Nonlinear Magnetoconvection in a Sparsely Packed Porous Medium

1. Introduction

Magnetoconvection in a porous medium uniformly heated from below is of considerable interest in geophysical fluid dynamics, as this phenomena may occur within the mushy layer of Earth′s outer core. Earth′s outer core consists of molten iron and lighter alloying element, sulphur in its molten form. This lighter alloying element present in the liquid phase is released as the new iron freezes due to supercooling onto the solid inner core. Hence we get mushy layer near the inner core boundary where the problem becomes convective instability in a porous medium [1]. The effect of geomagnetic field on the magnetoconvection instability is of interest in geophysics, particular in the study of Earth′s interior where the molten liquid Iron is electrically conducting, which can become convectively unstable as a result of differential diffusion.

In this paper we investigate the problem of magnetoconvection in a sparsely packed porous medium. The multiplicity of control parameters makes this system an interesting one for the study of hydrodynamic stability, bifurcation and turbulence [15]. Rudraiah [16] and Rudraiah and Vortmeyer [17] have studied both linear and steady nonlinear magnetoconvection in a sparsely packed porous medium using Brinkman model but they have taken effective viscosity μ_e same as fluid viscosity μ. However, experiments show that the ratio of effective viscosity μ_e to fluid viscosity μ takes the value ranging from 0.5 to 10.9 [18]. In Section 2, we write basic dimensionless equations in Boussinesq approximation for magnetoconvection in a sparsely packed medium by using a momentum equation with effective viscosity different from fluid viscosity. In Section 3, we study linear stability analysis. In Section 4.1, by using multiple-scale analysis of Newell and Whitehead [19], we derive two-dimensional nonlinear Ginzburg-Landau equation in complex amplitude A(X, Y, T) with real coefficients near the super critical pitchfork bifurcation. In Section 4.2, we show the occurrence of secondary instabilities such as Eckhaus instability and Zigzag instability. We have also considered the effect of Nusselt number on heat transport by magnetoconvection in a sparsely packed porous medium. In Section 5, we derive two nonlinear one-dimensional coupled Ginzburg-Landau type equations with complex coefficients near the onset of oscillatory convection at a supercritical Hopf bifurcation. Following Matthews and Rucklidge [20], we have dropped slow space dependence in X and obtained two coupled ordinary differential equations in A_1R and A_1L and discussed the stability regions of travelling and standing waves. By obtaining a one-dimensional Ginzburg-Landau equation in complex amplitude A(X, Y, T) with complex coefficients near a supercritical Hopf bifurcation, we have shown the condition for occurrence of Benjamin-Feir-type instability [21] for travelling and standing waves. In Section 6, we write conclusions of the paper.

2. Basic Equations

3. Linear Stability Analysis

3.2. Oscillatory Convection (ω² > 0)

For the oscillatory convection (ω ≠ 0) and from (28), R will be complex. But the physical meaning of R requires it to be real. The condition that R is real implies that imaginary part of (28) is zero, that is,

$$ (35)

where A₂ and A₃ are given by (30) and (31). For oscillatory convection ω² = −A₃/A₂ > 0 since A₂ > 0, for oscillatory convection A₃ < 0. For A₃ = 0, (35) implies that ω = 0 is a double zero corresponding to Takens-Bogdanov bifurcation point. For oscillatory convection, we have

$$ (36)

where $$. A necessary condition for ω² > 0 is

$$ (37)

However, this is not sufficient condition and one must have in addition

$$ (38)

At Takens-Bogdanov bifurcation point R_o(q_o) = R_s(q_s) = R_c(q_c), q_o = q_s = q_c, and ω² = 0 is a double zero at Q = Q_c(q_c) where

$$ (39)

The Takens-Bogdanov bifurcation point occurs when the neutral curves for Hopf and pitchfork bifurcation meet and only a single wave number is present, namely, q_o = q_s = q_c. If q_c > q_sc then for all q < q_c the first instability to set in is an oscillatory convection. The asymptotic values of q_c and q_sc for large Chandrasekhar number (Q → ∞) are

$$ (40)

