Nonlinear Magneto Convection in a Rotating Fluid due to Vertical Magnetic Field and Vertical Axis of Rotation

Double Diffusive convection is the process of mixing driven by the interaction of two fluid components. This can occur if the fluid consists of two or more constituents with disparate molecular diffusivity and they must contribute rival to the vertical density gradient [1]. The Double Diffusive convection problem has attracted considerable interest in the last few decades because of its wide range of applications, such as the disposal of waste material, groundwater contamination, chemical transport in packed-bed reactors, food processing and many other engineering applications. Double Diffusive convection is involved in the migration of moisture like transport in environment, magmas, groundwater and in fibrous insulation. The examples of Double Diffusive convection are magnetoconvection, Convection in Earth’s core and thermohaline convection [2].

The study of a magnetic field in a rotating fluid is more interesting in geophysics; Particularly in the research of Earth’s core. Convection in Earth’s outer core is strenuously affected by magnetic and rotational field. Drazin and Reid [3] had examined both experimental and theoretical results on thermal convection in a fluid layer in the presence and absence of a magnetic and rotational fields. The earliest work on magnetoconvection in rotating fluid was carried out by Eltayeb [4], Jones and Roberts [5, 6], Soward [7] and Tagare et al. [8]. Babu et al. [9, 10] had thoroughly investigated the nonlinear convection in the absence and presence of rotation and magnetic fields.

The number of components which must be varied for the bifurcation to occur is known as co dimension of a bifurcation. Pitchfork, Hopf, Takens-Bogdanov, Co dimension two bifurcation have been studied by Babu et al. [9, 10]. Ravi et al. [11] had investigated the stability properties of non-linear plane wave solutions of the form w=W(z)e^iqx+pt. The stability of these solutions has been investigated for both real and complex Landau-Ginzburg equations by using Multiscale analysis [12].

Skew-Varicose, Eckhaus and Zigzag instabilities are examples of long wavelength instabilities. These instabilities, originally identified in the stability theory for convection rolls. These instabilities can arise may be either in stationary convection or oscillatory convection. The Eckhaus instability is an important mechanism which can lead to a considerable change in the wave number. Zigzag instability is the most unstable instability.

Hopf bifurcation gives rise to Standing waves and Travelling waves when the Rayleigh number is increased [13]. Steady state convection can take place at large Rayleigh number. In the case of weak magnetic field Standing waves are stable and for stronger fields Travelling waves are stable [14]. In case of the rotating field, either Standing and Travelling waves are stable, based on the rate of rotation and Prandtl number [15-17].

In the present paper, nonlinear magneto convection in a rotating fluid due to vertical magnetic field and vertical axis of rotation is analyzed. Basic equations for the system, linear stability analysis, two-dimensional amplitude equation, occurrence of secondary instabilities and the transport of heat by convection through Nusselt number are discussed in subsequent sections. The system of nonlinear one-dimensional amplitude equations, Benjamin-Feir instability for Standing and Travelling waves are also examined.

For the linear stability analysis substitute w=W(z)e^i(lx+my)+pt in Lw=0, which is a linearized version of the Eq. (6), hence

$\begin{align}  & [\left( p-\left( {{D}^{2}}-{{q}^{2}} \right) \right)\left( {{D}^{2}}-{{q}^{2}} \right) \\ & {{[\left( \frac{{{\sigma }_{2}}}{{{\sigma }_{1}}}p-\left( {{D}^{2}}-{{q}^{2}} \right) \right)\left( \frac{1}{{{\sigma }_{1}}}p-\left( {{D}^{2}}-{{q}^{2}} \right) \right)-Q{{D}^{2}}]}^{2}} \\ & +Ta{{D}^{2}}\left( p-\left( {{D}^{2}}-{{q}^{2}} \right) \right){{\left( \frac{{{\sigma }_{2}}}{{{\sigma }_{1}}}p-\left( {{D}^{2}}-{{q}^{2}} \right) \right)}^{2}} \\ & +Ta{{D}^{2}}\left( p-\left( {{D}^{2}}-{{q}^{2}} \right) \right){{\left( \frac{{{\sigma }_{2}}}{{{\sigma }_{1}}}p-\left( {{D}^{2}}-{{q}^{2}} \right) \right)}^{2}} \\\end{align}$

$\begin{align} & +R{{q}^{2}}(\frac{{{\sigma }_{2}}}{{{\sigma }_{1}}}p-\left( {{D}^{2}}-{{q}^{2}} \right)) \\ & [\left( \frac{{{\sigma }_{2}}}{{{\sigma }_{1}}}p-\left( {{D}^{2}}-{{q}^{2}} \right) \right)\left( \frac{1}{{{\sigma }_{1}}}p-\left( {{D}^{2}}-{{q}^{2}} \right) \right)-Q{{D}^{2}}]]W(z)=0. \\\end{align}$     (9)

where, $D=\frac{\partial }{\partial z}$, q²=l²+m² and p is the growth rate of the disturbances. Substituting p=iω and W(z)=sinπz in to Eq. (9), we get

$R=\frac{{{k}_{1}}{{\omega }^{8}}+{{k}_{2}}{{\omega }^{6}}+{{k}_{3}}{{\omega }^{4}}+{{k}_{4}}{{\omega }^{2}}+{{k}_{5}}+i\omega \left( {{k}_{6}}{{\omega }^{6}}+{{k}_{7}}{{\omega }^{4}}+{{k}_{8}}{{\omega }^{2}}+{{k}_{9}} \right)}{{{q}^{2}}\left( {{k}_{10}}{{\omega }^{6}}+{{k}_{11}}{{\omega }^{4}}+{{k}_{12}}{{\omega }^{2}}+{{k}_{13}} \right)},$                         (10)

3.1 Stationary convection (ω=0)

For stationary convection substitute ω=0 in to Eq. (10),

${{R}_{s}}=\frac{\delta _{s}^{2}}{q_{s}^{2}}\left[ \delta _{s}^{4}+Q{{\pi }^{2}}+\frac{Ta{{\pi }^{2}}\delta _{s}^{2}}{\delta _{s}^{4}+Q{{\pi }^{2}}} \right],$      (11)

where, $q_{s}^{2}=l_{s}^{2}+m_{s}^{2}$and $\delta _{s}^{2}={{\pi }^{2}}+q_{s}^{2}$. The Rayleigh number R_S represents the stationary convection of Rayleigh number R. The critical value of R_S exists for q_sc, where

