicode[STIX]{x1D714}\;\cdot \;\cdot \,\unicode[STIX]{x1D6FE}}^{\;\;\unicode[STIX]{x1D6FD}0\,\cdot }$ does not depend on $\boldsymbol{k}$ , so that the tensor $\unicode[STIX]{x1D70E}_{\boldsymbol{k}\unicode[STIX]{x1D714}\;\cdot \;\cdot \,\unicode[STIX]{x1D6FE}}^{\;\;\unicode[STIX]{x1D6FD}0\,\cdot }$ does not differ from it. It is also invariant with respect to rotation of the coordinate plane $k_{x}$ – $k_{y}$ , and this justifies the fact that we have focused our analysis of formula (4.10) on the particular case $k_{y}=0$ .

The tensor (4.15) is known from the traditional linear hydrodynamic theory of plasma oscillations.

4.2 Natural frequency and polarization tensor of Langmuir waves

We first clarify the term long as applied to our Langmuir waves. Here we mean that the wave vector $k$ is small compared to the following three characteristic inverse lengths: $\unicode[STIX]{x1D714}_{p}/c$ , $|\unicode[STIX]{x1D714}_{Be}|/c$ and $\sqrt{|\unicode[STIX]{x1D714}_{p}^{2}-\unicode[STIX]{x1D714}_{Be}^{2}|}/c$ . The natural frequency of a Langmuir wave is then

(4.16) $$\begin{eqnarray}\unicode[STIX]{x1D714}_{\boldsymbol{k}}^{ls}=s\unicode[STIX]{x1D714}_{p}\left(1+\frac{1}{2}\frac{k_{\bot }^{2}c^{2}}{\unicode[STIX]{x1D714}_{p}^{2}}+\frac{1}{2}\frac{k_{\bot }^{2}c^{2}k_{z}^{2}c^{2}}{\unicode[STIX]{x1D714}_{p}^{4}}-\frac{1}{8}\frac{k_{\bot }^{4}c^{4}}{\unicode[STIX]{x1D714}_{p}^{4}}+O\left(\frac{k^{6}c^{6}}{\unicode[STIX]{x1D714}_{p}^{6}}\right)\right).\end{eqnarray}$$

Here $k_{\bot }$ denotes the wave vector component that is orthogonal to the leading magnetic field.

The ‘electric’ components of the polarization tensor are

(4.17) $$\begin{eqnarray}\!\!\!\left(\begin{array}{@{}c@{}}F_{\boldsymbol{k}x0}^{ls}\\ F_{\boldsymbol{k}y0}^{ls}\\ F_{\boldsymbol{k}z0}^{ls}\end{array}\right)=\left(\begin{array}{@{}c@{}}-{\displaystyle \frac{k_{x}k_{z}c^{2}}{\unicode[STIX]{x1D714}_{p}^{2}}}-\text{i}s{\displaystyle \frac{k_{y}k_{z}c^{2}}{\unicode[STIX]{x1D714}_{p}\unicode[STIX]{x1D714}_{Be}}}+O\left({\displaystyle \frac{k^{4}c^{4}}{\unicode[STIX]{x1D714}_{p}^{4}}}\right)\\ -{\displaystyle \frac{k_{y}k_{z}c^{2}}{\unicode[STIX]{x1D714}_{p}^{2}}}+\text{i}s{\displaystyle \frac{k_{x}k_{z}c^{2}}{\unicode[STIX]{x1D714}_{p}\unicode[STIX]{x1D714}_{Be}}}+O\left({\displaystyle \frac{k^{4}c^{4}}{\unicode[STIX]{x1D714}_{p}^{4}}}\right)\\ 1-{\displaystyle \frac{1}{2}}{\displaystyle \frac{(k_{x}^{2}+k_{y}^{2})c^{2}}{\unicode[STIX]{x1D714}_{p}^{2}}}+{\displaystyle \frac{7}{8}}{\displaystyle \frac{(k_{x}^{2}+k_{y}^{2})^{2}c^{4}}{\unicode[STIX]{x1D714}_{p}^{4}}}-{\displaystyle \frac{1}{2}}{\displaystyle \frac{(k_{x}^{2}+k_{y}^{2})k_{z}^{2}c^{4}}{\unicode[STIX]{x1D714}_{p}^{4}}}\left(3+{\displaystyle \frac{\unicode[STIX]{x1D714}_{p}^{2}}{\unicode[STIX]{x1D714}_{Be}^{2}}}\right)+O\left({\displaystyle \frac{k^{6}c^{6}}{\unicode[STIX]{x1D714}_{p}^{6}}}\right)\end{array}\right)\!,\end{eqnarray}$$

