3.1. Perturbation method and first-order solutions

Before commencing with the perturbative analysis, we first introduce the following new normalised variables, with the objective of simplifying the subsequent calculations

(3.1) \begin{eqnarray} \Delta n_{s}&\,=\,&\Delta n_+ + \Delta n_-,\;\;\; \Delta n_{d}=\Delta n_+ - \Delta n_-,\nonumber \\ \Delta v_s &\,=\,& v_++v_- \;\;\;\;\;\mbox{and}\;\;\;\;\; \Delta v_d = v_+ - v_-, \end{eqnarray}

where $ \Delta n_+ = n_+ - \eta$ and $\Delta n_- = n_- - 1$ . Thus, the basic (2.3) to (2.5) can be rewritten in terms of these new variables as

(3.2a) \begin{align} &\partial _t (\Delta n_d) + \frac {1}{2}\partial _x[(\eta +1)\Delta v_d+(\eta -1)\Delta v_s +\Delta n_s\Delta v_d+\Delta n_d\Delta v_s] = 0, \end{align}

(3.2b) \begin{align} &\partial _t (\Delta n_s) + \frac {1}{2}\partial _x[(\eta +1)\Delta v_s+(\eta -1)\Delta v_d +\Delta n_s\Delta v_s+\Delta n_d\Delta v_d] = 0, \end{align}

(3.2c) \begin{align} &\partial _t (\Delta v_s) +\frac {1}{4}\partial _x(\Delta v_s^2+\Delta v_d^2)=-(1-Z\mu )E, \end{align}

(3.2d) \begin{align} &\partial _t (\Delta v_d) +\frac {1}{2}\partial _x(\Delta v_s\Delta v_d)=(1+Z\mu )E, \end{align}

(3.2e) \begin{align} &\partial _x E=\Delta n_d - \delta _i\left [1-\exp \left (-\frac {\sigma \phi }{T}\right )\right ]+\delta _e\left [1-\exp \left (\frac {\phi }{T}\right )\right ]. \end{align}

Here, we have utilised the respective normalised expressions for the densities of the hot MB-distributed ions and electrons which are

(3.3) \begin{align} n_i&=\delta _i\exp \left (-\frac {\sigma \phi }{T}\right ), \end{align}

(3.4) \begin{align} n_e&=\delta _e\exp \left (\frac {\phi }{T}\right ). \end{align}

The normalisations for all the variables involved in (3.2a ) to (3.2e ), and (3.3) and (3.4) have been presented in table 1. Also, we have defined $\sigma =T_e/T_i$ , $T=Z_-T_ek^2/m_-\omega _-^2$ , $\delta _i = n_{i0}/n_{-0}$ and $\delta _e=n_{e0}/n_{-0}$ , such that $\eta =1+\delta _e-\delta _i$ in accordance with the quasineutrality condition.

Table 1. Normalisation scheme.

The spatio-temporal evolution of the excited DAWs is investigated by employing a simple perturbative expansion method where all the fluid-field variables $\psi (x,t)$ are expanded as $ \psi (x,t)= \sum _{j=0}^{\infty }\psi ^{(j)}(x,t)$ (Nayfeh Reference Nayfeh2000; Holmes Reference Holmes2013). Here, the $j$ th expansion term $\psi ^{(j)}$ is proportional to the $j$ th power of a small expansion parameter $\epsilon$ . We then insert this expansion into (3.2a ) to (3.2e ) and solve for successive orders of the perturbations in the fluid-field variables, subject to the following initial conditions:

(3.5a) \begin{eqnarray} \Delta n_d(x,0)=-(1+Z\eta \mu )\; \epsilon \cos x, \end{eqnarray}

(3.5b) \begin{eqnarray} \Delta n_s(x,0)=(1-Z\eta \mu )\;\epsilon \cos x, \end{eqnarray}

(3.5c) \begin{eqnarray} \Delta v_d(x,0)=\Delta v_s(x,0) = 0. \end{eqnarray}

Here, the small expansion parameter $\epsilon$ serves as the perturbation amplitude. In terms of the original unnormalised variables, these initial conditions are

(3.6a) \begin{eqnarray} n_{-}(x,0)=n_{-0}(1+\epsilon \cos kx), \end{eqnarray}

(3.6b) \begin{eqnarray} n_{+}(x,0)=n_{+0}(1-Z\mu \epsilon \cos kx), \end{eqnarray}

(3.6c) \begin{eqnarray} v_{+}(x,0) = v_{-}(x,0) =0. \end{eqnarray}

Note that, when we expand $\exp (\phi /T)$ and $\exp (-\sigma \phi /T)$ , we retain only the linear terms and ignore all the higher-order ones. In other words, we expand them as $1+(\phi /T)$ and $1-(\sigma \phi /T)$ , respectively. This is justified as $e\phi \ll T_i,T_e$ for DAWs.

On substituting the perturbative expansions of the fluid-field variables into (3.2a ) to (3.2e ), and collecting terms that are proportional to $\epsilon$ , yields the following first-order equations for all the variables involved:

(3.7a) \begin{eqnarray} \partial _t (\Delta n_d^{(1)}) + \frac {1}{2}\partial _x [(\eta +1)\Delta v_d^{(1)}+(\eta -1)\Delta v_s^{(1)}]=0, \end{eqnarray}

(3.7b) \begin{eqnarray} \partial _t (\Delta n_s^{(1)}) + \frac {1}{2}\partial _x [(\eta +1)\Delta v_s^{(1)}+(\eta -1)\Delta v_d^{(1)}]=0, \end{eqnarray}

(3.7c) \begin{eqnarray} \partial _t (\Delta v_s^{(1)}) = -(1-Z\mu )E^{(1)}, \end{eqnarray}

(3.7d) \begin{eqnarray} \partial _t (\Delta v_d^{(1)}) = (1+Z\mu )E^{(1)}, \end{eqnarray}

