2 Basic equations

We investigate the relaxed structures within a collisionless dusty plasma, taking into account three plasma species: electrons $(s_{e})$, ions $(s_{i})$ and negatively charged dust particles $(s_{d})$. Additionally, $Z_{s_{d}}$ denotes the charge state of the negatively charged dust particles, while the ions are assumed to be singly ionized. It is presumed that the plasma equilibrium condition is fulfilled, which can be expressed as

(2.1)\begin{equation} Z_{s_{d}}n_{s_{d}}+n_{s_{e}}=n_{s_{i}}, \end{equation}

here, $n_{s_{e}}$, $n_{s_{i}}$ and $n_{s_{d}}$ are the symbols to represent the equilibrium densities of the components: electrons, ions and dust particles, respectively. The masses assigned to the plasma constituents, encompassing electrons, ions and dust particles, are designated as $m_{s_{e}}$, $m_{s_{i}}$ and $m_{s_{d}}$, respectively. The normalized system of governing fluid equations, designed to elucidate the dynamics of the relaxed equilibrium structures, is presented as follows:

(2.2)\begin{gather} \frac{\partial}{\partial t}\left( \boldsymbol{U}_{s_{e}}-\boldsymbol{A}\right) +\boldsymbol{\nabla}\psi_{s_{e}}-\boldsymbol{U}_{s_{e}}\times\left( \boldsymbol{\nabla}\times\boldsymbol{U}_{s_{e}}-\boldsymbol{H}\right) =0, \end{gather}

(2.3)\begin{gather}\frac{\partial}{\partial t}\left( \boldsymbol{U}_{s_{i}}+M_{s_{i}}\boldsymbol{A} \right) +\boldsymbol{\nabla}\psi_{s_{i}}-\boldsymbol{U}_{s_{i}}\times\left( \boldsymbol{\nabla}\times\boldsymbol{U}_{s_{i}}+M_{s_{i}}\boldsymbol{H}\right) =0, \end{gather}

(2.4)\begin{gather}\frac{\partial}{\partial t}\left( \boldsymbol{U}_{s_{d}}-Z_{s_{d}}M_{s_{d} }\boldsymbol{A}\right) +\boldsymbol{\nabla}\psi_{s_{d}}-\boldsymbol{U}_{s_{d}} \times\left( \boldsymbol{\nabla}\times\boldsymbol{U}_{s_{d}}-Z_{s_{d}}M_{s_{d} }\boldsymbol{H}\right) =0, \end{gather}

where $\psi _{s_{e}}=-\phi +p_{s_{e}}+U_{s_{e}}^{2}/2$, $\psi _{s_{i}}=M_{s_{i} }\phi +(M_{s_{i}}/N_{s_{i}})p_{s_{i}}+U_{s_{i}}^{2}/2$ and $\psi _{s_{e} }=-Z_{s_{d}}M_{s_{d}}\phi +( M_{s_{d}}/N_{s_{d}}) p_{s_{d} }+U_{s_{d}}^{2}/2$, while the mass ratios of the plasma components are $M_{s_{d}}=m_{s_{e}}/m_{s_{d}}$ and $M_{s_{i}}=m_{s_{e}}/m_{s_{i}}$. In the given context, $\boldsymbol {U}_{j}$ $( j=s_{e},s_{i},s_{d})$ represent the electrons, ions and dust particles velocities, respectively, normalized by the Alfvén velocity $V_{A}=B_{0}/\sqrt {\mu _{0}n_{s_{e} }m_{s_{e}}}$ ($B_{0}$ is the arbitrary value of magnetic field and $\mu _{0}$ represents the permeability of free space). The pressure $p_{j}$ is scaled by $B_{0}^{2}/\mu _{0}$. The electric field $\boldsymbol {E}$ can be described as follows:

(2.5)\begin{equation} \boldsymbol{E}={-}\boldsymbol{\nabla}\phi-\frac{\partial\boldsymbol{A}}{\partial t}. \end{equation}

The vector potential $\boldsymbol {A}$ which can be defined as $\boldsymbol {H} =\boldsymbol {\nabla }\times \boldsymbol {A}$, where $\boldsymbol {H}$ is the magnetic field and the electrostatic scalar potential $\phi$ are normalized by $\lambda _{s_{e}}B_{0}$ and $\lambda _{s_{e}}B_{0}V_{A}$, respectively. The units of time and space are in terms of electron plasma period (inverse of electron gyrofrequency $\omega _{p}^{-1}$) and electron skin depth $\lambda _{s_{e}}$ (Iqbal & Shukla Reference Iqbal and Shukla2012), where $\omega _{p}^{-1}=V_{A}/\lambda _{s_{e}}$ and $\lambda _{s_{e}}=V_{A}/\omega _{p}=\sqrt {m_{s_{e}}/\mu _{0}n_{s_{e}}e^{2}}$. The plasma frequency (Miyamoto Reference Miyamoto1980) corresponding to $\omega _{p}$ is expressed as $eB_{0}/m_{s_{e}}$, where $e$ is the charge with a value of $1.6\times 10^{-19}\ C$. It is important to note that the effects of dust charge fluctuations, acting as a source of dissipation and contributing to the formation of shock structures, are not the primary focus of our present consideration. Upon using the curl operator on the dynamic equations presented in (2.2)–(2.4), we derive the subsequent set of vortex dynamic equations:

(2.6)\begin{gather} \frac{\partial}{\partial t}\left( \boldsymbol{\nabla}\times\boldsymbol{U}_{s_{e} }-\boldsymbol{H}\right) -\boldsymbol{\nabla}\times\left\{ \boldsymbol{U}_{s_{e} }\times\left( \boldsymbol{\nabla}\times\boldsymbol{U}_{s_{e}}-\boldsymbol{H}\right) \right\} =0, \end{gather}

(2.7)\begin{gather}\frac{\partial}{\partial t}\left( \boldsymbol{\nabla}\times\boldsymbol{U}_{s_{i} }+M_{s_{i}}\boldsymbol{H}\right) -\boldsymbol{\nabla}\times\left\{ \boldsymbol{U}_{s_{i}}\times\left( \boldsymbol{\nabla}\times\boldsymbol{U}_{s_{i} }+M_{s_{i}}\boldsymbol{H}\right) \right\} =0, \end{gather}

(2.8)\begin{gather}\frac{\partial}{\partial t}\left( \boldsymbol{\nabla}\times\boldsymbol{U}_{s_{d} }-Z_{s_{d}}M_{s_{d}}\boldsymbol{H}\right) -\boldsymbol{\nabla}\times\left\{ \boldsymbol{U}_{s_{d}}\times\left( \boldsymbol{\nabla}\times\boldsymbol{U}_{s_{d} }-Z_{s_{d}}M_{s_{d}}\boldsymbol{H}\right) \right\} =0, \end{gather}

and we ultimately reach the following equation:

(2.9)\begin{equation} \frac{\partial\boldsymbol{\varOmega}_{j}}{\partial t}-\boldsymbol{\nabla} \times\lbrack\boldsymbol{V}_{j}\times\boldsymbol{\varOmega}_{j}]=0, \end{equation}

which represents the compact form of (2.6)–(2.8). In this context, $\boldsymbol {\varOmega }_{j}( j=s_{e},s_{i},s_{d})$, ($\boldsymbol {\varOmega }_{s_{e}}=\boldsymbol {\nabla }\times \boldsymbol {U}_{s_{e} }-\boldsymbol {H}$, $\boldsymbol {\varOmega }_{s_{i}}=\boldsymbol {\nabla }\times \boldsymbol {U}_{s_{i}}+M_{s_{i}}\boldsymbol {H}$ and $\boldsymbol {\varOmega }_{s_{d} }=\boldsymbol {\nabla }\times \boldsymbol {U}_{s_{d}}-Z_{s_{d}}M_{s_{d}}\boldsymbol {H}$ ) denote the generalized vorticities, and $\boldsymbol {V}_{j}$

(2.10)\begin{equation} \boldsymbol{V}_{j}=\boldsymbol{U}_{j}, \end{equation}

represents the velocities of the plasma components. To complete the system, Ampere's law (Mahajan & Yoshida Reference Mahajan and Yoshida1998; Mahajan & Lingam Reference Mahajan and Lingam2020; Shatashvili et al. Reference Shatashvili, Mahajan and Berezhiani2016; Gondal et al. Reference Gondal, Iqbal, Khosa and Murtaza2017; Gondal & Iqbal Reference Gondal and Iqbal2020a; Gondal Reference Gondal2020b; Gondal & Iqbal Reference Gondal and Iqbal2020c) is applied, yielding the subsequent expression:

(2.11)\begin{equation} \boldsymbol{U}_{s_{i}}=\frac{\boldsymbol{1}}{N_{s_{i}}}\left( \boldsymbol{\nabla} \times\boldsymbol{H}+\boldsymbol{U}_{s_{e}}+Z_{s_{d}}N_{s_{d}}\boldsymbol{U}_{s_{d} }\right), \end{equation}

where the symbols $N_{s_{i}}$ and $N_{s_{d}}$ are defined as $N_{s_{i} }=n_{s_{i}}/n_{s_{e}}$ and $N_{s_{d}}=n_{s_{d}}/n_{s_{e}}$, the ratio of the ions to electrons and dust particles to electrons, respectively.

