Nonlinear Magnetoconvection in a Sparsely Packed Porous Medium

Abstract

Linear and weakly nonlinear properties of magnetoconvection in a sparsely packed porous medium are investigated. We have obtained the values of Takens-Bogdanov bifurcation points and codimension two bifurcation points by plotting graphs of neutral curves corresponding to stationary and oscillatory convection for different values of physical parameters relevant to magnetoconvection in a sparsely packed porous medium near a supercritical pitchfork bifurcation. We have derived a nonlinear two-dimensional Ginzburg-Landau equation with real coefficients by using Newell-Whitehead (1969) method. The effect of the parameter values on the stability mode is investigated and shown the occurrence of secondary instabilities namely, Eckhaus and Zigzag instabilities. We have studied Nessult number contribution at the onset of stationary convection. We have also derived two nonlinear one-dimensional coupled Ginzburg-Landau-type equations with complex coefficients near the onset of oscillatory convection at a supercritical Hopf bifurcation and discussed the stability regions of standing and travelling waves.

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.

Magnetoconvection in an electrically conducting fluid in a nonporous medium has been studied extensively [2–8]. However, magnetoconvection in a porous medium has not received any attention inspite of its application in geophysical fluid dynamics problems. Palm et al., [9] investigated Rayleigh-Benard convection problem in a porous medium. Brand and Steinberg [10, 11] investigated convecting instabilities in binary liquid in a porous medium; However, Plam et al. [9] and Brand et al. have made use of Darcy's law (−𝜈∇²𝑉 is replaced by 𝐾𝑉 where 𝐾 is the permeability of a porous medium. for nonporous medium 𝐾 is infinity). They have also not considered usual convective nonlinearity. It is well known that Darcy's law breaks down in situations where in other effects like viscous shear and inertia come into play. In fact Darcy's law is applicable to densely packed porous medium. An alternative to Darcy's equation is Brinkman equation and is of the form∇′𝜌′−𝜌′𝑔=−𝜇𝐾𝑉′+𝜇_𝑒∇′²𝑉′,(1) where 𝜇 is the fluid viscosity and 𝜇_𝑒 is the effective fluid viscosity. Brinkman model is valid for a sparsely packed porous medium wherein there is more window fluid to flow so that the distortion of velocity give rise to the usual shear force. Lapwood [13] was the first to suggest the inclusion of convective term (𝑉^⋅∇′)𝑉′ in the momentum equation and study the Rayleigh-Benard convection in a sparsely packed porous medium. Recently, Tagare and Benerji [14] have investigated the problem of nonlinear convection in a sparsely packed porous medium due to thermal and compositional buoyancy.

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

We consider an electrically and thermally conducting fluid saturating an infinite horizontal layer of a sparsely packed isotropic porous medium of depth 𝑑 with a uniform magnetic field 𝐻₀ in the vertical 𝑧-direction. This layer is heated from below, the upper and lower bounding surfaces of the layer are assumed to be stress-free. Physical properties of the fluid are assumed to be constant, except for the density in the buoyancy term, so that the Boussinesq approximation is valid. The temperature difference across the stress-free boundaries is Δ𝑇′ and the flow in the sparsely packed porous medium is governed by the Darcy-Lapwood-Brinkman model. The relevant basic equations are∇^⋅𝑉′=0,∇^⋅𝐻′=0,(2)𝜌^0𝜙1𝜕𝑉′𝜕𝑡′+𝜙1²−𝜇𝑉^⋅∇^𝑉′_𝑚4𝜋𝐻⁰′𝜕𝐻′𝜕𝑧^=−∇𝑝′+𝜇+_𝑚𝐻^⋅∇^𝐻′8𝜋||𝐻^||²+𝜇^𝑚𝐻₀4𝜋²𝐻^𝑧+𝜌′𝑔−𝜇𝐾𝑉′+𝜇_𝑒∇′²𝑉′,(3)𝑀𝜕𝑇′𝜕𝑡′+𝑉^⋅∇^𝑇′=𝜅∇′²𝑇′,(4)𝜙𝜕𝐻′𝜕𝑡′=∇′×𝑉^×𝐻^0𝑒_𝑧̂+∇′×𝑉^×𝐻^+𝜂∇′²𝐻′.(5) The fluid density 𝜌′ is described by𝜌′=𝜌^01−𝛼𝑇^−𝑇^𝑏,(6) where 𝛼=−𝜌^0−1(𝜕𝜌′/𝜕𝑇′) is thermal expansion coefficient and 𝜌^0 is mean fluid density. Here 𝑝′ is pressure, 𝑉′ is mean fluid velocity, 𝑇′ is temperature, 𝐻′ is magnetic field, 𝜙 is porosity, 𝑔 is acceleration due to gravity, 𝐾 is permeability of porous medium, 𝜇_𝑒 is coefficient of effective fluid viscosity, 𝜅 is thermal diffusivity, 𝜇_𝑚 is magnetic permeability, and 𝜂 is magnetic diffusivity. Equation (3) is known as Darcy-Lapwood-Brinkman equation and is valid for 0.8<𝜙<1. Givler and Altobelli [18] shown that the range of Λ=𝜇_𝑒/𝜇 varies from 0.5 to 10.9. 𝑀 is dimensionless heat capacity and is defined as the ratio of the effective heat capacity of the porous medium to the heat capacity (𝜌′𝐶_𝑝)_𝑓 of the fluid. In a nonporous medium, 𝜙=𝑀=Λ=1 and 𝐾→∞ and (3) reduces to Navier-Stokes equation. In this paper, for sparsely packed porous medium, we consider 𝑀=0.9,𝜙=0.9. The conduction state is characterized by𝑉^𝑠=0,𝑇^𝑠=𝑇^0−Δ𝑇^𝑑𝑧′,(7) and we take the temperature perturbation as 𝜃′=𝑇′−𝑇^𝑠. We use the scaling𝑥=𝑥^𝑢=𝑑,𝑢′𝑦=𝑦𝑑^,𝑧=𝑧𝑑^,𝑡=𝑀𝑑𝑡′²/𝜅,𝜃=𝜃′𝜅/𝑀𝑑,𝑣=𝜅/𝑣𝑀′𝑑,𝑤=𝜅/𝑤𝑀′𝑑,Δ𝑇′,𝑃=𝜌^0𝑀𝑃⁻²′𝜅²𝑑⁻²,𝐻=𝐻′𝜅𝐻₀/𝜂.(8) Here 𝑀𝑑²/𝜅 is thermal diffusion time in a porous medium. Using (6) and (8), we can write basic dimensionless equations for magnetoconvection in a porous medium as∇⋅𝑉=0,∇⋅𝐻=0,(9)1𝑀²𝜙Pr₁𝜕𝑉𝜕𝑡+𝜙1𝑉⋅∇𝑉−𝑄Pr²Pr₁𝐻⋅∇𝐻−𝑄𝜕𝐻=−∇𝑃𝜕𝑧𝑀Pr₁+𝑄2PPrr²₁||𝐻||²+𝑄𝐻_𝑧−𝑀1𝐷_𝑎+Λ𝑉𝑀∇²𝑉+𝑅𝜃𝑒_𝑧̂,(10)𝜕𝜃𝜕𝑡+𝑀1𝑉⋅∇𝜃=𝑤𝑀+∇²𝜃,(11)𝜙Pr²Pr₁𝜕𝐻𝜕𝑡−𝑀∇²𝐻=∇×𝑉×𝑒_𝑧̂+Pr²Pr₁∇×𝑉×𝐻.(12) The dimensionless parameters required for the description of the motion are Rayleigh number 𝑅=𝑔𝛼Δ𝑇𝑑³/𝜅𝜈, thermal Prandtl number Pr₁=𝜈/𝜅, magnetic Prandtl number Pr₂=𝜈/𝜂, Chandrasekhar number 𝑄=𝜇_𝑚𝐻²⁰𝑑²/4𝜋𝜌₀𝜈𝜂, and Darcy number 𝐷_𝑎=𝜅/𝑑². The Curl of (10) gives𝑀²1𝜙Pr₁𝜕𝜕𝑡+𝑀1𝐷_𝑎−Λ𝑀∇²𝜔−𝑄𝜕𝐽=𝑄Pr₂𝜕𝑧−𝑅∇×𝜃𝑒^𝑧̂Pr₁∇×−1𝐻⋅∇𝐻𝑀²𝜙²Pr₁∇×𝑉⋅∇𝑉,(13) where vorticity 𝜔=∇×𝑉, current 𝐽=∇×𝐻 and ∇∇××𝑉⋅∇𝑉=𝑉⋅∇𝜔−𝜔⋅∇𝑉,𝐻⋅∇𝐻=𝐻⋅∇𝐽−𝐽⋅∇𝐻.(14) The Curl of (13) in turn gives, after use of (9),𝑀²1𝜙Pr₁𝜕𝜕𝑡+𝑀1𝐷_𝑎−Λ𝑀∇²∇²−𝑅∇₂𝑉𝜃𝑒_𝑧̂−∇𝜕𝜃𝜕𝑧−𝑄𝜕𝜕𝑧∇²=1𝐻𝑀²𝜙²Pr₁−𝑄Pr₂∇×𝑉⋅∇𝜔−𝜔⋅∇𝑉Pr₁∇×𝐻⋅∇𝐽−𝐽⋅∇𝐻.(15) Now taking the scalar product of (12), (13), and (15) with 𝑒_𝑧̂, we get,𝜙PPrr²₁𝜕𝜕𝑡−𝑀∇²𝐻^𝑧−𝜕𝑤𝜕𝑧=PPrr²₁𝑒_𝑧̂⋅∇×𝑉×𝐻.(16)𝑀²1𝜙Pr₁𝜕𝜕𝑡+𝑀1𝐷_𝑎−Λ𝑀∇²∇²𝜔_𝑧+𝑄𝜕𝐽^𝑧=𝑄Pr₂𝜕𝑧−𝑅𝜕𝜕𝑥𝜃Pr₁𝑒_𝑧̂⋅∇×𝐻⋅∇𝐻−1𝑀²𝜙²Pr₁𝑒_𝑧̂⋅∇×𝑉⋅∇𝑉,(17)𝑀²1𝜙Pr₁𝜕𝜕𝑡+𝑀1𝐷_𝑎−Λ𝑀∇²∇²𝑤−𝑅∇^2ℎ𝜃−𝑄𝜕𝜕𝑧∇²=1𝐻_𝑧𝑀²𝜙²Pr₁−𝑄Pr₂𝑒_𝑧̂⋅∇×𝑉⋅∇𝜔−𝜔⋅∇𝑉Pr₁𝑒_𝑧̂⋅∇×𝐻⋅∇𝐽−𝐽⋅∇𝐻.(18) Geophysically acceptable velocities of propagating instabilities corresponding to geometric scalar variations occur only Pr₂/Pr₁>1 (where instabilities develop in ohmic diffusion timescale 𝑑²/𝜂), Pr₂/Pr₁=2 and 5, when the turbulent is present in the Earth's outer core. In the case of Pr₂/Pr₁≪1 the instabilities are extremely slow depending on the thermal diffusion timescale 𝑑²/𝜅. Using (11), (18), and (16) can be reduced to a formℒ𝑤=𝒩,(19) whereℒ=𝒟_𝜙𝒟_Pr1−𝑄𝜕²𝜕𝑧²𝒟∇²−𝑅𝑀∇^2ℎ𝒟_𝜙,(20)𝒩=𝑄𝒟∇²Pr₂Pr₁𝜕𝜕𝑧+𝒟𝒟_𝜙𝐻⋅∇𝑤−𝑉⋅∇𝐻_𝑧𝑒_𝑧̂⋅1𝑀²𝜙²Pr₁−𝑄Pr₂∇×𝑉⋅∇𝜔−𝜔⋅∇𝑉Pr₁−𝑅∇×𝐻⋅∇𝐽−𝐽⋅∇𝐻𝑀∇^2ℎ𝒟_𝜙𝑉⋅∇𝜃,(21) here𝒟=𝜕𝜕𝑡−∇²,𝒟^𝜙=𝜙Pr²Pr₁𝜕𝒟_Pr1𝜕𝑡−𝑀∇²,=1𝑀²𝜙Pr₁𝜕𝜕𝑡+𝑀1𝐷_𝑎−Λ∇_2ℎ𝑀∇²,=𝜕²𝜕𝑥²,∇²=𝜕²𝜕𝑥²+𝜕²𝜕𝑧².(22)