(3.7e) \begin{eqnarray} \partial _x E^{(1)}= \Delta n_d^{(1)}-\left (\frac {\sigma \delta _i+\delta _e}{T}\right )\phi ^{(1)}. \end{eqnarray}

In keeping with our goal of investigating the spatio-temporal evolution of DAWs, we aim to construct solutions to (3.7a ) to (3.7e ) that are separable, i.e. solutions that can be written as products of two functions, one which has an explicit dependence on $x$ , while the other has an explicit dependence on $t$ (standing wave solutions). Since, the initial (first-order) perturbation has been taken to be proportional to $\epsilon \cos x$ , we can write $\partial _x E^{(1)}=-\partial _x^2\phi ^{(1)}=\phi ^{(1)}$ , which allows us to write the following expression for $\phi$ :

(3.8) \begin{equation} \phi ^{(1)}=\gamma _1 \Delta n_{d}^{(1)}, \end{equation}

with $\gamma _1=\left (1+{(\sigma \delta _i+\delta _e)}/{T}\right )^{-1}$ . On using (3.7a ) to (3.7e ) along with (3.8), we obtain the following homogeneous differential equation for $\Delta n_d^{(1)}$

(3.9) \begin{equation} \partial _t^2(\Delta n_d^{(1)})+\Omega ^2\Delta n_d^{(1)}=0, \end{equation}

where $\Omega = \sqrt {\gamma _1(1+Z\eta \mu )}$ . Finally, on utilising the initial conditions given by (3.5a ) to (3.5c ), we arrive at the following expressions for the first-order perturbations in the fluid-field variables (first-order solutions):

(3.10a) \begin{eqnarray} \Delta n_d^{(1)}=-\epsilon a_1 \cos x \cos \Omega t, \end{eqnarray}

(3.10b) \begin{eqnarray} \Delta n_s^{(1)}=\epsilon a_2 \cos x \cos \Omega t, \end{eqnarray}

(3.10c) \begin{eqnarray} \Delta v_d^{(1)}=-\epsilon b_1 \sin x \sin \Omega t, \end{eqnarray}

(3.10d) \begin{eqnarray} \Delta v_s^{(1)}=\epsilon b_2 \sin x \sin \Omega t, \end{eqnarray}

(3.10e) \begin{eqnarray} \phi ^{(1)}=-\epsilon \gamma _1 a_1 \cos x \cos \Omega t, \end{eqnarray}

(3.10f) \begin{eqnarray} E^{(1)}=-\epsilon \gamma _1 a_1 \sin x \cos \Omega t, \end{eqnarray}

where we have defined $a_1 = 1+Z\eta \mu$ , $a_2=1-Z\eta \mu$ , $b_1=\Omega (1+Z\mu )$ and $b_2=\Omega (1-Z\mu )$ . As expected, the first-order solutions, given by (3.10a ) to (3.10f ), are purely standing wave solutions, obtained as the superposition of two travelling waves propagating in opposite directions. Expectedly, as of now the first-order equations show no indication of phase mixing, meaning we must extend our analysis to the second-order perturbations in the fluid-field variables.

3.2. Second-order solutions

Proceeding in a similar fashion as in §3.1, we can construct the equations for the second-order perturbations in the fluid-field variables by substituting the perturbative expansions of the variables into (3.2a ) to (3.2e ), and collecting terms that are proportional to $\epsilon ^2$ . Just as before, we aim for solutions that are separable, i.e. standing wave solutions. Owing to the convective nonlinearity of the plasma medium, the perturbations in the second-order analysis will be proportional to $\epsilon ^2\cos 2x$ . As a result, this allows us to write $\partial _xE^{(2)}=-\partial _x^2\phi ^{(2)}=4\phi ^{(2)}$ . Thus, we have

(3.11) \begin{equation} \phi ^{(2)}=\frac {\gamma _2}{4}\Delta n_d^{(2)}, \end{equation}

where $\gamma _2=\left (1+{(\sigma \delta _i+\delta _e)}/{4T}\right )^{-1}$ . Using the second-order equations so constructed along with (3.11), we obtain the following non-homogeneous differential equation for $\Delta n_d^{(2)}$ :

(3.12) \begin{eqnarray} \partial _t^2 ( \Delta n_{d}^{(2)})+\Omega _1^2 \Delta n_{d}^{(2)}&\,=\,&\frac {1}{4}(\eta +1)\;\partial _x^2\left [\Delta v_s^{(1)}\Delta v_d^{(1)}\right ]+\frac {1}{8}(\eta -1)\;\partial _x^2\left [\Delta v_s^{(1)^2}+\Delta v_d^{(1)^2}\right ]\nonumber \\ &&{}-\frac {1}{2}\partial _x\partial _t\left [\Delta n_s^{(1)}\Delta v_d^{(1)}+\Delta n_d^{(1)}\Delta v_s^{(1)}\right ], \end{eqnarray}

where $\Omega _1=\sqrt {\gamma _2(1+Z\eta \mu )}$ . On utilising the proper initial conditions, $\Delta n_d^{(2)}(x,0)=0$ and $\partial _t \Delta n_d^{(2)} (x,0)=0$ , we end up with the following expression for $\Delta n_d^{(2)}$ :

(3.13) \begin{align} \Delta n_d^{(2)}=\epsilon ^2\cos 2x\left [\frac {d_0}{\Omega _1^2}(1-\cos \Omega _1 t)+\frac {d_1}{\Omega _1^2-4\Omega ^2}(\cos 2\Omega t-\cos \Omega _1 t) \right ]\!. \end{align}

Once we have $\Delta n_d^{(2)}$ , it is straightforward to obtain the relevant expressions for the second-order perturbations in the remaining fluid-field variables (second-order solutions) from the second-order equations. These remaining second-order solutions turn out to be as follows:

(3.14a) \begin{align} \phi ^{(2)}&=\frac {1}{2}\epsilon ^2\cos 2x\;(e_0+e_1\cos \Omega _1 t +e_2 \cos 2\Omega t), \end{align}

