A theoretical investigation is presented to explain the formation and characteristics of relaxed equilibrium structures in a three-component dusty plasma within Saturn's atmosphere, composed of negatively charged dust particles, electrons, and ions. The Quadruple Beltrami equation is derived by utilizing the vortex dynamic equations along with the current density. Solutions for the higher Beltrami states are obtained in two different modes, a simple rectangular geometry and a coplanar rectangular geometry, to explore the characteristics of relaxed structures within the Saturn magnetosphere and its rings. The solutions are depicted through some plots by varying the Beltrami parameters and the densities of the plasma species. It is observed that only paramagnetic structures are formed in the coplanar geometry, while variations in the Beltrami parameters and plasma species densities significantly affect the magnetic characteristics of the relaxed structures in a simple rectangular geometry. This paper will provide an important contribution to understand the atmospheric vortical structures developed in different astronomical bodies that have double or more than double configurations, such as Saturn's rings, Jupiter magnetosphere, Uranus, Neptune, etc.

The presence of dust grains in various regions of space, including planetary rings, magnetospheres, comet tails, and interplanetary space, has been a subject of significant interest and research within the scientific community. Several recent studies were devoted to study the composition of plasmas of the giant planet Saturn. The planet is co-rotated with its magnetic field.1 In the inner Saturn magnetosphere, the electron density is about 200 cm 3 and the temperatures of the plasma species are T e = 0.5 8 and T i = 100 eV,2,3 while in the outer magnetosphere, the equatorial regions are more dense.4 The distribution of electron density in the Saturn magnetosphere (the second largest magnetosphere in our solar system5) is dependent on the distance from the axis of rotation.4 The plasma is co-rotated with the Saturn's field because most of the plasma is frozen in the magnetic field of its magnetosphere.6 It has been detected through CAPS ion spectrometer that7,8 the speed of this rotation attains its maximum limit by increasing the radial distance from the planet. The Saturn has co-rotational field as well as gravitational field and these fields interact with the debris in different ways. The debris are more influenced by the gravitational forces. The impact of these forces on these charged particles is truly dependent on the size, shape, and nature of the charge. In the E-ring of the Saturn, in situ observations,6 it has been found that the charged dust particles of different sizes are influenced by these both forces. Several observations9–11 explained the location of the charged dust particles in the E-ring. From the data, it has been explained that the formation of the E-ring around the Saturn is mostly based on the dust of the Enceladus12 (one of Saturn's moons). The density of massive dust grains is comparatively low about 10 7 cm 3. In 2008, the distribution and the properties of the dust particles in the E-ring have been observed,13 which explained that the distribution of dust particles follows the power law, and as a result, densities of the smaller dust particles sharply increase in the E-ring. Several investigations14,15 also explained the large population of smaller dust grains in the E-ring. Cassini Radio and Plasma Wave System (on March 2008)16 investigated a sharp drop out of the electron density in the planet E-ring. This decrease in the electron density is because of the attachment of electrons on the surface of the dust grains.

Chancia et al.17 identified the presence of numerous periodic dusty structures in Saturn's Roche limit, which is a region between A and F ring of about 3000 km. It has been observed that the oscillations of the magnetic field in the Saturn's magnetosphere17 are responsible for the formation of these structures. The atmospheric aerosol cloud structure18 was recaptured through high-resolution Cassini ISS images in the Saturn's southern hemisphere. During the 2013 Aurora Campaign from inside Saturn's magnetosphere,19 the identification of the interplanetary magnetic field structures20 has been presented. At Saturn's north polar region, many tiny discrete bright cloud structures21 have also been observed. Through Cassini/VIMS, the existence of a periodic structure like high-speed cyclonic hot vortex22 has been identified at Saturn's north pole. Several investigations reported the existence of such strong periodic cyclonic vortex structures.23–29 

Multiple research groups have studied the formation mechanism of vortex flow/coherent structures30 and their characteristics in dusty plasmas. The proportionality of local vorticity to the stream function31 describes the most fundamental relaxed state32 named as Beltrami state. The alignment of current density “ × B” to the magnetic field “B33 is given as
× B = λ B ,
where the λ is the constant known as Lagrange multiplier.34 The Beltrami magnetic field defines a zero divergence vector field.35 This equilibrium force-free state is the relaxation of a single fluid plasma explained by Woltjer34 and Taylor.36,37 The relaxed Beltrami structures are helical, twisted, and spiral in nature. Vortex structures like hurricanes38 are the example of the Beltrami state on the Earth's atmosphere. The development of helical flux tubes in the solar corona39 also describes the vortex structures. The formation of such structures (reversed field pinch,40 field reversal configuration41) are also useful for the controlled fusion programs. The formulation of relaxed magnetohydrodynamics (MHD) plasma was explained on the basis of variational principle.42 In the plasma dynamics, the vorticity and the magnetic field play a key role. Turner43 discussed the generalized vorticity for the ideal single fluid MHD plasma.

For the multispecies plasmas, it was observed the vorticity associated with each fluid acts as a constraint,44,45 which demonstrates a non-force-free relaxed state. For the two fluids (electrons and ions), a relaxation theory was proposed by Steinhauer and Ishida46,47 that described the minimization of the system energy while the helicities were constraints. This relaxation theory also predicted the development of pressure and vigorous flow which were absent for a single fluid plasma.

For a plasma system composed of two-fluid inertialess electrons and ions (Hall magnetohydrodynamic HMHD), Mahajan and Yoshida32,48 proposed a relaxation theory termed as Double-Curl-Beltrami equation. The mathematical formulation of Double-Curl-Beltrami state explored the two eigen values. These eigen values are described two relaxed states. The system investigated a fully diamagnetic structure and a complementary flow. A new framework describing the relaxation model for two-fluid plasmas was presented using variational principle.49 The system explained three invariants: energy, magnetic helicity, and generalized helicity containing vorticity. In two-fluid plasmas, high confinement boundary layers50 were discussed and high pressure was obtained. A mathematical model was derived to explain the catastrophic transformation. The model demonstrated the transformation of Double-Curl-Beltrami state into single Beltrami state employing two-fluid51,52 and three-fluid53 plasmas system. Moreover, the modeling for solar eruption54 has also been discussed using Double-Curl-Beltrami state.

The stability analysis55 for the single Beltrami state as well as for Double-Curl-Beltrami state56 is performed. Yoshida et al., investigated the stability conditions for the Beltrami states on the basis of constant of motion.57 Employing the analytical solutions of the Beltrami state, the mathematical modeling for high confinement boundary layer,50,58,59 spheromakes,60 and field reversed configuration61,62 have been studied which is useful for plasma fusion. The Beltrami flow has also been used to explore the spinning black hole63 accretion disk.64 

Bhattacharyya et al.65 extended the theory to new relaxed state know as Triple-Curl-Beltrami state by considering the inertial effects of both fluids in electrons and positron plasmas. The Euler–Lagrange equation for the Triple-Curl-Beltrami state can be derived by the addition of three separate single Beltrami flows. The equation is supported by three eigen values. The relaxation state for a non-relativistic multispecies dusty plasmas in the form of Triple-Curl-Beltrami state66 has been observed. The formation of multiscale structures in dense degenerate astrophysical plasmas has been studied by deriving another specialized higher-order equilibrium state, termed the Quadruple Beltrami state, by Shatashvili et al.67 These multiscale relaxed structures provide a richer toolkit to understand and study the complexities of space phenomena such as eruptions, quick bursts of outflow, and jet formation. Later on, this relaxed state was studied for a variety of plasma systems.68–71 Mahajan and Lingam72 discussed the relaxation of multispecies plasmas into a multi Beltrami state.