From the monotonic dependence of q_c and q_sc on Q, we may conclude that for Pr₂ > Pr₁, there exists a Q(Pr₁, Pr₂, M, Λ, ϕ, D_a) such that for Q < Q(Pr₁, Pr₂, M, Λ, ϕ, D_a) the onset of first instability will be stationary convection at pitchfork bifurcation while for Q > Q(Pr₁, Pr₂, M, Λ, ϕ, D_a) it will be oscillatory convection at Hopf bifurcation. Q(Pr₁, Pr₂, M, Λ, ϕ, D_a) and for Q = Q(Pr₁, Pr₂, M, Λ, ϕ, D_a), we have

$$ (41)

above condition (41) gives codimension-two bifurcation point. However, there is no simple formula to give Q(Pr₁, Pr₂, M, Λ, ϕ, D_a) at the codimension-two bifurcation point by assuming R as an independent variable, such kind of interesting result is not available in Chandrasekhar [2]. In Figures 1 and 2, each solid line stands for stationary convection (pitchfork bifurcation) and dotted line stands for oscillatory convection (Hopf bifurcation). In Figures 1 and 2, we have showed the effect of several physical parameters, like Q, Pr₁, Pr₂, Λ, M, ϕ, and D_a on the onset of both stationary convection and oscillatory convection when a physical parameter increases for the remaining fixed parameters, the onset of instabilities increases, that is, the onset of stationary convection and oscillatory convection inhibit when a parameter increases with the remaining fixed parameters.

4. Onset of Stationary Convection at Supercritical Pitchfork Bifurcation

4.1. Derivation of Two-Dimensional Nonlinear Ginzburg-Landau Equation Using Newell-Whitehead [19] Method

In this section the evolution of a general pattern is developed by means of a multiple scale analysis used by Newell and Whitehead [19]. A small amplitude convection cell is imposed on the basic flow. If this amplitude is of the size O(ϵ) then the interaction of the cell with itself forces a second harmonic and mean state correction of size O(ϵ²) and then in turn drives an O(ϵ³) correction to the fundamental component of the imposed roll. A solvability criteria for this correction yields the one-dimensional nonlinear Ginzburg-Landau equation of the complex valued amplitude A(X, Y, T) of the imposed disturbance with real coefficients. To simplify the problem we assume the formulation of cylindrical rolls with axis parallel to y-axis, so that y-dependence disappears from (19). The z-dependence is contained entirely in the sine and cosine functions, which ensures that stress-free boundary conditions are satisfied. We use the expansion parameter ϵ as

$$ (42)

For the values of R close to threshold value R_sc that is, ϵ ≪ 1, the structure of the slow length scales will be insensitive to ϵ, but a slow modulation in space and time is possible by making use of the band of the unstable solutions and linear growth rate is likely to saturate due to nonlinear effects. This behavior can be analyzed by writing solutions of (9)–(12) in power series ϵ as

$$ (43)

where f = f(u, v, w, θ, H_x, H_y, H_z) with the first approximation is given by the eigenvector of the linearized problem:

$$ (44)

where $$, here C.C. stands for complex conjugate, $$ is the critical mode for the linear problem at R = R_sc and q = q_sc. The complex amplitude A(X, Y, T) depends on the slow variables X, Y, Z, and T to be scaled by introducing multiple scales

$$ (45)

and these formally separate the fast and slow dependent variables in f. It should be noted that the difference in scaling in the two directions reflects the inherent symmetry breaking of instability which was chosen here with wave vector in x-direction. The differential operators can be expressed as

$$ (46)

with the assumption (46), the operators (20) and (21) are transformed into a set of linear in homogeneous equations. The solvability conditions for the latter yields the amplitude equation using (44) in the linear operator (20) can be written as

$$ (47)

where

$$ (48)

$$ (49)

$$ (50)

Similarly nonlinear term 𝒩 is given by

$$ (51)

substituting (47), (51), and (43) into (19), we get by equating the coefficients of ϵ, ϵ², ϵ³;

$$ (52)

$$ (53)

$$ (54)

Equation (48) gives the critical Rayleigh number for the onset of stationary convection

$$ (55)

In (53), 𝒩₀ = 0, ℒ₁w₀ = 0 and hence w₁ = 0. From equation of continuity we find that u₁ = 0. The relevant equations for θ₁ and $$ are

$$ (56)

form (56) and (44), we get

$$ (57)

Equation (12) gives relevant equation for $$ as

$$ (58)

From (58) and (44), we get

$$ (59)