(3.14b) \begin{align} E^{(2)}&=\epsilon ^2\sin 2x\;(e_0+e_1\cos \Omega _1 t+e_2 \cos 2\Omega t), \end{align}

(3.14c) \begin{align} \Delta v_s^{(2)}&=-\epsilon ^2\sin 2x \;(g_0t+g_1\sin \Omega _1 t+g_2\sin 2\Omega t), \end{align}

(3.14d) \begin{align} \Delta v_d^{(2)}&=\epsilon ^2\sin 2x\; (h_0t+h_1\sin \Omega _1 t +h_2\sin 2\Omega t), \end{align}

(3.14e) \begin{align} \Delta n_s^{(2)}&=\epsilon ^2 \cos 2x\; [\alpha t^2+c_1(1-\cos \Omega _1 t)+c_2(1-\cos 2\Omega t)]. \end{align}

The expressions for all the coefficients in the second-order solutions are presented below

(3.15) \begin{align} d_0&=-\frac {(\eta +1)b_1b_2}{4}+\frac {(\eta -1)(b_1^2+b_2^2)}{8}, \nonumber \\ d_1&=\frac {(a_1b_2+a_2b_1)\Omega }{2}+\frac {(\eta +1)b_1b_2}{4}-\frac {(\eta -1)(b_1^2+b_2^2)}{8}. \nonumber \\ e_0&=\frac {d_0\gamma _2}{2\Omega _1^2}, \;\;e_1=-\frac {d_0\gamma _2}{2\Omega _1^2}-\frac {d_1\gamma _2}{2(\Omega _1^2-4\Omega ^2)}, \;\; e_2 = \frac {d_1\gamma _2}{2(\Omega _1^2-4\Omega ^2)},\nonumber \\ g_0&=e_0(1-Z\mu )+\frac {1}{8}(b_1^2+b_2^2), \;\;g_1 = \frac {e_1}{\Omega _1}(1-Z\mu ), \nonumber\\ g_2 &= \frac {1}{2\Omega }\left [e_2(1-Z\mu )-\frac {1}{8}(b_1^2+b_2^2)\right ], \nonumber \\ h_0&=e_0(1+Z\mu )+\frac {1}{4}(b_1b_2), \;\; h_1 = \frac {e_1}{\Omega _1}(1+Z\mu ), \;\; h_2 = \frac {1}{2\Omega }\left [e_2(1+Z\mu )-\frac {1}{4}(b_1b_2)\right ], \nonumber \\ \alpha &= \frac {1}{2}[(\eta +1)g_0-(\eta -1)h_0], \;\; c_1 = \frac {1}{\Omega _1}[(\eta +1)g_1-(\eta -1)h_1], \nonumber \\ c_2 &= \frac {1}{2\Omega }\left [(\eta +1)g_2-(\eta -1)h_2-\left (\frac {a_1b_1+a_2b_2}{4}\right )\right ]. \end{align}

The second-order solutions clearly depict that, during the evolution of the excited DAWs, higher harmonics in both space and time are generated. This is a direct consequence of the inherent nonlinearity of the plasma medium. In addition to this, we also see that there now exists a new mode of oscillation with frequency $\Omega _1$ . This new mode arises due to the presence of the MB-distributed hot ions and electrons whose number densities are always dependent on $\phi$ , i.e. $n_{i,e}\equiv n_{i,e}(\phi )$ . Another important feature of the second-order solutions is the presence of direct current (DC) and secular terms, the DC terms being those that have no dependence on $t$ , while the secular terms are those that are proportional to $t$ or $t^2$ , and as such have an unbounded growth with evolving time. It must be stressed here that the presence of such secular terms, which are driven by terms like $\sim \partial _x^2 \left (\Delta v_s^{(1)^2}+\Delta v_s^{(1)^2}\right )$ and $\sim \partial _x^2 \left (\Delta v_s^{(1)}\Delta v_s^{(1)}\right )$ in the second-order equations, are in no way an artefact of the perturbation method employed. Rather, such secular terms serve as important signatures of phase mixing (Sen Gupta & Kaw Reference Sen Gupta and Kaw1999). Consider the secular term $\sim (\epsilon ^2 \alpha \cos 2x) t^2$ in the expression for $\Delta n_s^{(2)}$ . Physically, such a secular term is indicative of rapid bunching of plasma particles in space with time. Here, the constant $\alpha$ is given as

(3.16) \begin{eqnarray} \alpha = \frac {1}{2}[(\eta +1)g_0-(\eta -1)h_0], \end{eqnarray}

which, on using the expressions for $g_0$ and $h_0$ and simplifying appropriately, can be rewritten as

(3.17) \begin{equation} \alpha = \frac {Z\eta \mu \gamma _1}{2}(1+Z\mu ). \end{equation}

Thus, we notice that the presence and the subsequent effects of the secular term in $\Delta n_s^{(2)}$ on the evolution of DAWs in opposite polarity dusty plasmas is determined solely by the values of the plasma parameters $\eta$ and $\mu$ . In the limit $\eta \to 0$ , which implies the absence of positively charged dust grains, we end up with $\alpha \to 0$ . This indicates that there would be no secular terms in the second-order solutions in the presence of only single polarity dust grains. Alternatively, in the limit $\mu \to 0$ , implying that the positively charged dust grains are stationary, we again end up with $\alpha \to 0$ . Thus, in order to ensure the appearance of secular terms in the second-order solutions, it is not only necessary to have both positive and negative polarity dust grains, but also necessary to consider them both to be dynamical.

