2.1. Maxwell–Cattaneo diffusion

To derive the governing equations, it is necessary to consider the modifications from the M–C effect to be expected in an incompressible fluid moving with velocity $\boldsymbol{u}$ . As in HPE21, in which further details may be found, here we follow the formulation of Christov (Reference Christov2009), which leads to the equation

(2.1) \begin{align} \tau _{\scriptscriptstyle T}\left [ \frac {\partial Q_{\scriptscriptstyle T}}{\partial t}+ \boldsymbol{\nabla }\boldsymbol{\cdot }({\boldsymbol{u}} \, Q_{\scriptscriptstyle T})\right ]=- Q_{\scriptscriptstyle T} - K {\nabla} ^2 T, \end{align}

where $Q_{\scriptscriptstyle T}$ is the divergence of the heat flux and where we have assumed that $K$ is constant. As $\boldsymbol{u}$ is incompressible, the left-hand side of (2.1) can be written as the usual Lagrangian derivative of $Q_{\scriptscriptstyle T}$ . The equation for the salt flux may be obtained, analogously, as

(2.2) \begin{align} \tau _{\scriptscriptstyle S}\left [ \frac {\partial Q_{\scriptscriptstyle S}}{\partial t}+ \boldsymbol {\nabla }\boldsymbol{\cdot }({\boldsymbol{u}} \, Q_{\scriptscriptstyle S})\right ]=- Q_{\scriptscriptstyle S} - \kappa _{\scriptscriptstyle S} {\nabla} ^2 S. \end{align}

We note that the relation between salt flux and salt concentration differs from that between heat flux and temperature; hence the appearance of $K$ in (2.1) but $\kappa _{\scriptscriptstyle S}$ in (2.2).

2.2. Governing equations

As in HPE21, we consider a horizontal layer of an incompressible (Boussinesq) viscous M–C fluid, initially at rest, contained between two planes at $z=0$ (bottom) and $z=d \pi$ (top). The fluid has kinematic viscosity $\nu$ , and thermal and salt diffusivities $\kappa ,\kappa _{\scriptstyle {S}}$ . The density depends linearly on two components that diffuse at different rates. In equilibrium, the fluid is at rest, with temperature and salinity differences across the layer of $\Delta T$ and $\Delta S$ . The basic state temperature and salinity, $\overline T$ and $\overline S$ , are thus given by

(2.3) \begin{align} \overline T = T_0 + \Delta T (1 - z/d \pi ), \qquad \overline S = S_0 + \Delta S(1 - z/d \pi ), \end{align}

where $T_0$ and $S_0$ are representative values of temperature and salinity. For the perturbed state, with velocity ${\boldsymbol{u}}=(u,v,w)$ , we express the temperature and salinity by $T=\overline T + \widehat T$ , $S=\overline S + \widehat S$ . The density $\rho$ of the fluid obeys a linear relation of the form

(2.4) \begin{align} \rho =\rho _0 \left (1-\alpha _{\scriptscriptstyle T} \left (T-T_0 \right ) + \alpha _{\scriptscriptstyle S} \left ( S-S_0 \right ) \right ), \end{align}

where $\rho _0=\rho (T_0, S_0)$ , and $\alpha _{\scriptscriptstyle T}$ and $\alpha _{\scriptscriptstyle S}$ are (constant) coefficients of expansion. The crucial difference in the governing equations for the M–C system, in comparison with those of classical double-diffusive convection (with no M–C effects), is the replacement of the classical Fick’s law equations for $T$ and $S$ by the modified equations (2.1) and (2.2). On adopting the standard scalings for distance, time, velocity, heat flux, temperature, salinity flux, salinity and pressure of $d$ , $d^2/\kappa$ , $\kappa /d$ , $ \Delta T K/d$ , $ \Delta T$ , $\Delta S\kappa _{\scriptscriptstyle S}/d$ , $\Delta S$ and $\rho _0 \nu \kappa /d^2$ , and dropping the hats, the governing equations take the form

(2.5) \begin{align} \frac {1}{\sigma }\frac {D{\boldsymbol{u}}}{D t} &= -\boldsymbol{\nabla }\!p + {\textit{Ra}} T \hat{\boldsymbol{z}} -Rs S \hat{\boldsymbol{z}} + {\nabla} ^2 {\boldsymbol{u}}, \\[-10pt] \nonumber \end{align}

(2.6) \begin{align} \boldsymbol {\nabla }\boldsymbol{\cdot }{\boldsymbol{u}} &=0, \\[-10pt] \nonumber \end{align}

(2.7) \begin{align} \frac {DT}{D t} &= w - Q_{\scriptscriptstyle T}, \\[-10pt] \nonumber \end{align}

(2.8) \begin{align} 2 C_{\scriptscriptstyle T} \frac {DQ_{\scriptscriptstyle T}}{D t} &= - Q_{\scriptscriptstyle T} - {\nabla} ^2 T, \\[-10pt] \nonumber \end{align}

(2.9) \begin{align} \frac {D S}{D t} &= w - Q_{\scriptscriptstyle S}, \\[-10pt] \nonumber \end{align}

(2.10) \begin{align} 2 C_{\scriptscriptstyle S} \frac {D Q_{\scriptscriptstyle S}}{D t} &= - Q_{\scriptscriptstyle S} - \tau {\nabla} ^2 S, \\[9pt] \nonumber \end{align}

where the Rayleigh number ${\textit{Ra}}$ , the salt Rayleigh number $Rs$ , the Prandtl number $\sigma$ and the diffusivity ratio $\tau$ are defined by

(2.11) \begin{align} Ra= \frac {g\alpha _{\scriptscriptstyle T}\Delta Td^3}{\kappa \nu }, \quad Rs =\frac {g\alpha _{\scriptscriptstyle S}\Delta Sd^3}{\kappa \nu }, \quad \sigma = \frac {\nu }{\kappa }, \quad \tau =\frac {\kappa _{\scriptstyle {S}}}{\kappa }. \end{align}

With the Rayleigh numbers so defined, positive (negative) ${\textit{Ra}}$ is thermally destabilising (stabilising), whereas positive (negative) $Rs$ is solutally stabilising (destabilising).

These governing equations are the same as those derived in HPE21, where we examined the limiting cases of $C_{\scriptscriptstyle S}=0$ and $C_{\scriptscriptstyle T}=0$ . Our aim here is to understand the nature of any instability in between these two limits. Thus, rather than working with $C_{\scriptscriptstyle T}$ and $C_{\scriptscriptstyle S}$ directly, it is instructive to define two new parameters:

(2.12) \begin{align} \varGamma = C_{\scriptscriptstyle S} + C_{\scriptscriptstyle T}, \qquad \lambda = \frac {C_{\scriptscriptstyle S}-C_{\scriptscriptstyle T}}{C_{\scriptscriptstyle S}+C_{\scriptscriptstyle T}}. \end{align}

For certain parts of the analysis, it is also convenient to work with the quantities

(2.13) \begin{align} \lambda _1= \frac {C_{\scriptscriptstyle S}}{\varGamma } = \frac {1}{2}(1+\lambda ), \qquad \lambda _2 = \frac {C_{\scriptscriptstyle T}}{\varGamma } = \frac {1}{2}(1-\lambda ), \qquad \alpha =\frac {Cs}{C_{\scriptscriptstyle T}} = \frac {\lambda _1}{\lambda _2}. \end{align}

We shall be interested in the case of $\varGamma \ll 1$ (i.e. $C_{\scriptscriptstyle T}$ and $C_{\scriptscriptstyle S}$ both small); $\lambda$ varies between $-1$ ( $C_{\scriptscriptstyle S}=0$ ) and $+1$ ( $C_{\scriptscriptstyle T}=0$ ).

2.3. Linearisation and stability considerations

As in HPE21, we address the linear stability of the basic state given by (2.3) and (2.4), subject to the standard boundary conditions in which the horizontal boundaries are impermeable and stress-free, and on which the temperature and salinity are fixed. Thus,

(2.14) \begin{align} \frac {\partial u_x}{\partial z} = \frac {\partial u_y}{\partial z} = u_z = T = S = 0 \quad \textrm{on} \quad z=0, \pi , \end{align}

noting that $z$ is now dimensionless. We assume periodicity in the horizontal directions. In general, we may decompose the solenoidal velocity as

(2.15) \begin{align} {\boldsymbol{u}} = \boldsymbol {\nabla }\times \big ( \boldsymbol {\nabla }\times {\mathcal{P}} \hat{\boldsymbol{z}} \big ) + \boldsymbol {\nabla }\times {\mathcal{T}} \hat{\boldsymbol{z}} . \end{align}

The linearised form of (2.5) shows, however, that ${\mathcal{T}}$ decays for all parameter values, and thus only ${\mathcal{P}}$ is of relevance. Following the usual approach to the classical double-diffusive stability problem, we seek solutions to the linearised versions of (2.5)–(2.10) of the form

(2.16) \begin{align} {\mathcal{P}} \propto T \propto S \propto Q_{\scriptscriptstyle T} \propto Q_{\scriptscriptstyle S} \propto f(x,y) \sin mz\,\textrm{e}^{st}, \end{align}

where the planform function $f(x,y)$ satisfies

(2.17) \begin{align} {\nabla} _H^2 f = - k^2 f, \end{align}

with ${\nabla} _H^2$ being the horizontal Laplacian. For the classical problem, with no M–C effects (i.e. $\varGamma =0$ ), it is readily shown that the fundamental mode (i.e. $m=1$ ) is the most readily destabilised. Here we shall also restrict attention to the $m=1$ mode, but will discuss this assumption in § 6, in light of the results.

On substitution from (2.16) into the linearised forms of (2.5)–(2.11), we obtain, after some algebraic manipulation, a quintic dispersion relation for the growth rate $s$ :

(2.18) \begin{align} a_5 s^5 + a_4 s^4 + a_3 s^3 +a_2 s^2 + a_1 s + a_0 = 0, \end{align}

where

(2.19a) \begin{align} a_5 &= \frac {\varGamma ^2 (1-\lambda ^2) \beta ^2}{\sigma }, \\[-10pt] \nonumber \end{align}

(2.19b) \begin{align} a_4 &= \frac {2 \varGamma }{\sigma } \beta ^2 + \varGamma ^2 (1-\lambda ^2) \beta ^4 , \\[-10pt] \nonumber \end{align}

(2.19c) \begin{align} a_3 &= \frac {\beta ^2}{\sigma } + \frac {\varGamma }{\sigma } \left (1 + 2\sigma + \tau +\lambda (1-\tau ) \right ) \beta ^4 + \varGamma ^2 (1-\lambda ^2) \left (Rs-Ra\right ) k^2, \\[-10pt] \nonumber \end{align}

(2.19d) \begin{align} a_2 &= \left ( 1 + \frac {1+\tau }{\sigma }\right ) \beta ^4+ \varGamma \left ( 1+\tau + \lambda (1-\tau )\right ) \beta ^6 + 2 \varGamma \left (Rs-Ra\right )k^2, \\[-10pt] \nonumber \end{align}

(2.19e) \begin{align} a_1 &= \left (1 + \tau + \frac {\tau }{\sigma }\right ) \beta ^6 + Rs\big (1 + \varGamma (1+\lambda ) \beta ^2 \big ) k^2- {\textit{Ra}} \big (1 + \tau \varGamma (1-\lambda ) \beta ^2 \big ) k^2 , \\[-10pt] \nonumber \end{align}

(2.19f) \begin{align} a_0 &= \tau \beta ^8 + Rs \beta ^2 k^2 - \tau {\textit{Ra}} \beta ^2 k^2, \\[9pt] \nonumber \end{align}

with $\beta ^2 = k^2 + 1$ . As shown in HPE21, the symmetry of the system allows us to consider only the case of $\tau \lt 1$ .

In this paper, we shall concentrate on determining the conditions for the onset of instability; this may occur either as a direct mode (steady convection), in which case the growth rate $s$ passes through zero, or as an oscillatory mode, in which case, at onset, $s = \pm i \omega$ , with $\omega \in \mathbb{R}_+$ . It is traditional in studies of double-diffusive convection to treat ${\textit{Ra}}$ as the bifurcation parameter, although, mathematically, there is nothing to favour ${\textit{Ra}}$ over $Rs$ . For comparison with the existing literature, we shall maintain this tradition here. We shall denote ${\textit{Ra}}(k^2)$ at the onset of steady (oscillatory) convection by ${\textit{Ra}}^{(s)}$ ( ${\textit{Ra}}^{(o)}$ ), the minimum of ${\textit{Ra}}^{(s)}$ ( ${\textit{Ra}}^{(o)}$ ) over all wavenumbers as ${\textit{Ra}}^{(s)}_c$ ( ${\textit{Ra}}^{(o)}_c$ ), the associated value of $k^2$ by $k_{sc}^2$ ( $k_{oc}^2$ ), and, for the case of oscillatory onset, the associated value of $\omega ^2$ by $\omega _c^2$ . The lesser of ${\textit{Ra}}^{(s)}_c$ and ${\textit{Ra}}^{(o)}_c$ is the overall critical Rayleigh number, which we denote by ${\textit{Ra}}_c$ , with $k^2=k_c^2$ . We shall refer to the mode that first becomes unstable as ${\textit{Ra}}$ is increased as the preferred or favoured mode.