The solutions of Triple-Curl-Beltrami state in the presence of an internal conductor for the cylindrical geometry73 and for the rectangular geometry74 have been investigated. Internal conductor coil explores an innovative form of relaxation theory in the case of high pressure and flow. Yoshida et al.75 studied the plasma confinement for the internal coil device. In the cylindrical configuration,76 the solution for the case of single Beltrami state has been observed. The magnetic confinement for several devices based on internal coil77 was studied theoretically76,78 as well as experimentally.79 The confinement of a turbulent plasmas containing an internal coil80 or linear mirror device81 and the impact of biased electrodes82,83 were studied experimentally.

The present study exhibits the confinement of a magnetized three-component dusty plasmas (negatively charged dust grains, electrons, and ions) with an internal coil (in the form of a slab) device in the rectangular configuration. It is similar to the coplanar rectangular configuration. It is shown that dusty plasmas self-organized to a Quadruple Beltrami relaxed state, which is a combination of four individual solo Beltrami flows. The relaxed system is supported by four eigen values. The solutions of Quadruple relaxed Euler–Lagrange equation are derived in a linear rectangular geometry as well as in the coplanar internal conductor rectangular configuration. The magnetic characteristics of the relaxed structures are investigated for the possible role of the Beltrami parameters and the concentration of the plasma species. The graphs are plotted for the real plasma (Saturn's E-ring6,84) observation data. It is shown that the magnetic profiles exhibit only paramagnetic structures in a coplanar rectangular geometry.74 In a linear rectangular geometry, the characteristics of the magnetic structures are influenced by varying the Beltrami parameters and by changing the densities of the plasma components.

The aim of this current research is to explore the characteristics of relaxed structures within the Saturn magnetosphere and its rings. To achieve this, we have derived the solution of the relaxed state in both a simple rectangular geometry and a coplanar rectangular geometry. The coplanar solution is particularly relevant for describing the Saturn rings, which are concentric. This derivation of the coplanar solution with eight amplitudes (R1, R2, R3, R4 for the inner rectangular geometry, and S1, S2, S3, S4 for the outer rectangular geometry) is a novel contribution that has not been previously explored. These results are significantly important to understand the atmospheric vortical structures developed in different astronomical bodies such as Saturn's rings,85 Jupiter magnetosphere,80 near-Earth plasmasheet,86 Uranus, Neptune, ionosphere,87,88 etc. While acknowledging the curvature of planetary bodies, we chose coplanar rectangular geometry due to its simplicity. The double configuration of coplanar rectangular geometry introduces eight constants rather than four, adding complexity to the system. Therefore, compared to concentric cylindrical geometry, which is more complex, the use of rectangular geometry facilitates comprehension for readers.

The work in the manuscript is arranged as follows: in Sec. I, we introduce the problem whereas in Sec. II, the set of equations to explain the dynamics of the plasma species is described. The formalism of the relaxed equilibrium system is discussed in Sec. III. Section IV explains the constraints of the system. Two different sets of solutions of the relaxed Quadruple Beltrami state have been derived in Sec. V. The magnetic profiles of the vortex pattern on the basis of Beltrami parameters and the impact of density variation of the plasma species have been presented in Sec. VI. Section VII presents the generalized Bernoulli conditions. In Sec. VIII, we conclude the work.