3 Relaxed Beltrami equation in a dusty plasmas

By adhering to the Mahajan–Shatashvili (Shatashvili et al. Reference Shatashvili, Mahajan and Berezhiani2016) methodology, we derive the Beltrami condition, which is the steady-state condition of (2.9),

(3.1)\begin{equation} \varLambda_{j}\boldsymbol{\varOmega}_{j}=\boldsymbol{U}_{j}. \end{equation}

Expressed in terms of the plasma components – electrons, ions and dust particles – the equilibrium Beltrami condition can be characterized as

(3.2)\begin{gather} a_{s_{e}}\boldsymbol{U}_{s_{e}} =\boldsymbol{\nabla}\times\boldsymbol{U}_{s_{e} }-\boldsymbol{H}, \end{gather}

(3.3)\begin{gather}a_{s_{i}}\boldsymbol{U}_{s_{i}} =\boldsymbol{\nabla}\times\boldsymbol{U}_{s_{i} }+M_{s_{i}}\boldsymbol{H}, \end{gather}

(3.4)\begin{gather}a_{s_{d}}\boldsymbol{U}_{s_{d}} =\boldsymbol{\nabla}\times\boldsymbol{U}_{s_{d} }-Z_{s_{d}}M_{s_{d}}\boldsymbol{H}, \end{gather}

where the symbols $a_{s_{e}}$, $a_{s_{i}}$ and $a_{s_{d}}$ represent the vorticity to the flow ratio of the plasma particles – electrons, ions and dust particles, respectively. The inverse of these symbols are also termed as Beltrami parameters ($\varLambda _{j}=1/a_{j}$, where $j=s_{e}$, $s_{i}$ and $s_{d}$) (Gondal et al. Reference Gondal, Iqbal, Khosa and Murtaza2017). Obtaining the flow of electrons $\boldsymbol {U}_{s_{e}}$ is achieved by inserting (2.11) into (3.3),

(3.5)\begin{equation} \boldsymbol{U}_{s_{e}}=K_{1}\left( \boldsymbol{\nabla}\times\right) ^{2} \boldsymbol{H}-K_{2}\boldsymbol{\nabla}\times\boldsymbol{H}+K_{3}\boldsymbol{H}+K_{4} \boldsymbol{U}_{s_{d}}, \end{equation}

where $K_{1}=( a_{s_{i}}-a_{s_{e}}) ^{-1}$, $K_{2}=a_{s_{i}} /K_{1}$, $K_{3}=( 1+N_{s_{i}}M_{s_{i}}+Z_{s_{d}}^{2}N_{s_{d}}M_{s_{d} }) /K_{1}$ and $K_{4}=Z_{s_{d}}N_{s_{d}}( a_{s_{d}}-a_{s_{i} }) /K_{1}$. Substituting the value of $\boldsymbol {U}_{s_{e}}$ from (3.5) into (3.2) provides the expression for the flow of dust particles, which can be expressed as

(3.6)\begin{equation} \boldsymbol{U}_{s_{d}}=l_{1}\left( \boldsymbol{\nabla}\times\right) ^{3} \boldsymbol{H}-l_{2}\left( \boldsymbol{\nabla}\times\right) ^{2}\boldsymbol{H} +l_{3}\boldsymbol{\nabla}\times\boldsymbol{H}-l_{4}\boldsymbol{H}, \end{equation}

where $l_{1}=[ Z_{s_{d}}N_{s_{d}}( a_{s_{e}}-a_{s_{d}}) ( a_{s_{d}}-a_{s_{i}}) ] ^{-1}$, $l_{2}=( a_{s_{e} }+a_{s_{i}}) /l_{1}$, $l_{3}=( a_{s_{e}}a_{s_{i}}+1+N_{s_{i} }M_{s_{i}}+Z_{s_{d}}^{2}N_{s_{d}}M_{s_{d}}) /l_{1}$ and $l_{4}=( a_{s_{i}}+( a_{s_{e}}+a_{s_{i}}-a_{s_{d}}) Z_{s_{d}}^{2}N_{s_{d} }M_{s_{d}}+a_{s_{e}}N_{s_{i}}M_{s_{i}}) /l_{1}$. Employing the aforementioned (3.6) into (3.4) ultimately yields the relaxed equilibrium state, recognized as the quadruple relaxed Beltrami state,

(3.7)\begin{equation} \left( \boldsymbol{\nabla}\times\right) ^{4}\boldsymbol{H}-b_{1}\left( \boldsymbol{\nabla}\times\right) ^{3}\boldsymbol{H}+b_{2}\left( \boldsymbol{\nabla }\times\right) ^{2}\boldsymbol{H}+b_{3}\boldsymbol{\nabla}\times\boldsymbol{H} +b_{4}\boldsymbol{H}=0, \end{equation}

where

(3.8)\begin{gather} b_{1} =a_{s_{e}}+a_{s_{i}}+a_{s_{d}}, \end{gather}

(3.9)\begin{gather}b_{2} =a_{s_{e}}a_{s_{i}}+a_{s_{i}}a_{s_{d}}+a_{s_{d}}a_{s_{e}} +1+N_{s_{i}}M_{s_{i}}+Z_{s_{d}}^{2}N_{s_{d}}M_{s_{d}}, \end{gather}

(3.10)\begin{gather}b_{3} =N_{s_{i}}M_{s_{i}}\left( a_{s_{d}}+a_{s_{e}}\right) +Z_{s_{d} }^{2}N_{s_{d}}M_{s_{d}}\left( a_{s_{e}}+a_{s_{i}}\right) +\left( a_{s_{i} }+a_{s_{d}}\right) +a_{s_{e}}a_{s_{i}}a_{s_{d}}, \end{gather}

(3.11)\begin{gather}b_{4} =N_{s_{i}}M_{s_{i}}a_{s_{d}}a_{s_{e}}+Z_{s_{d}}^{2}N_{s_{d}} M_{s_{d}}a_{s_{e}}a_{s_{i}}+a_{s_{i}}a_{s_{d}}. \end{gather}

The equilibrium state, as indicated by (3.7), in a three-component dusty plasma is a composite of four distinct single equilibrium Beltrami states. The overall solution of (3.7) can be characterized in terms of the following eigen expression (Shatashvili, Mahajan & Berezhiani Reference Shatashvili, Mahajan and Berezhiani2019):

(3.12)\begin{equation} \boldsymbol{\nabla}\times\boldsymbol{B}_{\delta}=\lambda_{\delta}\boldsymbol{B}_{\delta}, \end{equation}

where the eigenfunction is $\boldsymbol {B}_{\delta }$ and the eigenvalue is $\lambda _{\delta }$. Here, $\delta =1,2,3,4$ describes four states having four individual eigenvalues. Equation (3.12) can also be interpreted as

(3.13)\begin{equation} \left( \operatorname{curl}-\lambda_{1}\right) \left( \operatorname{curl} -\lambda_{2}\right) \left( \operatorname{curl}-\lambda_{3}\right) \left( \operatorname{curl}-\lambda_{4}\right) \boldsymbol{H}=0, \end{equation}

by expressing the operator ($\boldsymbol {\nabla }\times$) as ‘curl’. In the above expression, $\boldsymbol {H}=\boldsymbol {H}_{1}+\boldsymbol {H}_{2}+\boldsymbol {H}_{3}+\boldsymbol {H}_{4}$ and $\lambda _{1}$, $\lambda _{2}$, $\lambda _{3}$ and $\lambda _{4}$ represent the four distinct eigenvalues. Solving (3.13) yields the same fourth-order equation as previously derived in (3.7),

(3.14)\begin{equation} \left( \boldsymbol{\nabla}\times\right) ^{4}\boldsymbol{H}-b_{1}\left( \boldsymbol{\nabla}\times\right) ^{3}\boldsymbol{H}+b_{2}\left( \boldsymbol{\nabla }\times\right) ^{2}\boldsymbol{H}+b_{3}\boldsymbol{\nabla}\times\boldsymbol{H} +b_{4}\boldsymbol{H}=0, \end{equation}

in the above expression, $b_{1}$, $b_{2}$, $b_{3}$ and $b_{4}$ are the constants, which can be read as in terms of eigenvalues,

(3.15)\begin{gather} b_{1} =\lambda_{1}+\lambda_{2}+\lambda_{3}+\lambda_{4}, \end{gather}

(3.16)\begin{gather}b_{2} =\lambda_{1}\lambda_{2}+\lambda_{2}\lambda_{3}+\lambda_{3} \lambda_{4}+\lambda_{4}\lambda_{1}+\lambda_{1}\lambda_{3}+\lambda_{2} \lambda_{4}, \end{gather}

(3.17)\begin{gather}b_{3} =\lambda_{1}\lambda_{2}\lambda_{3}+\lambda_{2}\lambda_{3}\lambda _{4}+\lambda_{3}\lambda_{4}\lambda_{1}+\lambda_{4}\lambda_{1}\lambda_{2}, \end{gather}

(3.18)\begin{gather}b_{4} =\lambda_{1}\lambda_{2}\lambda_{3}\lambda_{4}. \end{gather}

The values of $\lambda _{1}$, $\lambda _{2}$, $\lambda _{3}$ and $\lambda _{4}$ (Gondal et al. Reference Gondal, Iqbal, Khosa and Murtaza2017; Gondal & Iqbal Reference Gondal and Iqbal2020a; Gondal Reference Gondal2020b; Gondal & Iqbal Reference Gondal and Iqbal2020c) can be determined as $\lambda _{1}=( b_{1}+2r+2\delta ) /4$, $\lambda _{2}=( b_{1}+2r-2\delta ) /4$, $\lambda _{3}=( b_{1}-2r+2\delta ) /4$ and $\lambda _{4}=( b_{1}-2r-2\delta ) /4$, where $r=( \sqrt {b_{1} ^{2}-4b_{2}+4Y}) /2,$ $Y=( d-3u^{2}+3u\alpha _{1}) /3u,$ $u=\sqrt [3]{(q/2)+\sqrt {(q/2)^{2}+(d/3)^{3}}}$, $q=(9\alpha _{1}\alpha _{2}-2\alpha _{1}^{3}-27\alpha _{3})/27$, $d=(3\alpha _{2}-\alpha _{1}^{2})/3$, $\alpha _{1}=b_{2},$ $\alpha _{2}=b_{3}b_{1}-4b_{4}$ and $\alpha _{3}=b_{3} ^{2}+b_{1}^{2}b_{4}-4b_{2}b_{4}$. For $r\neq 0$, $\delta$ and $\xi$ are given as

(3.19)\begin{equation} \left.\begin{gathered} \delta =\sqrt{\frac{3}{4}b_{1}^{2}-r^{2}-2b_{2}+\frac{1}{4r}(4b_{1} b_{2}-8b_{3}-b_{1}^{3})},\\ \xi =\sqrt{\frac{3}{4}b_{1}^{2}-r^{2}-2b_{2}-\frac{1}{4r}(4b_{1} b_{2}-8b_{3}-b_{1}^{3})}, \end{gathered}\right\} \end{equation}

while for $r=0$, $\delta$ and $\xi$ read as

(3.20)\begin{equation} \left.\begin{gathered} \delta =\sqrt{\tfrac{3}{4}b_{1}^{2}-2b_{2}+2\sqrt{Y^{2}-4b_{4}}},\\ \xi =\sqrt{\tfrac{3}{4}b_{1}^{2}-2b_{2}-2\sqrt{Y^{2}-4b_{4}}}. \end{gathered}\right\} \end{equation}

The discourse presented in the above equations highlights that the eigenvalues are contingent on Beltrami parameters, plasma particle densities, the mass of plasma particles and the charge state of dust particles. These eigenvalues may exhibit either a real nature or a combination of real and a pair of complex conjugates. Therefore, by manipulating the Beltrami parameters and plasma parameters, we have the ability to alter the characteristics of the eigenvalues and subsequently influence the nature of the relaxed structures. In § 7, we delved into the analysis of velocity profiles by varying the Beltrami parameters and the densities of plasma species.