From the form of the flux equations (2.8) and (2.10) it is clear that M–C effects occur only in time-derivative terms, so these effects have no influence on the onset of steady convection. The value of ${\textit{Ra}}$ at the onset of steady convection is therefore given by

(2.20) \begin{align} Ra= Ra^{(s)} = \frac {Rs}{\tau } + \frac {\beta ^6}{k^2}. \end{align}

Since $\beta ^6/k^2$ is minimised when $k^2=1/2$ ,

(2.21) \begin{align} Ra^{(s)}_c = \frac {Rs}{\tau } + \frac {27}{4}. \end{align}

To determine the onset of oscillatory instability, we set $s = \pm i \omega$ , with $\omega \in \mathbb{R}_+$ , leading to the coupled equations

(2.22a,b) \begin{align} a_5 \omega ^4 - a_3 \omega ^2 + a_1 = 0, \qquad a_4 \omega ^4 - a_2 \omega ^2 + a_0=0. \end{align}

Since ${\textit{Ra}}$ and $Rs$ occur linearly in coefficients $a_0, a_1, a_2, a_3$ – and are absent in $a_4$ and $a_5$ – we can combine equations (2.22) to derive a cubic equation for $\omega ^2$ that does not involve ${\textit{Ra}}$ (or, alternatively, $Rs$ ), or eliminate $\omega ^2$ to derive an expression cubic in ${\textit{Ra}}$ and $Rs$ . In HPE21 the corresponding expressions were quadratic rather than cubic, and we made use of each. Here the expressions for the coefficients of the cubic in ${\textit{Ra}}$ are extremely lengthy, and so, with one exception, our primary mode of attack will be via the cubic equation for $\omega ^2$ , which takes the form

(2.23) \begin{align} b_6 \omega ^6 + b_4 \omega ^4 + b_2 \omega ^2 + b_0 = 0, \end{align}

where

(2.24a) \begin{align} b_6 &= \sigma \varGamma ^4 \big ( 1-\lambda ^2 \big )^2 \beta ^2, \\[-9pt] \nonumber \end{align}

(2.24b) \begin{align} b_4 &= \varGamma ^2 \big (2\sigma (1+\lambda ^2) + (1+\lambda )^2 \big ) \beta ^2 - \sigma \varGamma ^3 (1-\lambda ^2) \big (1+2 \tau +\lambda (1-2\tau ) \big ) \beta ^4, \\[-9pt] \nonumber \end{align}

(2.24c) \begin{align} b_2 &= (1+\sigma ) \beta ^2 - \varGamma \big ( \sigma + 2 \tau (1+\sigma ) + \lambda (2 \tau (1+\sigma )-\sigma ) \big ) \beta ^4 \nonumber \\ & \quad + \sigma \varGamma ^2 \big ( (\tau -1)\lambda ^2 -2 (1 + \tau )\lambda + \tau -1 \big ) Rs k^2, \\[-9pt] \nonumber \end{align}

(2.24d) \begin{align} & \quad + \sigma \tau \varGamma ^2 \big ( (\tau -2)\lambda ^2 - 2 \tau \lambda + 2 + \tau \big ) \beta ^6 , \\[-9pt] \nonumber \end{align}

(2.24e) \begin{align} b_0 &= \tau ^2 (1+\sigma ) \beta ^6 - \varGamma (1-\lambda ) \sigma \tau ^2 \beta ^8 - \sigma (1-\tau ) Rs k^2 + 2 \sigma \tau \varGamma \lambda Rs \beta ^2 k^2. \\[9pt] \nonumber \end{align}

For completeness, the cubic equation determining ${\textit{Ra}}^{(o)}$ is contained in the supplementary material.

We recap briefly the results for the classical double-diffusive problem; more details can be found in HPE21. Oscillatory instability can occur only in the first quadrant of the $(Rs, Ra)$ plane (i.e. $Rs$ and ${\textit{Ra}}$ both positive); this is often referred to as the diffusive regime. The marginal value of ${\textit{Ra}}$ for oscillatory motions (i.e. when $s = \pm i \omega$ , $\omega \in \mathbb{R}_+$ ) is given by

(2.25) \begin{align} Ra^{(o)} = \left ( \frac {\sigma +\tau }{1+\sigma }\right ) Rs + \frac {(1+\tau )(\sigma +\tau )}{\sigma } \frac {\beta ^6}{k^2}, \end{align}

provided also that $\omega ^2\gt 0$ , which translates to the condition

(2.26) \begin{align} Rs\gt \frac {\tau ^2(1+\sigma )}{\sigma (1-\tau )} \frac {\beta ^6}{k^2}. \end{align}

It follows from (2.26) that oscillatory motions are preferred at onset provided that

(2.27) \begin{align} Rs\gt \frac {27}{4} \frac {\tau ^2(1+\sigma )}{\sigma (1-\tau )}; \end{align}

from (2.25), the critical Rayleigh number for oscillatory convection is then given by

(2.28) \begin{align} Ra^{(o)}_c = \left ( \frac {\sigma +\tau }{1+\sigma }\right ) Rs + \frac {27(1+\tau )(\sigma +\tau )}{4\sigma }. \end{align}

In the third quadrant (i.e. $Rs\lt 0$ , ${\textit{Ra}}\lt 0$ ), instability always occurs as steady convection, with the critical Rayleigh number given by (2.21); this is typically referred to as the salt fingering regime.

In HPE21 we studied the role of the (small) M–C effects by considering different regimes for (large) $Rs$ ; this was achieved by scaling $Rs$ with either $C_{\scriptscriptstyle T}^{-n}$ (for $C_{\scriptscriptstyle S}=0$ ) or $C_{\scriptscriptstyle S}^{-n}$ (for $C_{\scriptscriptstyle T}=0$ ) and investigating the nature of the behaviour with increasing $n$ . In a similar vein, for the more general problem considered here, it is instructive to consider the regimes defined by $Rs =O(\varGamma ^{-n})$ . As discussed above, in the classical problem, it is in the first and third quadrants of the $(Rs,Ra)$ plane that double-diffusive effects come into play (in the second quadrant the fluid is top heavy, and hence unstable to overturning, whereas the fourth quadrant is a region of stability). Thus, with the inclusion of M–C effects, we examine separately the behaviour in the first and third quadrants in the $(Rs,Ra)$ plane.

3.1. $0 \lt n \leqslant 1$

When $Rs$ assumes $O(1)$ values, the problem is essentially that of classical double-diffusive convection. Qualitative changes from the classical problem first arise when $n=1$ ; to investigate this regime, we rescale both Rayleigh numbers with $\varGamma ^{-1}$ . For classical double diffusion, the critical wavenumber for oscillatory convection (and indeed steady convection also) is given by $k^2=1/2$ . To explore how the M–C effect can influence this picture, we consider the regime $\varGamma \ll 1$ , $Rs = \varGamma ^{-1} \widetilde {Rs}$ , ${\textit{Ra}} = \varGamma ^{-1} \widetilde {Ra}$ , where $\widetilde {Rs}$ and $\widetilde {Ra}$ are $O(1)$ , and with $k = O(1)$ . The details of the associated analysis, which is in the same spirit as for the endpoints $\lambda = \pm 1$ , though algebraically more messy, are contained in Appendix A. The leading-order expression for the onset of oscillatory convection ( $\widetilde {Ra}_0$ ) is given by (A3), namely

(3.1) \begin{align} \widetilde {Ra}_0 = \frac {(\sigma +\tau )}{(1+\sigma )}\widetilde {Rs} \qquad \mbox{or} \qquad Ra_0 = \frac {(\sigma +\tau )}{(1+\sigma )}Rs. \end{align}

We note that $\widetilde {Ra}_0$ is independent of both $\lambda$ and $k$ . Determination of the critical wavenumber comes at the next order (i.e. determining $\widetilde {Ra}_1$ ), where $\textrm{d} \widetilde {Ra}_1/\textrm{d} k^2 = 0$ gives rise to a quintic polynomial in $k^2$ (expression (A6), with coefficients (A5)). Figure 1(a–c) shows the evolution of the real parts of the roots of the quintic polynomial (A6) as $\widetilde {Rs}$ is increased for three values of $\lambda$ . With $Rs = 0$ , the only real positive root is at $k = 1/2$ (the classical case). For all values of $\lambda$ , the existing mode moves to higher $k$ . As for $C_{\scriptscriptstyle S}=0$ (i.e. $\lambda =-1$ ), for $\lambda$ close to $-1$ , two equal stationary points appear at a critical value of $Rs$ ( $Rs^*$ , say), which then separate with a further increase in $Rs$ ; $k^2$ remains negative (and thus unphysical) for the other two roots (see figure 1 a). As $\lambda$ increases, $Rs^*$ increases (it has moved off to the right in figure 1 b), with $Rs^* \to \infty$ as $\lambda \to \lambda ^*$ , where

(3.2) \begin{align} \lambda ^* = \frac {\sqrt {\sigma +\tau }-\sqrt {\tau (1+\sigma )}}{\sqrt {\sigma +\tau }+\sqrt {\tau (1+\sigma )}}. \end{align}

For $\lambda \gt \lambda ^*$ , there is only one real stationary point (as in figure 1 c). Thus, for $-1 \leqslant \lambda \lt \lambda ^*$ , the M–C influence is felt for $Rs = O(\varGamma ^{-1})$ by the emergence of two new stationary points in the oscillatory stability boundary. More generally, it can be seen that, for all $\lambda$ , there is a shift of the classical mode to higher wavenumbers. Figure 1(d–f) shows the oscillatory stability boundary with no approximations (calculated numerically), together with the zeroth-order approximation $\widetilde {Ra}_0$ , and with the first-order correction $\widetilde {Ra}_1$ ; the latter is essentially indistinguishable from the boundary of the full system. The oscillatory stability boundary for the classical problem, given by (2.25), is also shown. The three stationary points in figure 1(d) correspond to the three positive roots for $k^2$ at $\widetilde {Rs} = 15$ in figure 1(a). Whenever there is more than one minimum in the ${\textit{Ra}}^{(o)}$ vs $k^2$ curve (as in figure 1 d), the higher wavenumber mode is always preferred. We note also that since ${\textit{Ra}}^{(s)}_c \approx Rs/\tau \gt Ra_0$ (from (2.21)), the preferred mode is always oscillatory.

Figure 1.

(a–c) The real parts of the roots of (A6) for $k_s^2$ as a function of $\widetilde {Rs}$ for $ 0 \lt n \leqslant 1$ ; blue lines denote real roots, the red line denotes the real part of a conjugate pair; $\sigma = 1$ , $\tau = 0.1$ , with (a) $\lambda =-0.8$ , (b) $\lambda =0$ , (c) $\lambda =0.8$ . (d–f) Corresponding plots of the oscillatory stability boundary (grey solid line), together with the zeroth-order asymptotic expression (A3) (magenta dashed line) and the first-order correction (A5) (magenta squares); for the numerical results, $\varGamma = 10^{-3}$ , $Rs = 1.5 \times 10^4$ (corresponding to $\widetilde {Rs}=15$ ). For comparison, the cyan line shows ${\textit{Ra}}^{(o)}$ for the classical problem, given by (2.25).

3.2. $n=2$

It is helpful to begin by summarising the main findings from HPE21 of the $n=2$ regime for $\lambda = \pm 1$ . When $\lambda = -1$ ( $C_{\scriptscriptstyle S}=0$ ), ${\textit{Ra}}$ , as a function of $k^2$ , has two minima: a large wavelength mode with $k^2=O(C_{\scriptscriptstyle T})$ and a small wavelength mode with $k^2 = O(C_{\scriptscriptstyle T}^{-1})$ . The $k^2 = O(C_{\scriptscriptstyle T}^{-1})$ mode is always preferred; the onset of instability is oscillatory, with ${\textit{Ra}}_c = O(C_{\scriptscriptstyle T}^{-2})$ . When $\lambda = 1$ ( $C_{\scriptscriptstyle T}=0$ ), there is only one minimum of ${\textit{Ra}}$ as a function of wavenumber, with $k^2 = O(C_{\scriptscriptstyle S}^{-1/2})$ ; the onset of instability is oscillatory, with ${\textit{Ra}}_c = O(C_{\scriptscriptstyle S}^{-2})$ . Even for the limiting cases of $\lambda =\pm 1$ , analytical progress in the $n=2$ regime was limited, since the only simplification is that (at high wavenumbers) $\beta ^2 \approx k^2$ . For $-1 \lt \lambda \lt 1$ , where the behaviour is more complex, the $n=2$ regime is essentially accessible only to numerical study.

Figure 2.

Plots of $Ra^{(o)}$ versus $k^2$ for $\varGamma =10^{-3}$ , $Rs=10^6$ ( $n=2$ ), $\tau =0.5$ . In (a–c), $\sigma = 10$ , with (a) $\lambda =-0.8$ (black), $\lambda =-1$ (red); (b) $\lambda =0$ ; (c) $\lambda =0.8$ (black), $\lambda =1$ (red). In (d–f), $\sigma = 0.1$ , with (d) $\lambda =-0.8$ (black), $\lambda =-1$ (red); (e) $\lambda =0$ ; ( f) $\lambda =0.8$ (black), $\lambda =1$ (red). The cyan lines show ${\textit{Ra}}^{(o)}$ for the classical problem, given by (2.25).