Boundary Conditions
We assume that fluid is contained between 𝑧=0 and 𝑧=1, where 𝑧=0 corresponds to boundary of solid iron core with Earth's mushy layer and 𝑧=1 corresponds to boundary of Earth's mushy layer with Earth's outer liquid core. For perfectly conducting boundary with temperature, we have 𝜃=0,𝐻_𝑧=0on𝑧=0,𝑧=1∀𝑥,𝑦.(23) Also the normal component of the velocity would vanish on 𝑧=0, 𝑧=1, that is, 𝑤=0on𝑧=0,𝑧=1∀𝑥,𝑦.(24) However, there are two more conditions to be imposed on velocity depending on the nature of the surface. In this paper we consider free-free boundary conditions, that is, on surfaces the tangential stresses vanish, which is equivalent to 𝑃_𝑥𝑧=𝜇𝜕𝑢𝑃𝜕𝑧+𝜕𝜕𝑤𝑥=0,_𝑦𝑧=𝜇𝜕𝑣𝜕𝑧+𝜕𝜕𝑤𝑦=0,(25) where 𝜇=𝛾𝜌₀ is dynamic viscosity. Since 𝑤 vanishes for 𝑥,𝑦 on 𝑧=0, 𝑧=1, it follows that 𝜕𝑢/𝜕𝑧=𝜕𝑣/𝜕𝑧=0 on a free surface 𝑧=0,𝑧=1. Hence from equation of continuity we have 𝜕²𝑤/𝜕𝑧²=0 on 𝑧=0,𝑧=1 for all 𝑥, 𝑦. In this paper we have considered only the idealized stress-free conditions on the surface and vanishing of temperature fluctuations. Thus 𝑤=𝐷²𝑤=𝐷⁴𝑤=0 at 𝑧=0,1. 𝑤 and its even derivatives vanish at 𝑧=0 and 𝑧=1.

3. Linear Stability Analysis

We perform a linear stability analysis of the problem by substituting𝑤=𝑊(𝑧)𝑒^{𝑖𝑞𝑥+𝑝𝑡},(26) into linearized version of (19) is ℒ𝑤=0, and obtaining an equation𝐷²−𝑞²−𝑝𝑀𝐷²−𝑀𝑞²−𝑝𝜙Pr²Pr₁𝐷²×Λ−𝑞²𝑀𝐷²−𝑞²−1𝑀𝐷_𝑎−𝑝=−𝑅𝑞₂𝑀²𝜙Pr₁𝑊𝑀𝑀𝐷²−𝑀𝑞²−𝑝𝜙Pr²+𝑄𝐷₂Pr₁−𝑞²𝐷²−𝑞²−𝑝𝐷²𝑊.(27) We consider stress-free boundary conditions, then 𝑊=𝐷²𝑊=0 on 𝑧=0, 𝑧=1 for all 𝑥,𝑦. Thus we can assume 𝑊=sin𝜋𝑧.