On examining the expression for $E^{(2)}$ (and $\phi ^{(2)}$ as well), we notice the presence of a DC term, which signifies the existence of a non-zero ponderomotive force. This becomes evident when one averages the electric field over the oscillation time scale. The plasma particles respond to this non-zero ponderomotive force and undergo a spatial rearrangement, thereby rendering the plasma medium inhomogeneous. In such an inhomogeneous medium, the characteristic frequency of the relevant mode acquires a spatial dependence, and is no longer constant. What happens is that, as the DAWs self-consistently evolve in space and time, the plasma particles start to accumulate at certain locations while moving away from others, giving rise to spikes in the density profile. This is embodied by the presence of the secular term in $\Delta n_s^{(2)}$ . Thus, the excited DAWs eventually lose their coherency, and are said to be phase mixed.

3.3. Estimating the phase-mixing time

To obtain an estimate of the phase-mixing time, we first construct an approximate evolution equation for $\Delta n_d$ correct up to third order as

(3.18) \begin{equation} \partial _t^2 \Delta n_d+\Omega ^2\left [1+\frac {1+Z\mu }{2(1+Z\eta \mu )}\Delta n_s\right ]\Delta n_d \approx 0. \end{equation}

On substituting the leading-order secular term for $\Delta n_s$ , which is $\Delta n_s^{(2)}\approx (\epsilon ^2\alpha \cos 2x) t^2$ , (3.18) transforms to

(3.19) \begin{equation} \partial _t^2\Delta n_d +\Omega ^2(1+\mathcal{B}t^2\cos 2x)\Delta n_d \approx 0, \end{equation}

where

\begin{equation*}\mathcal{B}=\frac {\epsilon ^2\alpha }{2}\left (\frac {1+Z\mu }{1+Z\eta \mu }\right ).\end{equation*}

Using the initial conditions $\Delta n_d (x,0)=-a_1\epsilon \cos x$ and $\partial _t \Delta n_d(x,0)=0$ , we write the following Wentzel–Kramers–Brillouin (WKB) solution (Bender & Orszag Reference Bender and Orszag2009) for (3.19):

(3.20) \begin{equation} \Delta n_d \approx -a_1\epsilon \cos x \cos \left [\Omega t\left (1+\frac {\mathcal{B}t^2}{6}\cos 2x\right )\right ]. \end{equation}

Notice that, in accordance with our earlier discussion, the characteristic frequency of the dust acoustic mode has indeed acquired a spatial dependence, in addition to a dependence on $\eta$ and $\mu$ . This is what leads to the phase mixing of the excited DAWs.

The initial perturbation to the plasma at $t=0$ excites the primary mode of the DAWs. As time progresses, higher harmonics are generated. This happens at the expense of the electrostatic energy that was initially loaded into the primary mode, wherein it cascades irreversibly to the higher harmonics with evolving time. As a result, the primary mode suffers an eventual collisionless decay. This becomes evident when we expand (3.20) in terms of a Bessel series (Watson Reference Watson2006) as follows:

(3.21) \begin{eqnarray} \Delta n_d&\approx& -\frac {1}{2}a_1\epsilon \sum _{l=-\infty }^{+\infty } J_{l}\left (\frac {1}{6}{\mathcal{B}}{\Omega }t^3\right ) \left [\cos \left ({\Omega } t+\frac {l\pi }{2}\right ) \{\cos {(2l+1)x}+\cos {(2l-1)x} \} \right .\nonumber \\ &&{}\left . +\sin \left ({\Omega } t+\frac {l\pi }{2}\right ) \{\sin {(2l+1)x}+\sin {(2l-1)x} \} \right ].\nonumber \\ \end{eqnarray}

(3.21) shows that the amplitude of the primary mode ( $l=0$ ) varies as $\sim J_0(\mathcal{B}\Omega t^3/6)$ , which clearly indicates the decay of the primary mode as time progresses. Additionally, (3.21) depicts the subsequent generation of higher harmonics ( $l=1,2,\ldots$ ) as well. Finally, on using $J_0(\mathcal{B}\Omega t^3/6)\approx J_0(1)$ , we obtain an approximate expression for the phase-mixing time $\omega _-t_{mix}$ as

(3.22) \begin{equation} \omega _-t_{mix}=\frac {1}{\sqrt {\gamma _1}}\left [\frac {24\sqrt {1+Z\eta \mu }}{\epsilon ^2Z\eta \mu (1+Z\mu )^2}\right ]^{1/3}. \end{equation}

Note that, when $\eta \to 0$ (implying that there are no positive dust grains), the phase-mixing time becomes infinitely high, i.e. $\omega _-t_{mix} \to \infty$ . This is also the case when the limit $\mu \to 0$ (implying that the positive dust grains are at rest) is considered. Thus, in keeping with the discussion in §3.2, we note that, for the excited DAWs to phase mix, the presence of both positively and negatively charged dynamical dust grains in the plasma is crucial.

Here, we must point out an apparent asymmetry in the applicability of (3.22) in determining the phase-mixing time of DAWs in plasmas with single polarity dust grains. Up until now, the utilisation of proper limits has allowed us to successfully apply (3.22) to predict the phase-mixing time for DAWs in plasmas containing only negatively charged dynamical dust grains, which is found to be infinitely high. But it turns out that (3.22) cannot be used to do the same for DAWs in plasmas with only positively charged dust grains. To put it somewhat crudely, in the present analysis it is possible to switch off the positively charged dust grains (density and dynamics), but not the negatively charged ones. In other words, one cannot obtain the phase-mixing time for DAWs in plasmas with just positively charged dynamical dust grains from (3.22), by simply utilising the limits $\eta \to \infty$ and/or $\mu \to \infty$ . Evidently, this is due to the fact that, for such limits, the phase-mixing time becomes zero ( $\omega _-t_{mix} \to 0$ ), which bears no physical meaning. The reason behind this apparent asymmetry lies in the fact that the initial conditions chosen for the above analysis, given by (3.5a ) to (3.5c ) or (3.6a ) to (3.6c ), do not support such an operation and in the limits $\eta \to \infty$ and $\mu \to \infty$ , the initial conditions become meaningless. One must bear in mind that the situation where only the negatively charged dynamical dust grains are present is completely different from the one where it is just the positively charged dynamical dust grains. This is made evident by the stark contrast in the quasineutrality relation and the basic equations for the two cases.