and the ‘magnetic’ components are

(4.18) $$\begin{eqnarray}\left(\begin{array}{@{}c@{}}F_{\boldsymbol{k}yz}^{ls}\\ F_{\boldsymbol{k}zx}^{ls}\\ F_{\boldsymbol{k}xy}^{ls}\end{array}\right)=\left(\begin{array}{@{}c@{}}s{\displaystyle \frac{k_{y}c}{\unicode[STIX]{x1D714}_{p}}}\left(1-{\displaystyle \frac{(k_{x}^{2}+k_{y}^{2}-k_{z}^{2})c^{2}}{\unicode[STIX]{x1D714}_{p}^{2}}}\right)-\text{i}{\displaystyle \frac{k_{x}k_{z}^{2}c^{3}}{\unicode[STIX]{x1D714}_{p}^{2}\unicode[STIX]{x1D714}_{Be}}}\\ -s\frac{k_{x}c}{\unicode[STIX]{x1D714}_{p}}\left(1-{\displaystyle \frac{(k_{x}^{2}+k_{y}^{2}-k_{z}^{2})c^{2}}{\unicode[STIX]{x1D714}_{p}^{2}}}\right)-\text{i}{\displaystyle \frac{k_{y}k_{z}^{2}c^{3}}{\unicode[STIX]{x1D714}_{p}^{2}\unicode[STIX]{x1D714}_{Be}}}\\ \text{i}{\displaystyle \frac{(k_{x}^{2}+k_{y}^{2})k_{z}c^{3}}{\unicode[STIX]{x1D714}_{p}^{2}\unicode[STIX]{x1D714}_{Be}}}\end{array}\right)+O\left(\frac{k^{5}c^{5}}{\unicode[STIX]{x1D714}_{p}^{5}}\right).\end{eqnarray}$$

The tensor $\unicode[STIX]{x1D70E}_{\boldsymbol{k}\unicode[STIX]{x1D714}}^{\;\;\unicode[STIX]{x1D6FD}0\unicode[STIX]{x1D6FE}}$ has a simple spatial derivative,

(4.19) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}r^{\unicode[STIX]{x1D6FF}}}\unicode[STIX]{x1D70E}_{\boldsymbol{k}\unicode[STIX]{x1D714}}^{\;\;\unicode[STIX]{x1D6FD}0\unicode[STIX]{x1D6FE}}=\unicode[STIX]{x1D70E}_{\boldsymbol{ k}\unicode[STIX]{x1D714}}^{\;\;\unicode[STIX]{x1D6FD}0\unicode[STIX]{x1D6FE}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}r^{\unicode[STIX]{x1D6FF}}}\ln (n_{e}).\end{eqnarray}$$

Hence, not only can we omit the terms with derivatives of the conductivity tensor with respect to wave vectors in (3.27), but we can also make the substitutions