4 Unified flow

To determine the unified flow $\boldsymbol {U}$, we can employ the following momentum equation:

(4.1)\begin{equation} \boldsymbol{U}=\frac{\rho_{s_{e}}\boldsymbol{U}_{s_{e}}+\rho_{s_{i}}\boldsymbol{U}_{s_{i} }+\rho_{s_{d}}\boldsymbol{U}_{s_{d}}}{\rho}, \end{equation}

where $\rho$ is the mass density of the system which can be described as $\rho =\rho _{s_{e}}+\rho _{s_{i}}+\rho _{s_{d}}$. Here, $\rho _{s_{e}}$, $\rho _{s_{i}}$ and $\rho _{s_{d}}$ represent the density of electrons, ions and negatively charged dust fluids, respectively ($\rho _{s_{e}}=n_{s_{e} }m_{s_{e}}$, $\rho _{s_{i}}=n_{s_{i}}m_{s_{i}}$ and $\rho _{s_{d}}=n_{s_{d} }m_{s_{d}}$). By substituting the flow of ions from (2.11) into the aforementioned (4.1), we obtain

(4.2)\begin{equation} \boldsymbol{U}=c_{1}\boldsymbol{\nabla}\times\boldsymbol{H}+c_{2}\boldsymbol{U}_{s_{e}} +c_{3}\boldsymbol{U}_{s_{d}}, \end{equation}

where $c_{1}=\rho _{s_{i}}/\rho N_{s_{i}}$, $c_{2}=( \rho _{s_{e}} +\rho _{s_{i}}/N_{s_{i}}) /\rho$ and $c_{3}=( \rho _{s_{d}} +\rho _{s_{i}}Z_{s_{d}}N_{s_{d}}/N_{s_{i}}) /\rho$. By inserting the value of the electron flow $\boldsymbol {U}_{s_{e}}$ into the equation above, we obtain

(4.3)\begin{equation} \boldsymbol{U}=d_{1}\left( \boldsymbol{\nabla}\times\right) ^{2}\boldsymbol{H} -d_{2}\boldsymbol{\nabla}\times\boldsymbol{H}+d_{3}\boldsymbol{H}+d_{4}\boldsymbol{U}_{s_{d}}, \end{equation}

where $d_{1}=c_{2}K_{1}$, $d_{2}=c_{2}K_{2}-c_{1}$, $d_{3}=c_{2}K_{3}$ and $d_{4}=c_{2}K_{4}+c_{3}$. Now, by incorporating the value of the flow of negatively charged dust particles $\boldsymbol {U}_{s_{d}}$ from (3.6) into the above equation, we arrive at the ultimate expression for the unified flow, which can be read as

(4.4)\begin{equation} \boldsymbol{U}=e_{1}\left( \boldsymbol{\nabla}\times\right) ^{3}\boldsymbol{H} -e_{2}\left( \boldsymbol{\nabla}\times\right) ^{2}\boldsymbol{H}+e_{3} \boldsymbol{\nabla}\times\boldsymbol{H}-e_{4}\boldsymbol{H}, \end{equation}

where $e_{1}=d_{4}l_{1}$, $e_{2}=d_{4}l_{2}-d_{1}$, $e_{3}=d_{4}l_{3}-d_{2}$ and $e_{4}=d_{4}l_{4}-d_{3}$.

5 Physical invariants and the equilibrium state

In the realm of plasma dynamics, Mahajan's exploration of systems involving $S$ fluids reveals the existence of $S+1$ invariants (Mahajan Reference Mahajan2008; Mahajan & Lingam Reference Mahajan and Lingam2015). In the current research, the system under investigation features three distinctive components: electrons, ions and negatively charged dust particles. Consequently, the constraints for this tripartite plasma system encompass magnetofluid energy ($E$), the generalized helicity of electrons ($h_{s_{e}}$), the generalized helicity of ions ($h_{s_{i}}$) and the generalized helicity of negatively charged dust grains ($h_{s_{d}}$). By employing (2.6)–(2.8), these four integral invariants can be elegantly expressed as

(5.1)\begin{gather} E =\frac{1}{2}\left( \boldsymbol{U}_{s_{e}}^{2}+\frac{N_{s_{i}}}{M_{s_{i}} }\boldsymbol{U}_{s_{i}}^{2}+\frac{N_{s_{d}}}{Z_{s_{d}}M_{s_{d}}}\boldsymbol{U}_{s_{d} }^{2}+\boldsymbol{H}^{2}\right) , \end{gather}

(5.2)\begin{gather}h_{ s_{e}} =\frac{1}{2}\int_{v}\left( \boldsymbol{U}_{s_{e}}-\boldsymbol{A} \right) \boldsymbol{\cdot}\left( \boldsymbol{\nabla}\times\boldsymbol{U}_{s_{e}} -\boldsymbol{H}\right) {\rm d}v, \end{gather}

(5.3)\begin{gather}h_{s_{i}} =\frac{1}{2}\int_{v}\left( \boldsymbol{U}_{s_{i}}+M_{s_{i} }\boldsymbol{A}\right) \boldsymbol{\cdot}\left( \boldsymbol{\nabla}\times\boldsymbol{U}_{s_{i} }+M_{s_{i}}\boldsymbol{H}\right) {\rm d}v, \end{gather}

(5.4)\begin{gather}h_{s_{d}} =\frac{1}{2}\int_{v}\left( \boldsymbol{U}_{s_{d}}-Z_{s_{d}} M_{s_{d}}\boldsymbol{A}\right) \boldsymbol{\cdot}\left( \boldsymbol{\nabla}\times \boldsymbol{U}_{s_{d}}-Z_{s_{d}}M_{s_{d}}\boldsymbol{H}\right) {\rm d}v. \end{gather}

Another approach, termed the variational principle (Sudan Reference Sudan1979; Steinhauer & Ishida Reference Steinhauer and Ishida1997; Mahajan & Yoshida Reference Mahajan and Yoshida1998), can also be employed to recover the relaxed magnetized equilibrium state,

(5.5)\begin{equation} \delta\left( E-\mu_{s_{e}}h_{s_{e}}-\mu_{s_{i}}h_{s_{i}}-\mu_{s_{d}}h_{s_{d} }\right) =0, \end{equation}

where the symbols $\mu _{s_{e}}$, $\mu _{s_{i}}$ and $\mu _{s_{d}}$ represent the Lagrangian multipliers. By solving the aforementioned equations concurrently and treating $\delta \boldsymbol {A}$, $\delta \boldsymbol {U}_{s_{i}}$ and $\delta \boldsymbol {U}_{s_{i}}$, as independent parameters, we derive

(5.6)\begin{gather} \frac{1}{a_{s_{e}}}\left( \boldsymbol{H}-\boldsymbol{\nabla}\times\boldsymbol{U} _{s_{e}}\right) +\frac{N_{s_{i}}}{a_{s_{i}}}\left( M_{s_{i}}\boldsymbol{H} +\boldsymbol{\nabla}\times\boldsymbol{U}_{s_{i}}\right) +\frac{N_{s_{d}} }{a_{s_{d}}}\left( Z_{s_{d}}M_{s_{d}}\boldsymbol{H}+\boldsymbol{\nabla} \times\boldsymbol{U}_{s_{d}}\right) =\boldsymbol{\nabla}\times\boldsymbol{H}, \end{gather}

(5.7)\begin{gather}a_{s_{i}}\boldsymbol{U}_{s_{i}} =\boldsymbol{\nabla}\times\boldsymbol{U}_{s_{i} }+M_{i}\boldsymbol{H,} \end{gather}

(5.8)\begin{gather}a_{s_{d}}\boldsymbol{U}_{s_{d}} =\boldsymbol{\nabla}\times\boldsymbol{U}_{s_{d} }+Z_{s_{d}}M_{s_{d}}\boldsymbol{H}, \end{gather}

considering $\mu _{e}=-1/a_{s_{e}}$, $\mu _{s_{i}}=N_{s_{i}}/a_{s_{i}}M_{s_{i}}$ and $\mu _{s_{d}}=-N_{s_{d}}/a_{s_{d}}M_{s_{d}}Z_{s_{d}}$. Inserting (5.7)– (5.8) into (5.6), the resulting expression is

(5.9)\begin{equation} \boldsymbol{U}_{s_{i}}=\frac{1}{N_{s_{i}}}\left( \boldsymbol{\nabla}\times \boldsymbol{H}+Z_{s_{i}}M_{s_{i}}\boldsymbol{U}_{s_{i}}+\boldsymbol{U}_{s_{i}}\right). \end{equation}

It has been noted that the set of equations (5.7)–(5.9) obtained through the variational principle bears resemblance to our earlier set of equilibrium equations (3.2)–(3.4). This observation indicates that the self-organization of ordered structures can be discerned through the variational principle.

The Beltrami alignment illustrated in (2.4)–(2.3) imposes the subsequent generalized Bernoulli conditions, by replacing (2.6)–(2.8) into (2.2)–(2.4), which represent the macroscopic evolution equations of the plasma species, and setting the time derivative $\partial /\partial t$ equal to $0$ (indicating a steady state), we derive the subsequent equations:

(5.10)\begin{gather} \boldsymbol{\nabla}\left( \frac{1}{2}U_{s_{e}}^{2}+p_{s_{e}}-\phi\right) =0, \end{gather}

(5.11)\begin{gather}\boldsymbol{\nabla}\left( \frac{1}{2}U_{s_{i}}^{2}+\frac{M_{s_{i}}}{N_{s_{i}}}p_{s_{i} }+M_{s_{i}}\phi\right) =0, \end{gather}

(5.12)\begin{gather}\boldsymbol{\nabla}\left( \frac{1}{2}U_{s_{d}}^{2}+\frac{M_{s_{d}}}{N_{s_{d}}}p_{s_{d} }-Z_{s_{d}}M_{s_{d}}\phi\right) =0, \end{gather}

which articulate the equilibrium of all remaining potential forces. Upon integration of the aforementioned set of equations, we obtain

(5.13)\begin{gather} f_{s_{e}} =\tfrac{1}{2}U_{s_{e}}^{2}+p_{s_{e}}-\phi, \end{gather}

(5.14)\begin{gather}f_{s_{i}} =\frac{1}{2}U_{s_{i}}^{2}+\frac{M_{s_{i}}}{N_{s_{i}}}p_{s_{i} }+M_{s_{i}}\phi, \end{gather}

(5.15)\begin{gather}f_{s_{d}} =\frac{1}{2}U_{s_{d}}^{2}+\frac{M_{s_{d}}}{N_{s_{d}}}p_{s_{d} }-Z_{s_{d}}M_{s_{d}}\phi, \end{gather}

where the constants of integration are $f_{s_{e}}$, $f_{s_{i}}$ and $f_{s_{d}}$. The consolidated form of these equations can be described as

(5.16)\begin{equation} Constant=P+\tfrac{1}{2}\left( U_{s_{e}}^{2}+U_{s_{i}}^{2}+U_{s_{d}} ^{2}\right) -\left( 1-M_{s_{i}}+M_{s_{d}}\right) \phi{,} \end{equation}

where $P=p_{s_{e}}+ M_{s_{i}}p_{s_{i}}N_{s_{i}}^{-1}+Z_{s_{d}}M_{s_{d} }p_{s_{d}}N_{s_{d}}^{-1}$.