In order to be able to switch off the negatively charged dust grains and obtain the correct estimate for the phase-mixing time in plasmas with positively charged dynamical dust grains, we must consider an alternative set of initial conditions. This would yield a different expression for the phase-mixing time in comparison with (3.22), which would allow us to switch off the negatively charged dust grains by utilising the appropriate limiting conditions. But, this new set of initial conditions would also suffer from the same asymmetry as it would no longer allow us to switch off the positively charged dust grains. However, the final conclusion, i.e. that excited DAWs in dusty plasmas undergo phase mixing only if the plasma contains both positively and negatively charged dynamical dust grains, would remain unaltered.

3.4. Phase-mixing time of DAWs in electron-depleted dusty plasmas

A plausible scenario for a dusty plasma with opposite polarity dust grains is one where, during the charging of the dust grains, almost all the electrons have attached themselves onto the surface of the grains. Thus, the resulting plasma is almost entirely depleted of electrons, and now consists of just three major components, the positively charged dust grains, the negatively charged dust grains and the hot isothermal ions. Note that the keyword here is almost, since a complete depletion of the electrons is not possible. This is due to the fact that the minimum value for the ratio of the electron to ion equilibrium number density is given by the square root of the ratio of the electron to ion mass, when the electron and ion temperatures are approximately equal and the grain surface potential is zero (Shukla & Mamun Reference Shukla and Mamun2010; Tribeche & Merriche Reference Tribeche and Merriche2011). Thus, for electron-depleted dusty plasmas we have

(3.23) \begin{eqnarray} \frac {n_{e0}}{n_{i0}}=\sqrt {\frac {m_e}{m_i}}, \end{eqnarray}

where $m_e$ ( $m_i$ ) denotes the mass of the electrons (ions). If we consider the ions to be that of hydrogen ( $H^+$ ), then the electron to ion mass ratio has the value $m_e/m_i = 1/1836$ . On using (2.1) and (2.2), and normalising the electron and ion densities in (3.23) appropriately (see table 1), we obtain the minimum value of $\delta _e$ in an electron-depleted dusty plasma as

(3.24) \begin{eqnarray} \delta _e \approx 0.023\delta _i, \end{eqnarray}

which leads to the following quasineutrality condition:

(3.25) \begin{eqnarray} \eta +0.977\delta _i = 1. \end{eqnarray}

As it turns out, such electron-depleted plasmas can still support DAWs, and accordingly, the linear dispersion relation is found to be

(3.26) \begin{eqnarray} 1+\frac {1.023}{k^2\lambda _{Di}^2}=\frac {\omega _+^2 + \omega _-^2}{\omega ^2}, \end{eqnarray}

which in the long wavelength limit can be written as

(3.27) \begin{eqnarray} \omega = k C_{DA}\sqrt {1+Z\eta \mu }. \end{eqnarray}

Here, $C_{DA} = 0.9887 Z_-^{1/2}(n_{-0}/n_{i0})^{1/2}(T_i/m_-)^{1/2}$ is the dust acoustic speed. These DAWs will also undergo phase mixing, with the estimate for the phase-mixing time being given as

(3.28) \begin{eqnarray} \omega _- t_{mix}= \sqrt {1+\frac {\delta _i}{\beta _i}}\left [\frac {24\sqrt {1+Z\eta \mu }}{\epsilon ^2Z\eta \mu (1+Z\mu )^2}\right ]^{1/3}, \end{eqnarray}

where $\beta _i = Z_- k^2T_i/1.023 m_- \omega _-^2$ is a constant. The above estimate for the phase-mixing time has been directly obtained from (3.22), after having modified the expression for $\gamma _1$ as $\gamma _1=(1+\delta _i/\beta _i)^{-1}$ .

4.1. Variation of the phase-mixing time with $\eta$

Figure 1. Variation of phase-mixing time ( $\omega _- t_{mix}$ ) with $\eta \; (= n_{+0}/n_{-0})$ for (a) $\delta _i = 0.2$ , $\sigma = 50$ and $\mu = 0.01$ , (b) $\delta _i = 0.2$ , $\sigma = 50$ and $\mu = 100$ , (c) $\delta _e = 0.4$ , $\sigma = 50$ and $\mu = 0.01$ and (d) $\delta _e = 0.4$ , $\sigma = 50$ and $\mu = 100$ . Here, $Z = 10^{-2}$ and $\epsilon = 0.08$ .

In figure 1, we show the variation of the phase-mixing time with $\eta$ for four different cases. Recall that $\eta$ is the ratio of the equilibrium density of the positively charged dust grains to that of the negatively charged ones ( $\eta = n_{+0}/n_{-0}$ ). Thus, an increasing value of $\eta$ essentially signifies an increasing value of $n_{+0}$ in comparison with $n_{-0}$ . In figures 1(a) and 1(b), the variation in the phase-mixing time with $\eta$ is shown, whilst keeping $\delta _i$ fixed at $0.2$ . Following from the quasineutrality condition, such a consideration immediately constrains the values that $\eta$ can attain, as now we must have $\eta \gt 1-\delta _i$ (which is $\eta \gt 0.8$ in our case), in order to ensure that $\delta _e$ always remains positive. Thus, we have considered the variation in $\eta$ for the two plots to be in the range $0.805 \leqslant \eta \leqslant 25$ . On the other hand, in figures 1(c) and 1(d), we have kept $\delta _e$ fixed at $0.4$ , and have allowed $\delta _i$ to be modified with changing $\eta$ . This again imposes a constraint on the values that $\eta$ can attain, which turns out to be $\eta \lt 1+\delta _e$ (i.e. $\eta \lt 1.4$ in our case). Just as before, this constraint on $\eta$ follows from the quasineutrality condition, and the fact that $\delta _i$ is always positive. Accordingly, the variation in $\eta$ for the two plots in this case has been considered to lie in the range $0.05 \leqslant \eta \leqslant 1.39$ . Additionally, the variations in figures 1(a) and 1(b) with fixed $\delta _i = 0.2$ is shown for two different values of $\mu$ , i.e. $\mu = 0.01$ and $\mu =100$ , respectively. This has also been done for figures 1(c) and 1(d) with fixed $\delta _e = 0.4$ . The consideration of two different values for $\mu$ has been made in order to take into account the fact that dust grains with either polarity can be heavier than the other. Lastly, we have considered $\sigma = 50$ , $Z=10^{-2}$ and $\epsilon = 0.08$ , as well.