(4.20) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}(F_{\boldsymbol{k}0\unicode[STIX]{x1D6FD}}^{\unicode[STIX]{x1D70E}s})^{\ast }{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}r^{\unicode[STIX]{x1D6FF}}}}\unicode[STIX]{x1D70E}_{\boldsymbol{k}\unicode[STIX]{x1D714}}^{\;\;\unicode[STIX]{x1D6FD}0\unicode[STIX]{x1D6FE}}F_{\boldsymbol{ k}0\unicode[STIX]{x1D6FE}}^{\unicode[STIX]{x1D70E}s}=\unicode[STIX]{x1D70E}_{\boldsymbol{ k}\unicode[STIX]{x1D714}}^{\unicode[STIX]{x1D70E}s}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}r^{\unicode[STIX]{x1D6FF}}}}\ln (n_{e}),\\ (F_{\boldsymbol{k}0\unicode[STIX]{x1D6FD}}^{\unicode[STIX]{x1D70E}s})^{\ast }{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}r^{\unicode[STIX]{x1D6FF}}}}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D714}}}\unicode[STIX]{x1D70E}_{\boldsymbol{k}}^{\;\;\unicode[STIX]{x1D6FD}0\unicode[STIX]{x1D6FE}}F_{\boldsymbol{ k}0\unicode[STIX]{x1D6FE}}^{\unicode[STIX]{x1D70E}s}={\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D714}}}\unicode[STIX]{x1D70E}_{\boldsymbol{k}\unicode[STIX]{x1D714}}^{\unicode[STIX]{x1D70E}s}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}r^{\unicode[STIX]{x1D6FF}}}}\ln (n_{e}).\end{array}\right\}\end{eqnarray}$$

Calculations yield

(4.21) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70E}_{\boldsymbol{k}\unicode[STIX]{x1D714}}^{\unicode[STIX]{x1D70E}s} & = & \displaystyle \frac{\text{i}\unicode[STIX]{x1D714}_{p}^{2}}{4\unicode[STIX]{x03C0}}\left[\frac{1}{\unicode[STIX]{x1D714}}\left(1-\frac{k_{\bot }^{2}c^{2}}{\unicode[STIX]{x1D714}_{p}^{2}}+2\frac{k_{\bot }^{4}c^{4}}{\unicode[STIX]{x1D714}_{p}^{4}}-\frac{k_{\bot }^{2}k_{z}^{2}c^{4}}{\unicode[STIX]{x1D714}_{p}^{4}}\left(3+\frac{\unicode[STIX]{x1D714}_{p}^{2}}{\unicode[STIX]{x1D714}_{Be}^{2}}\right)\right)\right.\nonumber\\ \displaystyle & & \displaystyle \left.+\,\frac{k_{\bot }^{2}k_{z}^{2}c^{4}}{\unicode[STIX]{x1D714}_{p}^{4}}\frac{\unicode[STIX]{x1D714}-2s\unicode[STIX]{x1D714}_{p}+\left(\unicode[STIX]{x1D714}\unicode[STIX]{x1D714}_{p}^{2}/\unicode[STIX]{x1D714}_{Be}^{2}\right)}{\unicode[STIX]{x1D714}^{2}-\unicode[STIX]{x1D714}_{Be}^{2}}+O\left(\frac{(kc)^{6}}{\unicode[STIX]{x1D714}_{p}^{7}}\right)\right].\end{eqnarray}$$

We see that this feature is purely imaginary. Note that the vector

(4.22) $$\begin{eqnarray}F_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}s\unicode[STIX]{x1D6FD}0}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}k^{\unicode[STIX]{x1D700}}}(F_{\boldsymbol{k}\;\;\;\unicode[STIX]{x1D6FD}\;\cdot }^{\unicode[STIX]{x1D70E}s\,\cdot \;0})^{\ast }+\frac{1}{2}F_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}s\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6FE}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}k^{\unicode[STIX]{x1D700}}}(F_{\boldsymbol{k}\;\;\;\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6FE}}^{\unicode[STIX]{x1D70E}s})\end{eqnarray}$$

is also purely imaginary, so that the dependence of the phase of the wave polarization tensor on $\boldsymbol{k}$ does not affect the kinetics of wave drift in phase space (see (3.27)).