Substituting 𝑊=sin𝜋𝑧 and 𝑝=𝑖𝜔 into (27), we get𝑅=𝑀𝑞²𝐴₁+𝑖𝜔𝐴₂𝜔²+𝐴₃,(28)𝐴₁=𝒦𝑀𝛿⁶𝐷1_𝑎+Λ𝛿²+𝑀𝛿⁴×𝑄𝜋²−𝜔²Pr₂𝑀²Pr²¹−𝜔²𝑀𝜙Pr₁−𝜔²𝜙Λ+𝜔₂𝑀PPrr²₁Pr₂Pr₁PPrr²₁𝛿⁴𝜙²Λ×𝛿₄𝑀−𝑀𝜔𝜙²Pr²¹Λ𝜙+𝑀1Pr₁+𝑄𝜋²+𝛿𝜙₂𝜙Pr₂𝑀𝐷_𝑎Pr₁,(29)𝐴₂=𝒦𝛿²PPrr²₁²𝜙²Λ𝑀+𝑀𝜙²Pr₁+𝜙²Pr²²𝑀𝐷_𝑎Pr²¹,(30)𝐴₃=𝒦𝛿⁴𝑀Λ+𝜙P1r₁+𝑀𝐷𝛿²_𝑎+𝑄𝜋²𝑀−𝜙PPrr²₁,(31) where 𝒦=𝛿²(𝑀⁴𝛿⁴+𝜔²𝜙²Pr²²/Pr²¹)⁻¹ and 𝛿²=(𝜋²+𝑞²), from relation equation (30), 𝐴₂>0.

3.1. Stationary Convection (𝜔=0)

Substituting 𝜔=0 in (28), we get𝑅_𝑠=𝛿^2𝑠𝑞^2𝑠𝛿^2𝑠𝐷1_𝑎+𝛿^2𝑠Λ+𝑄𝜋²,(32) here 𝑅_𝑠 is the value of the Rayleigh number for stationary convection. The minimum value of 𝑅_𝑠 is obtained for 𝑞_𝑠=𝑞_sc. where2Λ𝑞𝜋⁶+3Λ+1𝜋²𝐷𝐷_𝑎𝜋𝑞⁴=Λ+𝑄𝜋²+1𝜋²𝐷𝐷_𝑎.(33) The wave number is identical to that for the single component fluid, while the threshold for the onset of stationary convection at pitchfork bifurcation is given by (34) with 𝑞_𝑠=𝑞_sc,𝑅_sc=𝛿^2sc𝑞²_sc𝛿^2sc𝐷1_𝑎+𝛿²_scΛ+𝑄𝜋²,(34) where 𝛿²_sc=𝜋²+𝑞²_sc. Thus the magnetic field inhibits the onset of stationary convection.

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,𝐴₂𝜔²+𝐴₃=0,(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 have𝜔²=−𝛿^4𝑜𝑀Λ+1/𝜙Pr₁+𝑀𝛿^2𝑜/𝐷_𝑎+𝑄𝜋²𝑀−𝜙Pr₂/Pr₁𝛿^2𝑜Pr₂/Pr₁²𝜙²Λ/𝑀+𝜙/𝑀²Pr₁+𝜙²Pr²²/𝑀𝐷_𝑎Pr²¹,(36)where 𝛿^2𝑜=𝜋²+𝑞^2𝑜. A necessary condition for 𝜔²>0 isPr₂Pr₁>1𝜙.(37) However, this is not sufficient condition and one must have in addition𝑄>𝛿^4𝑜𝑀Λ+1/𝜙Pr₁+𝑀𝛿^2𝑜/𝐷_𝑎𝜋²𝜙Pr₂/Pr₁−𝑀.(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) where𝑄=𝛿^4𝑐𝑀Λ+1/𝜙Pr₁+𝑀𝛿^2𝑐/𝐷_𝑎𝜋²𝜙Pr₂/Pr₁−𝑀,𝑞=𝑞^𝑐.(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→∞) are𝑞_𝑐⟶(𝜙Pr²−𝑀Pr₁)𝑄𝜋²(𝑀ΛPr₁+1/𝜙)^1/4,𝑞_sc⟶𝑄𝜋⁴𝑀2Λ^1/6.(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 have𝑅_ct=𝑅_oc𝑞_oc=𝑅_sc𝑞_scbut𝑞_oc≠𝑞_sc,(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.

(a)

(b)

(c)

(d)

Figure 1 

Numerically calculated marginal stability curves are plotted in (𝑅,𝑞)-plane for Pr1=1, Pr2=1.5, 𝐷𝑎=1500, Λ=0.85, 𝜙=0.9, and 𝑀=0.9,  (a)  𝑄=104, (b)  𝑄=106, (c)  𝑄=108, and (d)  𝑄=1010, then the onset of stationary convection and the onset of oscillatory convection increase (stationary convection represented by solid lines and oscillatory convection represented by dotted lines).

(a)

(b)

(c)

Figure 2 

Neutral curves for the stationary bifurcation (solid lines) and for the Hopf bifurcation (dashed lines) near the codimension two point for 𝑄=2000, 𝐷𝑎=1500, Λ=0.85, 𝜙=0.9, Pr1=1, and 𝑀=0.9, (a)  Pr2=1.27, (b)  Pr2=1.3, (c)  Pr2=1.35  𝑥-axis wave number, 𝑦-Rayleigh numbers 𝑅𝑠,𝑅𝑜.