Figure 2 shows ${\textit{Ra}}^{(o)}$ as a function of $k^2$ for a large and small value of $\sigma$ ( $\sigma =10$ in panels a–c and $\sigma =0.1$ in panels d–f), with $\tau =0.5$ , and for $\lambda = -0.8$ , $0$ , $0.8$ ; for comparison, the limiting values of $\lambda =-1$ and $\lambda =1$ are also shown, respectively, on the $\lambda = -0.8$ and $\lambda = 0.8$ plots. Note that since ${\textit{Ra}}^{(s)}_c \approx Rs/\tau = 2 \times 10^6$ , the oscillatory mode is preferred, and hence determines the overall stability boundary. For both values of $\sigma$ , the qualitative behaviour as $\lambda$ changes is the same. For $\lambda =-0.8$ , there are two minima in ${\textit{Ra}}^{(o)}$ , as for $\lambda =-1$ ; the preferred mode is at the larger wavenumber, with $k^2= O(C_{\scriptscriptstyle T}^{-1})$ . As $\lambda$ increases, the (non-preferred) minimum at small $k^2$ disappears and the preferred mode moves gradually to smaller $k^2$ . The minima of the $\lambda =0.8$ curves are located at slightly larger values of $k^2$ than those at $\lambda =1$ , where the preferred mode has $k^2 = O(C_{\scriptscriptstyle S}^{-1/2})$ . For comparison, ${\textit{Ra}}^{(o)}$ for the classical problem, given by (2.25), is also shown in figure 2; for the cases illustrated, it can be seen that the M–C effect is a destabilising influence.

Figure 3.

$\text{Plots of }Ra^{(o)}_c$ , $k_c^2$ and $\omega _c^2$ versus $\lambda$ for $\varGamma =10^{-3}$ , $Rs=10^6$ ( $n=2$ ). In (a–c), $\sigma = 10$ , $\tau =0.5$ ; in (d–f), $\sigma =0.1$ , $\tau =0.5$ ; in (g–i), $\sigma =10$ , $\tau =0.9$ ; in (j–l), $\sigma =0.1$ , $\tau =0.9$ . The blue dashed line shows ${\textit{Ra}}^{(o)}_c$ for the classical problem, given by (2.28).

Figure 3 shows ${\textit{Ra}}^{(o)}_c$ , $k_c^2$ and $\omega _c^2$ as a function of $\lambda$ for four pairs of $(\sigma , \tau )$ values. Again, ${\textit{Ra}}^{(s)}_c \approx Rs/\tau$ exceeds ${\textit{Ra}}^{(o)}_c$ and so the oscillatory stability boundary determines the overall boundary. As shown in HPE21, at both endpoints ( $\lambda =\pm 1$ ), ${\textit{Ra}}^{(o)}_c =O(\varGamma ^{-2})$ , though which of these is the more unstable depends on the values of $\sigma$ and $\tau$ (contrast, for example, figures 3 a and 3 d). In the displayed numerical results, the most stable configuration has a mid-range value of $\lambda$ . However, at very large $\sigma$ (not shown) and small values of $\tau$ , ${\textit{Ra}}^{(o)}_c$ essentially increases monotonically with $\lambda$ over the entire interval; furthermore, for very large $\sigma$ and $\tau$ close to unity, ${\textit{Ra}}^{(o)}_c$ increases for $-1 \lt \lambda \lesssim 0$ and then exhibits very little change over the range $0 \lesssim \lambda \lt 1$ . As already noted, the values of $k_c^2$ at the endpoints are asymptotically distinct. Thus, there is inevitably a decrease in $k_c^2$ from $\lambda =-1$ to $+1$ ; this may or may not be monotonic, depending on the values of $\sigma$ and $\tau$ . Conversely, the frequencies at $\lambda = \pm 1$ both scale as $\omega _c^2 = O(\varGamma ^{-2})$ , a scaling that is maintained throughout the range of $\lambda$ . Figure 3 also shows ${\textit{Ra}}^{(o)}_c$ for the classical problem, given by (2.28). It can be seen that for the cases illustrated, the M–C effect, save possibly for a small interval of $\lambda$ , is destabilising.

Figure 4.

$\text{Plots of }Ra^{(o)}_c$ , $k_c^2$ and $\omega _c^2$ versus $\lambda$ for $\varGamma =10^{-3}$ , $\sigma =1$ , $\tau =0.5$ . In (a–c), $Rs=10^9$ ( $n=3$ ); in (d–f), $Rs=10^{12}$ ( $n=4$ ); in (g–i), $Rs=10^{15}$ ( $n=5$ ). The asymptotic results (3.11) are marked as dashed red lines.

3.3. $n\geqslant 3$

Figure 4 shows ${\textit{Ra}}^{(o)}_c$ , $k_c^2$ and $\omega _c^2$ as functions of $\lambda$ for $n=3$ , $4$ , $5$ . The most striking feature of the $n \geqslant 3$ regime is the appearance of a narrow region close to $\lambda _1/\lambda _2= \tau$ (or $\lambda = \lambda _d = (\tau -1)/(\tau +1)$ ) in which both the critical wavenumber and its associated frequency are markedly smaller than at nearby values of $\lambda$ . This region corresponds to $\tau _{\scriptscriptstyle T} = \tau _{\scriptscriptstyle S}$ , as discussed in § 1, and we refer to it as ‘the dip’. The sharp minima in $k_c^2$ and $\omega _c^2$ are accompanied by a sharp maximum in ${\textit{Ra}}^{(o)}_c$ . The analysis below shows that the width of the narrow region is $O (\varGamma ^{(n-2)/2} )$ , as can be seen in figure 5, which shows a small range of $\lambda$ near $\lambda _d$ . Figure 6 shows ${\textit{Ra}}^{(o)}_c$ as a function of $k^2$ for values of $\lambda$ just to the left of the dip, in the dip itself, and just to the right of the dip. The transition of $k_c^2$ and $\omega _c^2$ in the dip is discontinuous: as $\lambda$ increases through $\lambda _d$ , the minimum at larger $k^2$ (figure 6 a, with $k_c^2=1.77 \times 10^8$ ) becomes higher than that at smaller $k^2$ , leading to a discontinuous change in the critical wavenumber (figure 6 b, with $k_c^2 = 1.22 \times 10^6$ ), before the minimum at large $k^2$ reasserts itself (figure 6 c, with $k_c^2 = 2.22 \times 10^8$ ).

Figure 5.

$\text{Plots of }k_c^2$ versus $\lambda$ , for a restricted range of $\lambda$ , with $\varGamma =10^{-3}$ , $\sigma =1$ , $\tau =0.5$ . In (a), $Rs=10^9$ ( $n=3$ ); in (b), $Rs=10^{12}$ ( $n=4$ ); in (c), $Rs=10^{15}$ ( $n=5$ ). In (b) the asymptotic result (3.25) is marked as a red line.

Figure 6.

$\text{Plots of }Ra^{(o)}$ versus $k^2$ with $\varGamma =10^{-3}$ , $Rs=10^{12}$ ( $n=4$ ), $\sigma =1$ , $\tau =0.5$ . In (a), $\lambda =-0.34$ (just to the left of the dip); in (b), $\lambda =-0.3335$ (in the middle of the dip); in (c), $\lambda =-0.33$ (just to the right of the dip).

3.3.1. Asymptotics for general $\lambda$

We can give accurate asymptotic descriptions of the behaviour of the marginal curves in the first quadrant when $\varGamma \ll 1$ and $n \gtrsim 3$ (the larger the value of $n$ the more accurate the results). To begin with, we consider the general case away from the dip, so that $\lambda _1/\lambda _2$ is not close to $\tau$ . Guided by the numerical results, we pose the following scalings:

(3.3) \begin{align} \begin{aligned} & Rs=\varGamma ^{-n}\widetilde {Rs}, \quad k^2=\varGamma ^{(1-2n)/3}\tilde k^2, \quad Ra^{(o)}=\varGamma ^{-(2+2n)/3}\widetilde {Ra}, \\ & \quad \omega ^2\sim k^2/2\varGamma =\varGamma ^{-(2+2n)/3}\tilde \omega ^2. \end{aligned} \end{align}

Note that ${\textit{Ra}}^{(o)}_c$ determined under the scalings (3.3) will be asymptotically smaller than ${\textit{Ra}}^{(o)}_c$ for the classical problem (in which, from (2.28), ${\textit{Ra}}^{(o)}_c$ and $Rs$ are of the same order).

When $n\gt 2$ , we can define the small parameter $\varepsilon =\varGamma ^{(n-2)/3}$ . We then make an expansion in powers of $\varepsilon$ , with the expansion carried as far as $\varepsilon ^2=\varGamma ^{(2n-4)/3}$ . Since $k^2$ is assumed large, we can replace $\beta ^2=k^2+1$ by $k^2$ , since the error in doing this is $O (\varGamma ^{(2n-1)/3} )= O (\varGamma \varepsilon ^2 )$ .

The leading terms in (2.23) suggest the ansatz

(3.4) \begin{align} \omega ^2=\frac {1}{\lambda _2}\left (\frac {\tilde k^2}{2}+\varepsilon p_1+\varepsilon ^2 p_2\right )\varGamma ^{-(2+2n)/3}. \end{align}

On substituting expression (3.4) into (2.23), we find, correct to $O(\varepsilon ^2)$ , that

(3.5a) \begin{align} b_6\omega ^6&=2\sigma \lambda _1^2\lambda _2^{-1} \big ( \tilde k^8+6 \varepsilon p_1\tilde k^6 + 12 \varepsilon ^2 p_1^2 \tilde k^4 +6\varepsilon ^2p_2\tilde k^6 \big ) \varGamma ^{(7-8n)/3}, \end{align}

(3.5b) \begin{align} b_4\omega ^4 &= \left (-2 \sigma \big ( \lambda _1^2\lambda _2^{-1} +2\tau \lambda _1 \big )\tilde k^8 - 8 \varepsilon \sigma \big ( \lambda _1^2\lambda _2^{-1} +2\tau \lambda _1 \big ) p_1 \tilde k^6 + \varepsilon ^2 \big ( \sigma +(1+\sigma )\lambda _1^2\lambda _2^{-2} \big )\tilde k^6\right . \nonumber \\ &\quad -\left .8\varepsilon ^2\sigma \big ( \lambda _1^2\lambda _2^{-1} +2\tau \lambda _1 \big ) \big (p_2\tilde k^2+p_1^2\tilde k^4 \big ) \right )\varGamma ^{(7-8n)/3}, \end{align}

(3.5c) \begin{align} b_2\omega ^2&= \left ( 2\sigma \tau ( \tau \lambda _2 + 2\lambda _1 ) \tilde k^8 + \varepsilon \big ( 2\sigma \big ( \tau \lambda _2 - \lambda _1^2 \lambda _2^{-1} \big )\widetilde {Rs}\tilde k^4 + 4 \sigma \tau ( \tau \lambda _2 + 2 \lambda _1 ) p_1 \tilde k^6 \big ) \right . \nonumber \\ & \quad + \varepsilon ^2 \left ( 4 \sigma \tau ( \tau \lambda _2 + 2\lambda _1 ) p_2 \tilde k^6 + 4 \sigma \big ( \tau \lambda _2 - \lambda _1^2\lambda _2^{-1} \big ) p_1 \widetilde {Rs} \tilde k^2 \right . \nonumber \\ & \left . \left . \quad - \left ( \sigma +2\tau \big ( 1+\sigma \big ) \lambda _1\lambda _2^{-1} \right ) \tilde k^6 \right ) \right ) \varGamma ^{(7-8n)/3}, \end{align}

(3.5d) \begin{align} b_0 &= \left ( -2\sigma \tau ^2\lambda _2 \tilde k^8 + 2 \varepsilon \sigma \tau (\lambda _1-\lambda _2) \widetilde {Rs}\tilde k^4+\varepsilon ^2 \tau ^2 (1+\sigma ) \tilde k^6 \right )\varGamma ^{(7-8n)/3} . \end{align}

By construction, the leading-order terms in (2.23) cancel. At $O(\varepsilon )$ we find that

(3.6) \begin{align} p_1=\frac {\lambda _1\widetilde {Rs}}{2 (\lambda _1-\tau \lambda _2) \tilde k^2 }, \end{align}

and at $O(\varepsilon ^2)$ we obtain

(3.7) \begin{align} 16 p_1^2 \sigma \left ( \lambda _1^2\lambda _2^{-1}-\tau \lambda _1 \right ) \tilde k^4 & + \left ( 4 p_2 \sigma \lambda _2^{-1} + (1+\sigma )\lambda _2^{-2}\right ) \left ( \lambda _1-\tau \lambda _2 \right )^2 \tilde k^6 \nonumber \\ &+ 4 p_1 \sigma \lambda _2^{-1} \left ( \tau \lambda _2^2 - \lambda _1^2 \right ) \widetilde {Rs}\tilde k^2=0. \end{align}

Finally, substituting for $p_1$ from (3.6) into (3.7) yields

(3.8) \begin{align} p_2=- \frac {1}{4}\left ( \frac {1+\sigma }{\lambda _2 \sigma }+\frac {2\lambda _1\left ( \lambda _1^2 + \tau \lambda _2^2 \right ) \widetilde {Rs}^2}{\left ( \lambda _1 - \tau \lambda _2 \right )^3\tilde k^6}\right ). \end{align}

We have now found the expression of $\omega ^2$ correct to order $\varepsilon ^2$ in expansion (3.4). To determine the corresponding value of the critical Rayleigh number, we substitute for $\omega ^2$ into either (2.22a ) or (2.22b ); expanding as before and cancelling a common factor of $\varGamma$ , we obtain the following expressions for the various terms in, for example, (2.22b ):