Finally,

(4.23) $$\begin{eqnarray}\left.\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D714}}\unicode[STIX]{x1D70E}_{\boldsymbol{k}\unicode[STIX]{x1D714}}^{\unicode[STIX]{x1D70E}s}\right|_{\unicode[STIX]{x1D714}=s\unicode[STIX]{x1D714}_{s\boldsymbol{k}}^{\unicode[STIX]{x1D70E}}(\boldsymbol{r},t)}=-\frac{\text{i}}{4\unicode[STIX]{x03C0}}\left[1-2\frac{k_{\bot }^{2}c^{2}}{\unicode[STIX]{x1D714}_{p}^{2}}+4\frac{k_{\bot }^{4}c^{4}}{\unicode[STIX]{x1D714}_{p}^{4}}-4\frac{k_{\bot }^{2}k_{z}^{2}c^{4}}{\unicode[STIX]{x1D714}_{p}^{4}}+O\left(\frac{(kc)^{6}}{\unicode[STIX]{x1D714}_{p}^{6}}\right)\right].\end{eqnarray}$$

For the purpose of the calculation, it is also convenient to use the following consequence of the wave dispersion law (4.16):

(4.24) $$\begin{eqnarray}\frac{\text{d}}{\text{d}x}\ln (n_{e})=\frac{2}{\unicode[STIX]{x1D714}_{p}}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D714}_{\boldsymbol{k}}^{l}}{\unicode[STIX]{x2202}x}\left[1+\frac{1}{2}\frac{k_{\bot }^{2}c^{2}}{\unicode[STIX]{x1D714}_{p}^{2}}+\frac{3}{2}\frac{k_{\bot }^{2}k_{z}^{2}c^{4}}{\unicode[STIX]{x1D714}_{p}^{4}}-\frac{1}{8}\frac{k_{\bot }^{4}c^{4}}{\unicode[STIX]{x1D714}_{p}^{4}}+O\left(\frac{(kc)^{6}}{\unicode[STIX]{x1D714}_{p}^{6}}\right)\right].\end{eqnarray}$$

Based on the above, we can derive the final equation of Langmuir wave drift.

4.3 Final equation of Langmuir wave drift

The rate of change of the spectral density of Langmuir waves due to their drift in phase space is given by

(4.25) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}n_{\boldsymbol{k}}^{l}}{\unicode[STIX]{x2202}t}=\frac{\text{d}\unicode[STIX]{x1D714}_{\boldsymbol{k}}^{l}}{\text{d}x}\frac{\unicode[STIX]{x2202}n_{\boldsymbol{k}}^{l}}{\unicode[STIX]{x2202}k_{x}}-\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D714}_{\boldsymbol{k}}^{l}}{\unicode[STIX]{x2202}k_{\unicode[STIX]{x1D6FF}}}\frac{\unicode[STIX]{x2202}n_{\boldsymbol{k}}^{l}}{\unicode[STIX]{x2202}r^{\unicode[STIX]{x1D6FF}}}-n_{\boldsymbol{k}}^{l}\frac{\text{d}\unicode[STIX]{x1D714}_{\boldsymbol{k}}^{l}}{\text{d}x}\frac{k_{x}c^{2}}{\unicode[STIX]{x1D714}_{p}^{2}}\left(\frac{3}{2}-\frac{5}{2}\frac{k_{\bot }^{2}c^{2}}{\unicode[STIX]{x1D714}_{p}^{2}}+\frac{5}{2}\frac{k_{z}^{2}c^{2}}{\unicode[STIX]{x1D714}_{p}^{2}}\right)+O\left(n_{\boldsymbol{k}}^{l}\frac{k_{x}k^{4}c^{6}}{l\unicode[STIX]{x1D714}_{p}^{5}}\right).\end{eqnarray}$$

The reader is reminded that the traditional weak plasma turbulence theory highlights a certain wavenumber density $N_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}}$ instead of our wave spectral density $n_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}}$ . Our predecessors believed that the use of the concept of wavenumber density provides descriptions of the evolution of wave spectra of turbulent plasmas that are maximally adequate to plasma realities.