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 as𝜖²=𝑅−𝑅^sc𝑅_sc.(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 as𝑓=𝜖𝑓₀+𝜖²𝑓₁+𝜖³𝑓₂+⋯,(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:𝑢₀=𝑖𝜋𝑞_sc𝐴(𝑋,𝑌,𝑇)𝑒^{𝑖𝑞sc𝑥}𝑣cos𝜋𝑧−c.c.₀,𝑤=0,₀𝜃=𝐴(𝑋,𝑌,𝑇)𝑒^{𝑖𝑞sc𝑥}sin𝜋𝑧+c.c.,₀=1𝑀𝛿²_sc𝐻𝐴(𝑋,𝑌,𝑇)𝑒^{𝑖𝑞sc𝑥}sin𝜋𝑧+c.c._𝑥0,=−𝑖𝜋²𝑀𝑞_sc𝛿²_sc𝐴(𝐻𝑋,𝑌,𝑇)𝑒^{𝑖𝑞sc𝑥}sin𝜋𝑧−c.c._𝑦0,𝐻=0,_𝑧0=𝜋𝑀𝛿²_sc𝐴(𝑋,𝑌,𝑇)𝑒^{𝑖𝑞sc𝑥}cos𝜋𝑧+c.c.,(44) where 𝛿²_sc=𝜋²+𝑞²_sc, here C.C. stands for complex conjugate, 𝑒^{𝑖𝑞sc𝑥}sin𝜋𝑧 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 scales𝑋𝑍==𝜖𝑥𝑧,,𝑌𝑇==𝜖𝜖^1/2₂𝑦,𝑡,(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𝜕𝜕𝑥⟶𝜕𝜕𝑥+𝜖𝜕𝜕𝑋,𝜕𝜕𝑦⟶𝜖^1/2𝜕𝜕𝜕𝑌,𝜕𝑧⟶𝜕𝜕𝑍,𝜕𝜕𝑡⟶𝜖²𝜕𝜕𝑇(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 asℒ=ℒ₀+𝜖ℒ₁+𝜖²ℒ₂+⋯,(47) whereℒ₀=−Λ∇⁸+1𝐷_𝑎∇⁶+𝑄∇⁴𝜕²𝜕𝑧²+𝑅_sc∇²𝜕²𝜕𝑥²,(48)ℒ₁=2𝜕²×𝜕3𝑥𝜕𝑋+𝜕𝜕𝑌²²𝐷_𝑎∇⁴−4Λ∇⁶+2𝑄∇²𝜕²𝜕𝑧²+𝑅_sc∇²+𝑅_sc𝜕²𝜕𝑥²,(49)ℒ₂=𝜕𝜕𝑇Λ+𝑀𝜙1Pr₁+𝜙Λ𝑀PPrr²₁∇⁶−1𝐷_𝑎+𝜙Pr²Pr₁1𝑀𝐷_𝑎∇⁴−𝑄∇²𝜕²𝜕𝑧²−𝑅^sc𝑀𝜙PPrr²₁𝜕²𝜕𝑥²+𝜕𝜕𝑋²²×3𝐷_𝑎∇⁴−4Λ∇⁶+2𝑄∇²𝜕²𝜕𝑧²+𝑅_sc∇²+𝑅_sc𝜕²+4𝜕₄𝜕𝑥²𝜕𝑥²𝜕𝑋²+1×−6Λ∇₄2𝜕𝜕𝑌²²+3𝐷_𝑎∇²+𝑄𝜕²𝜕𝑧²+𝑅_sc.(50) Similarly nonlinear term 𝒩 is given by𝒩=𝜖²𝒩₀+𝜖³𝒩₁+⋯,(51) substituting (47), (51), and (43) into (19), we get by equating the coefficients of 𝜖,𝜖²,𝜖³;ℒ₀𝑤₀=0,(52)ℒ₀𝑤₁+ℒ₁𝑤₀=𝒩₀,(53)ℒ₀𝑤₂+ℒ₁𝑤₁+ℒ₂𝑤₀=𝒩₁.(54) Equation (48) gives the critical Rayleigh number for the onset of stationary convection𝑅_sc=𝛿^2sc𝑞²_sc𝛿^4scΛ+1𝐷_𝑎𝛿²_sc+𝑄𝜋².(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 𝐻_𝑧1 are𝜕𝜕𝑡−∇²𝜃¹=𝑤𝑀¹−𝑀1𝑢⁰𝜕𝜃₀𝜕𝑥+𝑤⁰𝜕𝜃₀𝜕𝑧,(56) form (56) and (44), we get𝜃₁=−12𝜋𝑀²𝛿²_sc||𝐴||²sin2𝜋𝑧.(57) Equation (12) gives relevant equation for 𝐻_𝑧1 as𝜙PPrr²₁−𝑀∇²𝐻^𝑧1=𝜕𝑤¹𝜕𝑧+PPrr²₁𝜕𝜕𝑥𝑤⁰𝐻_𝑥0−𝑢₀𝐻_𝑧0.(58) From (58) and (44), we get𝐻_𝑧1=Pr²Pr₁⋅𝜋²4𝛿²_sc𝑞²_sc𝐴²𝑒^{2𝑖𝑞sc𝑥}+c.c..(59) Similarly we have 𝐻_𝑥1=0,𝐻_𝑦1=0. The solvability criterion of (54) gives the amplitude equation which can be written as𝜆₀𝜕𝐴𝜕𝑇−𝜆¹𝜕𝜕𝑋−2𝑞𝑖_sc𝜕²𝜕𝑌²²−𝜆₂𝐴+𝜆₃||𝐴||²𝐴=0,(60) where𝜆₀=Λ+1𝑀𝜙Pr₁+𝜙Pr²Pr₁Λ𝑀𝛿^6sc+1𝐷_𝑎+𝜙Pr²Pr₁1𝑀𝐷_𝑎𝛿^4sc+𝑄𝜋²𝛿²_sc−𝑅^sc𝑀PPrr²₁𝜆𝑞²_sc𝜙,₁=4𝑞²_sc6Λ𝛿^4sc+3𝐷_𝑎𝛿²_sc+𝑄𝜋²−𝑅_sc𝜆₂,𝜆₃=𝑅_sc𝑞²_sc𝛿²_sc,=𝑄Pr²²Pr²¹𝜋⁴𝑀²𝑞²_sc𝑞²_sc−𝜋²+𝑅^sc2𝑀²𝑞²_sc.(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 if𝑄<𝑅^sc𝑞⁴_sc2𝜋⁴𝜋²−𝑞²_sc⋅PPrr²¹²².(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 getd²𝐴d𝑋²+𝜆²𝜆₁1−𝜆𝜆³₂||𝐴||²𝐴=0,(63) since 𝜆₁>0, the solution of (63) is given by𝐴(𝑋)=𝐴₀tanh𝑋Λ₁,(64) where𝐴₀=𝜆²𝜆₃^1/2,Λ₁=2𝜆¹𝜆₂^1/2.(65)

Figure 3 

Above figure is plotted for 𝐷𝑎=1500, Λ=0.85, 𝜙=0.9, 𝑀=0.9, and Pr2=1.𝜆3 is the nonlinear coefficient of Ginzburg-Landau equation at the onset of stationary convection. The pitchfork bifurcation is supercritical if 𝜆3>0, subcritical if 𝜆3<0 and 𝜆3=0 on the curve.

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, as𝜆₀𝜕𝐴𝜕𝑡−𝜆¹𝜕𝜕𝑥−2𝑞𝑖_sc𝜕²𝜕𝑦²²𝐴−𝜖²𝜆₂𝐴+𝜆₃||𝐴||²𝐴=0.(66) In order to study the properties of a structure with a given phase winding number 𝛿𝑘, we substitute𝐴(𝑥,𝑦,𝑡)=𝐴₁(𝑥,𝑦,𝑡)𝑒^{𝑖𝛿𝑘𝑥},(67) into (66) and we obtain𝜆₀𝜕𝐴₁𝜕𝑡=𝜖²𝜆₂−𝜆₁(𝛿𝑘)²𝐴₁+2𝑖𝜆₁𝛿𝑘𝜕𝜕𝑥−2𝑞𝑖_sc𝜕²𝜕𝑦²𝐴¹+𝜆₁𝜕𝜕𝑥−2𝑞𝑖_sc𝜕²𝜕𝑦²²𝐴₁−𝜆₃||𝐴₁||²𝐴₁=0.(68) The steady-state uniform solution of (68) is𝐴₁=𝐴_1𝑜=𝜖²𝜆₂−𝜆₁(𝛿𝑘)²𝜆³⁻¹^1/2.(69) Let 𝑢(̃𝑥,𝑦,𝑡)+𝑖𝑣(̃𝑥,𝑦,𝑡) be an infinitesimal perturbation from a uniform steady-state solution 𝐴_1𝑜 given by (69). Now substituting 𝐴₁=𝐴_1𝑜=𝜖²𝜆₂−𝜆₁(𝛿𝑘)²𝜆³⁻¹^1/2+𝑢̃+𝑖𝑣,̃(70) into (68) and equating real and imaginary parts, we obtain 𝜆₀𝜕𝑢̃𝜕𝑡=−2𝜖²𝜆₂−𝜆₁(𝛿𝑘)²+𝜆₁𝜕𝜕𝑥²²+𝛿𝑘𝑞_sc𝜕²𝜕𝑦²−14𝑞²_sc𝜕⁴−2𝜆₁𝜕𝑦⁴𝑢̃𝛿𝑘−𝜆¹𝑞_sc𝜕²𝜆𝜕𝑦²𝜕𝜕𝑥₀𝑣̃,𝜕𝑣̃𝜕𝑡=2𝜆¹𝛿𝑘−𝜆¹𝑞_sc𝜕²+𝜆₁𝜕𝑦²𝜕𝜕𝑥𝑢̃𝜕𝜕𝑥²²+𝛿𝑘𝑞_sc𝜕²𝜕𝑦²−14𝑞²_sc𝜕⁴𝜕𝑦⁴𝑣.̃(71) We analyze (71) by using normal modes of the form𝑢𝑣̃̃==𝑈𝑉𝑒𝑒^𝑆𝑡_𝑆𝑡cos𝑞_𝑥𝑥cos𝑞_𝑦𝑦,sin𝑞_𝑥𝑥cos𝑞_𝑦𝑦.(72) Putting (72) in (71) we get, 𝜆₀𝑆+2𝜖𝜆²𝜆₂−𝜆₁(𝛿𝑘)²+𝜒₁𝑈+𝜆₁𝑞_𝑥𝜒₂𝑉=0,₁𝑞_𝑥𝜒₂𝑈+𝜆₀𝑆+𝜒₁𝑉=0.(73) Here 𝜒₁=𝜆₁[𝑞^2𝑥+(𝑞^2𝑦𝛿𝑘)/𝑞_sc+𝑞^4𝑦/4𝑞²_sc], 𝜒₂=(2𝛿𝑘+𝑞^2𝑦/𝑞_sc). On solving (73) we get,𝜆²⁰𝑆²++22𝑆𝜖₂2𝜆₀𝜖²𝜆₂−𝜆₁(𝛿𝑘)²+𝜆₀𝜒₁𝜆₂−𝜆₁(𝛿𝑘)²+𝜒₁𝜓₁−𝑞^2𝑥𝜆₁𝜒₂=0,(74) whose roots (𝑆±) are real. Here (𝑆±) is defined as(𝑆±)=−1𝜆²⁰2𝜆⁰±2𝜆₀𝜖²𝜆₂−𝜆₁(𝛿𝑘)²+𝜆₀𝜒₁𝜖²𝜆₂−𝜆₁(𝛿𝑘)²²+𝜆²¹𝑞^2𝑥𝜒²²^1/2.(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 get𝜆²⁰𝑆²++𝜆2₁𝑆2𝜆₀𝜖²𝜆₂−𝜆₁(𝛿𝑘)²+𝜆₀𝜆₁𝑞^2𝑥𝑞^2𝑥2𝜖²𝜆₂−3𝜆₁(𝛿𝑘)²+𝑞^2𝑥=0,(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 𝑞^2𝑥≤23𝜆₁𝛿𝑘²−𝜖²𝜆₂,(77) for this requires |𝛿𝑘|≥√(𝜖²𝜆₂/3𝜆₁), this condition defines the domain of Eckhaus instability. The above condition implies that the most unstable wave vector tends to zero, when |𝛿𝑘|→√(𝜖²𝜆₂/3𝜆₁).

