## Abstract

The analysis of part I, dealing with the morphological instability of a single interface in a fluid of infinite extent, is extended to the case of a Cartesian plume of compositionally buoyant fluid, of thickness 2*x*_{0}, enclosed between two vertical interfaces. The problem depends on six dimensionless parameters: the Prandtl number, *σ*; the magnetic Prandtl number, *σ*_{m}; the Chandrasekhar number, *Q*_{c}; the Reynolds number, *Re*; the ratio, *B*_{v}, of vertical to horizontal components of the ambient magnetic field and the dimensionless plume thickness. Attention is focused on the preferred mode of instability, which occurs in the limit *Re*≪1 for all values of the parameters. This mode can be either *sinuous* or *varicose* with the wavenumber vector either *vertical* or *oblique*, comprising four types. The regions of preference of these four modes are represented in regime diagrams in the (*x*_{0}, *σ*) plane for different values of *σ*_{m}, *Q*_{c}, *B*_{v}. These regions are strongly dependent on the field inclination and field strength and, to a lesser extent, on magnetic diffusion. The overall maximum growth rate for any prescribed set of the parameters *σ*_{m}, *Q*_{c}, *B*_{v}, occurs when 1.3<*x*_{0}<1.7, and is sinuous for small *σ* and varicose for large *σ*. The magnetic field can enhance instability for a certain range of thickness of the plume. The enhancement of instability is due to the interaction of the field with viscous diffusion resulting in a reverse role for viscosity. The dependence of the helicity and *α*-effect on the parameters is also discussed.

## 1. Introduction

In part I (Eltayeb *et al*. 2004), we considered the morphological instability of a vertical interface within an electrically conducting fluid of infinite extent. The fluid is composed of two components of differing density and the interface divides portions having differing amounts of the two components. Both portions have the same stabilizing thermal gradient and are threaded by a uniform applied magnetic field. In the present study, we investigate the effect of a second vertical interface parallel to the first so that a channel of finite thickness, 2*x*_{0}, is formed. To facilitate discussions, the channel is assumed to contain fluid that is compositionally more buoyant than the surrounding fluid, although the analysis is valid for the opposite case. We shall refer to this model as the *Cartesian plume*.

This study is motivated by a desire to understand better the morphological instability of plumes of compositionally buoyant material which may occur within Earth's outer core (Loper 1983, 1987; Moffatt 1989). The prototypical shape of such plumes is cylindrical and various aspects of plumes of this shape have been the subject of a number of experimental (Copley *et al*. 1970; Sample & Hellawell 1984; Bergman *et al*. 1997; Classen *et al*. 1999; Jellinek *et al*. 1999) and analytic (Loper 1987; Eltayeb & Loper 1997; Worster 1997; Chung & Chen 2000; Morse 2000) studies. Cylindrical plumes are difficult to analyse, particularly when subject to rotation and hydromagnetic effects, and studies of simpler configurations, which are more amenable to analytic solution, serve to provide insights into the nature of their morphological instabilities. In particular, Eltayeb & Loper (1994, 1997) have shown that the morphological instabilities of Cartesian and cylindrical plumes, in the absence of rotation and hydromagnetic effects, are qualitatively similar. The effect of rotation on these instabilities has been studied by Eltayeb & Hamza (1998), while the effect of hydromagnetic forces are investigated in part I and in this study.

Just as part I is a generalization of Eltayeb & Loper (1991), this study is a generalization of Eltayeb & Loper (1994), with the inclusion of hydromagnetic effects. The Cartesian plume has been shown to possess analytical solutions in the absence of hydromagnetic effects (Eltayeb & Loper 1994), and we shall see below that it does so also in their presence. The analytical solution allows us to examine the influence of the finite width of the plume together with the Lorentz force and magnetic diffusion. The morphological instability of the single interface depends on six dimensionless parameters—the Prandtl number, *σ*, the magnetic Prandtl number, *σ*_{m}, the Chandrasekhar number, *Q*_{c}, the Reynolds number, *Re*, together with *B*_{v} and *Γ*, which quantify the direction of the applied magnetic field:(1.1)where the ambient magnetic field is written in component form(1.2)*U* is a characteristic velocity, *L* is a characteristic length-scale, *ν* is the kinematic viscosity, *κ* is the thermal diffusivity, *η* is the magnetic diffusivity, *μ* is the magnetic permeability, *ρ*_{0} is the mean density and *B*_{0} is the total amplitude of the ambient magnetic field. The *x*-axis is normal to the interface, *z* is upward and *y* is horizontal and parallel to the interface (see figure 1). The analysis of part I included the effect of a small field normal to the interface (*Γ*≠0), but in the following analysis, this component is assumed to be zero, as its presence poses serious analytic difficulties, which will be discussed elsewhere. We thus assume that(1.3)

In the present study of the Cartesian plume, the dimensionless thickness of the channel, 2*x*_{0}, is another parameter of the problem.