Similarly we have $$. The solvability criterion of (54) gives the amplitude equation which can be written as

$$ (60)

where

$$ (61)

Equation (60) is two-dimensional, nonlinear time-dependent Ginzburg-Landau equation describing the effect of magneticfield in a sparsely packed porous medium near the onset of stationary convection at supercritical pitchfork bifurcation. Here λ₀ is always positive for Pr₂/Pr₁ < 1/ϕ and for any Q but if Pr₂/Pr₁ > 1/ϕ then λ₀ is positive only if Q < Q_c. Thus for supercritical pitchfork bifurcation λ₀ is always positive. For Pr₂/Pr₁ > 1/ϕ, λ₀ decreases as Q increases and becomes zero at Q = Q_c. λ₁ and λ₂ are always positive. λ₃ is positive only if

$$ (62)

The pitchfork bifurcation is supercritical if λ₃ > 0 and subcritical if λ₃ < 0. At λ₃ = 0, we get tricritical bifurcation point [22] (see Figure 3). Dropping the time dependence from (60), we get

$$ (63)

since λ₁ > 0, the solution of (63) is given by

$$ (64)

where

$$ (65)

4.2. Long Wavelength Instabilities (Secondary Instabilities)

The secondary Instabilities arising in nonequilibrium systems do not exhibit strict symmetries but may show spatially slow deformations of the cellular structures. Further, there are secondary instabilities like Eckhaus and Zigzag instabilities, such phenomena are studied using evolution equations for amplitudes which are slowly varying in time as well as in space. These envelope equations can be derived by the method of Newell and Whitehead [19]. The two-dimensional Ginzburg-Landau equation (60), can be written in fast variables x, y, t, and A(X, Y, T) = A(x, y, t)/ϵ, as

$$ (66)

In order to study the properties of a structure with a given phase winding number δk, we substitute

$$ (67)

into (66) and we obtain

$$ (68)

The steady-state uniform solution of (68) is

$$ (69)

Let $$ be an infinitesimal perturbation from a uniform steady-state solution A_1o given by (69). Now substituting

$$ (70)

into (68) and equating real and imaginary parts, we obtain

$$ (71)

We analyze (71) by using normal modes of the form

$$ (72)

Putting (72) in (71) we get,

$$ (73)

Here $$, $$. On solving (73) we get,

$$ (74)

whose roots (S±) are real. Here (S±) is defined as

$$ (75)

Solution S(−) is clearly negative, thus the corresponding mode is stable and if S(+) is positive then rolls can be unstable. Symmetry considerations help us to restrict the study of S(+) to a domain q_x ≥ 0,   q_y ≥ 0.

4.2.1. Longitudinal Perturbations and Eckhaus Instability

Inserting q_y = 0 into (75), we get

$$ (76)

since the roots are real and their sum always negative, the pattern is stable as long as both roots are negative, that is, their product is positive. The cell pattern becomes unstable when the product is negative, that is, when

$$ (77)

for this requires $$, this condition defines the domain of Eckhaus instability. The above condition implies that the most unstable wave vector tends to zero, when $$.

4.2.2. Transverse Perturbations and Zigzag Instability

Let us consider q_x = 0 into (75), we get

$$ (78)

where $$. The two eigenmodes are uncoupled and we have S(−),

$$ (79)

for one of them. The other is amplified when

$$ (80)

This implies that δk < 0, the above condition defines the domain of the Zigzag Instability. When δk → 0 from below the wave vector q_y of the instability also tends to zero while the growth rate varies as $$. We have studied the effect of magnetic field on long wavelength instabilities. We have observed that Eckhaus instability and Zigzag instability regions increases when Q increases (see Figure 4).