We examine the relaxed structures in a collisionless dusty plasma. The plasma under consideration consists of negatively charged dust particles (d), electrons ( e ) , and ions (i). Furthermore, Zd describes the charge state of negatively charged dust particles while the ions are singly ionized. The charge state of dust grain is denoted as Zd. It is assumed that the plasma equilibrium condition is satisfied which can be described as
Z d n d + n e = n i ,
(1)
where nd, ne, and ni represent the density of negatively charged dust particles, electrons, and ions. The masses of the plasma components dust particles, electrons, and ions can be read as md, me, and mi, respectively. As the charging of dust particles can be affected by the magnetic field,89 in the current system, we adhere to the Shukla–Mahajan work90 and do not account for magnetic field effects. Instead, we assume a constant dust charge to maintain charge neutrality. The normalized set of the governing fluid equations to explain the dynamics of the relaxed equilibrium structures is given by
t ( V d Z d M d A ) = V d × ( × V d Z d M d B ) ψ d ,
(2)
t ( V e A ) = V e × ( × V e B ) ψ e ,
(3)
t ( V i + M i A ) = V i × ( × V i + M i B ) ψ i ,
(4)
where ψ d = Z d M d ϕ + ( M d / N d ) p d + V d 2 / 2 , ψ e = ϕ + p e + V e 2 / 2 and ψ i = M i ϕ + ( M i / N i ) p i + V i 2 / 2. Here, M d = m e / m d and M i = m e / m i are used for the normalization of masses of the plasma species. V j ( j = d , e , i ) the flow of dust particles, electrons, and ions is scaled by V A = B 0 / μ 0 n e m e (B0 and μ 0 are the arbitrary value of magnetic field and the permeability of free space, respectively). The pressure p j is scaled by B 0 2 / μ 0,
E = ϕ A t .
(5)
Equation (5) defines the electric field E. The vector potential ( B = × A ) and the scalar potential ϕ are scaled by λ e B 0 and λ e B 0 V A, respectively. The coordinated of space and time are scaled by λ e = V A / ω c = m e / μ 0 n e e 2 and ω c 1 = V A / λ e ( ω c = e B 0 / m e is the corresponding plasma frequency91 and e = 1.6 × 10 19 C is the charge). By applying the curl operator on the dynamic equations (2)–(4), the following set of vortex dynamic equations are obtained:
t ( × V d Z d M d B ) = × [ V d × ( × V d Z d M d B ) ] ,
(6)
t ( × V e B ) = × [ V e × ( × V e B ) ] ,
(7)
t ( × V i + M i B ) = × [ V i × ( × V i + M i B ) ] ,
(8)
we finally arrive at the following equation:
Ω j t × [ U j × Ω j ] = 0 ,
(9)
which is the compact form of Eqs. (6)–(8). Here, Ω j ( j = d , e , i ) ( Ω d = × V d Z d M d B, Ω e = × V e B, and Ω i = × V i + M i B) are the generalized vorticities and
U j = V j
(10)
are the velocities of the plasma components. Here to close the system, Ampère's law48,67–71,92 is used and we obtained the following expression:
V i = 1 N i ( × B + V e + Z d N d V d )
(11)
containing V d , V e , V i, and B, where N d = n d / n e and N i = n i / n e are the densities ratio.
By following the methodology of Shatashvili et al.,67 we earn the Beltrami condition:
Λ Ω j = U j .
In the form of plasma components dust particles, electrons, and ions, the equilibrium Beltrami condition can be described as
× V d Z d M d B = a d V d ,
(12)
× V e B = a e V e ,
(13)
× V i + M i B = a i V i ,
(14)
where the parameters a d , a e , and ai are the ratio of generalized vorticity to the flow of the dust particles, electrons, and ions, respectively. These parameters are also defined the Beltrami parameters are the corresponding Beltrami constants. The flow of electrons V e is obtained by plugging Ve Eq. (11) into Eq. (14),
V e = I 1 ( × ) 2 B I 2 × B + I 3 B + I 4 V d ,
(15)
where I 1 = ( a i a e ) 1 , I 2 = a i ( a i a e ) 1 , I 3 = ( 1 + N i M i + Z d 2 N d M d ) ( a i a e ) 1, and I 4 = Z d N d ( a d a i ) ( a i a e ) 1. Inserting the value of V e from Eq. (15) into Eq. (13) gives the value of the flow of dust particles, we can be read as
V d = D 1 ( × ) 3 B D 2 ( × ) 2 B + D 3 × B D 4 B ,
(16)
where D 1 = [ Z d N d ( a e a d ) ( a d a i ) ] 1 , D 2 = ( a e + a i ) [ Z d N d ( a e a d ) ( a d a i ) ] 1 ,      D 3 = ( a e a i + 1 + N i M i + Z d 2 N d M d ) [ Z d N d ( a e a d ) ( a d a i ) ] 1, and D 4 = ( a i + ( a e + a i a d ) Z d 2 N d M d + a e N i M i ) [ Z d N d ( a e a d ) ( a d a i ) ] 1. Using Eq. (16) into Eq. (12) will eventually produce the relaxed equilibrium state which is known as Quadruple relaxed Beltrami state,
( × ) 4 B b 1 ( × ) 3 B + b 2 ( × ) 2 B + b 3 × B + b 4 B = 0 ,
(17)
where
b 1 = a e + a i + a d ,
(18)
b 2 = a e a i + a i a d + a d a e + 1 + N i M i + Z d 2 N d M d ,
(19)
b 3 = N i M i ( a d + a e ) + Z d 2 N d M d ( a e + a i ) + ( a i + a d ) + a e a i a d ,
(20)
b 4 = N i M i a d a e + Z d 2 N d M d a e a i + a i a d .
(21)
This relaxed state [Eq. (17)] of three-component dusty plasmas is the summation of four different single relaxed Beltrami states. The general solution of Eq. (17) can be described by the following eigen expression:93,
× S δ = λ δ S δ ,
(22)
where S δ is the eigenfunction and λ δ is the eigenvalue while δ = 1 , 2 , 3 , 4. The four relaxed Beltrami states have four distinct eigen values. Writing the operator ( × ) as “curl” Eq. (22) can also be read as
( curl λ 1 ) ( curl λ 2 ) ( curl λ 3 ) ( curl λ 4 ) B = 0 ,
(23)
where B = B 1 + B 2 + B 3 + B 4 and λ1, λ2, λ3, and λ4 are the four distinct eigen values. After solving the Eq. (23), we get the same fourth-order equation as we get earlier relaxed Eq. (17),
( × ) 4 B b 1 ( × ) 3 B + b 2 ( × ) 2 B + b 3 × B + b 4 B = 0 ,
(24)
where the constants b1, b2, b3, and b4 are
λ 4 + λ 3 + λ 2 + λ 1 = b 1 ,
(25)
λ 4 λ 3 + λ 4 λ 2 + λ 3 λ 2 + λ 4 λ 1 + λ 1 λ 3 + λ 2 λ 1 = b 2 ,
(26)
λ 3 λ 2 λ 4 + λ 3 λ 4 λ 1 + λ 4 λ 2 λ 1 + λ 3 λ 2 λ 1 = b 3 ,
(27)
λ 4 λ 3 λ 2 λ 1 = b 4 .
(28)
λ1, λ2, λ3, and λ4 can be figured out as68–71 
λ 1 = b 1 + 2 R + 2 δ 4 , λ 2 = b 1 + 2 R 2 δ 4 , λ 3 = b 1 2 R + 2 ξ 4 , λ 4 = b 1 2 R 2 ξ 4 ,
where R = ( b 1 2 4 b 2 + 4 Y ) / 2 , Y = ( d 3 u 2 + 3 u α 1 ) / 3 u , u = ( q / 2 ) + ( q / 2 ) 2 + ( d / 3 ) 3 3 , q = ( 9 α 1 α 2 2 α 1 3 27 α 3 ) / 27, d = ( 3 α 2 α 1 2 ) / 3 , α 1 = b 2 , α 2 = b 3 b 1 4 b 4 , and α 3 = b 3 2 + b 1 2 b 4 4 b 2 b 4. For R 0, δ and ξ are given as
δ = 3 4 b 1 2 R 2 2 b 2 + 1 4 R ( 4 b 1 b 2 8 b 3 b 1 3 ) , ξ = 3 4 b 1 2 R 2 2 b 2 1 4 R ( 4 b 1 b 2 8 b 3 b 1 3 ) ,
while for R = 0, δ and ξ read as
δ = 3 4 b 1 2 2 b 2 + 2 Y 2 4 b 4 , ξ = 3 4 b 1 2 2 b 2 2 Y 2 4 b 4 .
As the four eigen values are the roots of a quadratic equation
λ 4 b 1 λ 3 + b 2 λ 2 b 3 λ + b 4 = 0 ,
so the characteristics of these roots are dependent on the discriminant ( Δ ) of the general quadratic equation, which is given as
Δ = 256 b 4 3 192 b 1 b 3 b 4 2 128 b 2 2 b 4 2 + 144 b 2 b 3 2 b 4 27 b 3 4 + 144 b 1 2 b 3 b 4 2 6 b 1 2 b 3 2 b 4 80 b 1 b 2 2 b 3 b 4 + 18 b 1 b 2 b 3 3 + 16 b 2 4 b 4 4 b 2 3 b 3 2 27 b 1 4 b 4 2 + 18 b 1 3 b 2 b 3 b 4 4 b 1 3 b 3 3 4 b 1 2 b 2 3 b 4 + b 1 2 b 2 2 b 3 2 .
Figures 1 and 2 illustrate the three-dimensional plot to study the impact of Beltrami parameters (ad, ae, and ai) and densities of the plasma species (nd, ne, and ni), respectively, on the nature of eigen values (λ1, λ2, λ3, and λ4). These analysis are based on the real plasma (Saturn's E-ring6,84) observation data. The densities of the species and mass and charge of the dust particles of the Saturn's E-ring plasmas are n d = 10 4 10 5 m 3, n e = ( 2 7 ) × 10 7 m 3, Zd = 100, and m d = 4 ×  10 15 kg. Green regions in Figs. 1 and 2 demonstrate the real eigen values in terms of Beltrami parameters and densities of plasma species while the clear regions indicate those points when the two eigen values are real and the remaining two are complex conjugates.
FIG. 1.

3D plot to study the nature of the eigen values (λ1, λ2, λ3, and λ4) as a function of Beltrami parameters (ad, ae, and ai). Green regions indicate typical only real eigen values. White regions indicate two real eigen values and two eigen values are a pair of complex conjugate.

FIG. 1.

3D plot to study the nature of the eigen values (λ1, λ2, λ3, and λ4) as a function of Beltrami parameters (ad, ae, and ai). Green regions indicate typical only real eigen values. White regions indicate two real eigen values and two eigen values are a pair of complex conjugate.

Close modal
FIG. 2.

3D plot to study the nature of the eigen values (λ1, λ2, λ3, and λ4) as a function of density of plasma species (nd, ne, and ni). Green regions indicate typical only real eigen values. White regions indicate two real eigen values and two eigen values are a pair of complex conjugate.

FIG. 2.

3D plot to study the nature of the eigen values (λ1, λ2, λ3, and λ4) as a function of density of plasma species (nd, ne, and ni). Green regions indicate typical only real eigen values. White regions indicate two real eigen values and two eigen values are a pair of complex conjugate.