Our primary observation is that, in figures 1(a) and 1(b), for small values of $\eta$ the phase-mixing time is quite high, and as $\eta$ increases, the phase-mixing time gradually decreases. But this decrease happens only up to a certain value of $\eta$ , i.e. for $\eta = \eta _c$ , and on increasing the value of $\eta$ beyond $\eta _c$ , the phase-mixing time starts to increase. In other words, for $\eta =\eta _c$ , the phase-mixing time exhibits a minimum. In stark contrast to this, the variations in figures 1(c) and 1(d) depict that, with increasing $\eta$ , the initially high phase-mixing time decreases gradually, but exhibits no minimum. The reason behind such distinctly different variations in the two sets of plots lies in the fact that the quasineutrality condition for the plasma is profoundly influenced by whether we allow $\delta _e$ or $\delta _i$ to be modified with increasing $\eta$ , whilst keeping the other fixed. This becomes evident when we consider the expression for the phase-mixing time for the two cases. If we keep $\delta _i$ fixed then the quasineutrality condition dictates that, with increasing $\eta$ , $\delta _e$ increases as well. The expression for the phase-mixing time, given by (3.22), is then modified as

(4.1) \begin{equation} \omega _-t_{mix}=\left [\frac {24\sqrt {1+Z\eta \mu }}{\epsilon ^2Z\eta \mu (1+Z\mu )^2}\left (\alpha _i+\frac {\eta }{T}\right )^{3/2}\right ]^{1/3}, \end{equation}

where $\alpha _i = 1+{(\delta _i(\sigma + 1)-1)}/{T}$ . On the other hand, if we keep $\delta _e$ fixed, the quasineutrality condition dictates that, with increasing $\eta$ , $\delta _i$ decreases and the expression for the phase-mixing time is then modified as

(4.2) \begin{equation} \omega _-t_{mix}=\left [\frac {24\sqrt {1+Z\eta \mu }}{\epsilon ^2Z\eta \mu (1+Z\mu )^2}\left (\alpha _e-\frac {\sigma \eta }{T}\right )^{3/2}\right ]^{1/3}, \end{equation}

where, $\alpha _e = 1+{(\delta _e(\sigma + 1)+\sigma)}/{T}$ . Thus, (4.1) and (4.2) clearly show how the choice of considering either $\delta _i$ or $\delta _e$ to be fixed leads to vastly different quasineutrality situations for the plasma, which in turn affects the phase-mixing time of DAWs.

The secondary observation that we make is that a higher value of $\mu$ leads to lower overall values for the phase-mixing time. This is true irrespective of whether we keep $\delta _i$ or $\delta _e$ fixed, as seen from figures 1(a) to 1(d). A detailed discussion on this is presented in §4.2. Note that, for the variation with fixed $\delta _i$ in figures 1(a) and 1(b), we find that the value of $\mu$ also affects the value of $\eta _c$ . Indeed, this becomes evident when one writes the expression for $\eta _c$ as follows:

(4.3) \begin{equation} \eta _c = \frac {Z \mu \alpha _i T - 1 +\sqrt {1+Z^2 \mu ^2 \alpha _i^2 T^2+14 Z \mu \alpha _i T}}{4 Z \mu }. \end{equation}

Accordingly, for $\mu = 0.01$ , we have $\eta _c = 18.49$ in figure 1(a), while for $\mu = 100$ , we have $\eta _c = 5.75$ in figure 1(b). Note that (4.3) shows us that $\eta _c$ is also dependent on $\sigma$ , and as such, the variation of $\eta _c$ with $\mu$ and $\sigma$ is presented in figures 2(a) and 2(b), respectively. In figure 2(a), the variation in $\mu$ has been taken to lie in the range $0.01\leqslant \mu \leqslant 100$ , and we can clearly see that, with increasing $\mu$ , $\eta _c$ decreases continuously. On the other hand, figure 2(b) shows us that on increasing the value of $\sigma$ in the range $10 \leqslant \sigma \leqslant 200$ , the value of $\eta _c$ increases, almost linearly. Thus, the increasing values of the parameters $\mu$ and $\sigma$ have completely opposite effects on the value of $\eta _c$ , i.e. the former causes it to decrease monotonically while the latter causes it to increase monotonically.

Figure 2. Variation of $\eta _{c}$ with (a) $\mu$ for $\sigma = 50$ , and (b) $\sigma$ for $\mu = 10$ . Here, $\delta _i = 0.2$ , $Z = 10^{-2}$ and $\epsilon = 0.08$ .

Figure 3. Variation of phase-mixing time ( $\omega _- t_{mix}$ ) with $\eta \;(=n_{+0}/n_{-0})$ for (a) $\delta _i$ kept fixed at $0.2$ , and (b) $\delta _e$ kept fixed at $0.4$ . The variation in both (a) and (b) is shown for two distinct values of $\sigma$ , $(1)$ $\sigma = 50$ (red solid curve) and $(2)$ $\sigma = 150$ (blue dashed curve). In panel (b) a zoomed in version of the original plot around $\eta =1.202$ is embedded, in order to clearly bring forth the effect of $\sigma$ on the phase-mixing time. Also, we have considered $\mu =10$ , $Z = 10^{-2}$ and $\epsilon = 0.08$ .