$\begin{align}  & \delta _{sc}^{6}-3q_{sc}^{2}\delta _{sc}^{4}+Q{{\pi }^{4}} \\ & +\frac{Ta{{\pi }^{2}}\delta _{sc}^{2}}{\delta _{sc}^{4}+Q{{\pi }^{2}}}\left[ {{\pi }^{2}}-q_{sc}^{2}+\frac{2q_{sc}^{2}\delta _{sc}^{4}}{\delta _{sc}^{4}+Q{{\pi }^{2}}} \right]=0, \\\end{align}$       (12)

where, $q_{sc}^{2}=l_{sc}^{2}+m_{sc}^{2}$and $\delta _{sc}^{2}={{\pi }^{2}}+q_{sc}^{2}$ From Eq. (11) q=q_sc. Therefore

${{R}_{sc}}=\frac{\delta _{sc}^{2}}{q_{sc}^{2}}\left[ \delta _{sc}^{4}+Q{{\pi }^{2}}+\frac{Ta{{\pi }^{2}}\delta _{sc}^{2}}{\delta _{sc}^{4}+Q{{\pi }^{2}}} \right].$      (13)

1.png

Figure 1. Marginal stability curves (Solid lines represents stationary convection and dotted lines represents oscillatory convection) Ta=1000,${{\sigma }_{1}}=1.7$,${{\sigma }_{2}}=0.5$, (a)Q=118, (b)Q=120, (c)Q=121, (d)Q=122

2.png

Figure 2. Marginal stability curves (Solid lines represents stationary convection and dotted lines represents oscillatory convection)$Q=120$,${{\sigma }_{1}}=1.8$,${{\sigma }_{2}}=0.5$,$(a)Ta=1000$,$(b)Ta=2000$,$(c)Ta=3000$,$(d)Ta=3500$

3.png

Figure 3. Marginal stability curves (Solid lines represents stationary convection and dotted lines represents oscillatory convection)$Q=120, Ta=1000, {{\sigma }_{2}}=0.5$,$(a){{\sigma }_{1}}=1.42$,$(b){{\sigma }_{1}}=1.56$,$(c){{\sigma }_{1}}=1.70$,$(d){{\sigma }_{1}}=1.85$

4.png

Figure 4. Marginal stability curves (Solid lines represents stationary convection and dotted lines represents oscillatory convection)$Q=90, Ta=1000, {{\sigma }_{1}}=1.8$,$(a){{\sigma }_{2}}=0.5350$,$(b){{\sigma }_{2}}=0.5375$,$(c){{\sigma }_{2}}=0.5390$,$(d){{\sigma }_{2}}=0.5410$

3.2 Oscillatory convection (ω²=0)

For Oscillatory convection $\omega \ne 0$, the Eq. (10) is complex. The Rayleigh number is always real so that equate imaginary part to zero. i.e.

${{k}_{6}}{{\omega }^{6}}+{{k}_{7}}{{\omega }^{4}}+{{k}_{8}}{{\omega }^{2}}+{{k}_{9}}=0,$      (14)

where, k₆, k₇, k₈ and k₉ are given by relation (7). We get k₆>0, k₇<0, k₈>0 and k₉>0 for $1<{{\sigma }_{2}}<{{\sigma }_{1}}$ or ${{\sigma }_{2}}<{{\sigma }_{1}}<1$ and for some values of other physical components. So by Descarte’s rule there exist at least two positive roots of Eq. (14). From k₉=0, we get

$\begin{align}  & Ta{{s}_{3}}{{\pi }^{2}}{{\delta }^{6}}-{{Q}^{2}}{{s}_{3}}{{\delta }^{4}}-TaQ{{\pi }^{2}}{{\delta }^{2}}({{S}_{2}}+1) \\ & +{{Q}^{3}}({{s}_{2}}-1)=0. \\\end{align}$     (15)

At least three positive roots for Eq. (15) can be obtained. Oscillatory convection exists, if at least one positive root of Eq. (15) should exist. Critical Rayleigh number depends on Q and Ta for stationary convection where as Critical Rayleigh number depends on Q, Ta, ${{\sigma }_{1}}$ and ${{\sigma }_{2}}$ for oscillatory convection. At Takens-Bogdanov bifurcation point we have

${{R}_{c}}({{q}_{c}})={{R}_{s}}({{q}_{s}})={{R}_{o}}({{q}_{o}}) and {{q}_{c}}={{q}_{s}}={{q}_{o}}.$     (16)

The Figures 1-4 predict the relation between wave number and critical Rayleigh number. The solid and dotted lines represent the stationary (Pitchfork bifurcation) and oscillatory (Hopf bifurcation) convections respectively. In Figures 1-4, we have shown the effect $Q, Ta, {{\sigma }_{1}}$ and ${{\sigma }_{2}}$ on critical Rayleigh number. In Figures 1-4, ${{\sigma }_{1}}$ and ${{\sigma }_{2}}$ do not show any effect on solid lines which represent the stationary convection, since R_SC is independent of ${{\sigma }_{1}}$ and ${{\sigma }_{2}}$.

Now consider the region near the onset of oscillatory convection. Here y-axis is taken as the axis of cylindrical rolls, so that y-dependence vanishes from Eq. $Lw=N$. The z-dependence contained entirely in $sine$ and $cosine$ functions. We introduce $\epsilon $ as:

${{\epsilon }^{2}}=\frac{{{R}_{o}}-{{R}_{oc}}}{{{R}_{oc}}}\ll 1.$     (49)

We assume that the solution of the equation $Lw=0$, which satisfies boundary conditions is of the form.