Close modal
It has been investigated that72,94 in a plasma system, there will be S + 1 invariants, if a system is composed on S fluids. In this research work, the system has three components, namely, negatively charged dust grains, electrons, and ions. Therefore, the constraints for a three-fluid plasmas will be magnetofluid energy (E), generalized helicity of negatively charged dust grains (hd), generalized helicity of electrons (he), and generalized helicity of ions (hi). Using Eqs. (6)–(8), the four integral invariants can be written as
E = 1 2 ( V e 2 + N i M i V i 2 + N d Z d M d V d 2 + B 2 ) ,
(29)
h d = 1 2 v Ω d · curl 1 Ω d d v ,
(30)
h e = 1 2 v Ω e · curl 1 Ω e d v ,
(31)
h i = 1 2 v Ω i · curl 1 Ω i d v .
(32)
The relaxed magnetized equilibrium state can also be retrieved using another framework known as variational principle46,48,95
δ ( E μ d h d μ e h e μ i h i ) = 0 ,
(33)
where μd, μe, and μi are the Lagrangian multipliers. Solving the above equations simultaneously and considering δ A, δ V i, and δ V d, as an independent parameters, we obtain
× B = 1 a e ( B × V e ) + N i a i ( M i B + × V i ) + N d a d ( Z d M d B + × V d ) ,
(34)
a i V i = × V i + M i B ,
(35)
a d V d = × V d + Z d M d B ,
(36)
taking μ d = N d / a d M d Z d , μ e = 1 / a e, and μ i = N i / a i M i. Substituting Eqs. (35) and (36) into Eq. (34), we have
V i = 1 N i ( × B + Z d M d V d + V e ) .
(37)
It has been observed that the set of Eqs. (35)–(37) using variational principle are similar to our previous set of equilibrium Eqs. (12)–(14). It shows that the self-organization of the order structures can be observed through variational principle.
Equation (17) is a higher order relaxed state which is composed on four Beltrami states and can be presented as
B = C 1 B 1 + C 2 B 2 + C 3 B 3 + C 4 B 4 ,
(38)
where C1, C2, C3, and C4 represent the amplitudes of our four Beltrami states. Equation (17) represents the partial differential equation. ABC flow96 is the one of the solution of Beltrami state in the slab geometry. Following the ABC flow,96 the derivations of the solution of Eq. (17) are carried out for the two different modes: (i) simple rectangular geometry68,69,71 and (ii) a coplanar rectangular geometry (a rectangular conducing chamber in the Cartesian coordinate system).
Expressing the solution of Eq. (17) in terms of x, y, and z, we get
( B x B y B z ) = ( 0 C 1 sin ( λ 1 x ) C 1 cos ( λ 1 x ) ) + ( 0 C 2 sin ( λ 2 x ) C 2 cos ( λ 2 x ) ) + ( 0 C 3 sin ( λ 3 x ) C 3 cos ( λ 3 x ) ) + ( 0 C 4 sin ( λ 4 x ) C 4 cos ( λ 4 x ) ) ,
(39)
where C α ( α = 1 , 2 , 3 , and 4 ) are the constants (amplitudes) and the values of these constants can be obtained by applying the following boundary values | B y | x = x = g ̇ , | B z | x = 0 = h ̂ , | ( × B ) y | x = x = w, and | ( × B ) z | x = 0 = e ̈ in Eq. (39), we obtain
g ̇ = C 1 sin ( λ 1 x ) + C 2 sin ( λ 2 x ) + C 3 sin ( λ 3 x ) + C 4 sin ( λ 4 x ) ,
(40)
h ̂ = C 1 + C 2 + C 3 + C 4 ,
(41)
w = C 1 λ 1 sin ( λ 1 x ) + C 2 λ 2 sin ( λ 2 x ) + C 3 λ 3 sin ( λ 3 x ) + C 4 λ 4 sin ( λ 4 x ) ,
(42)
e ̈ = C 1 λ 1 + C 2 λ 2 + C 3 λ 3 + C 4 λ 4 .
(43)
Solving the above equations, we obtain
C 1 = Q 1 D , C 2 = Q 2 D , C 3 = Q 3 D , C 4 = Q 4 D ,
where
Q 1 = [ sin ( λ 3 x ) sin ( λ 4 x ) ( e ̈ h ̂ λ 2 ) + sin ( λ 2 x ) ( w g ̇ λ 2 ) ] ( λ 3 λ 4 ) + [ sin ( λ 3 x ) sin ( λ 2 x ) ( e ̈ h ̂ λ 4 ) + sin ( λ 4 x ) ( w g ̇ λ 4 ) ] ( λ 2 λ 3 ) + [ sin ( λ 2 x ) sin ( λ 4 x ) ( e ̈ h ̂ λ 3 ) + sin ( λ 3 x ) ( w g ̇ λ 3 ) ] ( λ 4 λ 2 ) ,
Q 2 = [ sin ( λ 3 x ) sin ( λ 1 x ) ( e ̈ h ̂ λ 4 ) + sin ( λ 4 x ) ( w g ̇ λ 4 ) ] ( λ 3 λ 1 ) + [ sin ( λ 3 x ) sin ( λ 4 x ) ( e ̈ h ̂ λ 1 ) + sin ( λ 1 x ) ( w g ̇ λ 1 ) ] ( λ 4 λ 3 ) + [ sin ( λ 1 x ) sin ( λ 4 x ) ( e ̈ h ̂ λ 3 ) + sin ( λ 3 x ) ( w g ̇ λ 3 ) ] ( λ 1 λ 4 ) ,
Q 3 = [ sin ( λ 1 x ) sin ( λ 4 x ) ( e ̈ h ̂ λ 2 ) + sin ( λ 2 x ) ( w g ̇ λ 2 ) ] ( λ 4 λ 1 ) + [ sin ( λ 1 x ) sin ( λ 2 x ) ( e ̈ h ̂ λ 4 ) + sin ( λ 4 x ) ( w g ̇ λ 4 ) ] ( λ 1 λ 2 ) + [ sin ( λ 2 x ) sin ( λ 4 x ) ( e ̈ h ̂ λ 1 ) + sin ( λ 1 x ) ( w g ̇ λ 1 ) ] ( λ 2 λ 4 ) ,
Q 4 = [ sin ( λ 3 x ) sin ( λ 1 x ) ( e ̈ h ̂ λ 2 ) + sin ( λ 2 x ) ( w g ̇ λ 2 ) ] ( λ 1 λ 3 ) + [ sin ( λ 1 x ) sin ( λ 2 x ) ( e ̈ h ̂ λ 3 ) + sin ( λ 3 x ) ( w g ̇ λ 3 ) ] ( λ 2 λ 1 ) + [ sin ( λ 2 x ) sin ( λ 3 x ) ( e ̈ h ̂ λ 1 ) + sin ( λ 1 x ) ( w g ̇ λ 1 ) ] ( λ 3 λ 2 ) ,
D = ( λ 4 λ 1 ) [ sin ( λ 4 x ) sin ( λ 1 x ) + sin ( λ 3 x ) sin ( λ 2 x ) ] ( λ 3 λ 2 ) + ( λ 4 λ 3 ) [ sin ( λ 4 x ) sin ( λ 3 x ) + sin ( λ 2 x ) sin ( λ 1 x ) ] ( λ 2 λ 1 ) + ( λ 2 λ 4 ) [ sin ( λ 2 x ) sin ( λ 4 x ) + sin ( λ 3 x ) sin ( λ 1 x ) ] ( λ 3 λ 1 ) .
The objective of this ongoing study is to investigate the properties of the relaxed structures within Saturn's magnetosphere and its rings. To accomplish this, we have determined the solution for the Quadruple Beltrami state in both a basic rectangular shape and a coplanar rectangular arrangement. The coplanar solution is especially important for describing Saturn's rings, which are concentric. This derivation of the coplanar solution with eight amplitudes (R1, R2, R3, R4 for the inner rectangular geometry and S1, S2, S3, S4 for the outer rectangular geometry) is a novel contribution. A coplanar rectangular geometry is the immersion of a rectangular chamber in another rectangular configuration. Assuming the length of this immersed rectangular chamber is x0 along the x-axis. For Eq. (17), the solution of such a region can be formulated as