The magnetic field has a profound effect on the instability of the single interface, with the nature of the effect depending on the orientation of the field. A horizontal magnetic field parallel to the interface has a stabilizing effect in the sense that the growth rate is reduced in magnitude, although the order of magnitude remains unchanged and the instability is not completely suppressed for any value of the parameters of the problem. On the other hand, the addition of a vertical component of the ambient field can counteract the influence of the horizontal field.

The study of the non-magnetic Cartesian plume (Eltayeb & Loper 1994) has shown that the finite width of the plume leads to an enhancement of the instability. The study of the magnetic single interface (part I) has shown that an inclined magnetic field can also enhance instability. Our purpose here is to examine the influence of the three factors of (i) finite thickness of plume, (ii) the Lorentz force and (iii) magnetic diffusion, all acting simultaneously. The analysis below will examine the influence of these factors on the preference of these modes in the space (*x*_{0}, *σ*, *σ*_{m}, *Q*_{c}, *B*_{v}) when the Reynolds number *Re* is small.

In §2, we briefly formulate the problem and present the solution. In §3, we discuss the stability results. As in the non-magnetic case (Eltayeb & Loper 1994), the morphological instabilities may be categorized as sinuous or varicose, depending on whether the symmetry of the solution in *x* is odd (*sinuous*) or even (*varicose*), and where necessary the symmetries will be distinguished by the parameter *P*, defined by(1.4)Also, the horizontal wavenumber *m* can be zero (with the convective rolls aligned with the *vertical*) or non-zero (with the rolls being *oblique*). Consequently, four types of modes can occur: varicose oblique (labelled *V*_{o}), varicose vertical (*V*_{v}), sinuous oblique (*S*_{o}) and sinuous vertical (*S*_{v}). These modes can further be characterized by the effect of the magnetic field; we shall find that, as in the case of a single interface, these can be non-magnetic modes modified by the field or new modes introduced by the presence of the field. We shall see that the finite width of the plume enhances the instability; as the thickness approaches zero, the growth rate approaches zero as the square power of the thickness of the plume but always remaining positive. Consequently, for fixed values of *σ*, *σ*_{m}, *Q*_{c}, *B*_{v} the growth rate of the preferred mode attains an overall maximum at a certain value of *x*_{0}. It is also found that the horizontal component of the field acting alone can enhance instability for some range of values of *x*_{0} and *Q*_{c}, in sharp contrast to the corresponding case of a single interface. The addition of a vertical component of field leads to further *enhancement* of instability in the sense that the growth rate is increased in value but remains of the same order of magnitude. It will be shown that while magnetic diffusion is always stabilizing, its influence on the sinuous and varicose modes depends on the other parameters. The destabilizing influences of magnetic field and finite thickness of plume can sometimes act in concert to enhance instability but the two factors can at other times oppose one another. In §4, we discuss the helicity and *α*-effect of the perturbations, and in §5, we include a few concluding remarks.

## 2. The basic equations and boundary conditions

Generalizing Eltayeb & Loper (1994), we shall consider an incompressible fluid of infinite extent flowing with velocity * u* in the presence of a stabilizing thermal gradient,

*γ*=d

*T*/d

*z*, and a uniform magnetic field

*. The density,*

**B***ρ*, depends on the temperature,

*T*and the concentration of light component,

*C*. The fluid has finite kinematic viscosity,

*ν*, thermal diffusivity,

*κ*and magnetic diffusivity,

*η*; material diffusivity is assumed to be zero, so that the distribution of

*C*can be specified. We shall assume that it has the top-hat profile(2.1)in the absence of deformation of the interfaces, where both

*C*

_{0}and are constant (see figure 1). Both the compositional distribution and the interfaces are material functions, moving with the fluid. The formulation is given in part I. Here, we state the main results noting that all variables used below are dimensionless. The non-dimensionalization process used the amplitude of the concentration, , and that of magnetic induction,

*B*

_{0}, as units of concentration of light material and magnetic induction, respectively, and(2.2)as units of length, velocity, time and pressure, respectively. The Boussinesq approximation, in which density variations are neglected except when they occur in the gravity term in the equation of motion, is adopted. As a consequence, gravity is the driving force for the instability and is represented by the terms

*C*and

*T*in equation (2.8).

The positions of the two interfaces are assumed to be slightly disturbed, with locations described by(2.3)for the varicose mode and by(2.4)for the sinuous (or meandering) mode, where c.c. is the complex conjugate and 0<*ϵ*≪1. This disturbance is accompanied by dimensionless convective perturbations of velocity, * u**, pressure,

*p**, magnetic field,

**, temperature,*

**b***T** and concentration of light material,

*C**, of the form(2.5)The perturbation equations are(2.6)(2.7)(2.8)(2.9)(2.10)(2.11)(2.12)(2.13)(2.14)and(2.15)in which(2.16)(2.17)and(2.18)The functions and represent the basic state vertical plume flow and temperature (see figure 1) and are given by(2.19)(2.20)in which(2.21)

As in part I, the following analysis is somewhat simplified by the introduction of *Q* and *δ* in place of *Q*_{c} and *B*_{v}. The parameter *Q* always occurs in the combination *δ*^{2}*Q* (or its square root) and such a combination is a local Chandrasekhar number based on the component of the magnetic field parallel to the perturbation wavenumber(2.22)where * k*=∇(

*my*−

*nz*). If the waves are aligned with the field, then

*.*

**k***=0, and the field has no influence on the instability. Note that , where is defined by eqn (4.9) in Eltayeb & Loper (1991), taking into account the difference in the sign of*

**B***n*between the two analyses. However, the results below will be presented using

*Q*

_{c}because it is independent of the direction of field and the wavenumbers.