${{w}_{0}}=\left[ \begin{align}  & {{A}_{1L}}{{e}^{i({{l}_{oc}}x+{{m}_{oc}}y+{{\omega }_{oc}}t)}} \\ & +{{A}_{1R}}{{e}^{i({{l}_{oc}}x+{{m}_{oc}}y-{{\omega }_{oc}}t)}}+c.c. \\\end{align} \right]\sin \pi z,$

Here ${{A}_{1L}}$ and ${{A}_{1R}}$ denotes the amplitude of left and right travelling waves of the roll, which depends on slow variables.

$X=\epsilon x,\quad Y={{\epsilon }^{\frac{1}{2}}}y,\quad \tau =\epsilon t,\quad T={{\epsilon }^{2}}t,$     (50)

and assume that ${{A}_{1L}}={{A}_{1L}}(X,\tau ,T)$, ${{A}_{1R}}={{A}_{1R}}(X,\tau ,T)$. We express the differential operators in the form of:

$\begin{align}  & \frac{\partial }{\partial x}\to \frac{\partial }{\partial x}+\epsilon \frac{\partial }{\partial X},\quad \frac{\partial }{\partial y}\to \frac{\partial }{\partial y}+{{\epsilon }^{\frac{1}{2}}}\frac{\partial }{\partial Y},\quad  \\ & \frac{\partial }{\partial t}\to \frac{\partial }{\partial t}+\epsilon \frac{\partial }{\partial \tau }+{{\epsilon }^{2}}\frac{\partial }{\partial T}. \\\end{align}$     (51)

We assume that the solution of basic Equations as power series in $\epsilon $,

$f=\epsilon {{f}_{0}}+{{\epsilon }^{2}}{{f}_{1}}+{{\epsilon }^{3}}{{f}_{2}}+\cdots ,$     (52)

where, $f=f(u,v,w,\theta ,{{H}_{x}},{{H}_{y}},{{H}_{z}})$ and the first approximation is given by eigenvector of the linearized problem:

$\begin{align}  & {{u}_{0}}=\frac{i\pi }{{{l}_{oc}}}[{{A}_{1L}}{{e}^{i({{l}_{oc}}x+{{m}_{oc}}y+{{\omega }_{oc}}t)}} \\ & +{{A}_{1R}}{{e}^{i({{l}_{oc}}x+{{m}_{oc}}y-{{\omega }_{oc}}t)}}-c.c.]\cos \pi z, \\\end{align}$

$\begin{align}& {{v}_{0}}=\frac{\pi T{{a}^{\frac{1}{2}}}}{i{{l}_{oc}}}[{{J}_{1}}{{A}_{1L}}{{e}^{i({{l}_{oc}}x+{{m}_{oc}}y+{{\omega }_{oc}}t)}} \\ & +{{J}_{2}}{{A}_{1R}}{{e}^{i({{l}_{oc}}x+{{m}_{oc}}y-{{\omega }_{oc}}t)}}-c.c.]\cos \pi z, \\\end{align}$

${{w}_{0}}=\left[ \begin{align}  & {{A}_{1L}}{{e}^{i({{l}_{oc}}x+{{m}_{oc}}y+{{\omega }_{oc}}t)}} \\& +{{A}_{1R}}{{e}^{i({{l}_{oc}}x+{{m}_{oc}}y-{{\omega }_{oc}}t)}}+c.c. \\\end{align} \right]\sin \pi z,$

$\begin{align} & {{\theta }_{0}}=[\frac{1}{\delta _{oc}^{2}+i{{\omega }_{oc}}}{{A}_{1L}}{{e}^{i({{l}_{oc}}x+{{m}_{oc}}y+{{\omega }_{oc}}t)}} \\ & +\frac{1}{\delta _{oc}^{2}-i{{\omega }_{oc}}}{{A}_{1R}}{{e}^{i({{l}_{oc}}x+{{m}_{oc}}y-{{\omega }_{oc}}t)}}+c.c.]\sin \pi z, \\\end{align}$

$\begin{align} & {{H}_{{{x}_{0}}}}=\frac{{{\pi }^{2}}}{i{{l}_{oc}}}[{{J}_{3}}{{A}_{1L}}{{e}^{i({{l}_{oc}}x+{{m}_{oc}}y+{{\omega }_{oc}}t)}} \\ & +{{J}_{4}}{{A}_{1R}}{{e}^{i({{l}_{oc}}x+{{m}_{oc}}y-{{\omega }_{oc}}t)}}-c.c.]\sin \pi z, \\\end{align}$       (53)

$\begin{align}  & {{H}_{{{y}_{0}}}}=\frac{i{{\pi }^{2}}T{{a}^{\frac{1}{2}}}}{{{l}_{oc}}}[{{J}_{5}}{{A}_{1L}}{{e}^{i({{l}_{oc}}x+{{m}_{oc}}y+{{\omega }_{oc}}t)}} \\ & +{{J}_{6}}{{A}_{1R}}{{e}^{i({{l}_{oc}}x+{{m}_{oc}}y-{{\omega }_{oc}}t)}}-c.c.]\sin \pi z, \\\end{align}$

$\begin{align}  & {{H}_{{{z}_{0}}}}=\pi [{{J}_{3}}{{A}_{1L}}{{e}^{i({{l}_{oc}}x+{{m}_{oc}}y+{{\omega }_{oc}}t)}} \\& +{{J}_{4}}{{A}_{1R}}{{e}^{i({{l}_{oc}}x+{{m}_{oc}}y+{{\omega }_{oc}}t)}}-c.c.]\cos \pi z. \\\end{align}$