As a matter of fact, this concept was elaborated for problems of homogeneous plasma wave turbulence. The definition of $N_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}}$ stems from the idea that the energy density of a turbulent wave field can be interpreted as the sum of the energies of individual wave quanta (see e.g. Tsytovich Reference Tsytovich1970). This clarifies the following relationship between the wavenumber density $N_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}}$ and the wave spectral density $n_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}}$ :

(4.26) $$\begin{eqnarray}N_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}}=\frac{2\unicode[STIX]{x03C0}^{2}}{\hbar \unicode[STIX]{x1D714}_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}}}\left.\frac{1}{\unicode[STIX]{x1D714}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D714}}(\unicode[STIX]{x1D714}^{2}(\unicode[STIX]{x1D716}_{\boldsymbol{ k}\unicode[STIX]{x1D714}}^{\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6FE}})^{H})\right|_{\unicode[STIX]{x1D714}=\unicode[STIX]{x1D714}_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}}}[n_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}}(F_{\boldsymbol{ k}\;\;\;0\unicode[STIX]{x1D6FD}}^{\unicode[STIX]{x1D70E}+\,})^{\ast }F_{\boldsymbol{ k}\;\;\;0\unicode[STIX]{x1D6FE}}^{\unicode[STIX]{x1D70E}+\,}].\end{eqnarray}$$

Here $(\pmb{\unicode[STIX]{x1D716}}_{\boldsymbol{k}\unicode[STIX]{x1D714}}^{\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6FE}})^{H}$ is the Hermitian component of the dielectric permittivity tensor, $(\unicode[STIX]{x1D716}_{\boldsymbol{k}\unicode[STIX]{x1D714}}^{\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6FE}})^{H}\equiv \unicode[STIX]{x1D6FF}^{\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6FE}}+4\unicode[STIX]{x03C0}\text{i}(\unicode[STIX]{x1D70E}_{\boldsymbol{k}\unicode[STIX]{x1D714}}^{\unicode[STIX]{x1D6FD}0\unicode[STIX]{x1D6FE}})^{A}/\unicode[STIX]{x1D714}$ .

As we have already noted, addressing the traditional wavenumber densities and understanding of their mutual influence lead away from highly informative nonlinear scenarios of weakly turbulent plasmas (recall failures of the golden rule). The use of wavenumber densities is even less intelligent in descriptions of wave drifts than in problems of nonlinear dynamics of weak wave turbulence of a homogeneous plasma. A major problem here is a generalization of the definition of plasma wave energy to inhomogeneous plasmas. Suppose that we have developed an adequate description of the energy density of the weakly turbulent wave field in an inhomogeneous plasma and that the distribution of the energy density is given by a function $W_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}}(\boldsymbol{r},t)$ . Then the understandings of geometrical optics yield the following continuity equation of wave energy drift in space and wave vectors:

(4.27) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}W_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}}}{\unicode[STIX]{x2202}t}=\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D714}_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}}}{\unicode[STIX]{x2202}r^{\unicode[STIX]{x1D6FF}}}\frac{\unicode[STIX]{x2202}W_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}}}{\unicode[STIX]{x2202}k_{\unicode[STIX]{x1D6FF}}}-\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D714}_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}}}{\unicode[STIX]{x2202}k_{\unicode[STIX]{x1D6FF}}}\frac{\unicode[STIX]{x2202}W_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}}}{\unicode[STIX]{x2202}r^{\unicode[STIX]{x1D6FF}}}.\end{eqnarray}$$

Here the wave natural frequency is considered as a Hamiltonian that governs the time transformations of the coordinates of the wave package (spatial position) and the conjugated momentum components (leading wave vector of the package). The equation neglects wave dissipation, which is consistent with our physical situation.