The continuity equation may be replaced by an equation governing the variation in pressure, obtained by forming Δ(2.14) and making use of (2.6)–(2.8), (2.14) and (2.15):(2.23)

Now (2.6), (2.8)–(2.10), (2.12) and (2.23) form a set of six equations for the six unknown variables *u*, *w*, *p*, *T*, *b*_{x} and *b*. Equations (2.7) and (2.11), and the associated variables *v* and *b*_{y} are decoupled and not of interest. However, note that the three-dimensionality of the solution is retained due to the presence of the factor *m* in (2.16) and (2.17).

The boundary conditions are the continuity of momentum, heat and magnetic field fluxes together with the condition that the interfaces are material surfaces, and that all the perturbation variables decay to zero away from the interface. In addition, the continuity of *v* and *w* can be used in (2.14) to show the continuity of *Du* across the interfaces. Similarly the continuity of *b*_{y} and *b* implies that of *Db*_{x} according to equation (2.15). Furthermore, equation (2.12) can be integrated across the interface making use of the continuity of *u*, *b*, *w* and to find that *Db* is also continuous across the interface. The relevant boundary conditions can then be written as:(2.24)The full solution to the problem is a linear combination of two solutions having opposite symmetries in *x* (see figure 2). In the even (varicose) solution *w*, *p*, *T*, *v*, *b*_{y}, *b* are even in *x* and *u*, *b*_{x} are odd. The reverse is true for the odd (sinuous) solution. We shall then restrict our analysis to the half-interval [0,∞) and use the following parity conditions at *x*=0:(2.25)for the varicose solution and(2.26)for the sinuous solution.

The variables in (2.6)–(2.15) are assumed to have the expansion(2.27)(2.28)in terms of the small parameter *Re*. Now the problem may be solved iteratively starting with the zeroth-order solution obtained by setting *Re*=0.

The stability of the interface is determined by the sign of the real part of the dominant term in the expansion for *Ω*. If Re(*Ω*_{r})<0 for all possible values of wavenumbers *m* and *n*, the interface is stable to harmonic perturbations. If, on the other hand, Re(*Ω*_{r})>0 for *any* pair of wavenumbers (*m*, *n*), then the plume is unstable.

The solution is obtained by substituting the expansions (2.27) and (2.28) into equations (2.6)–(2.15) and equating the coefficients of *Re*^{r}, (*r*=0, 1, 2, 3, …) to zero to obtain a hierarchy of systems of equations which can be solved seriatim. We will only need to consider the first two such problems in order to determine the growth rate to leading order.

The zeroth-order set of equations is obtained by setting the right-hand side to zero in equations (2.6)–(2.12) together with equations (2.14) and (2.15). The boundary conditions are those in (2.24). The solution may be expressed as(2.29)(2.30)where(2.31)

(2.32)The upper (lower) expression in the curly brackets refers, respectively, to the subinterval 0≤*x*<*x*_{0}(*x*>*x*_{0}) and *P* is defined by equation (1.4). The leading order contribution to the growth rate *Ω*_{1} is non-zero,(2.33)and the unstable waves must propagate. The phase speeds in the vertical and lateral directions are given, respectively, by(2.34)

The first-order equations (i.e. those terms having coefficients of *Re*^{1}) are non-homogeneous, with the homogeneous part being the same as the zeroth-order problem. The solution must obey a solvability condition that provides an expression for the second term, *Ω*_{2}, in the growth rate. The details of the derivation of the expression for *Ω*_{2} are given in appendix A. It can be written in the form(2.35)where *Ω*_{20}, *Ω*_{si}, *Ω*_{sm} are given in (A 26)–(A 28). The three terms on the right-hand side of equation (2.35) correspond to the contributions of thermal, viscous and magnetic diffusions.

The expressions (2.34) for the phase speeds and (2.35) for the growth rate will be discussed in §3 below. In that discussion we will revert to consideration of the externally imposed parameters *Q*_{c} and *B*_{v}, making use of equations (2.17) and (2.18).