(3.9a) \begin{align} a_4\omega ^4 &= \frac {\lambda _1}{\lambda _2} \tilde k^8+4 \varepsilon \frac {\lambda _1}{\lambda _2} p_1 \tilde k^6 + \varepsilon ^2 \left ( 4 \frac {\lambda _1}{\lambda _2}(p_2\tilde k^6+p_1^2\tilde k^4) + \frac {1}{2\lambda _2^2\sigma }\tilde k^6 \right ), \end{align}

(3.9b) \begin{align} a_2\omega ^2 &=\frac {1}{\lambda _2} \left [ (\tau \lambda _2+\lambda _1)\tilde k^8 + \varepsilon \left ( \widetilde {Rs}\tilde k^4+2(\tau \lambda _2+\lambda _1)p_1\tilde k^6 \right ) \right . \nonumber\\ & \quad \left . +\varepsilon ^2\left (2(\tau \lambda _2+\lambda _1)p_2\tilde k^6+2\widetilde {Rs} p_1\tilde k^2+\frac {1}{2}\left (\left(1+ \frac{1+\tau}{\sigma} \right)\tilde k^6-2\widetilde {Ra} \tilde k^4\right )\right )\right ] , \end{align}

(3.9c) \begin{align} a_0 &=\tau \tilde k^8+\varepsilon \widetilde {Rs}\tilde k^4-\varepsilon ^2\tau \widetilde {Ra}\tilde k^4. \end{align}

It is easily checked that both the $O(1)$ and $O(\varepsilon )$ terms in (2.22b ) cancel; the remaining terms lead to the relatively simple relation

(3.10) \begin{align} \widetilde {Ra}=\frac {\tilde k^2}{2\lambda _2}+\frac {\lambda _1^2\widetilde {Rs}^2}{(\tau \lambda _2-\lambda _1)^2\tilde k^4} \implies Ra^{(o)} = \frac {\varGamma ^{-1} k^2}{2\lambda _2} + \frac {\lambda _1^2 Rs^2}{(\tau \lambda _2 - \lambda _1)^2 k^4} . \end{align}

Minimising ${\textit{Ra}}^{(o)}$ with respect to $k^2$ determines the critical values $k_c^2$ and ${\textit{Ra}}^{(o)}_c$ as

(3.11) \begin{align} k_c^2=\left (\frac {4\lambda _1^2\lambda _2 \varGamma Rs^2}{(\tau \lambda _2-\lambda _1)^2}\right )^{\tfrac {1}{3}}, \qquad Ra^{(o)}_c=\frac {3}{2\lambda _2}\left (\frac {\lambda _1^2\lambda _2 \varGamma ^{-2} Rs^2}{2(\tau \lambda _2-\lambda _1)^2}\right )^{\tfrac {1}{3}}. \end{align}

The asymptotic results (3.11) are plotted in figure 4. As expected, the agreement with the full system improves with increasing $n$ ; for $n=4$ and $n=5$ , the asymptotic and numerical results are almost indistinguishable, except in the immediate neighbourhood of the dip, where expressions (3.11) diverge. In each case, the classical stability boundary is at a much higher value of ${\textit{Ra}}$ , and so is not shown.

It is interesting to relate the above results to those for $\lambda =\pm 1$ obtained in HPE21. From the expression for ${\textit{Ra}}^{(o)}_c$ in (3.11) we can calculate the position of the minimum in the ${\textit{Ra}}$ vs $\lambda$ curve, which is at $\lambda = -1 + 2 \sqrt {\tau /(1+\tau )}$ . We note that at this minimum, ${\textit{Ra}} = O(\varGamma ^{-(2+2n)/3})$ , whereas, when $\lambda =-1$ , different asymptotic scalings apply.

For $\lambda =-1$ ,

(3.12) \begin{align} {\textit{Ra}} \approx \varGamma ^{-1} \big(Rs/\tau \big)^{1/2} = O(\varGamma ^{-(1+n/2)}); \end{align}

hence, the smallest value of $\widetilde {Ra}_c$ always occurs at $\lambda =-1$ ( $C_{\scriptscriptstyle S}=0$ ). How then does the solution approach $\lambda =-1$ ? Assuming that the full solution varies slowly with $\lambda$ near $\lambda =-1$ , we can estimate where the internal solution given by (3.11) attains the ‘left-hand’ value. Assuming $\lambda _1$ is small (and hence $\lambda _2 \approx 1$ ), we find an intersection of the two solutions when $\lambda =-1+\delta \lambda _1$ where

(3.13) \begin{align} \delta \lambda _1 =O \big( \big( Rs/\tau \varGamma ^2\big)^{-1/4}\big); \end{align}

this is small when $ n \gtrsim 3$ .

Near $\lambda =1$ $(\lambda _2=0)$ , on the other hand, (3.11) implies that ${\textit{Ra}} =O ( \varGamma ^{-2/3}(Rs/\lambda _2)^{2/3} )$ as $\lambda _2 \to 0$ . However, the scaling for $\lambda =1$ gives

(3.14) \begin{align} {\textit{Ra}} \approx \sigma Rs/(1+\sigma ). \end{align}

Matching these two results is accomplished through a boundary layer of thickness $\delta \lambda _2$ , where

(3.15) \begin{align} \delta \lambda _2 =O \left ( (1+\sigma ^{-1})^{3/2}(\varGamma ^2 Rs)^{-1/2} \right ). \end{align}

3.3.2. Asymptotics near the dip

Near the critical value $\lambda _d=(\tau -1)/(\tau +1)$ different asymptotics apply, as can be seen from the numerical results, which give a much lower critical wavenumber. This much smaller wavenumber mode exists for a range of $\lambda$ , but the analysis is easier when $\lambda$ is close to $\lambda _d$ . For the time being, then, we set

(3.16) \begin{align} \lambda _1 = \lambda _2 (\tau + \gamma \kappa ), \end{align}

where we introduce the expansion parameter $\delta =\varGamma ^{(n/2-1)} \ll 1$ ( $n\gt 2)$ and write $\gamma =\delta /\lambda _2$ . Following numerical evidence, we write ${\textit{Ra}}=\varGamma ^{-n}\widetilde {Ra}$ , $k^2=\varGamma ^{-n/2}\tilde k^2$ (note that these differ from the scalings in § 3.3.1), and carry our analysis to $O(\delta )$ . As in § 3.3.1, since $k^2\gg 1$ we would like to approximate $\beta ^2 =1+k^2$ by $k^2$ . The error due to such an approximation is then of relative order $\varGamma ^{n/2}$ , while we need to carry the expansion only to $O(\delta ) \gg \varGamma ^{n/2}$ . Thus to this order we can write $\beta ^2=k^2$ throughout.

Keeping the two leading-order terms in the coefficients of (2.24) gives

(3.17a) \begin{align} b_6 &= 16\sigma \big ( \tau ^2+2\gamma \tau \kappa \big)\lambda _2^4\tilde k^2\varGamma ^{4-n/2}, \\[-9pt] \nonumber \end{align}

(3.17b) \begin{align} b_4 &= \left ( - 24\sigma \tau ^2 \tilde k^4-32\gamma \sigma \tau \kappa \tilde k^4 + 4\gamma \big(\sigma +(1+\sigma )\tau ^2\big) \tilde k^2\right ) \lambda _2^3 \varGamma ^{3-n}, \\[-9pt] \nonumber \end{align}

(3.17c) \begin{align} b_2 &= \left (12\sigma \tau ^2 \tilde k^6+ 4\sigma \tau (1 - \tau ) \widetilde {Rs} \tilde k^2 +8\gamma \sigma \tau \kappa \tilde k^6 - 8 \gamma \sigma \tau \kappa \widetilde {Rs}\tilde k^2 \right . \nonumber \\ &\quad \left .-\, 2\gamma (\sigma +2\tau ^2(1+\sigma )) \tilde k^4\right ) \lambda _2^2 \varGamma ^{2-3n/2}, \\[-9pt] \nonumber \end{align}

(3.17d) \begin{align} b_0 &= \left (- 2 \sigma \tau ^2 \tilde k^8 - 2\sigma \tau (1-\tau ) \widetilde {Rs} \tilde k^4 \right . \nonumber \\ & \quad \left .+\, \gamma \tau ^2 (1+\sigma ) \tilde k^6 +2 \gamma \sigma \tau \kappa \widetilde {Rs} \tilde k^4 - \gamma \sigma (1-\tau )\widetilde {Rs}\tilde k^2 \right ) \lambda _2 \varGamma ^{1-2n}. \\[9pt] \nonumber \end{align}

Note that in expressions (3.17), we have not expanded the $\lambda _2$ terms into their $O(1)$ and $O(\gamma )$ components. It turns out that there is significant cancellation of the $\lambda _2$ terms in (2.23) and we therefore defer to a later stage any $\lambda _2$ terms that remain.

It is easily verified that when $\gamma =0$ , equation (2.23) (with coefficients (3.17)) is solved by

(3.18) \begin{align} \omega ^2 = \frac {1}{2\lambda _2}\tilde k^2 \varGamma ^{-1-n/2} = \frac {1}{2\lambda _2} k^2 \varGamma ^{-n/2} . \end{align}

To obtain the next order correction to $\omega ^2$ , we thus write

(3.19) \begin{align} \omega ^2=\frac {1}{2\lambda _2}\left (\tilde k^2 +\gamma p\right ) \varGamma ^{-1-n/2}, \end{align}

where we still keep $\lambda _2$ unexpanded. From (2.23), at $O(\gamma )$ , there is considerable simplification; all of the factors of $\varGamma$ and $\lambda _2$ drop out, with the remaining terms determining $p$ simply as

(3.20) \begin{align} p = \frac {1}{2\tau } + \frac {\tilde k^2\kappa }{(1-\tau )}. \end{align}

We now use (2.22b ), correct to $O(\gamma )$ , to determine $\widetilde {Ra}$ , which we write as $\widetilde {Ra}=\widetilde {Ra}_0 + \gamma \widetilde {Ra}_1$ . At this stage we do need to pay heed to the separate $O(1)$ and $O(\gamma )$ terms in $\lambda _2$ ; thus, from (3.16), together with the relation $\lambda _1 + \lambda _2=1$ , we have

(3.21) \begin{align} \lambda _2^{-1}=(1+\tau )+\gamma \kappa . \end{align}

From (2.19b,d, f ), together with (3.19)–(3.21), retaining terms up to $O(\gamma )$ and dividing out a common factor of $\varGamma ^{-2n}$ , we obtain

(3.22a) \begin{align} a_4\omega ^4 &= \tau \tilde k^8 + \gamma \left (\left( 1 + \frac{1 + \tau}{2 \sigma} \right) {\tilde k}^6+\kappa \left (\frac {1+\tau }{1-\tau } \right )\tilde k^8 \right )\!, \\[-12pt] \nonumber \end{align}

(3.22b) \begin{align} a_2\omega ^2 &= 2\tau \tilde k^8+(1+\tau )\big(\widetilde {Rs}-\widetilde {Ra}_0\big) \tilde k^4 \nonumber \\ & \quad +\, \gamma \left ( \frac {1}{2} \left ( 3+\frac {1+\tau }{\sigma } \right ) \tilde k^6 + \frac {1+\tau }{2\tau } \left ( \widetilde {Rs}- \widetilde {Ra}_0 \right ) \tilde k^2 -(1+\tau )\widetilde {Ra}_1 \tilde k^4\right ) \nonumber \\ & \quad +\, \gamma \kappa \left (\frac {1}{1-\tau }\right ) \left ((1+\tau ) \tilde k^8 + 2 \left ( \widetilde {Rs}-\widetilde {Ra}_0 \right ) \tilde k^4\right )\!, \\[-12pt] \nonumber \end{align}

(3.22c) \begin{align} a_0 &= \tau \tilde k^8+\left ( \widetilde {Rs}-\tau \widetilde {Ra}_0 \right ) \tilde k^4 -\gamma \tau \widetilde {Ra}_1 \tilde k^4. \\[9pt] \nonumber \end{align}

On substituting from expressions (3.22) into (2.22b ), the $O(1)$ and $O(\gamma )$ terms give

(3.23) \begin{align} \widetilde {Ra}_0 =\tau \widetilde {Rs}, \qquad \qquad \widetilde {Ra}_1 = \frac {1}{2}\left (\tilde k^2+\frac {1}{\tau \tilde k^2}\big(1-\tau ^2\big)\widetilde {Rs}\right )+ 2 \kappa \widetilde {Rs}, \end{align}

or, in unscaled variables,

(3.24) \begin{align} Ra^{(o)} = \tau Rs + \gamma \left ( \frac {\varGamma ^{-n/2}}{2} \left ( k^2 + \frac {1}{\tau k^2}\big(1-\tau ^2\big) Rs \right )+ 2 \kappa Rs \right )\!. \end{align}

Thus ${\textit{Ra}}^{(o)}$ is minimised when

(3.25) \begin{align} k_c^4 = \frac {1}{\tau }\big(1-\tau ^2\big) Rs, \end{align}

which is independent of $\kappa$ at leading order. The excellent agreement between the asymptotic result (3.25) and the full numerical result is shown in figure 5. It should just be noted that in the dip, ${\textit{Ra}}^{(o)}_c$ ( $= \tau Rs$ at leading order, from (3.23)) and ${\textit{Ra}}^{(s)}_c$ ( $=Rs/\tau$ at leading order, from (2.21)) are of the same asymptotic order ( $O (\varGamma ^{-n} )$ ). However, the oscillatory mode is always preferred since $\tau \lt 1$ . Furthermore, ${\textit{Ra}}^{(o)}_c$ in the dip ( $= \tau Rs$ ) is also of the same asymptotic order as ${\textit{Ra}}^{(o)}_c$ for the classical problem ( $= (\sigma +\tau )/(1+\sigma ) Rs$ at leading order, from (2.28)). Since $\tau \lt 1$ , it follows that, even in the dip, ${\textit{Ra}}^{(o)}_c$ for the M–C problem with $n \geqslant 3$ is lower than that of the classical problem.