In figure 3, we have again shown the variation of the phase-mixing time as a function of $\eta$ , but for two different values of $\sigma$ , which are $(1)\; \sigma = 50$ (denoted as the red solid line) and $(2)\; \sigma = 150$ (denoted as the blue dashed line). Just as before, we have considered the two different cases of either keeping $\delta _i$ fixed at 0.2 (see figure 3 a), or keeping $\delta _e$ fixed at 0.4 (see figure 3 b), whilst allowing the other parameter to change with changing $\eta$ . As expected, in figure 3(a), the phase-mixing time exhibits a minimum. Note that, for smaller values of $\eta$ , the two curves corresponding to the two different $\sigma$ values are almost merged as one. In reality, however, the phase-mixing time for $\sigma = 150$ is lower than that for $\sigma = 50$ , but the difference is quite small in comparison with the overall variation with $\eta$ . As $\eta$ increases, we notice that the two curves start to deviate from one another, and as $\eta$ is increased further, the phase-mixing time gradually approaches its minimum value at $\eta = \eta _c$ , Here, we notice that, for the higher value of $\sigma = 150$ , the value for the minimum phase-mixing time is much lower in comparison with that for $\sigma = 50$ .

In figure 3(b), as $\delta _e$ has been kept fixed, we observe no minimum for the phase-mixing time. Here, we see that the two curves, (1) and (2), corresponding to two different values of $\sigma =50$ and $\sigma = 150$ , respectively, are completely merged as one. What has happened is that the overall variation in the phase-mixing time with $\eta$ is so large in comparison with the variation in it for two different $\sigma$ , that the two curves seem to have merged into a single one. In order to alleviate any confusion regarding this and to confirm that different values of $\sigma$ do indeed affect the phase-mixing time even in this case, we have embedded in the figure 3(b) a zoomed in version of the original plot at around $\eta = 1.202$ . This clearly shows that, just as before, a higher value of $\sigma$ leads to a lower phase-mixing time. We discuss the variation of the phase-mixing time with $\sigma$ in greater detail in §4.2.

4.2. Variation of the phase-mixing time with $\mu$ and $\sigma$

Figure 4. Variation of phase-mixing time ( $\omega _- t_{mix}$ ) with (a) $\mu \;(=m_-/m_+)$ for $\sigma = 50$ , and (b) $\sigma$ for $\mu = 100$ . Here, we have considered $\delta _i = 0.2$ , $\eta = 1.2$ , $Z = 10^{-2}$ , and $\epsilon = 0.08$ .

In this section we conduct a detailed parametric analysis of the variation of the phase-mixing time with $\mu$ and $\sigma$ . Recall that $\mu$ is defined as the ratio of the mass of the negatively charged dust grains to that of the positively charged ones (i.e. $\mu =m_-/m_+$ ), while $\sigma$ is defined as the ratio of the temperature of the MB-distributed electrons to that of the MB-distributed ions (i.e. $\sigma =T_e/T_i$ ). Thus, increasing values of $\mu$ and $\sigma$ signify an increasing value of $m_-$ and $T_e$ , respectively, in comparison with $m_+$ and $T_i$ , respectively.

The variation of the phase-mixing time with $\mu$ is depicted in figure 4(a) for $\delta _i=0.2$ , $\eta = 1.2$ , $\sigma = 50$ , $Z=10^{-2}$ and $\epsilon = 0.08$ . The range of values for $\mu$ has been chosen to be $0.01\leqslant \mu \leqslant 100$ , which ensures that the variation depicted as such takes into account the fact that it is possible for dust grains with either polarity to be heavier than the other, or they might even have equal masses. Figure 4(a) shows that the value of the phase-mixing time decreases very drastically for very small values of $\mu$ (i.e. for values corresponding to $m_+ \gg m_-$ ). This is represented by the almost vertical drop that the curve exhibits for values of $\mu$ that are very small. In order to bring out the variation in the phase-mixing time for such small values of $\mu$ , we have embedded a zoomed in version of the original plot in the figure. The zoomed in plot depicts the phase-mixing time at around $\mu = 1$ . After the initial drastic fall in phase-mixing time, we observe that, as the value of $\mu$ increases, the phase-mixing time continues to decrease, albeit the rate at which it happens slows down considerably. Thus, higher values of $\mu$ cause the excited DAWs to phase mix quicker. This is in agreement with the secondary observation made from figure 1. Physically, what happens is that higher values of $\mu$ cause the overall initial perturbations in the fluid-field variables, given by (3.5a ) to (3.5c ), to become higher as well, even though the perturbation amplitude is kept constant at $\epsilon = 0.08$ . This results in a quicker phase mixing of the excited DAWs.

In figure 4(b) we have plotted the variation of the phase-mixing time with $\sigma$ for $\delta _i=0.2$ , $\eta = 1.2$ , $\mu = 100$ , $Z=10^{-2}$ and $\epsilon = 0.08$ . Here, we have considered the variation in $\sigma$ to lie in the range $10 \leqslant \sigma \leqslant 200$ . Note that, since the electrons are much lighter than the ions, the electron temperature is always higher than the ion temperature. As such, we have $\sigma \gt 1$ always. From the figure, we see that, as $\sigma$ increases, there is a monotonic decrease in the phase-mixing time, which immediately corroborates our observation from figure 3. The reason behind this is that a higher temperature for the electrons means a higher value of the electron thermal pressure, which in turn leads to a higher value of the electron pressure gradient force ( $\sim T_e\partial _x P_e$ ). Thus, in order to maintain a MB-distribution for the electrons, a stronger electric field is required to nullify the higher value of the electron pressure gradient force. Such a stronger electric field leads the plasma medium to acquire a greater spatial inhomogeneity, which ultimately causes the excited DAWs to phase mix quicker.