( B x B y B z ) = ( 0 R 1 sin ( λ 1 x ) + S 1 cos ( λ 1 x ) R 1 cos ( λ 1 x ) S 1 sin ( λ 1 x ) ) + ( 0 R 2 sin ( λ 2 x ) + S 2 cos ( λ 2 x ) R 2 cos ( λ 2 x ) S 2 sin ( λ 2 x ) ) + ( 0 R 3 sin ( λ 3 x ) + S 3 cos ( λ 3 x ) R 3 cos ( λ 3 x ) S 3 sin ( λ 3 x ) ) + ( 0 R 4 sin ( λ 4 x ) + S 4 cos ( λ 4 x ) R 4 cos ( λ 4 x ) S 4 sin ( λ 4 x ) ) ,
(44)
where R α ( α = 1 , 2 , 3 , and 4 ) and S α ( α = 1 , 2 , 3 , and 4 ) are the amplitudes. By applying the appropriate boundary conditions which are | B y | x = x 0 = s ´ , | B z | x = x 0 = t , | ( × B ) y | x = x 0 = u ̃ , | ( × B ) z | x = x 0 = v , | ( ( × ) 2 B ) y | x = x 0 = w ̂ , | ( ( × ) 2 B ) z | x = x 0 = σ , | ( ( × ) 3 B ) y | x = x 0 = Y ̈, and | ( ( × ) 3 B ) z | x = x 0 = z ˇ, the following system of equations is obtained:
s ´ = R 1 sin ( λ 1 x 0 ) + S 1 cos ( λ 1 x 0 ) + R 2 sin ( λ 2 x 0 ) + S 2 cos ( λ 2 x 0 ) + R 3 sin ( λ 3 x 0 ) + S 3 cos ( λ 3 x 0 ) + R 4 sin ( λ 4 x 0 ) + S 4 cos ( λ 4 x 0 ) ,
(45)
t = R 1 cos ( λ 1 x 0 ) S 1 sin ( λ 1 x 0 ) + R 2 cos ( λ 2 x 0 ) S 2 sin ( λ 2 x 0 ) + R 3 cos ( λ 3 x 0 ) S 3 sin ( λ 3 x 0 ) + R 4 cos ( λ 4 x 0 ) S 4 sin ( λ 4 x 0 ) ,
(46)
u ̃ = λ 1 R 1 sin ( λ 1 x 0 ) + λ 1 S 1 cos ( λ 1 x 0 ) + λ 2 R 2 sin ( λ 2 x 0 ) + λ 2 S 2 cos ( λ 2 x 0 ) + λ 3 R 3 sin ( λ 3 x 0 ) + λ 3 S 3 cos ( λ 3 x 0 ) + λ 4 R 4 sin ( λ 4 x 0 ) + λ 4 S 4 cos ( λ 4 x 0 ) ,
(47)
v = λ 1 R 1 cos ( λ 1 x 0 ) λ 1 S 1 sin ( λ 1 x 0 ) + λ 2 R 2 cos ( λ 2 x 0 ) λ 2 S 2 sin ( λ 2 x 0 ) + λ 3 R 3 cos ( λ 3 x 0 ) λ 3 S 3 sin ( λ 3 x 0 ) + λ 4 R 4 cos ( λ 4 x 0 ) λ 4 S 4 sin ( λ 4 x 0 ) ,
(48)
w ̂ = λ 1 2 R 1 sin ( λ 1 x 0 ) + λ 1 2 S 1 cos ( λ 1 x 0 ) + λ 2 2 R 2 sin ( λ 2 x 0 ) + λ 2 2 S 2 cos ( λ 2 x 0 ) + λ 3 2 R 3 sin ( λ 3 x 0 ) + λ 3 2 S 3 cos ( λ 3 x 0 ) + λ 4 2 R 4 sin ( λ 4 x 0 ) + λ 4 2 S 4 cos ( λ 4 x 0 ) ,
(49)
σ = λ 1 2 R 1 cos ( λ 1 x 0 ) λ 1 2 S 1 sin ( λ 1 x 0 ) + λ 2 2 R 2 cos ( λ 2 x 0 ) λ 2 2 S 2 sin ( λ 2 x 0 ) + λ 3 2 R 3 cos ( λ 3 x 0 ) λ 3 2 S 3 sin ( λ 3 x 0 ) + λ 4 2 R 4 cos ( λ 4 x 0 ) λ 4 2 S 4 sin ( λ 4 x 0 ) ,
(50)
Y ̈ = λ 1 3 R 1 sin ( λ 1 x 0 ) + λ 1 3 S 1 cos ( λ 1 x 0 ) + λ 2 3 R 2 sin ( λ 2 x 0 ) + λ 2 3 S 2 cos ( λ 2 x 0 ) + λ 3 3 R 3 sin ( λ 3 x 0 ) + λ 3 3 S 3 cos ( λ 3 x 0 ) + λ 4 3 R 4 sin ( λ 4 x 0 ) + λ 4 3 S 4 cos ( λ 4 x 0 ) ,
(51)
z ˇ = λ 1 3 R 1 cos ( λ 1 x 0 ) λ 1 3 S 1 sin ( λ 1 x 0 ) + λ 2 3 R 2 cos ( λ 2 x 0 ) λ 2 3 S 2 sin ( λ 2 x 0 ) + λ 3 3 R 3 cos ( λ 3 x 0 ) λ 3 3 S 3 sin ( λ 3 x 0 ) + λ 4 3 R 4 cos ( λ 4 x 0 ) λ 4 3 S 4 sin ( λ 4 x 0 ) .
(52)
After some algebraic manipulation, we obtain the following set of equations:
R 1 = P 1 D 1 , R 2 = P 2 D 2 , R 3 = P 3 D 3 , R 4 = P 4 D 4 , S 1 = P 5 D 1 , S 2 = P 6 D 2 , S 3 = P 7 D 3 , S 4 = P 8 D 4 ,
where
P 1 = sin ( λ 1 x 0 ) [ ( λ 2 λ 4 ) ( Y ̈ u ̃ λ 4 2 + ( s ´ λ 3 λ 4 u ̃ λ 3 ) ( λ 3 + λ 4 ) )  − ( λ 2 2 + λ 2 λ 3 λ 3 λ 4 λ 4 2 ) ( w ̂ u ̃ λ 3 u ̃ λ 4 + s ´ λ 3 λ 4 ) ] + cos ( λ 1 x 0 ) [ ( λ 2 λ 4 ) ( z ˇ v λ 4 2 + ( t λ 3 λ 4 v λ 3 ) ( λ 3 + λ 4 ) )  − ( λ 2 2 + λ 2 λ 3 λ 3 λ 4 λ 4 2 ) ( σ v λ 3 v λ 4 + t λ 3 λ 4 ) ] ,
(53)
P 2 = sin ( λ 2 x 0 ) [ ( λ 1 λ 4 ) ( Y ̈ u ̃ λ 4 2 + ( s ´ λ 3 λ 4 u ̃ λ 3 ) ( λ 3 + λ 4 ) )  − ( λ 1 2 + λ 1 λ 3 λ 3 λ 4 λ 4 2 ) ( w ̂ u ̃ λ 3 u ̃ λ 4 + s ´ λ 3 λ 4 ) ] + cos ( λ 2 x 0 ) [ ( λ 1 λ 4 ) ( z ˇ v λ 4 2 + ( t λ 3 λ 4 v λ 3 ) ( λ 3 + λ 4 ) )  − ( λ 1 2 + λ 1 λ 3 λ 3 λ 4 λ 4 2 ) ( σ v λ 3 v λ 4 + t λ 3 λ 4 ) ] ,
(54)
P 3 = sin ( λ 3 x 0 ) [ ( λ 4 λ 2 ) ( Y ̈ u ̃ λ 2 2 + ( s ´ λ 1 λ 2 u ̃ λ 1 ) ( λ 2 + λ 1 ) )  − ( λ 4 2 + λ 1 λ 4 λ 1 λ 2 λ 2 2 ) ( w ̂ u ̃ λ 1 u ̃ λ 2 + s ´ λ 1 λ 2 ) ] + cos ( λ 3 x 0 ) [ ( λ 4 λ 2 ) ( z ˇ v λ 2 2 + ( t λ 1 λ 2 v λ 1 ) ( λ 2 + λ 1 ) )  − ( λ 4 2 + λ 1 λ 4 λ 1 λ 2 λ 2 2 ) ( σ v λ 1 v λ 2 + t λ 1 λ 2 ) ] ,
(55)
P 4 = sin ( λ 4 x 0 ) [ ( λ 3 λ 2 ) ( Y ̈ u ̃ λ 2 2 + ( s ´ λ 1 λ 2 u ̃ λ 1 ) ( λ 2 + λ 1 ) )  − ( λ 3 2 + λ 1 λ 3 λ 1 λ 2 λ 2 2 ) ( w ̂ u ̃ λ 1 u ̃ λ 2 + s ´ λ 1 λ 2 ) ] + cos ( λ 4 x 0 ) [ ( λ 3 λ 2 ) ( z ˇ v λ 2 2 + ( t λ 1 λ 2 v λ 1 ) ( λ 2 + λ 1 ) )  − ( λ 3 2 + λ 1 λ 3 λ 1 λ 2 λ 2 2 ) ( σ v λ 1 v λ 2 + t λ 1 λ 2 ) ] ,
(56)
P 5 = cos ( λ 1 x 0 ) [ ( λ 2 λ 4 ) ( Y ̈ u ̃ λ 4 2 + ( s ´ λ 3 λ 4 u ̃ λ 3 ) ( λ 3 + λ 4 ) )  − ( λ 2 2 + λ 2 λ 3 λ 3 λ 4 λ 4 2 ) ( w ̂ u ̃ λ 3 u ̃ λ 4 + s ´ λ 3 λ 4 ) ] sin ( λ 1 x 0 ) [ ( λ 2 λ 4 ) ( z ˇ v λ 4 2 + ( t λ 3 λ 4 v λ 3 ) ( λ 3 + λ 4 ) )  − ( λ 2 2 + λ 2 λ 3 λ 3 λ 4 λ 4 2 ) ( σ v λ 3 v λ 4 + t λ 3 λ 4 ) ] ,
(57)
P 6 = cos ( λ 2 x 0 ) [ ( λ 1 λ 4 ) ( Y ̈ u ̃ λ 4 2 + ( s ´ λ 3 λ 4 u ̃ λ 3 ) ( λ 3 + λ 4 ) )  − ( λ 1 2 + λ 1 λ 3 λ 3 λ 4 λ 4 2 ) ( w ̂ u ̃ λ 3 u ̃ λ 4 + s ´ λ 3 λ 4 ) ] sin ( λ 2 x 0 ) [ ( λ 1 λ 4 ) ( z ˇ v λ 4 2 + ( t λ 3 λ 4 v λ 3 ) ( λ 3 + λ 4 ) )  − ( λ 1 2 + λ 1 λ 3 λ 3 λ 4 λ 4 2 ) ( σ v λ 3 v λ 4 + t λ 3 λ 4 ) ] ,
(58)
P 7 = cos ( λ 3 x 0 ) [ ( λ 4 λ 2 ) ( Y ̈ u ̃ λ 2 2 + ( s ´ λ 1 λ 2 u ̃ λ 1 ) ( λ 2 + λ 1 ) )  − ( λ 4 2 + λ 1 λ 4 λ 1 λ 2 λ 2 2 ) ( w ̂ u ̃ λ 1 u ̃ λ 2 + s ´ λ 1 λ 2 ) ] sin ( λ 3 x 0 ) [ ( λ 4 λ 2 ) ( z ˇ v λ 2 2 + ( t λ 1 λ 2 v λ 1 ) ( λ 2 + λ 1 ) )  − ( λ 4 2 + λ 1 λ 4 λ 1 λ 2 λ 2 2 ) ( σ v λ 1 v λ 2 + t λ 1 λ 2 ) ] ,
(59)
P 8 = cos ( λ 4 x 0 ) [ ( λ 3 λ 2 ) ( Y ̈ u ̃ λ 2 2 + ( s ´ λ 1 λ 2 u ̃ λ 1 ) ( λ 2 + λ 1 ) )  − ( λ 3 2 + λ 1 λ 3 λ 1 λ 2 λ 2 2 ) ( w ̂ u ̃ λ 1 u ̃ λ 2 + s ´ λ 1 λ 2 ) ] sin ( λ 4 x 0 ) [ ( λ 3 λ 2 ) ( z ˇ v λ 2 2 + ( t λ 1 λ 2 v λ 1 ) ( λ 2 + λ 1 ) )  − ( λ 3 2 + λ 1 λ 3 λ 1 λ 2 λ 2 2 ) ( σ v λ 1 v λ 2 + t λ 1 λ 2 ) ] ,
(60)
D 1 = ( λ 1 λ 3 ) ( λ 1 2 λ 2 λ 1 λ 2 2 + λ 2 2 λ 4 λ 2 λ 4 2 + λ 1 λ 4 2 λ 1 2 λ 4 ) ,
(61)
D 2 = ( λ 2 λ 3 ) ( λ 2 2 λ 1 λ 2 λ 1 2 + λ 1 2 λ 4 λ 1 λ 4 2 + λ 2 λ 4 2 λ 2 2 λ 4 ) ,
(62)
D 3 = ( λ 3 λ 1 ) ( λ 2 2 λ 3 λ 2 λ 3 2 λ 2 2 λ 4 + λ 2 λ 4 2 λ 3 λ 4 2 + λ 3 2 λ 4 ) ,
(63)
D 4 = ( λ 4 λ 1 ) ( λ 3 2 λ 2 λ 3 λ 2 2 + λ 2 2 λ 4 λ 2 λ 4 2 + λ 3 λ 4 2 λ 3 2 λ 4 ) .
(64)