Expression (3.25) describes the depth of the dip. Its lateral extent in $\lambda$ can be estimated by equating the values of ${\textit{Ra}}^{(o)}_c$ for the outer solution, given by (3.11), with the value of ${\textit{Ra}}^{(o)}_c$ for the inner (dip) solution, which, to leading order, is given by ${\textit{Ra}} = \tau Rs$ , from (3.23). If we write $\lambda = \lambda _d + \delta$ then, to leading order, (3.11) gives

(3.26) \begin{align} Ra^{(o)}_c = 3 \left ( \frac {\tau Rs}{2 \varGamma \delta (1+\tau )}\right )^{2/3}. \end{align}

Equating expression (3.26) with $\tau Rs$ leads to the following estimate of $\delta$ , the half-width of the dip:

(3.27) \begin{align} \delta = \frac {3^{3/2}}{2 \varGamma (1+\tau ) (\tau Rs)^{1/2}}. \end{align}

The decrease in the width of the dip with increasing $Rs$ can be clearly seen in figure 5.

3.3.3. Low wavenumber modes

The analysis of § 3.3.2 shows that the preferred mode at the dip is a high wavenumber mode, with $k_c^2 = O (\varGamma ^{-n/2} )$ and ${\textit{Ra}}^{(o)}_c=O (\varGamma ^{-n} )$ . For completeness, it is worth noting that there is also a minimum in the $ (k^2, Ra^{(o)} )$ curve at very low wavenumbers. Guided by the numerical results, the relevant scalings for this low wavenumber mode are $k^2=O (\varGamma ^{n-1} )$ , $\omega ^2=O (\varGamma ^{-1} )$ , ${\textit{Ra}}^{(o)} = O (\varGamma ^{-n} )$ : we may thus write $k^2=\varGamma ^{n-1}\tilde k^2$ , $\omega ^2=\varGamma ^{-1} \tilde \omega ^2$ , $Rs = \varGamma ^{-n} \widetilde {Rs}$ , ${\textit{Ra}} = \varGamma ^{-n} \widetilde {Ra}$ . With these scalings, the dominant balance in the frequency equation (2.23) is between the $b_2 \omega ^2$ and $b_0$ terms, giving, to leading order,

(3.28) \begin{align} \tilde \omega ^2 = \left ( \frac {\sigma (1-\tau )}{1+\sigma } \right ) \widetilde {Rs} \tilde k^2. \end{align}

At leading order, (2.22b ) gives

(3.29) \begin{align} \widetilde {Ra}^{(o)}= \frac { \frac {2}{\sigma } \tilde \omega ^4 - \left ( 1 + \frac {(1+\tau )}{\sigma } + 2 \widetilde {Rs} \tilde k^2 \right ) \tilde \omega ^2 + \widetilde {Rs} \tilde k^2}{\tilde k^2 \left ( \tau - 2 \tilde \omega ^2 \right )}, \end{align}

which simplifies, using (3.28), to

(3.30) \begin{align} \widetilde {Ra}^{(o)}=\left ( \frac {\sigma +\tau }{1+\sigma }\right ) \widetilde {Rs} \implies Ra^{(o)}=\left ( \frac {\sigma +\tau }{1+\sigma }\right ) Rs. \end{align}

The $k^2$ dependence has dropped out at this order, so to determine $k_c^2$ we would need to go to higher order. However, this is unnecessary, since the value of ${\textit{Ra}}^{(o)}$ for the high wavenumber mode at the dip ( ${\textit{Ra}}^{(o)} = \tau Rs$ from (3.23)), although of the same order asymptotically as ${\textit{Ra}}^{(o)}$ given by (3.30) (i.e. $O ( \varGamma ^{-n} )$ ), is always lower (recalling that $\tau \lt 1$ ). Thus, in the first quadrant the low wavenumber mode is never preferred.

3.3.4. Varying $\tau$ and $\sigma$

In this subsection we consider, numerically, the effects of varying $\tau$ and $\sigma$ . Figure 7 shows ${\textit{Ra}}^{(o)}_c$ , $k_c^2$ and $\omega _c^2$ as functions of $\lambda$ for $n=3$ with $\sigma =1$ and for three different values of $\tau$ (the $\tau =0.5$ results are as plotted in figure 4 a–c). It can be clearly seen that increasing $\tau$ leads to the dip in $k_c^2$ becoming deeper and narrower, in accordance with expressions (3.25) and (3.27). The wider picture, taken across all $\tau$ , is provided by the surface plots in figure 8, which show ${\textit{Ra}}^{(o)}_c$ and $k^2_c$ as functions of $\lambda$ and $\tau$ ; the dip is the prominent valley in the $k_c^2$ surface.

Figure 7.

$\text{Plots of }Ra^{(o)}_c$ , $k_c^2$ and $\omega _c^2$ versus $\lambda$ for $\varGamma =10^{-3}$ , $Rs=10^9$ , $\sigma =1$ , with $\tau =0.2$ (red), $\tau =0.5$ (black), $\tau =0.8$ (blue).

Figure 8.

Surface plots of (a) ${\textit{Ra}}^{(o)}_c$ and (b) $k_c^2$ versus $\lambda$ and $\tau$ for $\varGamma =10^{-3}$ , $Rs=10^9$ , $\sigma =1$ . The range $0.05 \leqslant \tau \leqslant 0.95$ is plotted.

Figure 9.

$\text{Plots of }Ra^{(o)}_c$ , $k_c^2$ and $\omega _c^2$ versus $\lambda$ for $\varGamma =10^{-3}$ , $Rs=10^9$ , $\tau =0.5$ , with $\sigma =1$ (black), $\sigma =5$ (red), $\sigma =100$ (blue).

In terms of varying $\sigma$ , the general picture is unchanged between $\sigma =O(1)$ and $\sigma \gg 1$ , as illustrated by figure 9, which shows ${\textit{Ra}}^{(o)}_c$ , $k_c^2$ and $\omega _c^2$ as functions of $\lambda$ for $\sigma = 1$ , $5$ , $10$ . There are strong variations in $k_c^2$ and $\omega _c^2$ (but not ${\textit{Ra}}^{(o)}_c$ ) near $\lambda =-1$ to adjust to their limiting values at $\lambda =1$ (see § 4.2.3 of HPE21). For smaller $\sigma$ , the differences from $\sigma =O(1)$ become more marked. Figure 10 shows ${\textit{Ra}}^{(o)}_c$ and $k^2_c$ as functions of $\lambda$ and $\tau$ for $\sigma =0.075$ and $\sigma =0.01$ . In comparison with figure 8, it can be seen that decreasing $\sigma$ leads to a drastic change in the $k_c^2$ surface for $\lambda$ close to $-1$ . The sides of the valley are eroded, first of all to form just an island where higher $k_c^2$ modes are preferred (figure 10 b) before the valley disappears altogether (figure 10 d). The asymptotic analysis of § 3.3.1, performed under the assumption that $\sigma \gtrsim O(1)$ , and leading to the results (3.11), is clearly no longer valid. Indeed, one can see that the ordering adopted in § 3.3.1 breaks down once $\sigma$ is as small as $O(\varGamma ^{(n-2)/3})$ , i.e. when $\sigma = O(10^{-1})$ for the parameter values of figure 10. Analysing small $\sigma$ requires a separate asymptotic ordering (or orderings) along the lines pursued by Hughes, Proctor & Eltayeb (Reference Hughes, Proctor and Eltayeb2022) in their analysis of the M–C effect in rotating convection: as this would essentially be a further paper in itself, we have restricted our detailed attention here to the case of $\sigma \gtrsim O(1)$ .

Figure 10.

Surface plots of (a,c) ${\textit{Ra}}^{(o)}_c$ and (b,d) $k_c^2$ versus $\lambda$ and $\tau$ for $\varGamma =10^{-3}$ , $Rs=10^9$ . In (a,b), $\sigma =0.075$ ; in (c,d), $\sigma =0.01$ . The range $0.05 \leqslant \tau \leqslant 0.95$ is plotted.

4.1. $n=2$ : appearance of oscillations for smaller $\lambda$

We first look for the onset of oscillations in the third quadrant for $n=2$ , when $\lambda$ is sufficiently negative; in this case, the situation is similar to that described in HPE21 for $\lambda =-1$ . Provided $\sigma$ is sufficiently small, oscillations appear as a loop of marginal solutions, either above or below the steady branch, depending on the parameters.

We have performed a series of numerical calculations to establish the dependence of the critical values of ${\textit{Ra}}$ , $k^2$ and $\omega ^2$ on $\varGamma$ . Guided by these results, we deduce that the relevant scalings are ${\textit{Ra}}, Rs =O ( \varGamma ^{-2} )$ , $k^2 = O ( \varGamma ^{-1/2} )$ , $\omega ^2 = O ( \varGamma ^{-2} )$ : thus we write $Rs = \varGamma ^{-2} \widetilde {Rs}$ , ${\textit{Ra}} = \varGamma ^{-2} \widetilde {Ra}$ , $k^2 =\varGamma ^{-1/2} \tilde k^2$ , $\omega ^2 = \varGamma ^{-2}\tilde \omega ^2$ . At leading order, the coefficients of (2.19) then become

(4.1a) \begin{align} a_5 &= \big(1-\lambda ^2\big) \frac {\tilde k^2}{\sigma } \varGamma ^{3/2}, \\[-9pt] \nonumber \end{align}

(4.1b) \begin{align} a_4 &= \frac {2}{\sigma } \tilde k^2 \varGamma ^{1/2}, \\[-9pt] \nonumber \end{align}

(4.1c) \begin{align} a_3 &= \left ( \frac {\tilde k^2}{\sigma } +(1-\lambda ^2) (\widetilde {Rs}-\widetilde {Ra}) \tilde k^2 \right ) \varGamma ^{-1/2}, \\[-9pt] \nonumber \end{align}

(4.1d) \begin{align} a_2 &= 2 \left (\widetilde {Rs}-\widetilde {Ra}\right )\tilde k^2 \varGamma ^{-3/2} , \\[-9pt] \nonumber \end{align}

(4.1e) \begin{align} a_1 &= \left ( \widetilde {Rs}-\widetilde {Ra} \right ) \tilde k^2 \varGamma ^{-5/2}, \\[-9pt] \nonumber \end{align}

(4.1f) \begin{align} a_0 &= \left ( \widetilde {Rs} - \tau \widetilde {Ra} \right ) \tilde k^4 \varGamma ^{-3}. \\[9pt] \nonumber \end{align}

The dominant balance in (2.22b ) is then between the $a_4 \omega ^4$ and $a_2 \omega ^2$ terms, giving, at leading order,

(4.2) \begin{align} \tilde \omega ^2 - \sigma \left ( \widetilde {Rs} - \widetilde {Ra} \right ) = 0. \end{align}

Turning to the cubic equation for the frequency, (2.23), making the same scaling and keeping only the leading terms, leads to the reduced set of coefficients,

(4.3a) \begin{align} b_6 &= \sigma \left ( 1-\lambda ^2 \right )^2 \tilde k^2 \, \varGamma ^{7/2}, \\[-9pt] \nonumber \end{align}

(4.3b) \begin{align} b_4 &= \left (2\sigma (1+\lambda ^2) + (1+\lambda )^2 \right ) \tilde k^2 \, \varGamma ^{3/2} , \\[-9pt] \nonumber \end{align}

(4.3c) \begin{align} b_2 &= \left ( (1+\sigma ) \tilde k^2 + \sigma \left ( (\tau -1)\lambda ^2 -2 (1 + \tau )\lambda + \tau -1 \right ) \widetilde {Rs} \tilde k^2 \right ) \varGamma ^{-1/2}, \\[-9pt] \nonumber \end{align}

(4.3d) \begin{align} b_0 &= - \sigma (1-\tau ) \widetilde {Rs} \tilde k^2 \, \varGamma ^{-5/2}. \\[9pt] \nonumber \end{align}

Thus, (2.23) can be written at leading order as

(4.4) \begin{align} \begin{aligned} & \sigma \big ( 1-\lambda ^2 \big )^2 \tilde \omega ^6+\left (2\sigma (1+\lambda ^2) + (1+\lambda )^2 \right )\tilde \omega ^4\\ &\qquad +\, \left ((1+\sigma ) - \sigma \left ( (1-\tau )\lambda ^2 + 2 (1 + \tau )\lambda + 1 - \tau \right ) \widetilde {Rs}\right )\tilde \omega ^2 -\sigma (1-\tau ) \widetilde {Rs}=0. \end{aligned} \end{align}

In principle, (4.4) can be used to analyse the appearance of loops of marginal oscillations, as in HPE21, although analytical results are in general too elaborate to be useful. We can, however, look for the conditions for which one side of the loop crosses the steady branch, so that $\widetilde {Rs}=\tau \widetilde {Ra}$ , or, in view of (4.2),

(4.5) \begin{align} \tilde \omega ^2= - \frac {\sigma (1-\tau ) \widetilde {Rs}}{\tau }. \end{align}