4.2.2. Transverse Perturbations and Zigzag Instability

Let us consider 𝑞_𝑥=0 into (75), we get𝜆²⁰𝑆²++22𝑆𝜖₂2𝜆₀𝜖²𝜆₂−𝜆₁(𝛿𝑘)²+𝜆₀𝜒^𝑦1𝜆₂−𝜆₁(𝛿𝑘)²+𝜒^𝑦1𝜒^𝑦1=0,(78) where 𝜒^𝑦1=𝜆₁(𝑞^2𝑦𝛿𝑘/𝑞_sc+𝑞^4𝑦/4𝑞²_sc). The two eigenmodes are uncoupled and we have 𝑆(−),𝑆(−)=−2𝜖²𝜆₂−𝜆₁(𝛿𝑘)²−𝜆¹𝑞_sc𝛿𝑘𝑞^2𝑦−𝜆¹4𝑞²_sc𝑞^2𝑦<0,(79) for one of them. The other is amplified when 𝑆(+)=−𝜆₁𝑞^2𝑦𝛿𝑘+4𝑞𝑞^2𝑦_sc>0.(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 𝑞^2𝑦. 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).

Figure 4 

Numerically computed secondary instability regions of Eckhaus instability (E), Zigzag instability (Z), and stable regions (S) are plotted in (𝜆2/𝜆1,𝛿𝑞𝑠)-plane for 𝑄=2000, 𝐷𝑎=1500, Λ=0.85, 𝜙=0.9, 𝑀=0.9, Pr1=1, and Pr2=2. As |𝛿𝑞𝑠| increases then the secondary instability regions increases.

4.3. Heat Transport by Convection

The maximum of steady amplitude 𝐴 is denoted by |𝐴_max| which is given as||𝐴_max||=𝜖²𝜆₂𝜆₃^1/2.(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 as𝐻=−𝜕𝑇^total𝜕𝑧^_𝑧′=0.(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 asNu=1+𝜖²𝛿²_sc||𝐴_max||².(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.

(a)

(b)

Figure 5 

Graph (a) is plotted for 𝑄=1000 and graph (b) is plotted for 𝑄=3000 for the fixed values of 𝐷𝑎=1500, Λ=0.85, 𝜙=0.9, Pr2=1, Pr1=2, and 𝑀=0.9. in (Nu,𝑅/𝑅sc)-plane. In graphs (a) and (b), as 𝑅/𝑅sc increases then Nu increases.

5. Oscillatory Convection at the Supercritical Hopf Bifurcation

The existence of a threshold (critical value of Rayleigh number for the onset of oscillatory convection 𝑅=𝑅_oc) and a cellular structure (critical wave number 𝑞=𝑞_oc) are main characteristics of the oscillatory convection. In this section we treat the region near the onset of oscillatory convection. Here the axis of cylindrical rolls is taken as 𝑦-axis, so that 𝑦-dependence disappears from equation ℒ𝑤=𝒩. The 𝑧-dependence contained entirely in sine and cosine functions which ensure that the free-free boundary conditions are satisfied. The purpose of this section is to derive coupled one-dimensional nonlinear time-dependent Ginzburg-Landau type equations near the onset of oscillatory convection at supercritical Hopf bifurcation. We introduce 𝜖 as𝜖²=𝑅^𝑜−𝑅_oc𝑅_oc≪1.(84) We assume that𝑤₀=𝐴_1𝐿𝑒^{𝑖(𝑞oc𝑥+𝜔oc𝑡)}+𝐴_1𝑅𝑒^{𝑖(𝑞oc𝑥−𝜔oc𝑡)}+c.c.sin𝜋𝑧(85) is a solution to linearized equation ℒ𝑤=0, which satisfies free-free boundary conditions. Here 𝐴_1𝐿 denotes the amplitude of left travelling wave of the roll and 𝐴_1𝑅 denotes the amplitude of right travelling wave of the roll, which depends on slow space and time variables [23]𝑋=𝜖𝑥,𝜏=𝜖𝑡,𝑇=𝜖²𝑡,(86) and assume that 𝐴_1𝐿=𝐴_1𝐿(𝑋,𝜏,𝑇), 𝐴_1𝑅=𝐴_1𝑅(𝑋,𝜏,𝑇). The differential operators can be expressed as𝜕𝜕𝑥⟶𝜕𝜕𝑥+𝜖𝜕𝜕𝑋,𝜕𝜕𝑡⟶𝜕𝜕𝑡+𝜖𝜕𝜕𝜏+𝜖²𝜕𝜕𝑇.(87) The solution of basic equations can be sought as power series in 𝜖,𝑓=𝜖𝑓₀+𝜖²𝑓₁+𝜖³𝑓₂+⋯,(88) where 𝑓=𝑓(𝑢,𝑣,𝑤,𝜃,𝐻_𝑥,𝐻_𝑦,𝐻_𝑧) with the first approximation given by eigenvector of the linearized problem:𝑢₀=𝑖𝜋𝑞_oc𝐴_1𝐿𝑣𝑒^{𝑖(𝑞oc𝑥+𝜔oc𝑡)}+𝐴_1𝑅𝑒^{𝑖(𝑞oc𝑥−𝜔oc𝑡)}−c.c.cos𝜋𝑧,₀𝐻=0,_𝑦0𝜃=0,₀=1𝑀𝑒1₁𝐴_1𝐿𝑒^{𝑖(𝑞oc𝑥+𝜔oc𝑡)}+1𝑒^∗1𝐻×sin𝜋𝑧,_𝑥0𝐴_1𝑅𝑒^{𝑖(𝑞oc𝑥−𝜔oc𝑡)}+c.c.=−𝑖𝜋²𝑞_oc𝑒1₂𝐴_1𝐿𝑒^{𝑖(𝑞oc𝑥+𝜔oc𝑡)}+1𝑒^∗2𝐻×sin𝜋𝑧,_𝑧0𝐴_1𝑅𝑒^{𝑖(𝑞oc𝑥−𝜔oc𝑡)}−c.c.=𝜋1𝑒₂𝐴_1𝐿𝑒^{𝑖(𝑞oc𝑥+𝜔oc𝑡)}+1𝑒^∗2×cos𝜋𝑧.𝐴_1𝑅𝑒^{𝑖(𝑞oc𝑥−𝜔oc𝑡)}+c.c.(89) where 𝛿²_oc=(𝜋²+𝑞²_oc), 𝑒₁=(𝛿²_oc+𝑖𝜔_oc), and 𝑒₂=(𝑀𝛿²_oc+𝑖𝜔_oc𝜙Pr₂/Pr₁), here 𝑒^∗1 and 𝑒^∗2 are complex conjugate of 𝑒₁ and 𝑒₂.