In this configuration, additional boundary conditions are used as compared to the simple slab geometry. These extra boundary conditions lead to an extra degree of freedom. The whole system can be controlled very efficiently. By changing the magnetic field (a function of Beltrami parameter and densities ratio) in the inner slab, the nature of the structure can be easily controlled. The pressure inside the slab can be reduced by varying the length of the inner slab. Although, for the simple slab geometry, we cannot apply extra boundary conditions, in this way, there is difficulty to control the nature of the relaxed structure at the center of the system.

In this section, we will study the characteristics of the relaxed structures in a three-component dusty plasmas in both configurations (simple rectangular geometry and a coplanar rectangular geometry). We are examining the nature of the relaxed structures in two distinct configurations: a simple rectangular geometry and a rectangular geometry featuring an internal rectangular conductor. The simple rectangular geometry can be likened to a planet without any rings, such as Earth, or it can represent only the magnetosphere of a planet like Saturn. On the other hand, the configuration depicting the coplanar rectangular geometry is utilized to illustrate a planet with rings. In this setup, the inner rectangular box represents the magnetosphere, while the outer rectangular box represents the rings of the planet. To elucidate the underlying physics of the relaxed structures within both the central magnetosphere and its outer rings, we opted to utilize plasma parameters sourced from the Saturn ring system. Based on observational data, the Saturn ring6,84 system possesses the following parameter values: n d = 10 4 10 5 m 3 , n e = ( 2 7 ) × 10 7 m 3, Zd = 100, and m d = 4 × 10 15 kg. In this regard, we will mainly analyze the influence of Beltrami parameters and the plasma parameters, namely, the density variation of the dust particles on the behavior of self-organized structures. The graphs are plotted for x = 5 (the length of the rectangle) for simple rectangular geometry, whereas for the coplanar rectangular geometry x 0 = 2 (the length of inner rectangular slab along x-axis) and x = 5 (the length of outer rectangular slab along x-axis), the length of the rectangular is x = 5.