here $\begin{align}  & q_{oc}^{2}=l_{oc}^{2}+m_{oc}^{2}, \delta _{oc}^{2}=q_{oc}^{2}+{{\pi }^{2}}, {{J}_{5}}={{J}_{1}}{{J}_{3}}, {{J}_{6}}={{J}_{2}}{{J}_{4}}, \\ & {{J}_{1}}=\frac{\left( \delta _{oc}^{2}+\frac{1}{{{\sigma }_{1}}}i{{\omega }_{oc}} \right){{\left( \delta _{oc}^{2}+\frac{{{\sigma }_{2}}}{{{\sigma }_{1}}}i{{\omega }_{oc}} \right)}^{2}}}{{{\left( \delta _{oc}^{2}+\frac{1}{{{\sigma }_{1}}}i{{\omega }_{oc}} \right)}^{2}}{{\left( \delta _{oc}^{2}+\frac{{{\sigma }_{2}}}{{{\sigma }_{1}}}i{{\omega }_{oc}} \right)}^{2}}+{{Q}^{2}}{{\pi }^{2}}}, \\ & {{J}_{2}}=\frac{\left( \delta _{oc}^{2}-\frac{1}{{{\sigma }_{1}}}i{{\omega }_{oc}} \right){{\left( \delta _{oc}^{2}-\frac{{{\sigma }_{2}}}{{{\sigma }_{1}}}i{{\omega }_{oc}} \right)}^{2}}}{{{\left( \delta _{oc}^{2}-\frac{1}{{{\sigma }_{1}}}i{{\omega }_{oc}} \right)}^{2}}{{\left( \delta _{oc}^{2}-\frac{{{\sigma }_{2}}}{{{\sigma }_{1}}}i{{\omega }_{oc}} \right)}^{2}}+{{Q}^{2}}{{\pi }^{2}}},  \\ & {{J}_{3}}=\frac{1}{\delta _{oc}^{2}+\frac{{{\sigma }_{2}}}{{{\sigma }_{1}}}i{{\omega }_{oc}}}, {{J}_{4}}=\frac{1}{\delta _{oc}^{2}-\frac{{{\sigma }_{2}}}{{{\sigma }_{1}}}i{{\omega }_{oc}}} \\\end{align}$ and c.c stands for complex conjugate. By employing the definitions of Eq. (47) and Eq. (48) in to Eq. (6) and equating the coefficients of $\epsilon $, ${{\epsilon }^{2}}$, ${{\epsilon }^{3}}$ to zero, we get:

${{L}_{0}}{{w}_{0}}=0,$

${{L}_{1}}{{w}_{0}}+{{L}_{0}}{{w}_{1}}={{N}_{0}},$    (54)

${{L}_{2}}{{w}_{0}}+{{L}_{1}}{{w}_{1}}+{{L}_{0}}{{w}_{2}}={{N}_{1}},$

From the linear equation ${{L}_{0}}{{w}_{0}}=0$, we obtain the critical Rayleigh number. At $O({{\epsilon }^{2}})$, ${{N}_{0}}=0$ and ${{L}_{1}}{{w}_{0}}=0$ gives $\frac{\partial {{A}_{1L}}}{\partial \tau }-{{\nu }_{g}}\frac{\partial {{A}_{1L}}}{\partial X}=0$ and $\frac{\partial {{A}_{1L}}}{\partial \tau }+{{\nu }_{g}}\frac{\partial {{A}_{1L}}}{\partial X}=0$. Where ${{\nu }_{g}}=(\frac{\partial \omega }{\partial q}{{)}_{q={{q}_{sc}}}}$ is the group velocity and is real. Hence we get ${{u}_{1}}=0$. Similarly, the remaining first order solutions are:

${{u}_{1}}=0,$

$\begin{align} & {{v}_{1}}=\frac{\pi T{{a}^{1/2}}}{i{{l}_{sc}}{{\sigma }_{1}}}\{\frac{{{J}_{1}}}{4{{q}^{2}}+\frac{2i{{\omega }_{oc}}}{{{\sigma }_{1}}}}A_{1L}^{2}{{e}^{2i({{l}_{oc}}x+{{m}_{oc}}y+{{\omega }_{oc}}t)}} \\ & +\frac{{{J}_{2}}}{4q_{oc}^{2}-\frac{2i{{\omega }_{oc}}}{{{\sigma }_{1}}}}A_{1R}^{2}{{e}^{2i({{l}_{oc}}x+{{m}_{oc}}y-{{\omega }_{oc}}t)}}  \\ & +\frac{{{J}_{1}}+{{J}_{2}}}{4q_{oc}^{2}}{{A}_{1L}}{{A}_{1R}}{{e}^{2i({{l}_{oc}}x+{{m}_{oc}}y)}}+c.c.\}, \\\end{align}$

${{w}_{1}}=0,$

$\begin{align}  & {{\theta }_{1}}=\pi \{\frac{\delta _{oc}^{2}\left[ |{{A}_{1L}}{{|}^{2}}+|{{A}_{1R}}{{|}^{2}} \right]}{\delta _{oc}^{4}+\omega _{oc}^{2}} \\ & +\frac{{{A}_{1L}}A_{1R}^{*}}{\left( \delta _{oc}^{2}+i{{\omega }_{oc}} \right)\left( 2{{\pi }^{2}}+i{{\omega }_{oc}} \right)}{{e}^{2i{{\omega }_{oc}}t}}+c.c\}\sin 2\pi z, \\\end{align}$

$\begin{align} & {{H}_{{{x}_{1}}}}=\frac{\pi {{\omega }_{oc}}}{{{l}_{oc}}}\frac{\sigma _{2}^{2}}{\sigma _{1}^{2}}\frac{1}{\delta _{oc}^{4}+\omega _{oc}^{2}\frac{\sigma _{2}^{2}}{\sigma _{1}^{2}}} \\ & \left[ |{{A}_{1L}}{{|}^{2}}+|{{A}_{1R}}{{|}^{2}} \right]\sin 2\pi z, \\\end{align}$     (55)

${{H}_{{{y}_{1}}}}=0,$

$\begin{align} & {{H}_{{{z}_{1}}}}=2{{\pi }^{2}}\frac{{{\sigma }_{2}}}{{{\sigma }_{1}}}\{\frac{{{J}_{3}}}{4q_{oc}^{2}+2i{{\omega }_{oc}}}A_{1L}^{2}{{e}^{2i({{l}_{oc}}x+{{m}_{oc}}y+{{\omega }_{oc}}t)}} \\ & +\frac{2\delta _{oc}^{2}}{4{{q}^{2}}\left( \delta _{oc}^{4}+\omega _{oc}^{2}\frac{\sigma _{2}^{2}}{\sigma _{1}^{2}} \right)}{{A}_{1L}}{{A}_{1R}}{{e}^{2i({{l}_{oc}}x+{{m}_{oc}}y)}} \\\end{align}$