We expand the linear operator ℒ and nonlinear term 𝒩 as the following power seriesℒ𝒩=ℒ=₀𝜖₂+𝜖ℒ₁+𝜖²ℒ₂+⋯,𝒩₀+𝜖³𝒩₁+⋯,(90) substituting (87) and (88) into ℒ𝑤=𝒩, for each order of 𝜖, we getℒℒ₀𝑤₀=0,(91)₀ℒ𝑤₁+ℒ₁𝑤₀=𝒩₀,(92)₀𝑤₂+ℒ₁𝑤₁+ℒ₂𝑤₀=𝒩₁.(93) Hereℒ₀=𝒟_𝜙𝒟_Pr1−𝑄𝜕²𝜕𝑧²𝒟∇²−𝑅𝑀𝜕𝜕𝑥²²ℒ₁𝒟_𝜙,=𝜕𝜕𝜏ℱ¹+2𝜕²𝜕𝑥𝑋ℱ²ℒ₂,=𝜕ℱ¹𝜕𝑇+𝜕𝑥𝜕⁴²𝑋²𝑀∇²−𝒟_𝜙+𝑀𝒟𝒟_Pr1+Λ𝒟∇²−Λ𝑀𝒟𝒟^𝜙+Λ𝑀𝒟^𝜙∇²+𝑄𝜕²𝜕𝑧²+𝑅+𝜕²ℱ₂𝜕𝑋²+2𝜕²×𝒟𝜕𝑥𝑋𝜕𝜕𝜏_𝜙𝒟_Pr1−𝑀+𝜙Pr²Pr₁𝒟^Pr1∇²−𝜙Pr²Pr₁Λ𝑀+𝑀𝜙1Pr₁𝒟∇²+1𝑀²𝜙Pr₁𝒟𝒟_𝜙+𝜙Pr²Pr₁𝒟𝒟_Pr1−Λ𝑀+𝑀²1𝜙Pr₁𝒟^𝜙∇²−𝑄𝜕²𝜕𝑧²−𝜙𝑅Pr²+𝜕₂𝑀Pr₁𝜕𝜏²𝜙PPrr²₁𝒟_Pr1∇²+Pr²𝑀²Pr²¹𝒟∇²+1𝑀²𝜙Pr₁𝒟_𝜙∇²−𝑅𝑀∇^2ℎ𝒟_𝜙,(94) whereℱ₁=𝒟_𝜙𝒟_Pr1+𝜙Pr²Pr₁𝒟𝒟_Pr1+1𝑀²𝜙Pr₁𝒟𝒟_𝜙∇²−𝑄∇²𝜕²𝜕𝑧²−𝜙𝑅Pr²𝑀Pr₁ℱ₂∇^2ℎ,=𝒟𝒟_𝜙−𝒟_𝜙∇²−𝑀𝒟∇²𝒟_Pr1−Λ𝑀𝒟𝒟^𝜙∇²+𝑄∇²𝜕²𝜕𝑧²−𝑄𝒟𝜕²𝜕𝑧²−𝑅𝑀𝒟^𝜙+𝑅∇^2ℎ.(95) Equation (91) is linear problem. We get critical Rayleigh number for the onset of oscillatory convection by using the zeroth-order solution 𝑤₀ in (91). At 𝑂(𝜖²),𝒩₀=0 and ℒ₁𝑤₀=0 gives𝜕𝐴_1𝐿𝜕𝜏−𝑣^𝑔𝜕𝐴_1𝐿𝜕𝑋=0,𝜕𝐴𝜕𝜏^1𝑅−𝑣^𝑔𝜕𝐴_1𝑅𝜕𝑋=0,(96) where 𝑣_𝑔=(𝜕𝜔/𝜕𝑞)_𝑞=𝑞oc is the group velocity and is real. Hence from (92), we get 𝑤₁=0. From equation of continuity we find that 𝑢₁=0. Substituting the zeroth-order and first-order approximation into (56) and (58) we get,𝜃₁=−𝜋𝑀²||𝐴^1𝐿||²+||𝐴_1𝑅||²𝑣𝑡¹+𝑒₁2𝑒₄𝐽₁+𝑒^∗12𝑒^∗4𝐽^∗1sin2𝜋𝑧,₁𝐻=0,_𝑦1𝐻=0,_𝑥1=𝑖𝜋2𝑀𝑞_ocPr₂Pr₁𝑒1₂−1𝑒^∗2||𝐴^1𝐿||²−||𝐴_1𝑅||²𝐻sin2𝜋𝑧,_𝑧1=2𝜋²Pr₂Pr₁𝑒₂1𝑒₅𝐴^21𝐿𝑒^{2𝑖(𝑞oc𝑥+𝜔oc𝑡)}+1𝑒^∗2𝑒^∗5𝐴^21𝑅𝑒^{2𝑖(𝑞oc𝑥−𝜔oc𝑡)}+14𝑀𝑞²_oc𝑒1₂+1𝑒^∗2𝐴^1𝐿𝐴_1𝑅𝑒^{2𝑖𝑞oc𝑥}+c.c.,(97) where 𝑡₁=(1/4𝜋²)(1/𝑒₁+1/𝑒^∗1), 𝐽₁=𝐴_1𝐿𝐴^∗1𝑅𝑒^{2𝑖𝜔oc𝑡}, 𝑒₄=(4𝜋²+2𝑖𝜔_oc), and 𝑒₅=(4𝑀𝑞²_oc+2𝑖𝜙𝜔_ocPr₂/Pr₁) and 𝑒^∗4, 𝑒^∗5 and 𝐽^∗1 are complex conjugate of 𝑒₄,𝑒₅ and 𝐽₁, respectively.