On substituting for $\widetilde {Rs}$ from (4.5) into (4.4), we see that a factor of $\tilde \omega ^2$ may be cancelled, leaving the quadratic for $\tilde \omega ^2$ :

(4.6) \begin{align} \sigma \big(1-\lambda ^2\big)^2 \tilde \omega ^4 +\left ( \big(\lambda ^2+1\big)(1+2\sigma +\tau )+2\lambda \frac {1+\tau ^2}{1-\tau }\right )\tilde \omega ^2+(1+\sigma +\tau )=0. \end{align}

Equation (4.6) can have real positive solutions for $\tilde \omega ^2$ only if $\lambda$ is not too large. In fact, $\lambda$ must satisfy the following inequality (‘ $b^2\gt 4ac$ ’):

(4.7) \begin{align} \lambda ^2 \left ( \sqrt {1+\sigma +\tau }-\sqrt {\sigma } \right )^2 + \frac {2\lambda \big(1+\tau ^2\big)}{1-\tau } + \left ( \sqrt {1+\sigma +\tau }+\sqrt {\sigma } \right )^2\lt 0. \end{align}

After a little algebra, we find that this inequality for $\lambda$ is equivalant to

(4.8) \begin{align} \lambda \lt \frac {-(1-\tau )}{(\sqrt {1+\sigma +\tau }-\sqrt {\sigma })^2}. \end{align}

(The root of larger modulus is less than $-1$ , and thus impermissible.) We have to check that the value of $\lambda$ given by (4.8) is greater than $-1$ . Replacing the inequality by an equality and $\lambda$ by $-1$ , we find that

(4.9) \begin{align} \sqrt {\sigma (1+\sigma +\tau )}=\sigma +\tau \quad \textrm{and, hence,}\,\,\sigma =\sigma _c=\frac {\tau ^2}{1-\tau }, \end{align}

consistent with the results of HPE21. Thus if $\sigma \lt \sigma _c$ there will be a range of negative $\lambda$ , and an associated range of $\tilde \omega ^2$ , and hence of $\widetilde {Rs}$ , for which the onset of oscillations is coincident with the steady branch when $\widetilde {Ra}=\widetilde {Rs}/\tau$ . If $\sigma \gt \sigma _c$ , on the other hand, oscillations are not preferred until $n\gtrsim 3$ . The two different scenarios will be discussed further in § 4.3.

In HPE21 we showed that depending on the values of $\sigma$ and $\tau$ , the loop of oscillations appears either above the steady branch, in which case the transition to oscillatory convection occurs when the smaller of the two roots for $|\widetilde {Rs}|$ from (4.5) and (4.6) crosses the line of the steady branch; or below the steady branch, in which case oscillations are preferred just after the point of appearance. The transition between these two regimes occurs on a line ${\mathcal{L}}$ in $(\sigma ,\tau )$ space for each $\lambda$ ; for $\lambda =-1$ , this is given by expression (3.29) in HPE21.

For the case of general $\lambda$ , we can make some progress in finding this line by returning to (4.4), which can be written compactly as

(4.10) \begin{align} A\tilde \omega ^6+B\tilde \omega ^4+\big(C-D\widetilde {Rs}\big)\tilde \omega ^2-E\widetilde {Rs}=0, \end{align}

where

(4.11) \begin{align} \begin{aligned} A&=\sigma \big (1-\lambda ^2\big )^2,\,\,B=2\sigma (1+\lambda ^2) + (1+\lambda )^2 ,\,\,C=(1+\sigma ), \\ D&=\sigma \left ( (1-\tau )(1+\lambda ^2) + 2 \lambda (1 + \tau ) \right ), \,\, E=\sigma (1-\tau ). \end{aligned} \end{align}

If (4.10) has a double root, marking the appearance of the loop of oscillations, then we also have

(4.12) \begin{align} 3A\tilde \omega ^4+2B\tilde \omega ^2+(C-D\widetilde {Rs})=0. \end{align}

Combining (4.10) with (4.12) gives a further quadratic in $\tilde \omega ^2\text{:}$

(4.13) \begin{align} B\tilde \omega ^4+2(C-D\widetilde {Rs})\tilde \omega ^2-3E\widetilde {Rs}=0. \end{align}

We now seek the condition that the loop appears on the steady branch, and hence we can use expression (4.5) in (4.13). On cancelling a factor of $\widetilde {Rs}$ , we obtain the following explicit formula relating the value of $\widetilde {Rs}$ to the other parameters:

(4.14) \begin{align} \widetilde {Rs}=\frac {\tau \left (3\tau +2(1+\sigma )\right )}{\sigma \left ( (1+\lambda ^2)(1-\tau )(1+2(\tau +\sigma )) +2 \lambda (1+\tau +2\tau ^2) \right )} = \frac {F(\sigma , \tau )}{\sigma G(\sigma , \tau , \lambda )} \quad \textrm{say}. \end{align}

Since $\widetilde {Rs} \lt 0$ , this shows that $\lambda$ cannot be too large and, in particular, must be negative. On substituting for $\tilde \omega ^2$ and $\widetilde {Rs}$ from (4.5) and (4.14) into (4.12), we obtain the following complicated relation defining the transition line ${\mathcal{L}}$ :

(4.15) \begin{align} 3A \sigma (1-\tau )^2 F^2 - 2B \sigma \tau (1-\tau ) FG + \tau ^2 G \left ( \sigma C G - DF \right )=0. \end{align}

The line ${\mathcal{L}}$ is shown for $\sigma$ as a function of $\lambda$ for a number of values of $\tau$ in figure 11. Each of the curves, which have different fixed values of $\tau$ , denotes the value of $\sigma$ , as a function of $\lambda$ , for which the loop of oscillatory solutions appears precisely on the steady branch. Above the curve (i.e. at greater values of $\sigma$ ), the loop of oscillatory solutions is born above the steady branch; below the curve (i.e. at smaller $\sigma$ ), the loop of oscillatory solutions is born below the steady branch.

Figure 11.

$\text{Plot of }\sigma$ versus $\lambda$ for the line ${\mathcal{L}}$ , defined by (4.15), denoting the parameter values at which the loop of oscillatory solutions first appears on the steady branch. The curves show $\tau =0.9$ (top), $0.8$ , $0.7$ , $0.6$ , $0.5$ , $0.4$ , $0.3$ (bottom).

4.2. $n=2$ : appearance of oscillations for larger $\lambda$

As has been shown above, loops of oscillations cannot appear if $\lambda$ is too large. For larger values of $\lambda$ , branches of oscillations can appear via Takens–Bogdanov (T–B) bifurcations from the steady branch. This situation is similar to that described in HPE21 for $\lambda =1$ .

In contrast to the oscillation loops described above, the branches that appear for larger $\lambda$ have $k^2\sim \varGamma ^{-1}$ . The T–B points are characterised by $a_0=a_1=0$ in (2.19). Writing $k^2=\varGamma ^{-1}\tilde k^2$ , $Rs = \varGamma ^{-2} \widetilde {Rs}$ , ${\textit{Ra}} = \varGamma ^{-2} \widetilde {Ra}$ , and replacing $\beta ^2$ by $k^2$ since $k^2\gg 1$ , we obtain

(4.16) \begin{align} \tau \tilde k^4+\widetilde {Rs}-\tau \widetilde {Ra}=\left (1+\tau +\frac {\tau }{\sigma }\right )\tilde k^4+\widetilde {Rs}\big(1+(1+\lambda )\tilde k^2\big)-\widetilde {Ra}\big(1+\tau (1-\lambda )\tilde k^2\big)=0. \end{align}

Eliminating $\widetilde {Ra}$ between the two equations of (4.16) gives the following cubic for $\tilde k^2$ :

(4.17) \begin{align} \tau (1-\lambda )\tilde k^6-\tau \left (1+\frac {1}{\sigma }\right )\tilde k^4-2\lambda \widetilde {Rs}\tilde k^2+\widetilde {Rs}\left (\frac {1}{\tau }-1\right )=0. \end{align}

There are thus either one or three T–B points (noting that $\widetilde {Rs}\lt 0$ and that we must have $\tilde k^2\gt 0$ ). It is instructive to consider the condition under which two T–B points coincide; this results from setting the discriminant of the cubic equation  (4.17) to zero, giving

(4.18) \begin{align} 32 (1-\lambda ) \lambda ^3 S^2 + \left ( 4\lambda ^2 P^2 - 36 \lambda (1-\lambda ) PQ - 27 (1-\lambda )^2Q^2 \right ) - 4 P^3Q =0, \end{align}

where, to simplify notation, we have introduced $P=1+1/\sigma$ , $Q=1-1/\tau$ , $S=\widetilde {Rs}/\tau$ . The quadratic expression (4.18) for $S$ denotes the two values of $S$ (and hence $\widetilde {Rs}$ ) between which there are three T–B points. By seeking the condition for these two values of $Rs$ to be repeated, we may then identify the critical values of $\lambda$ at which the possibility of three T–B points disappears. Setting the discriminant of the quadratic (4.18) to zero gives

(4.19) \begin{align} \left ( 4\lambda ^2 P^2 - 36 \lambda (1-\lambda )PQ - 27(1-\lambda )^2 Q^2 \right )^2+512\lambda ^3(1-\lambda )P^3Q=0. \end{align}

On making the further substitutions

(4.20) \begin{align} C=\lambda P, \qquad D=(1-\lambda )Q, \qquad C=\Delta D, \quad \textrm{where} \quad \Delta = - \frac {\lambda (1+\sigma )\tau }{(1-\lambda ) \sigma (1-\tau )}, \end{align}

we obtain a (quartic) expression involving only $\varDelta$ , which, rather remarkably, factorises as

(4.21) \begin{align} 0=(4\Delta ^2-36\Delta -27)^2+512\Delta ^3=16(\Delta +9/2)^3(\Delta +1/2). \end{align}

On rearranging the expression for $\varDelta$ in (4.20), the two distinct roots of (4.21) thus give the following values of $\lambda$ for which there are coincident T–B points:

(4.22) \begin{align} \qquad \lambda _a=\frac {9\sigma (1-\tau )}{9\sigma +2\tau -7\sigma \tau }\qquad \textrm{and} \qquad \lambda _b=\frac {\sigma (1-\tau )}{\sigma +2\tau +\sigma \tau } . \end{align}

Recall that here we are interested in the third quadrant and so $Rs\lt 0$ ; this in turn requires (recalling $Q\lt 0$ ) that the coefficient of $S$ in (4.18) is positive. We find that only for $\lambda =\lambda _a\,(\Delta =-9/2)$ is this satisfied. Thus, we have the possibility of three T–B points (depending on the value of $\widetilde {Rs}$ ) for $1\gt \lambda \gt \lambda _a$ and one T–B point otherwise. (The value $\lambda _b$ denotes such a transition point in the first quadrant. However, oscillatory modes are favoured there, with the steady mode out of contention, as discussed in § 3; T–B points thus play no role in determining the overall stability boundary in the first quadrant.)

The role of the T–B points can be seen through numerical calculation of the steady and oscillatory boundaries for $\lambda \lt \lambda _a$ and $\lambda \gt \lambda _a$ . For illustration, we consider the case of $\sigma =0.2$ , $\tau =0.5$ , for which $\lambda _a=3/7 \approx 0.429$ . Figure 12 shows the stability boundaries for $\lambda =0.3$ , for which there can only be one T–B point. In figure 12(a), $\widetilde {Rs}=-10$ ; the oscillatory branch emerging from the T–B point always lies above the minimum of the steady branch (given by ${\textit{Ra}}^{(s)}_c=Rs/\tau$ ), and thus, steady convection is favoured. In figure 12(b), $\widetilde {Rs}=-31.48$ , the value at which the critical Rayleigh numbers for the (low wavenumber) steady and (high wavenumber) oscillatory modes are identical. In figure 12(c), $\widetilde {Rs}=-40$ ; although the T–B point itself lies above ${\textit{Ra}}=Ra^{(s)}_c$ , the oscillatory branch emerging from the T–B points dips down sufficiently that high wavenumber oscillatory modes are favoured.

Figure 12.

Plots of $Ra$ versus $k^2$ for the steady (red) and oscillatory (blue) stability boundaries, with $\varGamma =10^{-4}$ , $\sigma =0.2$ , $\tau =0.5$ , $\lambda =0.3$ . In (a), $Rs=-10^9$ ( $\widetilde {Rs}=-10$ ); in (b), $Rs=-3.148 \times 10^9$ ( $\widetilde {Rs}=-31.48)$ ; in (c), $Rs=-4 \times 10^9$ ( $\widetilde {Rs}=-40$ ).

For $\lambda \gt \lambda _a$ , where there is always a range of $\widetilde {Rs}$ for which there are three T–B points, we have identified two ways in which the oscillatory branches can evolve, dependent on the proximity of $\lambda$ to $\lambda _a$ . Figure 13(a–c) shows the stability boundaries for $\lambda =0.5$ , i.e. for $\lambda$ slightly greater than $\lambda _a$ . There are three T–B points in the range $-10.945 \lt \widetilde {Rs}\lt -9.868$ . Figure 13(a) shows the loop of oscillatory solutions created at $\widetilde {Rs}=-9.868$ , together with the branch of oscillatory solutions emerging from the higher T–B point. Note that steady solutions are preferred. At $\widetilde {Rs}=-10.945$ , shown in figure 13(b), the two oscillatory branches have merged, via coalescence of the upper two T–B points; again, steady solutions are still preferred. For $\widetilde {Rs} \lt -10.945$ , there is only one oscillatory branch; for sufficiently negative values of $\widetilde {Rs}$ , oscillatory solutions become preferred, as shown in figure 13(c).