$+\frac{{{J}_{4}}}{4q_{oc}^{2}-2i{{\omega }_{oc}}}A_{1R}^{2}{{e}^{2i({{l}_{oc}}x+{{m}_{oc}}y-{{\omega }_{oc}}t)}}+c.c.\}.$

Equating the coefficients of $\sin \pi z$ in ${{N}_{1}}-{{L}_{2}}{{w}_{0}}$ are equal to zero. We obtain:

${{\Lambda }_{0}}\frac{\partial {{A}_{1L}}}{\partial T}+{{\Lambda }_{1}}\left( \frac{\partial }{\partial \tau }-{{v}_{g}}\frac{\partial }{\partial X} \right){{A}_{2L}}-{{\Lambda }_{2}}\frac{{{\partial }^{2}}{{A}_{1L}}}{\partial {{X}^{2}}}$$-{{\Lambda }_{3}}{{A}_{1L}}+{{\Lambda }_{4}}|{{A}_{1L}}{{|}^{2}}{{A}_{1L}}+{{\Lambda }_{5}}|{{A}_{1R}}{{|}^{2}}{{A}_{1L}}=0,$      (56)

${{\Lambda }_{0}}\frac{\partial {{A}_{1R}}}{\partial T}+{{\Lambda }_{1}}\left( \frac{\partial }{\partial \tau }-{{v}_{g}}\frac{\partial }{\partial X} \right){{A}_{2R}}-{{\Lambda }_{2}}\frac{{{\partial }^{2}}{{A}_{1R}}}{\partial {{X}^{2}}}$$-{{\Lambda }_{3}}{{A}_{1R}}+{{\Lambda }_{4}}|{{A}_{1R}}{{|}^{2}}{{A}_{1R}}+{{\Lambda }_{5}}|{{A}_{1L}}{{|}^{2}}{{A}_{1R}}=0.$     (57)

$\begin{align}& {{\Lambda }_{0}}=\delta _{oc}^{2}\left( Q{{\pi }^{2}}+{{e}_{2}}{{e}_{3}} \right)\left[ Q{{\pi }^{2}}+2\frac{{{\sigma }_{2}}}{{{\sigma }_{1}}}{{e}_{1}}{{e}_{2}}+{{e}_{3}}\left( \frac{2{{e}_{1}}}{{{\sigma }_{1}}} \right) \right] \\ & -{{R}_{oc}}q_{oc}^{2}[\frac{e_{3}^{2}}{{{\sigma }_{1}}}+\frac{{{\sigma }_{2}}}{{{\sigma }_{1}}} (Q{{\pi }^{2}}+2{{e}_{2}}{{e}_{3}})] \\ & +Ta{{\pi }^{2}}{{e}_{3}}\left( 2{{e}_{1}}\frac{{{\sigma }_{2}}}{{{\sigma }_{1}}}+{{e}_{3}} \right), \\\end{align}$$\begin{align} & {{\Lambda }_{1}}=\frac{\delta _{oc}^{2}}{\sigma _{1}^{2}}{{e}_{1}}e_{3}^{2}+\frac{2}{{{e}_{1}}}\delta _{oc}^{2}{{e}_{3}}\left( Q{{\pi }^{2}}+{{e}_{2}}{{e}_{3}} \right) \\ & +Ta{{\pi }^{2}}\frac{{{\sigma }_{2}}}{{{\sigma }_{1}}}\left( {{e}_{1}}\frac{{{\sigma }_{2}}}{{{\sigma }_{1}}}+2{{e}_{3}} \right) \\ & +\frac{2}{{{e}_{1}}}\frac{{{\sigma }_{2}}}{{{\sigma }_{1}}}\left( {{e}_{1}}\delta _{oc}^{2}\left( Q{{\pi }^{2}}+2{{e}_{2}}{{e}_{3}} \right)+{{R}_{oc}}q_{oc}^{2}{{e}_{3}} \right) \\\end{align}$

$\begin{align}& +\frac{{{\sigma }_{2}}}{{{\sigma }_{1}}}{{e}_{2}}[{{R}_{oc}}q_{oc}^{2}\frac{{{\sigma }_{2}}}{{{\sigma }_{1}}} \\ & +\delta _{oc}^{2}({{e}_{1}}\left( {{e}_{1}}\frac{{{\sigma }_{2}}}{{{\sigma }_{1}}}+2{{e}_{3}} \right)+2Q{{\pi }^{2}})], \\ & {{\Lambda }_{2}}=Q{{\pi }^{2}}[{{R}_{oc}}-Q{{\pi }^{2}} \\ & -2({{e}_{1}}{{e}_{2}}+{{e}_{2}}{{e}_{3}}+{{e}_{1}}{{e}_{3}}+\delta _{oc}^{2}({{e}_{1}}+{{e}_{2}}+{{e}_{3}}))] \\& +{{R}_{oc}}[{{e}_{2}}(q_{oc}^{2}+2{{e}_{3}})+{{e}_{3}}(2q_{oc}^{2}+{{e}_{3}})] \\ & -2Ta{{\pi }^{2}}({{e}_{1}}{{e}_{2}}+{{e}_{3}})-\delta _{oc}^{2}[2{{e}_{3}}{{e}_{2}}({{e}_{2}}+{{e}_{3}}) \\ & +{{e}_{1}}(e_{1}^{2}+e_{3}^{2})+4{{e}_{1}}{{e}_{2}}{{e}_{3}}-{{e}_{3}}{{e}_{2}}({{e}_{3}}{{e}_{2}}+2{{e}_{1}}({{e}_{2}}+e_{3}^{3})), \\ & {{\Lambda }_{3}}=R_{oc}^{2}q_{oc}^{2}{{e}_{2}}\left( Q{{\pi }^{2}}+{{e}_{2}}{{e}_{3}} \right), \\\end{align}$