It would have been quite natural to use as the wavenumber density $N_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}}$ the ratio $W_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}}(\boldsymbol{r},t)/(\hbar \unicode[STIX]{x1D714}_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}}(\boldsymbol{r}))$ . Provided that the recipes of geometrical optics are sufficiently correct, we then would have got

(4.28) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}N_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}}}{\unicode[STIX]{x2202}t}=\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D714}_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}}}{\unicode[STIX]{x2202}r^{\unicode[STIX]{x1D6FF}}}\frac{\unicode[STIX]{x2202}N_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}}}{\unicode[STIX]{x2202}k_{\unicode[STIX]{x1D6FF}}}-\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D714}_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}}}{\unicode[STIX]{x2202}k_{\unicode[STIX]{x1D6FF}}}\frac{\unicode[STIX]{x2202}N_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}}}{\unicode[STIX]{x2202}r^{\unicode[STIX]{x1D6FF}}}.\end{eqnarray}$$

However, we have no adequate idea of the wave energy density. We have fairly logically developed the notion of the wave spectral density $n_{k}^{\unicode[STIX]{x1D70E}}(\boldsymbol{r},t)$ , but how to express the wave energy density $W_{\boldsymbol{k}}^{\unicode[STIX]{x1D70E}}(\boldsymbol{r},t)$ in its terms?

Suppose that the usual formula for the wave energy in homogeneous plasmas is suitable for inhomogeneous plasmas. Then relation (4.26) should have held. In our situation, it reduces to the equality

(4.29) $$\begin{eqnarray}N_{\boldsymbol{k}}^{l}=\frac{4\unicode[STIX]{x03C0}^{2}}{\hbar \unicode[STIX]{x1D714}_{p}}n_{\boldsymbol{k}}^{l}\left(1-\frac{3}{2}\frac{k_{\bot }^{2}c^{2}}{\unicode[STIX]{x1D714}_{p}^{2}}-\frac{5}{2}\frac{k_{\bot }^{2}k_{z}^{2}c^{4}}{\unicode[STIX]{x1D714}_{p}^{4}}+\frac{23}{8}\frac{k_{z}^{4}c^{4}}{\unicode[STIX]{x1D714}_{p}^{4}}+O\left(\frac{kc}{\unicode[STIX]{x1D714}_{p}}\right)^{6}\right).\end{eqnarray}$$

Correspondingly, equation (4.25) yields

(4.30) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}N_{\boldsymbol{k}}^{l}}{\unicode[STIX]{x2202}t}=\frac{\text{d}\unicode[STIX]{x1D714}_{\boldsymbol{k}}^{l}}{\text{d}x}\frac{\unicode[STIX]{x2202}N_{\boldsymbol{k}}^{l}}{\unicode[STIX]{x2202}k_{x}}-\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D714}_{\boldsymbol{k}}^{l}}{\unicode[STIX]{x2202}k_{\unicode[STIX]{x1D6FF}}}\frac{\unicode[STIX]{x2202}N_{\boldsymbol{k}}^{l}}{\unicode[STIX]{x2202}r^{\unicode[STIX]{x1D6FF}}}+\frac{1}{2}N_{\boldsymbol{k}}^{l}\frac{\text{d}\unicode[STIX]{x1D714}_{\boldsymbol{k}}^{l}}{\text{d}x}\frac{k_{x}c^{2}}{\unicode[STIX]{x1D714}_{p}^{2}}\left(1-3\frac{k_{\bot }^{2}c^{2}}{\unicode[STIX]{x1D714}_{p}^{2}}+3\frac{k_{z}^{2}c^{2}}{\unicode[STIX]{x1D714}_{p}^{2}}\right)+O\left(N_{\boldsymbol{k}}^{l}\frac{k_{x}k^{4}c^{6}}{l\unicode[STIX]{x1D714}_{p}^{5}}\right).\end{eqnarray}$$

We have got an equation that differs from (4.28) by an extra term on the right-hand side. It provides insight into the above-mentioned problems with the definition of the wave energy density. Note that the greater the $y$ -component of the magnetic field in the wave, the larger this term. It comprises some piece of the curl of the magnetic field in the wave that generates a piece of electric field in the wave. Undoubtedly, this suggests some contribution to the wave energy density that the relation (4.26) does not take into account.