4.3. Variation of the phase-mixing time in electron-depleted dusty plasmas

Figure 5. Variation of phase-mixing time ( $\omega _- t_{mix}$ ) in an electron-depleted dusty plasma, with (a) $\eta$ for $\mu = 100$ , and (b) $\mu$ for $\eta = 0.8$ . Additionally, here we have considered $\delta _e = 0.023\delta _i$ , $Z = 10^{-2}$ and $\epsilon = 0.08$ .

In §3.4, we discussed the possibility of phase mixing of DAWs in electron-depleted dusty plasmas, and presented an expression for the phase-mixing time as well, given by (3.28). In this section we analyse how this phase-mixing time varies with parameters $\eta$ , $\mu$ and $\beta _i$ (or, equivalently $T_i$ ), which are characteristic to electron-depleted plasmas. Following the discussion in §3.4, an electron-depleted dusty plasma cannot be completely depleted of electrons. Rather, there is a minimum value for the density of electrons in such plasmas. For our purposes, this value is considered to be $\delta _e=0.023\delta _i$ .

In figure 5(a), the variation of the phase-mixing time with $\eta$ is depicted. Here, we have considered $\mu = 100$ , $\beta _i = 1.35 \times 10^{-3}$ (which is equivalent to $T_i = 0.025$ eV), $Z=10^{-2}$ and $\epsilon = 0.08$ . Here, $\eta$ lies in the range $0.05\leqslant \eta \leqslant 0.99$ , which is in accordance with the constraint $\eta \lt 1$ . As this is an electron-depleted dusty plasma, meaning that the value of $\delta _e$ is fixed (at $0.023\delta _i$ ), the nature of the variation of the phase-mixing time is similar to the ones depicted in figures 1(c) and 1(d), i.e. with increasing $\eta$ , the phase-mixing time decreases gradually, and exhibits no minimum. In figure 5(b), we have shown the variation of the phase-mixing time as a function of $\mu$ , where $\mu$ lies in the same range as before, which is $0.01\leqslant \mu \leqslant 100$ . In addition to this we have considered $\eta = 0.8$ , $\delta _i = (1-\eta )/0.977 = 0.2047$ , $\delta _e = 0.023\delta _i$ , $\beta _i = 1.35 \times 10^{-3}$ , $Z=10^{-2}$ and $\epsilon = 0.08$ . The variation in this case is almost entirely similar to that depicted in figure 4(a). For small values of $\mu$ the phase-mixing time is very high, and it drastically drops in value for slight increments in $\mu$ . Similar to figure 4(a), here too we have zoomed in at around $\mu =1$ , in order to clearly bring forth the change in the phase-mixing time with small values for $\mu$ . As $\mu$ keeps on increasing, the phase-mixing time continues to decrease, with the rate of decrement becoming smaller with higher values of $\mu$ . As discussed in §4.2, the reason for such a variation can be realised from the fact that the value of $\mu$ greatly influences the initial perturbations to the fluid-field variables.

Figure 6 depicts the variation in the phase-mixing time of DAWs in electron-depleted dusty plasmas with $\beta _i$ , which is defined as $\beta _i=Z_-k^2T_i/1.023m_-\omega _-^2$ . Note that increasing $\beta _i$ essentially implies increasing values for the ion temperature $T_i$ , and hence the variation in the phase-mixing time has in reality been analysed as a function of $T_i$ . The variation in $\beta _i$ in the figure has been considered to be in the range $0.0541\times 10^{-3} \leqslant \beta _i \leqslant 1.62 \times 10^{-3}$ , which is equivalent to $0.001$ eV $\leqslant T_i \leqslant 0.03$ eV. Additionally, we have also considered $\eta = 0.8$ , $\delta _i = (1-\eta )/0.977 = 0.2047$ , $\delta _e = 0.023\delta _i$ , $\mu = 100$ , $Z=10^{-2}$ and $\epsilon = 0.08$ . As expected, we find that, for increasing values of $\beta _i$ , the phase-mixing time decreases monotonically. This is also similar to the variation depicted in figure 4(b). Just as before, the reason behind such behaviour of the phase-mixing time with $\beta _i$ is that, as $\beta _i$ increases, or as $T_i$ increases, the corresponding ion thermal pressure increases as well. This in turn leads to an increase in the pressure gradient force which then requires a higher electric field to balance it out, so as to maintain the MB-distribution for the ions. This increase in the electric field leads to an increase in the spatial inhomogeneity of the plasma, thereby ultimately reducing the phase-mixing time.

Figure 6. Variation of phase-mixing time ( $\omega _- t_{mix}$ ) with $\beta _i \;(=Z_-k^2T_i/1.023m_-\omega _-^2)$ in an electron-depleted dusty plasma for $\eta = 0.8$ , $\delta _i = (1-\eta )/0.977$ , $\delta _e = 0.023\delta _i$ , $ \mu = 100$ , $Z = 10^{-2}$ and $\epsilon = 0.08$ .

In figures 1 to 6, the phase-mixing time has been scaled using the oscillation period of the negatively charged dust grains ( $\omega _-^{-1}$ ). In the usual opposite polarity dusty plasmas, on considering certain particular values for the plasma parameters, say for $\eta = 0.2$ , $\delta _i = 0.2$ , $\sigma = 50$ and $\mu = 100$ , the phase-mixing time is found to be $\omega _- t_{mix} \approx 129$ . In the actual time scale, this is equivalent to approximately $0.23$ seconds. In terms of the period of the excited DAWs, the above phase-mixing time scales to $\omega t_{mix} \approx 16$ . Similarly, in electron-depleted dusty plasmas, the phase-mixing time turns out to be $\omega _- t_{mix} \approx 144$ , for $\eta = 0.8$ , $\delta _i = 0.2047$ , $\mu = 100$ and $\beta _i = 1.35 \times 10^{-3}$ . Again, this is equivalent to approximately $0.26$ seconds, and in terms of the DAW period it is $\omega t_{mix} \approx 15.67$ .