4.3. Heat Transport by Convection

The maximum of steady amplitude A is denoted by |A_max| which is given as

$$ (81)

Equation (81) is obtained from (64) with tanh(X/Λ₁) = 1. We use |A_max| to calculate Nusselt number Nu. To discuss the heat transfer near the neutral region, we express it through the Nusselt number is defined as Nu = Hd/κΔT, which is the ratio of the heat transported across any layer to the heat which would be transported by conduction alone. Here H is the rate of heat transfer per unit area and is defined as

$$ (82)

In (82), angular brackets correspond to a horizontal average. The Nusselt number Nu can be calculated in terms of amplitude A and is given as

$$ (83)

From (83), we get conduction for R ≤ R_sc and convection for R > R_sc. Since the amplitude equation is valid for λ₃ > 0, which is possible for R > R_sc (supercritical pitchfork bifurcation), thus we get Nu > 1 for R > R_sc. We get convection for Nu > 1 and conduction for Nu ≤ 1. In stationary convection Nu increases implies that heat conducted by steady mode increases. In the problem of double diffusive convection in porous medium with magnetic field, Nu depends on Pr₁,   Pr₂,   Λ,   M,   ϕ,   D_a, and Q. We have computed Nu for different values of Q, for some fixed values of other parameters and observed that Nu increases as Q decreases (see Figures 5(a) and 5(b)). This implies that magnetic field inhibits the heat transport. The parameters Pr₁, Pr₂, Λ, M, ϕ, and D_a show the same result as Q shows on Nu.

5. Oscillatory Convection at the Supercritical Hopf Bifurcation

5.1. Travelling Wave and Standing Wave Convection

To study the stability regions of travelling waves and standing waves, Coullet et al. [24]. we proceed as follows.

On dropping slow variable X from (102) and (103), we get a pair of first ODE′s

$$ (104)

$$ (105)

Put

$$ (106)

Then (104) and (105) take the following form

$$ (107)

$$ (108)

Consider $$ and $$ (we can write a complex number in the amplitude and phase form), where a_L = |A_1L|, ϕ_L = arg(A_1L) = tan⁻¹(Im (A_1L)/Re(A_1L)) and a_R = |A_1R|, ϕ_R = arg(A_1R) = tan⁻¹(Im (A_1R)/Re(A_1R)), here a_L, a_R, ϕ_L, and ϕ_R are functions of time T since A_1L and A_1R are functions of T. Thus a_L and a_R are positive functions. Substituting the definitions of A_1L and A_1R and β′ = β₁ + iβ₂, γ′ = γ₁ + iγ₂, δ′ = δ₁ + iδ₂ into (107) and (108) we get,

$$ (109)

$$ (110)

$$ (111)

$$ (112)

Equations (109) and (111) not contain phase term, so we take these two equations for the future discussions. We have (109) and (111) as

$$ (113)

since a_L and a_R are positive functions. Put

$$ (114)

Now we discuss the stability of equilibrium points of (114). We get four equilibrium points like (a_L, a_R) = (0,0) (conduction state), (a_L, a_R) = (a_L, 0) (a_L = amplitude of left travelling waves, here we get F₂ = 0, and we get one condition from F₁ = 0 i.e., $$), (a_L, a_R) = (0, a_R) (a_R = amplitude of right travelling waves, here we get F₁ = 0, and we get one condition from F₂ = 0 i.e., $$), and for a_L ≠ 0 and a_R ≠ 0 we get (a_L, a_R) = (−β₁/(γ₁ + δ₁), −β₁/(γ₁ + δ₁)) (this gives condition for standing waves. At standing waves we have A_L = A_R, so a_L = a_R). For the pair of (104) and (105), we do not get a_L ≠ a_R ≠ 0 (modulated waves). Now the Jacobian of F₁ and F₂ is given by

$$ (115)

If real parts of all eigenvalues of the Jacobian are negative at an equilibrium point, then that point is a stable equilibrium (Lyapunov′s theorem or principle of linearized stability). Some valuable conditions for travelling waves and standing waves are travelling waves are stable if β₁ > 0,   γ₁ < 0 and δ₁ < γ₁ < 0. Standing waves are stable if β₁ > 0,   γ₁ < 0 and (i) if δ₁ > 0, then −γ₁ > δ₁ > 0 and (ii) if δ₁ < 0, then −γ₁ > −δ₁ > 0.

The stability regions of travelling waves and standing waves are summarized in Figure 6. Here E is total amplitude and defined as $$. We do not distinguish between left travelling waves and right travelling waves. For rest state (steady state) E = 0, for travelling waves E = −β₁/γ₁, for standing waves E = −2β₁/(γ₁ + ς₁). Travelling waves are supercritical if γ₁ < 0 and standing waves are supercritical if γ₁ + ς₁ < 0. Figure 6(a) is drawn for stable travelling wave conditions and Figure 6(b) is drawn for stable standing wave conditions in (β₁, E)-plane. The symbols (−, −) and (+, +) in Figures 6(a) and 6(b) indicate that both roots of Jacobian are negative and at least one root is positive between two roots. In Figures 6(a) and 6(b), travelling wave solution and standing wave solution bifurcate simultaneously from the steady-state solution (β₁ ≥ 0 at this bifurcation point).