Figures 3 and 4 examine the variation in the magnetic profiles for the Beltrami parameters ae = 2.7, ai = 1.4, and ad = 0.9. Figure 3 describes the nature of the relaxed structures in a simple rectangular geometry whereas the behavior of the structure in a coplanar rectangular geometry is studied in Fig. 4. Both magnetic profiles (3) and (4) exhibit a paramagnetic trend, and all associated scale parameters have real values λ 1 = 2.257 08 , λ 2 = 0.443 826 , λ 3 = 1.399 09, and λ 4 = 0.9. However, the magnetic strength of the relaxed structures is significantly larger in the rectangular geometry with an internal conductor compared to the simple rectangular geometry. For another set of Beltrami parameters, where ae = 0.5, ai = 2.8, and ad = 0.9, Figs. 5 and 6 are plotted for a simple rectangular geometry and a coplanar rectangular geometry, respectively. Figure 5 illustrates that the magnetic field increases away from the center and reaches its maximum at the edges. In contrast, Fig. 6 demonstrates a drastic change in the behavior of the magnetic profile for the coplanar rectangular geometry, showing that the structure is paramagnetic, with the field strength being minimum at the edges. Although the scale parameters for these structures in both geometries are the same λ 1 = 2.799 82 , λ 2 = 0.9 , λ 3 = 0.250 088 + 0.968 308 i, and λ 4 = 0.250 088 + 0.968 308 i, the strength of the magnetic field for the relaxed structures in the rectangular geometry is higher than that of the simple rectangular geometry, similar to our previous case.

FIG. 3.

Plot of the magnetic field (B) for ae = 2.7, ai = 1.4, and ad = 0.9 for simple slab geometry. The scale parameters are λ 1 = 2.257 08 , λ 2 = 0.443 826 , λ 3 = 1.399 09, and λ 4 = 0.9.

FIG. 3.

Plot of the magnetic field (B) for ae = 2.7, ai = 1.4, and ad = 0.9 for simple slab geometry. The scale parameters are λ 1 = 2.257 08 , λ 2 = 0.443 826 , λ 3 = 1.399 09, and λ 4 = 0.9.

Close modal
FIG. 4.

Plot of the magnetic field (B) for ae = 2.7, ai = 1.4, and ad = 0.9 for slab geometry with an internal conductor. The scale parameters are λ 1 = 2.257 08 , λ 2 = 0.443 826, λ 3 = 1.399 09, and λ 4 = 0.9.

FIG. 4.

Plot of the magnetic field (B) for ae = 2.7, ai = 1.4, and ad = 0.9 for slab geometry with an internal conductor. The scale parameters are λ 1 = 2.257 08 , λ 2 = 0.443 826, λ 3 = 1.399 09, and λ 4 = 0.9.

Close modal
FIG. 5.

Plot of the magnetic field (B) for ae = 0.5, ai = 2.8, and ad = 0.9 for simple slab geometry. The scale parameters are λ 1 = 2.799 82 , λ 2 = 0.9 , λ 3 = 0.250 088 + 0.968 308 i, and λ 4 = 0.250 088 + 0.968 308 i. [Associated dataset available at https://doi.org/10.5281/zenodo.10632069] (Ref. 99).

FIG. 5.

Plot of the magnetic field (B) for ae = 0.5, ai = 2.8, and ad = 0.9 for simple slab geometry. The scale parameters are λ 1 = 2.799 82 , λ 2 = 0.9 , λ 3 = 0.250 088 + 0.968 308 i, and λ 4 = 0.250 088 + 0.968 308 i. [Associated dataset available at https://doi.org/10.5281/zenodo.10632069] (Ref. 99).

Close modal
FIG. 6.

Plot of the magnetic field (B) for ae = 0.5, 7, ai = 2.8, and ad = 0.9 for slab geometry with an internal conductor. The scale parameters are λ 1 = 2.799 82 , λ 2 = 0.9 , λ 3 = 0.250 088 + 0.968 308 i, and λ 4 = 0.250 088 + 0.968 308 i.

FIG. 6.

Plot of the magnetic field (B) for ae = 0.5, 7, ai = 2.8, and ad = 0.9 for slab geometry with an internal conductor. The scale parameters are λ 1 = 2.799 82 , λ 2 = 0.9 , λ 3 = 0.250 088 + 0.968 308 i, and λ 4 = 0.250 088 + 0.968 308 i.

Close modal

In the current context, the significance of Beltrami parameters is highlighted as they play a crucial role in transforming the nature of relaxed structures. For instance, in a simple rectangular geometry, varying the Beltrami parameters can switch the structure's behavior from paramagnetic (Fig. 3) to diamagnetic (Fig. 5), or vice versa. Specifically, when all scale parameters are real, the relaxed structures exhibit paramagnetism, while introducing a pair of complex conjugate scale parameters can induce diamagnetic behavior.

On the other hand, in a rectangular geometry with an internal conductor, the nature of relaxed structures tends to remain paramagnetic, irrespective of whether all associated scale parameters are real or if only two are real while the others form a pair of complex conjugate. Notably, a significant change observed from these plots is that when the scale parameters form a pair of complex conjugate and only two are real, the strength of the magnetic field doubles compared to when all scale parameters associated with the relaxed structures are real, as depicted in Figs. 4 and 6.

As the dust particles are massive in comparison to electrons and ions, so the concentration of dust particles in the plasmas is also very important to describe the nature of the relaxed structures. Figures 7–10 shows the plots for different densities of the plasma species keeping constant all other plasma parameters as for Figs. 3 and 6 and the Beltrami parameters are also constant. For the density of plasma species n e = 2 × 10 7, n d = 2 × 10 4, and n i = 2.2 × 10 7, the plots of the magnetic fields are shown in Figs. 7 and 8 for simple slab geometry and a coplanar rectangular geometry. The associated scale parameters are λ 1 = 2.067 51 , λ 2 = 0.484 689 , λ 3 = 1.2988, and λ 4 = 0.6. In both graphs, it has been observed the magnetic field is decreased by increasing the distance toward the edges, which depicts the paramagnetism trend.

FIG. 7.

Plot of the magnetic field (B) for the density of plasma species n e = 2 × 10 7 , n d = 2 × 10 4, and n i = 2.2 × 10 7 for simple slab geometry. The scale parameters are λ 1 = 2.067 51 , λ 2 = 0.484 689 , λ 3 = 1.298 8, and λ 4 = 0.6.

FIG. 7.

Plot of the magnetic field (B) for the density of plasma species n e = 2 × 10 7 , n d = 2 × 10 4, and n i = 2.2 × 10 7 for simple slab geometry. The scale parameters are λ 1 = 2.067 51 , λ 2 = 0.484 689 , λ 3 = 1.298 8, and λ 4 = 0.6.

Close modal
FIG. 8.

Plot of the magnetic field (B) for the density of plasma species n e = 2 × 10 7 , n d = 2 × 10 4, and n i = 2.2 × 10 7 for the slab geometry with an internal conductor. The scale parameters are λ 1 = 2.067 51 , λ 2 = 0.484 689, λ 3 = 1.2988, and λ 4 = 0.6.

FIG. 8.

Plot of the magnetic field (B) for the density of plasma species n e = 2 × 10 7 , n d = 2 × 10 4, and n i = 2.2 × 10 7 for the slab geometry with an internal conductor. The scale parameters are λ 1 = 2.067 51 , λ 2 = 0.484 689, λ 3 = 1.2988, and λ 4 = 0.6.

Close modal
FIG. 9.