Figure 13.

Plots of $Ra$ versus $k^2$ for the steady (red) and oscillatory (blue) stability boundaries, with $\varGamma =10^{-4}$ , $\sigma =0.2$ , $\tau =0.5$ . In (a–c), $\lambda =0.5$ ; in (d–f), $\lambda =0.8$ . In (a), $\widetilde {Rs}=-10.5$ ( $Rs=-1.05 \times 10^9$ ); in (b), $\widetilde {Rs}=-10.945$ ( $Rs=-1.0945 \times 10^9$ ); in (c), $\widetilde {Rs}=-15$ ( $Rs=-1.5 \times 10^9$ ); in (d), $\widetilde {Rs}=-10$ ( $Rs=-10^9$ ); in (e), $\widetilde {Rs}=-10.466$ ( $Rs=-1.0466 \times 10^9$ ); in ( f), $\widetilde {Rs}=-12$ ( $Rs=-1.2 \times 10^9$ ).

Figure 13(d–f) shows the stability boundaries for $\lambda =0.8$ , i.e. for $\lambda$ now rather greater than $\lambda _a$ . There are three T–B points in the range $-14.681 \lt \widetilde {Rs}\lt -4.490$ . We can see from figure 13(d) that the loop of oscillatory solutions created at $\widetilde {Rs}=-4.490$ and the branch of oscillatory solutions emerging from the higher T–B point have both developed a pronounced ‘hook’. Furthermore, by this value of $\widetilde {Rs}$ , oscillatory modes have already become preferred. As $\widetilde {Rs}$ is made more negative, the oscillatory branches meet when $\widetilde {Rs}=-10.466$ , as shown in figure 13(e); in contrast to the case illustrated in figure 13(a–c), the oscillatory branches merge before coalescence of the upper two T–B points. The oscillatory branches reconnect, as shown in figure 13( f) for $\widetilde {Rs}=-12$ . The upper two T–B points merge and disappear at $\widetilde {Rs}=-14.681$ .

4.3. Transition between steady and oscillatory onset for general $\lambda$

The foregoing results have shown that, for sufficiently large (negative) values of $Rs$ , oscillatory convection can be preferred. The occurrence of oscillations in this quadrant is due entirely to the M–C effect, since in the classical case only steady-state bifurcations are possible. When $\sigma \lt \sigma _c=\tau ^2/(1-\tau )$ (see (4.9)), the transition to oscillatory behaviour occurs for $|Rs|=O(\varGamma ^{-n})$ , where $\varGamma$ is generally between 2 and 3, while when $\sigma \gt \sigma _c$ , oscillatory behaviour is preferred only at values of $n \gtrsim 3$ for smaller $\lambda$ . The critical wavenumber of the oscillatory mode at the transition between steady and oscillatory modes is also much larger when $\sigma \gt \sigma _c$ .

Figure 14 gives a global picture of the dependence of the transition values on $\lambda$ for two cases: (i) $\tau =0.5$ , $\sigma =0.2$ ( $\sigma \lt \sigma _c$ ); (ii) $\tau =0.5$ , $\sigma =1$ ( $\sigma \gt \sigma _c$ ). In this figure the value of $|Rs|$ is represented by $n$ with $Rs=-\varGamma ^{-n}$ . The curves are constructed using an iterative process by which the critical values $k^2_c$ and ${\textit{Ra}}^{(o)}_c$ are calculated for a given $n$ and then $n$ is adjusted until ${\textit{Ra}}^{(o)}=Ra^{(s)}$ at $n=n_c$ , $k_c^2=k^2_{oc}$ . The figure thus incorporates the results of both §§ 4.1 and 4.2. It can be seen that in both cases the critical value $n_c$ climbs sharply as $\lambda$ approaches the ‘dip’ value of $\lambda _d = (\tau -1)/(\tau +1)$ ; $\lambda \approx \lambda _d$ is where anomalous behaviour of the critical wavenumber appears in the first quadrant. As we shall show in § 4.5, for $\lambda \approx \lambda _d$ in the third quadrant, steady onset is always preferred, so that there is in fact no dip in the critical oscillatory wavenumber in the third quadrant case.

Figure 14.

Plots of (a) $n_c$ and (b) $k_{oc}^2$ versus $\lambda$ for $\varGamma =10^{-4}$ , $\tau =0.5$ . The black lines denote $\sigma =0.2$ ( $\sigma \lt \sigma _c$ ), the red dashed lines $\sigma =1$ ( $\sigma \gt \sigma _c$ ).

4.4. $n \geqslant 3$ : high wavenumber

When $n\gt 2$ , and in particular when $n \gtrsim 3$ , oscillations are usually favoured except for particular values of the parameters. The favoured mode of oscillation can occur either at large or small wavenumber. In this subsection we investigate the high wavenumber modes; the low wavenumber modes will be discussed in § 4.5. In this high wavenumber regime, numerical results suggest the scaling $k^2 \sim \varGamma ^{1-n}$ ; we thus write $k^2 = \varGamma ^{1-n} \tilde k^2$ and take $\beta ^2=k^2$ at leading order.

The frequency is then determined from (2.24); with the above scaling for $k^2$ , and assuming that we are not particularly close to $\lambda = \pm 1$ , so that $1-\lambda ^2$ is not small, we find that the coefficients of (2.24) have $b_6 \sim \varGamma ^{5-n}$ , $b_4 \sim \varGamma ^{5-2n}$ , $b_2 \sim \varGamma ^{5-3n}$ , $b_0 \sim \varGamma ^{5-4n}$ . Thus, on rescaling $\omega ^2 = \tilde \omega ^2 \varGamma ^{-n}$ , and cancelling a common factor of $\sigma$ , we obtain the leading-order cubic equation for $\tilde \omega ^2$ :

(4.23) \begin{align} \big ( 1-\lambda ^2 \big )^2 \tilde k^2 \tilde \omega ^6 &- (1-\lambda ^2) \left (1+2 \tau +\lambda (1-2\tau ) \right ) \tilde k^4 \tilde \omega ^4 \nonumber\\ & + \tau \left ( (\tau -2) \lambda ^2 - 2 \tau \lambda + 2 + \tau \right ) \tilde k^6 \tilde \omega ^2 - (1-\lambda ) \tau ^2 \tilde k^8 =0. \end{align}

Note that the critical frequency does not depend on $\sigma$ at leading order. On introducing $\lambda _1,\lambda _2$ as in (2.13) and writing $X=\tilde \omega ^2/\tilde k^2$ , (4.23) simplifies to

(4.24) \begin{align} (2 \lambda _2 X -1) (2 \lambda _1 X - \tau )^2 = 0, \quad \mbox{so that} \quad X=\frac {1}{2 \lambda _2}, \quad X=\frac {\tau }{2 \lambda _1} \quad \textrm{(repeated)}. \end{align}

We now seek to determine ${\textit{Ra}}$ by using the coefficients $a_1,\ldots ,a_5$ defined in (2.12), and considering (2.22b ), namely $a_4 \omega ^4 - a_2 \omega ^2 + a_0 =0$ . To bring ${\textit{Ra}}$ into play at leading order suggests a priori that ${\textit{Ra}} \sim k^4 \sim \varGamma ^{2-2n}$ . However, for both values of $X$ in (4.24), we find that ${\textit{Ra}}=0$ in this scaling. This conclusion is supported by the numerical results. For example, with $n=4$ and $\varGamma = 10^{-4}$ , ${\textit{Ra}} \sim \varGamma ^{2-2n}$ would give ${\textit{Ra}}=O(10^{24})$ , whereas the computed value of ${\textit{Ra}}_c$ is actually $O(10^{18})$ .

We are thus led to the revised scalings $k_c^2 \sim \varGamma ^{1-n}$ , $\omega _c^2 \sim \varGamma ^{-n}$ , ${\textit{Ra}}_c \sim \varGamma ^{1-3n/2}$ , where the presence of $n/2$ in the exponent of $\varGamma$ for ${\textit{Ra}}_c$ derives from a half-power correction to $X$ , which, in turn, follows from the double root in (4.24). The value of ${\textit{Ra}}$ corresponding to the non-repeating root is much smaller in magnitude ( $O(\varGamma ^{-n})$ ), and hence we do not need to consider this root further. The corrections to $b_4$ , $b_2$ and $b_0$ have relative magnitude $\varGamma ^{n-2}$ , which we denote by $\varepsilon$ . Returning to using $\tilde \omega ^2$ and $\tilde k^2$ , and now keeping the zeroth- and first-order terms, we may express (2.23) in the form

(4.25) \begin{align} b_{60}\tilde \omega ^6 + (b_{40} + \varepsilon b_{41})\tilde \omega ^4 + (b_{20} + \varepsilon b_{21})\tilde \omega ^2 + b_{00} + \varepsilon b_{01} = 0, \end{align}

where

(4.26a) \begin{align} b_{60} &= \sigma \big ( 1-\lambda ^2 \big )^2 , \\[-9pt] \nonumber \end{align}

(4.26b) \begin{align} b_{40} &= - \sigma (1-\lambda ^2) \left (1+2 \tau +\lambda (1-2\tau ) \right ) \tilde k^2, \\[-9pt] \nonumber \end{align}

(4.26c) \begin{align} b_{41} &= 2\sigma (1+\lambda ^2) + (1+\lambda )^2 , \\[-9pt] \nonumber \end{align}

(4.26d) \begin{align} b_{20} &= \sigma \tau \left ( (\tau -2)\lambda ^2 - 2 \tau \lambda + 2 + \tau \right ) \tilde k^4, \\[-9pt] \nonumber \end{align}

(4.26e) \begin{align} b_{21} &= - \left ( \sigma + 2 \tau (1+\sigma ) + \lambda \left (2 \tau (1+\sigma )-\sigma \right ) \right ) \tilde k^2 \nonumber \\ &\quad +\, \sigma \left ( (\tau -1)\lambda ^2 -2 (\tau +1)\lambda + \tau -1 \right ) \widetilde {Rs}, \\[-9pt] \nonumber \end{align}

(4.26f) \begin{align} b_{00} &= - (1-\lambda ) \sigma \tau ^2 \tilde k^6 , \\[-9pt] \nonumber \end{align}

(4.26g) \begin{align} b_{01} &= \tau ^2 (1+\sigma ) \tilde k^4 + 2 \sigma \tau \lambda \widetilde {Rs} \tilde k^2. \\[9pt] \nonumber \end{align}

The correction to the repeated root is $O(\varepsilon ^{1/2})$ ; thus we write

(4.27) \begin{align} \tilde \omega ^2 = \frac {\tau \tilde k^2}{(1+\lambda )} + \varepsilon ^{1/2} \tilde \omega _1^2 = \tilde \omega _0^2 + \varepsilon ^{1/2} \tilde \omega _1^2. \end{align}

The $O(\varepsilon )$ terms in (4.25) give

(4.28) \begin{align} \sigma (1-\lambda ) \left ( (1+\lambda ) \tilde \omega _1^2 \right )^2 \left ( (1-\lambda ) \tilde \omega _0^2 - \tilde k^2 \right ) = - b_{41} \tilde \omega _0^4 - b_{21} \tilde \omega _0^2 - b_{01}, \end{align}

which, after some algebra, leads to

(4.29) \begin{align} \tilde \omega _1^4 = - \frac {\tau }{(1+\lambda )^3} \left (\tilde k^2 + (1+\lambda )\widetilde {Rs} \right ) = \frac {\tau }{(1+\lambda )^3} \left ( (1 + \lambda ) |\widetilde {Rs}| - \tilde k^2 \right )\!, \end{align}

since $Rs\lt 0$ . This expression is, remarkably, also independent of $\sigma$ .

Using (2.22b ), we can now determine the critical value of ${\textit{Ra}}$ . We know that at leading order, ${\textit{Ra}}$ does not appear. Then we may write, symbolically,

(4.30) \begin{align} \tilde a_4 = \tilde a_{40} + O(\varepsilon ), \quad \tilde a_2 = \tilde a_{20} + \varepsilon ^{1/2} \tilde a_{21} + O(\varepsilon ) , \quad \tilde a_0 = \tilde a_{00} + \varepsilon ^{1/2} \tilde a_{01} + O(\varepsilon ) , \end{align}

where the $\tilde a_{21}$ and $\tilde a_{01}$ terms are those containing ${\textit{Ra}}$ . After some algebra, the $O ( \varepsilon ^{1/2} )$ terms in (2.22b ) give

(4.31) \begin{align} \tilde k^4 \left ( (1-\lambda ) \tilde \omega _0^2 - \tilde k^2 \right ) (1+\lambda ) \tilde \omega _1^2 = 2 \widetilde {Ra} \tilde k^2 \tilde \omega _0^2 - \tau \widetilde {Ra} \tilde k^4. \end{align}

On substituting for $\tilde \omega _0^2$ and $\tilde \omega _1^2$ from (4.27) and (4.29), and rearranging, (4.31) gives

(4.32) \begin{align} \widetilde {Ra} =\widetilde {Ra}^{(o)}= - \frac { | (1-\lambda )\tau -(1+\lambda )|}{(1-\lambda )\left (\tau (1+\lambda )\right )^{1/2}} \left ( (1+\lambda ) |\widetilde {Rs}| \tilde k^4 - \tilde k^6 \right )^{1/2}. \end{align}

We note from (4.29) that there are positive and negative roots for $\tilde \omega _1^2$ ; in (4.32) we have chosen the appropriate solution for $\tilde \omega _1^2$ , dependent on whether $\lambda \lt \lambda _d$ or $\lambda \gt \lambda _d$ , to make $\widetilde {Ra}^{(o)}\lt 0$ , as required. From (4.32), $\widetilde {Ra}^{(o)}$ is minimised when

(4.33) \begin{align} \tilde k^2= \tilde k_c^2 = \frac {2(1+\lambda )|\widetilde {Rs}|}{3} \implies k_c^2= \frac {2 \varGamma (1+\lambda )|Rs|}{3}, \end{align}

giving

(4.34) \begin{align} \widetilde {Ra}^{(o)}_c = - \frac {2}{3^{3/2}} \left (\frac {1+\lambda }{1-\lambda }\right ) \left ( \frac { |(1-\lambda )\tau -(1+\lambda )| }{\tau ^{1/2}} \right ) |\widetilde {Rs}|^{3/2}\end{align}

or, equivalently,

(4.35) \begin{align} Ra^{(o)}_c = - \frac {2 \varGamma }{3^{3/2}} \left (\frac {1+\lambda }{1-\lambda }\right ) \left ( \frac { |(1-\lambda )\tau -(1+\lambda )| }{\tau ^{1/2}} \right ) |Rs|^{3/2}. \end{align}

In the scaling of this subsection, ${\textit{Ra}}^{(o)}_c$ vanishes (and so has a maximum) when $\lambda =\lambda _d$ , the critical dip value in the first quadrant. In § 4.5 we discuss whether a similar dip appears in the third quadrant.