$\begin{align}  & {{\Lambda }_{4}}=\left( Q{{\pi }^{2}}+{{e}_{2}}{{e}_{3}} \right)\{Q{{\pi }^{2}}\frac{\sigma _{2}^{2}}{\sigma _{1}^{2}}\frac{{{e}_{1}}}{{{e}_{4}}}[\frac{i{{\omega }_{oc}}{{\sigma }_{2}}}{{{\sigma }_{1}}} \\ & \left( (3{{\pi }^{2}}-l_{oc}^{2})-\frac{\delta _{oc}^{2}{{\pi }^{2}}}{2{{e}_{3}}{{l}_{oc}}} \right)-\frac{{{\pi }^{2}}{{e}_{3}}}{i{{l}_{oc}}}\frac{{{\pi }^{2}}-2l_{oc}^{2}}{2q_{oc}^{2}+i{{\omega }_{oc}}} \\& +\frac{\delta _{oc}^{2}{{\pi }^{2}}{{\omega }_{oc}}}{2{{l}_{oc}}}] \\\end{align}$

$\begin{align}  & +\frac{\sigma _{2}^{2}}{\sigma _{1}^{2}}\frac{Q\delta _{oc}^{2}{{\pi }^{4}}}{2q_{oc}^{2}+i{{\omega }_{oc}}}-\frac{{{R}_{oc}}{{e}_{3}}{{\pi }^{2}}q_{oc}^{2}\delta _{oc}^{2}}{\delta _{oc}^{4}+\omega _{oc}^{2}}\}+Ta{{e}_{2}}{{e}_{3}} \\ & [\frac{{{\sigma }_{2}}}{{{\sigma }_{1}}}\frac{{{e}_{5}}{{\pi }^{4}}}{4q_{oc}^{2}+2i{{\omega }_{oc}}}-\frac{{{\omega }_{oc}}}{{{e}_{4}}}\frac{\sigma _{2}^{2}}{\sigma _{1}^{2}}(i{{\pi }^{4}}+{{J}_{1}}\frac{{{\omega }_{oc}}}{2}) \\ & -\frac{1}{\sigma _{1}^{2}}\frac{{{J}_{1}}{{\pi }^{3}}}{2q_{oc}^{2}+i{{\omega }_{oc}}}] \\ & {{\Lambda }_{5}}=\left( Q{{\pi }^{2}}+{{e}_{2}}{{e}_{3}} \right)\{Q{{\pi }^{2}}\frac{\sigma _{2}^{2}}{\sigma _{1}^{2}}\frac{{{e}_{1}}}{{{e}_{4}}}[\frac{{{\omega }_{oc}}{{\sigma }_{2}}}{{{\sigma }_{1}}} \\ & \left( i(3{{\pi }^{2}}+l_{oc}^{2})-\frac{\delta _{oc}^{2}{{\pi }^{2}}}{2{{e}_{3}}{{l}_{oc}}} \right)-\pi \frac{\delta _{oc}^{2}}{q_{oc}^{2}}(2{{\pi }^{2}}-4l_{oc}^{2}-\pi \delta _{oc}^{2})] \\ & -\frac{{{R}_{oc}}{{e}_{3}}{{\pi }^{2}}q_{oc}^{2}\delta _{oc}^{2}}{\delta _{oc}^{4}+\omega _{oc}^{2}}\}+Ta{{e}_{2}}{{e}_{3}}\{\frac{{{\pi }^{3}}}{2q_{oc}^{2}}\frac{{{J}_{1}}+{{J}_{2}}}{{{\sigma }_{1}}} \\\end{align}$      (58)

$\begin{align}  & \left( 1+Q{{\pi }^{2}}{{J}_{3}}\frac{{{\sigma }_{2}}}{{{\sigma }_{1}}} \right)+\frac{{{\pi }^{4}}}{{{e}_{4}}}\frac{{{\sigma }_{2}}}{{{\sigma }_{1}}}[\frac{i{{\omega }_{oc}}}{2}\frac{{{\sigma }_{2}}}{{{\sigma }_{1}}}\left( {{J}_{5}}-Q\frac{{{\sigma }_{2}}}{{{\sigma }_{1}}} \right) \\ & +\frac{\delta _{oc}^{2}}{q_{oc}^{2}}\left( Q{{\pi }^{2}}\frac{{{\sigma }_{2}}}{{{\sigma }_{1}}}-\frac{J_{6}^{*}}{2} \right)]\}. \\\end{align}$

where, ${{\Lambda }_{0}}$, ${{\Lambda }_{1}}$, ${{\Lambda }_{2}}$, ${{\Lambda }_{3}}$, ${{\Lambda }_{4}}$ and ${{\Lambda }_{5}}$ are the complex coefficients in physical components ${{q}_{oc}}$, $Q$ and $Ta$. Here, $\begin{align}  & {{e}_{1}}=\delta _{oc}^{2}+i{{\omega }_{oc}}, {{e}_{2}}=\delta _{oc}^{2}+i{{\omega }_{oc}}\frac{1}{{{\sigma }_{1}}},  \\ & {{e}_{3}}=\delta _{oc}^{2}+i{{\omega }_{oc}}\frac{{{\sigma }_{2}}}{{{\sigma }_{1}}}, {{e}_{4}}=\delta _{oc}^{4}+\omega _{oc}^{2}\frac{\sigma _{2}^{2}}{\sigma _{1}^{2}}, \\\end{align}$ ${{e}_{5}}=2Q{{\pi }^{2}}J_{1}^{*}{{J}_{3}}\frac{{{\sigma }_{2}}}{{{\sigma }_{1}}}-Q\pi {{J}_{1}}(1+J_{3}^{*})\frac{1}{{{\sigma }_{1}}}-2i{{\pi }^{2}}J_{5}^{*}\frac{1}{{{e}_{3}}}.$