Equation (93) is solvable when ℒ₀𝑤₀=0, one requires that its right-hand side be orthogonal to 𝑤₀, which is ensured that if the coefficients of sin𝜋𝑧 in 𝒩₁−ℒ₂𝑤₀ are equal to zero. This implies thatΛ₀𝜕𝐴_1𝐿𝜕𝑇+Λ¹𝜕𝜕𝜏−𝑣^𝑔𝜕𝜕𝑋𝐴^2𝐿−Λ₂𝜕²𝐴_1𝐿𝜕𝑋²−Λ₃𝐴_1𝐿Λ+Λ₄||𝐴_1𝐿||²𝐴_1𝐿+Λ₅||𝐴_1𝑅||²𝐴_1𝐿=0,₀𝜕𝐴_1𝑅𝜕𝑇+Λ¹𝜕𝜕𝜏−𝑣^𝑔𝜕𝜕𝑋𝐴^2𝑅−Λ₂𝜕²𝐴_1𝑅𝜕𝑋²−Λ₃𝐴_1𝑅+Λ₄||𝐴_1𝑅||²𝐴_1𝑅+Λ₅||𝐴_1𝐿||²𝐴_1𝑅=0,(98) whereΛ₀=1𝑀²𝜙Pr₁𝑒₁𝑒₂+𝑒₂𝑒₃+𝜙Pr²Pr₁𝑒₁𝑒₃+𝑄𝜋²𝛿^2oc−𝑅^oc𝑞²_oc𝜙Pr₂𝑀Pr₁Λ₁,=𝛿²_oc𝑒³𝜙Pr²Pr₁+𝑒¹Pr₂𝑀²Pr²¹+𝑒²Λ₂𝑀²𝜙Pr₁,=4𝑞²_oc𝑒²𝑒₃+𝑀𝑒₃𝛿²_oc+Λ𝑒₁𝛿²_oc+Λ𝑀𝑒¹𝑒₂+𝑀𝑒₁𝑒₃+Λ𝑀𝑒²Λ𝛿²_oc+𝑄𝜋²−𝑅₃,=𝑅𝑀𝑞^2ocΛ₄𝑒₂,=−𝑄𝜋⁴𝑀PPrr²²²¹𝑒₁𝑒1₂−1−2𝑄𝜋₄𝑒^∗2Pr²²Pr²¹𝑒₁𝑒₅𝛿1²_oc+𝜋²−3𝑞²_oc𝑒^∗2+𝑀𝑅³Λ₅𝜋²𝑞²_oc𝑡₁𝑒₂,=𝑄𝜋⁴𝑀PPrr²²²¹𝑒₁𝑒1₂−1−𝑄𝜋₄𝑒^∗2𝑀𝑞²_ocPr²²Pr²¹𝑒₁𝜋²−𝑞²_oc1𝑒₂+1+𝑅𝑒^∗2𝑀³𝜋²𝑞²_oc𝑒₂𝑡¹+2𝑒₁𝑒₄.(99) Here 𝑒₃=(𝑖𝜔/𝑀²𝜙Pr+(Λ/𝑀)𝛿²+1/𝑀𝐷_𝑎). It should be noted that 𝐴_1𝐿 and 𝐴_1𝑅 are of order 𝜖 and 𝐴_2𝐿 and 𝐴_2𝑅 are of order 𝜖². If 𝜔_oc=0 in Λ₀,Λ₂,Λ₃, and Λ₄ then these expressions match with the coefficients 𝜆₀, 𝜆₁, 𝜆₂, and 𝜆₃ of Ginzburg-Landau equation at the onset of stationary convection. From (96), we get 𝐴_1𝐿(𝜉′,𝑇) and 𝐴_1𝑅(𝜂^,𝑇), where 𝜉^=𝑣_𝑔𝜏+𝑋,𝜂^=𝑣_𝑔𝜏−𝑋. Equations (98) can be written as2𝑣_𝑔Λ₁𝜕𝐴_2𝐿𝜕𝜂^=−Λ₀𝜕𝐴_1𝐿𝜕𝑇+Λ²𝜕𝐴_1𝐿𝜕𝑋²+𝜆₃𝐴_1𝐿2𝑣_𝑔−Λ₄||𝐴_1𝐿||²+Λ₅||𝐴_1𝑅||²𝐴^1𝐿,(100)Λ₁𝜕𝐴_2𝑅𝜕𝜂^=−Λ₀𝜕𝐴_1𝑅𝜕𝑇+Λ²𝜕𝐴_1𝑅𝜕𝑋²+𝜆₃𝐴_1𝑅−Λ₄||𝐴_1𝑅||²+Λ₅||𝐴_1𝐿||²𝐴^1𝑅.(101)

Let 𝜉′𝜖[0,𝑙₁], 𝜂′𝜖[0,𝑙₂] where 𝑙₁,𝑙₂ are periods of 𝐴_1𝐿,𝐴_1𝑅, respectively. Expansion (88) remains asymptotic for times 𝑡=𝑂(𝜖⁻²) only if an appropriate solvability condition holds. This condition obtained integrating (100) over 𝜂^ and (101) over 𝜉^, we getΛ₀𝜕𝐴_1𝐿𝜕𝑇=Λ²𝜕𝐴_1𝐿𝜕𝑋²+𝜆₃𝐴_1𝐿Λ−Λ₄||𝐴_1𝐿||²+Λ₅||𝐴_1𝑅||²𝐴^1𝐿,(102)₀𝜕𝐴_1𝑅𝜕𝑇=Λ²𝜕𝐴_1𝑅𝜕𝑋²+𝜆₃𝐴_1𝑅−Λ₄||𝐴_1𝑅||²+Λ₅||𝐴_1𝐿||²𝐴^1𝑅.(103)

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'sd𝐴_1𝐿d𝑇=ΛΛ³₀𝐴_1𝐿−Λ⁴Λ₀𝐴_1𝐿||𝐴_1𝐿||²−Λ⁵Λ₀𝐴_1𝐿||𝐴_1𝑅||²,(104)d𝐴_1𝑅d𝑇=ΛΛ³₀𝐴_1𝑅−Λ⁴Λ₀𝐴_1𝑅||𝐴_1𝑅||²−Λ⁵Λ₀𝐴_1𝑅||𝐴_1𝐿||².(105) Put𝛽^=Λ³Λ₀,𝛾^=−Λ⁴Λ₀,𝛿^=−Λ⁵Λ₀.(106) Then (104) and (105) take the following formd𝐴_1𝐿d𝑇=𝛽′𝐴^1𝐿+𝛾′𝐴_1𝐿||𝐴_1𝐿||²+𝛿′𝐴_1𝐿||𝐴_1𝑅||²,(107)d𝐴_1𝑅d𝑇=𝛽′𝐴^1𝑅+𝛾′𝐴_1𝑅||𝐴_1𝑅||²+𝛿′𝐴_1𝑅||𝐴_1𝐿||².(108) Consider 𝐴_1𝐿=𝑎_𝐿𝑒^𝑖𝜙𝐿 and 𝐴_1𝑅=𝑎_𝐿𝑒^𝑖𝜙𝑅 (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,d𝑎_𝐿d𝑇=𝛽¹𝑎_𝐿+𝛾₁𝑎_𝐿||𝑎_𝐿||²+𝛿₁𝑎_𝐿||𝑎_𝑅||²,(109)d𝜙_𝐿d𝑇=𝛽²+𝛾₂||𝑎_𝐿||²+𝛿₂||𝑎_𝑅||²,(110)d𝑎_𝑅d𝑇=𝛽¹𝑎_𝑅+𝛾₁𝑎_𝑅||𝑎_𝑅||²+𝛿₁𝑎_𝑅||𝑎_𝐿||²,(111)d𝜙_𝑅d𝑇=𝛽²+𝛾₂||𝑎_𝑅||²+𝛿₂||𝑎_𝐿||².(112) Equations (109) and (111) not contain phase term, so we take these two equations for the future discussions. We have (109) and (111) asd𝑎_𝐿d𝑇=𝛽¹d𝑎_𝑅𝑎_𝐿+𝛾₁𝑎^3𝐿+𝛿₁𝑎^2𝑅,d𝑇=𝛽¹𝑎_𝑅+𝛾₁𝑎^3𝑅+𝛿₁𝑎^2𝐿,(113) since 𝑎_𝐿 and 𝑎_𝑅 are positive functions. Putd𝑎_𝐿d𝑇=𝐹¹𝑎_𝐿,𝑎_𝑅,d𝑎^𝑅d𝑇=𝐹²𝑎_𝐿,𝑎_𝑅.(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^2𝐿=−𝛽₁/𝛾₁(=|𝐴_1𝐿|²)), (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., 𝑎^2𝑅=−𝛽₁/𝛾₁(=|𝐴_1𝑅|²)), 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 ⎛⎜⎜⎜⎝𝜕𝐹¹𝜕𝑎_𝐿𝜕𝐹₁𝜕𝑎_𝑅𝜕𝐹₂𝜕𝑎_𝐿𝜕𝐹₂𝜕𝑎_𝑅⎞⎟⎟⎟⎠.(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 𝐸=𝑎^2𝐿+𝑎^2𝑅. 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).