In these Figures 6(a) and 6(b), steady-state solution is stable for β₁ < 0 and unstable β₁ > 0. These figures show that for β₁ > 0 both travelling waves and standing waves are supercritical. When travelling waves and standing waves bifurcate supercritically then at most one solution among travelling waves and standing waves will be stable. Thus, for β₁ > 0 (Figure 6(a)) travelling waves are stable and (Figure 6(b)) standing waves are stable. In more detail we reproduce results of the stability analysis of equilibrium solutions in Figure 6(c), which is plotted in (γ₁, ς₁)-plane. From this figure we can observe that travelling waves are subcritical for γ₁ > 0 and standing waves are subcritical for γ₁ + ς₁ > 0. In Figure 7, We study the stability regions of travelling waves and standing waves at the onset of Hopf bifurcation. The stability regions of standing waves and travelling waves increases when Pr₂/Pr₁ increases for fixed parameters. For a fixed Pr₁ if we get initially travelling waves at the onset of oscillatory convection then they are replaced by standing waves as Q increases.

6. Conclusions

In this paper we have considered both linear and weakly nonlinear analysis of magnetoconvection in a sparsely packed porous medium in Earth′s outer core by using free-free (stress-free) boundary conditions. Even though free-free boundary conditions cannot be achieved in laboratory, one can use it in geophysical fluid dynamic applications to Earth′s outer core since they allow simple trigonometric eigenfunctions. Our goal is to identify the region of parameter values, for which rolls emerge at the onset of convection.

Following Chandrasekhar [2], we have described the stationary convection and oscillatory convection as curves R_s(q) and R_o(q, Pr₂) versus wave numbers. The critical wave numbers for stationary convection and oscillatory convection are $$. For the problem of magnetoconvection in a sparsely packed porous medium, we get Takens-Bogdanov bifurcation point and codimension-two bifurcation point. In the case of linear theory both marginal and overstable motions are discussed. In Figures 1 and 2, is shown that the effect of Chandrasekhar number and porous parameter is to make the system more stable. By drawing stability boundaries in the Rayleigh number plane it is shown that the effect of magnetic field and porous parameter is to decrease the region of stabilities. In the nonlinear equation (60), λ₀ = 0 gives the Takens-Bogdanov bifurcation point at q_s = q_sc and when λ₀ = 0, (60) is not valid. The pitchfork bifurcation is supercritical if λ₃ > 0 subcritical if λ₃ < 0. and we get tricritical point if λ₃ = 0. We have obtained from (60), long wave length instabilities, namely, Eckhaus and Zigzag instabilities. From (60) which is valid only for λ₃ > 0, we have calculated Nusselt number Nu and studied heat transport by convection. We have also derived two one-dimensional nonlinear coupled Ginzburg-Landau type equations, namely, (98) at the onset of oscillatory convection at supercritical Hopf bifurcation. We have computed stability regions of SW and TW at both Hopf bifurcation. The conditions for SW and TW are A_L = A_R and A_L = 0 or A_R = 0, respectively. TW exist if |A_L|² = −β₁/γ₁ > 0 and they are supercritical if γ₁ < 0. SW exist if |A_L|² = |A_R|² = −β₁/(γ₁ + δ₁) > 0 and SW are supercritical if γ₁ + δ₁ < 0. When both SW and TW are supercritical then at most one equilibrium solution is stable. At Takens-Bogdanov bifurcation point we get both TW and SW. By deriving one-dimensional Ginzburg-Landau equations with complex coefficients, namely, (116) and (117), we have shown the existence of Benjamin-Feir-type of instability for both TW and SW. Near the Takens-Bogdanov bifurcation point the conducting state becomes unstable against both stationary and oscillatory mode, that is, the real parts of two eigenvalues pass through zero simultaneously. This violates the assumption made for deriving amplitude equations (60) and (98). Instead a new equation, which is second order in time, has to be used near the Takens-Bogdanov bifurcation point.