## 3. The instability of the magnetic Cartesian plume

The phase speeds (2.34) and (2.33), which are independent of *σ* and *σ*_{m}, have been computed as functions of the parameters *Q*_{c}, *x*_{0} and *B*_{v} and the wavenumbers *m* and *n*. It was found that the lateral phase speed is very small everywhere except near *m*=0, and is negative for the varicose mode. The contours of the phase speed for the sinuous mode show that *U*_{y} is negative except in some small patches of small *n* where it is positive. For the varicose mode, when *B*_{v}=0, *U*_{z} is positive and its largest value occurs when *m*=0. As *Q*_{c} increases, *U*_{z} increases slowly, but an increase in *x*_{0} leads to appreciable reduction for *m*≠0. Except for small values of *x*_{0}, *U*_{z} for the varicose and sinuous modes have opposite signs as is small for *x*_{0}≥1.0.

The growth rate *Ω*_{2}, given by equation (2.35), is a complicated function of the parameters *m*, *n*, *x*_{0}, *σ*, *σ*_{m}, *Q*_{c}, *B*_{v}. Our purpose here is to identify the maximum growth rate (maximized over *m* and *n*) and delineate those regions in the five parameter space (*x*_{0}, *σ*, *σ*_{m}, *Q*_{c}, *B*_{v}) where the maximum is positive, leading to instability, and negative signifying stability. As there are two categories of modes (determined by *P*=1 for varicose and *P*=−1 for sinuous modes) we need to determine the maximum for each symmetry and then choose the larger of the two. In doing so, we shall denote the growth rate (2.35) by . The growth rate is maximized over the wavenumbers *m* and *n* by setting . The maximum for each mode will be denoted by and the associated wavenumber components are *m*_{m} and *n*_{m} and we shall refer to the associated parameter *δ*_{m}, which measures the inclination of the wave vector to the field lines. The larger of the two growth rates is denoted by *Ω*_{c} and the *preferred* (*critical*) *mode* of convection is defined by the maximum growth rate and the associated wavenumbers together with *δ* and is referred to as (*Ω*_{c}, *m*_{c}, *n*_{c}, *δ*_{c}). The results obtained for the single interface in part I provide a check on these results for *x*_{0}→∞ and those for *Q*_{c}=0 can be compared with those of the non-magnetic Cartesian plume (Eltayeb & Loper 1994). In view of the complexity of the problem in its entirety we will represent the most significant results graphically.

It is found that the influence of the parameters is very dependent on the symmetry (varicose or sinuous). The effect of the different parameters can be conflicting for some values and can act in concert at other values. In figure 3, we present a sample of the results for for some representative values of the parameters. In figure 3*a*,*b*, the dependence of the growth rate for the two symmetries of modes as functions of the thickness of the plume, *x*_{0}, is illustrated for sample values of *σ*, *σ*_{m}, *Q*_{c}, *B*_{v}. In all cases, the growth rate acquires a maximum at a finite value of the thickness of the plume and approaches the value of the single line interface when *x*_{0} is very large. For the sinuous mode, the growth rate increases from 0 at *x*_{0}=0.0 until it reaches a maximum before it decreases steadily to its asymptotic value at large values of *x*_{0}. In the case of the varicose mode, the growth rate increases from 0 at *x*_{0}=0.0 to a maximum and then decreases to a finite positive minimum before it increases slowly to its asymptotic value for large *x*_{0}. In the absence of viscous and magnetic diffusions, as in figure 3*a*, the sinuous mode is preferred except for small plume thickness and the influence of increasing the field inclination, as represented by *B*_{v}, is to reduce the growth rate for both modes. In figure 3*b*, the influence of increasing viscous diffusion is illustrated. It is evident that the effect of viscous diffusion on the two modes depends on the thickness of the plume. Viscous diffusion can be stabilizing or destabilizing depending on the mode and on the thickness of the plume. This interesting feature will be examined in more detail later. In figure 3*c*,*d*, we illustrate the growth rate as a function of *Q*_{c} for different sets of the parameters *x*_{0}, *σ*, *σ*_{m}, *B*_{v}. It is evident that the preference of either mode is a complicated function of the parameters. In all cases we see that magnetic diffusion, as represented by *σ*_{m}, is always stabilizing for both categories of modes. In figure 3*c*, *x*_{0}=2.0 and the influence of *σ* is stabilizing for the sinuous mode and destabilizing for the varicose mode. This leads to the preference of the sinuous mode for small *σ* and the varicose mode for large *σ* (see figures 4–8). The influence of the field inclination is more complicated. For every non-zero *B*_{v}, the growth rate of the sinuous mode decreases with *Q*_{c} and the rate of decrease increases with the increase in *B*_{v}. For the varicose mode, the growth rate can increase or decrease with the increase in *Q*_{c} depending on *B*_{v}. In figure 3*d*, *x*_{0}=3.5, the effect of viscous diffusion is always stabilizing. The growth rates of both categories of modes decrease with *Q*_{c} for all *B*_{v}, but the rate of decrease depends on the value of the inclination parameter. The sinuous mode is always preferred here. We will discuss these properties in more detail below.