6 Solutions of the field equation

Expressed as the linear sum of four Beltrami states, the quadruple Beltrami field (3.7) can be written as

(6.1)\begin{equation} \boldsymbol{H}=C_{1}\boldsymbol{H}_{1}+C_{2}\boldsymbol{H}_{2}+C_{3}\boldsymbol{H}_{3} +C_{4}\boldsymbol{H}_{4}. \end{equation}

In this context, the symbol $C_{1}$, $C_{2}$,$C_{3}$ and $C_{4}$ signify the amplitude four Beltrami states. Equation (3.7) denotes a partial differential equation, with ABC flow (Arnold & Khesin Reference Arnold and Khesin1998) being one of the solutions corresponding to the Beltrami state in the slab geometry. Subsequent to the ABC flow, the derivations for solving (3.7)) are conducted for two distinct modes: (i) a simple rectangular geometry (Gondal et al. Reference Gondal, Iqbal, Khosa and Murtaza2017; Gondal & Iqbal Reference Gondal and Iqbal2020a,Reference Gondal and Iqbalc); and (ii) a coplanar rectangular geometry, which involves a rectangular conducting chamber within the Cartesian coordinate system.

6.1 Solution of quadruple Beltrami state in a simple rectangular geometry

Expressing the solution of (3.7) in terms of $x$, $y$ and $z$, we get

(6.2)\begin{equation} \left(\begin{array}{@{}c@{}} H_{x}\\ H_{y}\\ H_{z} \end{array}\right) =\left(\begin{array}{@{}c@{}} 0\\ C_{1}\sin(\lambda_{1}x)+C_{2}\sin(\lambda_{2}x)+C_{3}\sin(\lambda_{3} x)+C_{4}\sin(\lambda_{4}x)\\ C_{1}\cos(\lambda_{1}x)+C_{2}\cos(\lambda_{2}x)+C_{3}\cos(\lambda_{3} x)+C_{4}\cos(\lambda_{4}x) \end{array} \right), \end{equation}

where the symbols $C_{\alpha }$ $( \alpha =1,2,3\textrm { and }4)$ signify the amplitude. The values of these constants can be determined by applying the following set of boundary conditions: $\vert H_{y}\vert _{x=X}=g_{1}$, $\vert H_{z}\vert _{x=0}=g_{2}$, $\vert (\boldsymbol {\nabla }\times \boldsymbol {H})_{y}\vert _{x=X}=g_{3}$ and $\vert (\boldsymbol {\nabla }\times \boldsymbol {H})_{z}\vert _{x=0}=g_{4}$ in (6.2), we obtain

(6.3)\begin{gather} g_{1} =C_{1}\sin(\lambda_{1}X)+C_{2}\sin(\lambda_{2}X)+C_{3}\sin (\lambda_{3}X)+C_{4}\sin(\lambda_{4}X), \end{gather}

(6.4)\begin{gather}g_{2} =C_{1}+C_{2}+C_{3}+C_{4}, \end{gather}

(6.5)\begin{gather}g_{3} =C_{1}\lambda_{1}\sin(\lambda_{1}X)+C_{2}\lambda_{2}\sin(\lambda _{2}X)+C_{3}\lambda_{3}\sin(\lambda_{3}X)+C_{4}\lambda_{4}\sin(\lambda _{4}X), \end{gather}

(6.6)\begin{gather}g_{4} =C_{1}\lambda_{1}+C_{2}\lambda_{2}+C_{3}\lambda_{3}+C_{4}\lambda _{4}. \end{gather}

Solving the above equations, we obtain

(6.7a-d)\begin{equation} C_{1}=\frac{L_{1}}{L_{5}},\quad C_{2}=\frac{L_{2}}{L_{5} },\quad C_{3}=\frac{L_{3}}{L_{5}},\quad C_{4}=\frac{L_{4}}{L_{5}}, \end{equation}

where

(6.8)\begin{align} L_{1} & =[\sin(\lambda_{3}X)\sin(\lambda_{4}X)(g_{4}-g_{2}\lambda_{2} )+\sin(\lambda_{2}X)(g_{3}-g_{1}\lambda_{2})](\lambda_{3}-\lambda_{4})\nonumber\\ & \quad +[\sin(\lambda_{3}X)\sin(\lambda_{2}X)(g_{4}-g_{2}\lambda_{4})+\sin (\lambda_{4}X)(g_{3}-g_{1}\lambda_{4})](\lambda_{2}-\lambda_{3})\nonumber\\ & \quad +[\sin(\lambda_{2}X)\sin(\lambda_{4}X)(g_{4}-g_{2}\lambda_{3})+\sin (\lambda_{3}X)(g_{3}-g_{1}\lambda_{3})](\lambda_{4}-\lambda_{2}), \end{align}

(6.9)\begin{align} L_{2} & =[\sin(\lambda_{3}X)\sin(\lambda_{1}X)(g_{4}-g_{2}\lambda_{4} )+\sin(\lambda_{4}X)(g_{3}-g_{1}\lambda_{4})](\lambda_{3}-\lambda_{1})\nonumber\\ & \quad +[\sin(\lambda_{3}X)\sin(\lambda_{4}X)(g_{4}-g_{2}\lambda_{1})+\sin (\lambda_{1}X)(g_{3}-g_{1}\lambda_{1})](\lambda_{4}-\lambda_{3})\nonumber\\ & \quad +[\sin(\lambda_{1}X)\sin(\lambda_{4}X)(g_{4}-g_{2}\lambda_{3})+\sin (\lambda_{3}X)(g_{3}-g_{1}\lambda_{3})](\lambda_{1}-\lambda_{4}), \end{align}

(6.10)\begin{align} L_{3} & =[\sin(\lambda_{1}X)\sin(\lambda_{4}X)(g_{4}-g_{2}\lambda_{2} )+\sin(\lambda_{2}X)(g_{3}-g_{1}\lambda_{2})](\lambda_{4}-\lambda_{1})\nonumber\\ & \quad +[\sin(\lambda_{1}X)\sin(\lambda_{2}X)(g_{4}-g_{2}\lambda_{4})+\sin (\lambda_{4}X)(g_{3}-g_{1}\lambda_{4})](\lambda_{1}-\lambda_{2})\nonumber\\ & \quad +[\sin(\lambda_{2}X)\sin(\lambda_{4}X)(g_{4}-g_{2}\lambda_{1})+\sin (\lambda_{1}X)(g_{3}-g_{1}\lambda_{1})](\lambda_{2}-\lambda_{4}), \end{align}

(6.11)\begin{align} L_{4} & =[\sin(\lambda_{3}X)\sin(\lambda_{1}X)(g_{4}-g_{2}\lambda_{2} )+\sin(\lambda_{2}X)(g_{3}-g_{1}\lambda_{2})](\lambda_{1}-\lambda_{3})\nonumber\\ & \quad +[\sin(\lambda_{1}X)\sin(\lambda_{2}X)(g_{4}-g_{2}\lambda_{3})+\sin (\lambda_{3}X)(g_{3}-g_{1}\lambda_{3})](\lambda_{2}-\lambda_{1})\nonumber\\ & \quad +[\sin(\lambda_{2}X)\sin(\lambda_{3}X)(g_{4}-g_{2}\lambda_{1})+\sin (\lambda_{1}X)(g_{3}-g_{1}\lambda_{1})](\lambda_{3}-\lambda_{2}), \end{align}

(6.12)\begin{align} L_{5} & =(\lambda_{4}-\lambda_{1})[\sin(\lambda_{4}X)\sin(\lambda_{1} X)+\sin(\lambda_{3}X)\sin(\lambda_{2}X)](\lambda_{3}-\lambda_{2})\nonumber\\ & \quad +(\lambda_{4}-\lambda_{3})[\sin(\lambda_{4}X)\sin(\lambda_{3} X)+\sin(\lambda_{2}X)\sin(\lambda_{1}X)](\lambda_{2}-\lambda_{1})\nonumber\\ & \quad +(\lambda_{2}-\lambda_{4})[\sin(\lambda_{2}X)\sin(\lambda_{4} X)+\sin(\lambda_{3}X)\sin(\lambda_{1}X)](\lambda_{3}-\lambda_{1}). \end{align}

6.2 Solution of quadruple Beltrami state in a rectangular geometry with an internal conductor

An internal conductor geometry involves embedding a rectangular chamber within another rectangular configuration. Let us assume the length of this submerged rectangular chamber is denoted as $x_{0}$ along the $x$-axis. The solution for (3.7) within this specific region can be expressed as follows:

(6.13)\begin{align} \left(\begin{array}{@{}c@{}} H_{x}\\ H_{y}\\ H_{z} \end{array}\right) & =\left(\begin{array}{@{}c@{}} 0\\ M_{1}\sin(\lambda_{1}x)+N_{1}\cos(\lambda_{1}x)+M_{2}\sin(\lambda_{2} x)+N_{2}\cos(\lambda_{2}x)\\ M_{1}\cos(\lambda_{1}x)-N_{1}\sin(\lambda_{1}x)+M_{2}\cos(\lambda_{2} x)-N_{2}\sin(\lambda_{2}x) \end{array} \right) \nonumber\\ & \quad + \left(\begin{array}{c} 0\\ M_{3}\sin(\lambda_{3}x)+N_{3}\cos(\lambda_{3}x)+M_{4}\sin(\lambda_{4} x)+N_{4}\cos(\lambda_{4}x)\\ M_{3}\cos(\lambda_{3}x)-N_{3}\sin(\lambda_{3}x)+M_{4}\cos(\lambda_{4} x)-N_{4}\sin(\lambda_{4}x) \end{array} \right). \end{align}

In this context, the symbols $M_{\alpha }$ $( \alpha =1,2,3\textrm { and }4)$ and $N_{\alpha }$ $( \alpha =1,2,3\textrm { and }4)$ signify the amplitude. Upon applying the relevant boundary conditions, namely $\vert B_{y}\vert _{x=x_{0}}=w_{1}$, $\vert B_{z}\vert _{x=x_{0}}=w_{2}$, $\vert (\boldsymbol {\nabla }\times \boldsymbol {B})_{y} \vert _{x=x_{0}}=w_{3}$, $\vert (\boldsymbol {\nabla }\times \boldsymbol {B})_{z}\vert _{x=x_{0}}=w_{4}$, $\vert (( \boldsymbol {\nabla }\times ) ^{2}\boldsymbol {B})_{y}\vert _{x=x_{0}}=w_{5}$, $\vert (( \boldsymbol {\nabla }\times ) ^{2}\boldsymbol {B} )_{z}\vert _{x=x_{0}}=w_{6}$, $\vert (( \boldsymbol {\nabla } \times ) ^{3}\boldsymbol {B})_{y}\vert _{x=x_{0}}=w_{7}$ and $\vert (( \boldsymbol {\nabla }\times ) ^{3}\boldsymbol {B} )_{z}\vert _{x=x_{0}}=w_{8}$, we derive the following system of equations:

(6.14)\begin{align} w_{1} & =M_{1}\sin(\lambda_{1}x_{0})+N_{1}\cos(\lambda_{1}x_{0})+M_{2} \sin(\lambda_{2}x_{0})+N_{2}\cos(\lambda_{2}x_{0})\nonumber\\ & \quad +M_{3}\sin(\lambda_{3}x_{0})+N_{3}\cos(\lambda_{3}x_{0})+M_{4}\sin (\lambda_{4}x_{0})+N_{4}\cos(\lambda_{4}x_{0}), \end{align}

(6.15)\begin{align} w_{2} & =M_{1}\cos(\lambda_{1}x_{0})-N_{1}\sin(\lambda_{1}x_{0})+M_{2} \cos(\lambda_{2}x_{0})-N_{2}\sin(\lambda_{2}x_{0})\nonumber\\ & \quad +M_{3}\cos(\lambda_{3}x_{0})-N_{3}\sin(\lambda_{3}x_{0})+M_{4}\cos (\lambda_{4}x_{0})-N_{4}\sin(\lambda_{4}x_{0}), \end{align}

(6.16)\begin{align} w_{3} & =\lambda_{1}M_{1}\sin(\lambda_{1}x_{0})+\lambda_{1}N_{1}\cos (\lambda_{1}x_{0})+\lambda_{2}M_{2}\sin(\lambda_{2}x_{0})+\lambda_{2}N_{2} \cos(\lambda_{2}x_{0})\nonumber\\ & \quad +\lambda_{3}M_{3}\sin(\lambda_{3}x_{0})+\lambda_{3}N_{3}\cos(\lambda _{3}x_{0})+\lambda_{4}M_{4}\sin(\lambda_{4}x_{0})+\lambda_{4}N_{4}\cos (\lambda_{4}x_{0}), \end{align}

(6.17)\begin{align} w_{4} & =\lambda_{1}M_{1}\cos(\lambda_{1}x_{0})-\lambda_{1}N_{1}\sin (\lambda_{1}x_{0})+\lambda_{2}M_{2}\cos(\lambda_{2}x_{0})-\lambda_{2}N_{2} \sin(\lambda_{2}x_{0})\nonumber\\ & \quad +\lambda_{3}M_{3}\cos(\lambda_{3}x_{0})-\lambda_{3}N_{3}\sin(\lambda _{3}x_{0})+\lambda_{4}M_{4}\cos(\lambda_{4}x_{0})-\lambda_{4}N_{4}\sin (\lambda_{4}x_{0}), \end{align}

(6.18)\begin{align} w_{5} & =\lambda_{1}^{2}M_{1}\sin(\lambda_{1}x_{0})+\lambda_{1}^{2}N_{1} \cos(\lambda_{1}x_{0})+\lambda_{2}^{2}M_{2}\sin(\lambda_{2}x_{0})+\lambda _{2}^{2}N_{2}\cos(\lambda_{2}x_{0})\nonumber\\ & \quad +\lambda_{3}^{2}M_{3}\sin(\lambda_{3}x_{0})+\lambda_{3}^{2}N_{3} \cos(\lambda_{3}x_{0})+\lambda_{4}^{2}M_{4}\sin(\lambda_{4}x_{0})+\lambda _{4}^{2}N_{4}\cos(\lambda_{4}x_{0}), \end{align}

(6.19)\begin{align} w_{6} & =\lambda_{1}^{2}M_{1}\cos(\lambda_{1}x_{0})-\lambda_{1}^{2}N_{1} \sin(\lambda_{1}x_{0})+\lambda_{2}^{2}M_{2}\cos(\lambda_{2}x_{0})-\lambda _{2}^{2}N_{2}\sin(\lambda_{2}x_{0})\nonumber\\ & \quad +\lambda_{3}^{2}M_{3}\cos(\lambda_{3}x_{0})-\lambda_{3}^{2}N_{3} \sin(\lambda_{3}x_{0})+\lambda_{4}^{2}M_{4}\cos(\lambda_{4}x_{0})-\lambda _{4}^{2}N_{4}\sin(\lambda_{4}x_{0}), \end{align}

(6.20)\begin{align} w_{7} & =\lambda_{1}^{3}M_{1}\sin(\lambda_{1}x_{0})+\lambda_{1}^{3}N_{1} \cos(\lambda_{1}x_{0})+\lambda_{2}^{3}M_{2}\sin(\lambda_{2}x_{0})+\lambda _{2}^{3}N_{2}\cos(\lambda_{2}x_{0})\nonumber\\ & \quad +\lambda_{3}^{3}M_{3}\sin(\lambda_{3}x_{0})+\lambda_{3}^{3}N_{3} \cos(\lambda_{3}x_{0})+\lambda_{4}^{3}M_{4}\sin(\lambda_{4}x_{0})+\lambda _{4}^{3}N_{4}\cos(\lambda_{4}x_{0}), \end{align}

(6.21)\begin{align} w_{8} & =\lambda_{1}^{3}M_{1}\cos(\lambda_{1}x_{0})-\lambda_{1}^{3}N_{1} \sin(\lambda_{1}x_{0})+\lambda_{2}^{3}M_{2}\cos(\lambda_{2}x_{0})-\lambda _{2}^{3}N_{2}\sin(\lambda_{2}x_{0})\nonumber\\ & \quad +\lambda_{3}^{3}M_{3}\cos(\lambda_{3}x_{0})-\lambda_{3}^{3}N_{3} \sin(\lambda_{3}x_{0})+\lambda_{4}^{3}M_{4}\cos(\lambda_{4}x_{0})-\lambda _{4}^{3}N_{4}\sin(\lambda_{4}x_{0}). \end{align}

Following certain algebraic manipulations, we acquire the subsequent set of equations:

(6.22)\begin{equation} \left.\begin{gathered} M_{1} =\frac{Q_{1}}{Q_{9}},\quad M_{2}=\frac{Q_{2} }{Q_{10}},\quad M_{3}=\frac{Q_{3}}{Q_{11}},\quad M_{4}=\frac{Q_{4}}{Q_{12}},\\ N_{1} =\frac{Q_{5}}{Q_{9}},\quad N_{2}=\frac{Q_{6} }{Q_{10}},\quad N_{3}=\frac{Q_{7}}{Q_{11}},\quad N_{4}=\frac{Q_{8}}{Q_{12}}, \end{gathered}\right\} \end{equation}

where

(6.23)\begin{gather}\begin{aligned} Q_{1} & =\sin(\lambda_{1}x_{0})\left[ \begin{array}{c} \left( \lambda_{2}-\lambda_{4}\right) \left( w_{7}-w_{3}\lambda_{4} ^{2}+\left( w_{1}\lambda_{3}\lambda_{4}-w_{3}\lambda_{3}\right) \left( \lambda_{3}+\lambda_{4}\right) \right) \\ -\left( w_{5}-w_{3}\lambda_{3}-w_{3}\lambda_{4}+w_{1}\lambda_{3}\lambda _{4}\right) \left( \lambda_{2}^{2}+\lambda_{2}\lambda_{3}-\lambda_{3} \lambda_{4}-\lambda_{4}^{2}\right) \end{array} \right] \nonumber\\ & \quad +\cos(\lambda_{1}x_{0})\left[ \begin{array}{c} \left( \lambda_{2}-\lambda_{4}\right) \left( w_{8}-w_{4}\lambda_{4} ^{2}+\left( w_{2}\lambda_{3}\lambda_{4}-w_{4}\lambda_{3}\right) \left( \lambda_{3}+\lambda_{4}\right) \right) \\ -\left( w_{6}-w_{4}\lambda_{3}-w_{4}\lambda_{4}+w_{2}\lambda_{3}\lambda _{4}\right) \left( \lambda_{2}^{2}+\lambda_{2}\lambda_{3}-\lambda_{3} \lambda_{4}-\lambda_{4}^{2}\right) \end{array} \right] , \end{aligned} \end{gather}

(6.24)\begin{gather}\begin{aligned} Q_{2} & =\sin(\lambda_{2}x_{0})\left[ \begin{array}{c} \left( \lambda_{1}-\lambda_{4}\right) \left( w_{7}-w_{3}\lambda_{4} ^{2}+\left( w_{1}\lambda_{3}\lambda_{4}-w_{3}\lambda_{3}\right) \left( \lambda_{3}+\lambda_{4}\right) \right) \\ -\left( w_{5}-w_{3}\lambda_{3}-w_{3}\lambda_{4}+w_{1}\lambda_{3}\lambda _{4}\right) \left( \lambda_{1}^{2}+\lambda_{1}\lambda_{3}-\lambda_{3} \lambda_{4}-\lambda_{4}^{2}\right) \end{array} \right] \nonumber\\ & \quad +\cos(\lambda_{2}x_{0})\left[ \begin{array}{c} \left( \lambda_{1}-\lambda_{4}\right) \left( w_{8}-w_{4}\lambda_{4} ^{2}+\left( w_{2}\lambda_{3}\lambda_{4}-w_{4}\lambda_{3}\right) \left( \lambda_{3}+\lambda_{4}\right) \right) \\ -\left( w_{6}-w_{4}\lambda_{3}-w_{4}\lambda_{4}+w_{2}\lambda_{3}\lambda _{4}\right) \left( \lambda_{1}^{2}+\lambda_{1}\lambda_{3}-\lambda_{3} \lambda_{4}-\lambda_{4}^{2}\right) \end{array} \right] , \end{aligned}\end{gather}

(6.25)\begin{gather}\begin{aligned} Q_{3} & =\sin(\lambda_{3}x_{0})\left[ \begin{array}{c} \left( \lambda_{4}-\lambda_{2}\right) \left( w_{7}-w_{3}\lambda_{2} ^{2}+\left( w_{1}\lambda_{1}\lambda_{2}-w_{3}\lambda_{1}\right) \left( \lambda_{2}+\lambda_{1}\right) \right) \\ -\left( w_{5}-w_{3}\lambda_{1}-w_{3}\lambda_{2}+w_{1}\lambda_{1}\lambda _{2}\right) \left( \lambda_{4}^{2}+\lambda_{1}\lambda_{4}-\lambda_{1} \lambda_{2}-\lambda_{2}^{2}\right) \end{array} \right] \nonumber\\ & \quad +\cos(\lambda_{3}x_{0})\left[ \begin{array}{c} \left( \lambda_{4}-\lambda_{2}\right) \left( w_{8}-w_{4}\lambda_{2} ^{2}+\left( w_{2}\lambda_{1}\lambda_{2}-w_{4}\lambda_{1}\right) \left( \lambda_{2}+\lambda_{1}\right) \right) \\ -\left( w_{6}-w_{4}\lambda_{1}-w_{4}\lambda_{2}+w_{2}\lambda_{1}\lambda _{2}\right) \left( \lambda_{4}^{2}+\lambda_{1}\lambda_{4}-\lambda_{1} \lambda_{2}-\lambda_{2}^{2}\right) \end{array} \right], \end{aligned}\end{gather}