Here${{A}_{2L}}=(\frac{\partial }{\partial \tau }+{{\nu }_{g}}\frac{\partial }{\partial X}){{A}_{1L}}$ and ${{A}_{2R}}=(\frac{\partial }{\partial \tau }-{{\nu }_{g}}\frac{\partial }{\partial X}){{A}_{1R}}$. ${{A}_{1L}}$, ${{A}_{1R}}$ and ${{A}_{2L}}$, ${{A}_{2R}}$ are of orders $\epsilon $ and ${{\epsilon }^{2}}$ respectively. From the Eqns. $\frac{\partial {{A}_{1L}}}{\partial \tau }-{{\nu }_{g}}\frac{\partial {{A}_{1L}}}{\partial X}=0$ and$\frac{\partial {{A}_{1L}}}{\partial \tau }+{{\nu }_{g}}\frac{\partial {{A}_{1L}}}{\partial X}=0$ we obtain ${{A}_{1L}}({\xi }',T)$ and ${{A}_{1R}}({\eta }',T)$. Where ${\xi }'={{\nu }_{g}}\tau +X$ ${\eta }'={{\nu }_{g}}\tau -X$. Eqns. (52) and (53) can be written as:

$\begin{align}  & 2{{v}_{g}}{{\Lambda }_{1}}\frac{\partial {{A}_{2L}}}{\partial {\eta }'}=-{{\Lambda }_{0}}\frac{\partial {{A}_{1L}}}{\partial T}+{{\Lambda }_{2}}\frac{\partial {{A}_{1L}}}{\partial {{X}^{2}}}+{{\Lambda }_{3}}{{A}_{1L}} \\ & -\left( {{\Lambda }_{4}}|{{A}_{1L}}{{|}^{2}}+{{\Lambda }_{5}}|{{A}_{1R}}{{|}^{2}} \right){{A}_{1L}}, \\\end{align}$      (59)

$\begin{align}  & 2{{v}_{g}}{{\Lambda }_{1}}\frac{\partial {{A}_{2R}}}{\partial {\eta }'}=-{{\Lambda }_{0}}\frac{\partial {{A}_{1R}}}{\partial T}+{{\Lambda }_{2}}\frac{\partial {{A}_{1R}}}{\partial {{X}^{2}}}+{{\lambda }_{3}}{{A}_{1R}} \\& -\left( {{\Lambda }_{4}}|{{A}_{1R}}{{|}^{2}}+{{\Lambda }_{5}}|{{A}_{1L}}{{|}^{2}} \right){{A}_{1R}}. \\\end{align}$      (60)

Let ${\xi }'\epsilon [0,{{l}_{1}}]$, ${\eta }'\epsilon [0,{{l}_{2}}]$ where ${{l}_{1}}$ is the period of ${{A}_{1L}}$ and ${{l}_{2}}$ is the period of ${{A}_{1R}}$ . Eq. (48) remains asymptotic for times $t=O\left( {{\epsilon }^{-2}} \right)$ only if an appropriate solvability condition holds. This condition derived by integrating Eq. (59) over ${\eta }'$ and Eq. (60) over ${\xi }'$, we obtain,

$\begin{align}  & {{\Lambda }_{0}}\frac{\partial {{A}_{1L}}}{\partial T}={{\Lambda }_{2}}\frac{\partial {{A}_{1L}}}{\partial {{X}^{2}}}+{{\lambda }_{3}}{{A}_{1L}} \\ & -\left( {{\Lambda }_{4}}|{{A}_{1L}}{{|}^{2}}+{{\Lambda }_{5}}|{{A}_{1R}}{{|}^{2}} \right){{A}_{1L}}, \\\end{align}$     (61)

$\begin{align} & {{\Lambda }_{0}}\frac{\partial {{A}_{1R}}}{\partial T}={{\Lambda }_{2}}\frac{\partial {{A}_{1R}}}{\partial {{X}^{2}}}+{{\lambda }_{3}}{{A}_{1R}} \\& -\left( {{\Lambda }_{4}}|{{A}_{1R}}{{|}^{2}}+{{\Lambda }_{5}}|{{A}_{1L}}{{|}^{2}} \right){{A}_{1R}}. \\\end{align}$     (62)

The Eqns. (61) and (62) are for the amplitudes of left and right moving waves respectively. These equations are known as one-dimensional coupled Landau-Ginzburg equations with original slow spatial coordinate and time.

5.1 Travelling wave and standing wave convection

On dropping slow variable X from Eqns. (61) and (62), we obtain a system of first order ODE’s.

$\begin{align}  & \frac{\text{d}{{A}_{1L}}}{\text{d}T}=\frac{{{\Lambda }_{3}}}{{{\Lambda }_{0}}}{{A}_{1L}}-\frac{{{\Lambda }_{4}}}{{{\Lambda }_{0}}}{{A}_{1L}}|{{A}_{1L}}{{|}^{2}} \\ & -\frac{{{\Lambda }_{5}}}{{{\Lambda }_{0}}}{{A}_{1L}}|{{A}_{1R}}{{|}^{2}}, \\\end{align}$     (63)

$\frac{\text{d}{{A}_{1R}}}{\text{d}T}=\frac{{{\Lambda }_{3}}}{{{\Lambda }_{0}}}{{A}_{1R}}-\frac{{{\Lambda }_{4}}}{{{\Lambda }_{0}}}{{A}_{1R}}|{{A}_{1R}}{{|}^{2}}-\frac{{{\Lambda }_{5}}}{{{\Lambda }_{0}}}{{A}_{1R}}|{{A}_{1L}}{{|}^{2}}.$    (64)

Put $\beta =\frac{{{\Lambda }_{3}}}{{{\Lambda }_{0}}},\quad \gamma =-\frac{{{\Lambda }_{4}}}{{{\Lambda }_{0}}}\quad and\quad \delta =-\frac{{{\Lambda }_{5}}}{{{\Lambda }_{0}}}.$

Then Eqns. (63) and (64) take the following form:

$\frac{\text{d}{{A}_{1L}}}{\text{d}T}={\beta }'{{A}_{1L}}+{\gamma }'{{A}_{1L}}|{{A}_{1L}}{{|}^{2}}+{\delta }'{{A}_{1L}}|{{A}_{1R}}{{|}^{2}},$    (65)

$\frac{\text{d}{{A}_{1R}}}{\text{d}T}={\beta }'{{A}_{1R}}+{\gamma }'{{A}_{1R}}|{{A}_{1R}}{{|}^{2}}+{\delta }'{{A}_{1R}}|{{A}_{1L}}{{|}^{2}}.$      (66)