As noted previously, the preferred mode is one of four modes: vertical sinuous, *S*_{v}, oblique sinuous, *S*_{o}, and the corresponding varicose modes, *V*_{v} and *V*_{o}. The influence of the different parameters on these four modes results in the preference of one mode for particular ranges of parameters. In view of the number of parameters and number of modes, we find it informative to represent the preference of modes in a regime diagram in the *x*_{0}−*σ* plane. This is illustrated in figures 4–8. In addition to the identification of the regions of preference of the different modes, we have also included the curve of the overall maximum growth rate, *Ω*_{max}, obtained by maximizing the critical growth rate as a function of *σ* and *x*_{0}, i.e. by locating the wavenumbers and growth rate satisfying . The curve of overall maximum growth rate is represented by a discontinuous curve in the regime diagrams. We note that it does not change very much with changes in *σ*, *σ*_{m}, *Q*_{c}, *B*_{v}.

In figure 4, the influence of increasing the amplitude of a horizontal field in the absence of magnetic diffusion is illustrated. Starting with the non-magnetic case of *Q*_{c}=0.0, *Q*_{c} is increased gradually and the associated evolution of the regime diagram in the *x*_{0}−*σ* plane is monitored. The influence of increasing *Q*_{c} here is to suppress the vertical varicose mode for small *x*_{0} and moderate *σ*. The vertical sinuous mode is suppressed for small *x*_{0} and moderate *σ* and extended for large *x*_{0} and moderate *σ*. The region of the oblique varicose mode remains almost unaffected by the increase in *Q*_{c}.

Some of the most interesting features of the stability of the problem are those brought about by the presence of magnetic diffusion and the vertical component of field. Figure 5 illustrates the evolution of the regime diagram with the increase of the field-inclination parameter, *B*_{v}, when magnetic diffusion is absent. It would have been expected that the presence of the vertical field will suppress the vertical modes as it will tend to align the waves with the inclined magnetic lines of force. This is found to be the case for the varicose mode, although even in this case only the vertical mode for small *x*_{0} is affected while the oblique varicose mode present for moderate *x*_{0} is unaffected. However, the effect on the sinuous mode is different. For small field inclination, the region in the *x*_{0}−*σ* plane where the vertical sinuous mode is preferred begins to shrink so that when *B*_{v} increases to about 0.02, *S*_{v} is present only in two small pockets: one for small *x*_{0} and another for large *x*_{0} (see figure 5*b*). As *B*_{v} increases further, the pocket for large *x*_{0} shrinks further until it eventually disappears, while the pocket for small *x*_{0} extends and moves towards the *σ*-axis and remains for all values of the inclination *B*_{v}. This latter vertical sinuous mode is magnetic in nature brought about by the presence of the field. When magnetic diffusion is present, the regime diagram evolves in the same way as regards the varicose modes but the sinuous mode evolves differently in the sense that the vertical sinuous mode is pushed further into regions of large *σ* for both small and large *x*_{0} (see figure 6).

The influence of magnetic diffusion in the absence of the vertical field is illustrated in figure 7. An increase in magnetic diffusion leads to a change in all of the regions (in the *x*_{0}−*σ* plane) of preference of the four modes *S*_{v}, *S*_{o}, *V*_{v}, *V*_{o}. In general, an increase in magnetic diffusion leads to an increased preference for vertical modes. The vertical sinuous mode region expands into areas of large *σ*. The region of preference of the sinuous oblique mode is pushed towards the *σ*-axis until it eventually disappears at some value of *σ*_{m} when *Q*_{c} is small but will persist as a thin region along the *σ*-axis for large *Q*_{c}. The increase in *σ*_{m} from 0 also leads to a new region of preference of a vertical varicose mode which appears as part of the left side of the region of preference of the oblique varicose mode. This new region of *V*_{v} eats away the oblique varicose mode as *σ*_{m} increases (see figure 7).

In the presence of the vertical field the influence of magnetic diffusion is, in general, dependent on *Q*_{c} (see figure 8). For small *Q*_{c}, the vertical sinuous mode present in the absence of diffusion is suppressed by the presence of magnetic diffusion. If, however, *Q*_{c} is large and the vertical sinuous mode is absent in the absence of diffusion, an increase of magnetic diffusion will lead to the appearance of a thin region of *S*_{v} adjacent to the *σ*-axis. While the oblique varicose mode present for small *x*_{0} is suppressed near the *σ*-axis, the oblique varicose mode for moderate *x*_{0} is unaffected by an increase in *σ*_{m}. The most prominent change in the regime diagram is the appearance of a vertical sinuous mode in a thin region near the *σ*-axis when *σ*_{m} increases beyond a certain value dependent on *Q*_{c} (see figure 8).