(6.26)\begin{gather}\begin{aligned} Q_{4} & =\sin(\lambda_{4}x_{0})\left[ \begin{array}{c} \left( \lambda_{3}-\lambda_{2}\right) \left( w_{7}-w_{3}\lambda_{2} ^{2}+\left( w_{1}\lambda_{1}\lambda_{2}-w_{3}\lambda_{1}\right) \left( \lambda_{2}+\lambda_{1}\right) \right) \\ -\left( w_{5}-w_{3}\lambda_{1}-w_{3}\lambda_{2}+w_{1}\lambda_{1}\lambda _{2}\right) \left( \lambda_{3}^{2}+\lambda_{1}\lambda_{3}-\lambda_{1} \lambda_{2}-\lambda_{2}^{2}\right) \end{array} \right] \nonumber\\ & \quad +\cos(\lambda_{4}x_{0})\left[ \begin{array}{c} \left( \lambda_{3}-\lambda_{2}\right) \left( w_{8}-w_{4}\lambda_{2} ^{2}+\left( w_{2}\lambda_{1}\lambda_{2}-w_{4}\lambda_{1}\right) \left( \lambda_{2}+\lambda_{1}\right) \right) \\ -\left( w_{6}-w_{4}\lambda_{1}-w_{4}\lambda_{2}+w_{2}\lambda_{1}\lambda _{2}\right) \left( \lambda_{3}^{2}+\lambda_{1}\lambda_{3}-\lambda_{1} \lambda_{2}-\lambda_{2}^{2}\right) \end{array} \right] , \end{aligned}\end{gather}

(6.27)\begin{gather}\begin{aligned} Q_{5} & =\cos(\lambda_{1}x_{0})\left[ \begin{array}{c} \left( \lambda_{2}-\lambda_{4}\right) \left( w_{7}-w_{3}\lambda_{4} ^{2}+\left( w_{1}\lambda_{3}\lambda_{4}-w_{3}\lambda_{3}\right) \left( \lambda_{3}+\lambda_{4}\right) \right) \\ -\left( w_{5}-w_{3}\lambda_{3}-w_{3}\lambda_{4}+w_{1}\lambda_{3}\lambda _{4}\right) \left( \lambda_{2}^{2}+\lambda_{2}\lambda_{3}-\lambda_{3} \lambda_{4}-\lambda_{4}^{2}\right) \end{array} \right] \nonumber\\ & \quad -\sin(\lambda_{1}x_{0})\left[ \begin{array}{c} \left( \lambda_{2}-\lambda_{4}\right) \left( w_{8}-w_{4}\lambda_{4} ^{2}+\left( w_{2}\lambda_{3}\lambda_{4}-w_{4}\lambda_{3}\right) \left( \lambda_{3}+\lambda_{4}\right) \right) \\ -\left( w_{6}-w_{4}\lambda_{3}-w_{4}\lambda_{4}+w_{2}\lambda_{3}\lambda _{4}\right) \left( \lambda_{2}^{2}+\lambda_{2}\lambda_{3}-\lambda_{3} \lambda_{4}-\lambda_{4}^{2}\right) \end{array} \right] , \end{aligned}\end{gather}

(6.28)\begin{gather}\begin{aligned} Q_{6} & =\cos(\lambda_{2}x_{0})\left[ \begin{array}{c} \left( \lambda_{1}-\lambda_{4}\right) \left( w_{7}-w_{3}\lambda_{4} ^{2}+\left( w_{1}\lambda_{3}\lambda_{4}-w_{3}\lambda_{3}\right) \left( \lambda_{3}+\lambda_{4}\right) \right) \\ -\left( w_{5}-w_{3}\lambda_{3}-w_{3}\lambda_{4}+w_{1}\lambda_{3}\lambda _{4}\right) \left( \lambda_{1}^{2}+\lambda_{1}\lambda_{3}-\lambda_{3} \lambda_{4}-\lambda_{4}^{2}\right) \end{array} \right] \nonumber\\ & \quad -\sin(\lambda_{2}x_{0})\left[ \begin{array}{c} \left( \lambda_{1}-\lambda_{4}\right) \left( w_{8}-w_{4}\lambda_{4} ^{2}+\left( w_{2}\lambda_{3}\lambda_{4}-w_{4}\lambda_{3}\right) \left( \lambda_{3}+\lambda_{4}\right) \right) \\ -\left( w_{6}-w_{4}\lambda_{3}-w_{4}\lambda_{4}+w_{2}\lambda_{3}\lambda _{4}\right) \left( \lambda_{1}^{2}+\lambda_{1}\lambda_{3}-\lambda_{3} \lambda_{4}-\lambda_{4}^{2}\right) \end{array} \right] , \end{aligned}\end{gather}

(6.29)\begin{gather}\begin{aligned} Q_{7} & =\cos(\lambda_{3}x_{0})\left[ \begin{array}{c} \left( \lambda_{4}-\lambda_{2}\right) \left( w_{7}-w_{3}\lambda_{2} ^{2}+\left( w_{1}\lambda_{1}\lambda_{2}-w_{3}\lambda_{1}\right) \left( \lambda_{2}+\lambda_{1}\right) \right) \\ -\left( w_{5}-w_{3}\lambda_{1}-w_{3}\lambda_{2}+w_{1}\lambda_{1}\lambda _{2}\right) \left( \lambda_{4}^{2}+\lambda_{1}\lambda_{4}-\lambda_{1} \lambda_{2}-\lambda_{2}^{2}\right) \end{array} \right] \nonumber\\ & \quad -\sin(\lambda_{3}x_{0})\left[ \begin{array}{c} \left( \lambda_{4}-\lambda_{2}\right) \left( w_{8}-w_{4}\lambda_{2} ^{2}+\left( w_{2}\lambda_{1}\lambda_{2}-w_{4}\lambda_{1}\right) \left( \lambda_{2}+\lambda_{1}\right) \right) \\ -\left( w_{6}-w_{4}\lambda_{1}-w_{4}\lambda_{2}+w_{2}\lambda_{1}\lambda _{2}\right) \left( \lambda_{4}^{2}+\lambda_{1}\lambda_{4}-\lambda_{1} \lambda_{2}-\lambda_{2}^{2}\right) \end{array} \right] , \end{aligned}\end{gather}

(6.30)\begin{gather}\begin{aligned} Q_{8} & =\cos(\lambda_{4}x_{0})\left[ \begin{array}{c} \left( \lambda_{3}-\lambda_{2}\right) \left( w_{7}-w_{3}\lambda_{2} ^{2}+\left( w_{1}\lambda_{1}\lambda_{2}-w_{3}\lambda_{1}\right) \left( \lambda_{2}+\lambda_{1}\right) \right) \\ -\left( w_{5}-w_{3}\lambda_{1}-w_{3}\lambda_{2}+w_{1}\lambda_{1}\lambda _{2}\right) \left( \lambda_{3}^{2}+\lambda_{1}\lambda_{3}-\lambda_{1} \lambda_{2}-\lambda_{2}^{2}\right) \end{array} \right] \nonumber\\ & \quad -\sin(\lambda_{4}x_{0})\left[ \begin{array} [c]{c} \left( \lambda_{3}-\lambda_{2}\right) \left( w_{8}-w_{4}\lambda_{2} ^{2}+\left( w_{2}\lambda_{1}\lambda_{2}-w_{4}\lambda_{1}\right) \left( \lambda_{2}+\lambda_{1}\right) \right) \\ -\left( w_{6}-w_{4}\lambda_{1}-w_{4}\lambda_{2}+w_{2}\lambda_{1}\lambda _{2}\right) \left( \lambda_{3}^{2}+\lambda_{1}\lambda_{3}-\lambda_{1} \lambda_{2}-\lambda_{2}^{2}\right) \end{array} \right] , \end{aligned}\end{gather}

(6.31)\begin{gather} Q_{9} =\left( \lambda_{1}-\lambda_{3}\right) \left( \lambda_{1} ^{2}\lambda_{2}-\lambda_{1}\lambda_{2}^{2}+\lambda_{2}^{2}\lambda_{4} -\lambda_{2}\lambda_{4}^{2}+\lambda_{1}\lambda_{4}^{2}-\lambda_{1}^{2} \lambda_{4}\right) , \end{gather}

(6.32)\begin{gather}Q_{10} =\left( \lambda_{2}-\lambda_{3}\right) \left( \lambda_{2} ^{2}\lambda_{1}-\lambda_{2}\lambda_{1}^{2}+\lambda_{1}^{2}\lambda_{4} -\lambda_{1}\lambda_{4}^{2}+\lambda_{2}\lambda_{4}^{2}-\lambda_{2}^{2} \lambda_{4}\right) , \end{gather}

(6.33)\begin{gather}Q_{11} =\left( \lambda_{3}-\lambda_{1}\right) \left( \lambda_{2} ^{2}\lambda_{3}-\lambda_{2}\lambda_{3}^{2}-\lambda_{2}^{2}\lambda_{4} +\lambda_{2}\lambda_{4}^{2}-\lambda_{3}\lambda_{4}^{2}+\lambda_{3}^{2} \lambda_{4}\right) , \end{gather}

(6.34)\begin{gather}Q_{12} =\left( \lambda_{4}-\lambda_{1}\right) \left( \lambda_{3} ^{2}\lambda_{2}-\lambda_{3}\lambda_{2}^{2}+\lambda_{2}^{2}\lambda_{4} -\lambda_{2}\lambda_{4}^{2}+\lambda_{3}\lambda_{4}^{2}-\lambda_{3}^{2} \lambda_{4}\right) . \end{gather}

In this arrangement, a greater number of boundary conditions is used compared with the uncomplicated slab geometry. These additional conditions introduce an additional degree of freedom, providing a more effective means of system control. The entire system can be managed with increased efficiency. Modifying the magnetic field, which is contingent on the Beltrami parameter and density ratio, within the inner slab allows for straightforward control over the structure's characteristics. Adjusting the length of the inner slab can reduce the pressure inside. In contrast, in the case of the simple slab geometry, the application of additional boundary conditions is not feasible, resulting in challenges in controlling the nature of the relaxed structure at the system's centre.