We note also that expression (4.35) does not hold for $\lambda =\pm 1$ and so different scalings must apply near these endpoints. The connections between the results of this subsection and the endpoint results from HPE21 are discussed below in § 4.6.

4.5. $n \geqslant 3$ : low wavenumber

In both the first and third quadrants, there is a minimum in ${\textit{Ra}}^{(o)}$ at very small $k^2$ . In general, this minimum is at a different asymptotic level to ${\textit{Ra}}^{(o)}_c$ at high $k^2$ and is not of interest. However, at the dip, it can be of the same order, and so we need to establish whether low wavenumber oscillations can be preferred (for the first quadrant low wavenumber modes, we showed in § 3.3.3 that this was never the case).

Guided by numerical results, we adopt the following scalings for $k^2,\omega ^2$ and $Rs$ :

(4.36) \begin{align} k^2 = \varGamma ^{n-2} \tilde k^2, \quad \omega ^2 = \varGamma ^{-2} \tilde \omega ^2, \quad Rs = \varGamma ^{-n} \widetilde {Rs}. \end{align}

From the frequency equation (2.23), $\tilde \omega ^2$ obeys the following leading-order equation:

(4.37) \begin{align} & \sigma \big ( 1-\lambda ^2 \big )^2 \tilde \omega ^6 + \left (2\sigma (1+\lambda ^2) + (1+\lambda )^2 \right ) \tilde \omega ^4 \nonumber \\ & \quad + \left ( (1+\sigma ) + \sigma \left ( (\tau -1)\lambda ^2 -2 (\tau +1)\lambda + \tau -1 \right ) \widetilde {Rs} \tilde k^2 \right ) \tilde \omega ^2 - \sigma (1-\tau ) \widetilde {Rs} \tilde k^2 = 0. \end{align}

In (2.22b ) the dominant balance is between the first two terms, giving

(4.38) \begin{align} \widetilde {Ra} = \widetilde {Rs} - \frac {\tilde \omega ^2}{\sigma \tilde k^2} . \end{align}

Expression (4.37) determines $\tilde \omega ^2$ (and hence $\widetilde {Ra}^{(o)}$ from (4.38)) as a function of $\tilde k^2$ . For the critical value ${\textit{Ra}}^{(o)}_c$ , where $\textrm{d} \widetilde {Ra} /\textrm{d} \tilde k^2 = 0$ , (4.38) implies that

(4.39) \begin{align} \frac {\textrm{d} \tilde \omega ^2}{\textrm{d} \tilde k^2} = \frac {\tilde \omega ^2}{\tilde k^2}. \end{align}

Expressions (4.37)–(4.39) may be combined to eliminate $\tilde \omega ^2$ , yielding a quadratic equation for $k_c^2$ , with one admissible root. However, this does not lead to a simple formula for the critical value of ${\textit{Ra}}$ . That being said, we can nonetheless make progress in addressing the key point of whether the oscillatory branch can ever reach the steady branch in the dip region (which would then suggest that oscillations can be preferred for some parameter values).

Substituting the critical value for the onset of steady convection ( $\widetilde {Ra} = \widetilde {Rs}/\tau$ ) into (4.38) gives

(4.40) \begin{align} \widetilde {Rs} = \frac {\tau }{\sigma (\tau -1)} \frac {\tilde \omega ^2}{\tilde k^2} . \end{align}

On substituting for $\widetilde {Rs}$ from (4.40) and for the dip value of $\lambda = (\tau -1)/(\tau +1)$ into (4.37), and simplifying, we then obtain the following quadratic for $\tilde \omega ^2$ :

(4.41) \begin{align} 16 \tau ^2 \tilde \omega ^4 + 4 \sigma (1+\tau ^2)(1+\tau )^2 \tilde \omega ^2 +(1+\sigma + \tau ) (1+\tau )^4=0. \end{align}

In principle, expression (4.41) gives the frequency of the oscillatory mode at the point where the oscillatory branch crosses the steady branch (at the dip parameter values). However, we can see that there are no real positive solutions for $\tilde \omega ^2$ . Hence, the oscillatory branch never reaches the steady branch and so steady modes are always preferred to low wavenumber oscillations in the dip region. Furthermore, numerical solutions and the results of § 4.4 demonstrate that at the dip values, high wavenumber oscillations can occur only at much smaller (negative) values of ${\textit{Ra}}$ than for steady convection. Thus, at the dip, the high wavenumber (oscillatory) mode gives way to the steady branch.

4.6. $n \geqslant 3$ : combined results across all wavenumbers

We have seen from the results above that when $n\gtrsim 3$ , the preferred mode of convection is oscillatory over a wide range of values of $\lambda$ . Away from the dip value $\lambda _d$ and the endpoints $\lambda =\pm 1$ , the dominant mode has a high critical wavenumber; as $\lambda$ approaches $\lambda _d$ , the critical value of ${\textit{Ra}}$ for oscillations increases, and for fixed $\lambda$ near the dip is $O(\varGamma ^{1-3n/2})$ . As $|\lambda -\lambda _d|$ becomes small, $O(\varGamma ^{n/2-1})$ , the (negative) value of ${\textit{Ra}}$ increases towards zero and the preferred mode of instability is steady convection, with a critical wavenumber of order unity and $|Ra| =O(\varGamma ^{-n})$ . There is also, in principle, an oscillatory convection mode at low wavenumber, which near the dip can exhibit a critical value comparable with that of classical steady convection, but this mode is never preferred.

The results for general $\lambda$ away from $\lambda _d$ cease to hold when $\lambda$ is close to $1$ or $-1$ . Indeed, from the results of HPE21, when $\lambda = \pm 1$ , ${\textit{Ra}}_c$ (and also $k_c^2$ and $\omega _c^2$ ) exhibits a different asymptotic ordering to that displayed in (4.35). At $\lambda =-1$ , the onset of instability is steady if $\sigma \gt \sigma _c = \tau ^2/(1-\tau )$ , with

(4.42) \begin{align} Ra^{(s)}_c \approx \frac {Rs}{\tau }, \end{align}

and oscillatory if $\sigma \lt \sigma _c$ , with

(4.43) \begin{align} Ra^{(o)}_c \approx \frac {(\sigma +\tau ) Rs}{\sigma }. \end{align}

At $\lambda =1$ , the onset of instability is always oscillatory, with

(4.44) \begin{align} Ra^{(o)}_c \approx -\varGamma ^2 Rs^2. \end{align}

The transition between the result (4.35), valid for general $\lambda$ apart from $\lambda = \pm 1$ and at the dip, and the endpoint results (4.42)–(4.44) is achieved through boundary layers close to $\lambda =\pm 1$ . As in § 3.3.1, we may estimate the width of the boundary layers by supposing that the endpoint solutions vary slowly with $\lambda$ , and then finding the value of $\lambda$ for which the interior value of ${\textit{Ra}}_c$ , given by (4.35), matches the appropriate endpoint value. Thus, the boundary layer close to $\lambda =-1$ ( $\lambda _1=0$ ) has width

(4.45) \begin{align} \delta \lambda _1 =O \left ( (\varGamma ^2|Rs|\tau ^3)^{-1/2} \right ) \ (\sigma \gt \sigma _c); \quad \delta \lambda _1 = O \left ( (\sigma +\tau ) (\sigma ^2\varGamma ^2|Rs|\tau )^{-1/2} \right ) \ (\sigma \lt \sigma _c). \end{align}

Finally, the boundary layer close to $\lambda =1$ has width

(4.46) \begin{align} \delta \lambda _2 =O \left ( (\varGamma ^2|Rs|\tau )^{-1/2} \right ). \end{align}

Figure 15.

$\text{Plots of }Ra_c$ , $k_c^2$ and $\omega _c^2$ versus $\lambda$ ; $\tau =0.5$ . In (a–c), $\sigma = 1$ , $\varGamma =10^{-3}$ , $Rs=-10^9$ ( $n=3$ ); in (d–f), $\sigma =0.2$ , $\varGamma =10^{-3}$ , $Rs=-10^9$ ( $n=3$ ); in (g–i), $\sigma =1$ , $\varGamma =10^{-3}$ , $Rs=-10^{12}$ ( $n=4$ ). Numerical results are shown as solid lines; black (magenta) denotes oscillatory (steady) solutions. The asymptotic results of § 4.4 are shown as dashed red lines.

Figure 15 gives an overview of the behaviour of the critical values of ${\textit{Ra}}$ , $k^2$ and $\omega ^2$ , calculated numerically, as functions of $\lambda$ in the third quadrant. The respective asymptotic results (4.35), (4.33) and (4.27) (combined with (4.29)) are also shown. The dip, in which the critical wavenumber is dramatically reduced and the preferred mode becomes steady for a range of $\lambda$ , can be clearly seen. In figure 15(a–c), $n=3$ and $\sigma =1 \gt \sigma _c =0.5$ ; close to $\lambda =-1$ , the onset of instability is thus steady. In figure 15(d–f), $n=3$ and $\sigma =0.2 \lt \sigma _c$ ; the onset of instability therefore remains oscillatory up to $\lambda =-1$ . Figure 15(g–i) is as for figure 15(a–c), but with $n=4$ . In comparison with the $n=3$ plots, two features are apparent: the width of the dip is markedly reduced and the agreement between the numerical and asymptotic results is dramatically improved.

Figure 16.

$\text{Plots of }Ra_c$ , $k_c^2$ and $\omega _c^2$ versus $\lambda$ ; $\tau =0.5$ , $\varGamma =10^{-3}$ , $Rs=-10^{12}$ ( $n=4$ ). (a–c) Region near $\lambda =-1$ for $\sigma =1$ ; (d–f) region near $\lambda =-1$ for $\sigma =0.2$ ; (g–i) region near $\lambda =1$ for $\sigma =1$ . Numerical results are shown as solid lines; black (magenta) denotes oscillatory (steady) solutions. The asymptotic results of § 4.4 are shown as dashed red lines.

Figure 16 shows the behaviour of the critical values of ${\textit{Ra}}$ , $k^2$ and $\omega ^2$ for $n=4$ close to $\lambda =-1$ and $\lambda =1$ , highlighting the thin transition regions between the solutions at the endpoint and in the interior. In figure 16(a–c), $\sigma \gt \sigma _c$ and there is a transition to steady onset in a region of width $\delta \lambda _1$ given by (4.45) (this thin transition region is too narrow to be seen in figure 15 g–i). In figure 16(d–f), $\sigma \lt \sigma _c$ ; onset remains oscillatory up to $\lambda =-1$ , but there is again a transition between the interior and endpoint solutions, now in a region of width $\delta \lambda _1$ again given by (4.45). Figure 16(g–i) shows the transition between interior and endpoint solutions over a region of width $\delta \lambda _2$ given by (4.46).

The surface plots of figure 17 show ${\textit{Ra}}_c$ and $k_c^2$ as functions of $\lambda$ and $\tau$ for $\sigma =1$ and $n=3$ . The $k_c^2$ surface shows clearly the two disconnected regions of oscillatory solutions at high values of $k^2$ , separated by a flat valley of steady solutions, with $k_c^2=0.5$ . For small $\tau$ , the ${\textit{Ra}}_c$ surface extending from $\lambda =-1$ is flat (independent of $\lambda$ ) before turning downhill as the oscillatory solution is preferred. For larger values of $\tau$ , there is a ‘bowl’ where high wavenumber oscillatory solutions are preferred; at such values of $\tau$ , with increasing $\lambda$ the surface again becomes flat after the bowl, with the steady mode preferred, before again turning downhill as the oscillatory solution is favoured.

Figure 17.

Surface plots of (a) ${\textit{Ra}}_c$ and (b) $k_c^2$ versus $\lambda$ and $\tau$ for $\varGamma =10^{-3}$ , $Rs=-10^9$ , $\sigma =1$ . The range $0.05 \leqslant \tau \leqslant 0.95$ is plotted.