The influence of magnetic diffusion and the presence of the vertical field on the growth rate of the preferred mode lead to some novel features of instability. Figures 9 and 10 depict the profiles of *Ω*_{c} as a function of the vertical field inclination *B*_{v} for different values of *Q*_{c}. In figure 9, the growth rate of the preferred mode of the oblique varicose mode is shown as a function of *B*_{v} for *σ*=5.0, *x*_{0}=1.8 and different values of *σ*_{m}. For small values of *σ*_{m}, as *B*_{v} increases from 0, the growth rate decreases gradually until it reaches a minimum, *Ω*_{c0}, at a value, *B*_{v0} of *B*_{v} before it increases again gradually to a maximum and then starts to decrease steadily with the increase of *B*_{v} (see figure 9*a*,*b*). The value *Ω*_{c0} is the same as that obtained in the absence of the field and the critical mode there is identical to that of the non-magnetic mode so that *δ*_{c}=0. For moderate and large *Q*_{c}, the growth rate *increases* as *B*_{v} increases from 0 until it reaches a maximum, *Ω*_{cm}, at *B*_{v}=*B*_{m1}, before it decreases to a minimum, *Ω*_{c0}, at *B*_{v}=*B*_{v0} after which it increases again to a maximum, *Ω*_{cm}, at *B*_{v}=*B*_{m2} and then decreases steadily with *B*_{v}. The local maximum values of the preferred growth rate at *B*_{v}=*B*_{m1}, *B*_{m2} are always identical whatever the value of *Q*_{c}, and *Ω*_{c0} is the same or greater than the corresponding value for *Q*_{c}=0. As *B*_{v} increases from 0, *δ*_{c} decreases steadily through positive values until *B*_{v} reaches *B*_{v0}. Here, *δ*_{c}=0 if *Q*_{c} is less than some value *Q*_{0}(*σ*,*σ*_{m}) dependent on *σ* and *σ*_{m}, in general. For *Q*_{c}>*Q*_{0}, *δ*_{c} jumps to a negative value and further increase in *B*_{v} leads to a steady decrease in *δ*_{c}, with the rate of decrease diminishing with increasing *Q*_{c}. If, on the other hand, *σ*_{m} is moderate or large, the preferred growth rate increases with field inclination reaching a maximum *Ω*_{c0} identical with that in the absence of the field before it starts to decrease steadily with *B*_{v}.

The values of *Ω*_{cm} and *Ω*_{c0} depend on *Q*_{c}, *σ* and *σ*_{m}. The remarkable result is that for small *σ*_{m}, *Ω*_{cm} *exceeds* the growth rate of the corresponding non-magnetic case and consequently the vertical field *destabilizes* the plume. Furthermore, if the magnetic field strength, as measured by *Q*_{c}, is in excess of a certain value *Q*_{0}(*σ*,*σ*_{m}), both *Ω*_{cm} and *Ω*_{c0} are greater than the growth rate in the absence of the field. However, an increase in magnetic diffusion leads to the suppression of *Ω*_{cm}, but *Ω*_{c0} remains for all non-zero values of *Q*_{c} and occurs for a value of *B*_{v} that depends weakly on *σ*_{m}. The wavenumbers and *δ*_{c} associated with these profiles are shown in figure 10. We note that the change of mode at *B*_{v0} is indeed associated with a drop in both vertical and horizontal wavenumbers, but this occurs in such a way that *δ*_{c} merely changes sign with the numerical value remaining the same. As *B*_{v} increases from 0, *δ*_{c} decreases steadily towards 0. For values of *Q*_{c} where there is no change of mode, *δ*_{c} decreases continuously into negative values. However, the cases in which there is a change of mode are associated with a jump in *δ*_{c} from positive to negative values (figure 11).

The simultaneous action of the vertical field and magnetic diffusion on the preferred mode depends on the category mode. This is illustrated in figure 10 for both varicose and sinuous modes in the regions where they are preferred. For values of *σ*_{m} less than about half *σ*, the varicose mode decreases from its value at *B*_{v}=0 to a minimum before it increases to a local maximum and then decreases steadily thereafter. For larger values of *σ*_{m}, the growth rate increases to a maximum before it starts to decrease steadily with *B*_{v} (see figure 10*a*,*c*). The sinuous mode behaves differently. The influence of the vertical field is potent only for small inclinations of the field and even then it does not appear to affect the vertical wavenumber if magnetic diffusion is not too large (i.e. *σ*_{m} is less than about half *σ*). As the magnetic field inclination is increased from 0, the critical growth rate will decrease slightly if *σ*_{m}=0 before a change of mode takes place, in which case the horizontal wavenumber is reduced to 0 while the vertical wavenumber remains almost unchanged (see figure 12). The new mode is associated with a growth rate that increases with inclination reaching a maximum before it gradually decreases to a minimum where another change of mode takes place. This new mode is associated with a relatively large value of *δ*_{c} (see figure 12*d*) and is magnetic in nature. Further increase in *B*_{v} leads to an increase in the growth rate to an overall maximum, *Ω*_{cc}, and thereafter decreases steadily. For moderate non-zero values of *σ*_{m}, similar behaviour occurs for the sinuous mode. The values of *B*_{v} between the two changes of mode are associated with a relatively large value of *δ*_{c}. As *σ*_{m} exceeds about half *σ*, the growth rate increases steadily with inclination reaching the same overall maximum occurring for *σ*_{m}=0 and thereafter decreases steadily with inclination. For values of *σ*_{m} close to *σ*, the growth rate again experiences a change of mode close to *B*_{v}=0 before it increases steadily reaching and following the values for moderate *σ*_{m} when *B*_{v} reaches values beyond about 0.2. This last change of mode is associated with a large drop in wavenumbers, as well as in *δ*_{c} (see figure 12). It then follows that the growth rate of the critical mode of the varicose mode is affected by the presence of magnetic diffusion at all inclinations of the magnetic field while that of the sinuous mode is affected only when the inclination is small.