(a)

(b)

(c)

Figure 6 

(a, b, c) are typical diagrams showing the stability of equilibrium solutions SS (steady state), SW (standing waves), and TW (travelling waves). On solid lines equilibrium solutions are stable and on dotted lines they are unstable.

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.

(a)

(b)

(c)

(d)

Figure 7 

Figures (a–d), are plotted for Pr2/Pr1=6,12,18, and 24, respectively, and for fixed parameters 𝐷𝑎=1500, Λ=0.85, 𝜙=0.9, and 𝑀=0.9. Stability regions of steady state (SS), travelling waves (TW), and standing waves (SW) are plotted (𝑄,Pr1)-plane.

5.2. Long Wavelength Instabilities for the Onset of Travelling Wave Convection (Benjamin-Feir Instability)

For right travelling wave 𝐴_𝑅(𝑋,𝑇)=𝐴(𝑋,𝑇) and 𝐴_𝐿(𝑋,𝑇)=0, for left travelling wave 𝐴_𝑅(𝑋,𝑇)=0 and 𝐴_𝐿(𝑋,𝑇)=𝐴(𝑋,𝑇). Thus for travelling waves we get a single amplitude equation from (102) and (103), given asΛ₀𝜕𝐴𝜕𝑇−Λ²𝜕²𝐴𝜕𝑋²−Λ₃𝐴+Λ₄||𝐴||²𝐴=0.(116) For standing waves 𝐴_1𝐿(𝑋,𝑇)=𝐴_1𝑅(𝑋,𝑇)=𝐴(𝑋,𝑇) and we get a single amplitude equation from (102) and (103), given asΛ₀𝜕𝐴𝜕𝑇−Λ²𝜕²𝜕𝑋²𝐴−Λ₃𝐴+Λ₄+Λ₅||𝐴||²𝐴=0.(117) Equation (117) possesses a family of planar wave solutions and solutions containing phase singular points, which describes weakly nonlinear wave phenomena [25]. We study the Benjamin-Feir instability of travelling waves from complex Ginzburg- Landau equation (116) can be written as𝜕𝐴𝜕𝑇=𝜉𝜕²𝐴𝜕𝑋²+𝛽𝐴+𝛾||𝐴||²𝐴,(118) where 𝜉=𝜉₁+𝑖𝜉₂,𝛽=𝛽₁+𝑖𝛽₂,𝛾=𝛾₁+𝑖𝛾₂. The phase winding solutions are obtained by substituting 𝐴=𝐴_𝑜𝑒^{𝑖(𝛿𝑞𝑜𝑋−𝛿𝜔𝑇)} into (118), and equating real and imaginary parts we get𝛿𝜔=𝜉₂||𝐴_𝑜||²=𝜉₁𝛿𝑞^2𝑜−𝛽₁𝛾¹⁻¹,𝛿𝑞^2𝑜−𝛽₂+𝛾₂𝛽₁−𝜉₁𝛿𝑞^2𝑜𝛾¹⁻¹.(119) Here 𝐴_𝑜 is constant and 𝛿𝑞_𝑜=𝑞𝑋−𝑞_oc. We consider a modulated solution in the form: 𝐴(𝑋,𝑇)=𝐴(𝑋,𝑇)𝑒^{𝑖(𝛿𝑞𝑜𝑋−𝛿𝜔𝑇)}. Substituting the modulated into (118) which gives𝜕𝐴𝜕𝑇=𝛾¹+𝑖𝛾₂𝛽¹−𝛿𝑞^2𝑜𝜉₁𝛾₁+||𝐴||²+𝛾₁𝐴+𝑖𝛾₂𝜕²𝜕𝑋²+2𝑖𝛿𝑞_𝑜𝜕𝜕𝑋𝐴.(120) It is possible to conduct a general investigation of the linear stability of 𝐴(𝑋,𝑇), but this is very difficult task, and therefore our primary concern here is to treat the stability of the uniformly oscillating solution 𝐴_𝑜. Inserting 𝐴=𝐴_𝑜+𝑢̃+𝑖𝑣̃ into (120) and equating real and imaginary parts we get𝜕𝑢̃𝜕𝑇=−2𝛽¹−𝛿𝑞^2𝑜𝜉₁𝑢̃+𝜉₁𝜕²𝑢̃𝜕𝑋²−2𝛿𝑞_𝑜𝜕𝑣̃−𝜉₂𝜕𝑋2𝛿𝑞^𝑜𝜕𝑢̃𝜕𝑣̃𝜕𝑋+𝜕𝜕𝑋𝑣²̃,(121)𝜕𝑇=−2𝛾²𝛽₁−𝛿𝑞^2𝑜𝜉₁𝛾₁𝑢̃+𝜉₁2𝛿𝑞^𝑜𝜕𝑢̃𝜕𝑋+𝜕²𝑣̃+𝜉₂𝜕𝑋²𝜕²𝑢̃𝜕𝑋²−2𝛿𝑞_𝑜𝜕𝑣̃𝜕𝑋.(122) Consider (𝑢,̃𝑣)̃=(𝑈,𝑉)𝑒^𝑆𝑇cos𝑞_𝑋𝑋 and 𝑆 in the growth rate of disturbances. Using solutions of 𝑢,̃𝑣̃, and 𝛿𝑞_𝑜=0 into (121) and (122) we get,𝑆+2𝛽₁+𝜉₁𝑞^2𝑋𝑈−𝑞^2𝑋𝜉₂𝑉=0,(123)𝑆+𝑞^2𝑋𝜉₁𝑉+2𝛽₁𝛾₂𝛾¹⁻¹+𝑞^2𝑋𝜉₂𝑈=0,(124) solving (123) and (124), we get𝑆²++2𝑞𝑆_2𝑋𝛽₁+𝜉₁𝑞^2𝑋+𝑞^2𝑋𝜉₁2𝛽₁+𝜉₁𝑞^2𝑋𝜉₂2𝛽₁𝛾₂𝛾¹⁻¹+𝑞^2𝑋𝜉₂.(125) There will be an instability only when a root of (125) is possible, that is,2𝛽₁𝜉₁+𝛾₂𝜉₂𝛾¹⁻¹+𝑞^2𝑋𝜉²¹+𝜉²²<0,(126)𝛽₁>0 when travelling waves or standing waves are stable. The instability of waves against long wavelength longitudinal modes is often called the Benjamin-Feir instability. Thus we get Benjamin-Feir instability for travelling waves when 𝜉₁+𝛾₂𝜉₂/𝛾₁<0. Similarly by considering (118) instead of (117) and proceeding in the same way we get Benjamin-Feir instability for standing waves when 𝜉₁+(𝛾₂+𝛿₂)𝜉₂/(𝛾₁+𝛿₁)<0.

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 𝑞_sc=𝑞_oc=𝜋/√2. 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.