Consider ${{A}_{1L}}={{a}_{L}}{{e}^{i{{\phi }_{L}}}}$ and ${{A}_{1R}}={{a}_{L}}{{e}^{i{{\phi }_{R}}}}$, where ${{a}_{L}}=|{{A}_{1L}}|$, ${{\phi }_{L}}=arg({{A}_{1L}})=\mathop{\tan }^{-1}\left( Im({{A}_{1L}})/Re({{A}_{1L}}) \right)$ and ${{a}_{R}}=|{{A}_{1R}}|$, ${{\phi }_{R}}=arg({{A}_{1R}})=\mathop{\tan }^{-1}(\frac{Im({{A}_{1R}})}{Re({{A}_{1R}})})$, here ${{a}_{L}}$, ${{a}_{R}}$, ${{\phi }_{L}}$ and ${{\phi }_{R}}$ are functions of time $T$. Clearly ${{a}_{L}}$ and ${{a}_{R}}$ are positive functions. Putting ${{A}_{1L}}={{a}_{L}}{{e}^{i{{\phi }_{L}}}}$, ${{A}_{1R}}={{a}_{L}}{{e}^{i{{\phi }_{R}}}}$ and ${\beta }'={{\beta }_{1}}+i{{\beta }_{2}}$, ${\gamma }'={{\gamma }_{1}}+i{{\gamma }_{2}}$, ${\delta }'={{\delta }_{1}}+i{{\delta }_{2}}$ into Eqns. (65) and (66) we get,

$\frac{\text{d}{{a}_{L}}}{\text{d}T}={{\beta }_{1}}{{a}_{L}}+{{\gamma }_{1}}{{a}_{L}}|{{a}_{L}}{{|}^{2}}+{{\delta }_{1}}{{a}_{L}}|{{a}_{R}}{{|}^{2}},$     (67)

$\frac{\text{d}{{\phi }_{L}}}{\text{d}T}={{\beta }_{2}}+{{\gamma }_{2}}|{{a}_{L}}{{|}^{2}}+{{\delta }_{2}}|{{a}_{R}}{{|}^{2}},$     (68)

$\frac{\text{d}{{a}_{R}}}{\text{d}T}={{\beta }_{1}}{{a}_{R}}+{{\gamma }_{1}}{{a}_{R}}|{{a}_{R}}{{|}^{2}}+{{\delta }_{1}}{{a}_{R}}|{{a}_{L}}{{|}^{2}},$      (69)

$\frac{\text{d}{{\phi }_{R}}}{\text{d}T}={{\beta }_{2}}+{{\gamma }_{2}}|{{a}_{R}}{{|}^{2}}+{{\delta }_{2}}|{{a}_{L}}{{|}^{2}}.$     (70)

Since the Eqns. (67) and (69) does not contain phase term, so that we consider these two equations for the further discussions. Let us consider the Eqns. (67) and (69) as:

$\frac{\text{d}{{a}_{L}}}{\text{d}T}={{\beta }_{1}}{{a}_{L}}+{{\gamma }_{1}}a_{L}^{3}+{{\delta }_{1}}a_{R}^{2},$     (71)

$\frac{\text{d}{{a}_{R}}}{\text{d}T}={{\beta }_{1}}{{a}_{R}}+{{\gamma }_{1}}a_{R}^{3}+{{\delta }_{1}}a_{L}^{2}.$      (72)

Since ${{a}_{L}}$ and ${{a}_{R}}$ are positive functions. Put,

$\frac{\text{d}{{a}_{L}}}{\text{d}T}={{F}_{1}}({{a}_{L}},{{a}_{R}}),\quad \frac{\text{d}{{a}_{R}}}{\text{d}T}={{F}_{2}}({{a}_{L}},{{a}_{R}}).$     (73)

Let us discuss the stability of equilibrium points of above Eq. (73). We obtain four equilibrium points, i.e.,

1). $({{a}_{L}},{{a}_{R}})=(0,0),$

2). $({{a}_{L}},{{a}_{R}})=({{a}_{L}},0),$

3). $({{a}_{L}},{{a}_{R}})=(0,{{a}_{R}}),$

4). for ${{a}_{L}}\ne 0$ and ${{a}_{R}}\ne 0$ we obtain $({{a}_{L}},{{a}_{R}})=\left( -\frac{{{\beta }_{1}}}{{{\gamma }_{1}}+{{\delta }_{1}}},-\frac{{{\beta }_{1}}}{{{\gamma }_{1}}+{{\delta }_{1}}} \right)$.

From these points we can say the conditions for Travelling Waves and Standing Waves. The conditions for TW and SW are ${{A}_{L}}={{A}_{R}}$ and ${{A}_{L}}=0$ or ${{A}_{R}}=0$.

Travelling Waves exist if $|{{A}_{L}}{{|}^{2}}=-\frac{{{\beta }_{1}}}{{{\gamma }_{1}}}>0$ and they are supercritical if ${{\gamma }_{1}}<0$. Standing Waves exist if $|{{A}_{L}}{{|}^{2}}=|{{A}_{L}}{{|}^{2}}=-\frac{{{\beta }_{1}}}{{{\gamma }_{1}}+{{\delta }_{1}}}>0$ and supercritical if ${{\gamma }_{1}}+{{\delta }_{1}}<0$.

Also, TW are stable if ${{\beta }_{1}}>0,{{\gamma }_{1}}<0$ and ${{\delta }_{1}}<{{\gamma }_{1}}<0$.

SW are stable if ${{\beta }_{1}}>0,{{\gamma }_{1}}<0$ and

(i) if ${{\delta }_{1}}>0$, then $-{{\gamma }_{1}}>{{\delta }_{1}}>0$,

(ii) if ${{\delta }_{1}}<0$, then $-{{\gamma }_{1}}>-{{\delta }_{1}}>0$.

8.png

Figure 8. Stability regions of Travelling waves (TW), Standing waves (SW) and steady state (SS) are plotted for ${{\sigma }_{2}}=0.5$, (a) $Ta=1000$, (b) $Q=1000$

The work reported in this paper is supported by SERB of India (Order No. SB/EMEQ-482/2014). The authors gratefully acknowledge the support thus received.