In an attempt to understand the mechanism leading to the enhancement of the growth rate by the magnetic field, we investigated the dependence of the individual contributions , , (see equation (2.35)) to the growth rate in detail. The destabilizing influence of the magnetic field can be traced down to the component of the growth rate due to viscous diffusion. In figure 13, we plot the maximum of and the associated wavenumbers *m*_{m}, *n*_{m} as functions of *Q*_{c} for the case *x*_{0}=1.8 in the absence of the vertical field. We find that initially increases with *Q*_{c} for both even and odd modes and that it is positive in both cases, but its values for the even mode are about seven times those for the odd mode. Although the thermal diffusion contribution has a maximum with *m*_{m}=0.0 and hence is aligned with the magnetic field, the maximum of the viscous contribution is inclined to the field. The magnetic diffusion contribution is negative for all *m* and *n* and has maximum 0 with *m*_{m}=0.0. We then conclude that the destabilization of the plume by the horizontal field is a result of the magnetic field tending to inhibit the stabilizing influence of viscosity.

In the presence of the vertical component of field, a similar detailed examination of the contributions , , to the growth rate reveals that the viscous contribution interacting with the vertical component is responsible for the enhancement of the growth rate. This is illustrated in figure 14. We see that increases with *B*_{v} as it increases from 0 for both modes. However, reaches a maximum and then decreases steadily, but slowly, with *B*_{v}, while increases slowly and then more rapidly for a short interval before it continues to increase slowly again. This change of pace with *B*_{v} in the case of the odd (sinuous) mode is due to a change of mode (see figure 14*b*,*c*). For both modes the inclined field tends to suppress the stabilizing influence of viscosity.

The two local maxima at *B*_{m1} and *B*_{m2} (see figure 9) are associated with values of *δ* of equal numerical value but different signs, because *Ω*_{c} depends on the magnetic field through the presence of *δ*^{2}. This means that they are associated with positions equally placed on either side of the direction *δ*=0. The result that the position *δ*=0 is not preferred, and indeed the system avoids it as it jumps from one side of *δ*=0 to the other, indicates that the interaction of viscous diffusion and the vertical component of the magnetic field can act in concert to destabilize the system through a wave that propagates at an angle to the magnetic field. The resulting growth rate is larger than that of the nonmagnetic system when the basic parameters are the same.

The perturbation equations (2.6)–(2.15) can be used to obtain an expression for the growth rate in terms of energy integrals (see Eltayeb & Loper 1991; eqns (4.40)–(4.42)). Omitting the details, we find that the growth rate depends on the three integrals (in addition to others)(3.1)representing the contributions of the basic state gradients to the growth rate. Here the superscript c refers to the complex conjugate. Computations of these integrals showed that *I*_{1}, representing thermal diffusion, is positive while *I*_{2}, representing magnetic diffusion, is negative signifying the stabilizing role of magnetic diffusion. The integral *I*_{3} representing the viscous diffusion contribution resulting from the interaction of the wave with the lateral variations of temperature can take positive and negative values. The enhancement of the growth rate by the magnetic field corresponds mainly to the positive values of this integral. It then follows that the instability is enhanced by the release of energy by the basic state temperature variations resulting from the rapid change in the composition of the light material across the interface.

## 4. Helicity and *α*-effect

The expressions for the helicity and *α*-effect (see equation (4.2) of part I) can be written in the form(4.1)

(4.2)

(4.3)

(4.4)Here *E*_{y}, *E*_{z} represent the components of the *α*-effect along the horizontal and vertical components of the ambient field while *E* represents the *α*-effect along the inclined field. The leading order *α*-effect and helicity are obtained by using equations (2.29) and (2.30). They are illustrated graphically in figures 15 and 16.

It is clear from equations (4.1)–(4.4) that the helicity and *α*-effect are both odd in *x* and we can therefore illustrate them in the half-interval [0,∞). The expressions (4.1)–(4.4) also show that the helicity is discontinuous across the interface, since *Dw*_{0} is discontinuous there, and both components of the *α*-effect are continuous across the interface.