Plot of the magnetic field (B) for the density of plasma species n e = 2 × 10 3 , n d = 2 × 10 5, and n i = 20002 × 10 3 for simple slab geometry. The scale parameters are λ 1 = 2.413 14 , λ 2 = 0.6 , λ 3 = 0.718 431 + 2.403 89 i, and λ 4 = 0.718 431 2.403 89 i.

FIG. 9.

Plot of the magnetic field (B) for the density of plasma species n e = 2 × 10 3 , n d = 2 × 10 5, and n i = 20002 × 10 3 for simple slab geometry. The scale parameters are λ 1 = 2.413 14 , λ 2 = 0.6 , λ 3 = 0.718 431 + 2.403 89 i, and λ 4 = 0.718 431 2.403 89 i.

Close modal
FIG. 10.

Plot of the magnetic field (B) for the density of plasma species n e = 2 × 10 3 , n d = 2 × 10 5, and n i = 20002 × 10 3 for the slab geometry with an internal conductor. The scale parameters are λ 1 = 2.413 14 , λ 2 = 0.6, λ 3 = 0.718 431 + 2.403 89 i, and λ 4 = 0.718 431 2.403 89 i.

FIG. 10.

Plot of the magnetic field (B) for the density of plasma species n e = 2 × 10 3 , n d = 2 × 10 5, and n i = 20002 × 10 3 for the slab geometry with an internal conductor. The scale parameters are λ 1 = 2.413 14 , λ 2 = 0.6, λ 3 = 0.718 431 + 2.403 89 i, and λ 4 = 0.718 431 2.403 89 i.

Close modal

By increasing the dust particle concentration n e = 2 × 10 3, n d = 2 × 10 5, and n i = 20002 × 10 3, the magnetic profiles (9) and (10) are plotted. For these figures, two of the scale parameters are real λ 1 = 2.413 14 and λ 2 = 0.6 and two of them are a pair of complex conjugate λ 3 = 0.718 431 + 2.403 89 i and λ 4 = 0.718 431 2.403 89 i. In a simple rectangular geometry, the magnetic field trend shifts from paramagnetic to diamagnetic as the density of dust particles increases. However, in a coplanar rectangular geometry, increasing the density of dust particles does not alter the magnetic field trend. Instead, it results in a significant 50-fold increase in the strength of the magnetic field. The findings are backed by experimental data detailed in the 2013 Aurora Campaign within Saturn's magnetosphere.19 These observations entail the detection of interplanetary magnetic field configurations.20 Moreover, the existence of periodic cyclonic hot vortex structures in the Saturn's ring observed through Cassini/VIMS.22 Several other experimental investigations23–29 also provide explanations for the existence of strong periodic cyclonic vortex structures.

These graphs reveal that in a simple rectangular geometry, the magnetic trend shifts from paramagnetism to diamagnetism when the scale parameters are real, while it remains consistent when there is a pair of complex conjugate parameters. However, in a coplanar rectangular geometry, the trend remains the same regardless of the scale parameters, but the magnitude of the magnetic field increases significantly when there is a pair of complex conjugate parameters. Additionally, increasing the concentration of dust particles leads to a change in the nature of the scale parameters from all real to only two real and two complex conjugates. Such results are useful to describe the underline physics of the Planetary rings. The trend of the magnetic profiles is 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 center which manifests the diamagnetic trend. This shows a clear distinction of the both geometries.

The Beltrami alignment shown in Eqs. (2)–(4) imposes the following generalized Bernoulli conditions, expressing the balance of all remaining potential forces:
( 1 2 V d 2 + M d N d p d Z d M d ϕ ) = 0 ,
(65)
( 1 2 V e 2 + p e ϕ ) = 0 ,
(66)
( 1 2 V i 2 + M i N i p i + M i ϕ ) = 0.
(67)
Integrating the above set of equations, we get
1 2 V d 2 + M d N d p d Z d M d ϕ = f d ,
(68)
1 2 V e 2 + p e ϕ = f e ,
(69)
1 2 V i 2 + M i N i p i + M i ϕ = f i ,
(70)
where fd, fe, and fi are the constants. These equations can be combined as
1 2 ( V d 2 + V e 2 + V i 2 ) ( M d M i + 1 ) ϕ + P = constant ,
(71)
where P = p e + M i p i N i 1 + Z d M d p d N d 1.

The relaxation of a magnetized three-fluid Saturn dusty plasma, consisting of negatively charged dust particles, electrons, and ions, has been investigated theoretically. The theoretical model is constructed by neglecting the gravity and rotation of Saturn as well as assuming uniform density.84,97,98 The Beltrami conditions are achieved by solving the momentum balance equations for the three components with Ampère's law. A Quadruple Beltrami states has been derived by considering three Beltrami parameters. The helicities of dust particles, electrons, and ions and the total energy serve as the four constraints in the present plasma system. Quadruple curl relaxed Beltrami state is a combination of four distinct single relaxed Beltrami states; therefore, the system has four different eigen values. The characteristics of these eigen values are examined as a function of Beltrami parameters and the densities of the plasma components. Furthermore, to investigate the characteristics of the relaxed structures, solutions are derived in two different modes: (i) for a simple rectangular mode (Saturn magnetosphere) and (ii) for a coplanar rectangular configuration (Saturn rings). The impact of Beltrami parameters and the densities of the plasma species are explored to understand the nature of the relaxed structures in Saturn's atmosphere.

Some of the important findings of this investigation are summarized as follows:

The coplanar geometry for the present plasma system may support the paramagnetic nature of the relaxed structures because it has been observed in the coplanar rectangular geometry that the magnetic field decreases with the distance for all cases (both sets of Beltrami parameters as well as for the both sets of densities variation). However, the magnetic strength of the relaxed structures is significantly influenced as the values of Beltrami parameters and the densities of the plasma species are varied. The increment in the concentration of the dust particle grows up the strength of the magnetic field.

Beltrami parameters have a meaning full impact on the magnetic characteristics of the relaxed structures in a simple rectangular geometry. It has been observed that the nature of the relaxed structures, transitioning from paramagnetism to diamagnetism or vice versa, can be transferred by varying the Beltrami parameters. Moreover, it has also been noted that, in the simple rectangular geometry, the higher density of the dust charged particle promotes the diamagnetic structures.

An important observation is that the presence of an internal rectangular chamber enhances the magnitude of the magnetic field. This result indicates that the double configuration (coplanar geometry) enhances the strength of the magnetic field. This outcome is supported by experimental observations presented in the 2013 Aurora Campaign from inside Saturn's magnetosphere,19 which include the identification of interplanetary magnetic field structures,20 as well as the existence of periodic structures in the Saturn's ring such as high-speed cyclonic hot vortex through Cassini/VIMS22 and strong periodic cyclonic vortex structures through several other investigations.23–29 

In the future, the present work can be extended by solving the solution of Quadruple Beltrami state in a simple cylindrical geometry and in a concentric cylindrical geometry to describe the curvature of the planet and its rings.

In this article, we investigate the characteristics of relaxed structures in two different configurations: one without an internal rectangular chamber and the other with an internal rectangular chamber. The internal conductor chamber describes the relevance of Saturn's magnetosphere and its concentric rings. Therefore, it is expected that the present work will also be useful for explaining the physics of relaxed structures in other astronomical objects that have double or more than double configurations, such as Jupiter, Uranus, Neptune, which are enveloped by their rings.

The authors have no conflicts to disclose.

S. M. Gondal: Conceptualization (lead); Data curation (lead); Formal analysis (lead); Funding acquisition (lead); Investigation (lead); Methodology (lead); Project administration (lead); Resources (lead); Software (lead); Supervision (lead); Validation (lead); Visualization (lead); Writing – original draft (lead); Writing – review & editing (lead).

The data that support the findings of this study are openly available in Zenodo at https://doi.org/10.5281/zenodo.10632069, Ref. 99.

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