7 Numerical results

The inherent magnetic field is a fundamental characteristic of a planet. Beyond influencing the electromagnetic conditions surrounding a planetary body, it acts as a crucial indicator of the internal structure and dynamics of the host planet (Cao et al. Reference Cao, Dougherty, Hunt, Provan, Cowley, Bunce, Kellock and Stevenson2020). The presence of a robust planetary-scale magnetic field is likely the result of dynamo action occurring within the planet. This process requires a large amount of electrically conductive fluid and involves rapid and intricate-moving fluid motions within the planet's interior. These movements help generate and maintain the magnetic field. In this current study, our focus is on investigating the dynamo action within Saturn. To achieve this, our numerical investigation relies on observational data from Saturn's E-ring (Wahlund et al. Reference Wahlund2009; Shohaib et al. Reference Shohaib, Masood, Jahangir, Siddiq, Alkhateeb and El-Tantawy2022), with the plasma parameters having the following specified values: $m_{d}=4\times 10^{-15}\ \textrm {kg}$, $n_{s_{e} }=(2 - 7)\times 10^{7}\, \textrm{m}^{-3}$, $n_{s_{d}}=10^{4} - 10^{5}\, \textrm{m}^{-3}$ and $Z_{d}=10^{2}$. In the present work, the velocities were normalized by a characteristic velocity, such as the Alfvén velocity $V_{A}$, and the distances were normalized by the electron skin depth $\lambda _{s_{e}}$. Thus, the plotted values are dimensionless.

7.1 Plasma flow and dynamo action

Dynamo processes are now widely accepted as the primary origin of magnetic fields in many astrophysical settings. This section will explore the features of the magnetic field and flow field of the relaxed structures within Saturn's E-ring a three-component dusty plasma (Wahlund et al. Reference Wahlund2009; Shohaib et al. Reference Shohaib, Masood, Jahangir, Siddiq, Alkhateeb and El-Tantawy2022), considering both configurations – simple rectangular geometry and coplanar rectangular geometry. The analysis will primarily focus on examining the impact of Beltrami parameters on the characteristics of self-organized structures. The graphs are generated with a fixed value of $x=5$, representing the length of the rectangle, in the case of simple rectangular geometry. Meanwhile, for coplanar rectangular geometry, the graphs are plotted with $x_{0}=2$, representing the length of the inner rectangular slab along the $x$-axis, and $x=5$, representing the length of the outer rectangular slab along the $x$-axis. Figures 1–2 exhibit the fluctuations in the magnetic and velocity profiles concerning the Beltrami parameter $a_{s_{e}}=25.7$, $a_{s_{i}}=0.1$ and $a_{s_{d}}=1.0$ and figures 3–4 are plotted for a different set of Beltrami parameters $a_{s_{e}}=0.7$, $a_{s_{i}}=1.4$ and $a_{s_{d}}=50.9$. Figures 1 and 3 depicting the magnetic field and velocity profiles are presented for the solution of the quadruple Beltrami state within a straightforward rectangular configuration and figures 2–4 illustrating the magnetic field and velocity profiles are generated to showcase the solution of the quadruple Beltrami state in a rectangular configuration featuring an internal rectangular slab.

Figure 1.

Magnetic field and velocity profiles are plotted for the solution of quadruple Beltrami state in a simple rectangular configuration. The Beltrami parameters are $a_{s_{e}}=25.7$, $a_{s_{i}}=0.1$ and $a_{s_{d}}=1.0$. The associated scale parameters are $\lambda _{1}=25.661$, $\lambda _{2}=0.050553$, $\lambda _{3}=1.0$ and $\lambda _{4}=0.0884166$.

Figure 2.

Magnetic field and Velocity profiles are plotted for the solution of quadruple Beltrami state in a rectangular configuration with an internal conductor. The Beltrami parameters are $a_{s_{e}}=25.7$, $a_{s_{i}}=0.1$ and $a_{s_{d}}=1.0$. The associated scale parameters are $\lambda _{1}=25.661$, $\lambda _{2}=0.050553$, $\lambda _{3}=1.0$ and $\lambda _{4}=0.0884166$.

Figure 3.

Magnetic field and velocity profiles are plotted for the solution of quadruple Beltrami state in a simple rectangular configuration. The Beltrami parameters are $a_{s_{e}}=0.7$, $a_{s_{i}}=1.4$ and $a_{s_{d}}=50.9$. The associated scale parameters are $\lambda _{1}=50.9$, $\lambda _{2}=1.3998$, $\lambda _{3}=0.350101+0.936942i$ and $\lambda _{4}=0.350101-0.936942i$.

Figure 4.

Magnetic field and velocity profiles are plotted for the solution of quadruple Beltrami state in a rectangular configuration with an internal conductor. The Beltrami parameters are $a_{s_{e}}=0.7$, $a_{s_{i}}=1.4$ and $a_{s_{d}}=50.9$. The associated scale parameters are $\lambda _{1}=50.9$, $\lambda _{2}=1.3998$, $\lambda _{3}=0.350101+0.936942i$ and $\lambda _{4}=0.350101-0.936942i$.

The magnetic field and flow field behave differently in these two cases, indicating distinct dynamo mechanisms.

7.1.1 Saturn's magnetosphere

Magnetic field < flow field: in this scenario, shown in figures 1 and 3, the magnetic field is weaker than the flow field. This suggests the presence of a fast dynamo. In such environments, where the flow dominates the magnetic field, turbulent or rapid fluid motions can stretch, fold and amplify the magnetic field efficiently, which is characteristic of a fast dynamo (Vainshtein & Rosner Reference Vainshtein and Rosner1991). Although altering the Beltrami parameters from $a_{s_{e} }=25.7$, $a_{s_{i}}=0.1$ and $a_{s_{d}}=1.0$ to $a_{s_{e}}=0.7$, $a_{s_{i} }=1.4$ and $a_{s_{d}}=50.9$ results in a transformation of the corresponding scale parameters from real $\lambda _{1}=25.661$, $\lambda _{2}=0.050553$, $\lambda _{3}=1.0$ and $\lambda _{4}=0.0884166$ to complex conjugate $\lambda _{1}=50.9$, $\lambda _{2}=1.3998$, $\lambda _{3}=0.350101+0.936942i$ and $\lambda _{4}=0.350101-0.936942i$, while the inherent characteristics of the magnetic and flow fields remain unchanged.

7.1.2 Saturn's E ring

Magnetic field > flow field: in this case, illustrated in figures 2 and 4, the magnetic field is stronger than the flow field, which may favour a slow dynamo (Soward Reference Soward1990). A stronger magnetic field leads to more organized and less turbulent interactions with the flow, resulting in slower amplification of the magnetic field. Figures 2 and 4 are plotted using the same Beltrami parameters as figures 1 and 3, respectively. Moreover, the magnetic field in figure 4 demonstrates a scale significantly exceeding that of the driving flow, it identifies a large-scale dynamo (Brandenburg Reference Brandenburg2009). The field strength is significantly higher here, not only due to the influence of the flow field, but also because of the internal conductor, resulting in a much stronger magnetic field than in the simple rectangular geometry. It indicates that altering the Beltrami parameters in the scenario of a rectangular configuration with an internal conductor can result in the dynamo transitioning between fast dynamo and slow dynamo, or vice versa.

7.2 Velocities profiles of the plasma species

Within this section, we showcase the velocity profiles of distinct plasma constituents – namely, electrons, ions and negatively charged dust particles – along with the unified velocity. The velocity profiles for electrons $\boldsymbol {V}_{s_{e}}$, ions $\boldsymbol {V}_{s_{i}}$ and negatively charged dust particles $\boldsymbol {V}_{s_{d}}$ are graphed in accordance with (3.5), (2.11) and (3.6), respectively. Meanwhile, the unified flow is illustrated through the useof (4.4). The analysis of the graphs is conducted for both solutions, involving a simple rectangular geometry and a rectangular configuration with an internal conductor. This examination involves the manipulation of Beltrami parameters and the densities of plasma species.

7.2.1 Influence of Beltrami parameters

Figures 5–6 are generated using identical Beltrami parameters $a_{s_{e}}=0.7$, $a_{s_{i}}=1.5$ and $a_{s_{d}}=50.9$. In figure 5, velocity profiles of plasma species are illustrated within a simple rectangular geometry, while figure 6 depicts the velocities of plasma components within a rectangular configuration featuring an internal conductor. The corresponding four scale parameters are denoted as $\lambda _{1}=50.9$, $\lambda _{2}=1.49979$, $\lambda _{3}=0.350104+0.9369271i$ and $\lambda _{4}=0.350104-0.9369271i$, comprising two real values and a pair of complex conjugates. Same Beltrami parameters $a_{s_{e}}=2.7$, $a_{s_{i}}=1.3$ and $a_{s_{d}}=0.9$ are employed in the creation of both figures 7 and 8. In figure 7, we showcase the velocity profiles of plasma species in a simple rectangular geometry, while figure 8 reveals the velocities of plasma components within a rectangular set-up featuring an internal conductor. The corresponding four scale parameters are articulated as $\lambda _{1}=2.25706$, $\lambda _{2}=0.443913$, $\lambda _{3}=1.29902$ and $\lambda _{4}=0.9$, all of which possess real values. In the scenario of a rectangular geometry with an internal conductor, it has been noted that when all the scale parameters, which are dependent on Beltrami parameters, are real, the unified flow reaches its peak magnitude (figure 8). Conversely, when the scale parameters form a pair of complex conjugates, the unified flow exhibits its minimum intensity (figure 6). This trend is reversed in the case of a simple rectangular geometry. This indicates that the Beltrami parameters exert a significant influence on the velocity of the plasma system and its constituent components.

Figure 5.

Velocities (velocity of dust particle $\boldsymbol {V}_{s_{d}}$, electrons $\boldsymbol {V}_{s_{e}}$, ions $\boldsymbol {V}_{s_{i}}$ and composite velocity $\boldsymbol {V}$) profiles are plotted for the solution of quadruple Beltrami state in a simple rectangular configuration. The Beltrami parameters are $a_{s_{e}}=0.7$, $a_{s_{i}}=1.5$ and $a_{s_{d}}=50.9$. The associated scale parameters are $\lambda _{1}=50.9$, $\lambda _{2}=1.49979$, $\lambda _{3}=0.350104+0.9369271i$ and $\lambda _{4}=0.350104-0.9369271i$.

Figure 6.

Velocities (velocity of dust particle $\boldsymbol {V}_{s_{d}}$, electrons $\boldsymbol {V}_{s_{e}}$, ions $\boldsymbol {V}_{s_{i}}$ and composite velocity $\boldsymbol {V}$) profiles are plotted for the solution of quadruple Beltrami state in a rectangular configuration with an internal conductor. The Beltrami parameters are $a_{s_{e}}=0.7$, $a_{s_{i}}=1.5$ and $a_{s_{d}}=50.9$. The associated scale parameters are $\lambda _{1}=50.9$, $\lambda _{2}=1.49979$, $\lambda _{3}=0.350104+0.9369271i$ and $\lambda _{4}=0.350104-0.9369271i$.

Figure 7.

Velocities (velocity of dust particle $\boldsymbol {V}_{s_{d}}$, electrons $\boldsymbol {V}_{s_{e}}$, ions $\boldsymbol {V}_{s_{i}}$ and composite velocity $\boldsymbol {V}$) profiles are plotted for the solution of quadruple Beltrami state in a simple rectangular configuration. The Beltrami parameters are $a_{s_{e}}=2.7$, $a_{s_{i}}=1.3$ and $a_{s_{d}}=0.9$. The associated scale parameters are $\lambda _{1}=2.25706$, $\lambda _{2}=0.443913$, $\lambda _{3}=1.29902$ and $\lambda _{4}=0.9$.