The profiles of the components of the *α*-effect show clear dependence on the parameters *σ*, *σ*_{m}, *Q*_{c}, *B*_{v} and they are associated with variations on the small scale of the plume. They are illustrated in figure 15. The amplitude of the sinuous mode is about half that of the varicose mode for both components. However, there are marked differences between the dependence of the varicose and sinuous modes on the parameters *σ*, *σ*_{m}, *Q*_{c}, *B*_{v}. For the varicose mode, both components show strong dependence on variations in *B*_{v}. An increase in *Q*_{c} leads to a reduction in the amplitude, while changes in *σ* or *σ*_{m} have little influence on the amplitude. For the sinuous mode, on the other hand, changes in *Q*_{c} or *σ* are associated with notable changes in amplitude, while changes in *B*_{v} and *σ*_{m} have little influence on the amplitude.

The profile of the helicity depends on the parity. For the varicose mode, the helicity decreases steadily from 0 at the centre of the plume until it reaches the interface where it suffers a positive jump. For *x*>*x*_{0}, the helicity for the varicose mode decreases rapidly to 0. Variations in the parameters *σ*, *σ*_{m}, *Q*_{c}, *B*_{v} have little effect on the profiles of the varicose mode. The most notable variations, albeit very small, occur near the interface within the plume. Here, an increase in *σ* reduces the amplitude while an increase in *σ*_{m} or *B*_{v} tends to increase the amplitude. In the case of the sinuous mode, the helicity increases from 0 at the centre of the plume reaching a maximum before it decreases to a negative value on the interface. At the interface, it experiences a positive jump and thereafter it decreases rapidly to 0. The profile of the helicity of the sinuous mode is weakly dependent on *σ*, *σ*_{m}, *Q*_{c}, but shows clear dependence on variations in *B*_{v}; both the local maximum within the plume and the jump across the interface increase with *B*_{v} (see figure 16).

## 5. Concluding remarks

The morphological instability of a planar plume of buoyant fluid, having thickness 2*x*_{0} in an infinite medium permeated by a magnetic field, has been studied. The maximum growth rate of the instability is identified for the space of parameters *x*_{0}, *σ*, *σ*_{m}, *Q*_{c}, *B*_{v}, and the results have been compared with the two related limiting cases of a single interface in the presence of a magnetic field (part 1) and the Cartesian plume in the absence of a field (Eltayeb & Loper 1994). These comparisons have revealed that the magnetic field and the finite thickness of the plume both have a profound effect on the stability problem.

Magnetic diffusion, quantified by *σ*_{m}, is always stabilizing, but the strength of the stabilizing influence depends on the symmetry or parity of the mode of instability, which may be either sinuous (odd) or varicose (even). An increase in the Prandtl number, *σ*, enhances the maximum growth rate for small and moderate values of *x*_{0} for both modes, but the precise range of influence depends on the parity of the mode. For given values of *x*_{0}, *σ*, *σ*_{m}, *Q*_{c}, *B*_{v}, the plume can be destabilized by an increase in the inclination of the field to the vertical so that an inclined field may make the plume more unstable than a horizontal field of the same strength. To clarify the differing effects of magnetic field and diffusion on the two categories of modes, the preference of the four modes (vertical sinuous mode, *S*_{v}, the oblique sinuous mode, *S*_{o}, and the two corresponding varicose modes, *V*_{v}, *V*_{o}) has been summarized in regime diagrams in the *x*_{0}−*σ* plane for different values of the parameters *σ*_{m}, *Q*_{c}, *B*_{v}. It is found that for any prescribed set of parameters *σ*_{m}, *Q*_{c}, *B*_{v}, the growth rate as a function of the plume thickness 2*x*_{0} and *σ* attains its maximum along a curve with *x*_{0} always occurring in the range 1.3<*x*_{0}<1.7. This overall maximum is sinuous for small *σ* and varicose for large *σ*.

As compared with the stability of the single interface studied in part I, it is found that the finite thickness of the buoyant fluid has the effect of enhancing the instability. This had already been shown to be the case in the absence of the field (Eltayeb & Loper 1994), and motivated an investigation of the influence of the field on the instability of the non-magnetic Cartesian plume. It has been shown that for certain values of the parameters *x*_{0}, *σ*, *σ*_{m}, *Q*_{c}, *B*_{v}, the magnetic field has a *destabilizing* influence. Detailed investigations of the growth rate revealed that the enhancement of instability is caused by viscous diffusion. We recall that viscous diffusion enhances instability in the case of the non-magnetic plume, and here we find that the magnetic field, whether horizontal or inclined to the vertical, can enhance further the destabilizing influence of viscosity. This is found to be the result of the transfer of energy from the basic state horizontal temperature variations to the wave. Such destabilizing influence affects both varicose and sinuous modes. However, the strength of its effect on either mode depends on the values of the parameters *x*_{0}, *σ*, *σ*_{m}, *Q*_{c}, *B*_{v}.

## Acknowledgments

We wish to thank the referees for their constructive comments which led to the improvement of the original version of the paper.

## Footnotes

As this paper exceeds the maximum length normally permitted, the authors have agreed to contribute to production costs.

- Received September 13, 2004.
- Accepted February 28, 2005.

- © 2005 The Royal Society