Figure 8.

Velocities (velocity of dust particle $\boldsymbol {V}_{s_{d}}$, electrons $\boldsymbol {V}_{s_{e}}$, ions $\boldsymbol {V}_{s_{i}}$ and composite velocity $\boldsymbol {V}$) profiles are plotted for the solution of the quadruple Beltrami state in a rectangular configuration with an internal conductor. The Beltrami parameters are $a_{s_{e}}=2.7$, $a_{s_{i}}=1.3$ and $a_{s_{d}}=0.9$. The associated scale parameters are $\lambda _{1}=2.25706$, $\lambda _{2}=0.443913$, $\lambda _{3}=1.29902$ and $\lambda _{4}=0.9$.

7.2.2 Influence of plasma species densities

In the atmosphere of Saturn, the densities of plasma species like electrons, ions and dust particles exhibit non-uniformity, fluctuating with changes in the magnetosphere's radius. Consequently, the velocities of plasma particles are significantly impacted by the density ratios among these species. Figures 9–18 depict variations in both the unified velocity and individual velocities of plasma particles, highlighting the influence of changing plasma particle densities and keeping constant all other parameters. Figures 9–13 are generated based on the quadruple Beltrami state solution within a basic rectangular geometry, while figures 14–18 are plotted from the solution of the quadruple Beltrami state in a rectangular configuration featuring an internal conductor. The examination of these graphs reveals that the magnitudes of velocities for smaller species such as electrons and ions remain constant when altering the densities of plasma species in both configurations – whether in a simple rectangular geometry or in a rectangular geometry with an internal conductor. Conversely, the velocity magnitude of dust particles is markedly influenced by the density ratio of plasma particles. It reaches its maximum, as shown in figures 9 and 14, when electron density is at its highest, $n_{s_{e}}=2\times 10^{9}$, and the density of negatively charged dust particles is at its lowest, $n_{s_{d}}=10$. Conversely, its magnitude diminishes shown in figures 13 and 18 when electron density decreases, $n_{s_{e}}=2\times 10^{6}$,and the density of dust particles increases, $n_{s_{d}}=10^{5}$. Since the unified velocity is a composite of the velocities velocities of electrons, ions and dust particles, the unified velocity attains its maximum value when the velocity strength of dust particles is at its peak, and conversely, it decreases when the velocity of dust particles diminishes. Furthermore, the total magnitude of the unified velocity is higher in the scenario of a rectangular geometry with an internal conductor compared with the velocity strength in a simple rectangular configuration. This observation indicates that the presence of an internal conductor has the potential to augment the velocity's intensity. This insight holds significance in elucidating the velocity dynamics of the plasma system within the Saturn magnetosphere and its rings. The trend of the magnetic profiles are strongly influenced by changing the Beltrami parameters or by varying the density of the dust particles in a simple rectangular geometry. In the above discussion for a simple rectangular configuration, it has been also observed that the associated scale parameters are real for the paramagnetic trend, but for complex scale parameters, the magnetic field becomes maximum moving away from the centre, which manifests the diamagnetic trend. This shows a clear distinction of the both geometries.

Figure 9.

Velocities (velocity of dust particle $\boldsymbol {V}_{s_{d}}$, electrons $\boldsymbol {V}_{s_{e}}$, ions $\boldsymbol {V}_{s_{i}}$ and composite velocity $\boldsymbol {V}$) profiles are plotted for the solution of quadruple Beltrami state in a simple rectangular configuration. The densities of plasma species are $n_{s_{e}}=2\times 10^{9}$, $n_{s_{i}}=2\times 10^{9}$, $n_{s_{d}}=10^{1}$ and $z_{d}=10^{2}$.

Figure 10.

Velocities (velocity of dust particle $\boldsymbol {V}_{s_{d}}$, electrons $\boldsymbol {V}_{s_{e}}$, ions $\boldsymbol {V}_{s_{i}}$ and composite velocity $\boldsymbol {V}$) profiles are plotted for the solution of the quadruple Beltrami state in a simple rectangular configuration. The density of plasma species are $n_{s_{e}}=2\times 10^{9}$, $n_{s_{i}}=2\times 10^{9}$, $n_{s_{d}}=10^{2}$ and $z_{d}=10^{2}$.

Figure 11.

Velocities (velocity of dust particle $\boldsymbol {V}_{s_{d}}$, electrons $\boldsymbol {V}_{s_{e}}$, ions $\boldsymbol {V}_{s_{i}}$ and composite velocity $\boldsymbol {V}$) profiles are plotted for the solution of the quadruple Beltrami state in a simple rectangular configuration. The density of plasma species are $n_{s_{e}}=2\times 10^{8}$, $n_{s_{i}}=2\times 10^{8}$, $n_{s_{d}}=10^{3}$ and $z_{d}=10^{2}$.

Figure 12.

Velocities (velocity of dust particle $\boldsymbol {V}_{s_{d}}$, electrons $\boldsymbol {V}_{s_{e}}$, ions $\boldsymbol {V}_{s_{i}}$ and composite velocity $\boldsymbol {V}$) profiles are plotted for the solution of the quadruple Beltrami state in a simple rectangular configuration. The density of plasma species are $n_{s_{e}}=2\times 10^{7}$, $n_{s_{i}}=2\times 10^{7}$, $n_{s_{d}}=10^{4}$ and $z_{d}=10^{2}$.

Figure 13.

Velocities (velocity of dust particle $\boldsymbol {V}_{s_{d}}$, electrons $\boldsymbol {V}_{s_{e}}$, ions $\boldsymbol {V}_{s_{i}}$ and composite velocity $\boldsymbol {V}$) profiles are plotted for the solution of the quadruple Beltrami state in a simple rectangular configuration. The density of plasma species are $n_{s_{e}}=2\times 10^{6}$, $n_{s_{i}}=2\times 10^{7}$, $n_{s_{d}}=10^{5}$ and $z_{d}=10^{2}$.

Figure 14.

Velocities (velocity of dust particle $\boldsymbol {V}_{s_{d}}$, electrons $\boldsymbol {V}_{s_{e}}$, ions $\boldsymbol {V}_{s_{i}}$ and composite velocity $\boldsymbol {V}$) profiles are plotted for the solution of the quadruple Beltrami state in a rectangular configuration with an internal conductor. The density of plasma species are $n_{s_{e}}=2\times 10^{9}$, $n_{s_{i}}=2\times 10^{9}$, $n_{s_{d}}=10^{1}$ and $z_{d}=10^{2}$.

Figure 15.

Velocities (velocity of dust particle $\boldsymbol {V}_{s_{d}}$, electrons $\boldsymbol {V}_{s_{e}}$, ions $\boldsymbol {V}_{s_{i}}$ and composite velocity $\boldsymbol {V}$) profiles are plotted for the solution of the quadruple Beltrami state in a rectangular configuration with an internal conductor. The density of plasma species are $n_{s_{e}}=2\times 10^{9}$, $n_{s_{i}}=2\times 10^{9}$, $n_{s_{d}}=10^{2}$ and $z_{d}=10^{2}$.

Figure 16.

Velocities (velocity of dust particle $\boldsymbol {V}_{s_{d}}$, electrons $\boldsymbol {V}_{s_{e}}$, ions $\boldsymbol {V}_{s_{i}}$ and composite velocity $\boldsymbol {V}$) profiles are plotted for the solution of the quadruple Beltrami state in a rectangular configuration with an internal conductor. The density of plasma species are $n_{s_{e}}=2\times 10^{8}$, $n_{s_{i}}=2\times 10^{8}$, $n_{s_{d}}=10^{3}$ and $z_{d}=10^{2}$.

Figure 17.

Velocities (velocity of dust particle $\boldsymbol {V}_{s_{d}}$, electrons $\boldsymbol {V}_{s_{e}}$, ions $\boldsymbol {V}_{s_{i}}$ and composite velocity $\boldsymbol {V}$) profiles are plotted for the solution of the quadruple Beltrami state in a rectangular configuration with an internal conductor. The density of plasma species are $n_{s_{e}}=2\times 10^{7}$, $n_{s_{i}}=2\times 10^{7}$, $n_{s_{d}}=10^{4}$ and $z_{d}=10^{2}$.

Figure 18.

Velocities (velocity of dust particle $\boldsymbol {V}_{s_{d}}$, electrons $\boldsymbol {V}_{s_{e}}$, ions $\boldsymbol {V}_{s_{i}}$ and composite velocity $\boldsymbol {V}$) profiles are plotted for the solution of the quadruple Beltrami state in a rectangular configuration with an internal conductor. The density of plasma species are $n_{s_{e}}=2\times 10^{6}$, $n_{s_{i}}=2\times 10^{7}$, $n_{s_{d}}=10^{5}$ and $z_{d}=10^{2}$.

8 Concluding remarks

The relaxation phenomenon in a magnetized three-fluid dusty plasma, consisting of negatively charged dust particles, electrons and ions, has been observed. The Beltrami conditions are attained by solving the momentum-balanced equations for the three components alongside Ampere's law. A quadruple Beltrami state is derived by considering three Beltrami parameters. The helicities of dust particles, electrons and ions, along with the total energy, serve as the four constraints in the current plasma system. The quadruple curl relaxed Beltrami state is a composite of four distinct single relaxed Beltrami states, resulting in the system having four unique eigenvalues. The solutions for the relaxed state have been obtained in two distinct configurations. One corresponds to a simple rectangular geometry, and the other involves a rectangular geometry featuring an inner rectangular slab acting as an internal conductor. We have established relationships describing the unified flow and derived the individual flows of plasma particles, including electrons, ions and negatively charged dust particles. Dynamo action has been identified in the Saturn plasma, using real plasma parameters from Saturn itself. Dynamo action is discerned by varying the Beltrami parameters, exhibiting small-scale dynamo behaviour in a simple rectangular geometry and generating large-scale dynamo effects in a rectangular configuration with an internal conductor. Additionally, it has been observed that the overall strength of the magnetic field reaches its maximum in the presence of an internal conductor in the rectangular geometry.

Velocity profiles have been graphed by systematically altering the Beltrami parameters and the density of plasma species. It has been noted that the densities of smaller species, namely electrons and ions, and the density of heavier particles, negatively charged dust particles, significantly impact the description of the plasma system's flow. The unified velocity increases with a higher electron concentration and lower dust particle abundance, and vice versa. Similarly, the flow of dust particles is influenced by changes in density ratio in a manner consistent with the unified flow, while the overall flow of smaller species – electrons and ions – remains unaltered. These findings are valuable for elucidating the characteristics of relaxed structures and dynamo action in the context of Saturn’s rings, planetary rings, rotating stars, pulsar magneto sphere and excitations in interstellar medium.